REFERENCES
Mos o he p e iously epo ed designs ollow as special cases o
ill L. B. Jackson, “An analysis o limi cycles due o mul iplica ion ounding
his gene al sec ion which is shown o exhibi minimum sensi i -
in ecu si e digi al (sub) il e s,” m P oc. 7 h Ann. Alle on Con . Ci cui
and Sys em Theo y, pp. 69-78, Oc . 1969.
i y. In ac , al hough some new ci cui s uc u es esul , he main
I21
J. L. Long and T. N. T ick, “An absolu e bound on limi cycles due o
objec i e o ou wo k is o ie all he ad hoc pape s abou
oundo e o s in digi al il e s,” IEEE T ans. Audio Elec &ous ., ol.
compensa ed ampli ie s oge he and o p o ide guidelines o
AU-21, pp. 27-30, Feb. 1973.
[31 T. L. Chang, “A no e on uppe bounds on limi cycles in digi al il e s,”
wha should be done o ob ain low-sensi i i y ac i e compensa ed
IEEE T ans. Acas ., Speech, Signal P ocessing, ol. ASSP-24, pp. 99- 100,
VCVs’s. As a consequence, he pape also gi es a amewo k o
Feb. 1976.
I41 D. C. Munson, J ., and B. Liu, “Na ow-band ecu si e il e s wi h e o
compa ing exis ing (and in some cases new) al e na i e designs
spec um shaping,” in P oc. IEEE In . Con . Acous ., Speech, Signal
o VCVs’s.
P ocessing, Washing on, DC, pp. 367-370, Ap . 1979.
PI
-, “Low-noise ealiza ions o na ow-band ecu si e digi al il e s,” II. ACTIVE COMPENSATION SCHEMES FOR MINIMUM
IEEE T ans. Acow ., Speech, Signal P ocessing, ol. ASSP-28, pp. 41-54, SENSITIVITY
Feb. 1980.
161
- “ROM/ACC ealiza ion o digi al il e s o poles nea he uni
ci cle,” IEEE T ans. Ci cui s Sys ., ol. CAS-27, pp. 147-151, Feb. 1980.
The a ailabili y o low-cos dual ampli ie s is one o he mos
[71 A. I. Abu-El-Haija, K. Shenoi, and A. M. Pe e son, “Digi al il e s nc-
impo an mo i a ions o he de elopmen o ac i e compensa-
u es ha ing low e o s and simple ha dwa e implemen a ion,” IEEE
ion schemes. All o he known con igu a ions use wo ope a-
T ans. Ci cui s Sys ., ol. CAS-25, pp. 593-599, Aug. 1978.
PI
E. I. Ju y, Theo y and Applica ion o he Z-T ans o m Me hod. New
ional ampli ie s and i appea s easonable o de i e s uc u es
Yo k: Wiley, 1964.
which only equi e a pai o ac i e elemen s.
I should be clea ha any VCVS based on wo ope a ional
ampli ie s is a second-o de ac i e-R il e which exhibi s a
low-pass cha ac e is ic, i.e., a wo-in eg a o s ci cui wi h a ans-
e unc ion gi en by
On he Ac i e Compensa ion o Ope a ional
Ampli ie Based VCVS
J. L. HUERTAS AND A. RODRIGUEZ-VAZQUEZ
B and and Schaumann [6] ha e shown ha all second-o de
ac i e-R il e s can be de i ed om he gene al s uc u e depic ed
in Fig. l(a). The ans e unc ion denomina o associa ed wi h
bo h possible ou pu s ua, and us*, is
A ~s ac -A uni ied ea men o he ac i e compensa ion o VCVS
ealized by ope a ional ampli ie s is p esen ed. I is a summa y o low- D(s)=2+s(c,GB,+y(JGB,)+GB,GB,(c,y,-cc,y,). (2)
sensi i i y ci cui s uc u es which can be ei he magni ude o phase
Rou ine calcula ions shown ha he condi ion o minimum
compensa ed. The pe o mance o such a ci cui s is discussed in de ail.
sensi i i y wi h complex poles is [6]
Expe imen al da a o some o hem a e also included. c(Jy(J = 0.
I.
