[John R. Brand (Auth.)] Handbook of Electronics Fo(BookZZ.org)

HANDBOOK OF

ELECTRONICS FORMULAS, SYMBOLS

AND DEFINITIONS

John R. Brand

~ ~~w~o~O~Jc~!~DA~A~~tJH~L~Ps Cs'l~~~I~r LONDON TORONTO MELBOURNE

Van Nostrand Reinhold Company Regional Offices: New York Cincinnati Atlanta Dallas San Francisco

Van Nostrand Reinhold Company International Offices: London Toronto Melbourne

Copyright © 1979 by Litton Educational Publishing, Inc. Softcover reprint of the hardcover 1st edition 1979

Library of Congress Catalog Card Number: 78-26242

ISBN-I3: 978-94-011-6999-8 DOl: 10.1007/978-94-011-6997-4

e-ISBN-I3: 978-94-011-6997-4

All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any meansgraphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems-without permission of the publisher.

Manufactured in the United States of America

Published by Van Nostrand Reinhold Company 135 West 50th Street, New York, N.Y. 10020

Published simultaneously in Canada by Van Nostrand Reinhold Ltd.

15 14 13 12 11 10 9 8 7 6 5 4 3 2

Library of Congress Cataloging in Publication Data

Brand, John R Handbook of electronics formulas, symbols, and

definitions.

Includes index. 1. Electronics-Handbooks, manuals, etc. I. Title.

TK7825.B7 621.381'02'02 78-26242

PREFACE The Handbook of Electronics Formulas, Symbols and Definitions has been compiled for engineers, technicians, armed forces personnel, commercial operators, students, hobbyists, and all others who have some knowledge of electronic terms, symbols, and theory.

The author's intention has been to provide:

A small, light reference book that may be easily carried in an attache case or kept in a desk drawer for easy access.

A source for the majority of all electronic formulas, symbols, and definitions needed or desired for today's passive and active analog circuit technology.

A format in which a desired formula may be located almost instantly without the use of an index, in the desired transposition, and in sufficiently parenthesized linear form for direct use with any scientific calculator.

Sufficient information, alternate methods, approximations, schematic diagrams, and/or footnotes in such a manner so that technicians and hobbyists may understand and use the majority of the formulas, and that is acceptable and equally useful to engineers and others very knowledgeable in the field.

iii

ACKNOWLEDGMENTS

Much of the material is this Handbook is based upon a small loose-leaf notebook containing formulas and other reference material compiled over many years. With the passage of time, the sources of this material have become unknown. It is impossible therefore to list and give the proper credit.

It is possible, however, to give richly deserved recognition to Juergen Wahl for his assistance during preparation of the manuscript. His suggestions, comments, proofreading, and checking of formulas for accuracy are all greatly appreciated.

Special thanks are due to my wife and family for their understanding and acceptance of long periods of neglect, without which this book would not have been possible.

iv

INTRODUCTION

All fonnulas in this Handbook use only the basic units of all terms. It is especially easy in this age of scientific calculators to convert to and from basic units.

Formulas in all sections are listed alphabetically by symbol with the exception of applicable passive circuit symbols, where, for a given resultant, all series circuit formulas are listed first, followed by parallel and complex circuit formulas.

If the symbol for an electronic term is unknown, a liberally cross-referenced listing of electronic terms and their corresponding symbols may be found in the appendix.

Symbols of all reactive magnitude terms in fonnulas have been consistently given the signs conventionally associated with them to maintain capacitive or inductive identity. In rectangular quantities, this also allows identification of the complex number as representing a series equivalent impedance/ voltage or a parallel equivalent admittance/current.

To prevent possible confusion, all symbols representing vector quantities in polar or rectangular fonn have been printed in boldface.

A number of fonnulas have the potential to develop a zero divisor. Conventional mathematics prohibits a division by zero, and calculators will overflow if this is attempted. However, fonnulas noted GD allow the manual conversion of the reciprocal of zero to infinity and the reciprocal of infmity to zero. Division by zero in fonnulas noted @ is prohibited.

Textbooks conventionally use italic (slanted) type for quantity symbols and roman (upright) type for unit symbols. However, this Handbook follows the example of almost all technical manuals, using roman type for both quantity and unit symbols.

v

Preface Introduction

CONTENTS

Section 1 Passive Circuits

1.1 English Letters 1.2 Greek Letters

Section 2 Transistors

2.1 Static Conditions 2.2 Small Signal Conditions

Section 3 Operational Amplifiers

3.1 Symbols and Definitions 3.2 Formulas and Circuits

Appendix

A Table of 5% Value Ratios B Electronic Terms and their Corresponding

iii v

1 185

201 221

253 277

321

Symbols 327

vii

SECTION ONE

PASSIVE CIRCUITS

1.1 ENGLISH LETTERS

A A = Symbol for ampere.

A = Basic unit of electric current.

A = Coulombs per second.

Ampere, Amplification etc.

A = 6.24· 1018 elementary charges per second (electrons or holes).

A = Unit often used with multiplier prefixes. pA = 10-12 A, nA = 10-9 A, IlA = 10-6 A rnA = 10-3 A, kA = 103 A, etc.

A = Symbol for area. Area is measured in various unit such as in2 , ft2 ,cm2 ,m2 etc.

Ah = Symbol for ampere-hours. One ampere-hour equals 3600 coulomb (C).

At or A = Symbol for ampere turn, the SI unit of magnetomotive force.

~ = Symbol for current amplification. See-Active Circuits

Av = Symbol for voltage amplification. See-Active Circuits.

a = Symbol for atto. A multiplier prefix for 10-18 .

a = Substitute for greek letter alpha. (Not recommended) See-a

a = Not recommended as a quantity symbol.

3

B B = Symbol for susceptance

Susceptance Definitions

B = The ease with which an alternating current of a given frequency at a given potential flows in a circuit containing only pure capacitive and/or inductive elements. The imaginary part of admittance. The reciprocal of reactance in any purely reactive circuit. The reciprocal of a pure reactance in parallel with other elements.

B = Magnitude of susceptance measured in mho (old) or siemens (new). Siemens (S) and mho (n-1 ) are equal.

B = IBI = Babsolute value = Bmagnitude B = Complete description of susceptance

B=B/±90° =O±jB =O-(±B)j

Be = Capacitive susceptance

Be = B /+90° = 0 + jB = 0 - (-B) j

BL = Inductive susceptance

BL = B /-90° = 0 - jB = 0 - (+B) j

Be = B magnitude identified as capacitive

BL = B magnitude identified as inductive

- B = B magnitude "given" the sign usually associated with capacitive quantities. - B = Be

+B = B magnitude "given" the sign usually associated with inductive quantities. +B = BL

±B = Identification of B as capacitive or inductive in many formulas.

±B = Identification of B as capacitive or inductive in the resultant of all formulas in this handbook.

4

Gl

B :c

Susceptance, ftI

.2 '" '" Series Circuits - Gl E a. ... .. a. 0 Gl c:(z I-

Be = -Xe/(~ + R;) CD@® Rs,Xc

BL = XL/eXt + R;) CD@® Rs,XL

±B = ±Xs/(X; + R;) <D@® Rs, ±XS

Be = -Xc/Z2 <D@® Xc,Z

BL = XL/Z2 CD@® XL,Z

±B = ±Xs/Z2 CD@® ±Xs,Z

±B = -Y~in(±8y)] CD@@ Y,±8y @@)

±B = ~in(±8z~ /Z CD@®

Z,±8 z @@)

BNotes:

<D B IS INTRINSICALLY A PARALLEL CIRCUIT QUANTITY. B DERIVED FROM A SERIES CIRCUIT IS THE EQUN ALENT PARALLEL CIRCUIT REACTANCE IN RECIPROCAL FORM.

@ Rs = Series R, Xs = Series X. ® B and X are magnitudes, however both B and X have been "given"

the signs usually associated with capacitive and inductive quantities. Be therefore "equals" -B, BL "equals" +B, Xc "equals" -X and XL "equals" + X. This allows direct identification of a reactive quantity derived from any formula in this handbook.

@) The form (±8) is used as a reminder that the sign of the phase angle determines the sign of B and therefore the identity of B as either capacitive or inductive.

5

Susceptance, B Parallel Circuits

(Bdt = (-Bdl + (-Bd2 --- + (-Bdn

-Bt = (-B1) + (-B2) --- + (-Bn)

(Bdt = (Bdl + (Bd2 - - - + (Bdn

+Bt = (+Bl) + (+B2) --- + (+Bn)

(Bdt = -w(CI + C2 -- - + Cn)

(Bdt = w-1(Li1 + 1.21 --- + r.;;-1)

(Bdt = (-Xdil + (-Xdil ---+ (-Xc);;1

-Bt = (-Xd-1 + (-X2rl --- + (-Xn)-1

(Bdt = (+Xdil +(+Xdi1 ---+(+Xd;;1

+Bt = (+Xd-1 + (+X2)-1 ---+ (+Xn)-1

I B I = The magnitude of the imaginary part of Y RECT

±B = The imaginary part of Y RECT

multiplied by -j.

BNotes:

® w = 2wf = angular velocity ® x-I = I/x

Q)

3 ell .2 VI - Q) c. ... c. 0 «2

® @

® @

®®@

®® ®@

®®@

®®@

®G> ®®

<Z> YRECT = G ±jlBI = G -j(±B) = G - (±B)j

'" E ... Q)

~

Bc

-B

BL

+B

C

L

Xc

-X

XL

+X

Y RECT

® IBI = magnitude of B without knowledge of vectorial direction. IBI therefore cannot be identified as either capacitive or inductive.

® (-j) • (-j) = +1, (-j). (+j) =-1 @) (x)t = total x = equivalent x

6

CII

B ::c

Susceptance, <11 .S:! fI) '" Parallel Circuits - CII E Co.., Co 0 ..

CII «2 I-

±Bt = BL - Be ® Be,BL

±Bt = (±BI) + (±B2) - -- + (±Bn) @ -B,+B

±Bt = (wLr i - (wC) ®® C L

±Bt = (wLd-I - (wCd+(wL2r I - (wC2 )-- ®@

IBI =vy2 - G2 ® G,Y

±B = -G ~an(±eY)J @® G,e y

@@ IBI = VZ-'1 - R-'1 @ Rp,Z

±B = ~an(±ez~ /R ®®

Rp,e z @

±Bt = XLI - :xcI ®® Xe,XL

±Bt = (±Xd-1 + (±X2rl --- + (±Xn)-I @ -X +X

±B = -Y ~in(±ey ~ ®® Y,e y @

±B = ~in(±ez~ /Z ®® z,e Z @

BNotes: @ x- 2 = 1/x2 @ Rp = parallel resistance @ If the admittance (Y) or the impedance (Z) and the associated

phase angle COy or 0 Z) are known, it is immaterial if the circuit configuration (i.e., series or parallel) is unknown.

7

B B = Symbol for bel. (Rarely used)

B = Ten decibels (dB) See-dB

B = Symbol for magnetic flux density.

Magnetic Flux Density, Bandwidth

B = The magnetic flux per unit area perpendicular to the direction of flux. (also known as magnetic induction)

B = Magnetic flux density measured in telsa (T), gauss (G), maxwell (Mx) per square centimeter and lines of flux per square inch.

telsa (T) = weber (Wb) per square meter gauss (G) = 10-4 telsa

maxwell (Mx) = lines of flux weber (Wb) = 108 maxwell (Mx) = 108 lines of flux

B = CPt A where A = cross sectional area of magnetic path. cP = Total magnetic flux in weber, maxwell

or lines of flux.

B = JlH where Jl = permeability H = magnetic force

B = Symbol for bandwidth (not recommended)

B = Symbol for bandwidth (not recommended)

BW or BW is the preferred symbol for bandwidth. See-BW

Bl = Symbol for unity gain bandwidth. (BW (Av=1»)

See-Active Circuits, Opamp

8

3dB Down Bandwidth BW

BW = Symbol for bandwidth

Bandwidth

Other symbols for or abbreviations of bandwidth include: B, 13, (f2 - f 1), B.W., BW, BW -3 dB

BW = The difference between the two frequencies of a continuous frequency band where the output has fallen to one half power. (- 3 dB is very close to one half power)

BW = Bandwidth expressed in hertz (Hz).

BW = (f2 - f1)-3 dB

BW= fr/Q

BW = (fr R)/XL(@fr)

BW = R/(21TL)

BW = (f2 - f1)-3 dB

BW= fr/Q

BW = (fr XC(@fr»)/R

BW = (21TRC)-1

BW(Av=l)-See-Active Circuits, Opamp

BW = Average bandwidth. Effective noise bandwidth. See also-BW Active Circuits, Opamp

BW Notes:

See-Q for frequency to bandwidth ratio. See-D for bandwidth to frequency ratio. See also-d Active Circuits.

9

c C = The symbol for capacitance.

Capacitance etc. Definitions

C = 1. In a system of conductors and dielectric or in a capacitor, that property which permits the storage of electrical energy.

2. The property which determines the quantity of electric charge at a given potential.

3. In a system of conductors (plates) and dielectric (insulator) or in a capacitor, the ratio of the quantity of electric charge to the potential developed.

C = Capacitance (also known as capacity) measured in farad (F) units unless noted.

[This extremely large unit is very seldom used except in formulas. The resultant of all capacitance formulas should be converted to more practical units such as microfarads (IlF) or picofarads (pF)]

C R:: 7 pF per sq in of parallel plates separated by 12 in of air.

C = The symbol for capacitor on part lists and schematics.

C = The symbol for coulomb (unit of quantity of charge) (Seldom used in electronics)

c = Obsolete symbol for cycles per second. [Use hertz (Hz)]

c = The symbol for the velocity of light or electromagnetic waves (physics). (Not recommended. Use v for velocity in electronics)

10

CIl

C ::c

Capacitance, C1I .2 en '" Series Circuits - CIl E c. .... c. 0 ...

CIl «2 I-

Ct =(C l 1 +C2"l ___ +C~l)-l CD C

Cx = (Ct1 - Cl1 )-1 eY Ct = w -1 [(Xd1 + (Xch ---+ (Xc)n] -1 CDeY Xc Ct = _w-1 [(-Xd + (-X2) --- + (-Xn)]-l Q) -X

C = D/(wRs) Series reactive element CDeY D Rs must be capacitive

C = (WRsQ)-l Series reactive element CD@ Q Rs must be capacitive.

C = [-wRs(tan Oz)] -1 Oz must be CDeY Rs Oz negative

C = [-wZ(sin Oz)] -1 Oz must be CDeY Z Oz negative

Series to Parallel Conversion CDeY Cp = [(w2 R;Cs) + C;l]-l @)

Cs Rs

CNotes:

CD C = Capacitance, D = Dissipation Factor, Q = Quality Factor, R = Resistance, Xc and -X = Capacitive Reactance, Z = Impedance, (J = Phase Angle, w = Angular Velocity Subscripts: C = capacitive, n = any number, p = parallel, s = series, t = total or equivalent, x = unknown

® x-I = l/x, w = 211"f Q) B and X are magnitudes, however both B and X are often "given"

the signs usually associated with capacitive and inductive quantities. In all formulas in this handbook -B = Be, -X = Xc, +B = BL and +X=XL

@ Equivalent capacitance varies with frequency.

11

CD

C :a

Capacitance, .~ '" '" Parallel Circuits - CD E a. ... .. a. 0 CD «2 I-

Ct = ~Bd 1 + (Bd2 - -- + (BdnJ / w <D@ Be

Ct = ~-B1) + (-B2) --- + (-Bn)]/-w ®® -B

Ct = C1 + C2 --- + Cn <D C

Ct = ~Xdl1 + (Xd2"l --- + (Xd~l]/w <D@ Xc Ct = K-X1)-1 + (-X2)-1 ---+(-Xn)-l]/-w ® -X

Cp = (wRpD)-l Parallel reactive element <D@ D,Rp known to be capacitive

Cp = [G(tan Oy)]/w Oy must be <D@® G,Oy positive

Cp = Q/(wRp) Parallel reactive element <D@ Q Rp known to be capacitive

Cp = (tan Oz)/(-wRp) Oz must be <D@ Rp,Oz negative

Cp = [Y(sin Oy)]/w Oy must be <D@® Y,Oy positive

Cp = (sin Oz)/(-wZ) Oz must be <D@ Z,Oz negative

Cp = ~t(sin OJ) /(wE)] OJ must be <D@® E I OJ positive

Parallel to Series Conversion

C = (w2C R2)-1 + C s p p p <D@@ Cp Rp

C Notes: ® B = Susceptance, E = rms Voltage, G = Conductance, 1= rms Current, Y = Admittance

12

G)

C :is

Capacitance .~ III III

Misc. Formulas - QI E a. ... ... a. 0 G) «2 I-

Cr = (w2L)-1 C required for resonance. CY® L Series or parallel circuits

C=Q/E C required for a charge ® E,Q of Q coulombs

C = Q2/(2W) W = work equiv. stored

energy in watt/sec ® Q,W Q = charge in coulombs

C=T/R C required for time con- ® R,T stant T and resistor R

C = (It)/E I = constant current ® C, E I E = voltage change after time t

Capacitance of two parallel plates (conductors) separated by an insulator (dielectric)

C = (Ak)/(4.45d) approx. pF

A = Useful area of each plate in square inches d = Spacing or distance between plates in inches k = Dielectric constant (Air = 1)

Capacitance of concentric cylinders (e.g., coaxial cable)

C = (7 . 354k)/ Uog(D/d~ pF per foot length

D = inside diameter of outside cylinder (inches) d = outside diameter of inside cylinder (inches) k = dielectric constant of material between cylinders

(Air = 1)

CNotes:

® Cr = Resonant Capacitance, E = de Voltage, I = de Current, L = Inductance, Q = Charge in coulombs, t = Time in sec., T = Time Constant, W = Work in joules

13

o D = The symbol for dissipation factor

Dissipation Factor Definitions

D = 1. The ratio of energy dissipated to the energy stored in dielectric material, in certain electric elements, or in certain electric structures. 2. The inverse of the quality factor Q. (also known as the storage or merit factor) 3. In certain electric elements or structures, the absolute value of the cotangent of the phase angle of the alternating current with respect to the voltage, the voltage with respect to the alternating current, the impedance, or the admittance.

D = A factor which usually has a numerical value of from zero to one and is expressed in either decimal or percentage form.

D = A factor most commonly associated with capacitor specifications or measurements, however may be used in all Q factor applications.

D ~ Power factor when D < . 1 D = A factor which is very useful for the calculation of equiva

lent series resistance. (Rs = DXc = D/(wC) = DXL = DwL)

DNotes:

<D B = Susceptance, C = Capacitance, G = Conductance, L = Inductance, Q = Storage Factor, Quality Factor or Merit Factor, R = Resistance, X = Reactance, Y = Admittance, Z = Impedance, (J = Phase Angle, w = Angular Velocity Subscripts: p = parallel, s = series

@ w = hf, x-l = l/x, x-t = 1/.JX, x-2 = l/x2

® Not valid for LC circuits.

14

0 Dissipation Factor, Series Circuits

D = l/Q

D = cotan (J Exception ®

D = wCsRs

D = V(ZWCs)2 - 1

D = Rs/(wLs)

D = V [Z/(wLs~ 2 - 1

D = Rs/Xs

D = ~Z/Rs)2 - 1J -t Exception ®

D = V(Z/Xs)'i - 1

Dr = wCsRs = Rs/(wLs)

Dr = Rs/Xc = Rs/XL

DNotes:

@ Xs may be Xc or XL but not (XL - XC) ® B may be Be or BL but not (BL - Be) ® cotan x = l/(tan x) G:> Dr = Dissipation Factor at Resonance,

Xc = Capacitive Reactance, XL = Inductive Reactance

CI)

:a co .~ fI) II)

- CI) E c. ... 0. 0 ...

CI)

<1:2 ~

<D Q

<D® (J

<D@ Cs Rs

<D@® Cs Z

<D@ Ls Rs

<D@® Ls Z

<D@ RsXs

<D@® Rs Z

<D@® Xs Z

<D@G:> Cs Ls Rs

<D@G:> Xc XL Rs

® If the resultant under the radical sign is negative, a mistake has occurred.

15

CI)

Dissipation Factor, 0 :is ftI

.!::! til til

Parallel Circuits - CI) E c. ... .. c. 0 CI) «2 I-

D = I/Q CD Q

D = cotan (J Exception ® CD® (J

D=G/B CD® B G

D = Y(Y/B)2 - 1 CD®® B Y

D = (RpWCp)-l CDQ) Cp Rp

D = V [Y/(wCp)] 2 - 1 CDQ)® Cp Y

D=Y(ZwCp) 2 - 1 CDQ)® Cp Z

D = [(Y/G)2 - U -t Exception ® CDQ)® G Y

D = (wLp)/Rp CDQ) Lp Rp

D = Y(YwLp)2 - 1 CDQ)® Lp Y

D= v~wLp)/ZJ2 - 1 CDQ)® Lp Z

D = Xp/Rp CD® Rp Xp

D= ~Rp/Z)2 - IJ -t Exception ® CDQ)® Rp Z

D = Y(Xp/Z)2 - 1 CD®® Xp Z

16

dB dB = The symbol for decibel

Decibel Definitions and Formulas

dB = 1. The standard logarithmic unit for expressing power gain or loss. 2. One tenth of a bel. (The basic bel unit is very seldom used) 3. A power ratio only-according to the original defInition and to a few purists. 4. A commonly used convenient unit for expressing voltage and current ratios. See-dB Note 2

Formulas for Defmitions 1,2, & 3

dB = 10 log (Po/Pi)

dB = 20 log (Eo/Ei) only when (Zo /0 0 ) = (ZJ 0 i)

dB = 20 log (lo/Ii) only when (Zo~ = C4/0i) dB = 20 log [(EoY~ cos Oi)/ (ErJZo cos 00 )]

dB = 20 log [(loYZo cos Oo)/(liY~ cos Oi~ Formulas for Defmition 4

dB = 10 log (Po/Pi)

dB = 20 log (Eo/&)

dB = 20 log (Io/Ii)

dB Notel1:

<D log = logarithm to the base 10, P = Power, E = rms Voltage, I = rms Current,6 = Phase Angle, Subscripts: i = Input, 0 = Output

@ When using defmition 4, it should be stated as dB voltage or current gain or loss, dB apparent power gain or loss, etc. ---, not as dB gain or loss or as dB power gain or loss.

® See also-dBm notes, dB editorial-opamp

17

dBm Power in dB Definitions and Formulas

dBm = Symbol for decibels referenced to one milliwatt.

dBm = Power level expressed in decibels above or below one milliwatt.

dBm = LP(mW)

dBm = V.U. (volume units) (sinewave only)

dBm = 10(log P) + 30

dBm = 10 Oog(1 000 P~

dBm = 10 Oog(E2/R)J + 30

dBm = 10 Uog(I2 R~ + 30

dBm = 10 Oog(EI cos e~ + 30

dBm = 10 Oog(eZ cos e~ + 30

dBm = 10 (log [(E2 cos e)/z J) + 30

dBm = 1000g(E2y cos e~ + 30

dBmNotes:

<D P = Power, E = dc or rms Voltage, I = dc or rms Current, Y = Admittance, Z = Impedance, () = Phase Angle, cos = cosine, log = Logarithm to the base 10.

® When using a calculator to obtain the log of a number smaller than one, the value of both the characteristic and the mantissa are likely to be different than the value obtained from log tables. The calculator value will have both a negative characteristic and a negative mantissa. This is the correct value to use. (Log tables always have a positive mantissa)

18

E E = Symbol for electromotive force (emf)

Voltage Definitions

(emf is more commonly called voltage or potential)

E = The electric force which causes current to flow through a conductor.

E = Potential measured in volts (V)

E = Edc or I Errns I E = Complete description of voltage

E = EpOLAR = ERECTANGULAR

E=E& =ER + (±Ex)j

ER = EjO° = ER + OJ

Ec = Ej-90° = 0 + (-Ex)j = 0 - jEx

EL =Ej+90° =O+(+Ex)j =O+jEx

ER = Ernagnitude identified as resistive or real

Ec = Ernagnitude identified as capacitive

EL = Ernagnitude identified as inductive

- Ex = Ec "given" the sign associated with capacitive quantities.

+Ex = EL "given" the sign associated with inductive quantities.

±Ex = Identification of Ex as capacitive or inductive in the resultant of many formulas.

e = The instantaneous value of voltage

Note: The symbol V is also used for voltage and predominates in active circuits. See-V, Active Circuits

19

Q)

E ::a

Voltage, ,~ '" '" ,t::

DC Circuits - Q) E ;:, Q) a. .. .. u a. a. 0 Q) ,~ > <r:z I- tJl-

Et = (±Ed + (±E2) --- + (±En) <D@ E

Et = Pt/I <D@ I P '" .. Et = (PI + P2 --- + Pn)/I '5

u .. Et = IRt i:J

<D@ I R '" Q)

Et = I(RI + R2 --- + Rn) ';: Q)

en Et = v'PtRt <D@ PR Et =v'(PI +P2 ---+Pn)(R I +R2 ---+Rn)

E = 11 /G1 = It/Gt <D@ GI :l

E = v'P1/G I = v'Pt/Gt <D@ GP '5 u ..

E = Ptll1 = Pt/lt <D@ I P i:J ]!

E = I1Rl = ItRt <D@ I R iii .. all

E=~=v'RtPt <D@ ~

RP

See-R, complex circuits ~

See-R, delta to Y conversion Q) UI - .. a. ,-

See also-I and P if necessary E G o ,~

Simplify circuit and use above formulas tJtJ

20

Transient Voltages, Voltage Ratios

t ec = E [1 - (f-l)RC]

ec = .6321 E when

ec = (It)/C t

ec = E/f RC

ec = .3679 E when

ec = E - [(It)/C]

Rt eL = E/f L

eL = .3679 E when

eE (E = Applied Voltage)

t = RC (I time constant)

(I = constant current)

(E = Initial Cap. Voltage)


(I = constant current)

(E = Applied Voltage)


eL = - L(di/dt) L(di/dt) = rate of current Change] in (ampere/seconds)

E=Q/C (Q = Charge in coulombs)

Eav = [(2v'Z)/1r] Erms = .9003 Erms

Eav = (2/rr) Epeak = .6366 Epeak

Epeak = (v'Z) Erms = 1.414 Erms

Ep_p = (2v'Z) Erms = 2.828 Erms

Erms = [rr/(2v'Z)] Eav = 1.111 Eav

Erms = Eeff

E.m. = (I/v'Z) Epeak = .7071 Epeak

Erms = [1/(2v'Z)] Ep_p= .3535 Ep_p

21

Capacitor Voltage DUring Charge thru Resistor

Capacitor Voltage During Discharge thru Resistor

Inductor Voltage DUring Energization thru Resistor

Inductor Voltage Developed By Current Change

Voltage Developed by Electric Charge

'" 0 .0:; .. a: QI ... .. .. "0 >

Notes E Notes

ENotes:

CD General B = Susceptance ®, C = Capacitance, e = Instantaneous Voltage, E = Voltage Magnitude or DC Voltage ®, E = Magnitude and Phase Angle of Voltage, f= Frequency, G = Conductance, I = Current, j = Imaginary Number ®, L = Inductance, P = Power, Q = Quantity of Electrical Charge, R = Resistance, X = Reactance ®, Y = Admittance, Z = Impedance, E = Base of Natural Logarithms ®, n = Ratio of Circumference to diameter of a circle ®, 0 = Phase Angle ®, w = Angular Velocity ®

@ Subscripts C = capacitive, E = voltage, I = current, L = inductive, n = any number, 0 = output, P = parallel circuit, r = (of or at) resonance, s = series circuit, t = total or equivalent, X = reactive, Y = admittance, Z = impedance

® Constants j = i j =.J=I j = 90° multiplier, E = 2.718+ E-I = .36788-, n = 3.1416- 2n = 6.2832-, W = 2nf W = 6.2832f

@) Algebra x-I = l/x, x-2 = 1/x2 , xt = .JX, x-t = 1/.JX, x=f = 1/.JX, I xl = absolute value or magnitude of x

® Trigonometry sin = sine, cos = cosine, tan = tangent, tan-I = arc tangent

® Reminders ±o --- use the sign of the phase angle ± X - - - -X identifies X as capacitive (XC)

+X identifies X as inductive (XL) ±B --- -B identifies B as capacitive (Be)

+B identifies B as inductive (BL) ±EX --- -EX identifies EX as capacitive (EC)

+EX identifies EX as inductive (EL)

22

Q)

E ::c

Voltage, co .!:! en en

Series Circuits - Q) E c. ... a. 0 ..

Q) «z I-

(Edt = (Ed 1 + (Ed2 ... + (Edn Ec

(Edt = (Edl + (Ed2 ... + (Edn CD@ EL

(ER)t = (ER)1 + (ERh ... + (ER)n ER

(±Ex)t = (Edl - (Edl + (Ed2 - (Ech CD@ Ec EL

(±Ex)t = (±E1) + (±E2) ... + (±En) ® -Ex +Ex

Et =v'E~ + EE CDQ)

Ec ER

Et=v'E~ +Et EL ER

(Edt = Iw -1(C11 + C21 ... + C~I) CD@ I C ®®

(Edt = Iw(Ll + L2 ... + Ln) CDQ)® I L

(ER)t = I(RI + R2 ... + Rn) CDQ) I R

(Edt = I ~Xdl + (Xch ... + (Xdn] CDQ) IXc

(-Ex)t=I~-Xd+(-X2)···+(-Xn)] ® I -X

(Edt = I ~Xdl + (Xd2 ... + (Xdn] CDQ) I XL

(+Ex)t = I ~+Xl) + (+X2)··· + (+Xn)] ® I +X

E= IZ CD I Z

Additional E magnitude formulas are included in E formulas starting on page 27.

23

GI

E ::c

Voltage, B en .- '" E Parallel Circuits - GI a. ... ~ a. 0 GI «z l-

E = Itl [(Bdl + (Beh --- + (Bdn] (DQ) It Be

E = IItl K-B1) + (-B2) --- + (-Bn)] I ®® It -B

E = Itl [(Bdl + (Bd2 --- + (Bdn] (DQ) It BL

E = It! [( +B.) + (+B2) - -- + (+Bn~ ® It +B

E = IItl rr±B1) + (±B2) --- + (±Bn~ I CDQ)

It ±B ®®

E=It/~(CI +C2 ---+Cn)] (DQ)

It Cp @

E = It/(G I + G2 ---+ Gn) (DQ) It G

E = Itw [(Lp)J:I + (Lp);l - - - + (Lp)~lll (DQ)

It Lp @®

E = It KRp)J:I + (Rp);l - - - + (Rp)~lll (DQ)

It Rp ® E = It [(Xc)J:1 + (Xd;l --- + (Xd~lll (DQ) It Xc

E = lIt [( _Xp}il + (_Xp);l --- + (-Xp)~llil ®® It -Xp

E = It KXdJ:1 + (XL);1 --- + (Xd~lll (DQ) It XL

E = It ~+Xp)J:I + (+XP)21 ___ +(+Xp)~lll ®® It +Xp

E = lIt [(±Xp)J:I + (±Xp);l - - -+ (±Xp)~llil CDQ)

It ±Xp ®®

E = I/Y CD I Y

E = IZ (D I Z

24

N en

Et =

{([

E1

cos

01J

+ [E

2 co

s 02

J --

-+ [

En

cos

On])

2

+ (

[E1

sin(

±O

l)J

+ [E

2 Si

n(±0

2)J

---+

[E

n Si

n(±

On)

J) 2}

1/2

Ot

= ta

n-1

[([E

1 si

n(±

OdJ

+ [E

2 sin

(±02~

---+

[En Sin

(±On~)

] ([

E1

cos

01J

+ [E

2 co

s 02

J --

-+ [E

n co

s O

nJ)

Seri

es S

um o

f EtL~..l,

E2L

!h, -

--E

n/O

n

Et

=

1([

E1

cos

01J

-

[E2

cos

02J)

2 +

([E

1 si

n(±

Ol)

J -~2

sin

(±02

~) 2

Ot

= t

an-1

[([

E1

sin(±Ol~ -

[E2 si

n(±0

2~) /

([

E1

COS

01J

-[E

2 CO

S 02

J)]

Ed

!..!

, E2~

Dif

fere

ntia

l

m

cnn

CD

0

:::!.

3 ll

'E.,

~~

~~

~ &t

C

Di

a ...

e:"

E Voltage & Phase I mportant Notes

1. It should be understood by the reader that the phase angle of voltage and current is the same one and only phase angle of a circuit or of a circuit element. The fact that current leads the voltage while the voltage lags the current in an inductive circuit means only that the signs of the voltage and current phase angles are different.

2. In a given circuit, the phase angle of voltage, current, impedance and admittance is the same one and only phase angle. The signs of the angle is the only difference. ±8E = -(±8!) = ±8 z = -(±8y ).

3. The voltage phase angle uses the current phase angle as a reference (0°) while the current phase angle uses the voltage phase angle as a reference (0°). Due to this fact, if the voltage phase angle is expressed, the current phase angle is 0° and if the current phase angle is expressed, the voltage phase angle is 0°. It should be obvious that the voltage and current phase angles cannot be used at the same time.

4. The same applies to rectangular form voltage and current. Rectangular form current cannot have an imaginary (reactive) component when the rectangular form voltage has an imaginary (reactive) component. The reverse, obviously, is also true.

5. Due to this confusing situation and the high probability of error, the author DOES NOT RECOMMEND THE USE OF POLAR OR RECTANGULAR FORM VOLTAGE OR CURRENT WHERE EACH USES THE OTHER AS A REFERENCE. THE USE OF THE GENERATOR AS THE PHASE REFERENCE IS RECOMMENDED.

6. The following polar and rectangular form voltage formulas are listed for reference only. Proceed to the Eo and vector algebra Eo formulas.

i6

Voltage & Phase, Series Circuits E

Resistive & Reactive Voltages In Series

E = The magnitude and phase angle of the voltage developed by current through a series circuit. (0 1 = 0°) See also-O

EpOLAR = E / ±OE ERECT = 1. The 0° and ±90° voltages which have a resul

tant equal to EpOLAR. 2. The voltages developed by current through series resistance and net reactance.

ERECT = ER + (±Ex) j CP

ERECT = (E cos 0E) + [E sin(±OE~ j 1i .~ til til

- CP E Q. .. .. Q. 0 CP <z I-

EpOLAR =VE~ + E~ /tan l(-Ec/ER) COG) U ~

ERECT = ER - jEc ®® ~ ~

EpOLAR =vEi + Et /tan-1(EL/ER) COG) ...l ~

ERECT = ER + jEL ®® ~ ~

COG) '" EPOLAR =VE~ + Elc /tan-1(-Ex/ER) ><

®® ~ ERECT =ER + (-Ex)j @ ~

~

COG) ...l

EpOLAR = VEi + Eh /tan-1(+Ex/ER) >< ®® ~

ERECT = ER + (+Ex) j @ ~ ~

EpOLAR = VEi + (EL - Ed'l'tan-1 [(EL - Ed/ER] COG) ...l ~

ERECT = ER + (EL - Ed j ®® u ~

ERECT = ER + (±Ex) j @ ~ ~

27

CD

E ::c

Voltage and Phase, .~ III III

Series Circuits - CD E Do ... ... Do 0 CD «z l-

E = I VR2 + (WC}-2 <D@@ ~

8E = tan-l (wCRr l @® u -E = I VR2 + (WL)2 <D@ ~

8E = tan-1 [(wL)/~ @® ...:I -E = P/(I cos 8z) <D@ <t>

+1

8E = ±8z = -(±8I) ®® Il.. -E = (IR)/(cos 8) <D@ <t>

+1

8E = ±8z = -(±8I) ®® ~ -E = I(IX)/(sin 8)1 <D@@ <t>

+1 -

8E = ±8z = -(±8I) ®® >< --E=IZ <t>

<D@® +1

8E = ±8 z = -(±8I) N -E = I VR2 + ~wL) - (wC}-lr <D@ ~

...:I 8E = tan-1 ([(wL) - (WC}-l] /R) @® u -E = IVR2 + (XL - Xd2 ...:I

<D@® >< 8E = tan-1 [(XL - Xd/~ ~~ See previous page for definitions, EpOLAR and ERECT

28

Voltage and Phase, Parallel Circuits

E Voltage and Phase When a parallel circuit is driven by a current source

E = The magnitude and phase angle of the voltage developed by the total current through a parallel circuit. (BIR = 0°) See also-B

EpOLAR = E j±BE ERECT = 1. The 0° and ±90° voltages which have a resultant

equal to EpOLAR' 2. The series equivalent voltages of a parallel circuit. 3. The voltages developed by current through the series equivalent of a parallel circuit.

ERECT = (E cos BE) + [E sin(±BE~ j G)

ERECT = (ER)s + [(±Ex)s] j ::is ftI

.2 (II (II

- G) E a. .... ... a. 0 G) «z l-

E = It /VG2 + (BL - Bd2 CD® 0 t:Q

BE = tan-1 [(BL - Bd/G] +1

®® .... ......

E = l(It sin B)/(BL - Bdl CD®® CI:> +1 t:Q

BE = -(±BI) = -(±By) ®G> +1 .... ......

E = It/VR 2 + ~wL) 1 - (wC)J2 CD@® ~ ~

BE == tan-1 (R [(wL)-l - (wC~) @)® u .... ......

E == l(It sin B)/~wLrl - (wC)] I CD@® CI:> +1 ~

BE = ±Bz = -(±BI) = -(±By) @)® u .... ......

E = l(It cos B)/GI CD®@) CI:> +1

BE = -(±BI) = -(±By) = ±Bz ®® 0

.... ......

29

Q)

Voltage and Phase, E j5

Parallel Circuits .~ ... ... - Q) E Q. ..... ..

With Current Source Q. 0 Q)

<cZ l-

E = Z YIk + (Ix - Ix P L c CD@ N

>< ...... OE = tan-1 [(Ix L - Ixc)/IR]

+1 ®® ~ ......

E = It YR 2 + (XLI - XC1)2 <D@@ ><: +1

OE = tan -1 [R(Xil - XCI ~ ®® p::: ... ......

c:t:> E=(ItR)cosO CD@ +1

p::: OE = ±Oz = -(±OI) = -(±Oy) ®® ... ......

E = lIt sin O/(XL1 - Xc?) I <D@@ c:t:> +1

@®® ><: OE = ±Oz = -(±OI) = -(±Oy) +1

(J) ... ......

E = It/Y CD@ >-c:t:> +1

OE = -(±Oy) ® >-......

E = ItZ <D@ N c:t:> +1

OE = ±Oz ® N ......

E = P/(It cos 0) CD@ c:t:> +1 ...

OE = -(±OI) = ±Oz ®® ...... ~

E = Y(PZ)/(cos 0) <D@ c:t:> +1

OE = ±Oz = -(±OI) ®® N ~

See previous page for definitions, EpOLAR, and ERECT.

30

Complex Voltages, Series & Differential

Et = V(ER)r + (Ex)r

E Ot = tan-l [(±Ex)t/(ER)t]

(ER)t = (E 1 cos 0 d + (E2 cos ( 2 ) - -

+ (En cos 0 n)

(±Ex)t = [El Sin(±Ol~ + [E2Sin(±02~ + [En sin(±8n~

Et = V(ER)r + (Ex)r

8t = tan-l [(±Ex)t/(ER)J

(ER)t = (El cos 0 d - (E2 cos ( 2 )

(+Ex)t = [El sin(+8l~ - [E2sin(±02)]

(ERECT)t = (ERECTh + (ERECT)2 ---+ (ERECT)n

ERECT = ER ± jEx = ER + (±Ex) j

(ERECT)t = ~ER)l + (ERh ---+(ER)n]

+ ~+EX)l +(±Exh---+(±Ex)n] j

(ERECT)t = (ERECT)l - (ERECTh

ERECT =ER ±jEx =ER + (±Ex)j

(ERECT)t = [(ER)l - (ER)~ + ~+EX)l - (±Exh]j

ENotes:

<D E~ = EpOLAR Q) ER = Eoo = + E = "Real" numbers (-ER = ElSOo) ® E = lEI = Epolar magnitude. ±EX = E±90o @)+EX = E+90o = EL = E sin(+O) ® EX = E-gOo = EC = E sin(-O)

31

Terms

El&

E2& Differential

(ERECTh

(ERECTh

(ERECT)n

(ERECT) 1

(ERECTh

Differential

Output

Eo = Eg ~R2/Rl) + IJ-l

Eo = (EgR2)/(R1 + R2)

OEo = OEg = 0°

Eo = (EgR)/VR2 + X~

Eo = Eg(cos 0Zi)

OEo = -(-OZi) = tan-1 (Xc/R)

Eo = (EgR)/VR 2 + xl Eo = Eg(cOS 0Zi)

OEo = -(+OZi) = tan-1 - (XL/R)

Eo = (EgXd/VR2 +X~

Eo = lEg (sin OZi)1

Voltage & Phase

OEo = -(-OZi) - 90° = ~an-l(Xc/R~ - 90° (Eo Lags)

Eo Notel1: <D Eg = Generator voltage, Zi = Input impedance, 8Eo = Phase angle of

output voltage ® XL = wL, Xc = (wC)-I, w = 21ft

® x-I = l/x, x-2 = l/x2 , x t =../X, x -t = 1/../X @) tan-I = arc tangent, sin = sine, cos = cosine

32

Eo = (EgXd/v'R2 + Xt

Eo = IEg(sin OZi)1

Output Voltage & Phase

OEo = -(+OZi) + 90° = 90° - ~an-l(XL/R~

OEo Leads OEg

Eo = (EgXc)/v'R2 + (XL - Xc)2

OEo = -(±OZi) - 90° = (tan-1 ~Xc - XL)/RJ) - 90°

OEo = -90° @ fp near 0° @ vIf, near -180° @ vhf

Eo = (EgXL)/v'R2 + (XL - Xc)2

OEo = +90° - (±OZi) = 90° - (tan-1 ~XL - XC)/~) OEo = +90° @ fro near 0° @ vhf, near 180° @ vif

33

LCR Filter Networks

Eo = (EgR)/Zi

Eo = Eg(cos BzD

BEo = -(±BzD


where Zi =VR2 + (XLI - XCI) 2 @

BZi = tan-I [R(Xcl - XLI~ -I @

BEo = 0° @ f" Lags BEg below f" Leads BEg above fr

Eo : (EgXd [R; + (XLs - Xd2J -t 5R ~ __

BEo - Bxc - BZi rf\ Eg L C r=: Eo

BEo = (-90°) - (±BZi) L I BEo = (-90°) - ~an-I ~XLS - Xd/RsJ)

where Rs = ~R/xD + R-IJ-I

XLs = ~XdR2) + XLIJ -I

Eo: (EgXL) [R; + (XL - Xcs)2J -~ SJRI---t--BEo -B XL - BZi $ c ~Eo

° '" Eg L BEo = +90 - (±BzD

BEo = +90° - (tan -I ~XL - Xcs)/RsJ)

where Rs = ~R/X~) + R -IJ -I

Xcs = ~XclR2) + Xc?J -I

34

LeR Filter

Networks

E = E [Z. (X-1 - X-I );,-1 o giL e ~

Eo = jEg(sin 8Zi)-lj @

8Eo = ±90° - (±8 Zi)


where Zi =v'R2 + (XLI - Xc?) 2 @

8Zi = tan -1 [R(Xr:1 - Xc?)J -1 @

8Eo = 00 @ fp Leads 8Eg below fp Lags 8Eg above fr

8Eo = 8Z2 - 8Zi

Zi = [R; + (XL - Xes)2J~, 8Zi = tan-1 ~XL - Xes)/RsJ

Z2 = (R-2 +X(?)-~,8Z2 = tan-1 (R/Xd

where Rs = [(R/X~) + R-1J -1

Xes = [(Xe/R2) + XC?]-l

8Eo = 8Z2 - 8Zi

Zi = [R; + (XLs - Xd2J~, 8 Zi = tan-1 ~XLS- Xd/RsJ

Z2 = (R-2 + Xr:2)-~, 8Z2 = tan-1 (R/Xd

where Rs = [(R/Xfj + R -Ill

XLs = [(XL/R2) + Xr:1l 1

35

Output LCR Filter Networks Voltage & Phase

Eo = [Eg(XL - Xd]/Zi

Eo = lEg (sin OZi)1

OEo = ±90° - (±OzD

where Zi = VR 2 + (XC Xd2

0Zi = tan-1 ~XL - Xd/R]

OEo = 00 @ fro Lags OEg below ff' Leads OEg above fr

Eo = (EgZ2)/Zi

OEo = OZ2 - 0Zi

OEo = (-OZ2) - (±OZi)

where Zi =VR2 + (XL - Xd2

0Zi = tan-1 ~XL - Xc)/R]

L

L

c

R

c

Z2 =VR2 +XtOZ2 = tan-l (-Xc/R)

Eo = (EgZ2)/Zi

OEo = OZ2 - 0Zi

OEo = (+OZ2) - (±OZi)

where Zi = VR2 + (XL - Xc)2

0Zi = tan-1 ~XL - Xc)/R]

Z2 = VR2 + xL OZ2 = tan-l (XL/R)

36

Eo = (EgZ2)/Zi

(JEo =(JZ2 - (JZi

(JEo = (-(JZ2) - (-(JzD

where Zi = V(RI + R2)2 + X~


(JZi = tan -I [- Xc/(RI + R2)]

Z2 =VR~ +xt (JZ2 =tan-I(-Xc /R2)

Eo = (EgZ2)/Zi

(JEo =(JZ2 - (JZi

(JEo = (+(JZ2) - (+(JZi)

where Zi = Vr::(R:-I-+---:R=-2--=-)"'--2 -+-=-=X"'-[

(JZi = tan- I [xLl(RI + R2 )J

(JEo = (JZ2 - (JZi

Eo = [Eg(R;2 + Xc~rt J/ ~R I + R2 J 2 + (XCii + Xc\s) -2J t

(JEo = [tan-I(R2/-XC2)] - (tan-I [(RI + R2S)/-(XCII + Xc\s)J)

where R2s = ~R2/Xb) + R;lll

37

Networks

Eo Output Driven By

Voltage & Phase Current Source

Eo = IgR F ~I

I· ~Eo OEo = 0°

Eo = IgXc

OEo = -900 (OEo Lags OIg)

Eo = IgXL

OEo = +90° (OEo Leads 0Ig)

Eo = IgZ

Eo = Ig yCR-"'2-+--:X'"""t

OEo = (-Oz) = tan-1(-Xc/R)

OEo Lags OIg

Eo = IgZ

Eo = Ig Y'"R--';2-+-X---"--r

OEo = (+Oz) = tan-l (XLiR)

(JEo Leads OIg

Note: -ClD- = Infinite impedance alternating current source

38

Networks Driven By Current Source


Eo IgZ

Eo=Ig(R-2+XC?)-~ ~Ig Ie !R rEo OEo=+(-Oz)=tan-I(R/-Xc) I - .

