THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
P ecise Implemen a ion o CDTA
Dalibo BIOLEK, Ji i VAVRA
Depa men o Elec ical Enginee ing, Facul y o Mili a y Technology, Uni e si y o De ence,
Kounico a 65, 662 10 B no, Czech Republic
dalib[email p o ec ed], ji i. a[email p o ec ed]
DOI: 10.15598/aeee. 15i5.2515
Abs ac . The Cu en Di e encing T ansconduc-
ance Ampli ie (CDTA) is a popula ac i e building
block o analog signal p ocessing. Since i is no a ail-
able as a comme cial IC, mos o he CDTA applica-
ions a e e i ied only ia simula ions o he CDTA is
implemen ed by comme cial ci cui s. The pape deals
wi h CDTA implemen a ion, emphasizing he accu acy,
linea i y, and dynamic ange o he p ocessed signals,
and low powe consump ion. Such a CDTA can be
use ul whe e e hese equi emen s a e mo e ele an
han he speed and bandwid h, ypically o implemen -
ing ha dwa e emula o s o mem is i e sys ems.
Keywo ds
Accu acy, CDTA, CDU, dynamic ange, imple-
men a ion, OTA.
1. In oduc ion
The Cu en Di e encing T ansconduc ance Ampli ie
(CDTA) in oduced in 2003 [1] o i s simpli ied e sions
Cu en Followe T ansconduc ance Ampli ie (CFTA)
[2] and Cu en In e ing T ansconduc ance Ampli ie
(CITA) [3] ha e become popula building blocks o
designing a ious linea and nonlinea analog ci cui s,
pa icula ly equency il e s [4], oscilla o s [5], wa e
gene a o s [6], igge s [7], and accu a e ec i ie s [8].
Since he CDTA is no cu en ly a ailable as an o -
he-shel de ice, mos o he p oposed CDTA applica-
ions a e checked only ia simula ions, using models o
bo h classical CMOS [8] o bipola [9] s uc u es and
also low- ol age low-powe echniques [10]. In mos
cases, howe e , he end s age is a design and compila-
ion o he co esponding SPICE code. The only ex-
cep ion known o he au ho s o his ex is an expe i-
men al chip, designed and manu ac u ed in 2007 in he
0.7 µm CMOS echnology [11], which does no comply
wi h oday’s demands o low powe consump ion and
linea o se ee cha ac e is ics o he in e nal blocks o
he Cu en Di e encing Uni (CDU) and Ope a ional
T ansconduc ance Ampli ie (OTA).
Less commonly, wi h he aim o comple ing he sim-
ula ions wi h expe imen al e i ica ion, he CDTA is
buil om o he o - he-shel ICs. Then he CDU can
be implemen ed ia wo Posi i e Cu en Con eyo s
o he 2nd Gene a ion (CCII+) [12], which a e pa
o he ansimpedance OpAmp AD844. The popula
LM13700 o simila ci cui s can be used as he sub-
sequen OTA s age. The so-called diamond ansis o
OPA860 is used in se e al designs [13]. Howe e , in all
he abo e cases, he linea ope a ion o he CDTA can
be accomplished only wi hin a na ow dynamic ange.
On he o he hand, i is desi able o ex end his egion
owa ds he limi s gi en by DC supply ails. As ega ds
he OTA, he only comme cial ype, MAX435(436)
wi h a gene ic linea ol age- o-cu en cha ac e is ic,
is no cu en ly a ailable. Mo eo e , none o he abo e
ICs is a low-powe , low-o se , o high-accu acy de ice.
The pape summa izes some esul s o designing he
CDTA, consis ing o he accu a e CDU and OTA wi h
ail- o- ail linea DC cha ac e is ics, and ope a ing in
a equency egion o up o hund eds o kHz. Since he
CDU is also designed on he basis o OTA, he linea
OTA, o ol age-con olled cu en sou ce, is he co e
o his modula concep .
2. CDTA
The basic e sion o he CDTA wi h schema ic diag am
as in Fig. 1(a) is a block wi h ou e minals. The p
and nlow-impedance e minals se e as inpu s o cu -
en s Ipand In. The high-impedance z e minal gen-
e a es a cu en Izequal o he di e ence be ween he
Ipand Incu en s, and he high-impedance x e minal
is in e nally in e connec ed wi h he z e minal ia he
OTA s age wi h he ansconduc ance gm. The co e-
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 824
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
sponding beha io al model o he CDTA is shown in
Fig. 1(b). Since he inpu pa o he CDTA, p o iding
he Izcu en as a di e ence be ween inpu cu en s,
is called Cu en Di e encing Uni (CDU), he CDTA
can be conside ed an in e connec ion o he CDU and
OTA.
