Applications of the Current Feedback Operational Amplifiers

Transcription

1 Analog Integrated Circuits and Signal Processing, 11, (1996) Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Applications of the Current Feedback Operational Amplifiers AHMED M. SOLIMAN Electronics and Comm. Engineering Dept., Cairo University, Egypt Received April 6, 1995; Revised May 10, 1996 Abstract. The current feedback operational amplifiers (CFOAs) are receiving increasing attention as basic building blocks in analog circuit design. This paper gives an overview of the applications of the CFOAs, in particular several new circuits employing the CFOA as the active element are given. These circuits include differential voltage amplifiers, differential integrators, nonideal and ideal inductors, frequency dependent negative resistors and filters. The advantages of using the CFOAs in realizing low sensitivity universal filters with grounded elements will be demonstrated by several new circuits suitable for VLSI implementation. PSPICE simulations using the AD844-CFOA which indicate the frequency limitations of some of the proposed circuits are included. Key Words: current feedback op amps 1. Introduction The current feedback operational amplifiers (CFOAs) also known as the transimpedance operational amplitiers are now commercially available in bipolar integrated circuits form from several manufacturers. [1-2]. Very recently a CMOS circuit configuration derived from the bipolar CFOA was described [3]. The CFOAs are now recognized for their excellent performance in high speed and high slew rate analog signal processing [4]. Recently current mode and voltage mode filters implemented from a single CFOA were given [5]. In this paper several applications of the CFOAs in realizing voltage amplifiers, integrators, inductors, frequency dependent negative resistors and filters are proposed. PSPICE simulation results are given which demonstrate the frequency limitations of some of the reported circuits. 2. The Current Feedback Op Amp The CFOA is a very versatile four-terminal active building block represented symbolically as shown in Fig. 1 and described by the following matrix equation: O loo / Iy i J Vy Iz = VZ (1) Vo Io The X terminal which is also defined as the inverting input terminal is characterized by a very low input impedance. The Y terminal which is also defined as the noninverting input has a very high input impedance. The two outputs Z and O exhibit a very high and a very low output impedance respectively. The CFOA is considered to be a cascade of a second generation current conveyor (CC II +) [6] and a voltage buffer. This paper concentrates on the applications of the CFOA with the output terminal Z (also known as the compensating pin) being available. It has been demonstrated in [5] that the CFOAs including the terminal Z are much more versatile than other existing CFOAs topologies, this fact will be clearly recognized from the applications of the CFOAs described in this paper. All the simulations included in this paper are based on using the PSPICE model for the AD 844 A/AD- CFOA in which the stray capacitance at the compensating pin Cz = 5.5 pf and Rz = 2.2 Mr2. The DC supply voltages used are -t-12v. In the following section the CFOA is used as the basic building block in realizing voltage amplifiers and integrators. 3. Voltage Amplifiers and Integrators The first building block considered here is the generalized three port voltage controlled voltage source (VCVS) shown in Fig. 2. The output voltage is given by: Vo = K(Vl - V2) (2-a)

