ELECTRONICS Formulas and Concepts jc

7/25/2019 ELECTRONICS Formulas and Concepts jc

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SEMICONDUCTOR THEORY

Atomic StructureDiameter of neutron = 10-13cm

Maximum number of electrons per shell or orbit

4,3,2,1

2 2

=

=

n

nNe

Letter designationK shell 1 O shell 5L shell 2 P shell 6M shell 3 Q shell 7N shell 4

Mass and Charge of different Particles

Particle Mass (kg) Charge (C)

Electron 31101096.9 19106022.1 Proton 27106726.1 19106022.1 +

Neutron 27106726.1 No charge

A = no. of protons + no. of neutrons

Z = number of protons or electrons

Where: A = Atomic mass or weight (A)Z = Atomic number (Z)

Note: Mass of proton or neutron is 1836 times thatof electron.

Energy Gap Comparison

Element No. of Valence

Electrons (Ve)

Energy

Insulator 8 > 5eVSemiconductor 4 Si = 1.1eV

Ge = .67eVConductor 1 0eV

At room temperature: there are approximately1.510

10 of free electrons in a cubic centimeter(cm3) for intrinsic silicon and 2.51013 forgermanium.

Diode Theory)( 0101 TTkVthVth TT +=

where: VthT1= threshold voltage at T1VthT0= threshold voltage at T0k = 2.5 mV/C for Gek = 2.0 mV/C for Si

The diode current equation

)1( = kd

T

kV

sd eII

Where: Id= diode currentIs= reverse saturation current or leakagecurrentVd= forward voltage across the diode

Tk= room temperature at K= C + 273

nk

11600=

for low levels of diode currentn = 1 for Ge and n = 2 for Si

for higher levels of diode currentn = 1 for both Si and Ge

Temperature effects on Is)( 01

01

TTk

sTsT eII =

Where: IsT1= saturation current at temperature T1IsT0= saturation current at room temperaturek = 0.07/CT1= new temperatureT0= room temperature (25C)

Reverse Recovery Time (Trr)

tsrr ttT +=

Where: Trr= time elapsed from forward to reversebias (ranges from a few ns to few hundredsof ps)Tt= transition timeTs= storage time

DC CIRCUITS 1

1 Coulomb = 6.241018electrons

By definition:A wire of 1 mil diameter has a cross-sectional area of 1 Circular Mil (CM)

1 mil = 10-3in1 in = 1000 milsAsquare= 1 mil

2

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L

A

L

A

RG

===

1

where: #= specific conductance or conductivity ofthe material in siemens/m or mho/m.

Note: The best is silver with 1.681024 freeelectrons per in3. Next is copperwith 1.641024freeelectrons per in3and then aluminumwith 1.61024free electrons per in3.

RIR

EIEE

t

Q

t

WP

22

=====

where: W = work in Joules (J)t = time in seconds (s)Q = charge in Coulomb (C)

Voltage Division Theorem

2 resistors in series with one

ERR

RV

21

11 += E

RR

RV

21

22 +=

Current Division Theorem

TIRR

RI

21

21 += TI

RR

RI

21

12 +=

Transformations or Conversations:

Delta (#) to Wye (Y)

= ____ _____Pr inRallofinRadjacentofoductRY

Wye (Y) to Delta (#)

YinROpposite

YinproductscrossofR

___

____=

Color Coding TableColor 1st

signifi-cant

2ndsignifi-cant

MultiplierToler-rance(%)

TempCoef

ppm/CBlack 0 0 100 20 0Brown 1 1 101 1 -33Red 2 2 102 2 -75Orange 3 3 103 3 -150Yellow 4 4 104 GMV -220Green 5 5 105 5 -330Blue 6 6 106 - -470Violet 7 7 107 - -750Gray 8 8 108 - +30White 9 9 109 - +500