(a) Schema ic symbol. (b) Beha io al model.
Fig. 1: CDTA.
CDTA as an ideal unc ional block can he e o e be
modeled by a se o equa ions:




Vp
Vn
Iz
Ix




=



0 0 0 0
0 0 0 0
1−1 0 0
0 0 gm0




·



Ip
In
Vz
Vx




.(1)
No e ha , o he sake o inc easing he CDTA uni-
e sali y, he numbe o i s cu en x e minals can
be inc eased, and he di ec ions o he co esponding
cu en s Ixcan be modi ied. On he o he hand, he
CDTA layou can be simpli ied o applica ions whe e
one o he po n e minals is no u ilized. Such single-
inpu CDTAs a e hen deno ed as CFTA (Cu en Fol-
lowe T ansconduc ance Ampli ie - wi h only he p
e minal) [2] and CITA (Cu en In e e T anscon-
duc ance Ampli ie - wi h only he n e minal) [3].
Ano he known modi ica ion o he CDTA is he
ZC-CDTA (Z-Copy CDTA). De ails a e gi en in [3].
3. CDTA Implemen a ion
The p oposed concep ion o he CDTA implemen a ion
is shown in Fig. 2. The cu en s Ipand Ina e sensed
by wo ixed esis o s R. The di e ence be ween hei
ol age d ops is ans o med o he ou pu cu en o
he OTA wi h he ansconduc ance gm= 1/R, and
he cu en lowing ou o he z e minal is equal o
he di e ence be ween he cu en s Ipand Iz. I he R
alue is chosen low enough (up o 100 Ω), hen hese
esis o s di ec ly de e mine he low-inpu esis ances
o pand n e minals. Such a design s a s om he
ac ha , pa icula ly o CMOS implemen a ions o
he CDTA, hese esis ances can be e en in kΩ alues
[5] and, in addi ion, hey a e nonlinea , and hus de-
penden on he signal. Fu he mo e, he CDU imple-
men a ion on he ansis o le el can be a he com-
plica ed and hus in oduce nonlinea i y, o se , and
equency dependence in o he CDU pa ame e s. The
ci cui complexi y is also esponsible o highe o al
powe consump ion. On he o he hand, he ci cui in
Fig. 2 does no su e om such p oblems. I bene i s
om he ac ha non-ze o inpu esis ances a e al-
lowed by de ini ion bu on a le el ha is accep able o
a gi en applica ion. The ac ha he impedances o
pand n e minals a e linea and o esis i e na u e up
o equency band, in which he pa asi ic capaci ances
o he OTA inpu s begin o ake e ec , can be u ilized
o applica ions which use he ixed inpu esis ances o
he CDU o gene a ing he ans e unc ions o il e s
[14].
Fig. 2: Concep ion o he CDTA implemen a ion.
I ollows om Fig. 2 ha he ol age o z e minal is
ans o med by ano he , now single-inpu linea OTA,
in o he cu en Ix.
Wi h espec o he equi ed linea i y, accu acy, and
high dynamic ange, he OTAs may be implemen ed
ia he Howland ol age-con olled cu en sou ce [15]
employed by accu a e ail- o- ail ope a ional o ins u-
men a ion ampli ie s. Connec ing addi ional OTAs o
he node zp o ides he CDTA wi h mul iple cu en
ou pu s. The di ec ion o cu en Ixcan be simply
changed ia e e sing he pola i y o he di e en ial
inpu o he Howland sou ce. The concep ion in Fig. 2
he e o e ep esen s a modula app oach when a ious
e sions o he CDTA, CFTA, o CITA can be ob ained
wi h he help o esis o s and OTAs.
The schema ic o he implemen ed OTA in Fig. 3
s a s om he well-known Howland cu en pump.
The ins umen a ion ampli ie AD8226 wi h p ese
uni y gain main ains i s ou pu ol age, which is a sum
o he inpu ol age Vin and he ol age a he REF e -
minal. Since he la e ol age is a bu e ed ol age o
he ou pu e minal, he ol age ac oss he esis o RS
is equal o Vin independen ly o he ol age a he ou -
pu e minal, and he cu en Iou is he e o e gi en by
he a io Vin/RS.
The conc e e ampli ie s in Fig. 3 we e selec ed
wi h espec o gi ing p e e ence o accu acy as is
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 825
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
Fig. 3: OTA as he Howland cu en sou ce. VS=±5V. Fo
RS= 100 Ω, he ansconduc ance gm= 10 mS.
speci ied in Sec. 1. Bo h ampli ie s ha e app ox-
ima ely he same 1.5 MHz bandwid h. Fo he dy-
namic ange o p ocessed signals, he limi ing ac o is
he maximum powe supply ol age o OPA735 (±6V).