INTB~DUCTION
Ful illing ha condi ion is equi alen o opening he local
The high- equency oll-o o he ope a ional ampli ie im- eedback loop o one in eg a o . Because he con igu a ion is
poses a p ac ical limi a ion o hei use ulness in RC-ac i e il e symme ical we may ake c0 = 0 and conside al e na i ely ei he
design. Wi h he ad en o low cos in e nally compensa ed dual we, o us2 as he ou pu o he compensa ed VCVS. Hence o h
ope a ional ampli ie s, he design o ac i e-compensa ed ol age-
con olled ol age sou ces (VCVS) is cu en ly he mos pop& we shall call ype-1 s uc u es he ci cui con igu a ions wi h u,,,
as he ou pu o he compensa ed ampli ie . Simila y, we shall
way o o e come such a limi a ion. In ha sense, many ad hoc call ype-II s uc u es he con igu a ions wi h u,,z as he ou pu .
ci cui con igu a ions ha e been epo ed o ealizing com- Equa ions (4) and (5) show, espec i ely, he ans e unc ion
pensa ed ampli ie s bu a clea s a emen o he connec ions o bo h ypes o s uc u es
among hese s uc u es is missing and i emains di icul o selec
anyone o hem o a speci ic applica ion. In pa icula , mos
ype-l: ,GB, + GBP,w, (4)
discussion ail o poin ou wo impo an conside a ions. The
i s one is a de ailed exposi ion o he di e en c i e ia which
can be used o achie ing an imp o emen o he ampli ie
ype-II: T,,(s)+ SC,% + GB,G&(w, + CRY,)
equency esponse. The second one being a discussion o he
I D(s)
sensi i i y p ope ies o he p oposed compensa ion schemes.
(5)
This pape p esen s a uni ied app oach o he gene a ion o
ac i e-compensa ed VCVs’s. S a ing om a second-o de ac i e- whe e o ensu e s abili y we mus impose he ollowing condi-
ions:
R low-pass sec ion, a p ocedu e is o mula ed o inding ci cui
implemen a ions o ealize an ampli ie ei he wi h ampli ude y. > 0 and c,y,<O.
(6)
(maximally la magni ude) o wi h phase ( --) 0) compensa ion. I is wo h no ing [6] ha all pa ame e s a e di e ences o
esis o a ios, i.e., e.g. (yi,c,)=xi-yi, wi h OG(xi,y,)Gl,
Manusc ip ecei ed Ap il 10, 1981; e ised Feb ua y 3, 1982.
The au ho s a e wi h he Depa men o Elec ici y and Elec onics, Uni e -
howe e , in o de o a oid la ge sensi i i y due o di e ence
si y o Se ille, Se ille, Spain,
e ec s, we shall choose ei he xi o y, always equal o ze o.
‘BW’ can be conside ed as a meaning ul measu e since o phase compensa-
Assuming ha , he ci cui in Fig. l(b) shows he gene al s uc u e
ion only ci cui s uc u es wi h Q 2 I a e o in e es .
o any low-sensi i i y compensa ed VCVS and p o ide de ini ion
009%4094/82/0700-0497$00.75 01982 IEEE
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-29, NO. i’, JULY 1982 497
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1982
"i
i
I
““1
(4
(b)
Fig. I. (a) Schema ic ep esen a ion o a gene al second-o de ac i e-R il e
(b) Gene al ci cui s uc u e o a low-sensi i i y compensa ed VCVS.
o he pa ame e s in e m o speci ic esis o a ios as ollows:
LYi gi 3
c,=---
I
i
gj ,i4 Jj'
i=1,2
jzz1
y;,L- Gi+3
3
2 Gj i Gj+Go
3 i=1,2
j=l j=4
Yo’ 6 GO
2 G,+Go
j=4
(7-a)
(7-b)
(7-c)
whe e ei he g,(G,) o g,+3(Gi+3) is always equal o ze o depending on he sign o ci( y,),
i
= 1,2. No e ha since he summa ions a e
pe o med by a esis i e node, he ollowing addi ional in e connec ion equi emen s can be de i ed:
ampli ie one : O~yo-~{(1--sgnyl)yl+(1--sgny2)y2)~1
ampli ie wo : O< -i{(l-sgnc,)c,+(l-sgnc,)c,} <I
(8-a)
(8-b)
(8-c)
(8-d)
which mus be conside ed when making a pa icula design.
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499
TABLE I
DESIGNEQUATIONSFORTYPE-IANDTYPE-II STRUCTURES;
PARAMETER b CORRESPONDSTO GB,/GB,
2
“”
phase
compensa ion
condi ion
magni ude
compensa ion
condi ion
III.