Gold - - 0.1 5 +100Silver - - 0.01 10 Bypass

None - - - 20 -

GMV = Guaranteed Minimum Value: -0%, +100%

Fifth band reliability color code

Color Failures during 1000

hours of operation

Brown 1.0%Red 0.1%Orange 0.01%Yellow 0.001%

Batteries

Battery life)(_

)(_

AdrawnAmperes

AhratinghourAmpere =

Cell Types and Open-Circuit Voltage

Cell Name Type Nominal Open-

Circuit VoltageCarbon-zinc Primary 1.5Zinc-chloride Primary 1.5Manganesedioxide (alkaline)

Primary orSecondary

1.5

Mercuric oxide Primary 1.35Silver oxide Primary 1.5Lead-acid Secondary 2.1Nickel-cadmium Secondary 1.25Nickel-iron(Edison cell)

Secondary 1.2

Silver-zinc Secondary 1.2Silver-cadmium Secondary 1.1Nickel metalhydride (NiMH)

Secondary 1.2

DIODES

Diode ApplicationsHalfwave Rectification

m

m

DC

VV

V 318.0==

PIV rating Vm

Fullwave Rectification

=t

rms dttVT

V0

2)(1

mDC VV 36.0= PIV rating Vm for bridge-type



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PIV rating 2Vm for center-tapped

Other Semiconductor DevicesZener Diode

)( 01 TTV

VT

Z

ZCC

=

where: TCC= temperature coefficient

T1T0= change in temperatureVZ= Zener Voltage at T0

Basic Zener Regulator

I. Viand RLfixed(a)Determine the state of the Zener diode by

removing it from the network and calculating thevoltage across the resulting open circuit.

(b)Substitute the appropriate equivalent circuit andsolve for the desired unknown.

II.

Fixed RL, variable Vi

L

ZSLi

R

VRRV

)(min

+= ZSRi VRIV += maxmax

III.Fixed Vi, variable RL

Zi

ZL

VV

RVR

=min

min

max

L

ZL

I

VR =

Varactor diode or Varicap diode

d

T

W

AC =

where: CT= transition capacitance which is due tothe established covered charges on either sideof the junctionA = pn junction areaWd= depletion width

In terms of the applied reverse bias voltage:

n

RT

TVV

kC

)( +=

where: CT= transition capacitance which is due tothe established covered charges on either sideof the junctionk = constant determined by thesemiconductor material and constructiontechniqueVT= knee voltageVR= reverse voltagen = for alloy junctions and $for diffusedjunctions

In terms of the applied reverse bias voltage:

n

T

R

T

V

V

CC

+

=

1

)0(

where: C(0) = capacitance at zero-bias condition

Also,

( )01 TTCCTC

O

C =

where: TCC= temperature coefficientT1 T0= change in temperatureC0= capacitance at T0

Photodiode

Joulesc

hhfW ;

==

where: W = energy associated with incident light

wavesh = Plancks constant (6.62410-34J-sec)f = frequency

1eV = 1.610-9J1 Angstrom () = 10-10m

Solar Cell

==

2

max

1)(

cm

WArea

P

P

P

i

O

where: (= efficiencyP0= electrical power outputPi= power provided by the light sourcePmax= maximum power rating of the deviceArea = in cubic centimeters

Note: The power density received from the sun atsea level is about 1000 mW/cm2

BIPOLAR JUNCTION TRANSISTOR

150001.0

150.0 ===base

total

width

widthRatio

Basic OperationRelationship between IE, IBand IC:

CBE III +=



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ICis composed of two components:

oritymajorityC III min+=

DC Transistor Parameters

E

C

tconsVcbE

C

I

I

I

I

=

==

tan

B

C

tconsVceB

C

I

I

I

I

=

==

tan

where: IE= emitter currentIB= base currentIC= collector current"= CB short-circuit amplification factor)= CE forward-current amplification factor

Relationship between "and $:

1+=

=

1

Stability Factor (S):

CO

C

COI

IIS

=)( Unitless

BE

CCO

V

IIS

=)( Siemens

= CCOI

IS )( Ampere

Small Signal AnalysisA.

