Using Ebers-Moll Equations to Evaluate the Nonlinear

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Retrospective Theses and Dissertations

1977

Using Ebers-Moll Equations to Evaluate the Nonlinear Distortion in Using Ebers-Moll Equations to Evaluate the Nonlinear Distortion in

Bipolar Transistor Amplifiers Bipolar Transistor Amplifiers

Amanollah Khosrovi University of Central Florida

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STARS Citation STARS Citation Khosrovi, Amanollah, "Using Ebers-Moll Equations to Evaluate the Nonlinear Distortion in Bipolar Transistor Amplifiers" (1977). Retrospective Theses and Dissertations. 346. https://stars.library.ucf.edu/rtd/346

USING EBERS-MOLL EQUATIONS TO EVALUATE THE NONLINEAR DISTORTION IN BIPOLAR TRANSISTOR AMPLIFIERS

BY

AMANOLLAH KHOSROVI B.S., ~ well Technological Institute, 1974

Lowell, Massachusetts

RESEARCH REPORT

a · al fulfillment of the requirements e o Master of Science in Engineering G duate Studies Program of the o lege of Engineering of

or "da Technological University

Orlando, Florida 1977

ABSTRACT

USING EBERS-MOLL EQUATIONS TO EVALUATE THE

NO I EAR DISTORTION IN BIPOLAR TRANSISTOR AMPLIFIERS

by

Amanollah Khosrovi

e ers- oll model , which is applicable to static

o a ic cond"tions , is used as a basis for

0 ,

0 0

a ~ p l e ethod for the evaluation of harmonic

enerated in bipolar transistor amplifiers .

eq ations are transformed into the desired

ac aurin Series expansion . A computer

•tte to provide numerical results of the

t ese predictions are compared to measured

·al es.

ACKNOWLEDGEMENTS

I _.is to acknowledge Dr . R. L. Walker for his

i ce d advisement in reviewing this research

r e 0

I 0 i to extend thanks to my committee members

• • E i c on and Dr . B. E . IVlathews . I am especially

0 • E . . ·a heMS for his assistance and

e co 0 au e entire school program .

iii

TABLE OF CONTENTS

c 0 lENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... . . iii T TS ••••••••••••••••••••••••.•••.••••.••.• i v

I . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

0 PROBLE . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17

TI-lE 'ffiTHOD •••••••.••••••••••••.•••• 18

o Basic Equations ............ 18 for Collector Current ... 22

LE •• · • · • · • • · • • • · • • • • • • • • • · · • · · • • • • 35

0 0 ERRORS •••••••••••••••.••••••• 47

CLUSI ON ••••••••••••••••••••.•••.•. 49

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... . .so D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -54

0 , •................•................... 57

iv

a

0

I. INTRODUCTION

Distortion In Transistor Amplifiers

I an ideal amplifier , the amplified output voltage

0

e ersion of the input voltage without any

ap . Ho ever, in practical amplifiers

difference in the waveshapes of

o ol a· e and the applied input

ere ce is called distortion. The

e ca sed either by the transistor itself

a d c1rc "tor by both . An important

o ·sto tion is the so- called "non­

o " ·h·ch ·esults from a non- linear

o · en u e instantaneous values of the

he amplified output current . Such a

1 uwed generally by the non- linearity of

c racteristics .

In c ses here distortion is present it is con-

en nt o express the response of an amplifier using a

pe iod·c ·nput waveform. A periodic input waveform may

be e·pressed in terms of its Fourier Series components in

the 10rm:

e

=

2

n=oo

e1 = E0 + 2:: E~ Sin(nw t + q, n)

n=1

.pl ·· tu e of n h harmonic in val ts .

( 1. 1)

E = . c. co ponent of the input voltage (in volts) .

~ = w =

=

0 (

=

0

1 of the nth harmonic .

requenc of the input signal ei .

c o the input signal ei .

e s ideal , the amplitude of all

appear at the input are multiplied by

n A, and the phase angles are

p oportional to their frequencies.

pl"fier , the amplified output wave-

po n ) be expressed in the form :

e0 = A Em1 Sin( \U t + 'f + cp1 )

+ A Em2 Sin(2 UJt + 2 'f + <P 2>

+ A Em) Sin() uJt + J 'f + c.p3) + (1 . 2)

Let W t + \f' = U.J t '.

hus ,

J

+ AEm2 sin ( 2 W t • + cp 2 J

+ A Em3 Sin ( 3 Wt ' + <fl 3 ) + ... (1 . J)

ro . (l.J) it can be seen that the amplified

t o e0 as the same waveshape as the applied

, /w,

,

ol e ei ' the only difference being an increased

e a t · e delay of the wave by an amount

0 1 there is no distortion .

o the amplifier differs from the

~s p ese t . Different types of distor­

in an amplifier , either separately

eo 1 , are :

plitude distortion

uenc distortion

3· P se distortion .

t ion

ase of amplitude distortion the voltage

the amplifier varies with the amplitude of the

n u , ·.e ., the amplifier output waveform has a

non-l ·ne relationship with the applied input voltage.

