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A study of negative-impedance converter compensation
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Authors Krohn, Howard Emil, 1939-
Publisher The University of Arizona.
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Download date 12/06/2018 07:01:21
Link to Item http://hdl.handle.net/10150/319655
A STUDX OF NEGATIVE-
IMPEDANCE COl'JVERTER C014PESSATIOE
by
Howard B* Krohn
A T h e s is S u b m it te d to t h e F a c u l t y o f t h e
DEPARTMENT OF ELECTRICAL ENGINEERING
In P a r t i a l F u l f i l l m e n t o f t h e R e q u ire m e n ts F o r t h e D eg ree o f
MASTER OF SCIENCE
In th e G ra d u a te C o l le g e
THE UNIVERSITY OF ARIZONA
1963
STATEMENT BY AUTHOR
T h is t h e s i s h a s been s u b m i t t e d i n p a r t i a l f u l f i l l m ent o f r e q u i r e m e n t s f o r an a d v an c ed d e g re e a t th e U n i v e r s i t y o f A r iz o n a and i s d e p o s i t e d i n 'th e U n i v e r s i t y L ib r a r y to be made a v a i l a b l e to b o r ro w e rs u n d e r r u l e s o f t h e / L i b r a r y . ' .. -
B r i e f q u o t a t i o n s from t h i s . t h e s i s , a r e a l l o w a b le w i t h o u t s p e c i a l p e r m i s s i o n , p r o v id e d t h a t a c c u r a t e a c k now ledgm ent o f s o u r c e i s made. R e q u e s ts f o r p e r m is s io n f o r e x te n d e d q u o t a t i o n from o r r e p r o d u c t i o n o f t h i s m a n u s c r ip t i n w hole o r i n p a r t may be g r a n t e d by th e h e ad o f t h e m a jo r d e p a r tm e n t o r t h e Dean o f t h e G ra d u a te C o l le g e when i n t h e i r ju d g m e n t ' t h e 'p ro p o se d u s e o f th e m a t e r i a l i s i n t h e i n t e r e s t s o f s c h o l a r s h i p . In a l l o t h e r i n s t a n c e s , h o w e v e rs p e r m i s s io n m u st be o b t a i n e d from th e a u t h o r .
APPROVAL BY THESIS DIRECTOR
T h is t h e s i s h a s b e e n a p p ro v e d on th e d a t e shown below g
{%LAWRENCE. P» HUBLSiiAN
A s s o c i a t e P r o f e s s o r of" E l e c t r i c a 1. tin g i n e e r in g
D ate
11
ABSTRACT
.A N e g a t iv e ' Im pedance C o n v e r te r (NIC) i s p r e ~ .
s e a t e d and . t h e n a n a ly z e d i n te rm s o f i t s h p a r a m e te r s <,
The NIC i s to be co m p e n sa te d so t h a t an im p e d a n c e ? Zjjj .
can be t r a n s f o r m e d a s c l o s e l y a s p o s s i b l e i n t o a
n e g a t i v e im pedance^ ~Z^». T h is Im pedance t r a n s f o r m s -
t i o n w i l l b e - p o s s i b l e i f t h e h p a r a m e te r s o f t h e NIC
a r e | h ^ i d h p g d 0 9 and h - ^ ^ p l ^ The a c t u a l h. p a r a
m e te r s o f an u n co m p en sa ted MIC a r e fo u n d to b e / d .1?
h l l » 0 , h 2i < Ix, and h g g 7 0 > o v e r a f r e q u e n c y r a n g e from
500cps to 20Kcps= I t i s fo u n d t h a t h g g and h ^l; c an b e
made a p p r o x im a te ly e q u a l to 0 an d 1% r e s p e c t i v e l y ^ - by
p r o p e r l y c h o o s in g th e b i a s i n g r e s i s t o r s and by a d d in g
c o m p e n sa tin g e le m e n ts to th e NIC® -
AGKN0WK3DGMEHT
The a u th o r w is h e s to e x p r e s s h i s s i n c e r e g r a t i t u d e
to D r* Lawrence P e . HUelsman f o r h i s lo n g h o u r s o f p a
t i e n c e and g u id a n c e »■ W ith o u t h i s h e lp t h i s t h e s i s c o u ld
n o t h a v e been p r e s e n te d *
t a b ib o f c o i m r a
GHAPIER 3
GHAPOSR k
>00000 0 60
> 0 0 0 0 0 0 0 0 0 0 6 0 0 6 0 0 6 0 0 0
0 0 0 0 0 6 0 OOO
X » » o 6 o o o o o e o » o » » e o e » o o e e o
GHIPYER 2 110 P r o p e r tie s» .« s.
2»-X" Bie Types o f l i e *s »»„ „ ®»»„
2«>2 H ecessary and S u f f ic ie n tGOndi felons. » o » o 6 e o o » o o » o o o e e 6 o o e
2 .3 The 1IC as a R eciprocal D e v ic e .
2.4.. Condensation- o f the lie,A P r a c t ic a l H1C<
3 .1 In tr o d u c tio n .. . . . . . . .
3 . 2 A P r a c t ic a l 32?IG C i r c u i t . . . . . . .
3®3 The'D arlington P air E quivalent CnroU-&»t... . . . . . . . . e . . . . . . . . . . . .
3 .4 A nalysis o f the Three T ransistor TKf 10 . . . . . . . . . . . . . . . . . . . . . . . . . . .
HIG Compi3nsati.on.. . . . . . . . . . . . . . . . . .
4a 1 production . . . . . . . . . . . . . . . o . . .
4 .2 The E ffe c t o f R , and Ry on hgg.
4 .3 Compensation fo r hgg 0 . . . . . . .
4 .4 Compensation fo r hg^ 1«,© 0 0 6 666
' 4 . 5 V e r i f i c a t i o n o f C om pensa tion P r o c e d u r e s <©0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0
0* O0i3.0 X.LLS XOZXS 6 6 0 6 0 0 0 0 0 6 6 0 0 0 0 0 6 0 6 0 6 0 9 0 0
©0 9 6 0 &6 0 a e 6 S > * 6 e 9 ©e 9 e » 6 0 6 6 6 0 0 6 ©6 ©6 ©6 6
C h a p te r 1
INTRODUCTION
In r e c e n t y e a r s th e n e e d l ia s a r i s e n f o r s m a l l e r
and m o re 'c o m p a c t n e tw o rk s . A n e tw o rk t h a t i s to be
com pact s h o u ld n o t c o n t a i n I n d u c to r s , s i n c e th e y hove
th e d i s a d v a n t a g e o f l a r g e s i z e and w e ig h t . The in d u c
t o r s fo u n d i n p a s s i v e n e tw o rk s a r e g e n e r a l l y u se d w i t h
r e s i s t o r s and c a p a c i t o r s to r e a l i z e com plex p o l e s .