OEo Lags OIg

Eo = IgZ

Eo = Ig(R-2 + XL2r~

OEo = +(+Oz) = tan-I (R/Xd

OEo Leads OIg

Eo = IgZ

OEo = +(±Oz)

Eo = IgZ

OEo = +(±Oz)

Eo = Ig v'R2 + (XL - Xc)2

OEo = tan -I ~XL - Xc)/R]

39

Networks Driven By Current Source

Eo = IgZ

8Eo = ±8z

Eo = Ig [R-2 + (XL - Xd-2l1-8Eo = tan-1 [R/(XL - Xc~

Eo = IgZ

8Eo = ±8z = ±90°

Eo = Ilg [XCll + (XL - xc2r 1l 11

@

Eo = IgZ

8Eo = ±8 z = ±90°

Eo = Ilg [XiJ + (XL2 - Xd-1]-11 @

8Eo = (±8 Zi ) + (-90°) - (±900)

Eo = Ig/[xcA + XCl2 - (XL/XC1 XC2 )]

See also-Z complex circuits, V complex circuits

Note@

If the reciprocal of zero is presented, Eo = 00.

40


E Voltage Vector Algebra

Vector Algebra AC Ohms Law

Eg = Eg I!t.... or Ig = Igl.2:...

E = IgZ = IgZ/O° + Oz = ±Oz

1= Eg/Z = Eg/Z/O° - Oz = -(±Oz)

Z = Eg/I = Eg/I/O° - OJ = -(±Od

Z = E/Ig = E/Ig /OE - 0° = ±OE

Addition and Subtraction of Rect. Quantities

El + E2 = E1(RECT.) + E2(RECT.)

= [ER + (±Ex) j]1 + [ER + (±Ex) j]2

= ~ERh + (ERh] + ~±Exh + (±EX )2] j

El - E2 = ~ER)1 - (ERh] + [(±Exh - (±Exh] j

I+Exl=EL I-Exl=Ec

11 + 12 = 11(RECT) + 12(RECT.)

= [IR - (±Ix) j]1 + ~R - (±Ix) jJ2

= [QRh + (IRh] - [(±IX )1 + (±Ixh] j

11 - 12 = [(IRh - QRh] - [(±Ixh - (±Ixh] j

I+Ixl = IL I-Ixl = Ic

Note: The rectangular current of a series circuit represents current through an equivalent parallel circuit.

Note: See ZRECT for addition and subtraction ofirnpedance

41

Ig = Ig I.!r Zj =Z

Zo = Z

Eo = Ig Z

Zj = ZI + Z2

Zo = [ZII + Zil]-1

Vo=VI+V2

Eo = (EgZ.)/Zj

Ig = Ig 10° Eg = IgZj

Zj = [Z3"1 + (Z2 + Z.)-lll

Zo = [ZII + (Z2 + Z3rll 1

V 0 = VI + (ViI + V3"1 r l

Eo = IgZI [1- (ZdZ3)]

42

Output Voltage Vector Algebra

Eg = Eg 10° Ig = Eg/Z j

Zj = Z4 + [Zi1 + (Z2 + Zd-1l 1

Zo = (Zi1 + [Z2 + (Zi1 + zi r 1l 1)-1

V 0 = VI + [ViI + (V 3 + V 4)-lll

Eo = Eg [1 - (Z4/Zj)JI [(Z2/Z1) + IJ

Ig = Ig ~ Eg = IgZj

Output Voltage Vector Algebra

Zj = [ZSl + (Z4 + [Zi1 + (Z2 + Zlr1l1)-~-1 Zo = [Zi1 + (Z2 + [Zi1 + (Z2 + Zlr1]-1 )-~-1 Vo = VI + (ViI + [V 3 + (ViI + Vi1r1]-1)-1

Eo = [lg(Zj - Z4)J I [(Z2/Z1) + IJ

43

Output Voltage

Source Conversions

Current Source to Voltage Source Conversion

Eg = IgZl

8Eg = 8Ig = 0°

Z2 = ZI at all frequencies

Eo2 = Eol at all frequencies

Voltage Source to Current Source Conversion

Ig = Eg/Z2

8 Ig = 8Eg = 0°

ZI = Z2 at all frequencies

Eol = Eo2 at all frequencies

Note: Eo may be loaded in any manner and the two outputs although changed will remain equal to each other.

44

Noise Voltage

eN(th) = Thermal noise (white noise) voltage of resistance. (Other symbols of thermal noise voltage include EN, ETH , EN(TH) , en, eN(TH), eN(y-;::), eN(yHz) , VN, VN(TH) etc) Note: Thermal noise voltage is always rms voltage regardless of symbol used.

eN(th) =V4kTK RBW

k = Boltzmann constant (1.38 • 10-23 JtK)

T K = Temperature in Kelvin. Cc + 273.15)

BW = Noise bandwidth in hertz. (Noise measured with infinite attenuation of frequencies outside of bandwidth)

eN(yHz) = Thermal noise per hertz. (per root hertz)

eN(yHz) = 1.283 • 10-10 v'R @ 25°C and 1 Hz bandwidth

EN(Ex) = The noise (1/f noise) voltage (rms) of a resistor in excess of thermal noise.

EN(Ex) = Resistor excess noise voltage (rms) in microvolts per volt of dc voltage drop per decade of frequency.

EN(Ex) = 10-6 Edc Uog-1(NI/20)]

NI = Noise Index in dB (a specification)

NI = +10 to -20 dB (carbon composition)

NI = -10 to -25 dB (carbon film)

NI = -15 to -40 dB (metal fIlm or wirewound)

EN Note: iog-1 = antilog10

45

f f = Symbol for femto.

f= A multiplier prefIx meaning 10-15 unit.

f = Symbol for frequency.

Femto, Frequency

f = The number of complete cycles per second of alternating current, sound, electromagnetic radiation, vibrations or certain other periodic events.

f = Frequency measured in hertz (Hz). (old cps)

fc = Crossover or cutoff frequency. (3 dB down)

fo = Oscillation, output or reference frequency.

fr = Frequency of resonance.

f=l/t (t=timeofonecycle)

f= ViA

Sound in Air

f ~ 1136/A (A in feet, @2S°C)

f~ 346.3/A (A in meters, @2S°C)

Electromagnetic waves including radio frequency and light in air or vacuum.

f ~ (9.83 • 108)/A (A in feet)

f ~ (3 . 108)/A (A in meters)

f = 1/(21TXc C)

f = XL /(21TL)

Notes: X = reactance, v = velocity, A = wavelength, C = Capacitance, L = Inductance

46

fc Crossover Frequency

fe = (21TCR)-1 'm ' R R Ehf L R Ehf

fe = 21TL Et ~:g RJf

R L j j

XL = Xc = R when f = fe

Z = R, (Elf + Ehf) = Eg when f = 0 to 00

L = R/(21Tfe)' C = (21TfeR)-1

EC(max) = EL(max) = Eg

IL(max) = IC(max) = (Eg/R)

t fe = (2V21TCR)-1

fe = R/(V21TL) '----+-_---">---4

R T XL = Xc =V2R when


L = R/(21TfeV2), C = (21TfeRV2)-1

EC(max) = EL(max) = 1.272 Eg

IL(max) = IC(max) = 1.029 (Eg/R)

feNotes:

C R Elf

j

<D C = Capacitance, E = rms Voltage Magnitude, E = Polar or Rectangular form of Voltage, 1= rms Current Magnitude, L = Inductance, R = Resistance, X = Reactance, Z = Polar or Rectangular form of Impedance

47

fe = (-..fi1TCR)-1

f = R e 21TL-..fi

XL = Xc = R/-..fi when f= fe

Crossover Frequency


L = R/ (21Tfe-..fi) , C = (y'21TfeR)-1

EC(max) = EL(max) = 1.029 Eg

IL(max) = Ic(max) = 1.272 (Eg/R)

3dS Down Frequency

fe = R/(21TL)

feNotes:

Cutoff Frequency

ITcr IT IT

® EC = Capacitor voltage, Eg = Generator voltage, EL = Inductor voltage, Ehf = High freq. voltage, Eif = Low freq. voltage, Ic = Capacitor current, IL = Inductor current

® x-I = llx @ 1r = 3.1416,.J2 = 1.414, 21r = 6.2832,..j21r = 4.443

48

Exponential Horn Formulas f' fFC fFC = Symbol for flare cutoff frequency.

Flare Cutoff

Frequency

fFC = In an exponential hom of infmite length, the frequency below which no energy is coupled through the hom.

fFC = .5 to .8 of the lowest frequency of interest in the usual exponential hom.

fFC = vj{l8.13 Q2A)

f FC =vj(18.13~) f FC = vj (18.13v'Q;")

fFC = (mv)j(41T)

fFcNotes:

22A, 22d, 22r = Length between points on the horn center line of double cross sectional area, double diameter, and double radius respectively. m = Flare constant = .6931/22A = .6931/v'QM v = Velocity of sound"" 13,630 in./sec, 1136 ft/sec, 346.3 meters/sec, 34,630 em/sec at 25°C

f' = The lowest frequency of "satisfactory" hom loading due to area of hom mouth.

f' = Frequency at which:

1. Mouth diameter equals! wavelength 2. Mouth circumference equals one wavelength 3. Mouth diameter equals t wavelength 4. Mouth diameter equals t wavelength 5. Mouth diameter equals ~ wavelength

Low frequency horns are almost always compromised to use criteria 1, 2 or 3. Wavelength-See A.

49

fo ~ (21TyLC)-1 fo ~ (21TRCJ'6)-1

(LC Oscillator)

Frequency of

Oscillation or Output

(Phase Shift Oscillator with three equal RC stages. May be phase lead or phase lag type)

(Wein Bridge Oscillator) (when Rl C1 = R2 C2)

See-Active Circuits

Output frequency of a electromechanical generator

fo = N pp Ct"s) (Number of pairs of poles times rev ./sec)

fr = Symbol for frequency of resonance.

Resonant Frequency Definitions

fr = 1. The frequency at which the circuit acts as a pure resistance. In a series circuit, the frequency at which the impedance is lowest. In a parallel circuit, the frequency at which the impedance is highest.

2. The frequency at which the inductive reactance equals the capacitive reactance.

Note: Defmition 2 is commonly used due to simpler mathematics, but in many low Q circuits, it is a poor approximation.

When Q is high, the difference between the defmitions is negligible.

50

Series Resonant Frequency

fr = (21Ty'LC)-1

@fr Z =0

Def. 1 & 2

XL =Xc, Q=oo

fr = (21Ty'LC)-1 Def. 1 & 2

@fr Z= R

8 z = 0° , XL = Xc,

fr = [(LC) - (L/R)2] -t /(21T) (LC>L2/R2)

fr ~ (21Ty'LC)-1

@fr Def. 1

Z = ~R/Xl) + R-1]-1 8z = 0°

Resonant Frequency, Series Resonance

C L o----j~

Ideal C & L

Q=XLlR

Def.l

Xc = ~XLlR2) + XLI] -1 Xc ~ XL

Q = Xc [(R/Xl) + R-1] Q ~ R/XL

frNotes:

<D C = Capacitance, L = Inductance, Q = "Q" Factor, R = Resistance, Rc = Resistance in capacitive circuit, RL = Resistance in inductive circuit, Xc = Capacitive reactance, XL = Inductive reactance, Z = Impedance, () z = Phase angle of impedance

51

Series Resonant

Frequency Resonant Frequency,

Series Resonance

fr = V(LC) 1 - (CR)-2/(2rr) Def. 1 exception = Fx

fr ~ (2rry'LC)-1

@ fr Definition 1 R

Z = [(R/X~) + R-1J-l

8z = 0°

XL = [(Xc/R2) + Xc?] -1 XL ~ Xc

Q = XL ~R/X~) + R -IJ Q ~ R/Xc

fr = I ~R~C)-1 - L -IJ / KL/RD - CJ / (2rr) Def. 1 exception = Fx

fr ~ (2rry'LC)-1 ~ @ fr (Definition 1) Rc RL

Z = KRc/~) + Rc?] -1 + ~RL/xD + RL1J-l

8z = 0°

[(XL/Rt) + XL1J = [(Xc/~) + Xc?]

Q ~ [XL (RL1 + RC?)J-l

52

Parallel Resonant

Frequency

Resonant Frequency.

Parallel Resonance

fr = (21TVLC)-1 Def. 1 & 2

@fro Z = 00, {jz = 0°

Xc =XL, Q=oo

f = r (21TVLC)-1 Def. 1 & 2

@fr,Z= R, {jz = 0°

fr = V(Lq-I - (R/L)2/(21T) Def. 1 exception = Fx

fr "'" (21TVLC)-1

@fr (jz = 0°

Z=(Xt/R)+R

Xc = (R2/XL) + XL

Q = ~Xt/R) + RJ/Xc

fr Notes:

Z"",Xt/R

Xc "",XL

Q"'" XL/R

@ x-I = l/x, xt = .JX, x-t = 1/.JX, x-2 = I/x2

IJ:

® Def. 1 = fr Defmition 1 (max or min Z plus fJZ = 0°) @ wr = Resonant angular velocity = 211fr ® Lp, RCp, RLp = Parallel equivalent quantities of series quantities

See also-Q, Z, Y

53

Parallel Resonant Frequency

Resonant Frequency,

Parallel Resonance

fr = ~LC) - (CR)2] -r 1(21t)

fr ~ (21ty'LC)-1

Def.l 1

(-xrT exception

@ fr Definition 1

(J z = 0°

z = (xt:/R) + R

XL = (R2/Xc) + Xc

Q = [(X~/R) + R]/XL

Z ~X~/R

XL~Xc

Q~ Xc/R

fr = V[e1 - (RUL~ I[L - (R~Ci! (21t) exception = Fx

fr ~ (21Ty'LC)-1

@ fr Definition 1

(J z = 0°

IJ Def. 1

Z = (~X~/Rc) + Rc] -1 + ~XURd + RL] -1) -1 ~RVXc) + xc] = [(RUXd + XL] Xc ~ XL

Q ~ [wrLpCRL~ + Rc~)] -1

Q ~ yLI [CCRL + Rc)2]/C21T)

Q ~ XC/CRL + Rc)

54

Frequency of

fr Acoustical Resonance

fr = v/(2Q + 1.6d)

Q = (v/2fr) - (d/1.25)

d = (v/1.6fr) - (1.25Q)

fr = v /(2Q + 1.8.JA)

Q = (v/2fr) - (.JA/1.11)

A = V(v/1.8fr) - (1.11Q)

fr = v/(4Q + 1.6d)

Q = (v/4fr) - (d/2.5)

d = (v/1.6fr) - (2.5ll)

fr = v / (4Q + 1.8.JA)

Q = (v/4fr) - (.JA/2.22)

d =V(v/1.8fr) - (2.22Q)

Pipe Notes:

<D A = Cross sectional inside area of pipe d = Inside diameter of pipe II = Length of pipe v = Velocity of sound in air

@ v = 13 630 inches per second @ 25° C v = 1136 feet per second @ 25° C v = 346.3 meters per second @ 25° C

CD

Pipes ::c I'll

and .S! VI - CD Q. ...

Tubes Q.O «2 <D

@

®

®

<D

@

@)

®

® Also has secondary resonances @ 2fr, 3fr, 4fr, 5fr, etc. @) Also has secondary resonances @ 3fr, 5fr, 7fr, etc. ® A, d, II and v must all use the same unit of linear measure

55

CD Q.

ii: c CD Q.

0

-'1:1 CD : Q. 0 ii:u '1:1'1:1 CD C Q. CD Q. CD s C cn-S

011

Frequency of

fr ::c .~ '" Acoustical - 011 c. ....

Resonance 0. 0 «2

fr = 2070 [A/V2] t ~

fr = 1948.7v'd/V <D .S! N .. CS

® !:o.c 01;j-c-

V = d(1948. 7/fr ] 2 @ .cCGIGI Eat:i "i GI &..c

d = V[fr/1948. 7] 2 :J:a:_ ~

fr """ 1424v'd/V (Assuming speaker ..... resonance is much @ -c GlCD

V""" d(1424/fr ] 2 5!)(~.= lower than box @ °oGloC

d """ V[fr/1424] 2 resonance) UCDc%~

fr """ 2070 [(.285A t + A 2)/V2] t @ j( GI :;:

V""" [20702v'.285At + A 2]/f; ® ..... CD -cGlGla: GI ~ C

A2 """ [V2(fr/2070)4] - .285At @ t:~:S~

o Co "'CD I1.CI)U_

fr """ 1713/ (.85d2 + Q) [(Vtld~) - (.251TQ)] @ "C ~ ... GI ~:!! t .. ",.-

Q """ [(1913d2)2/(f;V2)] - .85d2 ® :s .. CD..c o£c%~

Cabinet Notes:

<D A = Area of opening (port), d = Diameter of opening (port), V = Internal volume of sphere.

@ d = Diameter of speaker opening, V = Internal volume of cabinet (neglect speaker volume)

@ At = Area of speaker opening, A2 = Area of port, V = Internal volume of cabinet (neglect speaker volume)

@) d2 = Diameter of speaker opening and duct opening, V t = Internal volume of cabinet including duct volume. V2 = Internal cabinet volume excluding duct, Q = Duct length.

® xt = ffx, x4 = (x2)2 ® A, d, Q and V must all use the same unit of linear measure.

56

F F = Symbol for farad.

F = Basic unit of capacitance.

Farad,

Force etc

F = Capacitance required to store one coulomb of charge at one volt potential.

F = Extremely large unit. Seldom used without a prefix symbol.

F = tlF • 106 (Typewriter-use uF)

F = nF • 109 (just coming into usage in USA)

F = pF • 1012 (tltlF is not recommended)

Note: The prefix symbol m (milli) should not be used with F due to long time previous use of m with F to indicate microfarads.

F = Symbol for magnetic, electrostatic and mechanical force.

F = Magnetomotive force when units are in gilberts or ampere turns [gilbert = 1.257 ampere turns (At)]

F = 1/>6{ where I/> = total flux and 6{ = reluctance

Repulsive Electrostatic Force

F = 9 • 109 [QI Q2 /d2 ] dynes

QI, Q2 = charge in coulombs on two bodies

d = distance in cm separating two bodies

OF = Symbol for degrees Fahrenheit.

OF = Unit of temperature. (USA)

F, Fn-See-NF (Noise Figure)

Fp-See-pf (power Factor)

57

Conductance Definitions and gG DC Formulas, Mutual Conductance

G = Symbol for conductance.

G = The ease with which direct current flows in a circuit at a given potential. The ease with which alternating current at a given potential flows in a purely resistive circuit. The reciprocal of resistance in any purely resistive circuit. The reciprocal of a pure resistance in parallel with other elements. The real ~

part of admittance. The reciprocal of the equiva- .,8 lent parallel circuit resistance in a series circuit. ·2

~ G = Conductance in units of siemens (S). c [old unit mho (n -lor U) is still common usage in USA]

G = A parallel circuit quantity which may be used as easily in parallel circuits as resistance is used in series circuits.

G = Rp1 !..sr in terms of polar impedance.

G = l/R

G = I/E

G = P/E2

G = 12/P

Gt = (Rl + R2 - - - + Rn)-l Series Circuits

Gt = G1 + G2 - - - + Gn Parallel Circuits

Gt = Rl1 + Ri1 - - - + R;;-l Parallel Circuits

'" C1I "E E ... o u. CJ C

gm = Symbol for mutual conductance or transconductance. See-Active Circuits

gm = Alp/AEg (Vacuum Tubes)

58

Q)

G :c

Conductance, B en ._ en E Series Circuits - Q) 0..., ~ 0. 0 Q)

<cz I-

G = (Rl + R2 --- + Rn)-l <D<Y R

G = R/ [R2 + (WC)-2] <D<Y@ C R

G = (WCZ2)-l <D<Y@ C Z

G = wC(sin 8)2 <D@

C 8 @)~

G = I/ER CD@ ER I

G=P/E~ CD@ ER P

G = 12/P <D I P

G = R/[R2 + (wL)2] <D@ L R

G = (wL)/Z2 CD@ L Z

G = (sin 8)2/(wL) <D@

L 8 @)~

GNotes:

<D G IS INTRINSICALLY A PARALLEL CIRCUIT QUANTITY. G DERIVED FROM A SERIES CIRCUIT IS THE EQUIVALENT PARALLEL CIRCUIT RESISTANCE IN RECIPROCAL FORM.

<Y x-l = l/x, x-2 = l/x2

@ w = 211f = 6. 283f = angular velocity @) sin, cos, tan = abbr. for sine, cosine and tangent ® ER = Voltage developed by a resistance ® I xl = Absoiute value of x = Magnitude of x ~ 0 maybeoE,OI,Oy orOZ,B maybe Be or BL, X may be Xc or XL

59

II)

G :is

Conductance, ." .2 en '" Series Circuits - II) E c. ... ... c. 0 II)

<tz I-

G = R/[R2 + X~] CD ~

~

G= Xc/Z2 CD N ~

G = (sin 8Z)2/XC (D@)(1) "" u ><

G = R/[R2 + Xi] CD ~ ...l

><

G=XdZ2 CD N ...l

><

G = (sin 8)2/XL (D@)(1) "" ...l ><

G = R/ (R2 + ~wL) - (wC}-t] 2) ~

(D@® ....:I u

G = I ~wL) - (wC}-t]/Z2 1 (D@ N ®®

....:I u

G = I(sin 8)2/ ~wL) - (wert] I (D@® "" ®®(1)

....:I u

...l

G = R/[R2 + (XL - Xc)2] CD ><~ ~ ...l

G = I(XL - Xd/Z21 CD® ><N u ><

(D® ...l

G = I(sin 8)2/(XL - xc)I >< @C!) ~""

60

III

G ::a

Conductance, III .2 VI ...

Parallel Circuits - III E Q. ... Q. 0 ..

III «z I-

G = G1 + G2 - - - + Gn ® G

G=R11 +R2"1 ___ +R;;:1 ~® R

G=VY'-- B'- @ B Y

G =VZ-'J. - B'- ~@ B Z

G = IB/(tan 8)1 @®~ B 8

G = VY'- - (wC)'- @@ C Y

G = VZ-'i - (wC)'- ~@@ C Z

G = l(wC)/(tan 8)1 @@

C 8 ®~

G =P/E2 E P

G = Vy2 - (wL)-2 ~@@ LY

G=VZ-'i-(wL) 2 ~@@ L Z

G = I [(wL) (tan 8)J -11 @@

L 8 ®~

G=Vy'J. - X-'- ~~@ XY

G=VZ-'- - X-'- ~~@ XZ

61

GI

G ::c

Conductance, co .2 '" '" Parallel Circuits - GI E Q. ... .. Q. 0 GI «z I-

G = I [X(tan 0)] -11 @@('!) X 0

G = Y(cos 0) @('!) Y 0

G = (cos O)/Z @('!) Z 0

G =VY'- - (BL - Bc)'- @ Be BL Y

G = VZ 2 - (BL - Bc)2 @@ Be BL Z

G = I(BL - Bc)/(tan 0)1 @®G> Be BL 0

G= yy2 _ [(WL)-1 - (wC)] 2 @@@ C L Y

G= yZ-2 - [(wL)-1 - (wC)] 2 @@@ C L Z

G = I [(wL)-1 - (wC)]/(tan 0)1 @@@ C L 0 ®('!)

G = VY'- - (XiI - XC1)'- @@ Xc XL Y

G = VZ-'- - (XiI - XC1)'- @@ Xe XL Z

G = I(Xi1 - XC1)/(tan 0)1 @@ Xc XL 0 ®G>

GNotes:

® In a purely parallel circuit, the values of parallel reactances are not relevant to the value of G.

@ A negative resultant under the radical sign indicates an error.

62

H = Symbol for henry.

H = Basic unit of inductance.

H Henry Unit, Magnetic Field Strength

H = The inductance which develops one volt from current changing at the rate of one ampere per second.

H=mH -103

H = IlH - 106

H = Symbol for magnetic field strength.

H = Magnetomotive force per unit length. Magnetizing force. Magnetic intensity.

H = Gilberts per centimeter (CGS Oersteds).

H = Ampere turns per meter (SI Aim).

H = FQ where F = magnetomotive force Q = length of magnetic path

H = Bill where B = magnetic flux density Il = permeability

H = B when magnetic path is air

H = B when permeability (Il) = 1

Ampere turns per inch = .495 Oersteds Oersteds = 2 _ 02 Ampere turns per inch

63

h h = Symbol for hybrid parameter.

See-Active Circuits

h = Symbol for height.

h = Symbol for hour.

h = Symbol for planck's constant.

Hz Hz = Symbol for hertz.

Hybrid Parameter, Height, Hour

Hertz

Hz = The basic unit of frequency equal to one cycle per second.

Hz = Unit often used with multiplier prefIxes. kHz = 103 Hertz (kilohertz)

MHz = 106 Hertz (megahertz) GHz = 109 Hertz (gigahertz)

Hz = cps = cis

Hz = 3600 per second

Hz = 21T radians per second

Hz = Vectorial revolutions per second.

64

I : Symbol for electric current.

Current Definitions

I : 1. The movement of electrons through a conductor. 2. The rate of flow of electric charge.

I: Current in amperes (A). (Coulombs per sec.)

I : ±Ide or lae(effeetive)

leff: I rrns

lae: \1\ : labsolute value: Irnagnitude

0I : Phase angle of alternating current. I : Complete description of alternating current. I: IpOLAR or IRECTANGULAR(lpOLAR : IRECT)

IpOLAR : II!..!..: Vectorial current IRECT : (±IR ±Ix j): Complex number current

where ±IR : Current through a real or an equiv. parallel circuit resistance and where '::Ix : Current through a real or an equivalent parallel circuit reactance.

IRECT : Complex number form of current which expresses the 0° or 180° and the + 90° or - 90° vectors which have a resultant vector equal to I POLAR .

IRECT : IR - (±Ix) j in this handbook (one exception) whereby +Ix identifies Ix as inductive and -Ix identifies Ix as capacitive.

IRECT : Mathematical equivalent of resistive and reactive currents in parallel regardless of actual circuit configuration.

i : Instantaneous value of current. (exception: iN: rms noise current)

65

I=EG

I=P/E

1= E/R

I =yPfR

1= EdRl = E2/R2 = En/Rn

I

1= (El + E2 --- + En)/(Rl + R2 --- + Rn)

1= ~ = vP:;TR; = vP;JR;. 1=v'{Pl +P2 ---+Pn)/(Rl +R2 ---+Rn)

It = 11 + 12 - -- + In

It=EGt=E(G1 +G2 ---+Gn)

It =Pt/E = {PI +P2 ---+Pn)/E

It = E(Ri l + RZ1 - - - + R;.I)

It=~

It =v'pt (R11 + R21 ---+ R~l)

I Notes: <D General

Direct Current Formulas

B = Susceptance, C = Capacitance, e = Instantaneous Voltage, E = Voltage (dc or rms), f = Frequency, G = Conductance, i = Instantaneous Current, I = Current (dc or rms), L = Inductance, P = Power, Q = Quantity of Electric Charge, Q = Quality or Q Factor, R = Resistance, t = Time, T = Time Constant, X = Reactance, Y = Admittance, Z = Impedance, E = Base of Natural Logarithms, (J = Phase Angle,-Continued on page 67

66

• I Transient Currents, Current Ratios

1= Q/t (I produced by charge Q for t sec.) -t

i = (E/R) (e RC) (E = Applied voltage)

i = .36788 (E/R) @t = RC (one time constant)

1= (ecC)/t (constant current) -t

i = (E/R) (e RC ) (E = Initial voltage)

i = .36788 (E/R)@ t = RC (one time constant)

1= (E - ec) (C/t) (constant current)

-Rt

i = (E/R) (I - e r:-) (E = Applied Voltage)

i = .6321 (E/R) @ t = L/R (one time constant)

Ip.p = (2 y'2) Irms = 2.828 Irms

Ipeak = (","2) Irms = 1.414 Irms

lay = [(2 V2)/7r] Irms = .9003 Irms

lay = (2/1r) lpeak = .6366 Irms

Irms = [1r/(2 v'2~ lay = 1.111 lay

Irms = effective current = de equiv. current

Irms = (1/v'2) lpeak = .707 lpeak

Irms = [1/(2V2)] Ip.p = .3535 Ip.p

I Noter:

<D Continued

·1 ~ Coaa aa.c UU

... 411

S fI .- aa U.c !~ til .-uC

a .. -t:; o III ~ .& ;:, ... "0 411 C C _w

1r = Circumference to Diameter Ratio, w = Angular Velocity or Angular Frequency.

67

Q)

I ::c

Series Circuit .~ '" '" Current - Q) E a. .... .. a. 0 Q) «Z I-

1= Ee wC CDeYQ) Ee C

1= EL/(wL) CDeYQ) EL L

1= ER/R (DeY ER R

1= Ee/Xc (DeY Ee Xc

1= EL/XL CDQ) EL XL

I=EY CD EY

1= E/Z CD EZ

I =VP/R CD PR

1= PICE cos 6) <D® E P 6

I = (E cos 6)/R <D® E R 6

1= Ej yR2 + [(wL) - (wC) IJ 2 <D® E CL R

1= E/VR2 + (XL - Xc>2 <DQ) E XcXL R

1- VP/(Z cos 6) <D® P Z 6

I Notes:

Q) Subscripts C = capacitive, E = voltage, g = generator, I = current, L = inductive, n = any number, 0 = output, p = parallel, R = resistive, s = series, t = total or equivalent, X = reactive, Y = admittance, Z = impedance

Q) Constants j = .J=I, = 90° multiplier, = mathematical i, e = 2.718, e-I = .36788, 7r = 3.1416, 27r = 6.2832, w = 27rf

@) Algebra 1

x-I = l/x, x-2 = l/x2, x"2 =.,jX, x-t = 1/.,jX, x(-y/z) = l/x(Y/z), Ixl = absolute value or magnitude of x

68

Q)

I :E

Current, .~ <II <II

Parallel Circuits - Q) E 0. .... .. 0. 0 Q) «z I-

It = E(BCl + BC2 - - - + Bcn) <D@ E Bc

It = E(BLl + BL2 - -- + BLn) CDG> E BL

It = Ew(C I +C 2 ---+Cn) CD@@ EC

It = E(G I + G2 --- + Gn) <D@ EG

It = [E(Li l +L;I __ -+L~I~/W <D@

EL @® It=E(Ri l +R;I ___ +R~I) <D@@ ER

It = E(Xc:\ + Xc\ - -- + Xc~) CDG>@ E Xc

It = E(Xi:l + Xi:~ --- + Xi:~) CD@@ E XL

I=EY CD EY

1= E/Z CD EZ

1= E(BL - Be) CDG> E Bc BL

1= E(Xi:1 - XCI) CD@@ E Xc XL

1= E V G2 + (BL - Be)2 CD@ E Be BL G

I=EVR-2 + (Xi:I - XCI)2 CD@® E Xc XL R

I Notes:

® Trigonometry sin = sine, cos = cosine, tan = tangent, tan-1 = arc tangent

@ Reminders ±o -- - use the sign of the phase angle ±X, ±B, ±IX ---identifies X, B and IX as capacitive or inductive

-X, -B, -Ix are capacitive +X, +B, +IX are inductive

o Division by zero is prohibited

69

I Current & Phase Important Notes

1. It should be understood that the phase angle of voltage , current, impedance and admittance is the same, one and only, phase angle in a given circuit. The fact that the sign of the voltage or impedance phase angle differs from the sign of the current or admittance phase angle means only that if the current leads the voltage, the voltage must lag the current by the same angle.

2. ±Ol = -(tOE) = -(±Oz) = ±Oy 3. When using the phase angle of impedance (or admittance),

a phase angle always exists when the circuit is reactive. The phase angle of voltage and current however can only exist for one of the two at the same time. When the voltage phase angle exists, the current phase angle must be 0° and when the current phase angle exists, the voltage phase angle must be 0°. This is explained by the fact the voltage uses the current as a reference and the current uses the voltage as a reference.

4. The same situation exists with voltage and current in rectangular form. When ''imaginary'' current exists, the voltage must be EI.!t... or E + jO and when "imaginary" voltage exists, the current must be IL!t or 1+ jO.

5. For practical problems, the best method of minimizing confusion and errors is to use the phase angle of the generator as the 0° reference. If the generator is a current source, the phase angle of the total current is always 0° and if the generator is a voltage source, the phase angle of the total voltage is always 0°.

Note: The rectangular current of a series circuit driven by a voltage source represents the currents through an equivalent parallel circuit. The rectangular voltage of a parallel circuit driven by a current source represents the voltages across elements of the equivalent series circuit.

70

IRECT Series Circuit Definitions & Formulas I Current & Phase,

Series Circuits

I = The magnitude and phase angle of the current developed by a voltage applied to a circuit. (OEg = 0°) See also-O

IpOLAR = Ij±OI = I/-(±Oz)

IRECT = 1. The 0° and ±90° currents which produce a resultant equal to IpOLAR. 2. The current through resistance and net reactance in parallel. 3. The current through the parallel equivalent resistance and net reactance of a series circuit. (Note: Only one current is possible in a series circuit.)

IRECT = IR - (±Ix) j

IRECT = [I cos 01] - [- I sin(±OI~ j

IRECT = [I cos Oz] - [I sin(±Oz~ j Note: The above rectangular form is strongly recommended for most uses. The negative sign will always identify the complex quantity as current or admittance and as a parallel equivalent quantity. The use of 8 Z eliminates the double change of signs often needed and maintains the identity of the reactive quantity at all times.

Note: The rectangular form of current has been used by some as a substitute for rectangular admittance (VRECT) for solving series circuits in parallel. It should be noted that if the assumed Voltage is one, 'RECT and VRECT are identical in meaning and method except for the names of the quantities. When E = l/.ff:..., 'POLAR = VPOLAR, 'RECT = VRECT, IR = G, IC = BC' IL = BL, -IX = -B, +Ix = +B and ±Ix = ±B.

Note: Use formulas on following page to obtain 'POLAR then convert to 'RECT using above formulas.

71

IpOLAR and IRECT Series Circuits I Current & Phase,

Series Circuits

All Series Circuit I Fonnulas

IpOLAR = I/±OI .. c CD

IpOLAR = I/-(±Oz} .. ..

.. ::J C Col

IRECT = [I cos Oz] - [I sin(±Oz~ j !:t: .. ::J ::J Col

IR = I cos Oz U .!: .. Col c_

Ic = II sin(-Oz}1 CD CD

(Ic=-Ix) ii= > III .~ i GI

IL = I sin(+Oz} (IL = +Ix) 0' ::a W =' III

±Ix = I sin(±Oz} ii .~ .!::! ... = 0' -GI c.. ..

... E .. fW c..o

IRECT = IR - (±Ix) j III .. «Z ~ D. 0

N

1= P/(E cos Oz) (D<3>® "" ~ ~

N

1= (E cos Oz}/R (D<3>® "" ~ ~

N

1= E/Z (D "" N ~

I=E/y'R2 + [(wL)- (WC)-l]2 (D<3>@ ~ ...:l

±OZ = tan-1 ~wL) - (wC)-l]/R @)®@ u ~

~ 1= E/VR2 + (XL - Xd2 (D<3> ...l

>< ±OZ = tan-1 [(XL - Xd/R] ®® .;;

~

"" (D<3>@) ...l

1= I(E sin OZ}/(XL - Xdl ®® >< .;; ~

72

Current & Phase, Parallel Circuits I

Resistive & Reactive Currents In Parallel

I = The magnitude and phase angle of the current developed by the application of voltage to a parallel circuit. (8E = 0°)

IpOLAR = 1& = 1/8y = 1/-(±8z)

IRECT = 1. The 0° and ±90° currents which have a resultant equal to IpOLAR. 2. The resistive current and the reactive current in parallel.

IRECT = IR - (±Ix)j = [I cos 8zJ - ~ sin(±8z~ j

IRECT = [I cos 8IJ + [I sin(+8I~ j -IRECT = [I cos 8.a + [I sin(±8y ~ j Q)

:c .~ '" '" - Q) E c. ... .. 0. 0 Q)

<cz l-

1= (EG)/(cos 8y ) <D@® EG8y

1= PICE cos 8z) CD@® EP8 z

1= E/(R cos 8z ) CDQ)® ER8 z

I=EY CD EY8y

1= E/Z <D EZ8 z

I = VI~ + (IL - 1c)2 CD@

±81 = tan -1 [-(IL - Id/IRJ ®® IR Ic IL

1= VP/(Z cos 8z ) CDQ)® P Z 8z

73

I»

I ::c

Current & Phase, III .2 en en

Parallel Circuits - I» E c. ... c. 0 .. I»

<CZ l-

e.!)

1= EYG2 + (BL - Bc)2 <D@ .... ~

±OI = tan-I [-(BL - Bc)/GJ @® u ~

~

>-1= 1 [E(BL - Bc~ /(sin Oy)1 <D@ <:!:>

.... @® ~

±OI = ±Oy ® u ~

~

~

1= E y'R 2 + [(wL) I - (wC~ 2 <D@ ....:l

@@ ±OI = tan-I (-R [(wL)-1 - (wC~) ®®

u ~

<D@ N

1= (E [(wL)-1 - (wC~) /(sin Oz) <:!:>

@@ ....:l

±OI = -(±Oz) @® U ® ~

~

1= EyR 2 + (XLI - Xct)2 <DQ) .... @@ :><:

±OI = tan -I E- R(XLI - Xc? ~ u ® :><:

~

N

1= [E(XLI - Xc?~/[sinOzJ CD@ <:!:>

.... @)@ :><:

±OI = -(Hz) ®® u

:><: ~

74

I Parallel Complex Currents, Procedure Method

Procedure for those who are uncomfortable when working with rectangular form currents. Maintains positive identity of reactive currents.

Procedure:

1. Convert each 1/±01 to its equivalent parallel resistive current from:

IR = I cos 01 p

2. (IR )t = (IR )1 + (IR h --- + (IR )n p p p p 3. Convert each 1/±01 with a negative angle to its equivalent

parallel inductive current from:

ILp = I sin I-Oil

4. (IL )t = (IL )1 + (IL )2 --- + (IL )n p p p p 5. Convert each 1/±01 with a positive angle to its equivalent

parallel capacitive current from:

Ic = I sin(+OI) p

6. (Ic )t = (Ic )1 + (Ic )2 --- + (Ie )n p p p p 7. Convert totals back to a single polar form current from:

It = V(IRp),l + ~ILp)t - (Icp )t] 2

±Olt = tan-1 (- ~ILp)t - (Icp)t]/[IRp]t)

75

~

CJ)

It =

~ ~I1

cos

01

) +

(12

cos

O 2)

--- +

(In

COS

On)

J 2

+ (

[11

sin(±Ol~ +

~2 si

n(±0

2~ -

--+

[In

Sin

(±O

n)J)

2f 1

/2

OIt

= ta

n-1

[I s

in(±

O)J

t/ ~

cos

OJ t

SumofI1~' I

2&--an

dI

n&

It =

I~I1

cos

Od

-(1

2 co

s 02

~ 2

+ (~

1 si

n(±

OdJ

-[1

2 si

n(±0

2~)2

O

It =

tan

-1 ([

11 s

in(±Ol~

-~2

sin

(±02

~) /

~I1

COS

01

) -

(12

cos

02)J

Dif

fere

ntia

l o

f I1~ a

nd Id

!!.1..

S:.

."

CD

0 ..

..

... ::T

3

o c

CL

-II

) - Cl)

C')

c

0 3

3 f(O

'E.

c ~

:tC

')

CD

C

... ...

CD

...

::::I

CD

....

::::I

_ .....

2!.

... fA

-

Current and Phase, Complex

Circuits I IpOLAR = (EPOLAR)/(ZPOLAR)

II!.J. = (E/OE)/(Z/OZ)

I=Eg/Z,OI=Oo-OZ

IpOLAR = (EpOLAR) • (V POLAR)

I~ = (E&). (Y lOy)

1= Eg Y, 01 = 0° + Oy

{lRECT)t = {lRECT)l + {lRECT)2

Vector Algebra and/or Rectangular

Form Method

{lRECT)t = [IR + (±190o)jJ 1 + [IR + (±190o )jJ2

(IRECT)t = (IR1 + IR2)+ [(±190o)1 + (±190°)2] j

(IRECTh = [(lpOLAR)l]RECT + [{lPOLAR)2]RECT

{lRECT)t = [1 1fu]RECT + [ld~l]RECT {lRECT)t = [(11 COS 0 d + (12 COS 02~

+ [(11 sin ±Od + (12 sin ±02)] j

(IRECT)t = (lR)t + [(±190o)t] j

It = v'(lR)l + (±190o)l

(±Ol)t = tan-1 [(±190o)t!(lR)t]

17

'" E

~

10 = Ig/ ~R2/R1) + 1] 81 =8z =0°

o

10 = Ig / [R VR-2 + Xc2 ]

81 = 8z = tan-1(R/-Xc) o

10 = Ig/ [R VR-2 + Xi:2]

81 = 8z = tan-1(R/+Xd o

10 = Ig / [Xc VR-2 + Xc2 ]

81 = 8z + 90° o

810 = ~an-1(R/-Xc)] + 90°

10 = Ig/[XL VR-2 + Xi:2 ]

81 =8 z - 90° o

810 = &an-1(R/+XL~ - 90°

Output Current & Phase

Note: ---ID- = Infinite impedance current source

78


8::: :I;Z)/R "~ ,~ 'f +'" 10 = Ig/[Rv'R 2+(XLl_XC1)2]

8Ia = tan-l [R(XLI - Xc?)]

79

10 = (lgZ)/(XL - Xc)

81 = 8z - (±900) o


10 = Ig/ [(XL - Xc) v'R-2 + (XL - Xd 2 J

810 = (tan-1 [R/(XL - XdJ) - (±900)

10 = Eg/Xc

81 = -(-90°) = +90° o

10 = Eg/XL

81 = -(+90°) = -90° o

10 = Eg/Z

81 =-(±8z) o

10 = Eg/v'R2 + (XL - Xd2

810 = tan-1 [-(XL - Xd/RJ

80

'·9 of ,~ +" "~ ,! ,~ "FI'" ,.~ ,! 'f +'"

I CURRENT Vector Algebra

Vector Algebra AC Ohms Law

Eg = Egl..9:.... or Ig = Igl..9:....

1= Eg/Z = Eg/ZjO° - Oz = -(±Oz)

E = IgZ = IgZjOO + Oz = ±Oz

Z = Eg/I = Eg/IjOo - 01 = -(±Ol)

Z = E/Ig = E/IgjOE - 0° = tOE

Addition and Subtraction of Rect. Quantities (See also - ZRECT, Addition and Subtraction)

11 + 12 = 11(RECT) + 12(RECT)

= [IR - (±IX)jJl + [IR - (±Ix )jJ2

= [(IR)1 + (IRh] - ~±IX)1 + (±Ixh] j

11 - 12 = [(IR)1 - (IR)2] - ~±Ixh - (±Ixh] j

I+Ixl=IL I-Ixl=Ic

El + E2 = E1(RECT) + E2(RECT)

= [ER + (±EX)j]1 + [ER + (±EX)j]2

= [(ER)1 + (ERh] + [(±EX)1 + (±Exh] j

El - E2 = [(ER)1 - (ER)2] + [(±EX)1 - (±Exh] j

I+Exl = EL I-Exl = EC Note: The rectangular current of a series circuit represents current through an equivalent parallel circuit. The rectangular voltage of a parallel circuit represents voltage across elements of an equivalent series circuit.

81

Output Current Vector Algebra

Eg = EgLQ:.

Zj = Z

10 = Eg/Z

Ig = Ig/O°

Zj = [ZI1 + ZZ1]-1

Eg = IgZj

10 = [lgZd /Z1

Eg = ELQ:.