The implemen ed CDTA is supplied wi h ±5V. The
ampli ie s AD8226 and OPA735 p o ide low o se ol -
age (50 µV and 5 µV) wi h low d i (0.5 µV/◦C and
0.05 µV/◦C). The inpu di e en ial impedance o he
AD8226 is 800 MΩwi h a pa asi ic capaci ance o 2 pF,
hus i does no deg ade he ou pu esis ance o he
p elimina y CDU. Low inpu bias cu en (200 pA) o
he OPA735-based ol age bu e ep esen s a negligi-
ble e o in se ing he ou pu cu en .
OPA735 p o ides he ail- o- ail ou pu swing wi hin
50 mV o he ails. Mo e limi ing o he dynamic ange
o he OTA is he common mode inpu ol age ange.
Fo 5 V powe supply, he maximum common inpu
ol age o he OPA735 is ca 3.5 V. The AD8226 is
also speci ied as a ail- o- ail ou pu ampli ie , wi h
he inpu ange depending on he common and di -
e en ial ol ages and on he ol age a REF e minal.
Bo h ampli ie s p o ide he abili y o go wi h he inpu
ol age 0.1 V below he nega i e supply.
The high CMRR (mo e han 90 dB) and 0.01 %
accu acy o in e nal gain esis o s o AD8226 as well as
he high open-loop gain o OPA735 (130 dB) a e good
p e equisi es o a easonably high ou pu impedance
o he Howland cu en sou ce.
4. Analysis o Real E ec s
As ollows om Fig. 2, each sensing esis o R= 100 Ω
in he pand ninpu s is loaded wi h high-impedance
inpu o OTA wi h 2 pF pa asi ic capaci ance, which
p o ides a pa asi ic pole in he ans e unc ion wi h
a cu o equency o 800 MHz, hus absolu ely beyond
he CDTA bandwid h.
As ega ds he analysis o eal e ec s o he cu en
sou ce in Fig. 3, he e a e some esul s a ailable in
pape s dealing wi h a ious e sions o he Howland
pump [16]. Le us ocus on he DC analysis o he
key pa ame e , he in e nal esis ance o he cu en
sou ce Ri=Vou /Iou (see Fig. 4). I is use ul o s a
om he simpli ied schema ic o he di e ence ampli-
ie , which is a pa o he ins umen a ion ampli ie
AD8226. The nominal alue o he esis o s R1 o R4in
Fig. 4 is RN= 50 kΩ, and hei p ecision d ama ically
a ec s he in e nal esis ance. Suppose ha he open
loop gain o OPA1 is su icien ly high and he e o e i
does no in luence he p ecision o he ou pu cu en .
Then he ou pu esis ance can be subs an ially in lu-
enced by he pa ame e s o o he componen s wi hin
he eedback loop, namely by he esis ance RS, gain
Abo he bu e employing OPA2, and he common-
ol age gain Acm o OPA1. A ou ine analysis yields
he o mula o he in e nal esis ance:
Ri=RS
1−Ab
1 + Acm +R2
R1
1 + R4
R3
.
(2)
Fig. 4: Simpli ied OTA schema ic wi h deno ed di e ence am-
pli ie wi h OPA1 as a pa o he ins umen a ion
ampli ie AD8226 o analysing he ou pu esis ance
Vou /Iou .
In he ideal case, i.e. o Acm = 0,Ab= 1, and
R1=R2=R3=R4=RN, he in e nal esis ance is
in ini e.
Conside he a iances o he esis ances R1 o R4
abou he nominal alue RN= 50 kΩ:
Rk=RN+ ∆Rk=RN(1 + δk),
δk=∆Rk
RN
,
k= 1 . . . 4.
(3)
Equa ion (2) can hen be ew i en in he o m:
Ri=RS
1−Ab
1 + Acm +1 + δ2
1 + δ1
1 + 1 + δ4
1 + δ3
.
(4)
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 826
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
Le us pe o m he wo s -case analysis o he ampli-
ie AD8226, whose in e nal esis ances R1 o R4a e
made wi h an e o below 0.01 %, hus:
|δk|=δmax = 10−4,
k= 1 . . . 4.(5)
The wo s case, he lowes possible esis ance Ri,min,
occu s o δ1=−δ2=−δ3=δ4=δmax. Subs i u ing
his condi ion in o Eq. (2) and a anging yield:
Ri,min =RS
1−Ab(1 −δmax)Acm
2+1
1 + δmax .(6)
I he a ia ion o he ixed sensing esis o RSis no
conside ed, hen he in e nal esis ance depends on he
gains Acm and Ab. Since he common mode ejec ion
o he AD8226 is highe han 90 dB, Acm <3.16·10−5.