Hybrid Model

02221

01211

VhIhI

VhIhV

ino

ini

+=+=

If Vo= 0

in

i

I

Vh =11 ohms

If Iin= 0

0

12

V

Vh i= unitless

If Vo = 0

inI

Ih 021= unitless

If Iin= 0

0

011

V

Ih = siemens

where: h11= input-impedance, hih12= reverse transfer voltage ratio, hr

h21= forward transfer current ratio, hfh22= output conductance, ho

H-Parameters typical values

CE CB CC

hi 1k! 20! 1k!hr 2.510-4 310

-4 *1hf 50 -0.98 -50

ho 25+S 0.5+S 25+S

Comparison between 3 transistor configurations

CB CE CC

Zi low moderate highZo high moderate lowAi low high moderateAv high high lowAp moderate high low

shift none 180 none

B.

ReModelNote:Common Base : hib= re ; hfb= 1Common Emitter: )= hfe ; )re= hie

FIELD EFFECT TRANSISTORS

JFET2

1

=

P

GS

DSSDV

VII

P

DSSmo

V

Ig

2=

0

12

=

=

=

dsVgs

d

P

GS

P

DSS

mV

I

V

V

V

Ig

50 GSV where:Id= drain currentIdss= drain-source saturation currentVgs= gate source voltageVp= Vgs(off), pinch-off voltagegm= gfs, device transconductance

gmo= the maximum ac gain parameter of the JFET

MOSFET2)( THGSDS VVkI =

2/3.0 VmAk=

FET biasing



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DC bias of a FET requires setting the gate-sourcevoltage, which results in a desired drain current. Vggis used to reverse bias the gate so that Ig= 0.

POWER SUPPLY

Transformer

p

s

p

s

p

s

s

p

Z

Z

N

N

V

V

I

Ia ====

where: a = turns ratioVs= secondary induce voltageVp= primary voltageNs= no. of turns on the secondary windingsNp= no. of turns on the primary windingsIp= current in the primary windingsIs= current in the secondary windingsZs= impedance of the load connected to thesecondary winding

Zp= impedance looking into the primaryfrom source

RectifierHalfwave signal

mdc VV 318.0= 2m

rms

VV =

rmsdc VV 636.0= rmsVPIV 2= Ripple frequency = AC input frequency

Fullwave rectified signal (bridge type)mdc VV 636.0=

2m

rms

VV =

rmsdc VV 9.0= rmsVPIV 2= Ripple frequency = 2AC input frequency

Fullwave with center-tapped transformerVdc= 0.9Vrmsof the half the secondary

= 0.45Vrmsof the full secondary

= 0.637Vpkof half of the secondary= 0.637Vpkof the full secondary

PIV = 1.414Vrmsof full secondary

dc

r

V

rmsV

DC

ACr

)(==

22)( dcrmsr VVrmsV = where: r = ripple factor

Vr(rms) = rms value of the ripple voltage

Vdc= average value of the filters outputvoltage

mr VrmsV 385.0)( = halfwave rectified signal

mr VrmsV 308.0)( = fullwave rectified signal

Filter

32)(

3)()( ppVpVrmsV rrr ==

CR

V

C

I

fC

IrmsV

L

dcdcdcr

4.24.2

34)( ===

C

IV

fC

IV

ppVVV dcm

dc

m

r

mdc

17.4

42

)(==

=

%1004.2

%1004.2

%100)(

===CRCV

I

V

rmsVr

Ldc

dc

dc

r

where: Idc= the load current in mA

C = filter capacitor in +FRL= load resistance at the filter stage in k!Vm= the peak rectified voltageIdc= the load current in mAC = filter capacitor in +Ff = frequency at 60 Hz

RegulatorVoltage Regulation

%100..

=fload

floadnoload

V

VVRV

Stability factor (S)

in

out

V

VS

= (constant output current)

Improved series regulation

)( 22

21BEZo VV

R

RRV +

+=

INSTRUMENTATION

DC AmmeterRelationship between current without the

ammeter and current with the ammeter

mo

o

wom

wm

RR

R

I

I

+=

where: Iwm= current with meterIwom = current without meterRo= equivalent resistance



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Rm= internal resistance of ammeter