Zj = Z3 + [ZZ1 + ZI1] -1

Ig = Eg/Z j

10 = Eg/ (Zj [CZtlZ2) + 1J) IVA Notes:

CD Eg = Generator voltage Ig = Generator current Zj = Input impedance

Eo = Output voltage 10 = Output current Zo = Output impedance

82

II)

:E .2 III - II) a. .... a. 0 c:(z

IVA Notes

CD @

®

IVA Notes

CD @

®

IVA Notes

CD @

®

Output Cu rrent Vector Algebra

Ig = IgLQ:. Eg = IgZj

Zj = (Z41 + [Z3 + (Z21 + Zi1 r1ll)-1

10 = Ig[l- (Zj/Z4~/[(ZtlZ2)+ 1J

Eg = Eg/O° Ig = Eg/Z j

Zj = Zs + (Z41 + [Z3 + (Z21 + zi1r1l1)-1

10 = Ig (1 - [(Zj - Zs)/zJ) / [(ZtlZ2) + 1]

IVA Notes:

Q)

::a .~ a.!! a. 0 «z

IVA Notes

CD @

®

IVA Notes

<D @

®

@ Z, Zl' Z2, Z3, Z4 and Zs may represent resistances, capacitances, inductances, series circuits, parallel circuits, unknown circuits or any combination.

® All mathematical operations involving addition or subtraction must be performed in rectangular form. It is recommended that all mathematical operations involving multiplication or division be performed in polar form.

See also - Z, Vector Algebra See - Z to Z-l Conversion

83

• Thermal Noise

Current

i N(th) = Symbol for thermal noise current. (other symbols for thermal noise current are IN, ITH , IN(IH),iN,i th etc)

i N(th) = Thermal noise (white noise) current of resistance. (thermal noise current is always rms current regardless of symbol)

iN(th) = v' (4kTKBW)/R

k = Boltzmann constant (1.38' 10-23 JrK)

T K = Temperature in Kelvin Cc + 273.15)

R = Resistance generating thermal noise.

BW = Noise bandwidth in hertz. (Bandwidth with zero noise contribution from frequencies outside of bandwidth. SeeActive Circuits, Opamp, BWNOISE for correction factors for noise measurement with standard filters.)

iN(y'"ih) = Symbol for noise current per root hertz. [formerly called root cycle (y;:::)]

iN(th)(y'"ih) ~ 1.287 . 10-10 v'T7R @ room temperature (BWNOISE = 1 Hz)

Note: Above fonnulas do not include the excess noise current of resistors (that noise developed by dc voltage applied to resistors). Excess noise is l/f noise and may be of significance at frequencies below 1 KHz.

84

j J j = Symbol for Y-T

Imaginary Number, Joules

j = The imaginary unit of electrical complex numbers. The basic imaginary component of electrical rectangular form quantities. A unit identical to the mathematical imaginary unit i. A 90° indicator. A 90° multiplier. A mathematical quantity which rotates a number from the x axis (real numbers) to the y axis (imaginary numbers).

j = The imaginary unit used in all electronic calculations (instead of the mathematical unit i) to avoid confusion with the symbol for electrical current I or i.

j = Y-T = 1 /+90°

-j = -Y-T = 1/-90°

j2 = -1 = 1 j±180° or 1/+180°

_j2 = +1 = 11..st... j3 = -j = 1 /-90° or

j4 = + 1 = I I..st... or

J = Symbol for joules

1/+270°

1/+360°

J = A unit of work or work equivalent energy.

J = A unit of work equivalent to 1 watt· second.

J = A unit of work equal to .7376 foot· pounds.

J = A unit of work equal to . 102 kg • meters or 107 ergs (dyne' centimeters).

J = A unit of work equivalent to 9.478 • 10-4 Btu.

85

k Coupling Coefficient

k = Symbol for coupling coefficient (Capital K is sometimes used)

k = The ratio of mutual inductance to the square root of the product of the primary and the secondary inductances. The equivalent coupling coefficient provided by a discrete coupling element between two otherwise independent circuits.

k = M/v'Ll~

k = W~Mv'CI C2

k = LM ~LM + L1)(LM + ~)J-t

k = W~LMv'CIC2

k :::::: LM/ v'LI L2

k = - [(C IC ) + 1] -1 when 1 M c 1 =c, :+fl+:

Note: Circuits exhibit double peaks above critical coupling.

86

kK k = Symbol for kilo

(capital K is also used as a symbol for kilo)

Kilo, Dielectric Constant, Kelvin

k = A prefIx symbol meaning 1000. A multiplier prefIx used to indicate 103 units. Typical electronic uses include kilovolt (kV), kilowatt (kW), kilohertz (kHz) and kilohm (kQ). The symbol for kilohms (kQ) is often further abbreviated to k. In this form, it is most often capitalized. (K)

k = Symbol for dielectric constant (also K)

k = The capacitance multiplying effect of a specifIc material used as insulation between capacitor "plates" as compared to air. The ratio of capacitance of a given capacitor using a specifIc dielectric, to the capacitance of the capacitor using air as the dielectric.

k =1= A true constant since k varies somewhat with frequency, temperature, etc.

Typical dielectric constants

Air = 1 Paper = 2 - 3.5

Ceramics = 10 - 10 k+ Fiber = 5.0

Mica = 2.5 - 9.3 PVC =2.9

Polystyrene = 2.6 Waxes =2.3 - 3.7

k = Boltzmann constant = 1.3805 • 10-23 J/Ko (capital K is also used as a symbol for this constant)

K = Symbol for Kelvin (Kelvin temperature scale)

K = °c above absolute zero

K=oC+273.15

87

L L = Symbol for inductance. (self inductance)

Inductance Definitions

L = In an inductor, in a coil, in a transformer, in a conductor or in any circuit where a varying electric current is flowing; that property which induces voltage in the same circuit from the varying magnetic field at a polarity which opposes the change of electric current.

Note that RC circuits and active circuit "inductances" which produce a lagging current do not meet the above defmition and therefore cannot always perform in the same manner.

L = Inductance in henry (H) units. Note that the basic unit is of convenient size for

audio frequencies, but that millihenries (mH) and microhenries (J.LH) are more convenient at higher frequencies.

L = The symbol for an inductor on parts lists and schematics.

L = The symbol for inductive when used as a subscript.

L = One henry when a current change of one ampere per second develops one volt.

LM = Mutual Inductance (The symbol for mutual inductance is also M)

Ls = Series Circuit Inductance

Lp = Parallel Circuit Inductance

Q = Symbol for length

If = Abbreviation for low frequency

88

Q)

L ::c

Inductance, ca .!:! en en

Series Circuits Q. Q) E .... .. c. 0 Q) «2 I-

Lt = Ll + ~ - - - + Ln CD L

~ = Lt - Ll

Lt = [(Xdl + (Xd2 --- + (XdnJ/w CD

XL

Lt = ~+Xl)+(+X2)---+(+Xn~/w +X

L = R/(wD) Series reactive element CD DR must be inductive

L = (QR)/w Series reactive element CD Q R must be inductive

L = (R tan {}z)/w {}z must be a

CD R {}z positive angle

L = (Z sin {}z)/w {}z must be a

CD Z {}z positive angle

Series to Parallel Conversion

Lp = [R~/(W2Ls)J + (Rs/w) CD Ls Rs

Lp = [w(Z-1 sin {}z~ -1 ({}z must be @ Z (}z positive)

LNotes:

CD BL = Inductive susceptance, +B = Inductive susceptance, C = Capacitance, D = Dissipation Factor, E = rms Voltage, e = Instantaneous voltage, I = Current, LM = Mutual inductance, Lp = Parallel circuit inductance, Ls = Series circuit inductance, Q = Length, M = Mutual Inductance, N = Number of turns, n (subscript) = Any number, Q = Quality, Merit or Storage Factor, R = Resistance, r = Radius, T = Time constant, W = Work, XL = Inductive reactance, +X = Inductive reactance, Y = Admittance, Z = Impedance, () = Phase angle, w = Angular velocity = 21Tf, dildt = Current rate of change.

® x-I = llx, Ixl = Absolute value or magnitude of x

89

G)

L jS

Inductance, co .2 en '"

Parallel Circuits - G) E Q. ... .. Q. 0 G) «z I-

4 = [W(BLl + BL2 --- + BLn~-1 <D BL

4 = w-1 [(+B1) + (+B2) --- + (+Bn~ -1 0) +B

4= [L11 +1-21 ___ +1..;;"1J-1 <DO) L

4 = w-1 [(XL1)t + (XL1h --- + (XL1)nJ-1 <D XL

4 = w-1 [(+XI1) + (+Xi1) --- + (+X;;-1)J-1 0) +X

L= (DR)/w Parallel reactive element

<D D Rp must be inductive

L = I [wG(tan 8y )J -11 8y must be a <DO) G 8y negative angle

L= R/(wQ) Parallel reactive element

<D QR must be inductive

L = R/(w tan 8z ) 8z must be a <D R 8z positive angle

L = [wY sin 18y 11-1 8y must be a <DO) Y 8y negative angle

L = Z/(w sin 8z ) 8z must be a <D Z 8z positive angle

L = E/(wI sin 1(11) 81 must be a <DO) E I 81 negative angle


ls = [(w2Lp/R2) + Lp1]-1 <D Lp Rp

ls = (Z sin 8z)/w (8z must be positive) @ Z 8z

90

CII

L ::is rn

Inductance, CQ E .2 rn .. Misc. Formulas - CII CII a. ... a. 0 ~

<1:2

L = l/(wBd CD BL

Lr = 1/(w2C) C L required for resonance CD

Lr = Xc/w Xc

L = XL/w CD XL

L = (2W)/I2 (W = Work equivalent CD I W stored energy)

L= R/T (T = time constant) CD RT

L = -e/(di/dt) e = instantaneous voltage di

di/dt = rate of change in CD e -

ampere/seconds dt

L:::::: (rN)2 /(9r + lOQ) when magnetic path is air

r = radius to center of winding CD Q N r

N = number of turns Q = length of winding

Coupled Series Inductances

Lt = L1 + L2 + 2M (fields aiding) CD L1 ~M

Lt = L1 + ~ - 2M (fields opposing)

Coupled Parallel Inductances

Lt = [(L1 - M)-l + (~ - M)-l] -1 CD L1 ~ M

(fields opposing)

91

M Mega, Mutual Inductance

M = Symbol for mega (also meg).

M = A prefIx meaning one million. A multiplier prefix used to indicate 106 units.

Typical uses in electronics include megahertz (MHz), megawatt (MW), megavolt (MV) and megohm (MSl).

Note: Megohm is often contracted to Meg and Mn is often contracted to M.

M = Symbol for mutual inductance (The symbol LM is also used)

M = The equivalent inductance common to both the primary and secondary windings of a transformer. In a circuit with two discrete inductors coupled by magnetic fIeld interaction, the equivalent inductance common to both inductors.

M=kyLpLs

M= kNpNs

M = (Lta - Lto )/4

MNotes:

k = Coefficient of coupling. Lp = Primary inductance. Ls = Secondary inductance.

4a = Total inductance with primary and secondary windings connected series aiding.

40 = Total inductance with primary and secondary windings connected series opposing.

Np = Number of primary turns. Ns = Number of secondary turns.

92

m Flare Constant, Exponential Horns

m = Symbol for flare constant (flaring constant)

m = In acoustical horns, a constant used in formulas to determine the area, diameter or radius at any distance from the throat, e.g., A = Ao€mx or S = So€mx. In an exponential hom of infinite length, a constant used in formulas to determine the frequency (flare cutoff frequency fFe) below which no energy is coupled through the hom.

m = Flare constant expressed in units of inverse inches, inverse feet, inverse meters, etc.

m = .6931/Q2A

m= .6931/~

m = vi .4804/Q2d

m = vi .4804/Q2r

m = (2.3025/QT_M) [log(AM/Ao)]

m = (4.605/QT_M) [log(dM/do~ m = (41r)/AFC

m = (4mFc)/v

mNotes:

22A, 22d, 22r = Length between centerline points of double area, diameter and radius respectively. 2T-M = Throat to mouth length. A, AM, Ao, Ax = Area, Area of mouth, throat and at distance x from throat respectively. do and dM = Throat and mouth diameter. fFC and hFC = Flare cutoff frequency and wavelength. I: = Base of natural logarithms = 2.718. v = Velocity of sound.::: 13630 in/sec, 1136 ft/sec or 346.3 meters/sec @ 25°C

93

nN n = Symbol for nano

Nano, Number, Newton, Neper

n = Preftx symbol meaning 10-9 unit. One thousandth of a millionth unit.

Typical usage includes nanoamp (nA), nanovolt (nY), nanowatt (nW) and nanosecond (ns).

n = Symbol for an indefmite number

N = Symbol for number, number of turns, etc.

N = A pure number. Symbol for seldom used quantities where the natural symbol is in recognized use for another quantity. Np = Number of turns of primary winding of a transformer. Ns = Number of turns of secondary winding of a transformer. Npp = Number of pairs of poles in a motor or generator.

Np = (EpNs}/Es Ns = (EsNp}/Ep

Np = (IsNs}/Ip Ns = (IpNp}/Is

NLl = NLtyL1 /"4 (Tapped inductor turns)

Np = NsYZp/Zs Ns = NpYZs/Zp

NZI = NZtyZtlZt (Tapped secondary turns)

Npp = f/RPS (pairs of poles in a generator or sync. motor)

N = Symbol for newton (SI unit of force)

Np = Symbol for neper (logarithmic ratio unit)

Np = In yP2 /Pl = 8.686 dB

Np = In(E2/E1 } = In(I2/Id when impedances are equal

94

NF NI NF == Symbol for noise figure.

(noise figure is also known as noise factor)

Noise Figure,

Noise Index

NF == The ratio in decibels of device output noise to ideal device output noise with all conditions of operation specified.

See-Active Circuits

NI == Symbol for noise index.

NI == The ratio, in decibels, of rms microvolts of excess noise in a decade of frequency, to the dc voltage applied to a resistor.

NI == Noise index expressed in decibels (dB).

NI == 20 (log [(106EN(Ex»)/(V dC~)

( l(excess noise in microvolts rms)J~ NI == 20 log

(applied dc voltage)

NI == - 20 to + 10 dB carbon composition

NI == -25 to -10 dB carbon mm

NI == -40 to -15 dB metal fllm

NI == -40 to -15 dB wire wound

Notes:

<D Excess noise is noise in excess of thermal noise. @ Excess noise is Iff noise while thermal noise has equal output at all

frequencies. (white noise)

® (EN)EX = ""(EN)t - (EN)th (all voltages rms)

95

aO(/) Subscript Only Zero and Letter 0

0,0 = Subscript symbol for output, open circuit, zero time, zero current, characteristic, etc.

0= Output in Eo, 10 , Po, hob, hoe' hoc

0= Output in Cob, gos, Pob , Poe, Yoc ' Yos

o = Output in Cobo and Coeo (first 0)

o = Open circuit in Cobo and Coeo (last 0)

o = Open circuit in Cj bo' Cjeo

o = Open circuit in BV CBO, BV EBO, LV CEO

0= Open circuit in Ico, IcBo ,ICEO' lEBO

o = Open circuit in V CBO, V CEO, V EBO

o = Characteristic (impedance) in Zo

0= Oscillation (frequency) in fo

0= Resonant (frequency) in fo (fr is preferred)

o = Center (frequency of passband) in fo and Wo

0= Initial (at zero time) in Eo, etc.

o = Letter 0 in most printed material

o = Zero in most printed material

~ = The character used for many years to distinguish between zero and the letter O. Unfortunately, it has been used for both zero and the letter O. It also has been mistaken for the greek letter 1/>. The use of this character in formulas is not recommended.

96

Definitions p Definitions

P = Symbol for power

P = The rate at which energy is utilized to produce work. The rate at which work is done. The rate at which electrical energy is transformed to another form of energy such as heat, light, radiation, sound, mechanical work, potential energy in any form or any combination of any of the forms of energy.

P = Electrical power expressed or measured in watts (W)

Power is also expressed in dBm, microwatts (J,lW), milliwatts (mW), kilowatts (kW), megawatts (MW), etc.

Ppeak = Instantaneous peak power

P = Effective or average power

P = Ede • Ide = Erms • Irms (pure resistances only)

P =1= Eaverage • Iaverage

Psinewave = Power produced by sinewave voltage and current, not the wave shape of the power. (Power waveshapes are rarely used except for rectangular waves where the waveshapes of power, voltage and current are identical.)

P ae = P de in heating effect and all other transformations of electrical energy

P = Zero in all purely reactive circuits

P = Zero when the phase angle of the current with respect to the voltage equals ±90°

97

Power, DC Circuits p

Pt =Pl +P2 ---+Pn

P=E2G

P= EI

P = E2/R

P = 12/G

P= 12R

Pt =Pl +P2 ---+Pn

Pt = ~ER)l + (ERh - - - + (ER)n] I

Pt = ~ER)l + (ERh - - - + (ER)n] 2/(Rt )

Pt = E2/(Rl + R2 --- + Rn)

Pt = 12(Rl + R2 --- + Rn)

Pt =Pl +P2 ---+Pn

Pt = E2(G l + G2 ._- + Gn)

Pt = E(Il + 12 --- + In)

Pt =E2(Rll +R2'l ___ +R;;l)

Pt = IU(G l + G2 --- + Gn)

Pt =IU(Rl l +R2'l ___ +R;;l)

Note: G = l/R in all dc circuits

98

Power, DC Circuits

rA ;!::: :::J u ...

C3 rA Q)

".:: Q)

CI)

Power Ratios, Misc. Formulas

P peak = (ER)peak • (Ipeak)

P peak = (ER)~ak/R

p

P peak = (Ipeak)2 R (all series circuits)

Ppeak = 2 Paverage (sinewave)

P peak = P average (square wave )

Power Ratios, Misc. Formulas

Psquare = 2 Psine (with same Epeak or Ipeak)

P = (CE2)/(2t) Power from a capacitor charge for time t)

P = Wit (W = Work equivalent energy in joules or watt-seconds)

P = (U2)/(2t) (power for time t from energy stored in the field of an inductance)

PTH = Thermal noise power (any value resistance)

PTH = KBTKBW (Available PTH = PTH /4)

KB = Boltzmans constant = 1.38 • 10-23 JtK T K = Kelvin temperature, BW = Bandwidth

PWL = Power level in (acoustic) watts.

PWL = SPL + [20(1og r~ + _ 5 dB = dB above 10-12 watt (Freefield conditions) SPL = Sound pressure level in dB above 20 J1.N/m2, r = distance in feet

99

Power from p II>

E Dissipation or ..

Q)

Q Factor I-

P = EI cos(tan-1 D-1) E 1 D

P=Elcos(tan-1 Q) E 1 Q II>

P= [E2 cos(tan-1 D-l~/Z ....

EZD .:; u ..

P = [E2 cos(tan-1 Q)J/Z E Z Q C3

P=12Zcos(tan-1 D-1) «

1 Z D

P = 12 Z cos (tan -1 Q) I Z Q

P = [E cos(tan-1 D-1)r/[D(wC)-IJ E C D >-

P = Q(WL)-1 [E cos(tan-1 Q~ 2 II> -

E L Q Q) s:: . .:: 0 Q) ....

P = 12D(wC)-1 en .-

I C D Q) :l

:; .~ P = (I2 wL)/Q 1 L Q c.U

P = E2DwC ECD - >-Q)-

P = E2/(QwL) ELQ = s:: ~O

P= [I cos(tan-1 D-l~ 2/(wCD) d': .~

1 C D Q) :l

:; .~ P = wLQ ~ cos (tan -1 Q)J 2 1 L Q c.U

100

Gl

P ::c

Power, ~ ... . - ... Series Circuits - Gl E 0. ... .. 0. 0 Gl <cz I-

Pt=P1 +P2 ---+Pn <DGl P

Pt = I [(ER)l + (ERh - - - + (ER)J <DGl ER I

Pt = [(ER)l + (ER)2 - - - + (ER)n] 2/Rt <DGl ER R

Pt = I2(Rl + R2 --- + Rn) <DGl IR

P = EI pf CD E I pf

P = EI cos OI CDGl E I OI

P = (E pf)2/R CD E Rpf

P = (E cos Oz)2/R CDG) EROz

P = (E2 pf)/Z <D EZpf

P = (E2 cos 0z)/Z <DGl E Z Oz

P Notes:

CD Be = Capacitive susceptance, BL = Inductive susceptance, C = Capacitance, D = Dissipation Factor, E = rms or dc Voltage, Epeak = Instantaneous peak voltage, G = Conductance, I = rms or direct current, Ipeak = Instantaneous peak current, L = Inductance, pf = Power Factor, Q = Quality, Merit or Storage Factor, R = Resistance, t = Time, W = Work, X = Reactance, Y = Admittance, Z = Impedance, IJ = Phase angle, w = Angular velocity = 21Tf, tan = tangent, sin = sine, cos = cosine

101

CD

P :is

Power, co .2 en II>

Series Circuits - CD E a. .... .. a. 0 CD ctz I-

.... P = 12Z pf CD Po.

N .....

N

P = eZ cos 8z CD@ ."

N .....

P = (EZ-1)2 v' Z2 - [(wL) - (WC)-lr N

CD® ....:l U ~

CD@ ~~ P = (EZ-l)2v'Z:2 - (XL - XcP ® ~><

P = I(E pf)2 ~an(cos-l pf~ [(wL) - (WC)-l] -11 CD® ....:l u"" @® ~ Po.

P = ~E cos 8z)2 (tan 8z~ / ~wL) - (wC)-l] CD@ ....:l N

®® u." ~

CD@ '-

P = I(E pf)2 ~an(cos-l pf~ (XL - Xc)-ll ®@ ~~ ® ~><

CD@ N

P = KE cos 8Z)2 (tan 8Z~/(XL - Xc) ~~ ® ~><

P = 12 v'Z2 - [(wL) - (WC)-l] 2 N

CD® ....:l U .....

P = ev'Z2 - (XL - Xc):2 CD@ ~~ ..... ><

P = I ~2(XL - Xc~ /(tan 8z)1 CD@ N u."

® >< ...l ..... ><

102

CII

P ::c

Power, lJ '" .- <II E Parallel Circuits - CII c. .... a. 0 ..

CII «Z I-

Pt =P1 +P2 ---+Pn @@ P

P = E2(G1 + G2 --- + Gn) @@ EG

P = E2(Rt l + R2'1 --- + R~I) @@

ER ®

P = (IG)t!(GI + G2 - - - + Gn) @@ IG G

P = (IR); (Rt1 + R2"1 - - - + R~1 )-1 @@

IR R ®

P = Elt pf @@ E It pf

P = Elt cos OI @@ E It OI

P= E2Ypf CD EYpf

P = E2y COS Oy CD@ E Y Oy

P = (E2 pf)/Z CD EZpf

P = (E2 cos 0z)/Z CD@ E Z Oz

P Notes:

® Subscripts C = capacitive, E = voltage, G = conductance, I = current, L = inductive, n = any number, R = resistive, t = total or equivalent, X = reactive, Y = admittance, Z = impedance

® x-I = l/x, x-2 = 1/x2, Ixl = Absolute value or magnitude of x @) tan-1 = arc tangent

cos-1 = arc cosine

103

Q)

P :E

Power, ca .!::! en en

Parallel Circuits - Q) E c. ... .. 0. 0 Q)

<1:2 I-

P = (If pf)/Y CD It Ypf

P = (If cos {}y)/Y CD@ It Y {}y

P = IfZ pf CD@ It Z pf

P = If Z cos {}z CD@ It Z {}z

EBc P = E2yy2 - (BL - Bc)2 CD@ BL Y

ECL P = E2 y'Z-2 _ [(wL)-1 - (wC~ 2 CD® Z

CD@ EX<; P = E2yZ 2 - (XLI _ XCI)2 ® XL Z

CD@ EBc

P = I [E2(BL - Bd] / [tan (cos-I pf~ I ®@ ® BL pf

P = I [E2(BL - Bc~ /(tan {}y)1 CD@ EBc ®® BL {}y

CD@ ECL

P = IE2 [(wL)-1 - (wC~ [tan (cos-I pf~ -II ®@ ® pf

P = IE2 [(wL)-1 - (wC~ [tan {}z] -II CD@ ECL ®® {}z

P Notes: ® Division by zero, tangent of ±90° and purely reactive circuits are prohibited.

104

II>

P :c

Power, ClI .~ en en

Parallel Circuits - II> E ll. ... .. ll. 0 II> «2 t-

CD@ EXc p= ![E2(XL1 - XCI)]/[tan(cos-1 pf~! @@

0 XL pf

P = IE2(XLI - XCI) (tan BZ)-ll CD@ EXc ®® XL Bz

CD@ I Be P = (It y-l )2yy2 - (BL - Be)2 @ BL Y

CD@ P = (ItZ)2y'Z-2 - [(WL)-l - (wC)r ICLZ

®

CD@ IXc P = (ltZ)2YZ-2 - (XLI - XCI)2 @ XL Z

CD@ I Be P = ! [(I pf)2 tan(cos- l pO] /(BL - Bd! @@

® BL pf

P = ! [(I cos By)2 tan By] /(BL - Bd! CD@ I Be @0 BL By

P = ! ~I pf)2 tan(cos- l pf~ / ~wL)-1 - (wC~1 CD@ I CLpf

@0

P = ! ~I cos BZ )2 tan Bz] / ~wL)-1 - (wC~ ! CD@ ® 011CLBz

CD@ IXc P= ! [(I p02 tan (cos- l pf~/(XLI - Xc?)! @@

® XL pf

P = ! [(I cos BZ )2 tan BZ]/(XL1 - Xc:?)! CD@ IXc @0 XL Bz

105

P PF pf Pica, Power Factor

p = Symbol for pico (pronounced peeko).

p = PrefIx symbol meaning 10-12 unit. Replaces old J1J1 prefIx.

Typical usage includes picofarad (pF), picosecond (ps), picoampere (pA), and picowatt (pW).

PF = Symbol for power factor.

pf = Symbol for power factor. (other symbols for power factor include: Fp , cos e, PF, P.F. and pJ.)

pf = The ratio of actual power of an alternating current to apparent power. The ratio of power in watts to voltamperes. The cosine of the phase angle of alternating current with respect to the voltage.

pf = Power factor expressed as a decimal or as a percentage.

pf ~ The inverse of Q factor when Q > 7

pf = A measurement more often than a calculation.

pf = The cosine of the phase angle when the angle is positive or negative, when the phase angle is current with respect to voltage or voltage with respect to current and when the angle represents the phase of impedance or admittance.

pf = A decimal number between zero and one, or a percentage between a and 100.

pf = One in purely resistive circuits and zero in purely reactive circuits

pf = The ratio of resistance to impedance

106

GI

pf :a Power Factor, B '" .- '" E Series Circuits - GI a. .... .. a. 0 GI «2 I-

pf= cos () (0 = 0E, OI or Oz) Q) 0

pf= R/Z Q) RZ

pf= (R-2 [(wL) - (WC)-l] 2 + 1)-t Q)@ CLR

pf= 11- ([(wL)- (wC)-l]/ZY Q)@ CLZ

pf= P/(EI) Q) EIP

pf= (RI)/E Q) EIR

pf= (PZ)/E2 Q) EPZ

pf = P/(I2Z) CD IP Z

pf= (~XL - Xc)/R] 2 + 1) -t Q)@ RXc XL

pf= VI - [(XL - Xc)/Z] 2 Q) I Xc XL Z

pfNotes:

(j) Be = Capacitive Susceptance, BL = Inductive Susceptance, C = Capacitance, E = rms Voltage, G = Conductance, 1= rms Current, L = Inductance, P = Power, R = Resistance, Rp = Parallel Resistance, Xc = Capacitive Reactance, XL = Inductive Reactance, Y = Admittance, Z = Impedance, 0 = Phase Angle, w = Angular Velocity, w =211'f

107

CII

pf :c Power Factor, <II

.2 U) '" Parallel Circuits - CII E c. ... .. 0. 0 CII «2 I-

pf= cos 8 (8 = 8E , 81, 8y or 8z) @ 8

pf= G/Y @ GY

pf= Z/Rp CD Rp Z

pf = ([(BL - Bc)/G] 2 + 1) -t @@ Be BL G

pf= 11 - [(BL - Bc)/yr @ Be BL Y

pf= (EG)/I @ EIG

pf= P/(EI) @ EIP

pf= E/(IRp) @ E I Rp

pf= (PZ)/E2 CD EPZ

pf = ([Rp(Xr:1 - XCI)] 2 + 1) -t @@ Rp Xc XL

pf = F-[ZcXr:1 - XCI)] 2 @ Xc XL Z

pf Notes:

@ x-I = l/x, x-2 = 1/x2, x-t = 1/-/X, cos = cosine

108

Q Q Factor, Quality Factor

Q = Symbol for Q Factor, Merit Factor, Storage Factor, Energy Factor, Magnification Factor and Quality Factor. (All names refer to the same factor. "Q" Factor is preferred)

Q = 1. The ratio of energy stored to the energy dissipated in inductors, coils, tuned circuits, and transformers. (Dissipation Factor which is the inverse of Q is commonly used for capacitors and dielectrics). 2. The tangent of the phase angle of alternating current with respect to the voltage in inductors. 3. In inductors at a given frequency, the ratio of reactance to the equivalent series resistance.

Q = A number from zero to infinity. (usually between 10 and 100)

Q = A factor used to calculate equivalent series or parallel resistance and a factor used to predict the voltage or current magnification of LC resonant circuits.

Real or Equivalent Real or Equivalent Resistance in Series Resistance in Parallel '" ...

0 with Reactance with Reactance ... u

~

Q = (wLs)/Rs Q = Rp/(wLp) "C c

Q = (XL)s/Rs Q = Rp/(Xdp

'" Q = l/Ds Q = l/Dp ... 0 ...

Q = l/(wCsRs) Q=wCpRp . (j II:J Co

Q = (Xds/Rs Q = Rp/(Xdp II:J

(.)

109

Inductors or Capacitors in

Series or Parallel Q

i----l i----l oll~~

L ____ -.J L ____ ---1

: i~,rTl i~f-,~fl L- _____ ..J '- _____ ..J

01 = rp/wLp 02 = rp/wLp

Qt = (Li l + L2"I)/ [(LI Qd- l + (L2 Q2)-I]

: i~f~¥l i~f~Tl '- _____ ..J L- _____ ..J

D1 = (wCprp)-1 D2 = (wCprp)-1

Qt=(C I +C2)j[(CIDI)+(C2D2~

a Factor

Note: For series circuits C, D, L & Q must be Cs' Ds, Ls & ~. For parallel circuits C, D, L & Q must be Cp, Dp, Lp & Qp' See Q Notes ®&®

110

Series Resonant Circuits

Q=co

Z= 0, BW=O

fr = (21Tv'LCtl

Q Resonant Circuit Q Factor

C L

o--------j ~ R

~ Ideal C &L

Q = (Q;l + Ds)-l 1----, 1------, o----+--""r/lrs ______ LO.L,~_-I-'wr slr---i H-o

Q = fr/(f2 - fd-3dB

fr = (21Tv'LCtl

L ____ ~

Q = L/(Rv'LC)

Q = (21TfrL)/R

Q = fr/(f2 -:: fd-3dB

fr = 61Tv'LCtl

Q = v'LC ~RC)-l + (R/L~

Q R:: v'LC /(CR)

Q = (fr)DEF.l/(f2 - fd-3dB

Qs = wrL,/rs

(fr)DEF.l = ~LC) - (L/R)2r t/(21T)

L ___ S.--.J Ds == fsWrCs

::: II !J :;~ IJ

C

Q Notes: <D BW = Bl1ldwidth, C = Capacitance, D = Dissipation Factor, fr = Frequency of Resonance, L = Inductance, R = Resistance, X = Reactance

111

Series Resonant Circuits

Q = [(RC) + (L/R~ / v'LC

Q ~ L/(Rv'LC)

Q = (fr)DEF.d(f2 - f1)-3dB

Q

(fr)DEF.l = [(LC)-1 - (CRr2] ~/(27T)

Q = [Q;1 + Ds + (RVLsCs/L~-1 Q = L/ [VLsCs(R + fLs + fes)]

Q = fr/(f2 - f1)-3dB

fr = (27TVLsCsyl

fL = (WrLs)/Qs' fe = Ds/wrCs

Q = [wrLs/(RLS + Res)] Note ®

Q ~ [27TfrL(Ri:1 + R;:? )r1

QNotes:

Resonant Circuit Q Factor

Note@

CDCont. 1T=3.1416, w=Angular Velocity (21Tf) wr=Resonant Angular Velocity (21Tfr)

@ x-I = l/x, x~ =.JX, x-~ = 1/.JX ® D, Q, Land C do not have exactly the same value when capacitors

and inductors are measured in the parallel mode. LsCs' DsOs = Series mode.

112

Parallel Resonant Circuits

Q=oo

Z = 00, BW = 0

fr = (21Tv'LCtl

Note @)

Q = (Q;;l + Dpfl Note ®

Q = fr/(f2 - fd-3dB

fr = (21Tv'LpCpt Q = (R v'LC) /L

Q = R/(21TfrL)

Q

Q = fr/(f2 - f1 )-3dB = fr/BW

fr = (21Tv'LCtl

Q = v'(L/CR2) - 1 exception = Fx

Q = fr/(f2 - f1)-3dB = fr/BW

Resonant Circuit Q Factor

Ideal C & L

L- ___ ...J

Dp = (rcpwrCp)-l Q p = rLp/wrLp

Note ®

(fr )DEF.l = v'(LC) 1 - (R/L)~21T)

exception = Fx

QNotes:

® D, Q, L and C do not have exactly the same value when measured in the series mode. Dp, Qp, Lp and Cp = parallel mode.

113

Parallel Resonant Circuits

Q = V(L/CR2) - 1

Q = (fr)DEF.tf(f2 - f1)-3dB

Q Resonant Circuit a Factor

1

(fr)DEF.1 = rrLC) - (CR)2T'/(211') Il 1 exception = X-,

Q = [(Lp/CpQp) + Dp + (Lp/RvLpCp )T1 Note ® Note ® r------, r------,

Q = fr/(f2 - fd-3dB I

fr = (211'VLpCpt1 R :

Q = [wrLp(RL~ + Rc~~ -1

Q ~ 'ILl [C(RL + Rc)2]

Note ®

I '-- ___ ...J '-- ___ ...J

Dp = {wCprp)-1 Qp = rp/wLp

~L ~RL

(fr>OEF.1 = V[C-1 - (Rt/L)]/[L - (R~C)JA211') exception = Fx

QNotes:

® Cpo Dp. Lp & Qp = Parallel or equivalent parallel values. Cs• Ds. Ls & Qs = Series or equivalent series values.

@) rs = Equivalent series resistance derived from ~ or Ds. rp = Equivalent parallel resistance derived from Qp or Dp.

G> Def. 1 = Resonantfrequency deimition 1. - See fro (f2 - f1)-3dB = 3 dB down bandwidth (half power)

@ Ls = Equivalent series inductance. RLs = Equivalent series resistance of inductor resistor. Res = Equivalent series resistance of capacitor resistor.

® Lp. Rep & RLP = Parallel equivalent of series quantities.

114

Qq Q = Symbol for quantity of electric charge

Electric Charge

Q = Quantity of electric charge. The amount of excess electrons or the amount of holes (deficiency of electrons).

Q = Electric charge expressed in coulomb (C) units. (Many in electronics feel uncomfortable in using the symbol C for coulombs since the unit symbol C (coulombs) is seldom used and the capacitance symbol (C) is often used)

Q = Electric charge in units equal to 6.242 • 1018 electrons

Q = The product of current and time in ampere • seconds

Q=CE

Q=It

Q = (2W)/E

Q=v'2CW (W = work equivalent energy in joules or watt

seconds)

Charge of capacitor C, t seconds after application of voltage E to series RC circuit.

Q=EC[l- e;~J (e=ln base = 2.71828)

Q = Schematic Symbol for transistor. See - Active circuits

q = The electric charge of one electron or 1.6 . 10-19 coulombs. (symbol e is also used for q)

Notes: x(-y/z) = (x-1 )(y/z) = ~(x-l)Y

(your scientific calculator will perform correctly with a negative exponent)

115

Definitions and Notes

R = Symbol for resistance

R Definitions and Notes

R = That property which opposes the flow of electric current by the transformation of electrical energy into heat or other forms of energy. The total opposition to the flow of direct current at a given voltage. The non-reactive part of the total opposition to alternating current of a given voltage. The real part of impedance. The reciprocal of conductance in purely resistive or in dc circuits.

R = Resistance in units of ohms (0.). (0. = Greek letter capital omega) kn = 1000 ohms, Mn = 1,000,000 ohms. kn is often contracted to K and Mn is often contracted to M.

R = Parts list symbol for resistor.

R = Rioo in terms of polar impedance

R = R + jO in terms of rectangular impedance

RNotes:

CD B = Susceptance, C = Capacitance, D = Dissipation Factor, E = dc or rms voltage, f = Frequency, G = Conductance, I = rms or direct current, L = Inductance, P = Power, Q = Quality Factor, X = Reactance, Y = Admittance, Z = Impedance, ~ = Delta, (J = Phase Angle, 1T = Pi, w = Angular Velocity, n = Ohm

@ Subscripts: C = capacitive, E = voltage, I = current, L = inductive, n = any num· ber, p = parallel circuit, r = resonant, R = resistive, s = series circuit, t = total or equivalent, x = unknown, Y = admittance, Z = impedance 1, 2, 3 = fIrst, second, third, A, B, C = first, second, third counterparts

116

Resistance, R DC Circuits

Rt = RI + R2 - - - + Rn Rx = Rt - RI

Rt = [(ER)I + (ERh -- - + (ER)n]/I

Rt = (ER)l/(Pi + P2 --- + Pn)

Rt = ~ER)I + (ER)2 -- - + (ER)n] 2/pt

Rt = (PI + P2 --- + Pn)/I2

Rt = (G1 + G2 --- + Gnri

Rt = (RI R2)/(RI + R2)

Rt = (Ril + R2"I --- + R~Iri

Rx = (RI Rt)/(R1 - Rt )

Rx = (Rt l - Ri l )-1

Rt = E/ [(IRh + (IR)2 - - - + (IR)n]

Rt =E2/(Pi +P2 ---+Pn)

Rt = p t / ~IR)I + (IRh - - - + (IR)n] 2

RNotes:

® sin = sine, cos = cosine, tan = tangent @) x-I = l/x, x-2 = 1/x2, x-1 = 1/.JX

117

en E ... CD I-

R en

:!::

E I ::;, u ... U en

E P CD .~

CD CI)

I P

G

en :!::

R ::;, u ... U ]1 iij

E I ... (II

0..

EP

I P

Equivalent Resistance from D and Q REQUIV.

Equivalent Resistance

Rs = Equiv. Series Rp = Equiv. Parallel '" E ... Resistance Resistance CD

l-

Rs = D/(wC) Rp = (wCDr1 DC

Rs = wLD Rp = (wL)/D DL

Rs = XeD Rp =Xe/D D Xc

Rs = XLD Rp =XL/D D XL

Rs = (WCQ)-1 Rp =wCQ QC

Rs = (wL)/Q Rp = Q/(wL) Q L

Rs =Xc/Q Rp = Q/Xc Q Xc

Rs = XL/Q Rp = Q/XL Q XL

Series Resonant Circuits Parallel Resonant Circuits

Rs = [DcI(21TfrC~ Rp = ( [ 21TfrCDc] DQ

+ [(21Tf.L)/QL] + ~21TfrL)/QL])-1 C L

Rs = [DCXc(@f r )] Rp = ([D/Xc(@fr)] DQ

+ [(XL(@fr»/QL] + [(XL(@fr»/QrJ)-1 Xc XL

Special Note: fr = (21T..jLC)-1

118

GI

Resistance, R ::a I'CI

Series AC .2 U') III

- GI E c. .... .. Circuits c. 0 GI «z I-

R t =R1 +R2 ---+Rn <Y R

Rx = R t - Rl

R=VZ~ - (wC) ~ <D@)(1) C Z

R = I ~an Oz(wC)r11 <D<Y®

C Oz @®®

R= ER/I CD<Y ER I

R= E~/P CD<Y ER p

R=p/e CD I p

R = VZ~ - (wL)~ <DC!> LZ

R= (wL)/(tan Oz) CD<Y@C!> L OZ

R=VZ~ - xt CD<Y Xc Z

R = IXc/(tan Oz)1 CD<Y®® Xc Oz

R=vZ:2 - xi <D<Y XL Z

R = XL/(tan Oz) CD<Y@ XL Oz

R = Z cos Oz <D<Y® Z Oz

RNotes:

® Ixl = absolute value or magnitude ofx

119

Q)

Resistance, R :is III

Series AC .~ en - Q) a. ...

Circuits a. 0 <cz

R = VZ2 - [(wL) - (WC)-1]2 CD@)Gl

R = I [(wL) - (wC)-I]/(tan Oz)1 (DG)®@) ®Gl(g)

R = (E cos 0r)/I (DG>®

R = (Et cos 0d2/P CDG>®

R =VZ2 - (XL - Xc)2 (DG)

R= I(XL - Xc)/(tan Oz)1 (DG)® 0>®®


Rp = Z/(cos Oz) CDG>

Rp = [(XL - Xc); /Rs] + Rs ®

RNotes:

® Phase angle may be ° z or Oy also 0E - Or or Or - 0E ® Division by zero at resonance prohibited (tan 0° = 0) Gl w = 21Tf

120

fI)

E .. Q)

I-

CLZ

C L Oz

E 1 Or

Et P Or

Xc XL Z

Xc XL Oz

Z Oz

Xc XL R

CD

Resistance,

R j5 IV

Parallel .2 til '" - CD E a..., AC Circuits a. 0 ...

CD «2 I-

R= l/G CD R t = (G1 + G2 --- Gn)-1 @ G

Rx = (Gt - G1)-1 @)

R t = (Rl R2)/(R1 + R2)

Rt = (R11 + R21 )-1 CD R = (R-1 +R-1 ---+R-1)-1 t 1 2 n @ R

Rx = (Rl Rt )/(R1 - R t ) @)

R = (R-1 - R-1)-1 x t 1

R = [y2 - B2] -i- <D@) BY

R = I(tan O)/BI CD@® B 0

R = [y2 - (wC)2Ti- <D@)(]) CY

R= [Z-2 - (wC)2ri- CD@)(]) C z

R = I(tan O)/(wC)1 CD®®G> C 0

R= E/IR <D@ E IR

R= E2/P CD E P

R=P/I~ <D@ IR P

121

III

Resistance, R :is «I

Parallel .!:! VI ... - III E

AC Circuits a. ... ... a. 0 III <cz I-

R = [y2 - (wL)-2]-~ @@)~ LY

R = [Z-2 - (wL)-2]-1 @@)~ LZ

R = IwL(tan 6)1 @®

L 6 ®®~

R= [Z-2 - X-2r 1 @@) XZ

R = IX (tan 6)1 @®®® X 6

R = [Y cos 6y r 1 @@@@) Y 6y

R = Z/(cos 6z) @@®® Z 6z

R= [y2 - (BL - Bd2r~ @@@) Be BL Y

R = I(tan 6)/(BL - Bdl @@®

Be BL 6 ®®

R = (y2 - [(WL)-l - (WC)rr1 @@)~ CLY

R = (Z-2 - [(wL)-l - (WC)rr1 @@)~ CLZ

R = I(tan 6)/ [(wL)-l - (wC)] I @®@)

C L 6 ®~®

122

til

Resistance, R j5

Parallel .~ III ... - til E 0...- ...

AC Circuits 0. 0 CD c(z I-

R = EIicos B) (DG)®® E It B

R = P / [It( cos B)] (DG)®® It P B

R= [Z-2 - (XLi - Xci )2r1 CDG)@ Xc XL Z

R = I(tan B)/(XLi - Xci)1 CDG)®@

Xc XL B ®®~

til

Series Equivalent Resistance of a j5 CII

Parallel Circuit. [Parallel to Series .2 ... ... - til E

Conversion (Transformation)] 0...- ... 0. 0 til c(z l-

Rs = (cos By )/Y CDG)® Y By

Rs = Z cos Bz CDG)® Z Bz

Rs = G/ [G2 + (BL - Bc)2] CDG) Bc BL G

R = [R (X-i - X-i )2 + R-iri s p L C p (DG)@ Xc XL R

123

il, Y, 1T, T Network Resistance R Complex Network

Resistance

Series Circuits in Parallel

R t = [(R1 + R2 --- + Rn)i1 + (Rl + R2 --- + Rn)i1 --

+ (Rl + R2 --- + Rn)~lrl

Parallel Circuits in Series

R t = (Ri1 + Ril __ - + ~l)il + (Ri1 + Ril ___ + ~l)il ---

+ (Ri1 +Ri1 ___ +R~l)~l

Delta to Wye (~ to Y) Transformation

to

1T section to T section Transformation

Rl = (RARB)/(RA + RB + Rc) Notes Applicable to this

R2 = (RB Rc)/(RA + RB + Rc) page CD Q) @)

R3 = (RARc)/(RA + RB + Rc)

RA = [(R1R2) + (R2 R3) + (R1R3)]/R2

RB = [(R1R2) + (R2 R3) + (R1R3)]/R3

Rc = [(R1 R2) + (R2 R3) + (R1R3)]/R1

124

Yto~

or T to 1T

Transformation

Second s8 Siemen

s = Symbol for second

s = Basic unit of time. 9 192 631 770 transitions between the two hyperfme levels of the ground state of the cesium-133 atom.

s = 1012 ps, 109 ns, 106 p.s and 103 ms

s = 1/3600 of an angular degree (decimals preferred)

s = Symbol for spacing

S = Symbol for siemens

S = Basic SI unit of conductance (G), susceptance (B) and admittance (Y) [The mho (n-1 or U) predominates for this unit in the USA]

S = The reciprocal of resistance

S = l/n = mho

S = Abbreviation of signal (Sig is preferred).