The bu e employs he OPA735 wi h an open-loop
gain o 130 dB, hus A0= 3.16 ·106, and he e o e
Ab=A0(1 + A0)=0.999999684. These numbe s,
p ojec ed in o Eq. (6), demons a e he negligible im-
pac o Acm and Abo AD8226 and OPA735 on he
esul . Then:
Ri,min ≈RS
1−1−δmax
1 + δmax
=
=RS
21 + 1
δmax ≈RS
2δmax
.
(7)
Fo a maximum esis ance e o o 0.01 %, he
es ima ed lowes in e nal esis ance is 5000 RS, hus
500 kΩ. The measu emen s on implemen ed CDTA
con i m alues abo e 800 kΩ(see Sec. 5. ).
No e he high sensi i i y o he in e nal esis ance o
he bu e gain Abin Eq. (6). This goes o show he
impo ance o he high open loop gain o he OpAmp
which implemen s he bu e . Fo example, a low gain
o 10000 would esul in lowe ing he Ab o 0.9999
and dec easing he in e nal esis ance o he sou ce o
3300 RS. Tha is why one mus eckon wi h dec eas-
ing in e nal esis ance o highe equencies whe e he
OpAmp gain goes down.
On he o he hand, Eq. (4) can be used o de i -
ing he condi ion o DC (in)s abili y o he ci cui .
Fo a s able ci cui , he in e nal esis ance mus be
posi i e. The s abili y condi ion eads:
Ab<
1 + 1 + δ4
1 + δ3
1 + Acm +1 + δ2
1 + δ1
.(8)
Fo he Howland ci cui o be DC s able o all
possible a ia ions o he e o s (Eq. (3)), he con-
di ion (Eq. (8)) would ha e o be ul illed also o
−δ1=δ2=δ3=−δ4=δmax, o :
Ab<1−δmax
1 + δmax
1
1 + Acm
1−δmax
2
≈1−δmax
1 + δmax
.(9)
The la e simpli ied o mula holds o Acm 1.
Fo δmax = 10−4,Ab<0.9998, which would
ep esen he bu e implemen a ion by an OpAmp
wi h a non ealis ic low open-loop gain o 5000. The in-
e nal esis ance (Eq. (4)) would hen be e y low. On
he con a y, u ilizing an accu a e high-gain OpAmp
can, o un a o able sp ead o he esis ance alues o
AD8226, lead o ins abili y. I can be de i ed om
Eq. (9) ha , u ilizing he OPA735, he Howland ci -
cui would su ely be s able o any a bi a y esis i e
e o s wi h un ealis ically low δmax = 1.58 ·10−7. The
ci cui op imiza ion wi h he aim o maximizing Ri
and concu en ly p ese ing s able beha io is possible
ia expe imen al imming o esis ances, bu wi h he
ins umen al ampli ie buil om classical OpAmps
and accu a e esis o s [15].
The symbolic analysis o he ci cui in Fig. 4, made
wi h he linea one-pole models o bo h OpAmps,
p o ides he ollowing esul s ( he p oo s a e gi en in
he Appendix):
•The ou pu impedance exhibi s wo eal poles,
which do no depend on RSand on he impedance
le el RNo esis o s R1 o R4.
•The ou pu impedance is equal o Riacco ding o
Eq. (4) o ze o equency, and i con e ges o RS
o equencies ending o in ini y.
•The ou pu impedance con ains a capaci i e pa ,
which should be conside ed when designing appli-
ca ions which employ he implemen ed CDTA.
•The posi i e (nega i e) Riis he necessa y and
also su icien condi ion o he s abili y (ins abil-
i y) o he Howland ci cui .
5. Measu emen s on
Implemen ed CDTA
This sec ion summa izes he esul s o DC and
AC measu emen s on he CDTA, implemen ed
acco ding o Fig. 3 and Fig. 4 wi h he pa ame e s
Rp=Rn=RS= 100 Ω and VS=±5V. The
DC cha ac e is ics we e measu ed using he pa ame -
ic analyze B1500A and he equency esponses by
he ne wo k analyze E5061B, bo h made by Keysigh
Technologies, Inc.
Figu e 5, Fig. 6, and Fig. 7 show he CDU measu e-
men s.
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 827
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
Figu e 5 p o ides he DC cha ac e is ics Iz s Ip
wi h he cu en Inas a pa ame e , and Iz s Inwi h
he cu en Ipas a pa ame e . Bo h ypes o cha ac-
e is ics we e measu ed wi h he z e minal g ounded
(ze o RLload esis ance). Fo he p e minal, he cu -
en o se and posi i e and nega i e sa u a ion le els
we e 4µA, −14.5mA, and 12.5 mA. Fo he n e mi-
nal, hese alues we e 0.6 µA, −12.5mA, and 14.5 mA.