Accuracy Equation of an ammeter

wom

wm

I

Iaccuracy=

Percent of loading error

%100)1(% = accuracyerror Ammeter Shunt

fst

mfs

shII

RIR

=

shm

shm

shRR

RRRin

+=

t

mfs

in

insh

I

RI

I

VRin ==

where: Rsh= shunt resistanceIfs= full scale currentRm= meter resistance

It= total currentRinsh= input resistance of the shunted meterVin= voltage inputIin= current input

VoltmeterFor full scale current

Vfs= (Rs+ Rm)Ifs

m

fs

fs

s RI

VR =

Rin= Rs+ Rmwhere: Vfs= full scale voltageRs= series resistorRin= input resistance

Sensitivity of Voltmeter

fsIS

1=

fs

fs

inI

VR =

Voltmeter Loading Error

oin

in

wom

wm

RR

R

V

V

accuracy +==

oin

womin

wmRR

VRV

+=

Ohmmeter

o

ocfs

R

VI =

uo

oc

RR

VI

+=

uo

o

fs RR

R

I

ID

+==

where: Ifs= full scale currentVoc= open circuit voltageRo= internal resistance of ohmmeterD = meter deflectionRu= unknown resistance

AC Detection

fs

acI

S45.0= Sensitivity for a half-wave rectifier

fs

acI

S9.0= Sensitivity for a full-wave rectifier

DC BridgesWheatstone bridge ohmmeter

Bridge is balance if

4

3

2

1

RR

RR =

Attenuators

inoinso RRR = where: Ro= characteristic resistance

Rins= input resistance with output terminalsshortedRino= input resistance with output terminalsopen

L type or the voltage divider

21

2

RR

Rgain

+=

gainV

Vnattenuatio

out

in 1==

2

1

2

1

C

C

X

X

R

R=

1

221

R

CRC =

Symmetrical Attenuator

12

1

2 ; mRRR

Rm ==

Symmetrical T Analysis

mRR 2110 += m

mm

V

Va

out

in 211 +++==

Symmetrical Pi Analysis

m

RR

212

0+

= m

mm

V

Va

out

in 211 +++==



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Design Formulas for T Attenuator

oRa

aR

2

12

1

= oR

a

aR

1

12

+=

Design Formulas for T Attenuator

oR

a

aR

1

11

+

= oR

a

aR

1

222

=

Variable Attenuator

Analysis

R1= R0 12

1 +=R

Ra

Design

R1= R01

02 =

a

RR

0

3

1

R

aR

=

COMPUTER FUNDAMENTALS

r's complement(r

n)10 N

(r 1)s complement

(rn r

-m)10 N

Types of Binary Coding

Binary Coded Decimal Code (BCD)

DECIMAL DIGIT BCD Equivalent0 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001

Excess-3-code

DECIMAL DIGIT Excess-3

0 00111 01002 01013 01104 01115 1000

6 10017 10108 10119 1100

Gray Code (Reflected Code)

DECIMAL DIGIT Gray Code

0 0000

1 00012 00113 00104 01105 01116 01017 01008 11009 110110 1111

11 111012 101013 101114 100115 1000

DECIMAL 84-2-1 2421 Biquinary

5043210

0 0000 0000 01000011 0111 0001 01000102 0110 0010 0100100

3 0101 0011 01010004 0100 0100 01100005 1011 1011 10000016 1010 1100 10000107 1001 1101 10001008 1000 1110 10010009 1111 1111 1010000

OPERATIONAL AMPLIFIERS

VD= V+ V-

where: VD= differential voltageV+= voltage at the non-inverting terminalV-= voltage at the inverting terminal

c

d

A

ACMRR=

where: Ad= differential gain of the amplifierAc= common-gain of the amplifier



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Slew rate

pk

o Vft

VSR max2=

=

where: fmax= highest undistorted frequencyVpk= peak value of output sine wave

Differentiator

dt

dVRCV ino =

Integrator

dtVRC

V ino =1

Basic non-inverting amplifier

1

21R

Rgain +=

Basic inverting amplifier

1

2

R

Rgain =

LOGIC GATES

Boolean AlgebraPostulated and Theorems of Boolean algebra

11

1'

0

=+=+=+

=+

X

XXX

XX

XX

00

0'

1

===

=

X

XXX

XX

XX

(Commutative Law)

XYYX +=+ XYYX =

(Associative Law)

ZYXZYX ++=++ )()( ZXYYZX = )()(

(Distributive Law)YZXYZYX +=+ )(

))(()( ZYYXYZX ++=+

(Law of Absorption)

XXYXZX =++ )( XYXX =++ )(

(De Morgans Theorem)

'')'( YXYX =+ '')'( YXXY +=

Logic Family CriterionPropagation delay is the average transition delaytime for a signal to propagate from input to output.