An e. ample of a characteristic which results in ampli -

tude distortion is shown in Fig . 1 . 1 .

2. Frequency distortion

Frequency distortion occurs when the voltage gai n

4

c

Linear portion

input voltage

i . 1.1.Input-output characteristics of an Ideal

d Pra t cal Amplifier.

5

i h he equency of the applied input voltage .

T , the input oltage as given by Eq . (1 . 1) , ex-

pe ences a di ~ ent amplification factor dependent on

e e

e

0

J .

e r disto tion is primarily caused by the

nc of eac i e ele .. ents in the amplifier circuit .

e t

o a o

0

0

0

0

1 , plitude and frequency distortions may

c

e

o sl and may cause the output wave­

from the input waveform .

:as stated that for an ideal ampli ­

s o ld be either zero or propor-

e e c ~. If an amplifier has a phase

oportional to the frequency , phase 0

e distortion, like frequency

esults from the frequency depen­

cte istics caused by reactive elements

ssociated with the amplifier .

Sou~ces of Distortion

The sources of distortion in transistor ampli ­

r· rs a listed below:

1 . Non-constant spacing between constant- current

curves , particularly along the load line .

6

2 . on-l·near input resistance.

J . Too-l e a signal so that clipping occurs from

satur ation or cut off .

o em·n of the bias point with variation in

·ure, which causes clipping of large­

si al inputs.

t o de d- ase connection the constant-

,e e s are normally nearly equally spaced

e u

ol

0

e

0

e

e a

0

~·~~o-L.l. The presence of distortion

tra; sister amplifier by dra~::ing

g a plot of Ic versus IE . If the

~ ·ne up to the saturation region ,

o t · a f om non-equal spacing .

i tance of the transistor amplifier

n -dependent, distortion is present .

li ie~ this ·ill result in a distorted

_ d the output current will be an ampli -

o th"s distorted wave .

Di o ·an caused by clipping occurs when the

1 c· uses the amplifier to swing into saturation or

into the cut-off region . Both saturation and cut- off

will c u e the output to be clipped . If the allowable

po~. · e dissipation permits , the clipping may be remedied

by moving the operating point farther from the origin

(of the raph).

7

I he signal operates close to saturation or the

c - o f re ·on , distortion may be caused by the bias

poi s J. ing because of a change in ambient temperature.

a

0

h"rt rna· ause the signal to move into either the

0 he cut-off region .

eristics of the grounded emitter amplifier

o o as uniformly spaced as the grounded

e rounded emitter has inherently more

o be cor~ected to a certain extent

o ~ source resistance . Since the

o ·nput resistance is non- linear , as was

0

o ounded base amplifier , the source

ce osen to compensate for this non- linearity

o o o e e inherent distortion mentioned

o considerations are opposing ,

an optimum value of source resistance to

d·s~ortion . The clipping and bias sta­

pects of the grounded emitter are identical

o the grounded base.

Graphical Method For Calculating Distortion

The output current of a transistor amplifier may

be e.pressed by a Fourier Series of the form :

io

)

= Io

= I

=

=

0

00

+2 =

+ ff

e

c

8

2 r 0k Cos k w t

I 1 o s \.U t + I O 2 Cos 2 Wt ... ) ( 1 . 4)

e a e value , in amps , of i 0 when a . c .

is impressed at the input .

0 h . .

.armon~c ~n amps .

f" .al position of the a . c . load line

se the follo,ing procedure to deter-

1 . 2 sho san output character­

o . The intersections of the a . c .

ed a.c . load line (due to distor-

·stic curve corresponding to

, de er ·ne the quiescent operating point

Point A is the intersection of the

d . c . load line with the d . c . load line .

i t e harmonics higher than the fourth are

ne li · le , the output current can be written as :

+ r 03 Cos J uJ t + r 04 Cos 4 w t) .