No e he excellen linea i y o all he cha ac e is ics.
(a) Iz(Ip), pa ame e In.
(b) Iz(In), pa ame e Ip.
Fig. 5: CDU: DC cha ac e is ics.
The cha ac e is ics Iz s Ipand Inin Fig. 6 we e
measu ed o a ious loads connec ed o he z e mi-
nal. The sa u a ion le els now depend on he ol age
Vz, which appea s on he load due o he lowing cu -
en Iz. When d i ing he p e minal, he maximum
o se (3.16 µA) was measu ed o a load o 10 Ω. Fo
he n e minal, he maximum o se (0.4 µA) was o
RL= 1 kΩ.
Figu e 7 summa izes AC measu emen s on he CDU,
namely he Ampli ude (A) and Phase (P) equency
esponses o cu en gains om he po n e minal o
he g ounded z e minal. The cu o equency om
(a) Iz(Ip),In= 0, pa ame e RL.
(b) Iz(In),In= 0, pa ame e RL.
Fig. 6: CDU: DC cha ac e is ics.
Fig. 7: AC cha ac e is ics o CDU: Ampli ude (A) and Phase
(P) equency cha ac e is ics om pand n o z e minal.
he pand he n e minal is 275 kHz and 285 kHz, e-
spec i ely.
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 828

THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
The DC ansconduc ance cha ac e is ic, i.e. he
ou pu cu en Iou s he inpu ol age Vin, can be
easily ob ained om Fig. 6(a) a e a simple ans o -
ma ion o axes: Iou =Izand Vin =R·Ip, whe e
R= 100 Ω. Then, o RL= 0, he OTA sa u a es o
Vin ≈1.5V.
The DC in e nal esis ance o he OTA cu en ou -
pu is 883 kΩ. I was ead om he cha ac e is ic Iou
s Vou measu ed acco ding o Fig. 4.
The equency esponse o he OTA, i.e. he
ansconduc ance s equency, can be ob ained om
he cu es in Fig. 7, whe e he le el o 0 dB co esponds
o he alue gm= 1/R = 100 mS. The bandwid h is
525 kHz.
6. Benchma k Ci cui - CDTA
Biquad
The implemen ed CDTA was used in he 2nd o de
il e in Fig. 8, and he il e ope a ion was es ed
ia AC, DC, and ansien measu emen s. This ci -
cui was designed on he basis o simila il e wi h he
Cu en -Con olled Cu en Di e encing Bu e ed Am-
pli ie (CC-CDBA), published in [14], whe e he pa a-
si ic esis ances o pand n e minals o he CDU we e
used o implemen ing he equi ed ans e unc ions.
The ci cui in Fig. 8 p o i s om wo e ec s:
• he esis o s R1and R2a e pa s o he imple-
men ed CDTA,
• he ou pu bu e is also a pa o he CDTA
(OPA 735 in he Howland sou ce in Fig. 3).
Fig. 8: Biquad based on a single CDTA.
The ans e unc ions o he il e a e as ollows:
VLP
Vin
=NLP(s)
D(s),
VBP
Vin
=NBP(s)
D(s),
VHP
Vin
=NHP(s)
D(s),
(10)
whe e
D(s) = s2R2C1C2+sC2+gm,
NLP(s) = 1 + sC1(R2−R1)
R1
,
NBP(s) = −sC1,
NHP(s) = s2R2C1C2+s(C2−C1).
(11)
(a) LP.
(b) BP.
(c) HP.
Fig. 9: Ideal (KI,ϕI) and measu ed (KM,ϕM) equency e-
sponses o LP, BP, HP il e om Fig. 8.
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 829
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
The na u al equency and quali y ac o a e:
0=1
2π gm
R1C1C2
,
Q= gmR1
C1
C2
.
(12)
Fo he pa ame e s p oposed in he implemen a ion,
namely C1=C2= 100 nF, R1= 100 Ω,gm= 10 mS,
0and Qa e 15.9 kHz and 1, espec i ely. No e ha o
achie e low-pass o high-pass esponse, he condi ion
R1=R2o C1=C2mus be ul illed.
The measu ed equency esponses in Fig. 9 a e in
a good acco dance wi h he design objec i es. I ol-
lows om a compa ison o ideal and measu ed e-
sponses ha he implemen ed CDTA can sa is ac o ily
ope a e in his applica ion wi hin a equency ange o
up o hund eds o kHz. The pa asi ic low- equency
phase shi o he high-pass sec ion is caused by mis-
ma ched capaci ances C1= 110 nF and C2= 106 nF,
whose di e ence gene a es addi ional ans e ze o
acco ding o Eq. (11).