2PLHPHL

p

ttt

+=

where: tp= propagation delaytPHL= propagation delay high to lowtransitiontPLH= propagation delay low to hightransition

Power dissipationis the amount of power that an ICdrains from its power supply.

2)( CCLCCHCC

IIAVGI

+=

CCCCD VAVGIAVGP = )()(where: ICCH= current drawn from the power supply

at high level

ICCL= current drawn from the power supplyat low level

Noise Marginis the maximum noise voltage addedto the input signal of a digital circuit that does notcause an undesirable change in the circuit output.

Low State Noise Margin

OLILL VVNM = where: NM = Noise Margin

VIL= low state input voltage

VOL= low state output voltage

High State Margin

IHOHH VVNM = where: NM = Noise Margin

VIH= high state input voltageVOH= high state output voltage

Logic Swing

OLOHls VVV =

where: Vls= voltage logic swingVOH= high state output voltageVOL= low state output voltage

Transition Width

ILIHtw VVV = where: Vtw= voltage transition width

VIH= high state input voltageVIL= low state input voltage



10/22

TYPICAL CHARACTERISTICS OF IC LOGIC

FAMILIESIC

Logic

Family

Fan

out

Power

Dissipation

(mW)

Propagation

Delay (ns)

Noise

Margin

(V)

StandardTTL

10 10 10 0.4

Schottky 10 22 3 0.4

Lowpower

SchottkyTTL

20 2 10 0.4

ECL 25 25 2 0.2CMOS 50 0.1 25 3

LEVEL OF INTEGRATION

Level of Integration No. of gates per chip

Small Scale Integration(SSI)

Less than 12

Medium ScaleIntegration (MSI)

12 99

Large Scale Integration(LSI)

100 9999

Very Large ScaleIntegration (VLSI)

10000 99999

Ultra Large ScaleIntegration (LSI)

100000 or more

CAPACITOR/INDUCTOR TRANSIENT

CIRCUITS

CapacitorsThe Gauss TheoremThe total electric flux extending from a closedsurface is equal to the algebraic sum of the chargesinside the closed surface.

Q

Electric Flux Density

AD

=

where: D = flux density, Tesla (T) or Wb/m2,= electric flux, Weber (Wb)A = plate area, m2

Electric field strength or intensity (%)

d

V

Q

F==

where: -= field strength (N/C, V/m)

F = force (Newton)Q = charge (Coulomb)V = voltage across the plates (volt)d = distance between plates (m)

Coulombs Laws of Electrostatics

First Law:Unlike charges attract each other while like charges

repel.

Second Law:The force of attraction or repulsion betweencharges is directly proportional to the product of thetwo charges but inversely proportional to the squareof distance between them.

221

r

QkQF=

4

1=k 0 r=

Permittivity

A measure of how easily the dielectric will permitthe establishment of flux line within the dielectric.

D=

For vacuum,m

F129

0 10854.836

10

==

Capacitance

V

QC=

d

AnC )1( =

where: Q = chargeV = voltagen = number of platesA = plate aread = distance between plates

Relative Permittivity (Dielectric Constant) of

various dielectrics

Dielectric Material &r(Average value)

Vacuum 1.0Air 1.0006

Teflon 2.0Paper, paraffined 2.5

Rubber 3.0Transformer oil 4.0

Mica 5.0Porcelain 6.0Bakelite 7.0



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Glass 7.5Distilled water 80.0

Barium-strontium titanite(ceramic)

7500.0

Dielectric strength of some dielectric materials

Dielectric

Material

Dielectric Strength

(Average Value) in V/mil

Air 75Barium-strontiumtitanite (ceramic)