P o s i t i v e an d n e g a t i v e r e s i s t a n c e and c a p a c i t a n c e can
a l s o be u s e d to r e a l i z e com plex p o l e s . I n d u c t o r s may
t h e r e f o r e be e l i m i n a t e d f ro m n e tw o rk s t h a t a r e t o be
com p ac t, i f n e g a t i v e c a p a c i t a n c e and r e s i s t a n c e i s
a v a i l a b l e . The N e g a t iv e Im pedance C o n v e r te r (NIC) i s
a d e v ic e t h a t can be u s e d to o b t a i n n e g a t i v e c a p a c i t a n c e
and r e s i s t a n c e .
The NIC u se d i n t h e r e a l i z a t i o n o f n e g a t i v e c ap ac
i t a n c e and r e s i s t a n c e i s n o n - i d e a l i n th e s e n s e t h a t i t
d i f f e r s f rom t h e . t h e o r e t i c a l NIC. B ecause o f t h i s n o n -
i d e a l n e s s , d i f f i c u l t i e s a r e som etim es e n c o u n te r e d in
d e s i g n in g a c i r c u i t u s in g an NIC. I t i s t h e p u rp o s e o f
t h i s p a p e r to d e v e lo p m e th o d s , w h ich may be u s e d to make
a n o n - i d e a l NIC a p p e a r i d e a l i n a s p e c i f i e d f r e q u e n c y
1
2
ran g e*
The d e v e lo p m e n t o f t h i s p a p e r v ti .l l p r o c e e d a s
fo l lo w s g NIC th e o r y w i l l be d i s c u s s e d ; a g e n e r a l com
p e n s a t i o n m ethod w i l l be p r e s e n t e d ; a p r a c t i c a l NIC
c i r c u i t w i l l be p r e s e n t e d and a n a ly z e d ; and th e ETC
w i l l be c o m p e n sa te d b o th t h e o r e t i c a l l y and e x p e r im e n t
a l l y and th e r e s u l t s , com pared .
C h a p te r 2
H ie PH O PIH iES
2 .1 TIE TWO TYPES 0 ? m C C 1 >;i
/ill ITIG i s a t\70 p o r t d e v ic e t h a t e x h i b i t s t h e
f o l l o v i n g p r o p e r t y : I f o ne p o r t i s t e r m i n a t e d i n an
im pedance Z^, th e n th e im pedance s e e n lo o k in g i n t o th e
se co n d p o r t i s -k Z p , w here k i s some c o n s t a n t , g r e a t e r
t h a n z e r o , w hich may be f r e q u e n c y d e p e n d e n t .
C o n s id e r t h e im pedance t e r m i n a t e d two p o r t n e t
work shown i n F i g u r e 2 . 1 .
F i g u r e 2 .1 . An im pedance t e r m i n a t e d two p o r t n e tw o rk .
The r e l a t i o n be tw een V-,, V and I 0 may be
g iv e n i n te rm s o f t h e t r a n s m i s s i o n m a t r i x
A S
C D -Iz .( 2. 1)
w here A, E, C, and D a r e assum ed t o be r e a l
3
U sin g ( 2 . 1 ) , t h e im pedance s e e n lo o k in g i n t o p o r t
1 , when p o r t 2 i s t e r m i n a t e d i n im pedance i s fo u n d
t o be
zr.n = '1 ; 2u- ( 2 .2 )
i f A a D a 0•
From ( 2 .2 ) i t i s a p p a r e n t t h a t Z ^ w i l l e q u a l
-k Z L, w here k i s r e a l , i f e i t h e r B o r C, b u t n o t b o th ,
i s n e g a t i v e . T h e re a r i s e two c a se s* (1 ) I f B 0 and
C > 0 , th e n s' - | b | v 2 and s: - \C 112 . F o r t h i s c a s e
a s p e c i a l name i s g iv e n to t h e NIC. I t i s c a l l e d a
V o l t a g e N e g a t iv e Im pedance C o n v e r te r (VNIC). (2 ) I f
B > 0 and C < 0 th e n s' |b| V2 and Iq ~ \c\ Ip* For t h i s
c a s e t h e NIC i s c a l l e d a C u r r e n t N e g a t iv e Im pedance
Conv e r t e r ( INIC) .
2 . 2 SSCSSSAITi: A:ID SPFyXGIsra COIIDIxXOIIS^
The N e c e s s a ry and s u f f i c i e n t c o n d i t i o n s f o r a
two p o r t d e v ic e to be an NIC can be s t a t e d i n te rm s o f
t h e h p a r a m e te r s o f t h e two p o r t . The h p a r a m e t e r s o f
a two p o r t a r e d e f i n e d by t h e m a t r i x e q u a t io n
V, Vmi Uij. 3,"3 , k n ku
w here t h e h p a r a m e t e r s a r e a f u n c t i o n o f t h e com plex
f r e q u e n c y v a r i a b l e s .
U s in g ( 2 . 3 ) , t h e i n p u t im p e d an c e , Z ^ , o f t h e
im pedance t e r m i n a t e d tvjo p o r t n e tw o rk shown i n F i g u r e 2 .1
i s fo u n d to be (
z ,h = k " ' / Z u ( 2 .4 )
I f i s to e q u a l -kZjg, h n and h 22 m ust e q u a l
z e ro and h ^ ^ g i ^ u s t be g r e a t e r th a n z e r o . I f h]_i and
h 22 a r c z e r o , k r i l l be e q u a l to h ^ ^ g i * I t i s con
c lu d e d t h a t th e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s
f o r a two p o r t d e v ic e to be an IIIC a r e
t:t : ; »The two c a s e s g iv e n i n s e c t i o n 2 .1 may now be
i d e n t i f i e d . From ( 2 . b) i t i s a p p a r e n t t h a t h ^ 2 and h 2^
m ust h av e t h e same s i g n . V.'hen h - ^ and hg^ a r e n e g a t i v e ,
i s a n e g a t i v e m u l t i p l e o f V2 h e n c e th e two p o r t i s a
VIIIC. When h%2 and h 2% a r c p o s i t i v e , 1% h a s t h e same
s i g n a s I 2 t h e r e f o r e th e two p o r t i s an Ih lG .