7. Conclusion
Based on he concep o he Howland cu en pump,
he CDTA implemen a ion ia comme cial ICs is p o-
posed. A en ion is paid o he accu acy and maximum
dynamic ange o p ocessed signals. The measu emen s
con i med a high linea i y o DC cha ac e is ics o he
CDU and OTA. On he o he hand, he weak poin o
he Howland sou ce is he high sensi i i y o i s in e nal
esis ance o a ia ions o esis ances in he di e ence
ampli ie and he possibili y o uns able beha io , as
well as a sha p dec ease in in e nal impedance wi h
inc easing equency. These eal in luences a e ana-
lyzed in de ail. I is shown ha he bandwid h o his
impedance is he smalle , he highe he DC in e nal
esis ance ha can be adjus ed ia imming he esis-
o s in he di e ence ampli ie . A equency model o
he in e nal impedance is ound. I u ns ou ha he
bandwid h o he Holland sou ce is go e ned by i s DC
in e nal esis ance and by a capaci ance which is indi-
ec ly p opo ional o he bandwid h o he OpAmps
in he sou ce. I is he e o e ad isable o ocus on o e -
coming hese undamen al limi a ions in cons uc ing
he wideband con olled cu en sou ces, and sea ching
o mo e accep able ci cui solu ions p o iding highe
bandwid hs han he Howland sou ce can o e , hough
a he cos o lowe in e nal impedances.
Acknowledgmen
Fo he esea ch, he in as uc u e o he K217
Depa men , Uni e si y o De ence B no, was used.
Re e ences
[1] BIOLEK, D. CDTA - Building Block o Cu en -
Mode Analog Signal P ocessing. In: Eu opean
Con e ence on Ci cui Theo y and Design (EC-
CTD). K akow: IEEE, 2003, pp. 397–400.
ISBN 83-88309-95-1.
[2] HERENCSAR, N., J. KOTON, I. LATTEN-
BERG and K. VRBA. Signal-Flow G aphs o
Cu en -Mode Uni e sal Fil e Design Using Cu -
en Followe T ansconduc ance Ampli ie s (CF-
TAs). In: In e na ional Con e ence on Applied
Elec onics (APPEL). Pilsen: Uni e si y o Wes
Bohemia, 2008, pp. 69–72. ISBN 978-80-7043-654-
7.
[3] BIOLEK, D., R. SENANI, V. BIOLKOVA and
Z. KOLKA. Ac i e Elemen s o Analog Signal
P ocessing: Classi ica ion, Re iew, and New P o-
posals. Radioenginee ing. 2008, ol. 17, iss. 4,
pp. 15–32. ISSN 1210-2512.
[4] BIOLKOVA, V. and D. BIOLEK. Shadow il e s
o o hogonal modi ica ion o cha ac e is ic e-
quency and bandwid h. Elec onics Le e s. 2010,
ol. 46, iss. 12, pp. 830–831. ISSN 0013-5194.
DOI: 10.1049/el.2010.0717.
[5] KESKIN, A. U. and D. BIOLEK. Cu en mode
quad a u e oscilla o using cu en di e encing
ansconduc ance ampli ie s (CDTA). IEE P o-
ceedings - Ci cui s, De ices and Sys ems. 2006,
ol. 153, iss. 3, pp. 214–218. ISSN 1350-2409.
DOI: 10.1049/ip-cds:20050304.
[6] CHIEN, H.-C. Cu en -mode squa e wa e
gene a o s based on a single CDTA. In-
e na ional Jou nal o Elec onics. 2017,
ol. 104, iss. 9, pp. 1589–1605. ISSN 1362-
3060. DOI: 10.1080/00207217.2017.1312714.
[7] SRIVYSHNAVI, T. and A. SRINIVASULU.
A cu en mode Schmi igge based on Cu en
Di e encing T ansconduc ance Ampli ie . In: 3 d
In e na ional Con e ence on Signal P ocessing,
Communica ion and Ne wo king (ICSCN). Chen-
nai: IEEE, 2015, pp. 1–4. ISBN 978-1-4673-6823-
0.
[8] BIOLEK, D., E. HANCIOGLU and A. U.
KESKIN. High-pe o mance cu en di e encing
ansconduc ance ampli ie and i s applica ion in
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 830
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
p ecision cu en -mode ec i ica ion. AEU - In e -
na ional Jou nal o Elec onics and Communica-
ions. 2008, ol. 62, iss. 2, pp. 92–96. ISSN 1434-
8411. DOI: 10.1016/j.aeue.2007.03.003.
[9] TANGSRIRAT, W., T. DUMAWIPATA and
W. SURAKAMPONTORN. Mul iple-inpu
single-ou pu cu en -mode mul i unc ion
il e using cu en di e encing ansconduc-
ance ampli ie s. AEU - In e na ional Jou nal
o Elec onics and Communica ions. 2007,
ol. 61, iss. 4, pp. 209–214. ISSN 1434-8411.