75

Porcelain 200Transformer oil 400

Bakelite 400Rubber 700

Paper, paraffined 1300Teflon 1500Glass 3000Mica 5000

Energy stored

C

QCVE

22

1 22 ==

Capacitors in Series

n

T

CCCC

C1

...111

1

321

++++=

nT QQQQQ ===== ...321

Capacitors in Parallel

nT CCCCC ++++= ...321

nT QQQQQ ++++= ...321

Other capacitor configurations

Composite medium parallel-plate capacitor

++

=

3

3

2

2

1

1

0

rrr

ddd

AC

where: d1, d2and d3= thickness of dielectrics withrelative permittivities of .r1, .r2and .r3respectively

Medium partly air parallel-plate capacitor

=

r

ttd

AC

0

Cylindrical capacitor

910

log4.41

=

a

bC r

l

where: a = diameter of single core cable conductorand surrounded by an insulation of innerdiameter b

.r= relative permittivity of the insulation ofthe cablel = length of the cylindrical capacitor

Capacitance of an isolated sphere

C = 4!"rwhere: r = radius of the isolated sphere in a medium

of relative permittivity .r

Capacitance of concentric spheres

a.) When outer sphere earthed

)(4

ababC=

Where: a and b are radii of two concentric spheres.= permittivity of the dielectric between twospheres

b.) When inner sphere is earthed

)(4

2

ab

bC

=

InductorsInductance(L) is a measure of the ability of a coilto oppose any change in current through the coil andto store energy in the form of a magnetic field in theregion surrounding the coil.

In terms of physical dimensions,

l

ANL

2

= Henry

where: += permeability of the core (H/m)N = number of turns

A = area of the core (m2

)l = mean length of the core (m)

In terms of electrical definition,

di

dNL =



12/22

Faradays LawThe voltage induced across a coil of wire equals thenumber of turns in the coil times the rate of changeof the magnetic flux.

dt

dNein

=

where: N = number of turns of the coil

dt

d

= change in the magnetic flux

Lenzs Law

An induced effect is always such as to oppose thecause that produced it.

dt

dNein

=

Induced voltage by Faradays Law

dt

diLeL=

Energy stored

2

2

1LIWL =

Inductance without mutual inductance in series

nT LLLLL ++++= ...321

With mutual inductance (M)

a.) when fields are aiding

MLLLTa 221 ++=

b.)when fields are opposing

MLLLTo 221 +=

Total inductance without mutual inductance (M)

n

T

LLLL

L1

...1111

321

++++=

With mutual inductance (M)

a.) when fields are aiding

MLL

MLLL aT

221

221

)( +

=

b.)when fields are opposing

MLL

MLLL oT

221

221

)( ++

=

Mutual inductanceIt is a measure of the amount of inductive couplingthat exists between the two coils.

21LLkM=

4ToTa LLM

=

where: k = coupling coefficient

L1and L2= self-inductances of coils 1 and 2LTaand LTo= total inductances with mutualinductance

Coupling coefficient (k)

21LL

Mk=

1

21

___

_____

Lbyproducedflux

LandLbetweenlinkagefluxk=

Formulas for other coil geometries(a)LONG COIL

l

ANL

2

=

(b)SHORT COIL

d

ANL

45.0

2

+=

l

where: L = inductance (H)+= permeability (4/10-7for air)

N = number of turnsA = cross-sectional area of the coil (m2)l = length of the core (m)d = diameter of core (m)

(c)TOROIDAL COIL with rectangular cross-

section

1

22

ln2 d

dhNL

=

where: h = thicknessd1and d2= inner and outer diameters

(d)CIRCULAR AIR-CORE COIL

bR

RNL

1096

)(07.0 2

++=

l

22

bdR +=

where: L = inductance (+H)N = number of turnsd = core diameter, in



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b = coil build-up, inl = length, in

(e)RECTANGULAR AIR-CORE COIL

bC

CNL

109908.1

)(07.0 2

++=

l

where: L = inductance (+H)C = d + y + 2bd = core height, iny = core width, inb = coil build-up, inl = length, in