A s p e c i a l c a s e a r i s e s when h 22 = h q i = 0 and
h i p l i p i = 1 . Under t h e s e c o n d i t i o n s t h e im pedance se en
lo o k in g i n t o one p o r t i s th e e x a c t n e g a t i v e o f t h e im
p e d a n c e s e e n lo o k in g i n t o th e se co n d p o r t , i . e . , k » 1
and Z in = -Z-g. Under t h e s e c o n d i t i o n s th e NIG i s s a i d
to be i d e a l .
2 .3 2 : 3 I.IG AS A 'J -^ IC S 2
In t h i s s e c t i o n i t w i l l be shown t h a t t h e i d e a l
NIC i s a r e c i p r o c a l d e v i c e , i . e . , t h e im pedance se en
6
l o o k in g i n t o p o r t 1 when p o r t 2 i s t e r m i n a t e d i n ZT i sJjth e same a s th e im pedance se en l o o k in g i n t o p o r t 2 when
p o r t 1 i s t e r m in a t e d i n Z - .
L e t h-, 2ll21 = le From ( 2 . 4 ) , Zj_a , t h e im pedance
s e e n l o o k in g i n t o p o r t 1 when p o r t 2 i s t e r m i n a t e d i n
Z^, i s -Z-g, s i n c e h ^ = hgg = 0 , I f p o r t 1 i s t e r m in a t e
ed i n an im pedance Z^, th e n Z0 , t h e im pedance se e n lo o k
in g i n t o p o r t 2 i s
From ( 2 .6 ) i t i s a p p a r e n t t h a t Z0 = -Z ^ when
h n = h 22 = 0« S in c e i t h a s been shown t h a t = Z0 ,
i t i s c o n c lu d e d t h a t an i d e a l NIC i s a r e c i p r o c a l de
v i c e .
2 A coM?r:isAT'ij;T o r i u c 2
/m o u t s t a n d i n g f e a t u r e o f th e NIC i s i t s a b i l i t y
to co m pensa te f o r i t s own n o n - i d e a l n e s s u n d e r c e r t a i n
c o n d i t i o n s .
I f h i ^ » 0) h.22 £ 0 and h ^ g h p l = F, t h e i n p u t
im pedance o f t h e NIC i s
S h u n t in g p o r t 1 w i th a d m i t t a n c e r e s u l t s i n a
new in p u t im pedance Z * f •
From ( 2 .8 ) i t i s e a s i l y se e n t h a t Z^r 1 i s t h e
n e g a t i v e o f Z& when Xj. = h 2 2 » Under t h i s c o n d i t i o n th e
* -h ’/ Zu ( 2 .7 )
( 2. 8 )
NIC i s i d e a l .
I f *122 = 0 , h-[ i tf 0 , and hj_2^21 - f > th e n t l ie i n
p u t im pedance o f t h e NIC i s
^ M ~ I'M.I - Z.U ( 2 .9 )
A ddins im pedance Zg i n s e r i e s w i th w i l l change
th e i n p u t im pedance to Z ^ . f 1.
2 m = k i i - Zu- 2 i ( 2 .1 0 )
From ( 2 .1 0 ) i t i s s e e n t h a t t h e NIC w i l l be i d e a l
when Zg = s i n c e Z-jj-f * w i l l e q u a l -Z ^ .
From t h e above d i s c u s s i o n i t i s c o n c lu d e d t h a t an
NIC, w hich i s n o n - i d e a l i n t h e s e n s e t h a t e i t h e r h ^ i o r
h 22 i s z e r o , may be made i d e a l by t h e p r o p e r p l a c i n g o f
e x t e r n a l im pedances*
The c o m p e n sa t in g a c t i o n o f t h e NIC i s shown i n
F i g u r e s 2 .2 and 2 .3* In F i g u r e 2 .2 t h e a d m i t ta n c e se en
lo o k in g i n t o p o r t 2 i s + h ^ s i n c e h^ 1 - 0 . I f
= h 22 th e n e t e f f e c t o f t h e two a d m i t t a n c e s i s z e r o .
In F i g u r e 2 .3 , h 22 d 0 , t h e r e f o r e th e im pedance se e n
lo o k in g i n t o p o r t 1 i s h l l - Z 2-Z l,. I f h n = ^ ie I ni“
p e d a n c e l o o k in g i n t o p o r t 1 w i l l be -Z&. I n b o th c a s e s
th e two p o r t d e v ic e w i l l behave l i k e an i d e a l NIC.
3: d Sal NICi------------------------------------------------------------------------ -i i--------------— — -i II | M om X d s a l N I C I |
Ii
Port | * V,I
1-------II I_____________________ 1 1
F ig u r e 2 .2 C om p en sa tion f o r h^p>0 when b%]=0»
Z d £ a 1 M i c
h | ! 3 !ri 1. x r O
h i t-X;r j ' ' L ^
i _ ! xPort 1
Port
n - e ^
1— r
r d N I C
l \ l o n i d Cc. I M i c.
X d e,o I IM I C
=. h 3,1 - I
“ 1
-H
Z u
J J
F i g u r e 2 .3 C om pensa tion f o r 0 when l',o2 s 0 .
C h a p te r 3
A PRACTICAL BilC
3 * i httroeujction
I n C h a p te r 2 two ty p e s o f NIC? s $ th e INIC and th e
VHIGg w ere I n t r o d u c e d . O th e r a u t h o r s h a v e s h o r n t h a t
t h e H I C i s m ore d e s i r a b l e t h a n t h e VH1C =2 As a r e s u l t
m ost c i r c u i t s r e q u i r i n g 1 1 C f s h a v e u s e d t h e IN IC » The
r e m a in d e r o f t h i s p a p e r w i l l d e a l e x c l u s i v e l y w i t h t h e
INIC.