S = Symbol for standing wave ratio. (not recommended-use SWR or VSWR)

S = Symbol for cross-sectional area. (the preferred symbol is A)

s = Subscript symbol for series and secondary

s = Subscript symbol for source and short-circuited

125

t Time Definitions and Formulas

t = Symbol for time.

t = The duration of an event.

t = Time measured in seconds. (s or sec.) [time is expressed in picoseconds (ps), nanoseconds (ns), microseconds (J.ls) , milliseconds (ms), seconds (s), minutes (min.), hours (hr) etc]

t = l/f

t = (CE)/I

t = Q/I

Duration of one complete cycle of a periodic wave or of a periodic event.

Time required to charge capacitance C to voltage E with current I.

Time required to accumulate charge Q in a capacitance with current I.

Time required to charge capacitance C to voltage e through resistance R from source voltage E.

t = -RC(ln [1 - (e/E)])

Time required to discharge capacitance C through resistance R from voltage E to voltage e.

t = RC (In [1 - (e/E~-l)

Time after application of voltage E to series inductance L and resistance R for current to rise from zero to current i.

t = -LR-1 0n [1 - (iRE-I)])

tNotes:

x-I = l/x, In(x) = loge(x)

126

t T Temperature, Telsa, Tera

t = Symbol for "customary" temperature (int'l).

T = Symbol for Kelvin temperature (int'l).

T = Symbol for temperature (USA common usage).

Tc = Symbol for temperature in degrees Celsius CC).

T F = Symbol for temperature in degrees Fahrenheit CF).

T K = Symbol for temperature in Kelvin (K).

Tc = (TF - 32)/1.8

Tc = TK - 273.15

TF = 1.8Tc + 32

TF = 1.8TK - 459.67

TK =Tc +273.15

T K = °c above absolute zero

Temperature determination of copper wire and copper wire windings by resistance measurement.

T2 = [(R2/Rd(T I + 234.5)J - 234.5

Rl = Resistance at known temperature T 1

R2 = Resistance at unknown temperature T 2

T 1 , T 2 = Temperature in °c T = Symbol for telsa [SI unit of magnetic flux density

(magnetic induction)]

T = Symbol for tera (unit prefix meaning 1012 units)

127

T T C Time Constant, Temperature Coefficient

T = Symbol for time constant. [other symbols include: tc , Tc, TC, RC, script greek letter tau (1') etc.]

T = 1. The time required for a capacitance to discharge through a resistance to 36.8% of the initial voltage or for the current to fall to 36.8% of the initial current. 2. The time required for a capacitance to charge through a resistance to 63.2% of the final voltage or for the current to fall to 36.8% of the initial current. 3. The time required for the voltage developed by cutoff of current through an inductor to fall to 36.8% of the maximum value. 4. The time required after application of voltage for the current through a series connected inductance and resistance to rise to 63.2% of the final value. [The exact value of 36.8% and 63.2% is 100 e-1 and 100(1 - e-1)]

T = RC also (Mil) . (J.LF)

T=L/R

TC = Symbol for temperature coefficient (other symbols include a)

TC = In circuit elements or materials, a factor used to determine the changes in characteristics with changes in its temperature.

TC = A factor in decimal, percentage or parts per million form per degree temperature change. (temperature coefficient is almost always in 0c)

Notes: € = Base of natural logarithms (2.71828 ---), e-1 = lie. One part per

million = .0001%.

128

u Mu Substitute, Unit

u = Typewritten substitute for greek letter mu (p.) See-p.

u, U = Abbreviation of unit, ultra, etc.

v Velocity

v = Symbol for velocity

v = Rate of motion in a given direction. A vector quantity having both magnitude (speed) and direction with respect to a reference.

v = Velocity measured in various linear units per second.

v = fA (f = frequency, A = wavelength)

Velocity of sound in air

v~(1051 + l.ITF)ft/sec.

v ~ (331.4 + . 6T d meters/sec.

(1136 @ 77°F)

(346.3 @25°C)

Velocity of sound in fresh water

v = 1557 - [.245(74 - Td] meters/sec.

Velocity of electromagnetic waves in vacuum. (including light)

v = 2.997925 • 108 m/s (use symbol c for light)

129

v Volt, Voltage, Volume

v = Symbol for volt (unit of electromotive force)

V = Symbol for electromotive force (See-E for passive circuits. See also-V in active circuit sections)

V = The basic unit of electromotive force, potential or voltage. The electric force required to develop a current of one ampere in a circuit with an impedance of one ohm.

V = Unit often used with multiplier prefIxes

p.V == 10-6 V mV = 10-3 V

kV= 103 V MV= 106 V

V = ±Vdc or Vrms(magnitude) (exceptions noted)

VBE , Vee, VeE, etc.-See Active Circuits

V = Symbol for volume (cubic content)

V = The amount of space in three dimensions.

V = Volume measured in various units such as cubic inches (in 3), cubic feet (ft3), cubic centimeters (crn3) , cubic meters (m3), etc.

Volume required for Helmholtz resonator. (ported hollow sphere or box)

V = d[1948. 7 Ifr ]2 (d in x units, V in x3 units)

d = diameter of port.

fr = frequency of resonance in hertz

130

w W = Symbol for watt.

Watt, Work, Energy

W = Basic unit of electric power. A unit of power equal to a current of one ampere through a resistance of one ohm. (P = eR)

W = Unit often used with multiplier preflxes

JlW = 10-6 Watts mW = 10-3 Watts

kW = 103 Watts MW = 106 Watts

W = Symbol for work.

W = Symbol for energy. (Energy is potential work.) (The energy symbol E is rarely used in electronics thus avoiding confusion with emf symbol E.)

W = The product of power and time.

W = Work or energy in joule (J) units in electronics. (joules = watts· seconds) Other units include kilowatt hour (kWh), foot-pound (ft . Ibf), erg (erg) etc.

Energy stored in a capacitor charge

W= .5CE2

W = Q2/(2C)

W= .5QE

w = .5U2 Energy stored in the fleld of an inductance.

131

x Definitions

x = Symbol for reactance

X = That property of inductances and capacitances which opposes the flow of alternating current by storage of electrical energy. The imaginary part of impedance. The reciprocal of susceptance in purely parallel circuits. The non-resistive part of the total opposition to the flow of alternating current.

X = Reactance expressed in ohm (U) units.

X = Xmagnitude

Xc = Magnitude of capacitive reactance

XL = Magnitude of inductive reactance

- X = Reactance identified as capacitive, not a real negative quantity.

+X = Reactance identified as inductive, not a real positive quantity.

-X= I-Xl = Xc

+X= I+XI = XL

X = Complete description of reactance

Xc = Xc! -900 in terms of polar impedance

XL = XL! +900 in terms of polar impedance

Xc = 0 - jXc = 0 + (-Xc)j (rectangular impedance)

XL = 0 + jXL = 0 + (+Xd j (rectangular impedance)

Note that in (-Xc)j and (+Xdj, Xc and XL have become real negative and positive quantities with the same signs that are assigned to the magnitude quantities.

132

CD

X :c

Reactance, ,g '" General and Misc, - CD a. .. a. 0 «2

Xc = (WC)-l = 1/(21TfC)

XL = wL = 21TfL

X= Rs/D CD '" X=RsQ

.. @ ':;

u ® ..

X=Z when Rs = 0 U ® '" CD

XL - Xc = ±X '': CD en

I+XI = XL

I-Xl = Xc XL - Xc = 0 @ resonance

Xc = Bel = l/Be

XL = BLl = l/BL

Xc = (wC)-l = 1/(21TfC)

XL = wL = 21TfL CD '" ,'!:: :s

X=RpD @ u

® cJ X= Q/Rp ® :!

iii XLl - XCl = ±Xj;l = ±B @ J_

III a..

I+Xj;ll = XLl = BL

I_Xj;ll = XCl = Be

[XLl - XCl r l = 00 @ resonance

133

If

X :a

Reactance, IV .5:! fI) '" Series Circuits - cu E c. ... ... c. 0 cu <tz I-

(Xdt = w-l(C.l + Cil ... + C~l) CD@

-Xt = w-l(C.l + Cil ... + C~l) C

®®

(Xdt = w(Ll + ~ ... + Ln) @@ L

+ Xt = W(LI + L2 ... + Ln) ®®

(Xdt = (Xdl + (Xch ... + (Xc)n @@ Xc

-Xt = (-Xl) + (-X2) ... + (-Xn) ®® -X

(Xdt = (Xdl + (Xd2 ... + (Xdn <D@ XL

+Xt = (+Xl) + (+X2)··· + (+Xn) ®® +X

±X = (wL) - (WC)-l CD®® CL

IXI =VZ2 - R2 <D® RZ

±X = R [tan(±6z)] CD@

R6z @®

±X = XL - Xc <D@® Xc XL

±X = Z [sin(±6z)] CD@

Z6z @®

134

" x j5 Reacunce, .~ '" '" Series Circuits - " E a. ... ... a. 0 " «2 l-

I X I = V (E/I)2 - (P 112)'1 <D® EIP

IXI =V(E/I)'l - R2 <D® EIR

±X = (Ell) [-sin(±81)] <D@

E I 81 @@

±X = P-l(E cos 8)2 [tan(±8z)] <D@@

EP8z @®®

IXI = VZ'l - (P/I'l)2 <D® IP Z

±X = (P/12) [-tan(±81)] <D@

I P 81 @@


r 1 <D@@ ±Xp = Z[sin(±8 z)

®GI> Z8z

±Xp = ±X;1(±X; + ~) <D@@ Rs ±Xs @GI>

136

Q)

Reactance,

X :is ca

Parallel .~ en en - Q) E a. .... ...

Reactive Elements a. 0 Q) «2 I-

(Xdt = [(Bdl + (Beh --- + (Bdnr1 CD~ Be

(Xdt = [(-B1)+(-B2)---+(-Bn)f1 ® -B

(Xdt = [(Bdl + (Bd2 --- + (Bdn] ~ (D~ BL

(Xdt = [c+B1) + (+B2) - -- + (+Bn)fl ® +B

(Xdt=[w(CI +C2 ---+cn)f1 (D@ C ®

(Xdt = [w-1(L11 + 1.21 ---+ L~I)fl CD~ L ®

(Xdt = [(Xdl1 + (Xd2"1 --- + (Xd~l] -1 (D@ Xe

(Xdt = [c-Xl)-1 + (-X2)-1 ---+(-Xn)-lr1 ® -X

(Xdt = [(Xdl1 + (Xd2"1 --_+(Xd~lrl (D~ XL

(Xdt = [(+X1)-1 + (+X2)-1 --- + (+Xn)-I]-1 ® +X

(D@ ±X = [BL - Berl ®® Be BL

<a)

(D@ ±X = [(wL)-1 - (wC)r1 ®® CL

@

(D@ ±X = [XLI - X(;t r 1 ®® Xc XL

<a)

136

CII

X ::c

Reactance, '" .2 en CIt

Parallel Circuits - CII E c. ... ... 0. 0 CII <CZ I-

IXI = [y2 - G2rt CD® GY

±X = [-G tan (±8y )rl <D@® G8y @@GD

IXI = [Z-2 - R-2r t CD® RZ

±X = R[tan(±8z)]-1 CD@®

R8z @@GD

±X= [-Ysin(±8y~-1 <D@® Y8y @@GD

±X = [Z-I sin(±8z)]-1 <D@® Z 8z @@<iV

1

Ixi = [0/E)2 - G2]-, CD® EIG

±x = E[IL - lerl <D@®

E Ie IL @GD

Ixl = [0/E)2 - R-2r t <D® EIR

±x = - E[I sin(±8I )]-1 <D@® E 181 @<iV

137

CII

X 1i

Reactance, III .!: '" '"

Parallel Circuits a. OIl E ... .. Q. 0 011 «2 ...

IXI = [Z-2 - (p/E2)2r t CD® E P Z

±X = (E2/P) [tan(±6z)]-1 <D@® E P 6z @®@


±Xs = Z[sin(±6z)] <D@® Z 6z

±Xs = [C±Xp/R~) + (±Xpr1]-1 @®@ Rp ±Xp

X Notes:

<D General: B = Susceptance, C = Capacitance, D = Dissipation factor, E = Voltage, f = Frequency, G = Conductance, I = Current, L = Inductance, P = Power, Q = Q factor, R = Resistance, X = Reactance, Y = Admittance, Z = Impedance, IJ = Phase angle, w = Angular velocity, w = 21Tf, w = 6.283 --- f

CD Subscripts: c = Capacitive, E = Voltage, I = Current, L = Inductive, n = Any number, p = Parallel circuit, R = Resistive, s = Series circuit, t = Total or equiv., X = Reactive, Y = Admittance, Z = Impedance

® Mathematics: x-I = l/x, x-2 = 1/x2, xt = JX, x- t = l/JX, Ixl = Magnitude of x, 00 = Infinite

@ tan = tangent, sin = sine, cos = cosine, tan -1 = arc tangent, sin -1 = arc sine

® Reminders: ±B, ±X, ±IJ-use the sign of the quantity. I+BI = BL, I-BI = Be, l+xl = XL, I-xl = Xc· +IJ Z = Inductive circuit, -IJ Z = Capacitive circuit.

@ The reciprocal of zero may be manually converted to infmity. 00 • x = 00 when x *' 0, oo/x = 00 when x *' 00

The reciprocal of infinity may be manually converted to zero. o . x = 0 when x *' 00, O/x = 0 when x*,O

® Division by zero is prohibited.

138

y Y = Symbol for admittance

Admittance Definitions

Y = The total ease of alternating current flow at a given frequency and voltage. The reciprocal of impeddance. A quantity which in rectangular form is as useful for parallel circuits as impedance is for series circuits. The resultant of conductance and susceptance in parallel. The resultant of reciprocal resistance and reciprocal reactance in parallel.

Y = Admittance expressed in siemens (S) or mho (n-l )

units.

Y = IYI = YMAGNITUDE

Oy = Phase angle of admittance

YPOLAR = Y/±Oy = Z-l /-(±Oz)

YRECT = G - (±B)j = R;l - (±X;l)j

Y RECT = 1. The rectangular form of admittance 2. The complex number form of admittance 3. The mathematical equivalent of conductance

(G) and susceptance (B) in parallel 4. The mathematical equivalent of reciprocal resis

tance (R-1) and reciprocal reactance (X-I) in parallel.

Y RECT = An easy method of transforming a series circuit to a parallel equivalent circuit.

Y RECT = Complex quantity used to solve problems involving complex parallel circuits.

YRECT = A quantity that is identical to rectangular assumed current when the assumed voltage is one.

139

Notes y Notes

Y Notes:

<D General: B = Susceptance C = Capacitance D = Dissipation factor E = rms Voltage f = Frequency G = Conductance I = rms Current j = Imaginary number L = Inductance P = Power Q = Q factor R = Resistance X = Reactance Y = Admittance Z = Impedance € = Base of natural logarithms 1r = Circum. to diam. ratio o = Phase angle w = Angular Velocity

® Subscripts: C = capacitive L = inductive R = resistive X = reactive

® Constants:

E = voltage n = any number s = series circuit

Y = admittance

I = current p = parallel circuit t = total or equiv. Z = impedance

j = A = mathematical i = 90° multiplier € = 2.718 --- €-1 = .36788 ---1r=3.1416 21r=6.283---

w=21rf w=6.283---f

@) Algebra: x-I = l/x

x-t = 1/.JX x-2 = l/x2 xt = .JX

® Trigonometry: sin = sine

sin -1 = arc sine

® Reminders:

Ixl = absolute value or magnitude of x

cos = cosine cos-1 = arc cosine

tan = tangent tan-1 = arc tangent

± 0 - - - Use the sign of the angle ±X, ±Ix, ±EX' ±B --- + identifies the quantity as inductive

- identifies the quantity as capacitive (As terms in formulas, these quantities must be used as real positive or negative quantities)

<J) Cosine 0 : The cosine of either a positive or a negative angle is positive, therefore, cos Oz = cos Oy = cos 0E = cos 0I

140

Q)

y j5 Admittance, as

.~ en '" Series Circuits - Q) E 0. ... .. 0. 0 Q)

<2 I-

y= Z-l = liZ Q)@ Z

y = (R2 + [(wL) - (wC)-lrrt Q)@@ CL R

Y = I(Sin 8)/[(wL) - (wC)-l] I Q)@@ CL 8 ®G>®

y= liE Q) E I I

Y = [R2 + (XL - Xc)2]-' Q)@@ R XcXL

Y = (cos 8)/R Q)®G> R8

Y = i(sin 8)/(XL - Xc)i Q)@®

XCXL 8 G>®

Y = P/(E2 cos 8) Q)®G> E P 8

Y = (12 cos 8)/P Q)®G> I P 8

Y Notes:

ca> The reciprocal of zero may be manually converted to infInity. 00 • x = 00 when x * 0, oo/x = 00 when x * 00

The reciprocal of infInity may be manually converted to zero. o . x = 0 when x * 00, O/x = 0 when x * 0

® Division by zero is prohibited. A zero divisor will occur at resonance and/or in purely reactive circuits.

141

...

y j5 Admittance, .~ '" '" Parallel Circuits - III E 0. ...

~ 0. 0 III «2 I-

y = Z-l = l/Z <D@ Z

y = v'G2 + (BL - Bc)2 <D@ BCBL G

y = I(BL - Bc)/(sin By)1 <D<1>@

BCBL By @®

y= y'R-2 + [(wLrl - (wC)r <D®@ CL R

y = I [(wL)-1 - (wC)]/(sin Bz)1 <D<1>®

CL Bz @®®

y= I/E <D E I

Y = IG/(cos By)1 <D<1>@ G By

Y = v'R -'2 + (XLI - XCI)2 <D<1>@ R XCXL

Y= [RcosBz]-1 <D<1> R Bz @®

Y = P/(E2 cos Bz) <D<1>® P E Bz

Y = (If cos Bz)/P <D<1>® It P Bz

142

y Admittance and Phase, Series Circuits

Series Circuit Polar Admittance Formulas

YPOLAR = y / ±Oy = Z-I/_(±OZ)

Polar Impedance is Preferred

Series Circuit Rectangular Admittance Formulas

Special Note: Rectangular admittance is intrinsically a parallel circuit quantity. The rectangular admittance of a series circuit is the mathematical equivalent of reciprocal resistance and reciprocal reactance in parallel.

YRECT = G - (±B)j where I+BI = BL,

YRECT = Rpi - (±X-I)j where I+XI = XL,

YRECT = [Y cos Oy] - (Y sin [-(±Oy )]) j

YRECT = [Z-I cos Oz] - [Z-I sin(±Oz)] j

I-BI = Be I-Xl = Xc

G = Rpi = Y cos Oy = Z-I cos Oz Note (d)

±B = BL - BC = XLI - XCI p p

±B = Y sin [-(±Oy)] = Z-I sin(±Oz)

Note: Rectangular admittance is identical to rectangular current produced by a voltage of one except for the names of quantities. When E = 1, ipOLAR = YPOLAR, iRECT = YRECT, IR = G, IX = BL, IX = BC' ±Ix = ±B. L C

Note: The use of YPOLAR is not recommended unless used as a means of identification of a parallel quantity. Convert directly from ZpOLAR to YRECT See also-Z, (), G and B

143

CD

Admittance y :c ca .2 en '" and Phase, - CD E a. ... ~

Series Circuits c. 0 CD «z I-

Y = (R2 + [(wL) - (wC)-lrr t CD

/±8y = tan-1(- [(wL) - (WC)-l]/R) @

YRECT = G - (±B)j Q) ~

@) .....:l G = [(±Xi/Rs) + Rsrl ® u

±B = [(Ri/±Xs) + (±Xs)]-l ® <a>

±Xs = (wLs) - (WCs)-l

Y = [R2 + (XL - Xc)2r t CD

/±8y = tan-1 [-(XL - Xd/RJ @ ~

- R-1 (+X-1). Q) .... Y RECT - P - - p J @) ><:

R~l = Rs/[Ri + (±XJ2] ® ~ ±X~l = ±Xs/[(±Xs)2 + Ri] ®

<a> ±Xs = XL - Xc

y= Z-l, /±8y = -(±8z) CD @

YRECT = G - (±B) j ® N

<l::>

@) +1 G = Z-l COS 8z N

±B = Z-l sin (±8 z) ® ®

144

Admittance and Phase, Parallel Circuits

Note: G = Rj;l,

y ±B = (Xdj;l - (Xc)j;l

Y = VG2 + (BL - Bc)2

±Oy = tan-1 [-(BL - Bc)/G]

YRECT = G - (BL - Bc)j

±Oy = sin-1 [-(BL - Bc)/Y]

YPOLAR = Y!±Oy

YRECT = G - (±B) j

@ CI)

::c .~ <II <II

- CI) E a. .... .. a. 0 CI)

<tz I-

0 CD<Y ....l

@@ ~

u ~

>-CD<Y ....l

YRECT =Vy2 - (BL - Bc)2 - (BL - Bc)j @@ ~ u ~

Y = I(BL - Bc)/(sin Oy)1 >-CD<Y <:t:>

YRECT = (-(BL - Bc)![tan(±Oy)]) ....l

@@ ~

- (BL - Bc)j ~ u ~

Y= VR-2 + [(WL)-l - (wC)r CD<Y

±Oy = tan-1 (R[(wL)-l - (wC)]) ~

@@ .....:l

@@ U

YR~CT = R-1 - [(wL)-l - (wC)] j i

Y= Z-l

±Oy =sin-l(-Z~(wL)-l - (wC)]) CD<Y N @@ .....:l

YRECT = I' Z-2 - [(WL)-l - (wC)r @@ u

- [(wL)-l - (wC)] j

145


Note: G = Rj;l,

y ±B = (Xdj;l - (Xc)j;l

Y = 1 [(wL)-l - (wC)]! (sin Oy)1

YPOLAR = Y!±Oy

YRECT = G - (±B) j

<a> QI j5 <II .~ en '" - QI E c. ... .. 0. 0 QI «2 I-

YRECT = (_[(wL)-l - (wC)]![tan(±Oy)]) Q)<3> Q)@ >-

<:t>

- [(wLrl - (wC)] j @@ ~ u

0 Y = G/(cos Oy) Q)(Y >-

<:t>

YRECT = G - [-G tan(±Oy)] j @@ Co)

Y = YR .2 + (XLI - XC1)2 Q)(Y '"" >< ±Oy = tan-1 [-R(XLI - Xc?)] @@ u

>< YRECT = R -1 - (XLI - XCI) j

@ ~

Y = [R cos Ozr1 ±Oy = -(Hz) Q)<3> N

@® <:t>

YRECT = R-1 - [R-1 tan(±Oz)] j ® ~

Y= Z-l Q)(Y N

±Oy = sin-1 [-Z(XLI - XCI)] @@ '"" >< ® u

YRECT =yZ ~ - (XLI - X(1)2 - (XLI - X(1)j ><

146


y YPOLAR = Y/±8y

Y RECT = G - (±B) j

Note: G = Rpi , ±B = (Xdpi - (Xc)pl (jJ) GI

::a .~ ... '" - GI E

Y = I(XLI - X,;I)/(sin 8z)1 a. ... .. a. 0 GI «2 t-

Y RECT = ([XLI - x,?]/[tan(±8z)]) <D@ N

ct>

- (XLI - XCI) j @G> ...:I ><:

@® ~

Y RECT = [Ycos8y] - [-Ysin(±8y )] j <D@ >< ct>

G>® >-Y= Z-l, ±8y = -(±8z) <D@

N

@G> ct>

Y RECT = [Z-I cos 8z] - [Z-I sin(±8z)] j ® N

Y = It/E, ±8y = -(±8z) <D@ N

ct>

Y RECT = [(It/E) cos 8z] - [(It/E) sin(±8z)] j G>® -~

Y = P/(E2 cos 8z), ±8y = -(±8z) N <D@ ct>

Y RECT = (P/E2) - [(P/E2) tan(±8z)] j G>® j:L;

~

Y = (If cos 8z)/P, ±8y = -(±8z) <D@ N

ct>

Y RECT = [(It cos 8Z)2/p] - [Y sin(±8z)] j G>® j:L;

... -

147

Complex Networks In Parallel

Note: YRECT=G-(±B)j

y

The sign of(±B)j is real.

Parallel Complex Network Admittance

E ... CI)

I-

(Y RECT)t = (Y RECT)l + (Y RECT h -- - + (Y RECT )n

±Bt = ±Bl ± B2 --- ±Bn

(YRECT)t = Gt - (±Bt) j

(YRECT)l

(YRECTh

(YRECT)n

Y RECT Procedure applies to any circuit in parallel with others.

1. Convert each series and each parallel circuit to polar impedance using applicable formulas.

2. Convert each polar impedance to rectangular admittance from:

YRECT = [cos Oz/Z] - [sin(±Oz)/Z] j

3. The quantities inside the brackets represent G and ±B. Maintain the sign of B inside brackets. Do not simplify to ±jB.

4. Algebraically sum all ±B quantities. Sum all G quantities.

5. Convert to total polar impedance if desired from:

Zt = [Gr + (±Bt)2r t

(±Oz)t = tan-l [(±Bt)/Gt]

148

Conversions To Rectangular Admittance YRECT

ZPOLAR To V RECT

ZPOLAR = Z/±Oz, VRECT = G - (±B)j

VRECT = [Z-l eos Oz] - [Z-l sin(±Oz)] j

ZRECT To V RECT

ZRECT = Rs + (±Xs)j, VRECT = G - (±B) j

VRECT = [Rs/(±X~ + R~)] - [±Xs/(±X~ + R~)] j

Series R and X To VRECT

Rs = Series Rt, ±Xs = Series (XL - Xdt

VRECT = [Rs/(±X~ + R~)1 - [±Xs/(±X~ + R~)] j

Parallel R and X To V RECT

Rp = Parallel Rt , ±Xp = Parallel (XLl - xc;t)tl

VPOLAR To VRECT

VPOLAR = Y/±Oy, VRECT = G - (±B)j

VRECT = [YeosOy] - [-Ysin(±Oy)] j

149

Note@

Conversions From Rectangular Admittance

YRECT YRECT To ZPOLAR

Y RECT = G - (±B) j, ZPOLAR = Z/±8z

[ 2 2] -t / -1 ZPOLAR= G + (±B) tan [±B/G]

Y RECT To ZRECT

YRECT = G - (±B)j, ZRECT = Rs + (±Xs)j

ZRECT = [G/(±B2 + G2)] + [±B/(±B2 + G2)] j

YRECT To YPOLAR

YRECT =G- (±B)j, YPOLAR =Y/±8y

YPOLAR = VG2 + (±B)2 /tan-1 [-(±B/G)]

YRECT To Equiv. Series R and X

YRECT = G - (±B) j

Rs = G/(±B2 + G2), ±Xs = ±B/(±B2 + G2)

I-Xsi = Xc, I+Xsl = XL

YRECT To Equiv. Parallel R and X

YRECT = G - (±B)j

R = G-1 +X = +B-1 P ,- P -

I-Xpi = Xc, I+Xpl = Xc Note Gl>

150

y Vector Algebra AC Ohms Law

Eg = EgLQ:. or Ig = IgLQ:. (1 = 1 LQ:.)

E = Ig/V = Ig/Y /0° - (}y = -(±{}y)

I = Eg V = Eg Y /0° + {}y = ±{}y

V = l/Z = l/Z/O° - {}z = -(±{}z)

V = I/Eg = I/Eg/ {}I - 0° = ±{}I

V = Ig/E = Ig/E/O° - {}E = -(±{}E)

Z = l/V = l/Y/O° - (}y = -(±{}y)

ADMITTANCE Vector Algebra

Addition and Subtraction of Rectangular Admittance

VI + V2 = VI (RECT) + V2(RECT)

= [G- (±B)jJ. + [G- (±B)jJ2

= [GI + G2J - [(±BI) + (±B2)] j

'YI - V2 = [GI - G2J - [(±BI) - (±B2)J j

Gt = (Rj;I)t

±Bt = (±Xj;I)t

I-BI = Bc I+BI = BL

Bc = (Xc)j;1 BL = (Xdj;1 Note @

See also-Z, Vector Algebra See also-B, G, {}

151

ADMITTANCE Vector Algebra y

y = [y-l +y-l ___ +y-l]-l t 1 2 n

YVANotes:

Q)

:E .~ en 'Q..l!l 0. 0 «2

YVA

-(D -<J)

YVA

-(D -<J)

<D Admittance is a complex quantity requumg the mathematical operations of addition and subtraction to be performed in rectangular form. Rectangular form quantities may be multiplied like other binomials except that P = -1. Reciprocals or other division by rectangular form quantities requires the divisor to be rationalized by multiplication of both the divisor and the dividend by the conjugate of the divisor. (The conjugate of G - Bj is G + Bj). When using a calculator, it is easier to convert rectangular quantities to polar form for multiplication and division then reconverting to rectangular form for addition and subtraction.

152

ADMITTANCE Vector Algebra

Vi = V3 + (Vi1 + Vi1)-1

Vo = V1 + (Vi1 + V3"l)-l

Eo = Ig [1 - (V3/Vi)]/V1

y

Vi = V41 + [V3 + (Vi1 + Vi1)-1]-1

Vo = V1 + [Vi1 + (V3 + V4r 1]-1

Eo = Eg [1 - (VdV 4)] / ~(V 1 IV 2) + 1] YVANotes:

Ql

:c .~ 'Q.!l Q, 0 <CZ

@ Be or -B = Capacitive Susceptance, BL or +B = Inductive Susceptance, Eg = Generator Voltage *, Eo = Output Voltage *, G = Conductance (Parallel Circuit Reciprocal Resistance), Ig = Generator Current *, 10 = Output Current *, Rj;l = Parallel Circuit Reciprocal Resistance (Conductance), ±Xj;l = Parallel Circuit Reciprocal Reactance (Susceptance), Vi = Input Admittance *, V 0 = Output Admittance *, Zi = Input Impedance *, Zo = Output Impedance *, * = Vector (Phasor) characteristic.

153

z Z = Symbol for impedance

Impedance Definitions

Z = The total opposition to the flow of alternating current of a given frequency. A complex quantity having components of resistance and reactance. The ratio of applied alternating voltage to the alternating current flow through a circuit.

Z = Impedance expressed in ohm (il) units. Z = ZMAGNITUDE = IZI

e z = Phase angle of impedance Z = Complete description of impedance which in

cludes both magnitude and phase angle information.

ZPOLAR = Polar form of impedance = z/±e z ZPOLAR = The vectorial resultant of resistance (0°) and

reactance (±900). ZRECT = Rectangular form of impedance or the complex

number form of impedance. ZRECT = The 0° (resistance) and ±90° (reactance) vectors

in complex number form which have a resultant equal to polar impedance.

ZRECT = The mathematical equivalent of resistance and reactance in series. (The series equivalent of a parallel circuit)

ZRECT = R ± jX = R + (±X) j = R + (XL - Xc) j where 1+ Xl = XL and I-Xl = Xc

(ZRECT)-l = YRECT = G - (±B)j = Rj/ - (±X~l)j (ZRECT)-l = A parallel equivalent circuit. See-Y RECT

154

CD

Z :is

Impedance, Ia .2 en en

Series Circuits - CD E a. .. a.o ...

CD «Z t-

Z = VR:Z + [(wL) - (wCrI] 2 <D@)(]) CL R

Z = I [(wL) - (WC)-I]!(sin 8z )1 <D@Q) CL 8z @)G)®

Z = Ell CD E I

Z =..JR:Z + (XL - Xd:Z CDQ) R XCXL

Z = R/(cos 8z) <D@Q) R 8z

Z = I(XL - Xd/(sin 8z)1 CD@Q)® XCXL 8z

Z = ..JE~ + (EL - Ed2/I <DQ) ER ECEL I

Z = (E2 cos 8E )/P <D@Q) E P 8E

Z = P/(e cos 8l) CD@Q) I P 8l

ZNotes:

<D B = Susceptance, C = Capacitance, E = rms Voltage, G = Conductance, 1= rms Current, L = Inductance, P = Power, R = Resistance, X = Reactance, Y = Admittance, Z = Impedance, f) = Phase angle, w = Angular velocity

155

CD

Z ::c

Impedance, III .2 III <II

Parallel Circuits - CD E c. .. ... c. 0 CD <cz I-

z = y-l = I/Y <D® y

z = [G2 + (BL - Bd2r1 <D®® BCBL G

Z = 1 (sin Oy)f(BL - Bdl <D@®® BCBL Oy

Z = (R-2 + [(wL)-1 - (wC)]2r1 <D®c!) CL R

Z = I(Sin Oz)f [(wL)-1 - (wC)] I <D@® CL Oz ®(1)®

Z =EfI <D EI

Z = (cos Oy)fG <D@® GOy

I

Z = [R-2 + (XLI - XCI )2r' <D®® R XcXL

Z = R cos Oz <D@® R Oz

Z = I(sin 0 z)f(XLI - Xcl)1 <D@® XCXL Oz ®®

ZNotes:

@ cos = cosine, sin = sine, tan = tangent, cos- l = arc cosine, sin- l = arc sine, tan- l = arc tangent

156

III

I mpedance and

Z ::c

Phase, Single .~ '" '" - GI E Elements 8:l; ~

III «2 I-

ZPOLAR = Bel /-90° = 1-B-1 1 /-90° Q)®® Bc or-B

ZRECT = 0 + (-B-1)j ®~

ZPOLAR = Bi1 /+90° = 1+ B-1 1 /+90° Q)®® BL or+B

ZRECT =0+(+B-1)j ®~

ZPOLAR = (WC)-l /-90° Q)®~ C

ZRECT = 0 + (-wCr1 j

ZPOLAR = G-1!§:. Q)® G

ZRECT = G-1 + OJ ®CD

ZPOLAR = (wL) /+90° Q) L

ZRECT = 0 + (+wL) j CD

ZPOLAR = R /0° Q) R

ZRECT = R+ OJ ~

ZPOLAR = Xc /-90° = I-Xl /-90° Q)®~ Xc or-X

ZRECT = 0 + (-X)j

ZPOLAR = XL/+90° = I+XI/+90° Q)®~ XL or+X

ZRECT =O+(+X)j

ZNotes: ® Subscripts c = capacitive, E = voltage, I = current, L = inductive,

n = any number, p = parallel circuit, s = series circuit, t = total or equivalent, Y = admittance, Z = impedance

157

z/oz III

ZPOLAR ::i5

Impedance, .~ en en

Series - III E 0. ... .. 0. 0 II)

Circuits «z t-

Z = I R2 + [(wL) - (WC)-l] 2 <D@@ CL R

±Oz = tan-l ([(wL) - (WC)-l] /R) ®<1>®

±Oz = sin-l ([(wL) - (WC)-l J/Z) <D@@

CL Z ®C1>®

Z = I [(wL) - (WC)-lJ /(sin Oz)1 <DG)@

CL Oz ®G>®

Z =,yR2 + (XL - Xd2 <DG) R XCXL

±Oz = tan-l [(XL - Xd/R ] @®

IOzl = tan l[,yZ2 - R2/R] <D@@® RZ

ZNotes:

® x-l = l/x, x ~ = -IX, x-2 = l/x2, x-~ = l/-IX, Ixl = x magnitude or the absolute value of x

® Series resistance must equal zero. @) Series reactance must equal zero. <1> w = 211'f "" 6 .28f (f = frequency), j =...;:I = mathematical i = 90°

multiplier = imaginary quantity = y axis quantity = reactive quantity ® Reminders: ±I:/-Use the sign of the phase angle ±X, ±B-treat the

signs as real in all calculations except when converting to XL, XC, BL orBC· The signs of ± X and ± B identify these reactive quantities as inductive or capacitive. l+xl = XL, I-xl = XC, I+BI = BL, I-BI = Be XL, Xc, BL and Be are magnitudes, while ±X and ±B as used in formulas are "real" quantities.

158

z/Oz .. :c

Impedance, Z .2 .,. .,.

Series POLAR - .. E a. ... .. a. 0 .. Circuits «z I-

Z = R/(cos Oz) CDcrJ® R Oz

±Oz = sin-1 [(XL - Xd/Z] CD(])

XCXL Z ®®

Z = I(XL - Xd/(sin Oz)1 CD(])®® XCXL Oz

Z = E/I CD® ±Oz = ±OE ®@ E I OE

Z = VEi + (EL - Ed2/ I CD(]) ECEL ER I

±Oz = tan-1 [(EL - Ed/ER] ®®

Z = (E2 cos 0E)/P CD(])® ±Oz = ±OE ®@ E P OE

ZNotes:

® The phase angle of Z, Y, I and E (0 z, Oy, 01 and 0E) in a given circuit represent the same one and only one phase angle. ±O Z = ±OE = -(tOy) = -(±Ol)' The author does not recommend this use of OE and 01 where each uses the other as the reference phase. The author uses the generator Eg or Ig as the reference. See also-o

159

z/oz /II

ZPOLAR :is

Impedance, III .2 en '"

Parallel - /II E a. ... ... a. 0 /II

Circiuts «2 ~

Z = [G2 + (BL - Bd2r~ <Do)® BeBL G

±Oz = tan-1 [(BL - Bd/G] @®

Z=y-l <D@® ±Oz = sin-1 [(BL - Bd/Y] @® BeBL Y

Z = I(sin Oy)/(BL - Bdl <Do)® BeBL Oy

±Oz = -(±Oy) @®®

Z = (R-2 + [(wL)-l - (wC)] 2r~ Q)O)® CL R

±Oz = tan-1 (R[(wL)-l - (wC)]) @G>®

±Xp = [(wL)-l - (wC)r1 <D@®@ CL Z

±Oz = sin-1 [Z/±Xp] G>®(ij)

Z = I(Sin Oz)/ [(wL)-l - (wC)] I <D@® CL Oz @G>®

Z = (cos Oy)/G <DO) GOy

±Oz = -(±Oy) ®®

160

zfu CD

1i Impedance, Z (\I

.2 '" '" P~rall~1 PO LA R a. CD E .. .. CL 0 CD Circuits «z I-

Z = R cos 0z <Do)® R 0z

Z = I(sin 0z)/(Xil - x(?)1 <D@ XeXL °z ®@®

Z=y-l <D® ±Oz =-(±Oy) ®® Y Oy

Z = E/v'Ii + (IL - Id2 <DO) E IR lelL

±Oz = tan-l [(IL - Id/IR] ®®@

Z= Ell <D® ±Oz = -(±Ol) ®@ E I 01

ZNotes:

@ Mathematics and calculators do not allow a division by zero or infinity. In formulas noted @however, the reciprocal of zero may be manually converted to infinity and the reciprocal of infmity may be manually converted to zero. The following additional manual operations may also be performed as required:

x • 00 = 00 when x "" 0, x/oo = 0 when x "" 00

x . 0 = 0 when x "" 00, x/O = 00 when x "" 0 Ox = 0 when x "" 0, ooX = 00 when x "" 0

Calculators require the substitution of a very small number such as 1 0-30 for zero and of a very large number such as 1 030 for infmity to perform these operations. All very small resultants must then be accepted as zero and all very large resultants must be accepted as infinity. Extreme care must be exercised to avoid accidental violation of the listed exceptions whenever more than one zero and/or infinity appear in the same formula. The arc tangent of infinity may be obtained from a calculator by also substituting a very large number for infinity.

® Division by zero is prohibited. At circuit resonance, a zero divisor and a zero dividend will be presented. The division of zero by zero is always prohibited.

161

... Q)

N

Pola

r Im

peda

nces

In

Seri

es

Zt

= ~ (

[Zl

cOS(

8Z)1

] +

[Z2

cOS(

8Z)2

J --

-+

[Zn

cos(

8z)

nJ) 2

+ ([

Zl

sin(

±8

z)1]

+ [Z

2 si

n(±

8z)

2] -

--+

[Zn

sin(

±8

z)n

JYf

t

±8z

= ta

n 1 _ [(

[ Z,

'in(±

8z)

,] +

[Z, ~n

(±8 z

),]

... +

[z. ~

n(±8

z)nn ]

t

([Z

l cO

S(8Z

)1]

+ [Z

2 cO

S(8Z

)2]

---+

[Zn

cos

(8z)

nJ)

Unk

now

n Se

ries

Im

peda

nce

Zx =

V([Z

t cos

(8z)

t] -

[Zl

cOS(

8Z)1

J)2

+ ([

Zt

sin(

±8

z)t]

-[Z

l si

n(±

8z)

1]f

±8zx

= t

an-1

[([Z

t si

n(±

8z)

t] -

[Zl

sin(

±8

z)1

]) j([

Zt co

s(8z

)t]

-[Z

l C

OS(

8Z)1

J)]

N

3:,,"

-(1

)

~

0 ~ ~

,...

. ~.

:r3

CD

CD

o

c;e

.'"

e.-

I»

-a

I» =

0 Ii!

ii

.'" ..

Pola

r Im

peda

nces

in P

aral

lel

Zt =

[([

Zi1

cOS(

OZ)

l] +

[Zil

cO

S(O

Z)2]

---

+ [Z~

l CO

S(O

z)n]

/ I

+ ([

Zi1

sin

(±O

z)d

+ [

Zi1

sin(

±OZ

)2]

---+

[Z~lSin(±Oz)n]/J2

_ [([Z

" si

n (±

6 z)

,] +

[Zi'

sin

( ± 6 z h

] ..

. + [Z

~' si

n ( ±

6 zl

nJ) J

±OZ

= tan

1

t ([

Zi1

cOS(

OZ)

l] +

[Zil

cos

(Ozh

] --

-+ [Z~

l C

OS(

Oz)

nJ)

.

.... ~

N

Unk

now

n Pa

ralle

l Pol

ar I

mpe

danc

e

2 2

~ Zx

::: [

([Zt1

CO

S(O

Z)t]

-[Z

i1C

OS(

OZ

)1J)

+

([Z

t1S

in(±

Oz)

d-

[Zi1

sin(

±OZ

)1J)

J ~.

.,,-"V

±OZx

::: ta

n-l [

([Z

t1 sin

(±O

z)t]

-[Z

il s

in(±

OZ

)1J)

/([Z

t1 co

s(O

z)t]

-[Z

i1C

OS(

OZ)

1J)]

• 0

3 ..

i3

1&

a,c

..

!!

. Ii

iJ

"V

I 2-

• I:

z Procedure:

Parallel Circuits In Series, Procedu re Method

1. Convert each parallel circuit to polar form impedances using ZPOLAR, parallel circuit formulas.

2. Convert each polar impedance to equivalent series resistance and reactance from:

Rs = Z cos 8z ±Xs = Z sin(±8z )

[If your calculator has the polar to rectangular conversion feature (P-R), enter ZPOLAR as the polar coordinates. The calculator x axis output is Rs and the calculator y axis output is ±Xs]

3. Sum all Rs quantities and algebraically sum all ±Xs quantities. [If your calculator has multiple memories, sum all Rs quantities into one memory and all ±Xs quantities into a second memory.]

4. Convert the total equivalent series resistance and the total equivalent series reactance to total polar impedance from:

Zt = V(Rs); + (±Xs);

(±8z )t = tan-1 [(±Xs)t/(Rs)t]

[If your calculator has the rectangular to polar conversion feature (R-P), enter (Rs)t as the x coordinate and (±Xs)t as the y coordinate. Calculator output will be polar impedance coordinates]

164

z Procedure:

Series Circuits In Parallel, Procedure Method

1. Convert each series circuit to polar form impedance using ZPOLAR, series circuit formulas.

2. Convert each polar impedance to equivalent parallel reciprocal resistance and equivalent parallel reciprocal reactance. [Note: parallel reciprocal resistance is also known as conductance (G) and parallel reciprocal reactance is also known as susceptance (B)]

Rj;l = G = Z-l cos 8z

±Xj;l = ±B = Z-l sin(±8 z)

[If your calculator has the polar to rectangular conversion feature, enter Z-l and ±8 z as the polar coordinates. The calculator x coordinate output is Rj;l or G and the calculator y coordinate output is ±Xj;l or ±B.]

3. Sum all Rj;l (G) quantities and algebraically sum all ±Xj;l (±B) quantities. [If your calculator has multiple memories, sum all Rj;l (G) quantities into one memory and sum all ±Xj;l (±B) quantities into a second memory.]

4. Convert the total equivalent parallel reciprocal resistance [(Rj;l)t or Gt] and the total equivalent parallel reciprocal reactance [(±Xj;l)t or ±Bt] to total polar impedance from:

Zt = l/vG; + (±Bt)2 (±8z)t = tan-1 [±Bt/Gt]

[If your calculator has the rectangular to polar conversion feature (R-P), enter Gt as the x coordinate and ±Bt as the y coordinate. Calculator output will be Z-1(±8z ). Convert the magnitude to Z with the l/x key]

See also - Note Gi>

165

Polar Impedance Definitions and Vector Algebra

ZpOLAR = Z/±f)z

ZPOLAR

Z = Magnitude of impedance

if) z = "Phase" angle of impedance

±f)z = The vectorial resultant angle when the magnitude of series resistance is placed at 0°, the magnitude of inductive reactance is placed at +90° and the magnitude of capacitive reactance is placed at - 90°.

if) z = That angle which has a tangent equal to the series reactance divided by the series resistance; the reactance having a positive sign if inductive and a negative sign if capacitive.