9

p e ons for the five unknown coefficients , lOA '

r 01 , 02 , r03 , and r04 may be obtained by evaluating the

o e a

o. 0

c 00

0 0

0 . '

ol

Io

I a~

Iop

e dif erent points on the a . c . load line .

=

=

=

he a . c . input signal is assumed to be

inp ~ bias to be xc . The input signal is

+ 2 A X Cos w t c

0 i of t e time axis so that it

e val- e of the input signal .

w t as 0 , TTt J , lT/2 , J , andTI;

s o x are : xc + 2 ~x , xc + l:l x ,

- 2 6 The corresponding values

ed b lomax , l ao< , lop , lop , and

ted in Fig . (1 .2) .

ese values for UJ t and i 0 into Eq . (1 . 5) ;

i e s1multaneous equations may be written :

1oA + f2<1o1 + 1o2 + 1oJ + 1o4>

1oA + fi< 1o1 - tro2 - iiOJ - -l1o4>

1oA + f2< - 1o2 + 1o4>

00

0 .

10

x -AX c

The generalized output characteristics

~1) th d . c . load line , (2) the a . c . load line ,

( ) the . c . load line sho ing the shift due to

d. to t"on .

o e

11

Q5 = 10A + i2(- 101 - 2102 + 1oJ -~!04)

1om·n = 1oA + 12(-!01 + 102 - 10J + 1o4)

e e a o s ; the expressions for harmonic

d ·e ob ained as follo~s:

1 1omin) 1

+ Icy.g ) (1 . 6) = .., Io + + J<root. . ax

~-:- -( Io . ) 1 - IO{J ) (1 .7) = + J<ro~ J 0 n

~ ( 1omin) 1 (1 . 8) = 2 1op 0 a:·

Io . ) 1 Iop ) (1 .9 ) 0 = - -(I nu.n J ()o(

+ Io . ) - 1 + Ia/9 ) = J<Ia~ nun

1 + 2 1ap (1 .10)

ous harmonics are usually given as a per­

rna itude of the fundamental term . Thus,

e c n e harmonic for second , third , and fourth

h rmon c are defined by:

0

0

he e :

e - o 1

= I

I = - ol.

12

DJ = !02 X 100% 1o1

D = 1o4

X 10()% . 1o1

Ebers-Moll Equations

- all ations a e for low frequency

.e ra sistor and are derived 0 0

1 0 1 io . For pnp transistors ,

i ons are gi- en by equations 1 .11 and

( qv. I

- 1) - o( RICS (e CB KT - 1)

(1 . 11)

(eq .EB/ KT _ 1 ) + I (eqvCB/KT _ 1) ES CS

(1 . 12)

IES = Emitter saturation current when collector is

shorted to the base .

Ics = Collector saturation current when emitter is

1J

horted to the base .

O( F = Fo ~' rd short-circuit current gain .

D( = de rsed short- circuit current gain .

J .

fo constants depend on the diffusion

, t d"ffusion lengths , the equilibrium

oncentration , the area and the base

o s ade in deri ~ing these equations

0 ~

e s o is operated at low frequency , so

e · . ,e elements can be neglected .

o e terminal currents which change

c r iers stored in the transistor

c ed.

ndence of the base- width on junction

o t e s neglected , i . e ., base- width is

constant .

he transistor is subject to low- level

injections only .

5. Ohmic voltage drops at the contacts and in the

neutral regions are neglected .

Equations (1 . 11) and (1 . 12) apply to pnp transi s­

tors . The corresponding relationships for an npn tran-

14

i o e iren by:

0

lie

0

I

- ,

0

-q -qv I = -IES (e ~B/KT - 1) + o< RICS (e CB KT - 1 )

(1 . 13)

-q ( - qv I = o( I S ( e EB KT - 1) - Ics ( e CB KT - 1)

( 1 . 14)

l ,eq Y.ations, .'hich describe the large­

e i ·cs of the idealized transistor

a s ple and useful interpretation in

model that uses two idealized ex-

is oael is shown in Fig . 1.J . The

i erpretation in terms of the inter-

~ e ransistor . The emitter and col-

a eac be resol-ed into two components .

1 o ··s into the idealized diode is the

or -carrier injection at that junc-

o e component , which is provided by the

o· c , is the consequence of minority- carrier

c on · he other junction and transport across the

b Thus , the first component of emitter current

qv I I S (e KT - 1) results from diode action at the

emitter junction . qvCBIKT

The second current - ol RICS ( e

- 1) is the consequence of diode action at the collector

(

+ •

15

B

all static model for a pnp

-

I- I (eqv/ KT - 1) - s

•

i . 1 . (b) . Idealized pn- junction Diode symbol .

c

16

j ct · o d exis s because a fraction of that diode

en , c<R R' is transported across the base to the

e i t e , he e i contributes to the total emitter

r nt .