COMPENSATIONREQUIREMENTS
compensa ed design equi es ob aining he highes w,, and Q
-
Exp essions (4)-(8) allow conside able eedom in choosing he alues compa ible wi h (10).
di e en design pa ame e s. In ac , many choices a e possible A magni ude-o ien ed compensa ion scheme has o be di ec ed
and some o hem co espond o pa icula cases which ha e been owa ds maximizing he equency ange whe e he magni ude
eno ed elsewhe e 111-151. Hence, in o de o ake ad an age o esponse is la . F om (1) his esul s in
I -- --
ha eedom we will iden i y me p oblem o wha mu; be
op imized in a compensa ed VCVS. Basically, au ho s engaged
wi h ha ques ion ha e conside ed wo di e en app oaches,
emphasizing ei he magni ude-o ien ed o phase-o ien ed com-
WC/W0
Q=
(13)
J. 1+2g Wo’
pensa ion c i e ia.
A design based on he phase-compensa ion mus minimize he Fu he mo e, a condi ion may be de i ed consis ing in maxi-
low- equency phase e o [4]. Tha e o , om (l), is app oxi- mixing he 3-dB bandwid h o he ci cui . F om Geige ’s pape [l]
ma ely gi en by ha bandwid h is gi en wi h (13) by
BW=wo/w. (14)
Then a maximum o BW compa ible wi h (13) mus be looked
o .
F om (9) i is clea ha we mus impose he condi ion
Q2i
wo
in o de o educe he alue o Cp( w) o
In he phase-o ien ed case he pu pose o he design echnique
is wo old. The emaining phase-e o mus be educed o a
minimum while a he same ime ex ending as much as possible
he e ec i e bandwid h, BW’, which will be de ined as he
equency a which he gain alue is inc eased by 3-dB.’ A e
ou ine calcula ions we ge , using (10)
BW’= we/m. (12)
IV.
PRACTICAL CIRCUITS
Table I p o ides he design equa ions o ype-1 and ype-II
s uc u es. In wha ollows, we shall name hese equa ions in a
compac way by using a ma ix-like e minology. Fo ins ance,
he equa ion gi ing BW o he ype-II will be e med as (11-7).
These design equa ions ha e o be comple ed wi h he in e con-
nec ion equi emen s gi en by (7) and (8). Ma ched ope a ional
ampli ie s we e no assumed since bo h s uc u e ypes a e sensi-
i e o GB misma ch in a di e en way.
In ac , we can gene a e di e en compensa ed VCVs’s by
conside ing he combina ions o he sign o he pa ame e s in (7)
and (8) which a e compa ible wi h he equa ions on Table I as
well as wi h he s abili y condi ions gi en by (6). Also, o ype-II
s uc u es i is in e es ing o impose he addi ional cons ain :
sgn(w,) = w(c,y,) = sgnk, (15)
which a oids an unnecessa y limi a ion on he a ainable alue
o m, in Table I.
F om (11) and (12), i should be clea ha a “good” phase Figs. 2 and 3 show all he esul ing p ac ical ci cui s. In each
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1982
” 01
00 0 0
‘6 c /kb
M.C.: - =ko-1; G1=O; G2+m ; $ =/ + -1
g5 0
93 G5
“.C.: - =ko-1; G1+= ; G3=O; c = -kob
R2 0 I- - - -l
l+b +2kob
M.C.: $ = ,k, I; G4=0;
G4
: %=lk 1; G6=0; G
R5 P 0
Fig. 2. Ci cui con igu a ions o ype-1 compensa ed ampli ie s: CI.1; CI.2
[4] CI.3 [5]; CI.4 [4].
Pa ame e b co esponds o GB,/GB,.
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1982 501
G5 6 ‘6
4.c.: gl=o; g2- ; ” = q; ” = ko6-+-&
dko i
”
02
G3
:- G2
-kob-1
G3 G4
bk
M.C.: c, =ko-I; , s 0 -1
2 oajzl
n.c.: g4-0; *5+- ;
0
y-E&J
2 “02
=4
P.C.: 0
G4 %
cc.: c = -F
0 0
Fig. 3. Ci cui con igu a ions o ype-II compensa ed ampli ie s: CII. 1, CII.2
[I][31 CII.3 and CII.4. Pa ame e b co esponds o GB,/GB,.