In t h i s c h a p t e r a p r a c t i c a l INIC c i r c u i t i s p r e
s e n t e d . T h is c i r c u i t i s r e p r e s e n t e d by an e q u i v a l e n t
c i r c u i t and t h e n a n a ly z e d i n te rm s o f i t s h p a r a m e t e r s .
3 .2 A PRACTICAL INIC CIRCUIT .
The c u r r e n t and v o l t a g e a t p o r t s 1 an d 2 o f an
INIC a r e r e l a t e d by t h e e q u a t i o n s 8 Vq - (B| V2 and
I j d \ 0 \ Ip® I t i s e a s i l y se e n t h a t t h e INIC i s i d e a l
i f B » O s l o An i d e a l INIC may be r e a l i z e d u s i n g one
c u r r e n t c o n t r o l l e d s o u r c e . Such a c i r c u i t I s shown i n
F i g u r e 3 . 1 .
The H IC i n F i g u r e 3 .1 c an n o t be r e a l i z e d by
e x i s t i n g d e v i c e s . F i g u r e 3 . 2 shows a more u s e f u l INIC.
An e q u i v a l e n t c i r c u i t f o r t h i s IN10 i s shown i n F i g -
10
->•4
z X
F i g u r e 3 . 1 . An i d e a l E'JIC.
*r
F i g u r e 3 . 2 . A two t r a n s i s t o r INIC.
11
k -A A A - —
3 le t
ri>°— /\ / \ / '
•>J \ / \ t e
=c,
Zc =- r c // c c□ zc
cl To. Zc
cornr.ion e m i t t e r comiTion c o l l e c t o r
F i g u r e 3 . 3 . Tv/o t r a n s i s t o r e q u i v a l e n t c i r c u i t s .
■x,
V
xt
z \ / v \O i Z yA A y x /v
=C:•4--
Q |
1
K
F i g u r e 3«*+• A two t r a n s i s t o r BUG e q u i v a l e n t c i r c u i t .
a r e 3*4* The e q u i v a l e n t c i r c u i t was draw n, u s i n g th e
common e m i t t e r e q u i v a l e n t c i r c u i t o f F i g u r e 3 . 3 , u n d e r
t h e a s s u m p t io n t h a t t h e com ponent t r a n s i s t o r s p o s s e s s
t h e f o l lo w in g p r o p e r t i e s : (1 ) The e m i t t e r r e s i s t a n c e ,
r p , and t h e b a s e r e s i s t a n c e r%, a r e z e r o . (2 ) The
c o l l e c t o r c a p a c i t a n c e , Gc , i s z e ro and t h e c o l l e c t o r
r e s i s t a n c e , r G, i s i n f i n i t e . T hese a s s u m p t io n s w ere
made t o show t h e e f f e c t o f Z^, Z^, ^ j.* p on t h e
h p a r a m e te r s o f t h e INIC#
A n o d a l a n a l y s i s o f t h e c i r c u i t i n -F igure 3*4 i s
u s e d to f i n d t h e h p a r a m e t e r s o f t h e two t r a n s i s t o r
I I I G . The h p a r a m e t e r s a r e
From ( 3 .1 ) - (3*4) i t i s c o n c lu d e d t h a t t h e two
t r a n s i s t o r I I IG i s i d e a l when z 1* (Assum ing t h a t
r e = r ^ = 0 and Zc , t h e c o l l e c t o r Im pedance i s i n f i
n i t e ) . From (3*3) i t i s s e e n t h a t h ^ w i l l a p p ro a c h
(3*1)
V'i. X( 3 . 2 )
13
Zo / Zih as ^ 2 a p p ro a c h e s u n i t y . The BIIG w i l l t h e r e
f o r e be i d e a l when Z3 =■ Zi a n d ^ o = ! •
N o rm ally t h e cs o f a t r a n s i s t o r v a r i e s be tw een . 9 ?
and .9 9 • A c i r c u i t w i th an e f f e c t i v e ^ t h a t i s g e n e r
a l l y be tw een .9 9 and 1 was s u g g e s t e d by D a r l i n g t o n .
T h is c i r c u i t i s shown i n F i g u r e 3*b .
F i g u r e 3 .6 c o m p le te w i t h b i a s i n g e le m e n t s .
L i n v i l l h a s shown t h a t N IC 's h a v e c e r t a i n s t a b i l -
F i g u r c 3 .6 show t h a t p o r t 1 i s open c i r c u i t s t a b l e
t r a n s i s t o r Tp o f t h e two t r a n s i s t o r I . IC shown i n
F i g u r e 3 -2 to p ro d u c e an h p i n e a r e r u n i t y . T h is mod
i f l e d INIC c i r c u i t was f i r s t s u g g e s t e d by L a rk y .^
th e t l i r e e t r a n s i s t o r iNIC, i hown i n
i t y c r i t e r i o n . ^ T hese c r i t e r i a a p p l i e d to t h e INIC o f
(OCS) and p o r t 2 i s s h o r t c i r c u i t s t a b l e (SCS)
14
ft*f v v - D "
'
J
Z$
c
ort £
’i ^ u r e 3 . 6 . The t h r o e t r a n s i s t o r L.IG
3 .3 t:::.-: d a h l l . i ) _ g i . c j i t
A n r l y s i s oi‘ t h e t h r e e t r a n s i s t o r L.IG i s s i m p l i
f i e d by th e u s e 0 ' an e q u i v a l e n t c i r c u i t f o r t h e
D a r i inf.; to n T r a n s i s t o r P r i r . An e q u i v a l e n t c i r c u i t w i th
th e sa.’.ie c o n f i j u r a t i o n a s t h e con ion e l i t t e r e q u i v a l e n t
c i r c u i t sh o rn i n .? i j ire 3 -3 v i l l a l l ox: th e U1IC to be
a n a l y s e d u s i n j tx;o t r a n s i s t o r e q u i v a l e n t c i r c u i t s i n
s t e a d o ? t h r e e . In t h i s s e c t i o n such, an e q u i v a l e n t
c i r c u i t i s d e v e lo p e d .
G h m d h i h a s shoxrn t h a t t h e h p a r a m e te r s f o r t h e
co. 1 ion c . l i t t e r D a r l i : q to n T r a n s i s t o r P a i r c o n n e c t io n o f
I ?