0

0 I

e ba·e

0

0

o(

e c

e

II. T T ,1E T OF THE PROBLEM

p· ical ethod, described in Chapter I , to

e o ic content in the output of a bi-

s·o amplifier is very laborious and cumber­

. ~ · a , to determine the a . c . component of

t (co on emitter connection) assumptions

o ~ a .p e , Vbe is assumed negligible .

1 errors in the calculations .

~e normally inaccurate .

e o ~ect is to investigate a method

e ea sonable accuracy the harmon-

0 . .put val tage . The basis for

e s- all equations .

e ions the four parameters o( F '

C.;:, ( nom as the forward short- circuit

e erse short-circuit current- gain , emitter

ent rhen emitter is shorted to the base ,

1 ) must be kno' ·n.

These fou parameters are normally measured from

th ~ tran i to characteristic curves or taken from the

ve e d ta supplied by the manufacturer .

III. D I TION OF THE METHOD

A. Derivation of Basic Equation

) . 2 ) '

= - o<. c

0 0

0

(

0

ers- moll equations given by Eqs .

o tains :

qv / - 1)- ~RIGS (e CB KT - 1)

(J .1)

KT - 1) + Ics (eqvCB/KT - 1) .

(J .2)

o o an a plifier , the emitter-

·ased and the collector- base

d i sed :

c = ( C., VBE) <- 0 . 1 volt .

t 00 empe tu e 4~T/q = 0 . 1 volt .

hu , the term eqvCB/ KT in Eqs . () . 1} and (J .2) can

be con idered ne 0 li ibly small compared to unity . Hence

qs . ( . 1) and (J .2) reduce to :

19

= q '::B/ KT s (e - 1 > + ~ R1cs (J . J)

i = - o( I S ( e q ~B/KT - 1) - lcs . () .4)

o po· t on , onl the collector current

. ) . i 1 e studied , and the corres-

e •

0 . . • J . se . ·rch off ' s voltage

0 0

+

= (J .S)

. ( J . 5) and qj KT ~ L in Eq . ( J . 4) ,

0 0

(J . 6)

Now for transistor working in the normal active r egion

the ollowin equation holds :

B B

. • •

1 E

c

E

20

plifier circuit .

qv / Ico(e CB KT- 1)

--

Fig . ) . 2 . Equivalent Ebers-Moll model of Fig . J . l .

21

e o ~ . (J .6) yields :

o( F

- 1

0 :

A 1 = - o( ES

k ~ Lico

R' LvBB + D( 2 F s

nd k ~ LRs(1 - o( F)

) o( F

B.

22

i = () . 10)

Harmonic Equation For Collector Current

0 e e armonic content , Eq . (J . 10) will

a n Series Analysis . This

ppendix A. In this work only

e 0 1 CS ill be considered , as the

0 .. 0 ~at the higher order terms are

- L + k i ) o S 3 C can be expanded as :

1 + (k2 - Lv5 + kJiC)

(k2 - Lvs + k3iC)2 + 2

2LKJ

23

sic + 2k2kJiC k2 LJVJ + kJiJ + 2 s J c

6 2

2 - L

=

+ JL2v2(k s 2 + k~ic) 6

s> - 6k2Lv5k3

i 0 6

. () . 10) , one obtains :

kJ sic + k2kJic + ~

or , in terms of the descending powers of ic :

+ (

+ 1

0 0

24

1 3 - Lk1k3v5 + k1k2kJ

2 2

+ L k1k

2v8 - k1k2k

3Lv5 - 1) ic 2

2 L2 2k + 1 2 + s 1

2

J k LJVJ 2

+ k1k2 1 s k1Lk2v8

2 6 6 2

2 s

- 1cs) 0 () . 11) =

e aclaurin Series it is neces-

e ic successively with respect to

ppenidix A. The technique is as

·n i· in ~q . (J.11) ith respect to v5 yields :

25

) die

(-L ( . ) 1 ~ s a 5

+ ~c

( -

2 1 2 s = 0 (J . 12)

op t ng point , v5=o , ic=Ic (in this work

o in pain "11 be established) . So to obtain

di lu of the derivative d C at the operating point , vs

{J . 12);

v =0 s

substituting v5=o and ic=Ic in Eq .

