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University of Central Florida University of Central Florida
STARS STARS
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
Part of the Engineering Commons
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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 .