ZpOLAR = Impedance in a form where multiplication and division operations may be performed almost as easily as with ordinary numbers. (vector algebra)

(ZPOLAR)1 • (ZPOLARh = ZI Z2/(±f)Z)1 + (±f)zh

(ZPOLAR)I/(ZPOLAR)2 =ZdZd(±f)Z)1 - (±f)zh

EpOLAR = I (ZPOLAR) = IZ/O° + (±f)z)

IpOLAR = E/(ZpoLAR) = E/Z/O° - (±f)z)

YPOLAR = 1/(ZpoLAR) = liZ/0° - (±f)z)

±f)z = ±f)E = -(±f)l) = -(±f)y)

ZPOLAR = The form used most often as the final resultant when simplifying complex circuits. It is equally "correct" however for the final resultant to be ZRECT., YPOLAR or YRECT .. Both forms of Z are series equivalent while both forms of Yare parallel equivalent.

ZPOLAR = ZRECT = [YPOLAR] -1 = [YRECTrl

166

Rectangular Impedance Z Definitions, Sum R E C and Difference T

ZRECT = Rs + (±Xs)j Rs = Actual or equivalent total series resis

tance ±Xs = Actual or equivalent net series reac

tance where: 1+ Xsi = XL and 1-Xsi = Xc

±Xs = XL - Xc ZRECT = Rs + (±Xs) j regardless of actual circuit con

figuration. The rectangular impedance of a parallel circuit represents the equivalent series circuit. (Note: Equivalent circuit values will vary with frequency.)

ZRECT = Impedance in a form where multiple impedances in series may be summed as easily as multiple resistances and multiple reactances in series. The series connected impedances may be any combination of individual series, parallel and unknown circuits.

[ZRECThoTAL = The sum of the resistive quantities and the algebraic sum of the reactive quantities.

ZRECT = The form necessary to perform any mathematical operation involving the addition or subtraction of impedances.

ZRECT = The form used by some for all mathematical operations and for the final resultant. (Not recommended. Use ZPOLAR for all multiplication and division and for the final resultant)

ZRECT may be converted (transformed) at any time to ZPOLAR or [ZRECTrl using the appropriate formula. ([ZRECT]-l = YRECT)

ZRECT = ZPOLAR = [YpOLARrl = [YRECTrl

167

Rectangular Impedance

Notes

1. This handbook contains no circuit elements to ZRECT direct formulas. It is intended that all simple circuits should be converted to polar impedance and then converted as necessary to and from ZRECT and [ZRECTrl .

2. The author apologizes to readers with good working knowledge of YRECT for the use of [ZRECTr1 , however YRECT is a necessary part of this section, [ZRECT r 1 fits the format better and many engineers as well as most technicians are very uncomfortable with Y RECT'

3. The use of [ZpOLARrl or YPOLAR is not recommended. Do not confuse yourself or others by continually changing the signs of the angles. Convert directly from ZPOLAR to YRECT '

4. Use the rectangular forms Rs + (±Xs)j and G - (±B)j as shown and do not simplify. The plus sign will identify the complex quantity as impedance or voltage and as a series equivalent quantity while the minus sign identifies the complex quantity as reciprocal impedance, admittance or current and also as a parallel equivalent quantity. Note also that in this form the sign of the reactive quantity within the parentheses is real and does not change during inversions. This maintains identification of the reactive quantity as inductive (+) or as capacitive (- ) at all times.

5. If any reader is uncomfortable with all rectangular quantities, direct conversion of series resistive and series reactive quantities to equivalent parallel reciprocal resistive and reciprocal reactive quantities is recommended. This conversion and its reverse allows the Simplification of any series, parallel or series-parallel combination of impedances to a single impedance. This method is as fast as any other and there is less chance of error.

168

Reciprocal Rectangular [ Z ] 1 Impedance Definitions, CT -Sum and Difference R E

[ZRECTr1 = [Rs +{±Xs)jr1 =Rp1 - {±Xp1)j

The reciprocal of ZRECT. (intrinsically a series or series equivalent quantity) is intrinsically a parallel or parallel equivalent quantity.

[ZRECTr1 = YRECT = Rectangular admittance YRECT =G- {±B)j Seealso-YREcT

G = Total parallel conductance or the equivalent parallel conductance

±B = Total parallel susceptance or the equivalent parallel susceptance where: i+Bi = BL, i-Bi = Bc

±B= BL - Bc [ZRECT r 1 = Rp1 - {±Xp1)j regardless of actual circuit

configuration. The rectangular admittance of a series circuit represents the equivalent parallel circuit with the resistance and reactance in reciprocal form. (Note: Equiv. circuit values vary with freq.)

[ZRECT ]-1 = Reciprocal impedance in a form where complex quantities in parallel may be simplified as easily as multiple resistances and multiple reactances in series. The complex quantities in parallel may represent any combination of individual series, parallel or unknown circuit configurations.

[Z RECT ] fOTAL = The sum of the reciprocal resistances and the algebraic sum of the reciprocal reactances.

[ZRECTr1 may be inverted back to rectangular or polar impedance at any time using the appropriate formula.

[ZRECTr1 = [ZpOLARr1 = YRECT = YPOLAR

See also - Note GD

169

Rectangular Impedances In Series

[ZRECThoTAL = [ZRECTh + [ZRECTh --- + [ZRECT]n

Note: Rectangular impedance represents equivalent series resistance and reactance, however the actual circuit configuration may be series, parallel or unknown.

The first number in the rectangular impedance quantity is equivalent series resistance, the real part of impedance, the 00 component of impedance or the x axis coordinate of impedance.

The second number in the rectangular impedance quantity represents equivalent series reactance, the imaginary part of impedance, the ±90° component of impedance or the y axis coordinate of impedance.

When summing rectangular impedances, all resistive (Rs) components and all reactive (±Xs) components must be summed separately. The reactive components (±Xs) must also be summed algebraically. (Use the rectangular form Rs + (±Xs) j not Rs ± jXs)

[ZRECTlt = (Rs)l

[ZRECT h = (Rsh

[ZRECT]n = (Rs)n

+ (±XS)l

+ (±Xs)2

+ (±Xs)n

[ZRECThoTAL = (RshoTAL + (±XshOTAL j

Note: ZPOLAR = VCRs); + (±Xs); /tan-1 [(±Xs)t/(Rs)t]

ZRECT = [Z cos 8z] + [Z sin(±8 z )] j

170

Reciprocal Rectangular Impedances In Parallel

[ZRECT] i 1 = [ZRECTlil + [ZRECT]21 --- + [ZRECT]~l

[ZRECTrl = YRECT

[YRECT1t = [YRECTh + [YRECTb --- + [YRECT]n

[ZRECTr1= Rpl - (±Xpl)j

YRECT=G- (±B)j

[ZRECTlil = (Rpl)l

[ZRECT]21 = (Rpl)2

[ZRECT]~l = (Rpl)n

[YRECTh = G1

[YRECT]2 = G2

[YRECT]n = Gn

G = R-1 +B = +X-1 P' - - P

- (±Xpl)l

- (±Xp )2

- (±Xpl)n

- (±Bd

- (±B2)

- (±Bn)

- (±BhoTAL

Note: ZpOLAR = [Gf + (±Bt?r~/tan-l [±Bt/Gt]

[ZRECT] -1 = (Z-1 cos Oz] - [Z-1 sin(±Oz)] j

See also - Note GD

171

Conversions From Polar Impedance ZPOLAR

ZPOLAR To ZRECT

ZRECT = [Z cos Oz] + [Z sin(±Oz)] j

ZPOLAR To Y RECT or [ZRECT]-1

YRECT =G- (±B)j =Rj;1 - (±X;1)j

YRECT = [Z-1 cos Oz] - [Z-1 sin(±Oz)] j

[ZRECTr1 = YRECT

ZPOLAR To YPOLAR

YPOLAR = Z-1/_(±OZ)

ZpOLAR To Series R and X

Rs = Z cos Oz ±Xs = Z sin(±Oz)

I+Xsl =XL I-Xsl =Xc

ZPOLAR To Parallel Rand X

Rp=Z/(cosOz) ±Xp=Z/[sin(±Oz)]

I+Xpl =XL I-Xpl =Xc

See also - Note (jj)

172

Conversions

Conversion To Polar Impedance

Equiv. Series R and X To ZPOLAR

Z = YR; + (±Xs)2

±ez = tan-1 [±Xs/Rsl

Equiv. Parallel R and X To ZPOLAR I

Z = [R;2 + (±Xp)-2r2

±ez = tan-1 [Rp/±Xpl

ZRECT To ZPOLAR

ZRECT = Rs + (±Xs)j

Z =YR; + (±Xs)2

±ez = tan-1 [±Xs/Rsl

Conversions

[ZRECTr1 or YRECT To ZPOLAR

[ZRECTr1 = Y RECT = G - (±B)j

=Rj;l_(±Xj;l)j

Z = [G2 + (±B)2r-~ ±ez = tan-1 [±B/G]

See also - Note GD

173

Conversions From Z Rectangular RECT Impedance

ZRECT To ZPOLAR

ZRECT = Rs + (±Xs) j

ZPOLAR = v'R~ + (±Xs)2 !tan-1 [±Xs/Rs1

ZRECT To YPOLAR


[ 2 2J-t / -1 [ YPOLAR = Rs + (±Xs) tan -(±Xs/Rs)]

ZRECT To YRECT or [ZRECTr1


Conversions

YRECT = G - (±B)j = [ZRECTr1 = Rpl - (±Xpl)j

YRECT = [Rs/(±X~ + R~)] - [±Xs/(±X~ + R~)] j

ZRECT To Equiv. Parallel R and X

ZRECT = Rs + (±Xa)j

Rp = (±X~ + RD/Rs

±Xp = (±X~ + R~)/±Xs

I+Xpl=XL I-Xpl=Xc

See also - Note @

174

Conversions To Rectangular Impedance

ZPOLAR To ZRECT

ZPOLAR = Z j±B Z

ZRECT = [Z cos Bz] + [Z sin(±B z)] j

YPOLAR To ZRECT

YPOLAR = Y j±By

ZRECT = [y-l cosBy] + [_y-l sin(±By )] j

Y RECT or [ZRECTrl To ZRECT

Y RECT = G - (±B)j

ZRECT = [G/(±B2 + G2 )] + [±B/(±B2 + G2 )] j

Series R and X To ZRECT

ZRECT = Rs + (±Xs) j

Parallel R and X To ZRECT

Conversions

ZRECT = [(Rp/±X~) + Rplfl + [(±Xp/R~) + (±Xpl )r1 j

G and B to ZRECT

ZRECT = [G/(±B2 + G2 )] + [±B/(±B2 + G2 )] j

See also - Note @

175

Conversion From

Reciprocal Rectangular Impedance

[ZRECTr1 or Y RECT To ZPOLAR


ZPOLAR = [G2 + (±B)2r~ /tan-1 [±B/G]

[ZRECTr1 or Y RECT To ZRECT


ZRECT = [G/(±B2 + G2)] + [±B/(±B2 + G2)] j

[ZRECTr1 or Y RECT To YPOLAR


YPOLAR = VG2 + (±BP /tan-1 [-(±B/G)]

[ZRECT r 1 or Y RECT To Equiv. Series R and X

[ZRECTr1 = Y RECT = G- (±B)j

Rs = G/(±B2 + G2) ±Xs = ±B/(±B2 + G2)

[ZRECTr1 or Y RECT To Equiv. Parallel R and X


Rp = G-1 ±Xp = ±B-1

See also - Note @

176

Conversions To

Reciprocal [ Z ]-1 Rectangular R E CT Impedance

ZPOLAR To Y RECT or [ZRECTr1

ZPOLAR = Z/±8z


[ZRECTr1 = [Z-l cos 8z] - [Z-l sin(±8z)] j

ZRECT To Y RECT or [ZRECTr1


[ZRECTr1 = Y RECT = G- (±B)j

[ZRECTr1 = [Rs/(±X~ + R~)] - [±Xs/(±X~ + R~)] j

Series R and X To Y RECT or [ZRECT r 1

[ZRECTr1 =YRECT =G- (±B)j

[ZRECTr1 = [Rs/(±X~ + R~)I- [±Xs/(±X~ + R~)] j

Parallel R and X To Y RECT or [ZRECTr1

[z ]-1 = R-1 - (+X-1)' RECT p - p J

YPOLAR To Y RECT or [ZRECTr1

[ZRECTr1 = [Y cos 8y] - [-Ysin(±8y )] j

See also - Note @

177

z Rules of Vector Algebra

ZI • Z2 = Z1 Z2 1(±fJzh + (±8zh

Zt/Z2 = Zt/Z2 1(±8 zh - (±8 Z)2

(+1) • Z = Z 10° + (±8z) = ±8z

(+l)/Z = liZ 10° - (±8z) = -(±8z)

ZI + Z2 = [ZRECT] 1 + [ZRECT h ZI + Z2 = [Rs + (±XS)jJI + [Rs + (±Xs)jh

Impedance, Vector Algebra Rules

ZI +Z2 = [(RS)1 + (RS)2J + [(±Xsh +(±Xsh]j

ZI - Z2 = [(RS)1 - (RshJ + [(±XS)1 - (±XshJ j

Z+(+1)= [Rs + 1] + [±Xs]j

Z- (+1)= [Rs - 1] + [±Xs]j

ZI 1 + Z21 = [ZRECT] 11 + [ZRECT] 21

Zi1 + Z2 1 = [Y RECT ] 1 + [Y RECT ] 2

ZI1 + Z21 = [G - (±B)j] 1 + [G - (±B)j]2

ZI 1 + Z21 = [G1 + G2] - [(±Bh + (±Bh] j

Zi1 - Z21 = [G1 - G2] - [(±B)1 - (±B)2J j

Z-1 + (+1) = [G + 1] - [±B]j

Z-1 - (+1) = [G - 1] - [±B]j

See-Z Conversion Formulas

178

IMPEDANCE Vector Algebra z •

I ~! CL 0 <Z

VA-! ~-------[g----<> VA-2

VA-3 Zt=ZI+Z2---+Zn VA-S

Z2 = [Zil - Zil]-1

Y2=Yt- YI

ZNotes:

VA-! VA-2 VA-3 VA-S

VA-! VA-2 VA-3 VA-4 VA-S

VA-! VA-2 VA-3 VA-4 VA-S

VA-I Impedance is a complex quantity requiring the mathematical operation of addition and subtraction to be performed in rectangular form while multiplication and division operations are usually performed in polar form by treating the phase angle as an exponent. Impedances in rectangular form may be multiplied like other binomials, division however requires the divisor to be rationalized by mUltiplying the divisor and the dividend by the conjugate of the divisor (the conjugate of Rs + XJ = Rs - XJ). To eliminate this lengthy calculation, it is recommended that all multiplication and division be performed in polar form.

119

Input IMPEDANCE Vector Algebra

Zj etc. Eg = Eg I!t.... Zi = Z

10 = Eg/Z

Ig = Ig /0° Zi = [Z11 + Z21] -1

Eg = IgZi

10 = (Ig Zi)/ZI

Zi = Z3 + [Z21 + ZI1]-1

Ig = Eg/Zi

10 = Eg/(Zi [(ZI/Z2) + 1]) ZNotes:

~I"

ill Ig Z2 ~Io

V A-2 Eg = Generator voltage Eo = Output voltage Ig = Generator current 10 = Output current

Yo = Output admittance Y t = Total admittance Zi = Input impedance Zo = Output impedance Zt = Total or Equivalent impedance

180

III j5

.~ en - III a. ... a. 0 «2:

VA-1 VA-2 VA-3 VA-5

VA-1 VA-2 VA-3 VA4 VA-5

VA-1 VA-2 VA-3 VA4 VA-5

Input IMPEDANCE Vector Algebra

Ig = Ig fst....

Zj etc.

Zj = (Z41 + [Z3 + (Zi1 + Zi1 r 1r 1r1

Eg = IgZj

10 = Ig [1 - (Zj/Z4)]/ [(Zt/Z2) + 1]

Eg = Efst.... ~ = Zs + (Z41 + [Z3 + (Zi1 + Zi1r 1r 1 t1 Ig = Eg/~ 10 = Ig (1 - [(Zj - ZS)/Z4]) / [(Zt/Z2) + 1]

ZNotes:

CD

~ .- ... - CD a. ... a. 0 <1:2

VA-1 VA-2 VA-3 VA-4 VA-S


VA-3 Impedances Z, Zb Z2, Z3, Z4 and Zs may represent any resistance, reactance, series circuit, parallel circuit, unknown circuit or any circuit regardless of complexity or coniIguration.

VA4 Z-1=1/Z=Y, y-1=1/Y=Z VA-5 Z, Zj. Zo, Y, Yo, 10 and Eo all will vary with frequency except

purely resistive circuits.

181

IMPEDANCE Z Zo etc. Vector • Algebra I

Ig = Ig!!L

ill'" Zj = Z 19 Z

Zo = Z

Eo = IgZ

Eg = Eg!!L

Zj = Zl + Z2

Zo = [Zil + Z2 lr l

Yo =Yl +Y2

Eo = (EgZd/Zj

Ig = Ig!!L

Zj= [Z3 l +(Z2 +Zl)-lrl

Zo = [Zil + (Z2 + Z3)-lr l

Yo = Y 1 + (y2l + y 3l r l

Eg = IgZj

Eo = IgZl [1 - (Zd Z 3)]

182

III

:is IV .2 VI - III c. .. 0. 0 «z

VA-1 VA-2 VA-3 VA-S



IMPEDANCE Z Vector • Algebra I Zo etc.

Ig = Ig ~ Eg = IgZj

Zj = [Zsl +( Z4 + [Z31 + (Z2 + Z.)-l] -1 r 1J1

Zo = [ZI1 +( Z2 + [Z31 + (Z4 + Zsf1r 1 r 1J1

Vo = V1 + (V;:l + [V3 + (V41 + VS1)-1] -lr1

Eo = [1g(Zj - Z4)]/[(Z2/Z.)+ 1]

183

VA-1 VA-2 VA-3 VA-4 VA-5

VA-1 VA-2 VA-3 VA-4 VA-5

z IMPEDANCE ~ to Y Conversion

Delta (~) to Wye (Y) or Reverse Conversion Pi (1T) section to Tee (T) section or reverse

Transformation

~ or 1T section

Zl = (ZAZB)/[ZA + ZB + Zcl

Z2 = (ZBZe)/[ZA + ZB + Zc]

Z3 = (ZAZe)/[ZA + ZB + Zc]

ZA = [(Z l Z2) +(Z2 Z3) + (Z l Z3)]/Z2

ZB = [(Zl Z2) + (Z2 Z3) + (Zl Z3)] /Z3

Zc = [(Z l Z2) + (Z2 Z3) + (Z l Z3)]/Zl

Page Notes:

Y or T section

1. Technically, delta and wye diagrams should be drawn with only three terminals.

2. Convert all impedances and intermediate solutions to both polar and rectangular form. Perform all addition in rectangular form. Perform all multiplication and division in polar form.

184

PASSIVE

CIRCUITS

SECTION 1.2

GREEK

LETTERS

a to W Greek Alphabet

Spelling and Small Large Spelling and Small Large Pronunciation (Script) (Capital) Pronunciation (Script) (Capital)

alpha a A nu II N (al'fa) (nu) beta (3 B xi ~ Z (ba'ta) (zi) gamma 'Y r omicron 0 0 (gam' a) (om'i kron'

o'mi kron') delta 8 .d pi 7T n (del'ta) (pi) epislon € E rho p P (ep'sa Ion') (ro)

zeta t Z sigma a s ~ (zli'ta) (sig'ma)

eta 11 H tau T T (li'ta) (tou or taw)

(ouasinout)

theta e e upsilon v T (tha'ta) (up'sa Ion') iota t I phi !/J <P (i 0' ta) (fi or fe) kappa K K chi X X (kap'a) (ki)

lambda A A psi 1# 'II (lam'da) (si) mu J1. M omega w n (mu) (omeg'a

ome'ga) Note: For obvious reasons, capital greek letters ABE Z H I K M N 0 P T X are not used as electronic symbols.

186

a to 1} Greek Letters

a = Symbol for many different passive circuit quantities but no standardization has been achieved. See also-Active Circuits

~ = Symbol for many different passive circuit quantities but no standardization has been achieved. See also-Active Circuits

'Y = Symbol seldom used in electronics. Used for conductivity (G) in other fields.

l) = Symbol for loss angle.

l) = Ninety degrees minus the absolute value of the phase angle.

l) = 90° - lei l) = tan-1D

Note: The dissipation factor (D) of capacitors is specified by most USA manufacturers but the loss angle (6) or the tangent of the loss angle (tan 6) is specified by most foreign manufacturers .

.:l = Symbol for increment or decrement. (Still used for vacuum tubes, but small signal parameters such as hfe are used for semiconductors.)

e = Symbol for the base of natural logarithms.

e=2.718281828--- e-1 = .3678794412---

~ = Seldom used and no standardization of meaning.

1/ = Efficiency. See also-Active Circuits

187

lJ 8 = Symbol for phase angle.

Phase Angle Definitions

Note: Phi (tP) and other greek letters are also used as symbols for phase angle.

8 = 1. The angular difference in phase between a quantity and a reference.

2. The phase angle of voltage, current, impedance or admittance with respect to a reference.

3. The phase angle of voltage, current, impedance or admittance with respect to the phase angle of current, voltage, resistance or conductance.

4. The phase angle of voltage or impedance with respect to the phase angle of total current or with respect to 0°.

S. The phase angle of current or admittance with respect to the phase angle of total voltage or with respect to 0° .

8 = Phase angle measured and expressed in: 1. Decimal degrees

360° = one cycle or one revolution 2. Degrees, minutes, seconds

1 ° = 60' (minutes) I' = 60" (seconds)

3. Radians 21T radians = one cycle or one revolution

4. Grads 400 grads = one cycle or one revolution

8 = 0° when voltage and current are in phase. 0° when circuit is or acts as a pure resistance or conductance.

8 = ±9Q0 when circuit is or acts as a pure reactance or susceptance.

8 = +90° to -90° for all two terminal networks when 8 is angle of total voltage, total current, total impedance or total admittance.

188

() Phase Angle Definitions

+0 = Leading phase angle. Counterclockwise rotation of a vector. Earlier in time than 0° .

-0 = Lagging phase angle. Clockwise rotation of a vector. Later in time than 0° .

OE = 1. The difference in phase between the total voltage and the total current when the phase angle of the total current is placed at 0° .

2. The angular difference in phase between the total voltage and the current source. (Ig = Ig /0° = +Ig unless noted)

OEo = Output voltage phase with respect to the phase of the voltage or current input. (Voltage or current generator Eg or Ig = 0°)

01 = 1. The angular difference in phase between the total current and the total voltage when the phase angle of the total voltage is placed at 0° .

2. The angular difference in phase between the total current and the voltage source. (Eg = Eg 1st.... = +Eg unless noted)

010 = Output current phase with respect to the phase of the voltage or current input. (Input phase = 0° unless noted)

Page Note: The phase angles of E and I may be confusing. To prevent confusion, always calculate polar impedance fIrst and then assign zero degrees to the signal source. If the signal source is a voltage generator, 01 = -0 z and if the signal source is a current generator, OE = 0 z. Use vector algebra to determine the phase angles of circuit voltages and/or currents. e.g., when a voltage source is connected to a series circuit, 01 = -0 z, 0 E R = 0 f, 0 E L = 0 I + 90° , 0 E c = 0 I - 90° , 0 E t = 0° .

189

(J Phase Angle Definitions

8y = 1. The angular difference between the admittance and the conductance of a circuit. (The angle of conductance G = 0°)

2. The same angle as impedance except with opposite sign.

3. The same angle as the phase angle of the total current when the phase angle of total voltage is placed at 0°.

o z = 1. The angular difference between the impedance and the resistance of a circuit. (The angle of resistance R= 0°)

2. The same angle as the admittance except with opposite sign.

3. The same angle as the phase angle of the total voltage when the phase angle of total current is placed at 0°.

±8 E = -(±8I) but both may not coexist.

±81 = -(±8E ) but both may not coexist.

±8y =-(±8 z) [Y/±Oy=Z-l/_(±OZ)]

±OZ =-(±8y ) [Z/±OZ =y-l/_(±Oy)]

±8E = -(±8I) = -(±8y ) = ±OZ in all two terminal networks where the phase angle of either the total voltage or the total current is placed at 0°. (It should be understood that reactance, a component of impedance, is the cause of the difference in phase between the voltage and the current, that OE, 01, Oy and Oz are the same one and only phase angle from different reference points, that only one may be used at anyone time and that if Z or Y ap}(ears as a term in a formula the other term must be E 0° or IL!t.)

190

(J +OE = Inductive circuit phase angle of voltage

-OE = Capacitive circuit phase angle of voltage

+01 = Capacitive circuit phase angle of current

-0 1 = Inductive circuit phase angle of current

+Oy = Capacitive circuit phase angle of admittance

-Oy = Inductive circuit phase angle of admittance

+0 z = Inductive circuit phase angle of impedance

-0 z = Capacitive circuit phase angle of impedance

08 = +90° c

OE = -90° c

OG =0°

01 = +90° c

OR =0°

Ox = -90° c

1~ = +1

1/+90° = y'-l" = 0 + Ij

1/-90° = - y'-l" = 0 - Ij

1/±180° =-1

191

Phase Angle Definitions

Phase Angle, () ... Series Circuits E ..

Q)

I-

10EI = 1011 = 10yi = 10zi = tan-1 [D-1 ] Ds

10E I = 101 I = lOy I = 10 z I = tan -1 Q Qs

±OE = ±oz = tan-1 [±Ex/ER] ER ±Ex

±OE = ±oz = tan- 1 [±X/R] R ±x

10EI = 1011 = 10yi = lozl = cos-1 [R/Z] RZ

±OE = ±oz = sin-1 [±X/Z] ±x Z

±OE = ±oz = tan-1 [(EL - Ed/ER] ER Ec EL

±OE = ±o z = tan -1 [(XL - Xc)/R] R Xc XL

±OE = ±OZ = sin-1 [(XL - Xd/Z] XCXL Z

(±Oz)t = tan-1 [(±Xs)t!(Rs).]

(±Xs)t = [Zl sin (±Ol)] + [~sin(±~)] Zt/±Ol

Zd±02

(Rs)t = (Zl cos 01) + (Z2 cos O2)

192

Phase Angle, (J ParaNel Circuits

IOEI = lOll = 10yi = 10zi = tan-1 [D-1]

IOEI = 1011 = 10yi = IOzl = tan-1 Q

tOy = ±Ol = tan-1 [-(±B)/G]

±Oy = ±Ol = sin-1 [-(±B)/Y]

±Ol = ±Oy = tan-1 [-(±Ix)/IR]

IOEI=IOd=IOyl=IOzl=cos-1 [G/Y]

±Oz = tOE = tan-1 [Rp/±Xp]

leE 1= 1011 = lOy 1= 10z 1= cos-1 [Z/Rp]

±ez = ±OE = sin-1 [Z/±Xp]

Page Notes: I+B 1= BL = (Xl?)p I-BI = Bc = (XC)p

I+Xpl = (XL)p = BLI

I-Xpl = (Xc)p = Bi:?

193

'" E .. • ~ Dp

Qp

±B G

±B Y

±Ix IR

GY

R ±X

RZ

±X Z

Phase Angle, (J to K '" Parallel E ..

Q)

Circuits I-

±8y = tan- l [-(BL - Bd/G] Be BL G

±8y = sin- l [-(BL - Bd/Y] Be BL Y

±81 = tan- l [-(IL - Id/IR] Ie IL IR

±8 z = tan- l [R(XLI - XCI)] R Xc XL

±8 z = sin-l [Z(XLI - XCI)] Xc XL Z

(±8 z )t = tan- l [±BtlGtl

±Bt = [Zil sin (±81)] + [Zil sin (±82 )] Zz/±82

Zt/±81

Gt = (Zil cos 8d + (Zi l cos 82 )

L = Seldom as a symbol due to similarity to english letter i.

K = Seldom as a symbol due to similarity to english letter k.

194

A = Symbol for wavelength.

Wavelength Definitions & Formulas

A = 1. In a periodic wave, the distance between points of corresponding phase of two consecutive cycles. 2. The length of one complete cycle of a periodic wave.

A = Wavelength measured and expressed in various units of distance such as inches, feet, centimeters or meters.

A = v/f where v is the velocity of the wave in the medium through which it is traveling. f = frequency of wave. Note: In physics, the symbol c is used for the velocity of light.

Wavelength of Sound in Air

A ~ 1136/f feet @ 25° C

A ~ 346.3/f meters @ 25° C

A ~ (I051 + 1.1 T F )/f feet @ std pressure

A ~ (331.3 + .6 T d/f meters @ std pressure

Wavelength of Electromagnetic Waves

A ~ (9.8 • 108 )/f feet

A ~ (3 • 108 )/f meters

A = (2.997 93 • 108 )/f meters (in vacuum)

195

J.I. = Symbol for micro.

Micro, Mu Factor, Permeability

J.I. = Prefix meaning 1/1,000,000. 10-6 multiplier prefix for most basic units. J.l.V = 10-6 Volts, J.l.A = 10-6 Amperes, J.l.F = 10-6 Farads, J.l.H = 10-6 Henries.

J.I. = Symbol for mu factor.

J.I. = Amplification factor (voltage) in vacuum tubes.

J.I. = dEp/dEg (with Ip constant)

J.I. = gmrp Eg = grid voltage, Ep = plate voltage gm = mutual conductance (transconductance) rp = dynamic plate resistance.

J.I. = Term not used with semiconductors.

See-Active Circuits Av

J.I. = Symbol for magnetic permeability.

J.I. = The magnetic equivalent of electrical conductivity. The magnetic conductivity of a material compared to air or vacuum.

J.I.=B/H B = Flux density in gauss H = Magnetizing force in oersteds

11&

rp = Symbol for total magnetic flux.

Total Magnetic Flux, Complementary Angles

rp = The total measure of the magnetized condition of a mag-netic circuit when acted upon by a magnetomotive force.

rp = The total number of lines of magnetic flux.

rp = Maxwells or lines of flux units (CGS).

rp = Weber units (SI).

rp = F/'YI rp=FP

rp = BA

rp= BA

1 maxwell (Mx) = 1 line of flux

1 maxwell (Mx) = 10-8 Weber (Wb)

where F = magnetomotive force 'YI = reluctance P = permeance

where B = flux density in gauss A = cross sectional area of magnetic path

in cm2 .

where B = flux density in lines per in2

A = cross sectional area of magnetic path in in2

rp = Alternate symbol for phase angle.

rp = Symbol for complementary angle.

rp = Complement of phase angle 8.

rp=±90o-8 (rp+8=±900)

197

w Angular Velocity Definitions & Formulas

W = Symbol for angular velocity or angular frequency.

W = The rate at which an angle changes in radians per second. The angular change in a uniformly rotating system measured in radians per second. (27T radians = 3600 , 27T ra-dians per second = rps = rls = Hz) Terms

W = 27Tf= 6.2831853 --- f

w= BclC

Resonant Angular Velocity

We = V(LC)-l

fe Definition 1

we = [(LC) - (L/RiJ -t 1

(-x)-' exception

198

f

Be C

BL L

C Xc

Wr = V(LC)-l - (CR)-2 Fx exception

fr definition 1

wr ~ V(LC)-l

Resonant Angular Velocity

Wr = V[(R~C)-l - L-1]/[(L/Ri)- C] Fxexception

fr definition 1

wr ~ V(LC)-l

Wr = V(LC)-l - (R/L)2 Fx exception

fr deftnition 1

wr ~ V(LC)-l

1 1

Wr = [(LC) - (CR)2] -2 (-x)-' exception

fr definition 1

wr ~ V(LC)-l

Wr = V[C-1- (Rt/L)]/[L- R~C] fr definition 1 Fx exception

Wr ~ V(LC)-l

199

]]

n n = Symbol for olnu.

Ohm Definitions

n = 1. The basic unit of resistance, reactance and impedance.

2. That resistance which will develop a current of one ampere from an applied potential of one volt.

3. That reactance or impedance which will develop a steady state rms current of one ampere from an applied sinewave potential of one volt rms.

4. The resistance of a uniform column of mercury 106.3 cm long weighing 14.4521 g at a temperature ofO°C.

n = Unit often used with multiplier prefixes.

pn = 10-6 olnus

mn = 10-3 olnus

kn = 103 olnus

Mn = 106 olnus

Gn = 109 olnus

Tn = 1012 olnus

Note: kn is frequently contracted to K Mn is frequently contracted to M Megohm is frequently contracted to Meg

n = A real (positive or 0°) quantity when a unit of resistance.

n = A magnitude or a complex quantity when a unit of reactance or impedance.

200

SECTION TWO

TRANSISTORS

2.1 STATIC (DC)

CONDITIONS

A to E a - See-hFB or a

AI = Static current amplification (seldom used)

Av = Static voltage amplification (seldom used)

BVCBO -See-V(BR)CBO

BVCEO - See-VCEO(SUS)

BV CER - See-V CER (SUS)

BVCES - See-VCES(SUS)

BVCEV - See-VCEV(SUS)

BV CEX - See-V CEX (SUS)

BVEBO - See-V(BR)EBO

E - See-V for dc transistor voltages See also-V, Opamp See also-E, Passive Circuits

DC Transistor Symbol Definitions

E = The original symbol for the electric force originally known as electromotive force. This force is now known as voltage, potential or potential difference. The voltage symbol E has been superseded by V for dc transistor voltages and for all operational amplifier voltages.

ESfb - (second breakdown energy) See-ISfb

Note: The tenn second breakdown energy (ES/b) has never been appropriate for static transistor conditions since continuous power at any level converts to infinite energy.

203

h Static (DC) Hybrid Parameters

hFB = Seldom used common-base static forward-current transfer ratio.

hFB = DC alpha (a)

hFB = Ie/IE

hFB = hFE/(hFE + 1)

hFE = Common-emitter static forward-current transfer ratio at a specified collector current, collector voltage and junction temperature.

hFE = DC beta (13)

hFE = Ie/IB

hFE = (IE/IB) - 1

hFE = [(IE/Ie) - lr1

hFE(INV) = Seldom used hFE when collector and emitter leads are interchanged.

hIE = Seldom used common-emitter static input resistance.

hNotes:

The DC counterparts of hfe, hib, hob, hoc, hoe, hrb, hre• and hre are very seldom used.

hFE usually has a different value than hfe measured under the same conditions.

hIE and hIB will have a much higher value than their small signal counterparts measured under the same conditions.

204

I IB = DC base current.

Ic = DC collector current. See-I Note <D IE = DC emitter current.

IBB = Base supply current.

Icc = Collector supply current.

Ico - See-ICBO

lEE = Emitter supply current.

lEO - See-lEBO

DC Transistor Current Definitions

ICBO = DC collector to base leakage current at a specified voltage and temperature with emitter open. @ @

ICEO = DC collector to emitter leakage current at a specified voltage and temperature with base open. @ @

leER = DC collector to emitter leakage current at a specified voltage and temperature with a specified base to emitter resistance. See-I Notes @@

ICES = DC collector to emitter leakage current at a specified voltage and temperature with the base and emitter shorted. See-I Notes @ @

ICEV = DC collector to emitter leakage current at a specified voltage and temperature with a specified base to emitter reverse bias voltage. See-I Notes @ @

ICEX = DC collector to emitter leakage current at a specified voltage and temperature with a specified base to emitter circuit. See-I Notes @ @

lEBO = DC base to emitter reverse bias leakage current at a specified voltage and temperature with open collector. See-I Notes @@

205

I IF = Forward bias dc current.

DC Transistor Currents, Second Breakdown Current

IN = Noise current. See-IN, Passive Circuits See also-NF

10 = DC output current. See also-Passive Circuits

IR - See-I, Passive Circuits

IS/b = Symbol for second breakdown current.

IS/b = The collector current at which second breakdown occurs at a specified collector voltage, case or junction temperature and pulse duration.

Second breakdown occurs when the combination of voltage, current, temperature, time and a current constriction within the transistor produces spot heating sufficient to thermally maintain or increase the collector current regardless of base bias. In the usual transistor circuit, if second breakdown has been allowed to occur, transistor failure will also occur due to excessive spot junction temperature.

Second breakdown is not the same as thermal failure where failure may be predicted from low voltage thermal resistance calculations. Second breakdown may occur at positive, zero, or negative base bias.

Circuit design should be such that the manufacturers second breakdown specifications are not exceeded under worst case conditions. Alternately, the second breakdown characteristics of transistors may be measured with special non-destructive procedures.

206

Static (DC) Transistor Currents

IB = Ie/hFE

IB = IE/(hFE + 1)

IB = VBE/hIE

IB ~ [log-1 (VBE/.06)]/(lOI3hFE)

Ie = hFEIB

Ie = IE - IB

Ie = (hFE VBE)/hIE

Ie ~ [log-1 (VBE/.06)] (5 • 1O-16hFE)

Ie ~ 10-13 [log-1 (VBE/.06)]

IE = Ie + IB

IE = (hFE + I)IB

IE = [VBE(hFE + 1)J/hIE

IE ~ (5 • 10-16) (hFE + 1) [log-1 (VBE/.06)]

I Notes:

u ::ci .~ <II a.s a. 0 «z

@)

®

®

@)

®

®

@)

®

®

<D The subscript of Ie is a capital letter for DC. It is often difficult to distinguish between a capital and a lower case C subscript. Ie (lower case) is rrns collector current and ie (upper case) is instantaneous total collector current.

207

Static (DC) .!!

18 Ie IE .c

Transistor .~ '" - CD

Currents 8:~ «2

R j CD

Ie ~ 10-13 [log-1 (VBB /.06)] @ + ®

IB = IclhFE + ® IE = Ie + IB ®

®

Ie = [hFE(VBB - VBE)] /R2 CD

VBE ~ .06 [log (1013 Id] @

® VBE ~.6 R2 ®

+ - ® IB = Ie/hFE

B VBB ®

IE = Ie + IB G>

Ie = [hFE(Vee - VBE)]/R2 CD

VBE ~ .06 [log (10 13 Id] R j @

R2 ® VBE ~.6 ®

IB = Ie/hFE ® ®

IE = Ie + IB G>

I Notes:

@ The standard specified temperature is 25°C ® Transistor leakage currents have a temperature dependent com

ponent and a voltage dependent component. @) hFE' VBE and hIE are temperature, current and voltage dependent. ® log x = loglO x, log-1 x = antilog x = lOx

208


IE = (Vee - VBE)/(R 1 + [R3/(hFE + 1)]) VBE ~ .06 [log (lO l3Id] ~ .6

IB = IE/(hFE + 1)

Ie = IE - IB

lee = IE

IE = (Vee - VBE)/(R1 + Rs + [R3/(hFE + 1)]) VBE ~ .06 [log (lO l3Id] ~.6

IB = IE/(hFE + 1)

Ie = IE - IB

lee = IE

Ie = [Vee - (VBERx)]/(RI + [(R 1 +R3)/hFEJ) VBE ~ .06 [log (lO l3Id] ~ .6

Rx = [(R1 + R3)/R4] + 1

18 = Ie/hFE

IE = Ie + IB

Ice = IE + (VBE/R4 )

209

QI

li .~ ... - QI Q,~ Q, 0 «Z

<D @

® @)

® ® cr>

CD @

® @)

® ® CD


VBE ~.6

IB = Ie/hFE

IE = Ie + IB

Ie = [hFE(VBB - VBE)]/(R2 + [Rs(hFE + 1)]) VBE ~ .06 [log (l013Id]

VBE ~.6

IB = IclhFE

Ie = [hFE(VX - VBE)]/(Rx + [Rs(hFE + 1)]) VBE ~ .06 [log (1013Id]

VBE ~.6

Vx = Vee/[(R2 /R4 ) + 1] Rx = (Ri"1 + R41 )-1

IB = Ie/hFE

IE = Ie + IB

210

c

E

+ vee

III :is B .- '"

- III a. ... a. 0 «2

<D Q)

® @)

® ® (2)

<D Q)

® @)

® ® (2)

<D Q)

® @)

® ® (2)


IB = IclhFE

IE = Ie + IB

lee = IE + (VBE /R4)

Ie = [Vee - (VBERxl)]/ [RX2 + (RX3 /hFE)]

VBE ~ .06 [log (I0 13Ic)] ~.6

RXl = [(Rl + R3)/R4] + 1

RX3 = RX2 + R3

IB = IclhFE

IE=Ie+IB

lee = IE + [(IERs + VBE)/R4]

I Notes:

RS

II)

J5 .~ lit - II) Co ... Co 0 «z

® "Exact fonnulas" apply to silicon, gennanium, npn, pnp, small signal and power transistors. (Exact fonnulas are not really exact since hFE will vary somewhat with collector current, collector voltage and temperature.)

(1) The VBE of silicon transistors varies with temperature at the rate of approximately -2.2 m V per 0 C.

211


Ie = [Vee - (VBERxI)]/[Rx2 + (RX3 /hFE)]

VBE ~ .06 [log (l 013 Id] ~ .6

RXI = (RI G4) + (R3G4) + 1

RX2 = (RIRs G4) + (R3R SG4) + RI + Rs

RX3 = RX2 +R3

RI = RIA + RIB R3

RII

G4 = 1/R4

~: IB = IclhFE R4

IE = Ie + IB Rs

Icc = IE + [(IERs + VBE)/~]

Ve = Vee - (Icc RIA) - (IcRl1 )

Page Notes:

1. RIA, RIB, G4, Rs and/or Ru may equal zero.

RIA

+ vee

-

RIB

2. R4 must be manually converted to G4 since conventional mathematics and calculators will not allow division by zero or infinity.

3. R4 may equal infinity. When R4 = 00, G4 = o. 4. Reverse power supply polarity and emitter arrow for pnp

transistors.

5. The effect of varying collector voltage upon collector current has been assumed to be negligible.

212

<C ...I ~ ~ a: o II.

... Z w a: a: ~ CJ

CJ C a: g en Cii Z <C a: ... ...I

~ a: w > Z ~

L to r LVCEO - See-VCEO(SUS)

LVCER - See-VCER(SUS)

LVCES - See-VCES(SUS)

LVCEV - See-VCEV(SUS)

LVCEX - See-VCEX(SUS)

Static (DC) Definitions

n = Region of transistor where electrons are the majority carriers.

npn = Transistor type having two n regions and one p region. (positive polarity Vcc and VBB)

p = Region of transistor where holes are the majority carriers.

pnp = Transistor type having two p regions and one n region.

(negative polarity Vcc and VBB)

Pc = Collector power dissipation

Pc = VCElc

PD = Device power dissipation. See-PT

PT = Total power dissipation of transistor.

PT = (VcE1d + (VBEIB)

rB = T equivalent static internal series base resistance.

rc = T equivalent static internal series collector resistance.

rE = T equivalent static internal series emitter resistance.

rCE(SAT) = Collector to emitter saturation resistance.

213

Ro = Symbol for thermal resistance. (old symbol was 6)

Thermal Resistance

Ro = 1. The opposition to the transfer of thermal energy which develops an increase in temperature at the thermal energy source. 2. The ratio of temperature rise in degrees Celsius to the power dissipated in watts.

ROCA = Case to ambient (usually air) thermal resistance. (formerly 6C- A or 6CA)'

Rocs = Case to (heat) sink thermal resistance. (formerly 6c -s or 6cs)

ROJA = Junction to ambient (usually air) thermal resistance. (formerly 6J -A or 6JA)

RoJC = Junction to case thermal resistance. (formerly 6J -c or 6Jc)

ROJT = Junction to tab thermal resistance. (formerly 6J -T or 6JT)

ROSA = (Heat) sink to ambient (usually air) thermal resistance. (formerly 6s -A or 6SA)

ROTS = Tab to (heat) sink thermal resistance. (formerly 6T -s or 6TS)

Ro = Thermal resistance expressed in °c per watt.

ROJA ~(TJ - TA)/(VcEIc)

T = Temperature in °c P = Power in watts

ROJA = (TJ - TA)/[(VCEIc) + (VBEIB)]

ROJA = ROJC + Rocs + ROSA

214

R to T RB = External series base resistance.

Re = External series collector resistance.

RE = External series emitter resistance.

RL = Load resistance.

Rs = Source resistance.

RBC - See-ReB

RBE = External base to emitter resistance.

ReB = External collector to base resistance.

ReE = External collector to emitter resistance.

REB - See-RBE

REC - See-ReE

T A = Ambient temperature.

DC or Static Definitions

Tc = Case temperature. (Tc meaning "temperature in °C" is not used for semiconductors since temperature is given in °c unless noted.)

TJ = Junction temperature.

TJ = TA + (Pt R6JA)

TJ = TA + [Pt (R6SA + R6CS + R6JC>] TL = Lead temperature.

Ts = (Heat) sink temperature.

T T = Tab temperature.

TSTG = Storage temperature.

215

v VB = Base voltage.

VBB = Base supply voltage.

VBC - See-VCB

DC Transistor

Voltage Symbol Definitions

VBE = Base to emitter forward bias voltage

YBE(ON) = Base to emitter forward bias voltage with normal collector to base reverse bias voltage.

YBE(ON) ~ (.06 [log (1013IdJ) - [.0022 (TJ - 27)]

VBE(SAT) = Base to emitter forward bias voltage with collec· tor in saturation. (typically, saturation occurs when the collector to base junction becomes for· ward biased)

V(BR)CBO = Collector to base breakdown voltage with emitter open·circuited.