(f) MAGNETIC CORE COIL (no air gap)

c

ANL

l

2012.0=

(g)MAGNETIC CORE COIL (with air gap)

c

g

ANL

ll +

=2012.0

where: L = inductance (+H)N = number of turnsA = effective cross-sectional area, cm2

cl = magnetic path length, cm

gl = gap length, cm

0= magnetic permeability

DC Transient CircuitsCircuit

Element

Voltage

across

Current

flowing

R iRv= R

vi=

L

dt

diLv= vdt

Li =

1

Cidt

CC

qv ==

1

dt

dvCi=

Response of L and C to a voltage sourceCircuit Element @ t = 0 @ t = '

L open shortC short open

RL Transient Circuit

Storage Cycle:

=

=

tt

L

R

eR

Ee

R

Ei 11

R

L=

=

tL

R

R eEv 1t

L

R

L Eev

=

Decay Phase:

tt

L

R

eR

Ee

R

Ei

==

TR

L

= RRRT += 1

RC Transient Circuit

Charging Cycle:

( ) RCt

eECqECq

+= 0

=

RC

t

eECq 1 with q0= 0

RC

t

eR

Ei

= RC

t

R Eev

=

=

RC

t

C eEv 1 RC=

Discharging Phase:

RC

t

C Eev

= RC=

RLC Transient Circuits

Conditions for series RLC transient circuit:

(1)@ t = 0, i = 0(2)@ t = 0, Ldi/dt = E

Current equations

Case 1 Overdamped case

whenLCL

R 1

2

2

>

then

trtreCeCi 21 21 +=

21 CC = L

EC

22 =

+=1r =2r

L

R

2=

LCL

R 1

2

=

Case 2 Critically damped case

whenLCL

R 1

2

2

=

then

)( 21 tCCei t +=



14/22

01=C L

EC =2

L

R

2=

Case 2 Underdamped case

whenLCL

R 1

2

2

fr, Z is inductive. 2= (I Lags E).

Quality Factor (Q) of a resonant circuit:

RofpowerActive

CorLeitherofpoweractiveQ

___

______Re=

C

L

RR

X

R

XQ CL

1===

Resonant Rise in Voltage

QEVV CL ==

Bandwidth (BW) is the range of frequencies overwhich the operation is satisfactory and is takenbetween two half-power (3dB down) points.

QfffBW r== 12

If Q 310; then frbisects BW

21BW

ff r= 22BW

ff r+=

Parallel ResonanceA. Theoretical Parallel Resonant Circuit

Characteristics of parallel resonance1. At resonance, BL= BC, XL= XC, IL= IC.2. At resonance, Z is maximum. Z = RP.3. At resonance, ITis minimum. IT= IRP.4. At resonance, Z is resistive. 2= 0 (I in phase

with E).5. At f < fr, Z is inductive. 2= (I Lags E).6. At f > fr, Z is capacitive. 2= + (I Leads E).

Q of a Theoretical circuit:

L

CR

X

R

X

RQ P

C

P

L

P ===

Resonant Rise in tank current

CLTk IIQII ===tan

Bandwidth (BW)

Q

fffBW r== 12

B. Practical Parallel Resonant Circuit

Equivalent Theoretical Circuit



20/22

Impedance transformation:

Q of Equivalent Theoretical Circuit

Leq

P

X

RQ=

Q of Practical Circuit

S

L

R

XQ=

Resonant frequency (practical circuit)

L

CR

LCf Sr

2

12

1=

; if RS= 0;

LCfr

2

1=

2

2

12

1

Q

Q

LCfr +

=

; if Q 310;LC

fr2

1=

Total Impedance Z

SS RQQRZ22 )1( += if Q 310

MAGNETISM AND MAGNETIC

CIRCUITS

MagnetismCurie temperature (Pierre Curie) the criticaltemperature such that when ferromagnets are heatedabove that temperature their ability to possesspermanent magnetism disappears.