F i g u r e 3 o a r e :
1 + k,', Ax',( 3 . ? )
Hxx = ( 3. 3)Ku k%i_
w hore ; h ’ a r e t h e c a r aon c o l l e c t o r porn l e t c r s o f
T r a n s i s t o r A, h 1 f a r e t h e c o :n o n c o l l e c t o r p a r a l e t o r s
o f T r a n s i s t o r . , and & 1' and ^ ' a r e t h e d e t e r n i n e n t sr-1
o f th e IV and hV » - . i a t r i c e s , r e s p e c t i v e l y .
The D n r l i n ^ t o : . T r a n s i s t o r P a i r e q u i v a l e n t c i r c u i t
p a r a m e te r s may bo c : p r e s s e d i n t e r is o f t h e common e m it
t e r D a r i i n ' to n T r a n s i s t o r P a i r h p a r a m e t e r s . They a r e
r« ‘ - y t <»•”
z c - - l n . — ( 3 . i o )to Xz
, | ( I + Mx.i Mix- IA,, j— ( 3 .1 1 )
16
- . ( 3 . 12)A ~I -
Tlie h p a r a m e te r s o f a t r a n s i s t o r c o n n e c te d i n
th e co.'.imon c o l l e c t o r c o n f i g u r a t i o n a r e
hu ^ rk i . *~C.I ~ ^ (3 .1 3 )
u - ' ( 3 .1 ^ )
hi- - — ( 3 .1 5 )I -
™ ^ r r m i ( 3 - 16)
i f i t i s assum ed t h a t r c ( l - o t ) r G#
S u b s t i t u t i n g ( 3 0 ) - ( 3 .3 ) and ( 3 .1 3 ) - ( 3 .1 6 )
i n t o ( 3 *9 ) - (3 -1 2 ) r e s u l t s i n t h e f o l lo w in g e q u a t io n s
w here i t h a s been assum ed t h a t r Q and r ^ a r e much
l e s s th a n r @ ( l - oO • -
( 3 *17)
re - ' 'h a ( i - 4 -fc4 ( 3 .1 8 )
17
. (3 -2 0 )
The Darlington T ransistor Pair connection of
Figure 3*5 may now be represented by the common em itter
equ ivalent c i r c u i t shown in Figure 3*3 where ZQf r Ql
r b and o( 0 are given by ( 3 . 1 7 ) - ( 3 . 20 ) .
3• 4 ANALYSIS OF THE THREE TRANSISTOR INIC
An equivalent c i r c u i t for the three t r a n s is to r
XL'JIC sh ow in Figure 3 .6 may be drawn using the r e s u lt s
given in s e c t io n 3*3* Such an equ ivalent c i r c u i t i s
shown in Figure 3*7.
F i g u r e 3-7* A t h r e e t r a n s i s t o r XNIC e q u i v a l e n t c i r c u i t .
T h is e q u i v a l e n t c i r c u i t was drawn u n d e r th e a s
su m ptio n that r@ z r% - 0 .
% Vs v4A / V AA A A --------
V
18
C o n d u c ta n c e s £ 5 , g ^ , and gy a r e b i a s i n g e l e m e n t s . I t
i s e a s i l y se e n from th e e q u i v a l e n t c i r c u i t t h a t i s
z e ro and. h^n i s 1 . The r e m a in in g h p a r a m e t e r s , h g j and
h p 2 a r e fo u n d by w r i t i n g node e q u a t i o n s . They a r e
y3
where i t i s assumed that p = 1 .
liquation ( 3 . 22 ) in d ic a te s that hpp i s a fun ction
o f the b ias in g r e s i s t o r s Rg, and Ry. I t i s apparent
from ( 3 . 2 2 ) that h -o w i l l be reduced in magnitude i f
Y tg£ = Y^gy. Tliis requ ires tliat g& be n early equal to
gy s in ce i s g en era lly chosen approximately equal to
Y4 . Any change in b ia sin g r e s i s t o r s Rg and Ry w i l l
not a f f e c t h p l s in ce ( 3 . 2 1 ) shows that h 2i i s not a
fu n ction o f b iasin g r e s i s t o r s .
Chapter b
22JIC COMPENSATION
4-.1 INTRODUCTION
Chapter 3 introduced a p r a c t ic a l INIC» This
c h a p te r -m il show.how the INIC i s compensated so i t
appears n early id e a l over a given frequency range. For
the purpose of th is paper the frequency range from
500c p s . to 20Kcps was s e l e c t e d . The.compensation w i l l ,
be accomplished by; ( 1 ) adjusting the b iasin g r e s i s t o r s
so R5 - Hr?? ( 2 ) p lacing an admittance equal to h22
across port 1 and (3 ) by p lacing a r e s i s t o r a n d a ca
p a c ito r across Zi* so that h2% approaches un ity .
k*2 THE EFFECT OF % AND Ry ON h 22
I t was shown in se c t io n 3 .4 that h22 i s a func
t io n o f b iasin g r e s i s t o r s R5 and Ry. Using (3<>22). i t
i s p o ss ib le to c a lc u la te h2 2 (jw) for the BIIC of F ig
ure 4 ,1 . (Assuming R5 and Ry have the va lues 32K and
48K r e s p e c t iv e ly to s a t i s f y b iasing requirem ents.)
T a b le s .4 .1 and 4 .2 w i l l a id in. the c a lc u la t io n s .
Table 4 .1 g ives the measured va lues o f Yc (jw) and cd
for the three t r a n s is to r s which were used in the c ir c u i t
shown in Figure 4 .1* Table 4 .2 g iv e s the values of the
19
20_
composite Tc (jw) and <^sfor the Darlington Transistor
P air . The values in Table 4 .2 were ca lcu la te d from
Table 4 .1 using (3 .17) and (3 .2 0 ) .
R<2 0 u f
— I h "
A
l lK
Figure 4 .1 . A p r a c t ic a l CIIC c i r c u i t .
Figure 4 .2 shows a p lo t o f the Re h ^ ( jw ) and
the Ini hggt j v ) . The c i r c u i t shown in Figure 4 .3 was
used to measure hooQ w ). The General Radio Z-Y Bridge
shown in Figure 4 .3 w i l l measure p o s i t iv e or n eg a tiv e
admittance or i ipedance over a frequency range from
20cps to 20Kcps.