V(BR)CEO - See-VCEO(SUS)

V(BR)CER - See-VCER(SUS)

V(BR)CES - See-VCES(SUS)

V(BR)CEV - See-YCEV(SUS)

V(BR)CEX - See-VCEX(SUS)

V(BR)EBO = Emitter to base breakdown voltage with collector open·circuited.

Vc = Collector voltage.

VCB = Collector to base voltage.

VCBO = See-V(BR)CBO

Vcc = Collector supply voltage.

VCE = Collector to emitter voltage.

216

v VCEO - See-VCEO(SUS)

DC Transistor Voltage Symbol Definitions

VCEO(SUS) = Collector to emitter sustaining voltage with base open.

VCER - See-VCER(SUS)

VCER(SUS) = Collector to emitter sustaining voltage with specified base to emitter resistance.

VCES - See-VCES(SUS)

VCES(SUS) = Collector to emitter sustaining voltage with base to emitter short-circuit.

VCEV - See-VCEV(SUS)

VCEV(SUS) = Collector to emitter sustaining voltage with specified base to emitter voltage.

VCEX - See-VCEX(SUS)

VCEX(SUS) = Collector to emitter sustaining voltage with specified base to emitter circuit.

Note:

VE = Emitter voltage.

VEB = Emitter to base reverse bias voltage.

VEBO - See-V(BR)EBO

VEE = Emitter supply voltage.

VRT = Reach through voltage (certain old transistors only).

Collector to emitter breakdown voltage of almost all present production transistors is measured at a current above the negative resistance region where the voltage is sustained over a wide range of current and is therefore called sustaining voltage.

217

DC Transistor Voltages

VB = VBE ~.6

VBE ~ .06 [log(I013Id]

Ve = Vee - [hFE(Vee - VBE)(R1/R2 )]

VB = VBE ~.6

CD :c .~ '" - CD 0. .. 0. 0 <cz

I·<D I·@ I·® I·®

- I·® I·® 1-(2)

I·<D I·@

- I·® VBE ~ .06 [log(1013Ic)]

Ve ~ Vee/ ([hFE(RdR3)] + I)

Ve = [(Vee - VBE)/ ([(hFE + I)(RdR3)1+ I)J + VBE

I·® I·® I·® 1-(2)

VE = [Vee - VBE ] / [(Rd[Rs(hFE + 1)]) + IJ

Ve = Vee - RlhFE([Vee - VBE]/[R2 + Rs(hFE + 1)]) I·<D I·@

VB = VE +VBE R2

VBE ~ .06 [log (1013Ic )]

VBE ~.6

218

E

+

I·® I·®

Vee I·® - I·®

1-(2)

DC Transistor V V V Voltages BeE

Ve = Vee - ([R1hFE(VX - VBE)]/[Rx + RsChFE +0])

VE = [Vx - VBE1/(Rx/[RsChFE + 0])

VB = VE + VBE ~ VE + .6 R2

VBE ~ .06 [log (l 0 13 Ie)] ~.6

Vx = Vee/[CR2/R4 ) + 1] Rx = [R21 + R41 1-1

Ve = Vee - (IeeRd

VB = VE + VBE ~ VE + .6

VBE ~ .06 [log (l0131c)] ~ .6

IB = Ic/hFE

IE = [ldhFE + O]/hFE ~ Ie

Icc = IE + [(IERs + VBE)/R4 ]

RS

Ie = [Vee - CVBERX1)J![Rx2 + CRX3 /hFE)]

RX1 = [CR1 + R3)/R4 ] + 1

RX2 = R1 + Rs + [CR1Rs + R3Rs)/R4 ]

219

III :is

.2 '" - III a. ... a. 0 «2

HI) I-@ I-® I-@ I-@ I-@) I-G)

I-<D I-@ I-® I-@ I-@ I-@) 1-(2)

ex to (J a = Greek script letter alpha.

a = Static (DC) alpha.

Static (DC) Definitions & Formulas

Note: Although "DC alpha" is still verbalized, the equivalent hybrid parameter symbol hFB has almost completely superceeded (i as the accepted written symbol. See-hFB

a=hFB

a ~ Common base static forward current transfer ratio. See, hFB Note: (i and hFB are seldom used with modern transistors since specifications are in the common emitter form hFE'

a = Ie/IE = hFE/(hFE + 1)

a = (hFEIB)/IE = Ic/[IB(hFE + 1)] {3 = Greek script letter beta.

13 = Static (DC) beta. Note: DC beta is often verbalized, but the equivalent hybrid parameter symbol hFE is used on all specifications and most other written or printed usage. See-hFE

13 = hFE

13 = Common emitter static forward current transfer ratio at specified Ic , VCE and TJ • See-hFE

13 = Ie/IB = (IE/IB) - 1 = [(IE/Ie) - 1 r 1

13 = [a- 1 - 1] -1 = [hF~ - 1]-1

(J = Greek letter theta = Obsolete symbol for thermal resistance. See-Ro

220

TRANSISTORS

SECTION 2.2

Small Signal

Conditions

a to C Small-Signal Low Frequency

Definitions

a = Substitute symbol for a (not recommended)

Ai = Small-signal current amplification. (small-signal current gain)

Ai = The ratio of output current to input current

Ai = a = hfb when circuit is common base with output ac shorted.

~ = (3 = hfe when circuit is common emitter with output ac shorted.

Av = Small-signal voltage amplification. (small-signal voltage gain)

Av = The ratio of output voltage to input voltage

Cc = Collector to case capacitance.

Cb'c = Collector to base feedback capacitance.

Ccb = Collector to base feedback capacitance.

Cob - See-Cobo

Coe - See-Coeo

Cibo = Common base open-circuit input capacitance.

Cieo = Common emitter open-circuit input capacitance.

Cobo = Common base open-circuit output capacitance.

Coeo = Common emitter open-circuit output capacitance.

223

.!! Small-Signal

A· .c

Current .~ III Low-Frequency Amplification - CD a. ..

Common Base a. 0 I «2

~ = io/ig E c

~ = ic/ie ic ........

<D ~ = a (alpha)

~io ® @)

~ = hfb ® ~ = hfe/(hfe + 1) ®

~~1

~ = io/ig c ic~

<D ~ = ic/ie RL ~ io ® ~ R; 1 @)

[ 1 ]-1 ®

~ ~ hie (lloeRL + 1) + 1 ® (accuracy typically> 4 digits)

RE E C

Ai = io/ig ic~ <D

~ io ®

~ = ic/ie RL @)

Ai R; 1 ®

[ -1 r 1 ®

~ ~ hfc (hoeRL + 1) + 1

A Notes:

<D -<D- is the graphic symbol for an alternating current generator (infinite impedance) or any very high impedance signal source.

224

Small-Signal Low Frequency Common Collector

~ = io/ig = ie/ib

Ai = hfc

~ = hee + 1

Ai = io/ig = ie/ib

Ai ~ hee

Ai = io/ig = ie/ib

Ai ~ hfe

A· I

Ai ~ hfe [hoeCRL + Rc) + 1]

Current

Amplification

Ai = ([hee - (hoeRd]/[hoe(RL + Rc)+ 1J) + 1

A Notes:

CD @

@)

0)

®

CD ® @)

0)

®

CD ® @)

0)

®

@ --e- is the graphic symbol for an ac voltage generator (zero impedance) or any very low impedance signal source.

® Formulas apply to silicon, germanium, npn and pnp bipolar transistors. Emitter arrows and the power supply polarity (if shown) must be reversed for pnp transistors.

@) Small-signal parameters will vary with temperature as well as with dc bias currents and voltages.

® Small-signal parameters if specified by the manufacturer seldom have maximum or minimum limits and may vary widely. The relationships of parameters, however, will hold very closely to the formulas.

225

Small-Signal Low Frequency Common Emitter

Ai = io/ig = ic/ib

~ = (3(beta)

Ai = hfe

~ = io/ig = ic/ib

Ai ~hfe

~ = io/ig = ic/ib

~~hfe

A· I

Ai ~hfe/[hoe(RL + RE)+ 1]

Current Amplification

~ = [hfe - hoeRE]/[hoe(RL + RE) + 1]

Ai = ic/ib

Ai ~ [(RL/RF ) + hi~] -1

Ai ~ hfe [RL(RFI hfe + RFI + hoe) + 1] -1

226

CD

~ ._ <II

- CD ca. ... ca. 0 «z

(j)

@

@)

® ®

(j)

@

@)

® ®

CD

Small-Signal

Av :D

Voltage .~ '" Low Frequency Amplification - CD a. ....

Common Base a. 0 <CZ

Ay = eo/eg = ec/ee

Ay ~ 37IcRL

~ CY

Ay ~ (hfeRd/hie B: RL

®

Ay ~ [hieh~(Ri:l + hoe) - hre]-1

@

"" eg ®

[hiehil hoe - hre + 1] ® A =

y [hiehil(Ri:1 + hoe) - hre]

Ay = eo/eg

~ Ay = ec/ee B

RL E CY RB

Ay ~ (hfeRd/(hie + RB) "" eg @

Ay ~ [hil(hie + RB) (Ri:1 + hoe) - hre] -1 @)

®

[(hie + RB) hilhoe - hre + 1] ® A =

y [hil(h1e + RB) (Ri:1 + hoe) - hre]

eo

C

B E CY RL

Ay = eo/eg RF @

RB @ "" eg

Ay ~ RL/[RE + hil(hie + RB)] ®

Ay ~ [h~(~e + RB + hfeRE) (Ri:1 + hoe) - hre r 1 ®

227

Small-Signal Voltage

Low Frequency Common CoJlector Av Amplification

Ay = eo/eg

Ay = ee/eb

Ay~1

B ~ Rl ~ 'V eg

Ay = [hie(hoe + RLl)(hfe + 1)-1 - hre + 1]-1

Ay = eo/eg

Ay~ 1

Ay~ 1 when Rg «(hfeRd and RL» (hre/hoe)

E

Ay = ([ (hie + Rg)(hoe + RL1) (hfe + 1)-1] - hre + 1)-1

Ay = eo/eg

Ay ~ 1

Ay = (Common emitter Ay)

• RL(hii + 1) RL~ 'V eg

See-Common emitter formulas

A Notes:

e0 2

RU

RL

CII ::c .~ ... ~$ Q. 0 «2

a> ® @)

® ®

a> ® @

® ®

® Most small-signal parameters are drastically different from the dc parameters due to the nonlinear nature of transistors. A diode or transistor junction which develops .6 volts from 1 rnA forward current has a dc resistance of 600 ohms according to ohms law,

228


Voltage Amplification

Av ~ 37IcRL

Av ~ (hfeRd/hie

Av = [hiehi'~(RLl + hoe) - hrerl

Av = eo/eg

Av ~ Rd.0271(? + Rghi'~rl

Av ~ (hfeRd/(hie + Rg)

Av = [hi'~(Rg + hie )(RL1 + hoe) - hre]-1

Av = eo/eg

Av ~ RL/RE

Av ~ (hfeRd/[hie + (hfeRE)]

Av = [hi';hie(RLl + tt;,e)

+ RERLI (hCe + 1) + hi'~ h~e - hre ]-1 tt;,e = (RE + ~!)-1

A Notes:

+

CI)

:c .~ ... - ., a. ... a. 0 !C(Z

@

® @

® ®

~

® @

® ®

® Cont. but if a small ac signal is superimposed upon the dc and measured, the ac resistance will be found to be about 26 ohms. This small-signal resistance (r) is often verbally expressed as impedance (z), but admittance (y) is used at frequencies where internal capacitances are significant.

229



eo --__._--,

Av = eo/eg C RL

Av R! RL [RE(Rghie1 + 1)r1 E +

Av ~ hfeRL [hie + Rg + hfe REr 1

Av = (hi: RBX(Ri:1 + h~e)

vee

+ hi: REX [Ri:1(hfe + 1) + h~eJ)-1 RBX = Rg + [hie - hreh~!(hfe + 1)] REX = RE + hreh~!

h~ = (h~! + RE)-1

Av = eo/eg

Av R! 38IdRi:1 + RF1)-1

Av ~ hi~1 [hfe (Ri:1 + RF1)-1]

Av = (hiehiH Ri:1 + RF1 + hoe] - hre r1

RE

Av = eo/eg

Av R! RF/Rg

'0 -..---.----,

Av = Avx ([RgRF1 (Avx + 1)] + 1) -1 '" ,:'

Avx ~ 38IdRi:1 + RF1)-1 L-__ -4-----'

Avx = (hiehi: [Ri:1 + RF1 + hoe] - hre)-1

230

Q)

:is

.~ '" - Q) c. .. 0. 0 «2

Q)

® @)

® ®

Q)

® @)

® ®

Q)

® @)

® ®

Small·Signal Low Frequency

Common Emitter

Ay = eo/eg

Voltage

Amplification

Ay = ([RBXhii(RLI + GCX)] +

+ REXhii [Gcx + RLl(hfe + 1)J)-1 fe = hreh;;! fc =h;;!(hfe + 1)

fb = hie - [hreh;;!(hfe + 1)] RBX = Rg + [(RF fb)(RF + fb + fc)-I]

R - R + [(fb + fc)(RF + fb + fcrl] EX - E fe + ~~......:..:~~~---=~....:!.

(hfe + 1)

Rex = (RFfc)(RF + fb + fc)-1

GCX = ([Rex(hfe + 1)-1] + REXfl

231

ell

:is .~ en - ell a. ... a. 0 «Z

~ ® @

® ®

e Small-Signal Voltage Definitions

e = Symbol for emitter. (small signal subscript)

e = Small-signal voltage. (rms or instantaneous)

eb = Small-signal base voltage.

ec = Small-signal collector voltage.

ee = Small-signal emitter voltage.

eg = Generator voltage.

ej = Input voltage.

eN = Noise voltage (rms).

eN = Thermal noise voltage or equivalent input total transistor noise voltage.

eN(v'Hz) = Noise voltage per root hertz. (BW = 1 Hz or eN /VSW for white noise )

eN(s) = Transistor shot noise (white noise) voltage.

eN (th) = Thermal noise voltage. (white noise voltage of an ideal resistance at a specified temperature)

eN (TR) = Transistor noise voltage.

eN(l/f) = l/f noise voltage of a transistor. (Resistor l/f noise is known as excess or current noise)

eo = Output voltage.

ep = Peak voltage.

es - See-eN (s)'

et = Total or equivalent voltage.

el/f - See-eN(l/f).

232

eb =:e. ighie when Av < 50

eb = igZj See-Zj

ec =:e.-hfeigRL

ec = -AvigZj See-Av,Zj

eb = eg

ec =:e.-371c egRL when Av< 50

eb =:e.eg/[(Rs/hie) + 1J eb = eg/ [(Rs/Zj) + 1]

ec =:e. -(eghfeRd/(Rs + hh,l) when Av< 50

SmaUSignal Voltages

rue

eb +

. vc~ 19

ec = -(egAjRd/(Rs + Zj-l) See-Aj, Zj

eNotes:

GI

~ .- ... "Q.$ Q. 0 «z

CD @

Q)

@

@

@

Q)

@

@

®

<D -([)- is the graphic symbol for an alternating current generator. (an infinite impedance signal source)

® Transistors must be biased into an active region. Q) Approximations apply to high beta silicon small-signal transistors

while exact formulas apply to all bipolar transistors.

233

eb = eg

ec ~-(egRd/RE when RE» (371d-1

ec = -egAy See-Ay

ee ~ eb when RE» (37Id-1

eb ~ eg/ [Rs(Rsl + h~l) + 1] eb = eg/[Rs(Rsl + Zi1) + 1] ec ~ -37Ic ebRL

ec = -ebAy See-Avo Zj

ee = 0

eb = eg

ec ~-(eghfe) [hje(Rc? + RL1)]-1

ec = -(egAj) [Zj(I~c? + RL1)] -1 eb

ee = 0 See-Aj, Zj

eNotes:

SmallSignal Voltages

+

CII

:is .~ c.! a. 0 «z

@

® @)

® ®

@ Formulas apply to pnp transistors as well as the npn transistors shown. Reverse emitter arrow and power supply polarity for pnp transistors.

® Transistor parameters will vary with collector voltage and temperature as well as collector current.

® The resistance of base bias resistors must be included in all calculations where the generator source resistance is of significance.

234

Noise Voltage

Thermal noise voltage

(eNh/Hz~ th = V4KB T K Rs Thermal noise voltage per root hertz

eN(s) = re V2qIBBW Transistor shot noise (white noise)

(eN (y'Ifz») s = re~ Transistor shot noise per root hertz

eN(l/f) = V(eN)tR - (eN); Both same bandwidth I

Total Equivalent Input Noise Voltage

(eNh = ([Rs(iNhR] 2 + (eN)iR + (eNh1t)t

(all same BW)

Wideband Total Noise Voltage Output

eN (OUT) = Av([Rs(iNhR] 2 + (eN)h + (eN)~)t (all same BW)

Total Spot Noise Voltage Output

eN(OUT) = Av([RS(iNhR] 2 + (eN)h + 4KsTKRst

all noise terms are for 1 Hz BW at same frequency

eN Notes: Kll = Boltzmann constant (1.38 • 10-23 J/K); TK = Kelvin temperature (,"C + 273.15); q = Charge of electron (1.6 • 10~1~; BW = Bandwidth, See-Opamp, BWNOISE; re = Internal transistor dynamic emitter resistance; re"" .027/Ie; (iN)TR = Transistor noise current equivalent input); Av = stage voltage amplification; IB' Ie = dc base and collector currents.

235

F to Qme F - See-NF, See also-F, Passive Circuits

FN - See-NF

Small-Signal Definitions

fe = Symbol for cutoff frequency. (The half power or 3 dB down frequency)

See-fe,Opamps

fT = Gain-bandwidth product. The frequency at which the common emitter small-signal forward current transfer ratio falls to unity. (dc biased for large signal)

ft = Same as fT except biased for small-signal.

fab - See-fhfb

fae - See-fhfe

fhfb = Common base small-signal forward current transfer ratio cutoff frequency with output ac shorted.

fhfe = Common emitter small-signal forward current transfer ratio cutoff frequency with output ac shorted.

Gpb = Common base small-signal average power gain.

Gpe = Common emitter small-signal average power gain.

Gpe(eonv) = Common emitter conversion gain.

gme = Common emitter small-signal transconductance.

236

tlfb tlfc tlfe

Small-Signal Forward Current Ratios

hfb = Common base small-signal forward current transfer ratio with output ac shorted.

hfb = Small signal alpha (a)

hfb = ic/ie

hfb = hfe/(hce + 1)

hcc = Common collector (emitter follower) small-signal forward current transfer ratio with output ac shorted.

hcc = ie/ib

hcc = hCe + 1

hCe = Common emitter small-signal forward current transfer ratio with output ac shorted.

hCe = Small-signal beta (13).

hCe = ic/ib

hCe = a/(1 - a)

hCe = hCb/(l - hCb )

hCe = hcc - 1

hCe = hfbhcc

hCe = Common emitter current gain when output is ac shorted.

hCe = (ichie)/ebe when output is ac shorted.

237

Small-Signal Input Impedance

hib = Common base small-signal input impedance with output ac shorted.

hib = ee/ie

hib = hie/(hfe + 1)

hib = re + [rb/(hfe + 1)] hib ~ 1/(37Ic)

hie = Common collector (emitter follower) small-signal input impedance with output ac shorted.

hie = hie (since emitter is ac shorted)

hie = Common emitter small-signal input impedance with output ac shorted.

hie = eb/ib

hie = hie

hie = hib(hfe + 1)

hie = rb + re(hfe + 1)

hie ~ hfe /(37Ic)

hie ~ (26. 7hfe )/(lOOOIc)·78

Approximations apply to small-signal silicon transistors.

238

Small-Signal Output Admittance

hob = Common base small-signal output admittance with input ac open-circuited.

hob = ic/ec when emitter is ac open-circuited (constant current emitter supply)

hob = Yob (Yob is generally used at high frequencies)

hob ~ hoe/(hfe + 1)

hob = r + rb

hob = ([~!(hfe + 1)] + [14e - hre~!(hfe + 1)])-1

hoc = Common collector (emitter follower) small-signal output admittance with input ac open-circuited.

hoc = hoe (since input is open-circuited)

hoe = Common emitter small-signal output admittance with input ac open-circuited.

hoe = ic/ec when base is ac open~ircuited (constant current base supply)

hoe = Yoe (Yoe is generally used at high frequencies)

hoe = hoc

hoe = (hfe + l)/rc

hoe ~ 20 J.LS (50 kU)-l when Ic ~ 1 rnA, Y CE > 5 Y, T ~ 25°C

239

Small-Signal Reverse Voltage Ratios

hrb = Common base small-signal reverse voltage transfer ratio with input ac open-circuited.

hrb = ee/ee when emitter is ac open-circuited (constant current emitter supply)

hrb = rb/(rb + re)

hrb ~ [hiehoe(hee + 1)-1] - hre

hrb = [([hiehoe(hee + 1rl] - hretl + 1r1

hre = Common collector small-signal reverse voltage transfer ratio with input ac open-circuited.

hre =l-hre

hre = Common emitter small-signal reverse voltage transfer ratio with input ac open-circuited.

hre = eb/ee when base is ac open-circuited (constant current base supply)

hre = [re(hee + 1)] Ire hre ~ 1.33 • 10-6 hee when Ie ~ 1 rnA and VeE> 5 V

Note: hre is very VeE sensitive at low voltage. hre typically is very nonlinear over large variations of Ie.

240

•

I I - See-I, Static (DC) Symbols

See also-I, Passive Circuits

i = Small-signal current.

ib = Small-signal base current.

ic = Small-signal collector current.

ie = Small-signal emitter current.

ig = Small-signal generator (source) current.

iin = Small-Signal input current

iN = Noise current.

Small-Signal Current Definitions

iN (TRANSISTOR) = That portion of the input equivalent internal transistor noise which is proportional to the external resistance in shunt with the input (source resistance)

iN Notes:

<D iN(TRANSISTOR) does not include the thermal noise or the excess noise currents of the effective external source resistance.

@ iN(TRANSISTOR) may be included in the eN(TRANSISTOR) (TOTAL) if a source resistance (RS) has been specified.

® Much of the confusion regarding noise voltages and noise currents results from the difficulty in proper identification of the symbols for the various noise voltages and noise currents.

@) See also-eN, NF; See also-Vn, in, Opamps;See also-EN, IN, Nth, NI, Passive Circuits

io = Small-signal output current.

ip = Small-signal peak current.

241

ib = ig

ic = ighfe

ie = ig(hfe + 1)

ib = ig

ic ~ ighfe

ic = igAi See-Ai

ie ~ ig(hfe + 1)

• •

ie = ig(Ai + 1) See-Ai

ib ~ eg/hie

ib = eg/Zi See-Zi

ic ~ (eghfe)/hie

ic = (egAy)/RL See-Ay

ie = [eg(Ay + 1)]/RL See-Ay

iNotes:

Small-Signal Transistor Currents

Q)

::c fj ._ rn - Q) a. ... a. 0 ~2

<D ~ is the graphic symbol for an infmite impedance alternating current generator. (any very high impedance source)

@ Transistors must be biased into an active region. ® Approximations apply to high beta small-signal silicon transistors.

Exact formulas apply to all bipolar transistors.

242

• •

ib ~ eg/[ (hfeRE) + hie]

ib = eg/~ See-~

ic ~eg/RE when RE» (37Id-1

ic = (egAy)/RL See-Ay

ie ~ic when hfe > 100

ie = [eg(Ay + 1)] /RL See-Ay

ib ~ eg/(Rs + hie)

ib = eg/(Rs + Zi)

ic ~ (eghfe)/(RS + hie)

•

ic = (egAi)/(Rs + Zi) See-Ai> Zi

ie ~ ic when hfe > 100

ic ~eg/RE when RE »(37Id-1

io ~(eg/RE) [(RL/Rd + 1r1

io = ic/[(RL/Rd+ 1]

iNotes:

• SmallSignal Currents

CD :s .~ ... - CD Q.~ Q. 0 «z

@

® @

® ®

@) All formulas apply to pnp transistors as well as the npn transistors shown. Reyerse emitter arrow and power supply polarity for pnp transistors.

@) Transistor parameters will yary with collector yoltage and temperature as well as with collector current.

® The resistance of base bias resistors must be included properly in all calculations where the signal source resistance is of significance.

243

NF Noise Figure

NF = Symbol for noise figure (also known as noise factor)

(other symbols include F and F N )

NF = 1. The ratio (usually in decibels) of the output signalto-noise power to the input signal-to-noise power. 2. The ratio in decibels of the total output noise to that

portion of the output noise generated thermally by the input termination resistance.

NF = 20 Dog (endenR'~

NF = 20 (log [enO/(enR'Av)J)

where enR' = enR/ [cRs/ri) + IJ and enR = v'4KB TKBW

See alsO-eN(out) and BWNOISE , Opamp

NF Notes:

<D Av = Voltage amplification, BW = Average bandwidth (rms bandwidth), eni = Input equivalent total noise voltage, eno = Output noise voltage, enR = Thermal noise voltage of source resistance, KB = Boltzmann constant 0.38 . 10-23 J/"K), ri = Transistor input resistance, RS = Source resistance (the total effective resistance presented to the transistor input), TK = Kelvin temperature (oC + 273.15), log = Base 10 logarithm.

G> The standard noise temperature (TN) of the source resistance is 290 K (16 .85°C) if unspecified.

See also - eN Notes

244

Vv v = The unit symbol for volt.

See-V, Passive Circuits

V = The quantity symbol for voltage. See-V, Static (dc) Parameters See also-V, Opamp

Definitions

Note: A defmite trend exists towards the elimination of E and e as symbols for voltage. At present, E and e predominate in passive circuits, V and v predominate in operational amplifiers, V has superseded E in dc transistor parameters and e predominates for ac transistor parameters.

v = Symbol for small signal voltage. See-e See also-V, Opamp

Vb - See-eb

Vc - See-ec

ve - See-ee

Vg - See-eg

vi - See-ei

VN - See-eN (VN-See-V,Opamp)

Vo - See-eo

Vp - See-ep

Vs - See-eN (s)

Vt - See-et

Vl/f - See-eN(l/f)

246

Small Signal Low Frequency

Common Base

Zi:::; Ij(37Id

Zi = re + rb(hfe + 1)-1

Zi = hib

Zi = hiej(hfe + 1)

Zi:::; Ij(37Id (when Ay < 50)

Zi ~ hiej(hfe + 1) (when Ay> 50)

z· I

Zi = (hoeRL + l)j(hoe RL + 1 + hfe )

Small Signal

Z· Low Frequency Common Collector I

Zi:::; hfeRL

Zi ~ hie + RL(hfe + 1)

Zi = hie + [(hfe + l)j(hoe + Ri:1)]

246

B

Input Impedance

RL

Input Impedance

E

+

I»

:E ~ .- '" - I» 0..., 0. 0 «2

® @)

@

® ®

® @)

@

® @

®

I» :E .~ '" - I» 0..., 0. 0 «2

® @)

@

® @

(2)

ZNotes:

<D -(])- is the graphic symbol for an infinite impedance alternating current generator. (an ac current source) In practice, any very high impedance source of current may be substituted.

® ~ is the graphic symbol for a zero impedance signal generator. (an ac voltage source) In practice, any very low impedance signal source may be substituted.

® Approximations apply to high beta, small signal, silicon transistors. Exact formulas apply to all bipolar transistors.

@) Formulas apply to pnp as well as the npn transistors shown. Reverse emitter arrow and power supply polarity for pnp transistors.

@) All internal dynamic resistances (rb, rc' re) vary with operating conditions. Primarily, re varies with emitter current while rc varies primarily with temperature and collector voltage. Usually, rb is assumed to be a non-varying resistance.

@) All biasing resistors connected in shunt with an input are effectively in parallel with the input impedance. The equivalent resistance of all parallel quantities must be used in all calculations where the source resistance becomes significant. Z; = (Zi(k) + R.1 + R21)-1

(1) x-I = l/x ® In the usual circuit where the collector is capacitor coupled to a

load, the series collector resistor and the load resistance are effectively in parallel and the net parallel resistance should be used in all ac calculations. RL = (R.1 + R21)-1

® Base biasing is not shown but transistors must be biased into an active region.

@ Collector bias and base bias circuits are not shown, however the transistors must be biased into an active region.

247

Small-Signal Low Frequency Common E mitte,

~ ~ hfe/(37Ic)

Zi = fb + fc(hfe + 1)

Zi = hie

Zi ~ hfe /(371c) (when Av < 50)

Zi ~hie (when Av < 50)

z· I

Zi = hie + hrehfeh~~ [(RLh~ + 1)-1 - 1]

Zi ~ hfe [(371c)-1 + RE] (when Av < 50)

Zi ~ hie + hfeRE

[(hfe + l)(RE + fe)] z

Input Impedance

RL

RL

Z - ' i - ([hii(fcRI:? + 1) + 1] -1 + 1) RE

-

fb = hie - hreh~~ (hfe + 1)

fc = h~~(hfe + 1)

fe = hreh~~

248

@

@

® ® (2)

@

@

® ® C!> ® ®

® @

® ® C!> ® (2)

Small-Signal

Low Frequency

Common Base

Zo = l/hob

Output

Impedance

Zo = ([h~~(hfe + 1)] + [hie - hreh~~(hfe + 1)])-1

Zo ~ 200kn (when Ie ~ 1 rnA)

Zo ~ h~~ + h~~(50Iehiehi! - 1)-1 ~ e, L...------"+--------6

Zo = h~~ ([(hiehoe/hfehre) - 1] -1 + 1)

Zo < RB + (hfe/hoe)

Zo = ~~ + hfe~~ ([(RB + fb)/(Rg + fe)] + 1)-1 fe = hreh~~

249

III

:is .~ ... - III a. ... a. 0 C:(Z

CD ® @)

® ® G> @

@

® @

® ® CD @>

@

® @)

® ® G> @>

Small-Signal Low Frequency Common Collector

Zo ~50kn (when Ic ~ 1 rnA, VCE > 5)

Zo = l/hoc

Zo = l/hoe

Zo ~ 1/(37Ic)

Zo =:::: hie/hfe

Zo = Ie + Ib(hfe + 1)-1

Zo = hie/(hfe + 1)

Zo ~ (37Ic)-1 + (Rg/hfe )

Zo =:::: (Rg + hie)/hfe

Zo = Ie + [(Rg + Ib)/(hfe + 1)] Zo = (hie + Rg)/(hfe + 1)

250

Output Impedance

E

c

Q)

:E .~ U)

- Q) c. .... c. 0 <tz

CD ® ® ® ® @

@

® ®

Zo ®

c

E

c

® ~ @

@

® ® ®

Zo ® ~ @


Zo ~ 50kD (when Ie ~ 1 rnA

Zo ~ fc/(hfe + 1)

Zo = fc(hfe + 1)-1 + fe

Zo = l/hoe

Zo ~ 200kD (when Ie ~ 1 rnA)

Output Impedance

Zo ~h~! + h~!(50Iehiehf-l - 1)-1 "'----" __ -+.:::-_----<

Zo = h~! ([(hiehoe/hfehre) - 1] -1 + 1)

Zo < Rg + (hfe/hoe)

Zo = (h~! + hfeh~!)/ ([(Rg + fb)/(RE + fe)] + 1) fe = hreh~!, fb = hie - fe(hfe + 1)

251

Q)

::c B ._ en

- Q) a. .... a. 0 «2

af3 a = Greek script letter alpha.

Small-Signal Current Ratios

a = Symbol for small signal common base forward current transfer ratio with output ac shorted. Note: Although alpha predominates as the "oral symbol," the equivalent hybrid parameter hfb has almost completed superseded 01 as the accepted written symbol. See-hfb

a = hfb a = hfe/(hfe + 1) or (hi~ + 1)-1

a = icfie when ec and eb = 0

a~1

a < 1 exception-very early point contact transistors

{3 = Greek script letter beta.

{3 = Symbol for small signal common emitter forward current transfer ratio with output ac shorted. Note: Although beta predominates as the "oral symbol," the equivalent hybrid parameter hfe has almost completely superseded {3 as the accepted written symbol. See-hfe

{3 = hfe

{3 = ic/ib when ec and ee = 0

{3 = (ichie)/eb when ec and ee = 0

(3 = hfb/(hfb - 1) or (h~ - 1)-1

(3 = (ie/ib) - 1 = [(ie/ic) - 1 r 1

252

SECTION

THREE

OPERATIONAL

AMPLIFIERS

3.1 DEFINITIONS

A-See-Av a - See-a (alpha)

ACL - See-AVCL AOIFF - See-Avo

A(fo) - See-Avo

A Opamp

Symbol

Definitions

AI = Large signal current amplification (gain). Also dc current gain in direct coupled circuits.

Ai = Small signal current amplification (gain). AlAC = Alternating current amplification (gain). AIDC = Direct current amplification (gain).

Am = Gain margin. The reciprocal of the open-loop voltage amplification at the lowest frequency at which the open-loop phase shift is such that the output is in phase with the inverting input. See also-Om

Ao -See-Avo AOL - See-AVOL Av = Large signal voltage amplification (gain). Also dc volt

age gain in direct coupled circuits. Av = Small signal voltage amplification (gain).

AVAC = AC voltage amplification (gain). Avo = Large signal differential voltage amplification (gain).

Avoc = DC voltage amplification (gain). AVCL = Large signal closed-loop voltage amplification. The

large signal voltage gain of an opamp stage with inverse feedback. Applies also to dc voltage gain in direct coupled circuits. This symbol is used in place of Av only when the meaning would otherwise be confusing. See also-Av

255

AB Opamp Symbol Definitions

Avcl = Small-signal closed-loop voltage amplification. The small-signal voltage gain of an operational amplifier stage with inverse feedback applied. See alsoAv

Avo = Midband voltage amplification. The voltage amplification at the midband or reference frequency (fo).

AvoL = Large-signal open-loop voltage amplification. The large-signal voltage gain of an operational amplifier before application of inverse feedback.

Avol = Small-signal open-loop voltage amplification. The small-signal voltage gain of an operational amplifier (opamp) before application of inverse feedback.

B = See-BW, See also-B, Passive Circuits. Bl = See-BW(Av = 1)

BOM = Maximum output swing bandwidth. BW = Bandwidth

BW(-3dB) = Half power or 3 dB down bandwidth BW(-3dB) = fo/Q (bandpass fllters) BW (Av = 1) = Unity gain bandwidth. The range of frequencies

within which the open-loop voltage amplification is greater than unity. Unity gain bandwidth is also known as gain-bandwidth product but, is only approximately equal to actual gain-bandwidth product. (See-GBW, fT). The unity-gain bandwidth is equal to the product of the small-signal closed-loop voltage amplification (Avel) and the closed-loop flat-response bandwidth only when the open-loop voltage gain is inversely proportional to frequency in the frequency range between the top bandpass frequency and the unity-gain frequency.

256

BWNOISE Noise Bandwidth

BWNOISE = Bandwidth used to compute noise output. (other symbols include: B, B, BW, BWn )

BWNOISE = Noise bandwidth with zero noise contribution from frequencies above or below bandwidth limits

BWNOISE = Noise bandwidth measured with filters having nearly rectangular response curves. ("cliff" or "brick wall " filters)

Effective Noise Bandwidth from zero to the 3 dB Down Frequency using Butterworth Filters

BWNOISE = 1.57 BW -3dB 6 dB per octave filter

BWNOISE = 1.11 BW-3dB 12 dB per octave filter

BWNOISE = 1.05 BW-3dB 18 dB per octave filter

BWNOISE = 1.025 BW -3dB 24 dB per octave filter

BWNOISE = BW -3dB 00 dB per octave filter

Notes:

6 dB per octave = 20 dB per decade (first order filter)

12 dB per octave = 40 dB per decade (second order filter)

dB per decade = 3.333 (dB per octave) dB per octave = .3 (dB per decade)

257

BCD BW(Av= 1) = [Avcd/[BWcd

BW cl = Small signal flat response bandwidth.

BWcl :::: [BW(Av = 1)]!Avcl BWNOISE - See preceding page

BW p = Power bandwidth. See also-PBW

BWp = SR![1TVopp ]

CB = Bypass capacitor. Bootstrap capacitor.

Cc = Coupling capacitor.

CJ, C j, CIN , Cin = Input capacitance.

Opamp Symbol Definitions

CMRR = Common mode rejection ratio. The ratio of differen-tial voltage gain to common mode voltage gain.

Co, Co, Cout = Output capacitance.

Cp = Parallel capacitance.

CT , Ct = Total capacitance.

D-See-THD

d = Damping factor. (other symbols include ex and 0) The reciprocal of the Q factor in most applications. A symbol used in high and low pass fIlter formulas where the 3 dB down definition of Q factor is not applicable. Note: Nearly everyone understands the meaning of Q factor regardless of the difficulty with an all encompassing definition. See-Q

d= l/Q

258

dB Editorial

The decibel (dB) has a long history of use as a voltage or current ratio regardless of the original, and still the only completely unqualified, definition of the term as a power ratio.

The lack of an official sanction for the use of the decibel as a voltage or current ratio has created the ridiculous situation where, input and output resistances are "assumed" to be equal when it is known that large differences exist, specifications use mV/V, mV/V, V/V, V/mV and dBV for voltage ratios, and the same engineer who uses the decibel for voltage ratios on a daily basis, switches to a "more technically correct" form when writing a specification or a paper. (Some opamp manufacturers do specify certain voltage ratios in decibels)

The time has arrived for the official definition of the decibel to be expanded to include voltage and current ratios. There can be absolutely no misunderstanding of the expanded definition as long as the decibel is stated as a power, voltage or current ratio, so you may start using this very useful tool now, without waiting for an official definition.

There will be a few ultra conservatives who will insist that the decibel can only be used correctly as a power ratio but they will be a very small minority. Some of this same group will insist that the word gain means only power and although completely understandable the use of the term "voltage gain" is technically incorrect.

The use of any of the words gain, amplification or decibels without reference to power, voltage or current is not recommended since all three would have to be interpreted as power in the absence of an adjective.

259

DE Opamp Symbol Definitions

dB = Decibel. A logarithmic ratio of power, voltage or current. See-dB editorial on preceding page. See also-dB, Passive Circuits

dB = 10 [log (P ofPi)]

dB = 20 [log (VO!Vi)]

dB = 20 [log (lo/Iin)]

dBf= Power in decibels referenced to one femtowatt. (fW = 10-15 W)

dBm = Power in decibels referenced to one milliwatt.

dBre - See-dBREF

dBREF = Reference level in decibels.

dBV = 1. Voltage in decibels referenced to one volt. 2. Voltage ratio in decibels. (not recommended)

E - See-V See also-E, Passive Circuits

e - See-V See also-e, Transistors and e, Passive Circuits

eg, ej, ein -See-Vg, Vj, Vs

EN , En, eN, en etc - See-V n

- See also-eN, NF, Transistors

- See also-EN, NI, Passive Circuits Note:

The transition from E to V as the quantity symbol for voltage is complete in this opamp section. The symbol E was used exclusively in the passive circuit section while the transistor section used V for dc voltages only. It is expected that eventually the symbol V will replace E for all electronic usage.

260

FG Opamp Symbol Definitions

F = Noise factor. Noise factor is also known as noise figure (NF). F may represent the average or the spot noise factor. See-NF, Transistors

F = Average noise factor.

fl - See-B!, BW(Ay = 1)

F(f) = Spot noise factor.

fc = Cutoff frequency. The frequency at which the output falls to one-half power or 3 dB down from maximum.

fIN, fin = Input frequency

fo = Reference, center, midband, resonant, oscillation or output frequency.

fp = Frequency of pole. (poles and zeros)

fr = Resonant frequency.

fT' ft = Unity gain frequency. The frequency at which the open-loop voltage gain falls to unity. Has exactly the same meaning as BW(Ay = 1) in all integrated circuit opamps. See-BW(Ay = 1)

fz = Frequency of zero. (poles and zeros)

G = Conductance See-G, Passive Circuits

GBW = Gain-bandwidth product. The product of the small signal voltage amplification (Ay) and the bandwidth (BW). See-BW(Ay = 1)

GBW ~ or = BW(Ay = 1) or fT

Depending upon the exact definition.

261

G H I Gm = Large-signal forward transconductance.

gm = Small-signal forward transconductance.

Gp = Large-signal power gain.

Gp = Small-signal power gain.

Ho = Function at reference frequency. (fo)

1+ = Positive dc supply current

1- = Negative dc supply current

IA = Amplifier dc supply current

IABC = Amplifier bias current.

IB = Bias current

Icc = Positive dc supply current

ID = Device dc supply current

ID+ = Device positive dc supply current

ID- = Device negative dc supply current.

IDG = Non-inverting input grounded current.

IDa = Non-inverting input open current.

lEE = Device negative dc supply current.

Ig = Small-signal generator (source) current.

lIB = Input bias current

lIN, lin = Input signal current

262

Opamp

Symbol Definitions

I Opamp

Symbol Definitions

Ira = Input offset current. The difference between the bias currents into the two input terminals of an opamp with the output at zero volts.

I Ira I = The magnitude of input offset current. See also-Ira

In = Device equivalent-input noise current. That component of device total equivalent-input noise which varies with the external source resistance and therefore is properly represented by an infinite impedance current source in parallel with the input terminals.

In = v'1~s + I~f in - See-In

Inf = Device equivalent-input l/f noise current. That part of In which has a spectral density which is inversely proportional to frequency.

InR = Thermal (white) noise current of resistance See-IN, Passive Circuits

Ins = Device equivalent-input shot (white) noise current.

10 = Large signal output current.

10 = Small signal output current.

10+ = Large signal positive swing output current.

10 - = Large signal negative swing output current.

lapp = Peak to peak output current.

los = Short-circuit output current. The maximum output current available from the device with the output shorted to ground or either supply.

263

I J K Ip = Large signal peak current

Ip, Ipk' Ipeak = Small signal peak current Is = 1. Source current. See-Ig , lin

2. Shot noise current. See-Ins Isc -See los

IT, It, iT = Total current ITH = Threshold current.

J, j - See-J, j, Passive Circuits


K = 1. Kelvin temperature. Cc + 273.15) 2. Voltage gain See-Ay 3. Any constant.

k = 1. Any constant 2. Boltzmann constant.

kB = Boltzmann constant. (1.38 • 10-23 JtK) kcMR = Common mode rejection ratio. See-CMRR kSVR = Supply voltage rejection ratio. The absolute

value of the ratio of change in supply voltage to the change in input offset voltage. The reciprocal ofPSRR or PSS. See also-PSRR, PSS, ksvs

kSVR = I.:lVCcl.:lVIOI ksvs = Supply voltage sensitivity. The absolute value of

the ratio of change in input offset voltage to the change in supply voltages producing it. The reciprocal ofksvR. See Also-PSRR, PSS, kSVR

ksvs = I.:lVIO/.:lVccl

264

LMN L = 1. Inductance. See-L, Passive Circuits


2. Level. Signal level in decibels with respect to a noted reference level.

mAdc = Direct current milliampere.

MAG = Maximum available (power) gain.

MUF = Maximum usable frequency.

mW;oC = Milliwatt per degree Celsius.

Mil, M = Megohm

N= 1. Noise. See also-Vn, In. 2. Noise power. See-PN , Passive Circuits 3. Number. A pure number or a ratio.

NF - See-F, See also-NF, Transistors

NI - See-NI, EN(EX) , Passive Circuits.

Np - See-PN , Passive Circuits

Nth - See-PN , Passive Circuits, See also-VnR

nV /v'Hz, nV/(Hz)~, nV /..F- = Nanovolts per hertz or nanovolts per root hertz or nanovolts per root cycle. The spot noise voltage in nanovolts. The noise voltage in nanovolts for a bandwidth of one hertz at a specified frequency.

n V /v'Hz = (V n(nv»/0Wi only when the noise voltage has constant spectral density. (only when the noise voltage is white noise)

265

OP os, OS = Overshoot

Pc = 1. (Device) power consumption.


2. Collector power dissipation. See-Pc, Transistors

PD = 1. Device power dissipation 2. Power dissipation.

PF, p J. = Power factor. See-pf, Passive Circuits

pF = Picofarad. (10-12 farad)

Pi, PIN, Pin = Input power. PN = Noise power. See-PN, Passive Circuits.

Po = Output power.

PSRR = Power supply rejection ratio. The absolute value of the ratio of the change in input offset voltage to the change in power supply voltage producing it. This ratio is usually in I1V IV or in dB. When all are given in decibels and disregarding the sign of the decibel ratio, KSVR' K svs , PSS, PSRR, VSRR, I~Vccl~Vlol and I~Vlo/~Vccl are all equal. It is hoped that the industry will soon standardize on only one of these symbols.

PSRR = See-PSRR

PSS = Power supply sensitivity. See-PSRR

PSS± - See-PSS

PSS+ = Positive power supply sensitivity. See-PSRR

PSS- = Negative power supply sensitivity. See-PSRR

PT , Pt , Ptot = Total power.

266

QR Opamp Symbol Definitions

Q = Q factor. In simple bandpass filters, the ratio of the resonant frequency to the 3 dB down bandwidth. In highpass or lowpass filters where the 3 dB down definition is not applicable, the reciprocal of the damping factor (d). See also-Q, Passive Circuits. Note: The Q factor is also known as the merit, quality, storage, magnification and energy factor. There is no known simple definition of Q which will encompass all of the applications. The general meaning of the term appears to be understood but the exact meaning, except in a few applications, is open to interpretation.

Q = l/d

Q = fo /BW(-3dB)

Q = fr /BW(-3dB)

QL = Loaded Q factor.