2 266 Soliman Figs. 3(c), (d) represent similar simulations for the inverting amplifier of Fig. 2(c). It is seen that the phase is independent of R1 since the inverting input X is at virtual ground which eliminates the effect of the stray capacitance Cx on the amplifier characteristics. The differential voltage integrator is shown in Fig. 4. The circuit has the advantage of using a single resistor and a single grounded capacitor, where as in the conventional op amp balanced time constant integrator two resistors and two capacitors are required. The output voltage Vo is given by: where Fig. 1. The symbolic representation of the CFOA. RE K = -- R1 (2-b) It is seen that the circuit provides equal gain for the noninverting input V1 and the inverting input Vz, besides this gain is controlled by varying the resistor R1 without affecting the bandwidth which is controlled by the grounded resistor R2. These properties are not achievable with the VCVS using conventional op amps in which both resistors are floating, the noninverting voltage gain equals the inverting voltage gain plus one and the bandwidth depends on the voltage gain and the gainbandwidth of the op amp. Fig. 2(b) represents the noninverting VCVS which has infinite input impedance and gain equals to K which can be less than one. The inverting VCVS shown in Fig. 2(c) has a finite input impedance given by R1 which must be taken much larger than the input resistance of the CFOA which is typically of the order of 65f2. Of course the realization of an inverting amplifier with infinite input impedance and with grounded resistors requires two CFOAs as shown in Fig. 2(d). Figs. 3, (b) represent the PSPICE simulations of the magnitude in db and the phase in degrees for the noninverting amplifier shown in Fig. 2(b) with Rz = 20 Kf2 and RI = 5,10,15 and 20 Kf2. From the simulations it is seen that the 3 db frequency is very close to the theoretical value which is given by f3 db = MHz. At frequencies above 100 KHz, 27r R2Cz -- the phase characteristics depend on R1, this is due to the zero which resulted from the stray input capacitance at port X. where Vo ---- W~ (V~ - V2) S 1 w0 = -- CR (3-a) (3-b) Figs. 4(b) and 4(c) represent the noninverting and the inverting integrators which are obtainable from Fig. 4 by setting Va = 0 and V] = 0 respectively. Fig. 4(d) represents the infinite input impedance inverting integrator using a grounded resistor. Figs. 5, (b) represent the magnitude and phase for the noninverting integrator shown in Fig. 4(b) with R = 5, 10, 15 and 20 Kf2 and C = lne The magnitude error is given approximately by Cz/C and equals to 0.55%. Similar simulations for the inverting integrator of Fig. 4(c) are given in Figs. 5(c), (d). It should be noted that the phase for the inverting integrator equals to 90 ~ over a wide frequency range and is independent of the magnitude of R (due to the virtual ground at X). It is also possible to realize differential integrators with infinite input impedance at both inputs using two CFOAs as shown in Figs. 6 and 6(b) where Vo in both cases is given by eqn. (3). Of course replacing the capacitors in Fig. 6 by resistors, one obtains the voltage instrumentation amplifiers, based on their well known current conveyor version [7-8]. 4. Inductor and FDNR Realizations In this section the realizations of nonideal and ideal grounded inductors using the CFOA are considered Series L-R Circuits Fig. 7 represents a series L-R circuit based on the circuit given in [9].

3 Applications of the CFOAs 267 Vi. Y -vo Vl. V2 o =% R1 (b) RI Vi., Y -_Vo Vi 9 ~ m ---. Vo (c) (d) Fig. 2. The differential VCVS. (b) The noninverting VCVS. (c) The inverting VCVS. (d) The infinite input impedance inverting VCVS. The input impedance Zi is given by: Zi = scr1r2 -I- (R1 + R2) (4) Thus it is seen that the circuit realizes a series L-R circuit, with L = CR1Rz and R = Rt + R2. Another circuit which employs also four resistors, a single capacitor together with the CFOA is shown in Fig. 7(b). Its input impedance is given by: Zi = scr1r2 + R1 sc[r2 - RI(K - 1)] + 1 (g-a) Thus it is seen that the necessary condition to realize a series L-R circuit is given by: In this case L = CR1R2 and R = R1, that is for equal R1 and R2 this circuit has double the Q factor of the circuit of Fig. 7. Again this circuit is derived from the inductor circuit given in [ 10]. Fig. 8 shows the simulation results of the magnitude and the phase of the input impedance of the circuit of Fig. 7 with R1 = R2 = 1 KS2, C = 1 nf and R = 1 Kf2. Fig. 8(b) shows similar simulations for the circuit of Fig. 7(b) with R1 = R2 = 1Kf2, C = 1 nf, R = 10Kf2 and K = Parallel L-R Circuits R2 K = 1 + R--7 (5-b) In this section two new circuits are given, each circuit employs a single CFOA, a single capacitor and two

4 268 Soliman 20 T... Gain in Db. 15 ~ : Rl=Sk lo ~ $' : Rl=10k Rl=iSk \ 9 9 \ \ Rlw20k I I R2=20k r... r loohz 1.0KHz IOKHz 100KHz 1.0MHZ IOMH: o e 9 9 vdb(5) Od T... -2od ~ \L\ Rl~20k -40d ~ \ \\ \\ Rl=lOk -6o~ ~ R2-20k \ Rl~Bk IOOHz I.OIC~Iz 10KHz 100KHz 1.0MHz. o v * vp(5) (b) Fig. 3. The magnitude characteristics of the VCVS of Fig. 2(b). (b) The phase characteristics of the VCVS of Fig. 2(b).