21
T ab le 4 .1
Freq.Kcps
= .975 Tp)=><2 ** • 971 T3 l=<3 - .965
Y c j ( u 2ho) Ycgfuwho) Yc^Cmnlio)
.5 .30 j .0 5 .13 +- 3 .04 .2 4 + j .0 41 .31 -4- j . l l .19 4. 3 .0 9 .2 5 + J .0 83 .3 3 + j .2 4 .21 4-3.27 .2 6 j . 2 55 . 3 5 + j . 4 4 . 2 2 4. 3 .4 5 .27 i. j . 4 l7 .36 +. j .7 1 .23 4- 3 .64 .2 0 4. j .6 3
10 .39 * j l .0 3 .2 5 4- 3 .9 1 .29 -+.J.3012 .42 * j l . 2 1 .2 6 4- 3 1 .0 9 • 31 4- 5 .9615 .47 4- j 1 .51 .2 9 4- 3 1 .3 6 .33 -4 d i .2 017 •51 + 3 1 .7 3 • 32 4. 31 .55 .36 + 3 1 .3 520 • 55 3 2 .0 3 .3 6 4. 3 1 .3 2 .39 ^ 3 1 .6 0
T a b le 4 . 2
Freq. Kcp s
Tq1o(0 = .993
Yc ( itiho)
• 5 .19 4- 3 .041 .2 0 4-3-093 .2 1 4- 3 • 285 . 2 2 4- 3 .477 .2 4 4. 3 .0 6
10 .2 6 4. 3 .9 412 .2 8 4- 31.1315 .3 1 4. 3 i . 4u17 .3 3 4. 3 1 .6 020 .37 +- 3 1 .8 8
fr
4MKQ4di ^e d
23
Figure 4.3* A c i r c u i t fo r the measurement o f hgg*
A comparison o f the ca lcu la te d Re h 22 (jw) and
the measured Be both o f which are shown in Fig
ure 4 .2 shows th a t the values d i f f e r by no more than 5$*
The measured Im h Pp(jw) and the ca lcu la te d Im h 2p(jw)
are seen to d i f f e r by as much as 2%. Although th is
i s a la rg e percentage d if fe r e n c e , the magnitude o f the
q u a n tit ie s involved are o f the order o f the stray
capacitance, thus the r e s u l t s seem reasonable.
The e f f e c t o f h22 on the impedance seen
looking in to port 1 when port 2 i s terminated in
impedance may be seen using (2 * 4 ). Assuming that
h n = 0 and h ^ ^ g l = (2 .4 ) becomes
Figure 4 .4 shows a p lo t o f the Bn Zj_n ( jw) when
ZL(jw) i s 5*00K. Zi n (jw) was ca lcu la te d using (4 .1 )
and the curves o f Figure 4 .2 . The measured value o f
ZjLn (jw) was found using the c i r c u i t shown in F ig
ure 4 .J .
y ic<//tor\ e.a u f e d
-
2?
z - yBr.cU (; W l r Por t } Z
2t ,
Figure 4») . A c i r c u i t for the measurement of Z n .
Both Z 0 and are i : roadmces accurate to I f .
vruen the d if fe r e n c e between Z0 (jv:) and Z^(jw) i s l e s s
than IK, the Z-Y bridge w i l l icasure Z'(jw) d ir e c t ly .
Choosing Z 0 ( j w ) > Z j ( j w ) w i l l make Z1( jw) p o s i t iv e and
thus prevent o s c i l l a t i o n s . Using the c i r c u i t o f
Figure 4 . p n-j_n (jw) i s found to bo
2 m C a = 2 z() u/\ - 2p ( 2 )
The curves of Figure 4 .4 in d ic a te that the meas
ured and ca lcu la ted values o f Ini 2^n (jw) agree quite
w e l l . The ca lcu la ted value o: Zi n (jv;) using the ex
perim entally determined value of jw) i s nearer the
observed value o f jw) than the ca lcu la ted value of
Zin (jw) found using the ca lcu la ted value of UggCjw).
The ca lcu la te d value of Re Z^Cjw) i s r e la t iv e ly
c 3n sta n t and ran, es from 4. -OK a t pOOcos to 4 . 74k a t
20Kcps.
26
'S ection suggested a p o s s ib le p a r t ia l compen
sa t io n for non-zero I122 by s e t t in g equal to Zi gy*
Since Z3 and are chosen approximately e q u a l , gg rnust
equal gy for the compensation to be p o s s ib le . I f E5 i s
changed from 32K to 4BK, the operating p o in ts o f the
tr a n s is to r s in Figure 4 ,1 w i l l change only s l i g h t l y .
The change in operating p o in ts w i l l not cause a n o t ic e
ab le change in the tr a n s is to r parameters hence the values
given In Tables 4 .1 and 4 .2 may be used for the new ca l
cu la t io n s in which B5 z Ky = 48k.
Figure 4 .6 shows a p lo t of the Lm hggCjw) and the
Re h2 2 (dw) when R<$ s By # 48k.'. The Be hppCjw) when
R^is 48k i s about one tenth the Re b ^ U w ) when 1% i s
32K. The measured and ca lcu la te d values of the
Re hggCjw) vary by a maximum of 20fL This discrepancy
i s probably due to lo s s of accuracy of the Z-Y bridge - -
as the - admittance being measured becomes sm all. F ig- •
ure 4 .6 shows that the Im hggtjw) i s not g r e a t ly
a f fe c te d by p lacing R5 - Ry.
E ar lier in th is se c t io n it . was assumed th a t "
h n - 0.. In se c t io n 3 = 4 i t was found that h%i i s
zero. This r e s u l t was v e r i f i e d experim entally over the
same- frequency range used for the preceding measure--
rnents using the c i r c u i t shown in Figure 4.7»
Measured c y td u1< rt*4
i
u
28
P d %a/ r c
Figure 4.7* A c ir c u i t fo r the measurement o f h]j_.
4 .3 COMPENSATION FOR h ?p A 0
In se c t io n 2 .4 i t was shorn that an NIC can
compensate for i t s non-zero i f h-^ f s zero.
Since the IN10 o f Figure 4 .1 has an parameter o f
zero i t i s p o s s ib le to e lim in ate by p lacin g an ad
m ittance equal to hgg across port 1. For s im p l ic i ty
th is admittance c o n s is t s o f a r e s i s t o r in p a r a l le l
with a cap acity Csh. Rg^ i s used to e lim inate the
Re hogCj’'r) v ir ile CQu e l im in ates the e f f e c t o f the
Im hggC j v ) .