Q 0 = Q factor at center or reference frequency (fo).

Qu = Unloaded Q factor.

R = Resistance See-R, Passive Circuits

r = Small Signal (dynamic) resistance. Any resistance of a semiconductor device which may be non-linear and therefore produce a different value between dc and small signal measurements.

RF = Feedback Resistor

Rg = Generator resistance. See-Rs

267

R Opamp Symbol Definitions

RJ = 1. Input resistor. (Not recommended) 2. Large signal input resistance. (Not recom

mended)

~ = Small signal input resistance. See also-Z j

rj = Device small signal input resistance.

rjd = Device differential input resistance.

RIN = Large signal input resistance.

R jn -See-Rj

rjn -See-rj

RL = Load resistance.

Ro = Large signal output resistance.

Ro = Small signal output resistance. See also-Z o

ro = Device small signal output resistance.

RoPT = Optimum resistance. e.g. Rs(OPT) = V n/In

ROUT' Rout -See-Ro, Ro

Rp , Rp = Parallel resistance.

rp = Dynamic plate resistance (vacuum tube) (anode resistance (ra) is also used).

Rs = Source resistance.

Rs = Series resistance.

RT, Rt = Total resistance.

Rth - See-Re, Transistors

Re - See-Re, Transistors

268

8T S = 1. Sensitivity.

2. Signal. See-sig s = Laplace transform function.

S+ - See-PSS+ S- - See-PSS-

S± - See-kSYS' PSRR, PSS


sig = Signal. Any electrical, visual, audible or other indication used to convey information.

SIN = Signal to noise ratio. SR = Slew rate. The closed-loop average-time rate-of

change of output voltage for a step-signal input. A specification used to determine the maximum combination of frequency and peak-to-peak output signal without the distortion associated with rise and fall time.

SR = 1T PBW V OPP

SR(Ay = 1) = Slew rate when closed-loop voltage amplification is unity.

T = 1. Temperature. (oC unless noted) 2. Time constant. See-T, Passive Circuits 3. Time. See-t 4. Loop gain. (AYOL/Aycd

t = 1. Time. Time or period in seconds 2. Temperature. See-T

TA = Ambient temperature. The average temperature of the air in the immediate vicinity of the device.

TC = Temperature coefficient. Tc = Case temperature.

269

T Opamp Symbol Definitions

TClIo = Temperature coefficient of input offset current. The ratio of the change in input offset voltage to the change in free-air temperature when averaged over a specified temperature range.

TClIo = I [(I1O)1 - (I1O)2]/[(TA)1 - (TAh] I TCylO = Temperature coefficient of input offset voltage. The

ratio of the change in irlput offset voltage to the change in free-air temperature when averaged over a specified temperature range.

TCylO = I [(VlOh - (V1O)2]/[(TA)1 - (TA)2] I tf = Fall time. The time required for the trailing edge of

an output pulse to fall from 90% to 10% of the final voltage in response to a step function pulse at the input.

THD = Total harmonic distortion.

THD = v'V~ + V~ - - - + V~/V 1 where Viis a sine-wave input signal (fundamental) and V2 through Vn are the 2nd through nth harmonic respectively.

Thigh = High temperature.

TK = Kelvin temperature. CC + 273.15)

T L = Lead temperature.

T10w = Low temperature.

tos = Time of output short-circuit.

tp, tpd = Pulse duration

tPLH - See-tr

270

TUV Opamp

Symbol

Definitions

tr = Rise time. The time required for an output voltage step to rise from 10% to 90% of the final value.

tsetlg = Settling time. See-ttot

Tstg = Storage temperature.

tTHL - See-tf

ttot = Total response time. (Settling time) The time between a step·function change of the input signal level and the instant at which the magnitude of the output signal reaches for the last time a specified level range.

U = Teletypewriter or computer printer substitute for greek letter mu (f.1).

u = Typewriter substitute for greek letter mu (f.1).

v = Symbol for the voltage quantity as well as for the volt unit.

VA = DC or rms large signal voltage.

Va = Small signal rms signal voltage.

v A = Instantaneous large signal voltage.

va = Instantaneous small signal voltage.

+V = Any positive dc voltage.

- V = Any negative dc voltage.

V+ = Positive polarity power supply voltage.

271

v v - = Negative polarity power supply voltage.

V AC - See-V AC (V AC or V ac = unit-symbol)

VAC = Alternating current voltage.

Opamp

Symbol Definition

V BB = Base power supply voltage or base bias voltage.

Vcc = Collector supply voltage. (positive polarity in all present IC opamps)

+Vcc = Positive polarity collector supply voltage.

VCM = Common mode voltage.

VDC - See-V DC 01 OC or V dc = unit symbol)

VDC = Direct current voltage.

VEE = Emitter supply voltage. (negative polarity in all present IC opamps)

- VEE = Negative polarity emitter supply voltage.

V g = Generator (signal) rms voltage.

VI = Input voltage range.

Vi = Input (signal) rms voltage.

Vi = Instantaneous input voltage.

VIC - See-VICM

VICM = Common mode input voltage.

VICR = Common mode input voltage range.

Vm = Differential input voltage.

272

v VIDR = Differential input voltage range.

VIN = Large signal input voltage.

Yin = Small signal input voltage.


VIO = Input offset voltage. The de voltage that must be applied between the input terminals to force the quiescent de output to zero.

IVIOI = The magnitude ofVIO . See-VIO

VIOR = Input offset voltage adjustment range.

VIR = Input voltage range. See also-VI

V n - See following page

Vo = Large signal output voltage.

V 0 = Small signal output voltage.

vo = Instantaneous large signal output voltage.

Vo = Instantaneous small signal output voltage.

VO(CM) = Common mode output voltage.

V OM = Maximum output voltage.

V OM +, V OM+ = Maximum positive output voltage.

V OM-, VOM- = Maximum negative output voltage.

Voo = Output offset voltage.

Voos -See-Voo

V opp = Peak to peak output voltage.

Vop-p, VO(p-p) - See-Vopp

273

V n = 1. Any rms noise voltage


2. The equivalent-input rms noise voltage of that part of the device total noise which is independent of source resistance. Notes: 1. The other parts of total equivalent-input noise voltage (II ni) are the voltages developed by device noise current through the source resistance and that developed thermally by the source resistance. 2. Noise voltages vary with bandwidth. Wide band noise may be any bandwidth but is usually specified for a 10.7 kHz bandwidth. Narrow-band noise voltages are for a bandwidth of 1 Hz and usually are specified in nV. (nV /$z)

v~ = The mean square noise voltage

V n f = 1 If rms noise voltage

V ng = Generator (noise generator) rms noise voltage

V ni = The total equivalent input rms noise voltage

Vni = Vno/Av r-~~------------------

Vni =v'BW(V~ + InRS + 4KBTKRs) See-BWNOISE

V no = The total output rms noise voltage

Vno = AvVni

V nR = Source resistance (Rs) rms thermal noise voltage.

V ns = 1. Device rms shot noise voltage 2. See-VnR

V nt = 1. Any rms thermal noise voltage 2. See-VnR

V nT = Device total equivalent-input rms noise voltage including V nand (lnRS)

274

vz VOR = Output voltage range.

VOUT - See-Yo

Vp , Vpk , Vpeak = Peak voltage.

Vp _p = Peak to peak voltage.

Vps = Power supply voltage.

VQ = Quiescent voltage.

Vs = 1. Signal voltage 2. Source voltage 3. Supply voltage

+Vs, Vs+ = Positive polarity supply voltage.

- Vs , Vs- = Negative polarity supply voltage.

Zi = Small signal closed-loop input impedance.

Opamp

Symbol Definitions

Zi = Device small signal open-loop input impedance.

Zi+ = Small signal closed-loop non-inverting input impedance.

Zi+ = (riAyol)/A ycl

Zi- = Small signal closed-loop inverting input impedance.

Zi- ~ Series input resistor R.

Zi- = R + [RF/(A Y01 + 1)] zic = Device common mode input impedance. The parallel

sum of the small signal open-loop impedance between each input terminal and ground.

275

z to 0 Zid = Device differential input impedance.

Zo = Small signal closed-loop output impedance.

Zo = zo/[(AyodAycd + 1] Zo = Device small signal output impedance.


Zod = Differential output impedance. (opamps with differential output)

a - See-d etc.

ano - See-TCllO ayIO - See-TCylO

~IIO/~T - See-TCllO

~VCcl~VIO -See-kSYR etc.

~Vlo/~T - See-TCyIO

~VIO/~VCC - See-kSYS etc.

8 - See-d etc.

8m = Phase margin. The absolute value of the open-loop phase shift between the output and the inverting input at the frequency at which the modulus of the openloop amplification is unity.

tPm - See-8m

We = Cutoff (- 3dB) angular velocity (angular frequency).

Wo = Reference angular velocity (angular frequency).

Wr = Resonant angular velocity (angular frequency).

276

OPERATIONAL AMPLIFIERS

SECTION 3.2

FORMULAS

AND

CIRCUITS

DCor Low Frequency, Large or Small Signal

AI = lo/Ig

AI = -RF/RL

Ai = io/ig

Ai = -RF/RL

AI = lo/Ig

AI = RdRL

Ai = io/ig

Ai=Rl/RL

AI = lo/Ig

AI = IRL/IRl

AI = (VoRd/(ViRd

AI = (RdRd [(RF /R2 ) + 1]

Precision Voltage-to-Curren t

Converter

(Bilateral)

AI = lo/Iin

10 = Vg /R2

AI = Rl/R2

279

Current Amplification

RL ~Io

Vo

~Io

10 k

m ::is .~ .,. - m a. ... a. 0 «z

CD @

@

®

CD @

@

®

CD ® @

@

®

Deor Low Frequency, Large or Small Signal

Av = VolVJN

Av = VOIVIN

Av = VolVs

Av = VolVs

Av ~ (RF/RB) + 1

error typically extremely small at low frequencies

Av v~

~ vin 0

Av = [(RF/RB) + 1) / [(Rs/zd + 1]

Zi = ri[(AvOL/Avcd + 1]

280


F ~ 0

Vo

~ 0

Q)

:E .~ ... - Q) a. .. a. 0 «2

CD @ @

@

®

CD @

®

CD ® @

@

®

Vo CD ...--_--0 ®

@

®

Small Signal Voltage Gain At Any Frequency

when (Avfin) < .1 BW(Av=l)

or when (Avol @fin» 100 Av

Av = - [Rpl(RB + RBk-1) + k-1]-1

where k = Avol @ fin

when (Avfin) <.1 BW(Av=l)

or when (Avol @ fin) > 100 Av

Av= [Rpl(RB +RBk-1)+k-1r 1 +1

where k = Avol @ fin


0»

:c .~ en - 0» c. ... C. 0 «z

CD ® @

@

@

@ @

CD ® @

@

@

Note: Above formulas are for small-signal output only. The output may be limited by device slew rate at frequencies above 5 kHz and at any output level greater than small-signaL

See-SR, PBW, VOPP

281

DC or Low Frequency, RMS or I nstantaneous, Small or Large Signal

Av(_) = - VO/Vl

Av(+) = VO /V2

Av(_) = - RF/RB

Av(+) = [(RF/RB) + 1]/[(Rl /R2 ) + 1]


Vo =(V2 [(RF/RB) +1]/[(Rl /R2)+ 1])

- [(vlRF)/RB]

Vo = Instantaneous output voltage

Av(_) = Av (+) when R2 /Rl = RF/RB

When Av(_) = Av(+)

Vo = (V2 - vd (RF/RB)

VO(rms) = (V2 (rms) - Vl(rms») (RF/RB)

(()V2 = ()Vl)

VO(rms) = (V2 (rms) + Vl(rms») (RF/RB)

(()V2 ± ()Vl = 180°)

VO(DC) = (V2 (DC) - Vl(DC») (RF/RB)

(assuming zero offset)

Opamp Notes:

CI)

:c CIS .~ en - CI) a. ... a. 0 <tz

CD ® @

@

@

@

<D 0 is the graphic symbol for an operational amplifier ~ (opamp) with differential inputs. The minus input

(the inverting input) develops an inverted (180°) output while the plus input (the non-inverting input) develops a non-inverted (0°) output.

282

II)

::c DC or Low

Frequency, RMS or Instantaneous, Small or Large Signal

Voltage B Amplification ~ :;

Co 0 <tz

Low Gain from Inverting Input and High Gain from Non-inverting Input.

AY2 = VO/V2

AY1 = -RF/RB

AY2 = [RF(Ri31 + Rx1)] + 1

Rx = [Ri;t(AY2 - 1) - Ri31rl

Vo = (V2 Ay2 ) - (VI Ayd

v = instantaneous voltage

360° Phase Shifter with Flat Freq. Response (0° midband, ±180° @ DC, ±180° @oo freq.)

Ay = IVO/V1NI

Ay = IRF/RBI

when

Rx = [RFl(2RIR21 + 2) - Ri3lrl and XCI = RI @ fo and XC2 = R2 @ fo

(fo = midband and 0° phase shift freq.) Note: Two of above circuits with different capacitor values will maintain a 90° phase differential ± 1.5° over > 20 to 1 frequency range. (e.g. Resistors per above, (CI)1 = .03 /-IF, (C2)1 = .015 /-IF, (C t )2 = .0068 /-IF, (C2h = .0033 /-IF)

283

CD @)

@

@

@

@

CD ® @

@

@

@

DC or Low Frequency, Large or Small Signal


Positive and Negative Feedback (decreases input impedance)

- [1 + (RdR2)] Av = --------

[(RB R1)/(R2RF )] - 1

when divisor is positive

Positive and Negative Feedback [negative input impedance (negative irnmittance)]

(input voltage produces reverse current through source)

Av =Vo/Vs

Av = 1 +

1 + (RdRs) when divisor is positive

Rs must be resistive over entire unity gain bandwidth of op-amp and divisor must be positive to prevent oscillation or latch-up.

284

CD :E .~ '" - CD a. .. a.o <cz

<D @)

@ @

@)

<D ® @

@)

DC or Low Frequency,

Large or Small Signal

Ay = -Vo/Vs

Ay =Vo/Vs

Opamp Notes:


R2 R3 R4

Vo

R2 R3 R4

Vo

GI

15 III

.!:! III

C. GI .... Co 0 «2:

<D ® @)

@

@

@

<D ® @ @

Q) -(])- is the graphic symbol for an infinite impedance alternating current generator or any very high impedance signal source. If the signal source resistance is too low to be ignored, consider the source resistance to be in parallel with an ideal alternating current generator. (Note: The graphic symbol for a direct current source is~)

® -e- is the graphic symbol for a zero impedance signal generator or any very low impedance signal source. If the signal source resistance is too high to be ignored, consider the source resistance to be in series with an ideal voltage generator.

Opamp notes continued at YOpp.

285

Av Voltage Amplification

Required Passband Gain for VCVS 12 dB per Octave Low Pass Filter

Av = Va/Vs

Ava = (RF/RB) + 1

QI

:is .~ ... - QI 0. .. 0. 0 <cz

CD AV(RQD) = 1 + [C2Cil(R2Ril + 1)] - [d y'R2Ri1C2Cil ] ®

® AV(RQD) = 2 + (R2/Rd - (d y'R2/R1 ) when C1 = C2 ®

AV(RQD) = 1 + (2C2/C 1) - (dy'C2/C1 ) when Rl = R2 G) @

AV(RQD) = 3 - d when C1 = C2 and Rl = R2 @

d = V2 for flattest response (Butterworth) @ d = 2 for isolated two section RC response @ d = 1.731 for best transient response (Bessell) @ d = 1.045 for 1 dB hump (I dB ripple @

Chebyshev) d = .895 for 2 dB hump (2 dB ripple


Chebyshev)

fo = [21Ty'RIR2CIC2rl

fc = fo when d = y'2 1] 1 fc =fo[k+(k2 + 1)2"2 k= 1- 2/d2

286

Av Voltage Ampl ification

Required Passband Gain for VCVS 12 dB per Octave High Pass Filter

Ay = VoNs

AyO = (RF/RB ) + 1

AY(RQD) = 1 + [RIRil(C1Cil +1)] - [dYCICilRlR21]

AY(RQD) = 1 + (2R1/R2) - (dyR1/R2) whenC l = C2

AY(RQD) = 2 + (C1/C2) - (dyCdC2) when Rl = R2

as ::c .~ ... - as a. ... a. 0 <cz

CD ® ® ® (J)

@ AY(RQD) = 3 - d when C1 = C2 and Rl = R2 @

d = VI for flattest response (Butterworth) @ d = 2 for isolated two section RC response @) d = 1.731 for best transient response (Bessell) @ d = 1.045 for 1 dB hump (1 dB ripple



Chebyshev)

fo = [27TyRIR2CIC2rl

fc = fo when d = VI 1] 1

fc = fo/[k + (k2 + 1)2" k = 1 - 2/d2

287


Multiple Feedback Bandpass Filter

Av =Vo/Vs

CD :a .~ ... a.~ Co 0 «2

CD Vo Q)

Let C1 = C2

Avo = RF/(2Rd

Q = V.5 Avo

fr = [21TC1 VRFRlrl

fr = 500/ [1TC1 VRFRI ] (CQ,tF), R(kO»

Higher Input Impedance, Lower Gain Version

Av = Vo/Vs

Let C1 = C2 , Let RIB = lORlA

Avo = RF/(2RlB)

Q = VII R2 /4RlA

fr = [21TC1 VRFRlB/11 r l

fr = 166.9/[C1 VRFRIA ] (C(.uF) , ~O»

288

@

@

@

@

CD Q) @

@

@

@

BW-3dB 3dB Down Bandwidth

BWC- 3dB) ~ [v'2 BWCAy=l)]/ Aycl

See also-PBW for large signal

BWC-3dB) ~ (RB/RF) v'2 BWCAy=l) '------4--------'



BWC-3dB) = f2C-3dB) - flC-3dB)

BWC-3dB) = fr/Q

Let Cl = C2

BWC- 3dB) = [1TC2R2r l

fr = [21TC2 vlRl R2 rl

Ayo = R2 /(2Rl )

Q = .5 vlR2/Rl

289

AvoI> 100 at highest frequency of interest.

II> :is .~ III - 011 c. ... C. 0 <tZ

<D ® @

@

@

<D ® @

@

<D ® @

@

@

@

d d = I/Q

Damping Factor

Note: The general meaning of Q is understood by almost everyone, however an exact definition that is applicable to all circuits is very elusive. The inverse of Q (d or 0<) is generally used for all high and low pass type circuits when Q is less than 2.

Low Pass and High Pass Filters

d = 2 for isolated RC sections response.

d = 1.731 for best delay, transient or pulse response. (Bessell response.)

d = VI for maximally flat and exact 3 dB down cutoff or crossover frequency. (Butterworth response -Recommended for most applications.)

d = 1.045 for 1 dB peak or hump near the cutoff frequency. (1 dB ripple Chebyshev response.)

d = .895 for 2 dB peak or hump near the cutoff frequency. (2 dB ripple Chebyshev response.)

d = .767 for 3 dB peak or hump near the cutoff frequency. (3 dB ripple Chebyshev.)

d > 0 in all non- oscillating circuits.

290

Q)

j5

.~ en - Q) c,+, c, 0 <t2

@)

@)

d = 2 "';C2 /C1 when RI = R2 and Av = 1

(RF = 0 and/or RB = 00)

291

d VCVS High Pass Filter

d= l/Q

[RiteCt + C2 )] - [(C2 RF)/(Rt RB)] d=~--~~~~~--~

v'(C t C2 )/(Rt R2 )

Avo = (RF/RB ) + 1

fc = 21Tv'Rt R2 Ct C2 when

Damping

Factor

d= V2

d = 2v'Rt /R2 when Ct = C2 and Av = 1

(RF = 0 and/or RB = 00)

292

III ::c .~ '" - III c.. ... c.. 0 «2

<D ® ® ® (j) @

@

@

@

@

Small Signal

fc = [BW(-3dB)]/ [Av(lf)]

fc ~ [y'2 BW(AV=l)]/[Av(lf}]

fc ~ [y'2 BW(AV=l)] [RB/RF]


fc = [BW(-3dB)]/[Av(lf)]

fc ~ [y'2 BW(AV=l)]/[ Av(lf)]

fc ~ [y'2 BW(AV=l)] [(RB/RF) - 1]


Note: To increase fc: 1. Use opamp with greater unity gain bandwidth. 2. Decrease circuit gain. 3. Use multiple decreased gain stages.

293

Cutoff (-3 dB)

Frequency

Small Signal Output

CD ::a .g ... - CD a. .. a. 0 <1:2

<D ® @

@

(\)

<D ® @

@

Unity Gain VCVS Low Pass Filter

Let Rl = R2

Let CI fC 2 =

1 for two isolated RC sections response 1 t for best transient response. (Bessell) 2 for flattest response (Butterworth)

Cutoff (-3 dB) Frequency

3.3 for 1 dB hump (1 dB ripple Chebyshev) 4.7 for 2 dB hump (2 dB ripple Chebyshev) 6.8 for 3 dB hump (3 dB ripple Chebyshev)

fc ~ [21TRI yCIC2r I (exact when CI fC2 = 2)

Unity Gain VCVS High Pass Filter

Let CI = C2

Let R2 fRl = Use low pass C ratios above.

fc ~ [21TCl yRIR2 r l (Exact when R2fRI = 2)

294

CD :g .~ en - CD a. ... a. 0 «2

CD ® ® ® <'D @ @

@

@

@

CD ® ® ® <'D @ @

@ @

Free Gain VCVS Low Pass Filter

Let C1 = C2, Let Rl = R2

Passband gain (Ayo) = (RF/RB ) + 1

Let Ayo = See below

Cutoff (-3 dB)

Frequency

CD

:is .~ <II - CD a. .. a. 0 <tz

<D ® ® ® (1)

® @

@ 1.0 for two isolated RC sections response @

1.27 for good transient response (Bessell) @

1.586 for flattest response (Butterworth) @)

1.955 for 1 dB hump (1 dB ripple Chebyshev) 2.105 for 2 dB hump (2 dB ripple Chebyshev) 2.233 for 3 dB hump (3 dB ripple Chebyshev)

fc ~ [21TRI Cd-1 (Exact when Ay = 1.586)

Free Gain VCVS High Pass Filter

Let C1 = C2, Let Rl = R2

Use all low pass formulas above.

295

<D ® ® ® (1)

@

@

@

@)

Cutoff (-3 dB)

Frequency

12 dB per Octave Multiple Feedback Butterworth Response Low Pass Filter (Recommended version with gain of 4)

Let C2 =.1 C1

Let R2 = 4 Rl

Let R3 =.8 Rl

fc = [v'2 1TR3C1]-1

Avo =4

d = v'2 (Q = lid)

12 dB per Octave Multiple Feedback Butterworth Response High Pass Filter

(Recommended unity gain version)

Let C2 = C1

Let C3 = C1

Let R2 = 4.5 Rl

fc = v'2 j(61TRl C1 )

Avo = 1

d = v'2 (Q = lid)

Note that 3 capacitors are required. VCVS filters with same response require only 2.

296

cu ::c .~ '" - cu a. ... a. 0 <cz

CD ® ® (j)

@

@

@

0.>

CD ® ® (j) @

@

Cutoff (-3 dB)

Frequencv

12 dB per Octave Multiple Feedback Butterworth Response Low Pass Filter.

Let R3C1 =

[V21Tfc]-1 Let R2 =

R3(Avo + 1)

Let R1 = R3(Avh + 1)

Let C2 = Ct/[2(Avo + 1)]

fc = [V2 1TR3C1] -1

Avo = R2 /R1 , R3 = (Rt1 + R21 )-1

d= V2 (Q= I/V2)

12 dB per Octave Multiple Feedback Butterworth Response High Pass Filter

Let k = 2Avo + 1

Let R1C3 =

[V2 21Tfckr1

Let C1 = C3

Let C2 = CdAvo

Let R2 =.5 R1k2

fc = [21TC1 ~r1

Avo = C1 IC2 , d = V2 (Q = I/V2)

297

II ::a .~ ... - . 8:0 «2

<D ® ® <Z> @

@

@ @

<D ® ® <Z> @

@

@

Cutoff (-3 dB)

Frequency

18 dB per Octave VCVS Butterworth Response Low Pass Filter with Gain of Two.

Let CI = C2 = C 3 = C

Let RF = RB ~ 7 R

Let RI = 1.565 R

Let R2 = 1.467 R

Let R3 = .435 R

fc = [21TRC]-I

Avo =2

d= V2 (Q= 1/V2)

Let RI = 3.6 k

Let R2 = 3.3 k

Let R3 = 1.0 k

Let RF = RB = 15 k

fc = [14,337 C]-I (COl F) = 69.75/fc)

Avo =2, d= V2

298

CD :E .~ <II - CD c. ... 0. 0 <cz

CD ® ® ® (j)

@

@

@

@


18 dB per Octave VCVS Butterworth Response Unity Gain High Pass Filter

Let C1 = C2 = C3 = C

Let R3 = 4.94 R

Let R2 = .282 R

Let Rl = .718 R

fc = [21TC(RIR2R3}~]-1 fc = [21T RC]-1

d=V2 (Q=I/V2)

Avo = 1

Let R3 = 68 k

Let R2 = 3.9 k

Let Rl = 10. k

fc = [86,970 C] -1 (C(~F) = 11.5/fc)

d = V2 (Q = 1/..[2)

Avo = 1

299

III :c .2 III - III a. .. a. 0 «2

CD ® ® ® <V @

@

@ @


Bessell Response VCVS Low Pass Filter (Best pulse, delay and transient response)

12 dB per Octave (Voltage Gain = 1.269)

R .269 R

24 dB per Octave (Total Voltage Gain = 1.91)

R .084 R R .759 R


R .041 R R .364 R R 1.023 R

C t, fc = [21TRC]-1

300

CD :B .2 en - CD a. ... a. 0 «z

<D ® ® ® Q)

@

@

@

@

Cutoff (-3 dB)

Frequency

Bessell Response VCVS High Pass Filter (Best pulse, delay and transient response)

12 dB per Octave (Voltage Gain = 1.27)

R .269 R


R .084 R R .759 R


R .041 R R .364R R 1.023 R

301

011

:is .~ ... - 011 a. ... a. 0 «Z

<D ® ® ® <2> @

@

@

Definitions

fo = 1. Frequency of oscillation. (steady state) 2. The frequency at which oscillation first occurs.

(before the signal amplitude reaches a non-linear region of operation)

fo (LC) = The series resonant frequency or the frequency of maximum circulating current regardless of input and output connections. (Not necessarily the frequency of maximum output)

fo(Rc) = The frequency where the capacitive reactance equals the resistance.

Passive or Active Bandpass Circuits

fo = The frequency of maximum output.

fo = The frequency of resonance (fr)

Passive or Active Bandstop Circuits

fo = The frequency of minimum output

fo = The frequency of antiresonance also known as frequency of resonance (fr)

Multisection Passive or Active Bandpass Circuits

fo = Center of pass band

Multisection Passive or Active Bandstop Circuits

fo = Center of stop band

fo = Reference frequency

302

Active Low Pass Circuits

Definitions and Formulas

fo = The frequency of maximum change in output developed by a change of feedback amplitude. (Not necessarily the frequency of maximum output or the cutofffrequency)

fo = The cutoff (-3 dB) frequency (fe) only when damping factor (d) equals Vi or when Q equals 1/Vi. (when output is Butterworth response)

fo = wo/21T

fo = fel ([1 - .Sd2] [(1 - .Sd2)2 + 1] 1) t fo = fp Y 1 - .Sd2 when (1 - .Sd2) > 0

Note: fp = frequency of peak output. Response has no peak when d:> .j2.

fo = [21T V~R-l R-2 C-1-C-2 r1

Active High Pass Circuits

fo = The frequency of maximum change in output developed by a change of feedback amplitude.

fo = fe only when d = Vi (Butterworth response)

fo = wo /21T

fo = fe([1 - .Sd2] [(1- .Sd2)2 + 1] t) 1

fo = fp/y1 - .Sd2 when (1 - .Sd2) > 0 Notes: 1. fp = frequency of peak output

2. Response has no peak when d:>.j2

fo = [21TyRIR2CIC2TI

303

Wein Bridge Oscillator

fo = [21rRC]-1

when C 1 = C2 = C Rl = R2 = R, RF/RB > 2

Let Rl/R2 = C1/C2

RF/RB = 2RdR2 minimum

Oscillation Frequency

fo = [21rR1Cd-1 = [21rR2C2r 1 = [21ryR1R2C1C2t Oscillator Using Low Pass VCVS Circuit

when: C 1 = C2 = C Rl = R2 = R RF/RB > 2

fo = [21ryR1R2C1 c2l 1

when RF/RB > (C2/C 1)[(R2/Rd + 1]

Notes: 1. Loop gain must be only slightly over unity if low distortion sinewave

output is desired 2. Potentiometers above should have reverse log tapers if scale is on

housing and log tapers if scale is on knob. (reverse audio and audio tapers may be substituted for reverse log and log tapers)

304

Let R3 =

[Rll + Ri"l r 1

Function Generator

Oscillation Frequency

Triangle Output

~--------------~

Quadrature Oscillator (Sinewave generator with 0° and 90° outputs)

C3

Note:

C,

820 pF Cosine (90°) Output

Sine (0°) Output

1. Four phase output (0°, 90°, 180°, 270°) can be obtained by the addition of two phase inverters.

2. Zener diode voltage ratings should total < 2/3 total supply voltage.

305

Notch Filter (unity passband gain)

Let R2 = Rl

Let R3 = Rl/2

Let C2 = Cl

Let C3 = 2C l

fo= [27TR1Cd-l

Let R1C l = R2C2

Let R3 = (Ril + R;:l r l

Let C3 = Cl + C2

fo = [27TR1Cd-l

Notch Frequency

Filter with Notch and Bandpass Outputs

R

300 K

fo = [27TRC]-1

Q = RdR, AV(BP) = RdR2

306

fr = Resonant frequency

Resonant Frequency

fr = The frequency at which a system will respond with maximum amplitude when driven by an external constant amplitude sinewave signal.


Let C1 =C2

fr = [27TC 1 v'Rl R2fl

Avo = R2/(2Rd

Q =v'R2/4Rl

High Zin, Low Av , High Q Version

Let C1 =C2

CD ::a .~ '" - CD c. ... c. 0 «z

CD ® @

@

@

<!)

CD ®

Vo ® ~--~--~----~

Let Rl = (RiA + Ri~rl

fr = [27TR1Cd-1

Avo = R2/(2R1B)

Q = .5 v'R2(R1A + Ri~)

307

@

@

@

PBW

PBW = BWp = f(max)

Power Bandwidth

PBW = In circuits where the low limit bandwidth is zero, the maximum frequency which may be used at a specified peak-to-peak output without the distortion associated with slew rate. (e.g. a sinewave becoming triangular)

PBW = SR/(7TVOpp)

f(max) = SR/(7TVOpp)

V OPP(max) = SR/(7Tf(max»)

SR(min) = 7Tf(max)VOPP

See also - SR, V opp, V OM , MUF

See also - BW (Av = 1) for small signal frequency limitations.

308

Q Multiple Feedback Bandpass Filter

Q = fr /(BW(3dBDOWN»)

Let C1 = C2

Q = .5 v'RF/Rl

Q = .25 v' Ava

Ava = RF/(2R1) = 2Q2

fr = [21TC 1v'RFR l r 1

fr = 500/[1TC1v'RFR1]

Q

Factor

Higher Zin, Higher Q and Lower Av Version

Let C1 = C2

Q = .5 v'R3(R" + R21)

fr = [21TC 1 v'R3/(R/ + R.21)r1

Ava = R3/2R l

R2 = [C4Q2/R3) - R11r1

309

CD :E .~ II> - CD a. ... a. 0 «z

Q)

@

@

@

@

@

CD @

@

@

@

@

Vni =Vno/Av

Equivalent Input Total Noise Voltage

V ni = Total equivalent input rms noise voltage including: 1. Device equivalent input noise voltage (V n) 2. The product of the device equivalent input noise cur

rent (In) and the sum of the effective source resistances at both inputs.

3. The thermal noise voltage (VnR) of the effective source resistances at both inputs.

Note: All three noise voltages have components of l/f noise as well as constant spectral density (white) noise. The device white noise component is shot noise and the white noise of resistance is thermal noise. The l/f noise of resistances is excess noise or current noise. White noise voltage may be easily calculated from a spot noise voltage by multiplying by the square root of the noise bandwidth but l/f noise or noise having a significant l/f noise component must be averaged over the total bandwidth by the rms method. l/f noise is usually neglected at frequencies above 1 kHz and is often assumed to have straight line response between the 100 Hz and 1 KHz spot noise measurement points.

Vni = Vno when Av = 1

Vni =VBW[V~ + I~R§ + 4KBTKRsl

KB = 1.38 • 10-23

TK = °c + 273.15

310

V ni = YBw[vi + v~ + V~]

where VI = Vn Vs = 0

V2 = InRX

Rx = [RFI + (Rs + RBr1]-1 V3 =y4kBTKRx

Equivalent Input Total Noise Voltage

kB = Boltzmann constant (1.38 • 10-23 JtK)

T K = Kelvin Temperature. CC + 273.15)

Vni = Vno/Av

V ni = YBW [Vi + V~ + V~]

where VI =Vn, V2 = InRx

Rx = Rs + (RF1 + Ri31rl

V3 =y4kBTKRx

kB = Boltzmann constant (1.38 . 10-23)

T K = Kelvin temperature. Cc + 273.15)

311

Noise

Voltage

Output

Vno = Ay v'BW ·VcV~~-+~(=R-s-+~R~B~)~2(~I~~-+~4~kT~K-R~;~)-+~4~k-BT~K-(~R~s-+~R~B)

Notes:

V n = Equivalent input spot noise voltage of opamp at a given frequency. (usually given in nVjyHZ at 1 kHz)

In = Equivalent input spot noise current of opamp at a given frequency. (usually given in pAjyHZ at I kHz)

kB = Boltzmann constant

kB = 1.38 . 10-23 JtK

T K = Temperature in Kelvin

TK =oC+273.15

Ay = Closed loop circuit voltage amplification (Avcl or Vo/Vs)

Av = RFj(Rs + RB)

Rs = Source resistance

1. Formula does not include opamp l/f noise. (opamp l/f noise usually is insignificant above 1 kHz)

2. Formula includes thermal noise of all external resistances, but does not include resistor excess noise (current noise or l/fresistor noise)

3. Noise measurements require a bandwidth correction factor for all except rectangular response curves. See-BWN01SE

312

Let BW = 10 kHz

Let T:::::: 27°C

V = 100 v'V2 + I2R2 + 1 656 • 10 20R no n n s· S

'" Vs = 0

V no = Ay v'BW v'V~ + I~(Rs + RX)2 + 4kB T KRS

Noise Voltage Output

V n = Equivalent input spot noise voltage of opamp at a given frequency. (usually given in nV/.../Hi at 1 kHz)

In = Equivalent input spot noise current of opamp at a given frequency. (usually given in pA/.../Hi at 1 kHz)

TK = Kelvin temperature (C + 273.15)

kB = Boltzmann's constant (1.38 • 10-23 )

Rx = RFIIRB = (Ri + R"jl r 1

Ay = (RF/RB) + 1 See-Preceding page notes. See also-BWNOISE

313

Ps C(,

'" Vs .

r - - - .---.---1

Add capacitor C unless +Vcc is well filtered.

I

:~: c I

'~7 v

V O(DC) = Vcc /2 when Rl = R2

VO(DC) = Vcc/[ (RdR2) + 1]

VO(AC) = -AyVs

VO(AC) = -(VSRF)/(RB + Rs)

VO(DC) = Vc cl2 when Rl = R2

VO(DC) = Vcc/ [(RdR2) + 1]

VO(AC) ""='AyVs

Output Voltage

VO(AC) = V s [(RF/RB ) + I ]/[Rs(Ri1 + R2"l) + I]

Note: +Ycc to Rl must be well filtered. IfRS is low, such as the output impedance of another opamp stage, a large coupling capacitor (C2) may provide proper filtering.

314

GI

:c .~ en - GI c. .... C. c «z

Q)

® @)

@

Q)

® @

~c %VS +VBB

VO(DC) = +VBB ± VOO See-VOO

VO(AC) = AyVs

VO(AC) = V s [(RF/RB) + 1J / [(Rs/Rl) + 1]

Output Voltage

Note: Maximum undistorted output is obtained when VO(DC) = Vcel2. Typically, VBB must be between +2Vand (VCC - 2V) to prevent saturation.

~c %VS

CD :c .2 <II - CD Co ... Co 0 «z

<D @

@

<D @

VO(DC) = (V2 [(RF/R1) + 1]) - [VI (RF/Rd ± VOO] @

VO(AC) = AyVs

VO(AC) = Vs [RF/(RB + Rs)J

Note: Maximum undistorted output is obtained when VO(DC) = V cel2. Typically, VIand V 2 each must be between +2V and (V CC - 2) to prevent saturation.

315

Voo Output Offset Voltage (input voltage(s) = 0)

R1

~ R2

+ R3

-VEE

Output Offset Voltage

6 Vo

9

Output from input offset voltage (VIO) only (lIO = 0, lIB = 0)

Voo = VIO(Av + 1)

Voo = VIO [(R2/Rd + 1]

QI

1i .~ '" - QI a. ..... a. 0 «2

Output from input offset current (lIO) only @

(VIO = 0, R3 = [Ri1 + R2"l rl)

Voo = lIOR3(Av + 1)

Voo = lIO R3 [(R2/Rd+ 1]

Output from bias current (lIB) only (lIO = 0, VIO = 0)

Voo = lIB [R3(Av + 1)- R2]

Voo = lIB [R3(R2/R1 + 1) - R2]

Total Output Offset Voltage

Voo = [VIO(Av + 1)] + lIB [R3(Av + 1) - R2]

± [IIO R3(Av + 1)]

316

Vopp

v opp = SR/(21Tf) when V 0 is limited only by SR

V OPP < Total supply voltage

Maximum Peak to Peak Output Voltage

Typically V OPP > 2/3 total supply voltage when output load resistance is 2 kn or higher.

Note: A sinew ave input signal is transformed into a triangular wave output signal by the effects ofSR at outputs above VO(MAX poP).

Resistor RF is effectively in parallel with the output load resistance RL.

Opamp Notes:

@) A negative resultant for AV or Vo indicates that a phase inversion has taken place. (output 1800 out of phase with the input)

® VCVS is the abbreviation for voltage controlled voltage source. VCVS high pass and lowpass ftlters are characterized by high input impedance, low output impedance and by non-inverted passband output.

317

Low Frequency I nput Impedance

Zi~RB

Zi = RB + [RF/(AYOL + 1)]

Input Impedance

Zi = RB + [ RF /([IOg-I(AYOL(dB)/20)] + I)J

Zi ~ [fiAyod /AYCL

Z. =...!.f i (.o....A-,Y,-"OO!;:LC-.+_1-'.) 1 (RF/RB) + 1

Zi = fi(IOg-1 [(AYOL(dB) - AYCL(dB»/20] + 1) AYCL(dB) = 20(log [(RF/RB) + 1 J)

Opamp Notes:

® 12 dB per octave filters are also known as second order fIlters.

QI

jS

.~ en - QI c. .... C. 0 <cz

CD @

@

@

@

@

CD @

@

@

@

(J) 6 dB per octave equals 20 dB per decade, 12 dB per octave equals 40 dB per decade, 18 dB per octave equals 60 dB per octave etc.

® Ixl = The magnitude or the absolute value of x ® log x = log 10 x, log-Ix = antiloglO x = lOx

@ x-I = l/x, x t =-IX @ Source resistance, if significant, must be considered as an additional

resistance in series with the inpu t.

318

z = fO

o (Ayod Aycd + 1

Zo ::O:(fo RF )/(RBAyod

Z = _ fo

o [(RBAYOd/RF] + 1

AYOL = log-l [AYOL(dB)/20]

Opamp Notes:

Output Impedance

CD :is III .i! en 'C.$ Q. 0 «z

<D ® ® @

@

@ When supply voltage connections are not shown, a split supply is assumed with Vee positive with respect to common (ground) and VEE negative with respect to common (ground).

@ Low gain Butterworth or Bessel response VCVS fJIters are relatively insensitive to value changes except for the cutoff frequency. 5% tolerance resistors and 10% tolerance capacitors are recommended for most applications.

@> Free gain Butterworth VCVS fJIters may be empirically determined by simultaneously adjusting the gain just under that which produces a slight hump while adjusting R1, R2, C1 and/or C2 for the proper 3 dB down frequency.

@ See-Appendix A for table of ratios possible with standard 5% value components.

319

APPENDIX A RATIOS AVAILABLE

FROM

COMPONENT VALUES 5% Component Values

**10 **33 11 36

*12 *39 13 43

**15 **47 16 51

*18 *56 20 62

**22 **68 24 75

*27 *82 30 91

* * are also 10% and 20% values * are also 10% values

Above values are available over the range of .1 ohm to 10 megohms in resistors.

5% capacitor values are not as available and demand a much greater premium than resistors and are not recommended. Resistor values may be changed to accept 20% value (not 20% tolerance) capacitors in almost all RC circuits. 10% tolerance capacitors and 5% tolerance resistors (7.5% overall) are recommended for most applications.

