Precise Implementation of CDTA

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