The values o f Cg% and Rsh that reduce the over
a l l va lu e of hp2 (dv) to zero are found from Figure 4 .6 .
Figure 4 .6 shows that the Ro hp._(jw) i s not constant,
th erefo re a s in g le shunt r e s i s t o r can not compensate
t h e Re hnnCJv) at a l l freq u en cies . A good compromise
to the problem o f compensating the Re h^gC jv) over t h e
s p e c i f ie d frequency range i s obtained i f Rg^ i s chosen
equal to the Re hg2( jw) a t lOKcps. A t lOKcps t h e
29
c a lcu la te d value of the Re h2?(jv ) i s •97uli1io th erefore
^sh should be 1.03M eg. The ca p a c it iv e component o f hpp
i s n early constant with frequency. Using the ca lcu la ted
Irn h^pCjw) va lue , GS21 i s found to be 43pf.
I f h 9 p(jv) i s neasurod d ir e c t ly and Cgh and Rsh
varied , i t i s found that hpoUV) becomes zero nhcn
i s ?6pf an . Rgn i s 1. Oil leg. The ca lcu la te d end
measured values of Rgil agree favorab ly . The d if fe r e n c e
between the ca lcu la te d and observed values of i s
probably clue to stray capacitance present in the c i r
c u i t .
4 . 4 C )I IP h i SAT ION FOR h?1 < 1
In se c t io n 4 .2 Z-j^Cjv) was ca lcu la ted assuming
that h^2^2l - S ince the measured and ca lcu la te d
values of were in q u ite good agreement, i t apucared
that t i l ls was a v a l id assumption. Using (3*21) i t can
be shown that h^ph^l w i l l be s l i g h t l y l e s s than u n ity
(assuming that Zn - Z^). By properly choosing Zo and
^4> h-iphog can be adjusted to u n ity . The value o f ZLh
in terms of Z-> that w i l l produce h1^ 2 1 = can be
found by s e t t in g ( 3 . 2 1 ) equal to 1 ( s in c e hjo r D *
2 r ; V c a w , g ( ^ 3 )
I f h^gh^i i s to be un ity and Zn i s a IK r e s i s t o r
then from (4 .3 ) i t i s found that Z4 should be a IK
r e s i s t o r shunted by a l3 p f cap acitor .
30
4.5 v^giJATXOi: o? co:y ;a/.Ti3,i pnoccpuaKsAn INIG block diagram complete w ith compensating
elements i s shown in "igure 4 .d . (C and iny be
adjusted u n t i l h^2^2l u n ity * ) Terminating port 2
in an impedance and measuring Zj_n (jw) for various
stages o f compensation w i l l v e r i fy how the INIC becomes
id e a l r s compensation elements are added.
to be 23 pf an a 26 5K r e s p e c t iv e ly . I f Cg, i s made larger
than 23p f th e c i r c u i t o s c i l l a t e s .
Figure 4 .9 shows th a t , without compensation, the
Re Si n (jv ) v a r ies between -4 ./6 K and - 4 . 74k. With
compensation the Re ( jw) w i l l be -5*0UK over the
Figure 4 .3 . A co nansatod UIIC
s how Z ( jw) aoproaches -p.OOK,
co m p e n sa tin g elements arc added.
IQKcos by sa t t in
by adjustin': Cg and Rg a t lOKcps, once the e f f e c t o f
hop had been e lin in : ted , u n t i l Z±n became approximately
equal to -p.UDK. The values fo r Gg and Rg were found
32
s p e c i f ie d frequency range. I t Is a lso seen th a t 5 w ith
no compensation, the Im Zi^ w i l l be 155 ohms a t ZOKcps.
With compensation the Im Z-j w i l l be 15 ohms at the
same frequency.
I t I s In te r e s t in g to note the e f f e c t the compen
sa tin g elements have on Z ^ , when i s r e a c t iv e .
Figure 4 .10 shows a p lo t o f Z.-LnUw) fo r a r e a c t iv e term
in a t io n . At 20Kcps the d es ired 'in p u t i s -5«00K - j800„
Without compensation w i l l be -4*74 - j 530 a t 20Kcps« •
With compensation Z_._ w i l l be -4.99K - 3760 a t the same-LIIfrequency. For th is case .Cs^ was reduced to 4 lp f to
prevent o s c i l l a t io n s w hile 0 %, was increased to 34pf.
and Rsh remained at 265K and l.OlMe'g. r e s p e c t i v e l y . .
Table 4 .3 g ives Zj_n ( jw ), before and a f te r compen
sa t io n , for various values of Z%( jw) • Using Table 4.3,
i t i s seen that the HUG appears n ear ly id e a l a f t e r
com pensation." The IKIC- i s not e x a c t ly id e a l , mainly
because the Re h ^ Q v ) i s not zero over the e n t ir e
frequency range. At lOKcps the Re hgg i s zero however
i t i s 25umho a t 5u0 cps and .2umho 8t 20Kcps.
32
T ab le 4 .3
Freq. Kcp s Kofcns
-Zj_n (no com p.) Kohms
*"zin (com p.) Kohms• 5 . 5 0 0 ' .>00 .5 0 0
1 .5 0 0 .5 0 0 .5003 .5 0 0 .500 .5005 .5 0 0 . 500 .5 0 07 .5 0 0 .5 0 0 . 500
10 .5 0 0 .5 0 0 . 50012 .5 0 0 .500 .50 015 .5 0 0 • 500 .50017 .5 0 0 .4-99 - j .0 0 7 .5 0 020 .5 0 0 .497 - j . 003 .5 0 0
.5 5 .0 0 4.76 5 .0 01 5.uo 4.76 - 3 .0 1 0 p. 003 5 . JO 4.70 - 3.023 5 .0 05 5 . JO 4.70 - j .o 4 0 5 .0 07 5 .0 0 4 .7 5 - j .0 5 7 >.00 - 3 .0 0 2
10 5 .0 0 4 .7 5 - 3 .0 8 0 5 .0 0 - j .0 0 712 5 .0 0 4 .7 5 - 3.091 5 .0 0 - 3 .0 0 815 5 .0 0 4 .7 5 - 3.117 5 .0 0 - 3 .0 1 317 5 .0 0 4 .7 5 - 3 .132 ■ 5 .0 0 - 3 .0 1 420 5 .0 0 4 .74 - j .1 5 5 5 .0 0 - 3 .0 1 5
.5 1 0 .0 0 8.93 1 0 .0 11 1 0 .0 0 8 .9 2 - 3 .0 2 0 1 0 .0 03 1 0 .0 0 * 9 2 — 3 • 0 JO 1 0 .0 05 1 0 .0 0 3 . 9 2 - 3 .1 4 0 1 0 .0 0 - 3 .0 0 57 1 0 .0 0 8 .9 2 - 3 .2 0 0 1 0 .0 0 - 3 .0 0 7
10 1 0 .0 0 8 .9 1 - 3 .2 0 0 1 0 .0 0 - 3 .0 1 012 1 0 .0 0 3 .90 - j .3 3 5 1 0 .0 0 - 3 .0 1 515 1 0 .0 0 8 .3 9 - j .4 1 4 1 0 .0 0 - i .01317 1 0 .0 0 8 .30 - j .4 7 0 9 .9 9 - 3 .0 2 020 1 0 .0 0 8.70 - 3 .5 7 0 9 .9 9 - j .0 2 7
33
Table 4 .3 (C ont.)