321

RATI OS 5%Comp~nent Value Ratios

Ratio Values Ratio Values Ratio Values

9.38 150/16 8.24 750/91 6.96 390/56 9.23 120/13 8.23 510/62 6.94 430/62 9.23 360/39 8.20 82/10 6.92 270/39 9.22 470/51 8.18 180/22 6.91 470/68 9.17 110/12 8.18 270/33 6.88 110/16 9.17 220/24 8.15 220/27 6.83 82/12 9.17 330/36 8.13 130/16 6.83 560/82 9.15 430/47 8.00 120/15 6.82 75/11 9.15 750/82 8.00 160/20 6.82 150/22 9.12 620/68 8.00 240/30 6.81 620/91 9.11 510/56 7.69 100/13 6.80 68/10 9.10 91/10 7.69 300/39 6.80 510/75 9.09 100/11 7.68 430/56 6.67 100/15 9.09 200/22 7.67 330/43 6.67 120/18 9.09 300/33 7.66 360/47 6.67 160/24 9.07 390/43 7.65 390/51 6.67 180/27 9.07 680/75 7.58 91/12 6.67 200/30 9.03 560/62 7.58 470/62 6.67 220/33 9.01 820/91 7.56 620/82 6.67 240/36 9.00 180/20 7.50 75/10 6.50 130/20 9.00 270/30 7.50 120/16 6.47 330/51 8.89 160/18 7.50 150/20 6.43 360/56 8.89 240/27 7.50 180/24 6.38 300/47 8.67 130/15 7.50 270/36 6.32 430/68 8.46 110/13 7.50 510/68 6.31 82/13 8.46 330/39 7.47 560/75 6.29 390/62 8.43 430/51 7.47 680/91 6.28 270/43 8.39 470/56 7.45 82/11 6.27 470/75 8.37 360/43 7.41 200/27 6.25 75/12 8.33 100/12 7.33 110/15 6.25 100/16 8.33 150/18 7.33 220/30 6.25 150/24 8.33 200/24 7.27 160/22 6.22 510/82 8.33 300/36 7.27 240/33 6.20 62/10 8.30 390/47 7.22 130/18 6.18 68/11 8.29 680/82 7.06 360/51 6.15 240/39 8.27 91/11 7.02 330/47 6.15 560/91 8.27 620/75 7.00 91/13 6.11 110/18 8.24 560/68 6.98 300/43 6.11 220/36

322

RATI as 5%Component Value Ratios


6.07 91/15 5.16 470/91 4.41 300/68 6.06 200/33 5.13 200/39 4.40 330/75 6.00 120/20 5.13 82/16 4.39 360/82 6.00 180/30 5.12 220/43 4.35 270/62 5.93 160/27 5.11 240/47 4.33 130/30 5.91 130/22 5.10 51/10 4.31 56/13 5.89 330/56 5.09 56/11 4.31 220/51 5.88 300/51 5.06 91/18 4.30 43/10 5.81 360/62 5.00 75/15 4.29 240/56 5.77 75/13 5.00 100/20 4.29 390/91 5.74 270/47 5.00 110/22 4.27 47/11 5.74 390/68 5.00 120/24 4.26 200/47 5.73 430/75 5.00 150/30 4.25 51/12 5.73 470/82 5.00 180/36 4.25 68/16 5.69 91/16 4.85 160/33 4.19 180/43 5.67 68/12 4.85 330/68 4.17 75/18 5.64 62/11 4.84 300/62 4.17 100/24 5.64 220/39 4.82 270/56 4.17 150/36 5.60 56/10 4.81 130/27 4.14 91/22 5.60 510/91 4.80 360/75 4.13 62/15 5.58 240/43 4.77 62/13 4.10 82/20 5.56 100/18 4.76 390/82 4.10 160/39 5.56 150/27 4.71 240/51 4.07 110/27 5.56 200/36 4.70 47/10 4.02 330/82 5.50 110/20 4.69 75/16 4.00 120/30 5.47 82/15 4.68 220/47 4.00 300/75 5.45 120/22 4.67 56/12 3.97 270/68 5.45 180/33 4.65 200/43 3.96 360/91 5.42 130/24 4.64 51/11 3.94 130/33 5.36 300/56 4.62 180/39 3.93 220/56 5.33 160/30 4.58 110/24 3.92 47/12 5.32 330/62 4.56 82/18 3.92 51/13 5.29 270/51 4.55 91/20 3.92 200/51 5.29 360/68 4.55 100/22 3.91 43/11 5.24 430/82 4.55 150/33 3.90 39/10 5.23 68/13 4.53 68/15 3.88 62/16 5.20 390/75 4.44 120/27 3.87 240/62 5.17 62/12 4.44 160/36 3.85 150/39

323

Ratio

3.83 3.79 3.78 3.75 3.73 3.73 3.72 3.70 3.67 3.66 3.64 3.63 3.62 3.61 3.60 3.60 3.58 3.57 3.55 3.55 3.53 3.53 3.50 3.49 3.44 3.42 3.41 3.40 3.40 3.40 3.37 3.33 3.33 3.33 3.33 3.31 3.30 3.30

RATIOS 5% Component

Value Ratios

Values Ratio Values Ratio Values

180/47 3.29 270/82 2.80 56/20 91/24 3.27 36/11 2.79 120/43 68/18 3.25 39/12 2.78 75/27 75/20 3.24 220/68 2.78 100/36 56/15 3.23 200/62 2.77 36/13 82/22 3.21 180/56 2.77 130/47

160/43 3.20 240/75 2.76 91/33 100/27 3.19 150/47 2.75 33/12 110/30 3.18 51/16 2.73 30/11 300/82 3.14 160/51 2.73 82/30 120/33 3.13 47/15 2.70 27/10 330/91 3.13 75/24 2.69 43/16

47/13 3.11 56/18 2.68 150/56 130/36 3.10 62/20 2.68 220/82 36/10 3.09 68/22 2.67 200/75

270/75 3.08 120/39 2.65 180/68 43/12 3.06 110/36 2.64 240/91

200/56 3.04 82/27 2.61 47/18 39/11 3.03 91/30 2.60 39/15

220/62 3.03 100/33 2,58 62/24 180/51 3.02 130/43 2.58 160/62 240/68 3.00 30/10 2.56 100/39

56/16 3.00 33/11 2.56 110/43 150/43 3.00 36/12 2.55 51/20 62/18 3.00 39/13 2.55 56/22 82/24 2.97 270/91 2.55 120/47 75/22 2.94 47/16 2.55 130/51 51/15 2.94 150/51 2.54 33/13 68/20 2.94 200/68 2.53 91/36

160/47 2.93 220/75 2.52 68/27 91/27 2.93 240/82 2.50 30/12

100/30 2.90 180/62 2.50 75/30 110/33 2.87 43/15 2.48 82/33 120/36 2.86 160/56 2.45 27/11 130/39 2.83 51/18 2.44 39/16 43/13 2.83 68/24 2.44 200/82 33/10 2.82 62/22 2.42 150/62

300/91 2.82 110/39 2.42 220/91

324

R AT I 0 S 5% Component Value Ratios


2.40 24/10 2.08 75/36 1.78 91/51 2.40 36/15 2.07 56/27 1.77 39/22 2.40 180/75 2.07 62/30 1.77 110/62 2.39 43/18 2.06 33/16 1.76 120/68 2.35 47/20 2.06 68/33 1.76 160/91 2.35 120/51 2.00 20/10 1.74 47/27 2.35 160/68 2.00 22/11 1.74 68/39 2.34 110/47 2.00 24/12 1.74 75/43 2.33 56/24 2.00 30/15 1.74 82/47 2.33 91/39 2.00 36/18 1.73 130/75 2.33 100/43 2.00 150/75 1.72 62/36 2.32 51/22 1.98 180/91 1.70 51/30 2.32 130/56 1.96 47/24 1.70 56/33 2.31 30/13 1.96 100/51 1.69 22/13 2.30 62/27 1.96 110/56 1.69 27/16 2.28 82/36 1.95 39/20 1.67 20/12 2.27 68/30 1.95 43/22 1.67 30/18 2.27 75/33 1.95 160/82 1.65 33/20 2.25 27/12 1.94 91/47 1.65 150/91 2.25 36/16 1.94 120/62 1.64 18/11 2.21 150/68 1.92 75/39 1.64 36/22 2.20 22/10 1.91 130/68 1.63 39/24 2.20 33/15 1.91 82/43 1.63 91/56 2.20 180/82 1.89 51/27 1.62 110/68 2.20 200/91 1.89 68/36 1.61 82/51 2.18 24/11 1.88 30/16 1.61 100/62 2.17 39/18 1.88 62/33 1.60 16/10 2.16 110/51 1.87 56/30 1.60 24/15 2.15 43/20 1.85 24/13 1.60 75/47 2.14 47/22 1.83 22/12 1.60 120/75 2.14 120/56 1.83 33/18 1.59 43/27 2.13 51/24 1.83 150/82 1.59 62/39 2.13 100/47 1.82 20/11 1.59 130/82 2.13 160/75 1.80 18/10 1.58 68/43 2.12 91/43 1.80 27/15 1.57 47/30 2.10 82/39 1.80 36/20 1.56 56/36 2.10 130/62 1.79 43/24 1.55 51/33 2.08 27/13 1.79 100/56 1.54 20/13

325

Ratio

1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.47 1,47 1.47 1.47 1.47 1.46 1.46 1,45 1,45 1.44 1.44 1.44 1.43 1.43 1.42 1.42 1.38 1.38 1.38 1.36 1.36 1.35 1.34 1.34 1.34 1.33 1.33 1.33 1.33 1.33 1.33

RATIOS 5% Component

Value Ratios

Values Ratio Values Ratio Values

15/10 1.32 62/47 l.l1 62/56 18/12 1.32 82/62 l.l1 91/82 24/16 1.32 120/91 l.l0 11/10 27/18 1.31 47/36 1.10 22/20 30/20 1.31 51/39 1.10 33/30 33/22 1.30 13/10 1.10 43/39 36/24 1.30 39/30 1.10 56/51 22/15 1.30 43/33 1.10 68/62 75/51 1.30 56/43 1.10 75/68 91/62 1.25 15/12 1.10 100/91

100/68 1.25 20/16 1.09 12/11 110/15 1.25 30/24 1.09 24/22 82/56 1.23 16/13 1.09 36/33

120/82 1.23 27/22 1.09 47/43 16/11 1.22 22/18 1.09 51/47 68/47 1.22 33/27 1.09 82/15 39/27 1.22 62/51 1.08 13/12 56/39 1.22 100/82 1.08 39/36 62/43 1.21 47/39 1.07 16/15 43/30 1.21 68/56

130/91 1.21 75/62 1.00 ALL 47/33 1.21 82/68 51/36 1.21 91/75 18/13 1.21 110/91 This listing of all 22/16 1.20 12/10 possible ratios 33/24 1.20 18/15 between 10 and 15/11 1.20 24/20 1 may also be 30/22 1.20 36/30 used for all other 27/20 1.19 43/36 possible ratios by 75/56 1.19 51/43 moving the proper 91/68 1.19 56/47 decimal points.

110/82 1.18 13/11 16/12 1.18 39/33 20/15 1.15 15/13 24/18 1.13 18/16 36/27 1.13 27/24 68/51 1.11 20/18

100/75 1.11 30/27

326

APPENDIX B ElECTRONICS TERMS AND THEIR SYMBOLS

This is an alphabetical listing of passive, bipolar-transistor and operational-amplifier (opamp) linear-circuit electronics terms with their corresponding symbols. Included also, are selected electronic, magnetic, acoustic, electrical, mechanical, mathematical and physical terms with their corresponding symbol, abbreviation, sign or acronym.

An attempt has been made to include present common usage (USA), traditional and recognized standard symbols, however, the preferred symbol (listed last) is often the author's projection of present trend, personal preference or arbitrary selection and does not necessarily represent an accepted industry standard.

This listing is intended as a reference source of electronics symbols, but may also be used to locate formulas having unfamiliar resultant symbols. It should be noted, however, that several different symbols may be shown for a given term and that the last-listed symbol is not always the one used in the formula and definition sections, since the last-listed symbol may be the author's projection of present trend.

Textbooks and scientific journals conventionally use italic (slanted) type for quantity symbols, however, this handbook follows the example of almost all technical manuals where roman (upright) type is used for both quantity and unit symbols. Unit symbols are clearly indicated as such in this appendix.

Common electronics abbreviations should be written without periods and generally in lower case letters as listed, however, certain abbreviations are capitalized and certain others are capitalized when used as a noun.

An asterisk is used to indicate schematic letter symbols. No attempt has been made to include terms or symbols as

sociated with computing systems, control systems, digital systems, digital devices, non-linear circuits, non-linear devices, vacuum tubes or field effect transistors.

327

a

about equal to absolute temperature

(quantity) (unit)

absolute value (of x) See also-magnitude

absolute zero temperature

acceleration, angular acceleration, linear acoustic

angular frequency angular velocity attenuation coefficient damping coefficient frequency impedance loudness level mechanical impedance period resonant frequency reverberation time sound power sound power level

To 0:

a

sound pressure PWL,Lp

P sound pressure level

SPL,Lp sound velocity specific impedance wavelength

c, v Zs A

admittance input magnitude output

Y Yin,Yi

vector admittance, transistor

(hybrid parameters) output

common base common collector common emitter

admittance, transistor, (y parameters) common base

forward transfer input output reverse transfer

common emitter forward transfer input output

!Y!,Y Yo

Y,V

Yfb Yib

Yob

Yre

Yfe

Yie

Yoe reverse transfer Yre

alpha (greek letter) 0:

alpha, transistor small signal 0:, hfb static (dc) a, hFB

alpha cutoff frequency fOtb alternating current AC, ac ambient temperature tA , T A

American wire gage AWG ampere (unit) A ampere-hour (unit) A • h, Ah

328

ampere-squared-seconds (unit) 12t

ampere-turn (unit of magnetomotive force)

A - t,A,At ampere per meter (unit of

magnetic field strength) At/m,A/m

amplification (quantity) A (unit) dg, (numeric), dB See also-gain

amplification, dc or large signal

current Al power (gain) Gp

voltage Ay small signal

current Ai power (gain) Gp

voltage Ay amplification factor

(vacuum tube) p. amplitude modulation AM angle, loss 5 angle, phase cp, ()

admittance (}y current (}I

impedance (}z voltage CPE, cfJv, (}E, (}y

angle, phase margin

angle, plane

angle, solid n angular frequency w angular velocity w antilogarithm (of x)

base 10 19-1, lOx, 10g-1 base e eX, eX, In-1 common 19-1, lOx, log-1 natural eX, eX, In-1

antiresonant frequency fo, fr apparent power

(quantity) S'Ps ' VA (unit) VA

approximately equal to R::

arc cosine arccos, cos-1 hyperbolic arcosh, cosh-1

arc cosecant arcsec, sec-1 hyperbolic arsech, sech-1

arc cotangent arccot, coCI

hyperbolic arcoth, coth-1 arc secant arcsec, sec-1

hyperbolic arsech, sech-1

arc sine arcsin, sin-1 hyperbolic arsinh, sinh-1

arc tangent arctan, tan-1 hyperbolic artanh, tanh-1

area A area, cross-sectional S, A atmosphere atm attenuation coefficient Q

atto (unit prefix for 10-18) a audio frequency a-f automatic frequency

control AFe

329

automatic gain control AGe

average current noise current

iN,in,in,IN,In,In noise voltage

EN,En,eN,en,Vn,Vn power (long term average)

P,Pav power (short term or one

cycle average) P voltage Eav' Vav

bandwidth 3dB down

half power

noise

b

(f2 - f1), B, B3, BW,BW-3dB

(f2 - f1 ), B, B3, BW, BW-3dB

B, BW, D, Bn, BW, BWn unity gain

B1 , BW(Av= 1)

*base (transistor) B base 10 logarithm

Ig, IOgI0' log base of natural logarithms

E,e,€ base € logarithm

10&, In

*base capacitor (transistor) CB

base current (transistor) small signal Ib static (dc) IB

base resistance (transistor) external RB internal

small signal static (dc)

base spreading resistance (transistor) rbb', rbb

base supply voltage (transistor)

base-to-emitter voltage (transistor) V BE active V BE(ON) saturated V BE (SA T)

base voltage (transistor) VB bel See-decibel beta (greek letter) beta, transistor

small signal static (dc)

bias current, input (op amp)

Boltzmann constant *bootstrap capacitor breakdown, second

(transistor) current energy

(3

330

breakdown voltage (transistor) collector-to-base

emitter open BVcBo , V(BR)CBO

collector-to-ernitter base-emitter

circuit BVcEx , VCEX(SUS)

resistance BV CER, V CER(SUS)

shorted BV CES, V CES(SUS)

voltage BV CEV , V CEV(SUS)

base open BV CEO, V CEO(SUS)

ernitter-to-base, collector open

BVEBo , V(BR)EBO

breadth (width) b British thermal unit Btu broadband

noise current fn, In noise voltage

EN, En, en, V n voltage gain,

common emitter transistor GvE

Brown and Sharpe wire gauge (American wire gage) AWG

*bypass capacitor CB

capacitance parallel resonant series

c

capacitance, transistor

C Cp,Cp Co,Cr

Cs,Cs

collector-to-base Ceb collector-to-case Ce ernitter-to-base Ceb feedback CFB , Cb'e input, common base Cib output, common base Cob

open circuit Cobo capacitive

current reactance susceptance voltage

-Ix,Ic -X,Xc -B,Bc

-Ex, -Vx, E c , Vc

*capacitor C bootstrap CB

bypass CB

coupling Cc feedback CFB , CF

carrier frequency fe case temperature tc , Tc cathode-ray tube CRT Celsius temperature

(quantity) toc, t, Tc (unit) °c

cent ¢ centi (unit prefIx for 10-2 ) c centigrade See-Celsius

331

centimeter (unit) cubic (unit) square (unit)

centimeter-gram-second (unit system) cgs, CGS

characteristic impedance Zo charge, electric Q charge, elementary

e, q (charge of electron) closed-loop voltage

amplification (op amp) AYCL,Ay

coefficient, attenuation coupling damping

a k o

temperature a, TC *collector (transistor) C collector current (transistor)

small signal I c static (dc) Ic

collector cutoff current (transistor) base open ICEO base-to-emitter

circuit ICEX resistance ICER short ICES voltage ICEY

collector dissipation (transistor ) Pc

collector efficiency (transistor) 77, 77c

collector resistance (transistor) external Rc internal, T equiv.

collector supply voltage (transistor, op amp)

collector voltage (transistor) small signal static (dc)

common base (transistor) forward current

VCC

transfer ratio hfb input impedance hib output admittance hob reverse voltage

transfer ratio hrb common collector (transistor)

forward current transfer ratio hfc

input impedance hic output admittance hoc reverse voltage

transfer ratio hrc common emitter (transistor)

forward current ratio small signal hfe static (dc) hFE

input impedance hie output impedance hoe reverse voltage

transfer ratio hre

332

common emitter (transistor) constant, voltage gain Gve acceleration of free fall g

broadband GvE Boltzmann k,kB common logarithm dielectric k,kd

19, loglQ, log gravitational G common mode (op amp) Planck h

input voltage VICM time T, T range VICR See also-coefficient and

rejection ratio CMRR factor voltage VCM conversion gain Ge

complex quantity conversion (phasor quantity) transconductance admittance Y,V &,gme

~

current I, I cosecant cosec impedance Z,Z hyperbolic cosech voltage E,V,E,V cosine cos

conductance G hyperbolic cosh conductance, mutual gm cotangent cot

See also-transconductance hyperbolic coth large signal Gm coulomb (unit) Q

conductance, transistor *coupling capacitor Cc (real part of y coupling coefficient k parameters) critical ke common base critical

forward transfer gfb angular frequency we input gib angular velocity we output gob coupling coefficient ke reverse transfer grb frequency fe

common emitter wavelength Ae forward transfer gfe crossover input gie angular frequency We output goe angular velocity we reverse transfer &e

333

crossover current frequency fc second breakdown ISlb

wavelength Ac vector (phasor) 1,1 cubic units current,opamp

centimeter cm3 bias IB foot cu ft, ft 3 device inch cu in, in 3 negative supply meter m3 1-, 10 -, lEE yard cu yd, yd3 non-inverting input

current I grounded lOG alternating lAC, lac' I open 100 average lav positive supply capacitive +jlx,lc 1+,10+, Icc direct loc, Ide, I input effective leff, Irms , I bias lIB generator Ig offset lIO inductive -jlx,IL signal lIN, lin input lin, Ii noise, equivalent input instantaneous i l/f Inf lagging -jlx device In leading +jlx shot Ins magnitude I noise, thermal noise of noise iN, IN, in, In input resistance InR output output

small signal 10 large-signal 10 large signal 10 maximum 10M

peak Ipk , ip, Ip negative swing 10 -peak-to-peak Ip_p peak-to-peak lopp

phasor 1,1 positive swing 10+ polar form 1 POLAR shorted los rectangular form IRECT small-signal 10 root-mean-square Irms , I

334

current, transistor base, small-signal Ib base, static (dc) Is collector, small-signal Ie collector, static (dc) I C

emitter, small-signal Ie emitter, static (dc) IE

current, transistor collector cutoff base-emitter

circuit ICEX

resistance ICER

shorted ICES

voltage ICEV

current, transistor emitter cutoff collector open lEBO

customary temperature t cutoff

angular frequency we angular velocity we frequency f e wavelength Ae

cycle, duty See-duty factor

cycles per second cps, cis, Hz See also-hertz

d

damping coefficient 0 damping factor IX, 0, d deci (unit prefix for 10-1) d

decibel (ratio unit for power, voltage and current) dB

decibel level See-level decilog decimal point degree deka (unit prefix for 10)

(rare in USA) delay time delta (greek letter)

capital script

depth device under test diameter

dg

o

da

~

o d

DUT d

inside db din, ID outside do, dout , OD

dielectric constant k, kd dissipation

collector (transistor) Pc

device power total

dissipation factor distance distortion

intermodulation total harmonic

IM,IMD THD

direct current DC, dc double pole (switch)

double throw DPDT single throw DPST

drain See-FET literature

335

duty cycle See-duty factor

duty factor FD , D,DF,df dynamic resistance r

See-vaccum tube literature See also-internal smallsignal resistance (transistor and opamp)

dyne (CGS unit) dyn

effective bandwidth

e

B, BW, BWNOISE , BWeff , S, BW, Bn, BWn

current (ac)

power radiated power voltage (ac)

Eeff , Erms ' Vrms , E, V See also-equivalent

and total efficiency 1/

electric charge Q electromotive force

See also-voltage elementary charge

emf, E, V

(charge of electron) e, q *emitter (transistor) E

breakdown voltage

BV EBO, V (BR)EBO

emitter (transistor) *capacitor CE resistance, external RE resistance, internal

small signal re static (dc) rE

*emitter resistor RE energy e,E,W

second breakdown ES/b epsilon (greek letter) s,c equal

approximately "" identically not f,* very nearly - --,

equivalent (of x) Xequiv, XT, Xl' X =

Note: The resultant of formulas is the equivalent quantity.

equivalent series resistance ESR

erg (CGS unit) erg eta (greek letter) 1/

exa (unit prefix for 1018 ) E excess noise voltage

EEX, EN(EX), v nR(EX) f

factor damping dissipation energy See-quality flare (flaring)

a,D, d D

m

336

factor magnification

See-quality merit See-quality noise (transistor)

(noise figure) F, NF, F n power cos 8, PF, pf, Fp Q Q quality Q storage See-quality

Fahrenheit temperature (quantity) t, tOF, TF (unit) of

fall time tf farad (unit) F feedback * capacitor CFB , CF * resistor RFB , RF

transfer ratio (3 femto (unit prefix for 10-15 )

f field effect transistor FET field strength, electric

(quantity) (unit)

field strength, magnetic

E Vim

(quantity) H, H (unit) De, At/m, A/m

figure, noise (transistor) (noise factor) F, NF, Fn

flare (acoustic horn) cutoff frequency factor

fFe FF,m

flux density, magnetic (quantity) B

G,T (unit) flux, total magnetic

(quantity) (unit)

foot (unit) cubic (unit) square (unit)

force

CP, cp Mx,Wb

" ft cu ft, ft3 sq ft, ft2

electromotive emf, E, V magnetizing See

magnetic field strength

magnetomotive (quantity) (unit)

mechanical (quantity) (unit)

F, j=, Fm A· t,A, At

forward current (semiconductor)

forward current transfer ratio common base

F kgf, lbf, N

IF

hfb

common collector hfc

common emitter small·signal hfe

static (dc) hFE forward transfer

admittance common base Yfb common emitter Yfe

337

frequency f frequency modulation FM angular W function F, f carrier fe critical fe 9

critical angular we gain (amplification) crossover fe current

crossover angular We large-signal AI cutoff fe small-signal Ai cutoff angular we margin rf>m,Om deviation fd voltage Doppler shift fD large signal Av extremely high ehf small signal Ay

flare cutoff fo,fe,fFC transistor Gye high hf broadband GvE

input fj, fin gain (power) intermediate i-f large signal Gp

low If small-signal Gp lowest satisfactory transistor

horn loading f' common base, maximum usable MUF large signal GpB

midband fo common base modulation fm small signal Gpb oscillation fose, fo common emitter pulse repetition fp large-signal GpE

reference rref, to common emitter resonant fo, fr small-signal Gpe resonant, angular wo,wr gain-bandwidth superhigh shf product GBW transition, transistor fT' ft opamp (unity gain ultra high uhf frequency) very high vhf B1 , BW(Ay = 1)

very low vlf transistor (transition frequency) ft> fT

338

gamma (greek letter) 'Y high frequency gate See-FET literature extremely (30-300 GHz) gauss (CGS unit) G ehf

generator current ig,Ig super (3-30 GHz) shf

generator voltage eg, E g, Vg ultra (300 MHz-3 GHz) giga (unit prefix for 109 ) G uhf

(pronouced jiga) very (30-300 MHz) vhf

gilbert (CGS unit) Gb horn, acoustic gram (CGS unit) g flare cutoff gravitational frequency fo' fe, fFe

acceleration g flaring factor FF,m acceleration, lowest frequency for

standard go satisfactory loading f' constant G horsepower (unit) hp

greater than (x) >x hour (unit) h not :1>x hour, ampere (unit) or equal to ;;;:'X A-h,Ah

grid See-vacuum tube hybrid parameter (transistor) literature forward current ratio

small signal h common base hfb

harmonic distortion, common collector hfc total THD common emitter hfe

heater See-vacuum tube static (dc) literature common emitter hFE

heatsink temperature ts , Ts hybrid parameter hecto (unit prefix for 102 ) (transistor)

(rare USA) h input impedance height h common base hib henry (unit) H common collector hie hertz (unit) Hz common emitter hie high frequency (3-30 MHz)

hf

339

hybrid parameter (transistor) output admittance

common base hob common collector hoe common emitter hoe

reverse voltage ratio common base hrb common collector hre common emitter hre

idling current Ii> Iq idling current drift ~Ii> ~Iq

imaginary number i, j imaginary part of (x) 1m x imaginary part of transistor

y parameters common base

forward transfer admittance

input admittance output admittance reverse transfer

admittance common emitter

forward transfer admittance

input admittance output admittance reverse transfer

admittance

±bfb

±bib ±bob

±bfe

±bie ±boe

impedance characteristic input magnitude mechanical output parallel phasor polar form primary rectangular form scalar secondary series vector (phasor)

impedance, opamp small signal input

Z Zo

Zin, Zi Z

Zm Zo

Zp-'.Zp Z,Z

ZPOLAR Zp

ZRECT Z

high frequency Zi common mode Zie

low frequency Rio ri output Zo

impedance, transistor small signal input, high frequency

common base Z ib common emitter zie

input, low frequency common base hib common collector hie common emitter hie

output Zo See also-admittance

340

inch (unit) in cubic (unit) square (unit)

increment indefinite number index, noise inductance

cu in, in3

sq in, in2

Ll

mutual parallel primary resonant secondary series

induction, magnetic See-magnetic field

strength inductive

n NI

L M

Lp,Lp Lp

Lr Ls

Ls,Ls

current reactance susceptance voltage

-jIx,+lx,IL

+X,XL

-jB, +B, BL

+Ex , +Vx, EL , VL

*inductor L *inductor, mutual LM infinity 00

infra-red IR input admittance Yin,Yi

transistor common base Yib common emitter Yie

input capacitance transistor

common base Cib , Cibo common emitter

Cie , Cieo input equivalent noise

(opamp and transistor) current total voltage

input frequency input impedance

in, In enj, Vni

en, Vn

fi' fin

opamp zin, zi common mode Zic

input impedance, transistor common base hib common collector hic common emitter hie

high frequency zie low frequency rie

input offset current (opamp) 110

input offset voltage (opamp) VIO

input power input resistance

opamp differential

transistor common base

Pin, Pi Rin,Ri

Rj, ri

rid

Rib, Re(hib ), rib common emitter

R ie , Re(hie), rie

341

instantaneous current peak current peak power peak voltage

epk, ep' vpk, vp power voltage

P e, v

*integrated circuit Ie intermediate frequency i-f intermodulation 1M intermodulation distortion

IM,IMD internal resistance, opamp

input Rj, rj output Ro, ro

internal resistance, transistor (T equivalent) base rb collector emitter re

intrinsic standoff ratio (unijunction transistor) 1/

inverse See-arc, negative reciprocal or reverse

joint army-navy specification JAN

joule (unit) W • s, Ws, J

k

kelvin (unit) K

kelvin temperature (thermodynamic

temperature) T, T K

kilo (unit prefix for 103 ) K, k knot(unit) kn

lambda (greek letter) capital A script A

lead tern perature t L, T L

leakage coefficient a leakage current IL

transistor See-cutoff current

leakage inductance primary secondary

length less than (x)

or equal to not

level (in decibels) current

ref. 1 pA power

L's, Is Vp,lp L's' Is

Q

<x <x «x

ref. 1 mW dBm, Lp/mw ref. 1 fW Lp/fW

sound power ref. 1 pW PWL, Lp/pw

sound pressure ref. 20 ppa/m2

SPL, Lp/20 J.LPa

342

level (in decibels) voltage

ref. IV dBV, Ly / y

ref. IVp _p dBv, LyfVp_p

light amplification by stimulated emission of radiation LASER, laser

light dependent resistor LDR light emitting diode LED line (of magnetic flux) (unit)

See-Maxwell liter (unit) load admittance load impedance load resistance *load resistor loaded Q logarithm

base 10 base €

common natural

loss angle

Ig, log 10, log 10ge,1n

Ig, log 10, log 10ge,1n

{j

lot tolerance percent defective LTPD

If vlf

low frequency very

m

magnetic field strength (quantity) H, H (unit)

Gb/cm, Oe, At/m, A/m

magnetic flux (quantity) (unit)

magnetic flux density

cJ>, </>

MX,Wb

(quantity) B (unit) G, T

(magnetic) permeability (quantity) J.I. (unit) G/Oe, (numeric)

(magnetic) reluctance (quantity) R,6t (unit) A/Wb, At/Wb

magnetizing force See-magnetic field strength

magnetomotive force (quantity) J, F, Fm (unit) A • t, A, At

magnitude (of x) Ixl magnitude of

admittance IYI,Y capacitive reactance Xc capacitive susceptance Be current impedance inductive reactance inductive susceptance input offset current

II 1,1 IZI,Z

XL BL

(opamp) 11101,110 input offset voltage

(opamp) IVIOI, VIO reactance X susceptance B

343

magnitude of medium frequency voltage lEI, lVi, E, V (300 kHz-3 MHz) mf

mega (unit prefix for 106 ) M merit factor Q

See also-quality factor

magnification factor (Q factor or quality factor)

Q margin, gain margin, phase mark See-sign mass maximum (device)

available gain

meter (unit) m cubic (unit) m 3

square (unit) m2

m mho (unit) mho, S, U, n-1

See also-seimens MAG micro (unit preftx for 10-6) J.l.

output current 10M mile (unit) mi peak-to-peak lapp square (unit) mi2

output swing bandwidth mile per hour (unit)

output voltage usable frequency

maxwell (CGS unit) mean -time-between

failures mean-time-to-failure mean-time-to-ftrst

failure mechanical

efftciency energy force impedance power pressure torque work

BaM VOM MUF

Mx

MTBF MTTF

MTTFF

1]

E,W F

mph,mijh milli (unit preftx for 10-3) m milli-inch (unit) mil mode, common

rejection CMR rejection ratio CMRR

mouth area (acoustic horn)

mu (greek letter) *mutual capacitor mutual conductance

(transconductance) transistor

SM,AM J.l.

CM gm

Zm common emitter gme P large-signal Gme , gME p mutual inductance M T *mutual inductor LM W mutual impedance ZM

344

n

nano (unit prefix for 10-9) n naperian logarithm loge, In natural logarithm loge, In natural resonant

frequency fn negative negative quantity

See-specific quantity "negative reactance" -X, Xc negative supply (opamp

or npn transistor) current lEE voltage VEE

neper (power ratio unit) Np net parallel susceptance

(BL - Be), ±B net series reactance

(XL - Xc), ±X neutralizing capacitor CN newton (unit) N *no connection NC noise current

average (broadband) in, In

spot (1 Hz BW)

. in, In, In/V'HZ nOIse current, device

equivalent input average (broadband) in, In spot (1 Hz BW)

in, In, In/V'HZ

noise, excess (quantity)

EEX, EN(EX)' VnR(EX) (unit) IN/Vdc

noise factor (quantity) (unit)

noise figure See-noise factor

noise index

F,NF, Fn dB

(quantity) NI (unit) dB

noise power N, P n noise, resistance

See-thermal noise noise temperature TN noise, thermal

current power voltage

noise voltage

iN, In(th), InR Nth, Pn(th), PnR eN, En(th) , VnR

average (broadband) en, en, En, Vn

spot (1 Hz BW)

. en, Vn, en/JJiz' Vn/V'HZ nOIse voltage, deVIce

equivalent input l/f Enf, enf, Vnf average (broadband)

en, Vn, En, en, Vn shot e" s' ens, Vns spot (1 Hz BW)

345

noise voltage, device equivalent input total (Vn, InRS and V nR ) Eni , eni, V ni s

noise voltage output

Eno ' eno ' Vno *non-polar (capacitor) NP *normally closed

(contact) NC *normally open

(contact) NO number

definite imaginary indefinite pairs of poles poles primary turns secondary turns turns turns ratio

o

n,N N

i, j n

Npp

Np

Np

Ns N,N t

n, Np/s

oersted (CGS unit) Oe ohm (unit) n omega (greek letter)

capital n script w

on -off ratio See-duty factor

open loop voltage amplification (opamp )

AVOL

operating temperature topn T OPR

operational amplifier op amp, opamp

operational transconductance amplifier OT A

optimum resistance oscillation frequency output admittance output admittance,

transistor h parameters

common base common collector common emitter

y parameters common base Yob common emitter Yoe

output capacitance Cout , Co output capacitance,

transistor common base

open circuit common emitter

open circuit output current

Cout , Co Cob

Cobo Coe

Coeo 10

output current (opamp) maximum peak-to-peak shorted output

output frequency

10M

lopp los

fout, fo

346

peak voltage output impedance circuit Zo epk. ep• Epk • Ep. Vpk. Vp opamp transistor

Zo peak-to-peak Zo current Ip_p

See also-output admittance

output power Po output resistance

circuit Ro opamp transistor

See also-output conductance

output voltage Eo. Vo output voltage. opamp

maximum (peak) peak -to-peak

overshoot

parallel capacitance impedance inductance resistance

p

YOM Vopp

OS. os

CP.Cp ZP.Zp Lp.Lp Rp.Rp

parameters. hybrid See-hybrid parameters

passband voltage amplification Avo

peak current Ipk • ip. Ip peak inverse voltage PlY peak reverse voltage PRY peak power p. P pk. P p

voltage Ep_p• Vp_p peak-to-peak (opamp)

current voltage

percent period

IOpp Vopp

%

period. time permeability (magnetic) permeance (magnetic) permittivity

T

(dielectric constant) k. kd peta (unit prefix for 1015) P phase angle rf>. 0

admittance rf>y • Oy current rf>I. 01 impedance rf>z. Oz voltage rf>E. rf>v. 0E. OV

phase margin rf>m. Om phasor quantities

admittance polar rectangular

current polar rectangular

impedance polar rectangular

Y

YPOLAR YRECT

I

IpOLAR IRECT

Z

ZPOLAR ZRECT

347

phasor quantities voltage E, V

polar EpOLAR rectangular E REeT

phi (greek letter) cjJ

pi (greek letter) 'IT

pico (unit prefix for 10-12)

(pronounced "peeko") p Planck constant h plate See-vacuum tube

literature *plug (male connector) P polar

admittance Y/()y, YPOLAR current

impedance

voltage

I~, IpOLAR

Z/()z, ZPOLAR

E/()E, EpOLAR

V/()v, VPOLAR pole frequency

(poles and zeros) positive positive quantities

See-specific quantity positive supply, opamp

current ID+' lee voltage VD+> Vee

positive supply, transistor npn

current voltage

pnp current voltage

lee Vee

lEE VEE

potential See-voltage pound (unit) lb pound per square inch psi power P power amplifier PA power factor cos(), PF, pf, Fp power gain Gp

transistor, large-signal common base GpB common emitter GpE

transistor, small signal common base Gpb common emitter Gpe

power, device PD power dissipation PD power, effective radiated

ERP power input Pin, Pi power level (quantity)

reference 1 fW reference 1 mW

power level (unit) reference 1 fW reference 1 mW

power level, acoustic

dBf dBm

reference 1 pW PWL, Lp/pw power output Pout , Po power, radiated PR power ratio (unit) dB power, signal S, Ps power, total PT, Pt

348

preftx, unit multiplier atto (10-18)

centi (10-2)

deci (10-1)

deka (10) exa (10 18)

femto (10-15)

giga (109)

(pronounced jiga) hecto (102)

kilo (10 3)

mega (106)

micro (10-6)

milli (10-3)

nano (10-9)

peta (10 15)

pico (10-12)

(pronounced peeko) tera (1012)

primary

q

a Q factor c quality assurance d quality control

da E quality factor f quantity of charge

G (quantity) (unit)

h quench frequency k quiescent current

quiescent voltage M

m radian (unit) n radius P P radiated power

effective T radiation efficiency

radiation resistance radio detection and

Q QA QC

Q

rad

ranging RADAR, radar current Ip impedance Zp voltage E V p' p

printed circuit PC printed circuit board PCB printed wiring board PWB programable unijunction

transistor PUT psi (greek letter) 1/1 public address (system) PA pulse energy test PET

radio frequency rf r-f radio frequency choke RFC radio frequency interference

random noise See-thermal noise

rate, repetition (frequency)

ratio (of x to y)

RFI

f x/y, x:y

349

ratio (unit) current, voltage or power (numeric), dB other (numeric)

ratio, power supply rejection (opamp) PSRR

ratio, transistor forward current transfer small signal

common base htb common collector hfc common emitter hfe

static (dc) common emitter hFE

ratio, transistor reverse voltage transfer

common base hrb common collector hrc common emitter hre

n, Np/s

X -X,Xc

ratio, turns reactance

capacitive inductive parallel series

reactive

+X,XL

±Xp, Xp ±Xs, Xs

current ±Ix, Ix power P q' var voltage ±Ex , ±Vx, Ex, Vx

real part of (x) Re (x)

real part of transistor admittance common base

forward transfer Re (Yfb), gfb

input Re (Yib), gib output

Re (hob), Re (Yob), gob reverse transfer

Re (Yrb), grb common emitter

forward transfer Re (Yfe), gfe

input Re (Yie), gie output

Re (hoe), Re (Yoe), goe reverse transfer

Re (Yre), gre rectangular form

admittance Y RECT current I RECT impedance ZRECT voltage ERECT, VRECT

reference re, ref angular frequency wo angular velocity wo current I ref frequency fo voltage Eref, V ref

reluctance (magnetic) tR reluctivity (magnetic) V,I1-1

350

repetition rate (frequency)

resistance device input device output generator input output parallel series source

resistance, opamp input output

resistance, transistor input

common base

f R

ri ro

Rg Rin,Ri

Rout> Ro Rp,Rp

Rs,Rs Rs

Ri,ri Ro, ro

hib , Re (hib ), rib common emitter

hie, Re (hie), rie output See also- ro

output conductance resistance, transistor,

saturation rCE(SAT) resistive current IR resistive voltage ER, VR resistivity p *resistor R * base (transistor) * collector (transistor) * emittor (transistor) * feedback

RB

Rc RE RF

resonant angular frequency angular velocity capacitance frequency inductance wavelength

reverberation time

Wo,Wr

wo,wr Co, Cr

fo, fr

Lo' Lr Ao, Ar

TRVB , T, T60 reverse current IR reverse transfer

admittance (transistor) common base y rb common emitter

reverse voltage reverse voltage, peak reverse voltage transfer

ratio (transistor) common base common collector common emitter

Yre VR

PRY

revolutions per minute (unit) second (unit)

rho (greek letter)

r/min, rpm rps, r/s

p rise time tr root-mean-square rrns

S

saturation SAT saturation resistance

(transistor) rce(SAT)

351

scalar See-magnitude sigma (greek letter) screen See-vacuum tube capital L

literature script S, a second (angle unit)

/I

signs and marks second (time unit) s absolute value II second breakdown addition +

(transistor) approaches -current ISfb ampersand & energy ESfb and &

secondary angle L current Is apostrophe impedance Zs asterisk * turns Ns at @

voltage Es'Vs because sectional area S,A braces { } sensitivity S brackets [ ] sensitivity, power supply breve

(opamp) PSS caret series cent ¢

aiding inductance LSA circumflex capacitance Cs , Cs colon impedance Zs, Zs comma inductance Ls, Ls congruent "'" opposing inductance Lso dagger t reactance Xs , ±Xs' Xs decimal point resistance Rs , Rs degree 0

short-circuit output current difference (opamp) los directly proportional ex:

shot noise See-noise division siemens (unit) S dollar $

See also-mho double dagger :j: em dash en dash

352

signs and marks signs and marks equal to number #

approximately ~ paragraph ~ congruently ~ parallel II identically parentheses ()

not ¢ partial differential a nearly :::::: percent % not =j=, oF period very nearly !..- ~ plus + ~,==

equivalent -"- plus or minus ::1::, ± "V

exclamation mark positive + factorial positive or negative ::1::, ±

greater than > pound # not :l> prime or equal to L,~ double (second) "

hyphen triple (third) II' inch " proportion infmity 00 proportional, directly a:

integral f question mark ? less than < quotation marks "" ,

not <t radical sign v' or equal to ::;;.,~ ratio

macron second " mean value sectional symbol § minute semicolon minus solidus multiplication X . subtraction , negative therefore not varies as a:

equal to =j=,oF viculum

greater than :l> virgule / identical ¢ signal S, sig less than <t

353

signal generator current impedance resistance voltage

signal, large See-specific quantity

signal level See-level

signal power Ps signal, small

See-specific quantity signal source

current voltage

signal-to-noise ratio

Is Es , Vs

SIN silicon controlled rectifier

SCR silicon controlled switch SCS silicon unilateral switch SUS sine sin

hyperbolic sinh sinew ave power Psine

single pole (switch) double throw SPDT single throw SPST

single sideband SSB sink temperature

(heatsink) ts, Ts small-signal

See-specific quantity *socket (receptacle or

female connector) S

sound navigation and ranging SONAR, sonar

sound power P sound power level,

ref. 1 pW PWL, Lp/pw sound pressure p sound pressure level,

ref. 20 t.tPa/m2 SPL, Lp/2o J./Pa

source current Is impedance Zs resistance Rs voltage Es , Vs

source (field effect transistor) See-FET literature

spacing s speed

See also-velocity light c sound c,v

spot noise See-noise square units

centimeter foot inch meter mile yard

square wave power standing wave ratio

power voltage

cm2

sq ft, ft 2

sq in, in2

m2

sqmi, mi2

sq yd, yd2

SWR S, VSWR

354

t static transistor parameter See-specific parameter

storage factor See-quality factor

sum

tangent hyperbolic

tau (greek letter) l:: television

tan tanh

T

TV summation super high frequency supply voltage sensitivity

l:: temperature shf ambient

(opamp) PSS, ksvs susceptance

capacitive inductive

susceptance, transistor (imaginary part of y parameters) common base

forward transfer

B

±jbtb, ±btb , btb input ±jbib , ±bib , bib output ±jbob, ±bob, bob reverse transfer

±jbrb , ±brb , brb common emitter

forward transfer ±jbfe , ±bfe , bfe

input ±jbie , ±bie , bie output ±jboe, ±boe, boe reverse transfer

±jbre , ±bre , bre sustaining voltage

See-voltage ·switch S,SW

case Celsius centigrade

See-Celsius

tA, TA te , Te

toe, t, Te , T

coefficient a, TC Fahrenheit t, tF, T F junction tJ, TJ Kelvin T, TK lead tL, TL noise Tn, TN sink (heat) ts, Ts tab tT, TT

tera (unit prefix for 10 12) T tesla (magnetic unit) T thermal conductance Ge thermal conductivity A thermal noise

See-noise thermal resistance theta (greek letter)

capital script

threshold current throat area

(acoustic hom) time

B,Re

e B

355

*transformer *transistor

T Q, TR

time constant time, delay time, fall tf transistor parameters time of one cycle T See-specific parameter time,

periodic T phase propagation tl/> pulse duration tp rise tr

reverberation T RVB, T, T 60

storage T s' ts, tSTG total t TOT

torque T total (also meaning effective

or equiValent) admittance capacitance conductance current dissipation harmonic distortion impedance inductance

YT , Yt

CT,C t

GT,G t

IT, It Pt , PT

THD ZT,Zt ~,Lt

power Pt , PT resistance RT, R t

susceptance BT, Bt

time tToT voltage ET, VT, E t , Vt

transadmittance See-admittance

transconductance See also-mutual

conductance

transistor-under-test TUT transmission loss

(attenuation) (quantity) a (unit) (numeric), dB

*tube, vacuum V turn(s) n, N

ampere (magnetic unit)

primary ratio secondary

u

A' t, A, At Np

n, Np/s

Ns

ultra-high-frequency ultra-violet

uhf UV

unijunction (transistor) UJT unipolar transistor

(field effect tranSistor) FET

unknown capacitance current impedance inductance resistance voltage

unloaded Q

Cx

Ix Zx Lx Rx

Ex, Vx Qu

356

v voltage (quantity) *vacuum tube V average Eav' Vav vacuum tube voltmeter capacitive Ec , Vc

VTVM dc Edc ' Vdc variable frequency effective Erms ' Vrms , E, V

oscillator VFO gain (amplification) vector See also-phasor Ay,Av

admittance Y generator Eg, Vg current inductive EL, VL impedance Z input Ein , Yin, Ej, Vi voltage E, V instan taneous e,v

velocity See also-speed peak ep, vp (quantity) v level Ly (unit) ftls, mls output Eo, Vo

velocity of light peak Epb Vpb Epo Vp See-speed peak-to-peak Ep_p, Vp_p

velocity of sound c,v polar E/8E, EpOLAR very high frequency

V/8y , VpOLAR (30-300 MHz) vhf power supply Eps , Vps very low frequency primary Ep, Vp

(3-30 kHz) vlf rectangular

very nearly equal to "'" ERECT, VRECT volt (unit) V resistive ER, VR ac VAC, V AC, Vac, root-mean -square average Vav

Ermso Vrms , E, V dc VDC, V DC, V dc Es , Vs

peak V pk source

voltage controlled peak-to-peak V p_p

oscillator VCO root-me an-square V rms voltage controlled

voltage (quantity) resistor VCR

ac Eac , Vac amplification Ay,Av

357

voltage, transistor (general) base base supply base-to-emitter

active saturated

collector collector supply collector-to-base

VB VBB VBE

VBE(ON) VBE(SAT)

Vc VCC VCB

emitter open V CBO collector-to-emitter VCE

base-emitter circuit resistance short voltage

base open emitter emitter supply

emitter-to-base open collector

voltage, transistor, breakdown collector-to-base

emitter open

VCEX VCER VCES VCEV VCEO

VE VEE

BVcBo, V(BR)CBO collector-to-emitter

base-emitter circuit

BV CEX, V CEX(SUS)

voltage, transistor breakdown

base-emitter resistance

BV CER, V CER(SUS) base- emitter

short BV CES, V CES(SUS)

base open

BV CEO, V CEO(SUS) emitter-to-base

collector open

BVEBo, V(BR)EBO voltage, transistor,

sustaining LV, V(SUS) collector-to-emitter

base-emitter resistance

LVCER, VCER(SUS) base-emitter short

LVCES , VCES(SUS) base-emitter voltage

LVCEV , VCEV(SUS) base open

LVCEO , VCEO(SUS) voltage, working WV voltampere

(apparant power) (quantity) S, Ps ' VA (unit) VA

volt-ohm meter YOM volume (cubic content) V volume unit (similar to dBm)

vu,VU

358

w

watt (unit) watthour (unit) wattsecond (unit)

Goule) wavelength

W W·h, Wh W·s,Ws

J A

Wb W

weber (magnetic unit) weight

See also-mass white noise See-noise width (breadth) wire gage (gauge)

American British standard steel

b

AWG SWG

StlWG

wirewound (resistor) WW work

(quantity) W (unit) KWh, W • s, J

working voltage WV wye connection Y

xyz

xi (greek letter) capital :E: script ~

zener (semiconductor) current Iz impedance Zz voltage Vz

zeta (greek letter) t

359

Documents

[John R. Brand (Auth.)] Handbook of Electronics Fo(BookZZ.org)