DOI: 10.1016/j.aeue.2006.04.004.
[10] KHATEB, F. and D. BIOLEK. Bulk-D i en
Cu en Di e encing T ansconduc ance Ampli ie .
Ci cui s, Sys ems, and Signal P ocessing. 2011,
ol. 30, iss. 5, pp. 1071–1089. ISSN 1531-5878.
DOI: 10.1007/s00034-010-9254-9.
[11] BIOLEK, D., A. U. KESKIN and V. BI-
OLKOVA. G ounded capaci o cu en mode sin-
gle esis ance-con olled oscilla o using single
modi ied cu en di e encing ansconduc ance
ampli ie . IET Ci cui s, De ices &Sys ems. 2010,
ol. 4, iss. 6, pp. 496–502. ISSN 1751-8598.
DOI: 10.1049/ie -cds.2009.0330.
[12] TLELO-CUAUTLE, E. In eg a ed Ci cui s o
Analog Signal P ocessing. 1s ed. New Yo k:
Sp inge , 2013. ISBN 978-1-4614-1383-7.
[13] KHATEB, F., J. VAVRA and D. BIOLEK.
A No el Cu en -Mode Full-Wa e Rec i ie Based
on One CDTA and Two Diodes. Radioenginee ing.
2010, ol. 19, iss. 3, pp. 437–445. ISSN 1210-2512.
[14] ONER, S. E., M. KOKSAL and M. SAG-
BAS. Elec onically Con ollable Biquads Using
Single CDBA. In: IEEE In e na ional Sympo-
sium on Ci cui s and Sys ems (ISCAS). Kos:
IEEE, 2006, pp. 3333–3336. ISBN 0-7803-9389-9.
DOI: 10.1109/ISCAS.2006.1693339.
[15] Texas Ins umen s. A Comp ehensi e S udy o
he Howland Cu en Pump. In: Texas In-
s umen s: Applica ion Repo [online]. 2013.
A ailable a : h p://www. i.com/li /an/
snoa474a/snoa474a.pd .
[16] YAZDANIAN, H. and M. M. SAMANI. Cha -
ac e is ics o he Howland cu en sou ce o
Bioelec ic Impedance Measu emen s Sys-
ems. In: 20 h I anian Con e ence on Biomed-
ical Enginee ing (ICBME). Teh an: IEEE,
2006, pp. 189–193. ISBN 978-1-4799-3232-0.
DOI: 10.1109/ICBME.2013.6782216.
Abou Au ho s
Dalibo BIOLEK ecei ed he M.Sc. deg ee in
Elec ical Enginee ing om B no Uni e si y o Tech-
nology, Czech Republic, in 1983, and he Ph.D. deg ee
in Elec onics om he Mili a y Academy B no, Czech
Republic, in 1989, ocusing on algo i hms o he sym-
bolic and nume ical compu e analyses o elec onic
ci cui s wi h a iew o he linea con inuous- ime and
swi ched il e s. He is cu en ly wi h he Depa men
o EE, Uni e si y o De ence B no (UDB), and wi h
he Depa men o Mic oelec onics, B no Uni e si y
o Technology (BUT), Czech Republic. His scien i ic
ac i i y is di ec ed o he a eas o gene al ci cui
heo y, equency il e s, mem-sys ems, and compu e
simula ion o elec onic sys ems. He has published
o e 400 pape s and is he au ho o se e al books
on ci cui analysis and simula ion. A p esen , he is
p o esso a BUT and UDB in he ield o Theo e ical
Elec ical Enginee ing. P o . Biolek is a Senio
Membe o he CAS/COM Czech Na ional G oup o
IEEE. Cu en ly he also se es as an Associa e Edi o
o Elec onics Le e s.
Ji i VAVRA was bo n in 1983. He ecei ed
he M.Sc. and he Ph.D. deg ees a he Facul y o
Elec ical Enginee ing and Communica ion om B no
Uni e si y o Technology, Czech Republic, in 2007 and
in 2012, espec i ely. His scien i ic ac i i y is di ec ed
o he a ea o analog signal p ocessing and mem is i e
sys ems.
Appendix A
Conside he ollowing ans e unc ions K1and K2
o he OPA1 and OPA2 ampli ie s in he model om
Fig. 4:
K1=ω1
s,
K2=Ab
1 + s
ω2
.(13)
He e, K1is a model o high-gain OpAmp (a pa o
he ins umen a ion ampli ie ) wi h he bandwid h ω1
co esponding o he bandwid h o uni y-gain ins u-
men a ion ampli ie , and K2models he ol age bu e
in Fig. 4, whe e Ab≈1and ω2is he cu o equency,
which co esponds o he GBW o he u ilized OpAmp.