26

2 die L =0

~ + 2J{J + k1k2kJ

- 1) 3 1 2 dv3 s

) ,., + ( - 1 2kJL ) Ic + { - Lk1

)

2

= 1 , and collecting terms con-=0

27

+ Lk1 Lk k

2]

+ Lk k + l 2 1 2 2

0

!2 + c LklkJIC + k1k2k3LIC

!2 + 2 2 c (klkJ + k1k2kJ) Ic

k1k~L} + 2

{J . lJ) 2

-1J .. ? ...... + 1k2k

3 .)

e

= = constant coefficient of the =

s o as indicated in Appendix A.

= o c rent (d . c . ) at operating point ,

o s ants given by Eqs . {J .?) , (J .8) , and

( J .

s· e differentiation must be taken of Eq .

(J . ) o e A2 , the harmonic coefficient of the

n Se ies e~pansion .

Di ntiating once again yields ,

v

[·

J

2 +

28

d. k LJ - + - ) + L2k1 - 1 2vs

d s

= 0

op

nd u titut·

ing point , i . e ., when v5=0 ,

d2. ~c

2 dv5 v =0 s

= A2

yields ,

29

s e final equation for A2 :

- (

2 L · 2Ic 1 2 - L 1 - k1k

{J .15)

h

=

e

oll o

d2. ~c

d 2

s=o

JO

= a coefficient of the Maclaurin

o as indicated in Appendix A, and Ic =

(d . c . ) at the operating point .

e i lds c . ( J . 16) :

d2 ' di d. 2 ~- + ~-c 2i _Q + ( ~c)

d .. C dv dv s s s

[ dJ. d2. d2. die 1 ~c + 2ic

l.c + vs

l.c vs2ie J

dv2 2 dv5 dv5 s dv5

die d2. die 2 d2. ~e

2 + 4ie ~e

+ 2v52- 2 + (- ) 2 dv5 dv5 dv5 dv5

die die ] dJ. l.e

(k1kJ + kl k2k) + 4-- + dvJ dv5 dv5 s

( - 3 )

J

J J =

31

(3 .16)

= ) , 5=0 (at the operating point) ,

i el ds Eq . ( 3 . 1 7 ) .

f> - (k1k~ + k1k2k~) (JA1A2)

+ A~) + )Lk1k~2 - JA1L2

k1kJ

+ kL3

(3 . 17)

0 ed· 0

oe . '

i

+ vs

32

d4. ~c

4 = 4 dv5

, the Maclaurin Series

dJ. die d2. d2. +

~c 2ie 2

~e ic

~e

3 -+ 2 2 d s dv5 dv8 dv8

c die die dJ. die 2 ~e --+ 2i -

dvJ + 6( - )

d s C dv dv8 s s

di dJ. +·4i _Q

J.e c e dv dvJ

s s

dJ. d. d2. d2. d. dJ. c I ~e ~c ~e J.e ~e 1e

+ -+ dv2 2 +- J J dv8 dv8 dv8 dv8 dv8 s

d' dJi d2. d2. 3[ d4. ~e J.e k 1Lk) . J.e

+ 2 2 2 ) - 2 vs21e 4 d s d s dv5 dv8 dv8

d). die dJ. dJ. c

2ic J.e

+ 6ic J.e

2-+ dvJ dvJ dv5 dv8 s s

+ 0

dlc

s

33

d2. d2. d. d3· di~ , ~c ~c ,. 1 c 1 c

_.J + ovs 2 2 + ovs - J d-s

d2. c

d-2 s

- 1)

+

dv5 dv5 dv5 dv5

] die d2ic

+ (k1k3 + k1k2kJ + 12 - 2 dv5 dv5

d4. ~c

+ (-Lk1k3

) dv4

s

2 21kJ [ d4.

? l.c 4 + s dv5

d . d3· 1 c 1 c --+ ) =0 d s dv~

[ vs d4. dJ. l.c l.c

4 + dvJ dv5 s

d3· d2. l.c l.c 2v5 +

dv3 2 s dv5

= A4 , v5 = o and ic = Ic (at the

o t n point) and rearranging the terms yields

. {J .18):

6

)4

J ( 2 6 2 2 2 J C .1 J + 3IcA2 + A1A2) - (k1kJ + k1k2kJ)

( ? 2

1 ~J + J 2 + 4k1Lk) (ICAJ + JA1A2 ) + 4AJ (LklkJ)

1 · J + (k1k2kJ ) 4AJ

= ----------------------------------------------

2 I

().18)

o e ~ . (A.4) from the Appendix A,

=

Co

().19)

= - alue of the input signal. The values of

e co i t A1

, A2

, A3

, and A4 are evaluated from

o · alues of the Ebers-Moll parameters ( o< F , o< R, ,

, nd Ic ); the harmonics can be determined from Eq.