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502 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL.CAS-29, NO. 1, JULY 1982
TABLE II(a)
PARAMETERVALUESFORPHASECOMPENSATION; k,,>O;
PARAMETER b CORRESPONDSTO GB,/GB*
Pi cui .oa a
Fe e* signs-and
in e conec ion
equi emen s. Op imum design pa ame e alues
YI
Y2 <I 2 m
cII.2
kl,Y2)‘0
(C2.Y,F0
Y, 9, = 1
TABLE II(b)
PARAMETERVALUESFORPHASECOMPENSATION; k,<O;
PARAMETER b CORRESPONDSTO GB,/GB,
Ci cui , pa *
me e sipns and
ince conec ion
equi emen s Op imum design pa ame e alues
I
cI.3
(c1.c2)‘0
(Y,.Y2)‘0
Cl + c2 = 1
Yo-Y,-Y2= I
cI.4
Y2>0
k,.c2,Y1)<0
Cl + c2 =-1
Yo-!, =a
yoke
Yo-Y,-1
yoke -lk,l
l+lkol
(Y1,Y*)>O
k,.c2 n
, 4 2 =-1
Y, + Y2 = 1
cII.4
c2>o
(C,‘Y,.Y2)<0
Yo-Y,-Y2' 1
Y'Ik lb
l-Y2 1 (l+ L)
l+ kol l-Yoby, b yokob
-1
l+lko -1
-l-c, ----W~kohob+Y~lko~b)
l+lkol
1 -!-
l+lkol
ci cui he ac ual esis o a ios mus be choosen in o de o ul ill
A. Phase Compensa ion
ei he he phase o he magni ude compensa ion condi ion. Nex , Phase compensa ion condi ions ha e been applied o bo h
we will conside he pa icula choices o bo h compensa ion ype-1 and ype-II s uc u es. The esul s a e summa ized in
condi ions. Table II(a) o k, > 0 and Table II(b) o k. ( 0. A ew ema ks
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL.CAS-29,NO.l,JULY
1982 503
TABLE III(a)
PARAMETERVALUESFORMAGNITUDECOMPENSATION;
k,>O;
PARAMETER
b
CORRESPONDSTO
GB,/GB,
TABLE III(b)
PARAMETERVALUESFORMAGNITUDECOMPENSATION;
k,<O;
PARAMETER
b
CORRESPONDSTO
GB,/GB2
Ci cui
“o iln”m ‘ie+ipn na ame e alues
Y
Y
0 Y1 2 Cl ‘2
a e no ewo hy:
1) No e om (I-2) and (11-2) ha 1 c2/ and ] y2 1 ha e o be
selec ed as high as possible in o de o a ain a maximum alue
o w,. Hence, he in e coMec ion equi emen s in ol ing such a
pa ame e ha e been chosen as uni y whene e i does no gi e
ise o a con adic ion wi h he design equa ions.
2) The alues o he di e en pa ame e s a e lis ed as unc ions
o y. and k,. In mos cases, he designe is ee o choose he
pa icula alue o y. bu , om he exp essions o BW’ and $J
(see Table I), i should be clea ha y. mus be selec ed as small
as possible in o de o op imize he compensa ion.
3) An addi ional cons ain on he minimum alue o y. is
de i ed om a mo e de ailed s abili y analysis including he
ope a ional ampli ie ’s second pole [7]. I esul s in
w2 lW22 2
GB, W2:
” GB,(
w21+ w22)
+ y” GB, ( w2, + w*2)2
’ ICZYZI (16)
whe e w2i is he loca ion o he i h (i = 1,2) ope a ional ampli-
ie ’s second pole.
B. Magni ude Compensa ion
Table III(a) and III(b) depic he esul s o applying he
magni ude compensa ion condi ions o bo h ypes o s uc u es.
Fo e e y ci cui , he pa ame e alues allowing o a ain a
maximum BW a e shown. These pa icula alues ha e been
de i ed by a compu e -aided nume ical analysis on he c, and y,
alues.
V.
DISCUSSION
Fo con enience, we shall conside sepa a ely wo cases de-
pending on he sign o k,. A numbe o conclusions ollow o
bo h o hem.