The symbolic analysis o he in e nal impedance Zi
o he cu en sou ce leads o he o mula:
Zi=RS
a2s2+a1s+a0
a2s2+a1s+a0−b0
,(14)
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 831
THEORETICAL AND APPLIED ELECTRICAL ENGINEERING VOLUME: 15 |NUMBER: 5 |2017 |DECEMBER
whe e
a0=ω1ω2R1(R3+R4),
b0=ω1ω2AbR3(R1+R2),
a1=ω2Sp+Abω1R1(R3+R4),
a2=AbSp,
Sp=R1R3+R1R4+R2R3+R2R4.
(15)
Fo s= 0, Eq. (14) yields Eq. (2) o DC esis ance Ri
while neglec ing he common-mode gain Acm:
Ri=RS
a0
a0−b0
=RS
1−Ab
1 + R2
R1
1 + R4
R3
.
(16)
The de e minan o he quad a ic equa ion in he
denomina o o Eq. (14) is always posi i e:
a2
1−4a2(a0−b0) =
= (ω2Sp−Abω1R1(R3+R4))2+
+4A2
bSpω1ω2R3(R1+R2)>0.
(17)
The ci cui he e o e exhibi s wo di e en eal poles,
and he cu en esponse o he s eps in he load o
he OTA inpu will be mono onic wi hou o e shoo s.
The su icien condi ion o he s abili y, i.e. o bo h
poles o be nega i e, a e posi i e coe icien s a2,a1, and
a0-b0in he denomina o o Eq. (14), o a0> b0. As is
ob ious om Eq. (16), i is equi alen o he condi ion
Ri>0. The posi i e DC in e nal esis ance he e o e
gua an ees he s abili y.
I ollows om Eq. (14) ha o high equencies,
hus o s→ ∞, he impedance alls o RS. The
way his happens is gi en by he Zimodel in Fig. 10,
which can be de i ed by he ollowing a angemen o
Eq. (14):
Zi=RS+1
1
Req
+sCeq +s2Deq
,(18)
whe e Req =Ri−RS,
Ceq =a1
b0RS
,
Deq =a2
b0RS
,
(19)
a e pa ame e s o he esis o , capaci o , and F e-
quency Dependen Nega i e Resis o (FDNR) in
Fig. 10. Fo low equencies, he impedance is
RS+Ri−RS=Ri. I he equency inc eases, he
esis o Ri−RSis g adually shun ed by he capaci o
Ceq whose suscep ance inc eases p opo ionally wi h
he equency. As a second-o de e ec , he nega i e
conduc ance o shun ing FDNR, inc easing wi h he
squa e o equency, dec eases he o al esis i i y o
he pa allel con igu a ion. Howe e , i does no a ec
he inal d op o he impedance o his pa allel subci -
cui owa ds ze o, when Zi ends o RS. As he sim-
ula ion con i ms, Ceq is he dominan elemen o he
equency dependence o Zi, and he FDNR only mod-
i ies his dependence in he equency a ea whe e he
impedance is al eady supp essed. Fo p ac ical com-
pu a ion, o example o es ima ing he bandwid h o
he Zi, he schema ic in Fig. 10 can be simpli ied ia
emo ing he FDNR.
Fig. 10: Impedance model o he Howland ci cui .
Conside ing ha R1 o R4a e accu a e esis o s,
hus R1≈R2≈R3≈R4≈R, and ha Ab≈1, hen
he capaci ance Ceq om Eq. (19) can be es ima ed as:
Ceq ≈1
RS2
ω1
+1
ω2.(20)
Fo he Howland pump wi h AD8226 and OPA735 and
wi h RS= 100 Ω, he equi alen capaci ance is abou
3 nF. I may no be a p oblem i his sou ce is used
in he OTA s age o he CDTA since he cu en e -
minal xis usually connec ed o a low-impedance node.
A wo se case can occu wi h he OTA as a pa o he
CDU: a g ounded capaci o is equen ly connec ed o
he co esponding z e minal. Then i s capaci ance
should no be less han ca 10 ·Ceq.
I ollows om Fig. 10 ha he cu o equency o
he impedance is go e ned by he cu o equency ωc
o he co esponding RC cell, hus:
ωc≈1
RiCeq
≈RS
Ri
1
2
ω1
+1
ω2
.(21)
I e eals he disad an age o he Howland ci cui : he
highe he in e nal esis ance Riwill be adjus ed ia
op imizing he esis i e ne wo k R1 o R4, he lowe
he bandwid h will be, and he close o he ins abili y
egion he ci cui will be.
c
2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 832