( J .19).

0

0

0

IV. N MERICAL EXAMPLE

shovn in Figure 4 .1 . To check the

i entally measured values of VBB ' RB '

. e e substituted in Equation (J .19)

t , and a program was run on the

e program is shown in Appendix B.

es of c<. R, o( F, IES , and Ics were found

0 . •

pter III, for the normal active

to nt Ic is given by :

a d in"tion ,

[VeE = canst . 0 l normal gain region

= o( F ll ~ F 1- ~ F

( 4 .1 )

J6

=10KJ1. B

= · .535

• . 1 . The basic amplifier circuit •

37

, {J F can be found by noting the average spacing

o he o 1-emi tter output curves in the normal region .

e 1" ' o( F can be calculated from Equation (4 . 1) .

e p n of the common-emitter output curves in

o c i e region as found using a curve tracer .

1 e sur,ements ere taken the average value

0 a o p ed and was used in the experiment .

(J = 11.5

= fJF

1 = 0 . 92 + (A F

e ermined by noting the average

0 o -e "tter output curves in the inverse

0

[ CE = const . ) 0 l = ~1 __ 1_o(_R

·n erse-gain region

ut he , for a more accurate result , the emitter and

collector leads were interchanged (collector grounded) ,

and by noting the average spacing of output curves in the

0

s 10

)8

on of the inverted connection , the value of ~ R

d as :

o(. R =

1- ci.. R 1.5

saturation current parameters , IES

e e ed most accurately by direct

o ransi stor . The technique is to let

0 , · ·c· reduces the Ebers- Moll equation

es o Ic or different values of VCB were

o T n Ics as determined by plotting a graph of

Ic us VCB for VCB~0 . 1 , and by extrapolating back

To determine the corresponding reverse parameter

ICS ' on setting VcB=O the emitter current becomes:

39

qVEJ!KT IE = IES (e - 1)

A en ioned in chapter III , the four parameters

ollo ·ng equation.

(4 . 2)

, ·e need to know only either Ics or

e i e ·.as performed to determine Ics . Then

) .as plotted and the value of Ics was

o t intercept of the ordinate . Then using

( , I~s as calculated .

= 8.9J X 10- 9 A

Th e.:perimental results , which were obtained

u in H lett-Packard Model J02A wave analyzer (Fig . 4 . 2)

are shown in Table 4 . 1(a) and 4 .1(b) . The results using

40

t . 0.

0 r-i J

0

.J .4

vCB in volt-..

or

or

Fi0

• 4.1. The graph log of Ic versus vcB ·

HP

0

2A

Sig

nal

Gen

erat

or

c·

cu

Und

e T

W

ave

An

aly

zer

Fig

. 4

.2.

Har

mon

ic D

isto

rtio

n M

easu

rem

ent.

(

CE

0 c

0 c

3rd H

m

onic

s

v ee

in

(d.e

.)

eel

n in

er-

in

in p

er-

rms

vo

lts

in v

olt

s o

lt

rms

e n

ta

e rm

s ee

nta

ge

vo

lts

of

V eel

vo

lts

of

V

1 ee

+=

-l\

)

0.9

212

28.0

11

.6

0.5

8 4

. 96%

0

.084

O

.?J%

1.2

783

28.1

5 1

.14

6.?4

%

0.2

J 1.

J4%

)

0 0

vs

Ic

( . c

.)

01

0 J

d m

onic

in

( u

) u

) (p

e k

· lu

e)

VRL

v v

rms

vo

lts

Ic =

RL

1c1

=

i2

c2

=

Ic

.J =

ce

x

2 R

~

\.....

)

0.9

212

2.1

0 rnA

1

.10

rnA

0.0

54 r

nA

0.0

08

rnA

1.2

783

2.0

9 rn

A 1

.60

rnA

0.1

1 rn

A 0

.022

rnA

44

the measured para~e~ ers i n t he co mputer program a re

sho~n in Table 4.2. h co mparison of th e harmonic ralues

1s ;iven in Ta bl e 4.~.

0 Q

_ I

0

0 c

s 3

rmon

ic

in

pea

k

(d.c

.)