A. k,>O
F om a phase compensa ion poin o iew he ci cui CI.2
exhibi s pe o mance supe io o ha o ci cui CI. 1. I is due o
he ac ha o he same y. he o me allows o ob ain a highe
(lowe ) alue o m($), de ined in Table I, han he la e . I mus
be ema ked ha Reddy’s ci cui [4] is a special case o ci cui
CI.2, o y. = l/k,. The maximum alue o m is ob ained o he
ci cui CII.2 [3]. Ne e heless, he ci cui CI.2 allows o a ain a
lowe phase e o and highe BW’ han CII.2 whene e y. o he
o me is selec ed, espec i ely, o be
A he o he hand, he maximum BW is ob ained o ci cui
CII.2 in acco dance wi h Geige ’s pape [l]. Mo eo e , i is
impo an o no e ha ci cui s CI. 1 and CII. 1 ha e in ini e inpu
impedance, and in addi ion ob aining a BW alue which is no
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1982
42
J
40
.
38
.
36
-
0
34
32
.
30
-
28
.
26
24
_
22
.
20
-
18
.
16.
14.
12.
10
8-
6-
4-
2 10 20 30 40 50 60 70 100 200 300 400 500 800 1000 (KHz)
(4
PHASE
T(jw)
.”
-20
-30
x
-40
-50
-60
-70
-80
-90
-100.
-110. L
-lZO-
-130-
-140'
-150-
-160-
-170- 00 0
-180 - 10 20 30 40 50 60 70 100 200 300 400 500 800 1000
(b)
Fig. 4. Ci cui CII.2 compensa ed ampli ie o k, = 10.9. (a) Magni ude
cha ac e is ics. (b) Phase cha ac e is ics. con inuous ace lines co espond o
he heo e ical esponses.
+ co esponds o he measu ed esponse o he uncompensa ed ampli ie .
* co esponds o he measu ed esponse o he magni ude compensa ion.
D co esponds o he measu ed esponse o he phase compensa ion.
(KHz)
oo a om he one ob ained by he Geige ’s ci cui (see Table CI.4 allows lowe phase e o s han CII.4. The o me has been
9. p e iously epo ed by Reddy o a = 1, a de ined in Table II(b)
B. k,<O
[4]. Howe e , o magni ude compensa ion, he maximum BW is
ob ained om ci cui CII.4.
Fo phase compensa ion, he ci cui s CI.4 and CII.4 yield he Finally, i is in e es ing o no e ha he phase compensa ion
maximum alue o m, de ined in Table I. Mo eo e , he ci cui condi ion emains s ill alid o ype-1 s uc u es no ma e wha
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CAS-29,N0.
7,JULY
1982
505
42~
40.
38. D
36.
34.
92
30.
28.
26.
24.
22.
20.
18.
16-
14.
12-
10.
8-
6-
4-
2 L
10 20 30 40 50 ho 70 100 200 300 400 500 800 1000
(4
PHASE
T(jw)
-180
-170
-160
-150
-140
-130
-120
-110
-100
-90
-80
-70
-60
-50
-40
IO 20 30 40 50 6n 7n 10'1 200 3no 4 )o 500 8on im
03
Fig. 5. Ci cui CII.4 compensa ed ampli ie o k, = - 10.9. (a) Magni ude
cha ac e is ics. (b) Phase cha ac e is ics.
(KHz)
(KHz)
he deg ee o misma ch. This is an ad an age when compa ed
wi h phase-compensa ed ype-II ci cui s. Qui e in he con a y,
o magni ude compensa ion, bo h ypes a e sensi i e o he
misma ch, as can be seen on Tables I and III.
VI.
EXPERIMENTALRE~ULTS
As an example o he pe o mance o he di e en ci cui s, we
show he magni ude and phase cha ac e is ics o he CII.2 and
CII.4 conside ing bo h magni ude- and phase-compensa ion cases.
Fig. 4 depic s he magni ude and phase esponse o he ci cui
CII.2 wi h
k,
= 10.9. Al e na i ely, Fig. 5 depic s he co espond-
ing cha ac e is ics o he ci cui CII.4 wi h
k,
= - 10.9. Fo bo h
ci cui s we ha e used ~A747 dual ope a ional ampli ie s.
ACKNOWLEDGMENT
The au ho s wish o hank o he e iewe s o hei sugges ions
o imp o e he p esen a ion o his wo k.
REFERENCES
[I] R. L. Geige , “Ampli ie s wi h maximum bandwid h,” IEEE T ans.
Ci cui Sys ., ol. 24, pp. 510-512, Sep . 1977.
[2] A. M. Soliman and M. Ismail, “Ac i e compensa ion o ope a ional
ampli ie s,” IEEE T ans. Ci cui Sys ., ol. 26, pp. 112-l 17, Feb. 1979.
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