Curie temperatures of ferromagnetsFerromagnet Temperature (C)

Iron (Fe) 770Nickel (Ni) 358Cobalt (Co) 1130Gadolinium 16

Alloys commonly magnetized

Alloy Percentage Content

Permalloy 22% Fe, 78% NiHipernik 40% Fe, 60% Ni

Perminvar 30% Fe, 45% Ni, 25% CoAlnico 24% Co, 51% Fe

Coulombs Laws

First LawThe force of attraction or repulsion between twomagnetic poles is directly proportional to theirstrengths.

Second First LawThe force of attraction or repulsion between twopoles is inversely proportional to the square of thedistance between them.

2

21

r

mm

kF= (Newtons, N)

where:4

1=k 0 r=

Magnitude of the Force

sinBIlF= (Newtons, N)where: B = flux density (Wb/m2)

I = current (A)l = length of conductor (m)2= angle between the conductor and field

Magnitude of the flux surrounding a straight

conductor

r

RIllog14= (Maxwells, Mx)

where: I = current (A)l = length of conductor (ft)R = radius to the desired limiting cylinderr = radius of the conductor

The force between two parallel conductors

721 102 =d

lIIF (Newtons, N)

where: l = length of each conductor (m)d = distance between conductors (m)I1= current carried by conductor AI2= current carried by conductor B



21/22

Magnitude of the flux between two parallel

conductors

r

rdIl

)(log28

= (Maxwells, Mx)

where: I = current (A)l = length of conductor (ft)r = radius of each conductor (m)d = distance of the conductors from center to

center (m)

Magnetic Circuits

AB

=

where: B = Flux density in Tesla (T)4= Flux lines in Webers (Wb)A = Area in square meters (m2)

Note: 1 Tesla = 1 Wb/m2

Permeability

m

Hor

meterAmpere

Weber

= 70 104

Note: += +0; +r= 1 5nonmagnetic+< +0; +r< 1 5diamagnetic+> +0; +r> 1 5paramagnetic+>> +0; +r>> 1 5ferromagnetic (+r3100)

A

L

=

where: = reluctanceL = the length of the magnetic pathA = the cross-sectional area

Note: The t in the unit A-t/Wb is the number of turnsof the applied winding.

Different units of Reluctance ( )

a.)Weber

turnAmpere b.)

Maxwell

turnAmpere

c.)Maxwell

Gilbert d.)

Weber

Gilbert

Note: 1 Weber = 1108maxwells1 Gilbert = 0.7958 ampere-turns1 Gauss = 1 maxwell/cm2

Ohms Law for Magnetic Circuits

Opposition

CauseEffect=

Then,

=

where: = reluctance = magnetomotive force, mmf (Gb or At) = flux (Weber or Maxwells)

Comparison bet. Magnetic and Electric Circuits

Electric Circuits Magnetic Circuits

Resistance, R (!) Reluctance, (Gb/Mx)Current, I (A) Flux, 4(Wb or Mx)

emf, V (V) mmf, (Gb or At)

Total reluctance in series

nT +++= ...21

Total reluctance in parallel

nT ++

+

=

1

...111

21

Total flux in series

nT ==== ...21

Total flux in parallel

nT +++= ...21

Energy stored

2

2

1=mW Joules

Magnetomotive force (mmf, )NI= Ampere turns, At

NI4.0= Gilberts, Gb

mmf of an air gap

0

dBmmf= Ampere-turns

Tractive force or lifting force of a magnet

=

0

2

2

1

ABF Newtons



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Magnetizing Force (H)

l

=H

l

NIH=

Note: The unit of H is At/m

Permeability the ratio of flux density to themagnetizing force.

H

B

=

B and H of an infinitely long straight wire

r

IB

2=

r

IH

2=

Steinmetzs Formula of Hysteresis Loss

6.1mh fBW = 3m

J

where: (= hysteresis coefficient

f = frequencyBm= maximum flux density

Amperes Circuital Law

The algebraic sum of the rises and drops of themmf a closed loop of a magnetic circuit is equal tozero; that is, the sum of the mmf rises equals the sumof the mmf drops around a closed loop.

0= (for magnetic circuits)

Source of mmf is expressed by the equation

NI= (At)

For mmf drop,

= (At)

A more practical equation of mmf drop

lH= (At)

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