0 c

-rn

A in

rnA

in

per-

rms

val

ue

(Pe

k c

nta

e

(Pea

k ce

nta

ge

vo

lts

of

V8

in A

mp

Ic1

n

rnA

v

a t

) 0

Ic1

v

alu

e)

of

rc1

+=-

\.1\

0.9

212

1.)

02

6

2.J

9

1.2

7

0.0

64

5.

06%

0

.006

4 0

.51%

1.2

783

1.8

090

2.J

9 1

. 71

0.1

30

7.4

8%

0.0

17

1.0

9%

0

J

3

46

TABLE 4 . .)

S - 0 THE RESULTS OF THE EXPERIMENTAL

D COMPUTER METHOD

c

( . .

l.C

. c. )

onic

d armonic

d Harmonic

Experimental

ethod

2.100 mA

1.100 mA

0 . 054 mA

0 . 008 mA

2.09 mA

1.60 mA

0 . 11 mA

0 . 022 mA

Computer

Method

2 . .)941 mA

1.2700 rnA

0 . 0640 rnA

0.0064 mA

2.)941 mA

1.707 mA

0.1J rnA

0.017 mA

EST! TED SOURCE OF ERRORS

~~~~~~~~~o~d: The Hewlett-Packard Model J02A wave

e

0

·o 0

e

fo determining the harmonics has a

0 of less than 1%. The sensitivity of the

0 JO ,. v to JOO volts. Since the harmonic

sing the Ebers- ~oll equation , and

0 e not expected to approach the

t the experimental results have , these

e a ill be considered the standard for

_c rae of this method depends on (1)

o e bers-1oll model, (2) accuracy of the

e or measuring the voltages), and (J)

o e ssumptions and approximations made in

o he expression for harmonic evaluation .

o sl s~ated in the derivation of the Ebers­

the effect of the space-charge-layer

. on o( F, o( R, IES and Ics has been ignored .

cond , ~ F and o( R were considered to be independent of

u r nt ~n the model. In actual practice this is not the

e. So these assumptions will cause some error in the

results. Experience has shown that these errors are of

48

1 o o na nitude of 5%.

In de · in the computer method two approximations

e e (1) VCB~0 . 1 volt , where for the most part ,

0

a d- of this voltage is much greater than 0 . 1

e ~a can be considered small . (2) The expan­

o p ( 2 - L S + KJiC) was terminated after the 4th

e

e

0

0

circumstances , this could introduce

o·e er , previous experimental work

oi as and load networks are linear

ics higher than the Jrd do not

t changes in the waveshape . In

o e literature and experience with this

0 e o der of 10~ can be expected

e

0

·I. SU~~RY ArD CONCLUSION

bas·c objective of this work was to obtain a

o e 1 a in5 the harmonic distortion generated in

0

-ansistor amplifiers . Ebers- Moll equations

e o

0 f(

e - o 1

0

~ ... - ,- ···-1 .

basic equations for developing this

tions ··ere then transformed into a

.e ~ claurin Series expansion .

o · e .alue of harmonics that will be

en applied input signal , knowing the

~ I s and Ics (the four parameters of

a · ~ s), can be determined .

a 2N12J4 transistor , the four

ed e~perimentally , and harmonics

the derived method . The computer

·n in ppendix B. An experiment wa s

onics v·ere measured applying the two

. 1 the esul s of the two methods (computer and

· ent 1) were compared as shown in table 4 . J .

n u.wary , considering the experimental errors and

th assu ptions made to derive the computer method , the

results a e in excellent agreement .

( ) =

= ) :

PPENDIX A

e e sa function Z=f(x) , it can be expanded

) + ~ t ( 0 ) + X 2 f" ( Q) 2!

n :r(n) (0) + • • • + X n!

e e i 1 es are taken at point x=O .