Freq.Kcps Kofeas
“ ■-'in (no coup.) Koluas
“Zin (comp.) Kohms
• ? $ .0 0 + j .0 2 0 4 .71 t .0 1 2 5.00 4 j .0201 5.00 t j .0 4 0 4 .71 + .027 5.00 4 ,5.0333 5 .0 0 4. j • 120 4.71 + .030 5.00 4 j .1145 5 .0 0 + j .2 0 0 4 .71 * .1 2 0 ? .o 0 4 3.1367 5 .00 4. j .2 3 0 4 .72 + .177 5 .0 0 4 3-263
10 5 .0 0 4 j .4 0 0 4 .72 + .2 6 0 4 .99 4 3 - 3 7 512 5 .0 0 + j . t o o 4 .72 * .3 1 5 4 .99 3-4-5015 5 .0 0 4 j .6 0 0 4 .73 + .390 4 .99 *- 3-57017 5 .0 0 + j .680 4 .73 * .450 4 .9 9 4 3-64420 5 .00 4 j .8 0 0 4 .7 4 + .5 3 0 4.99 t 3-760
/J-X2
ETOC
'
CD OC CO C
P- 7 ~ to >. V 6M
C o 6#
PgL Klml At 2&6 #K) 14 =H?
-fr^ci Y&*cy
C h a p te r 5
CONCLUSIONS
I t h a s ' been shown t h a t an IHIC w i l l t r a n s f o r m an
im pedance i n t o a n e g a t i v e im p e d a n c e . - Z y i f t h e h .
p a r a m e te r s o f . t h e IHIC a r e h ^ = h g g - 0 and h^ ^hp^ - 1
E q u a t io n s hav e been d e v e lo p e d , f o r t h e u n c o m p e n sa te d .
IHIC, w h ich show t h a t h j j s 0 , h^g = I? h p ^ < l a n d
hgp > 0 o T hese e q u a t io n s w ere u s e d to co m p en sa te th e -
IHIC so t h a t hpp and h^ p became 0 and 1 r e s p e c t i v e l y .
C om p ensa tion o f th e IHIC by e x p e r im e n ta l means showed
t h a t t h e c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s a g re e d
f a v o r a b l y .
M easurem ent o f th e IN IC 1s i n p u t im pedance h a s
shown t h a t hgg- h a s a much g r e a t e r e f f e c t on t h e INIC
-than h jpo At 20Kcps th e i n p u t im pedance was fo u n d to
be " 4 „ 7 lhE t b e f o r e c o m p e n sa t io n and -4 - .98% t j2 7
a f t e r h ^ g co m p en sa tio n * A f t e r h ^ 0 c o m p e n sa t io n th e
i n p u t im pedance be case- "*5»00K ^ j 15» (F o r t h e s e m eas
u re m e n ts Zj^ s $100%) =
An a r e a f o r f u r t h e r s tu d y m ig h t be th e i n v e s t ! - :
g a t i o n o f INIC s t a b i l i t y . I t was m e n t io n e d t h a t th e
INIC became u n s t a b l e f o r c e r t a i n l o a d t e r m i n a t i o n s i f
3?
t h e oom p-ensatlag e a p a e i t o r s became to o l a r g e » I t sh o u ld
be p o s s i b l e to develop- e q u a t io n s w h ich e x p la in , t h i s I n
s t a b i l i t y .
M o t h e r a r e a f o r f u r t h e r s t u d y c o u ld be t h e ex
t e n s i o n o f c o m p e n s a t io n m e th o d s p r e s e n t e d I n t h i s
p a p e r* I n t o f r e q u e n c y r a n g e s g r e a t e r t h a n gOKcps,
B I B L IO G R A P H Y
Lim.dry'j M» R» % Ui-Iegat±ve Bnpedanee C o n v e r t e r C i r c u i t s ~ Som e^Basic R e l a t i o n s and L i m i t a t i o n s ^ ^ ____ ^ r a n s a p t lo n s on C l r o n l t I h e o r Ty ft V o l . 3 " U ?‘Septem EeFi PP- 1 3 2 -1 3 9 /
2 . L a r k y , A. I . , MH e g a t i v e Im pedance C o n v e r t e r s , n'I 0 Ro £U I r a n s a c t i o n s on C i r c u i t T h e o ry , V o l . ^3- '^ ,
. Septe2nSer>*"l9 57, PP® .
3« Larky,. A. I . , $8H .egative Im pedance C o n v e r te r D esign ,S t a n f o r d E l e c t r o n i c s L a b o r a to r y Techn i c a l R e p o r t
• Ho. 11 o S t a n f o r d , C a l i f . , O c to b e r ,
4 . L i n v i l , J . G . , ^ T r a n s i s t o r n e g a t i v e Im pedance
J u n e , 1953 , , i> n . '7 2 7 -7 2 9 •
5» G h a n d h i , s . K . , “D a r l i n g t o n Compound C o n n e c t io nf o r T r a n s i s t o r s , r I . R. B . T ran s a c t i ons on C i r c u i t T h e o ry , V o l . CT-4, Ho. 3 s SeptemBef7' s P
38