is a ction ic=f{v8) , {Eq . J . 10) , we can

)

o bout the operating point (v8=0 ,

v =0 s v =0 s

v =0 s (A . J)

but ·0

= C + ic(d .c. component+ a . c . component) but

Ic=f(O) in Eq . (A . J) . So Eq . (A . J) reduces to

( . .

0 =

here

51

omponent)

v =0 s

di and A for _Q

f &a J 4 dv5

. , 0 0 ins:

=

o e s are retained .

0 w

1 + Cos 2 UJ t 2

v =0 s

v =0 s

v =0 s

v =0 s

=

0

52

0 J V-J t = Cos 3 t + )Cos t 4

Co 4 \U t = Cos 4 w t + 8Cos2 w t - 1 8

= Cos 4\IJt + 4cos 2UJt + 3 8

al e in above equation for ic yields :

A2

: A vJ o w + ~- + 4 Cos 2Wt + ~ Cos J W t

0

A 4 m

w t + 24x8 3 ·

ou th and higher order terms (terms

os 4 UJ t and higher) yields :

(A .4)

53

Co e · c · ents of Cos w t , Cos 2 UJ t and Cos 3 wt in

P e eses represent fundamental , second , and third

ic espectively . The last term (in the brackets)

l e e ectified a . c . component . Hence , the first ,

0 i d 1armonics are given by :

J = A1v + AJ;m

4 +

A4vm c 2 = 48

J m

cJ =

0 0 =

APPENDIX B

I I e p o r mming data were taken as follows :

L = L = I = )8 .46 volts- 1 (assumed)

= · 5.35 ol (Fig . 4 .1)

= = 10600 ohms (Fig . 4 .1)

= = -9 ( a e J?)

= = 1 -9 amperes (page J9)

0 = 0 = 33 " 10-9 amperes (page .39)

= = J .6 .. 10-9 amperes (page J9)

. J = c = 0 t i d from eq . () . 10) for V =0 . s

D D

55

PROGRAM FOR EVALUATION OF HARMONIC I TORSION IN TRANSISTOR AMPLIFIER

) ) .

I

-. OI*RSt AF / AF

. 0 =2 .O*R) / 27 . 0

3· )/27 . 0) . / J . O)

_(52 )**(1.0/ J .0))*( - 1 .0)

C/ 2.0+(1 . 0+T2)*(1 . 0+TJ*QC)

/ 2.0+(1 . 0+T2))+(T1*TJ)*(1 . 0+T2

1 J ' ) () . L*( C*AL(2)+AL(1)*AL(1) - TJ*{J . O* ) ( )+AL(1)*AL(1)*AL(1)) - J .O*AL(1)*AL(2)

- ( . ) .0 ABL*T1*TJ*(AL(2) - AL(1)*ABL+AL(2)* ( J .O) )

I D . L~TJ*TJ*{QC*AL(J)+J . O*AL(1)*AL(2)) -T1 * ( J -' . 0) * ( AL ( J) * C * AL ( 1 ) *4 . 0+ 6 . 0* AL ( 1 ) * AL ( 1 ) *

{. ) . 0* *A ( 2) *AL( 2)) -T1 *TJ*TJ*( 1 . 0+T2) *( 4 . 0* (1) · A ( )+ . O*AL(2)*AL(2))+ABL*T1*TJ*(4 . 0*AL(J)

- . 0 ABL*AL(2)+4 . 0*AL{J)*T2) A ( )= N t AD1 005 =1 , V =V (J) _IC=1 . 0*AL(1)*VM(J)+AL{J)*({VM(J)**J . 0) / 8 . 0)

BIC=(AL(2)*(VM(J))*(VM(J)) / 4 .0)+AL(4)*((VM(J)) **4 . 0)/ 48 . 0

CIC=AL{J)*((VM(J))**J . 0) / 24 . 0 DIC={(((VM(J))**2 .0)*AL(2)) / 4 . 0+(AL(4))*((VM(J)**4 . 0)/

64 . 0)-2 . )94

56

5

= .JO

. - 0 . - 02 , - 0 . - OJ, = 0 . - 0

( . ) = . 3 1

= . JO

-o .- . = =

. ) = .J

3

5

LIST OF REFERENCES

, Paul ., and Zemanian , Armen H. t o ·cs. McGraw- Hill Electrical and

Electronic ~ngineering Series . New York : ·-ill Book Company , Inc ., 1961 .

o . , • S. Principles of Radio Engineering . New o craw-Hill Book Company , Inc ., 19)6 .

. , and Searle , Campbell L. Electronic r~nc~ples Physics, Models, and Circuits . New

John Wiley & Sons, Inc., 1969 .

, o , d Halkias , Christos c. Integrated ~lec~ronics: Analog and Digital Circuits and S stems New York: McGraw- Hill Book Company ,

72.

, 1 P ., and Riddle , Robert L · ransistor Physics and Circuits . 2nd Ed .

lewood Cliffs N J.: Prentice- Hall , Inc .

oothroyd , A. R.; Angleo , E . J ., , Pa E . ; and Pederson , Donald 0 .

ementary Circuit Properties of Transistors . ol. J. New York: John Wiley & Sons Inc .