5 \ % \\ Applications of the CFOAs 269 2O T... Gain in Db. zs" ~ ~ Rl=10k s" \ \,,.... ",\ \ '~ : R2=20k " \", i' \ \\ 100Hz i. 0KHz 10KHz 100KHg i. 0MHz 10MHI a * v ~ vdb(5) (c) Phase 200d ~... RI= 5k,10k, 15k,20k i6oa 9 7\... \ *,\ 9 ~,t\ 12oa R~=20k \ \ \ 10OHz 1.0KHZ 10KHz IO0KHZ 1.0MHz 10MHz (d) Fig. 3. (c) The magnitude characteristics of the VCVS of Fig. 2(c). (d) The phase characteristics of the VCVS of Fig. 2(c).

6 270 Soliman Vi. Y -Vo V1 9 V2 " --Vo R m T c vi~ R vi 9 > (b) -S % :v~ I C m (c) Fig. 4. The differential integrator. (b) The noninverting integrator. (c) The inverting integrator. (d) The infinite input impedance inverting integrator. (d) resistors only, that is two resistors less than the circuits of Fig. 7. For the circuit of Fig. 9 the input admittance is given by: 1 1 l Yi -- scr1r~ + ~ + R-2 (6) It is seen that the circuit realizes L = CR1Rz in parallel with R = R~R2 This circuit is equivalent to RI + R 2 " the two well known circuits using the op amp and the CC II as the active building blocks [11-12]. This circuit however has the advantage of using a grounded capacitor. A second parallel L-R circuit is shown in Fig. 9(b). A necessary condition for this circuit is to have R2 = R1, in this case it is clear that the circuit realizes a parallel L-R circuit with L = CR1R2 and R = R1, that is it has double the Q factor of the inductor circuit of Fig. 9, when equal resistors are used. This circuit however is very sensitive to the resistors ratio which must be unity (due to the cancellation of two terms in the denominator of the Yi). Practically R2 must be taken equals to R1 + Rx for proper operation of the circuit. Fig. 10 shows the simulation results of the magnitude and the phase of the input impedance of the circuit of Fig. 9 with R1 = R2 = 1 Kf2 and C = 1 ne Fig. 10(b) shows similar simulation results for the circuit of Fig. 9(b) with R1 = K~2, R2 = R1 + Rx = 1 K~2 and C = 1 nf. It is worth noting that the input resistance of this circuit at DC equals to (R2 - R1), and a phase error of 180 ~ can be noticed in the simulations if R1 is taken equals to R2 without compensating the effect of Rx. As pointed out before the design equation for R2 should be modified to take the effect of Rx into account. The sim-

7 ... //y Applications of the CFOAs 271 R=SK I i i o~ R=20k " ~ C=lnf 100Hz 1. OKHz 10KHz 100KHz 1.0MHz o *, ~ vdb(5) (a~ -40d T... ii Phage....,,,. -food " C=lnf d , T r ~ Hz 1.0KHZ 10KHz 100KHz 1.0MHz 10MH: * 9 * vp{5) (b) Fig. 5. The magnitude characteristics of the integrator of Fig. 4(b). (b) The phase characteristics of the integrator of Fig. 4(b).

8 272 Soliman Db ' R=5k r I o~ c=inf loohz I. 0XHz 10KHz 100KHz 1.0MHz = * 9 A vdb(5) (c/ ~o~ T T-&U... i 2 9 " 9 9 '\ : \ ~oo~-:'.\\... ~, R=Sk,10k,15k,20k I C=lnf SOd ~ d , ~ ~ ~ HZ 1.0KHz 10gHz IOOKHz 1.0m~z 10HH = * 9 A vp(s) (d) Fig. 5. (c) The magnitude characteristics of the integrator of Fig. 4(c). (d) The phase characteristics of the integrator of Fig. 4(c).

9 Applications of the CFOAs 273 Vl o. Y X Z 0 + R vo V2 o, V1 9 Z O _Vo v C C V2e -Vo (b) Fig. 6. Two equivalent differential integrators with infinite input impedance at both inputs.

10 274 Soliman R2 C Zi < Zi )- R1 I R It c A w Fig. 7. Series L-R circuits. (b) ulation results shown in Fig. lo(b) with R2 = R1 + Rx, are in good agreement with the theoretically expected ones Ideal lnductor Circuits Here two new and very attractive ideal inductor circuits are reported. The first circuit has the advantage of using two grounded capacitors[13], where as the second circuit has the advantage of using a single capacitor. For the inductor circuit of Fig. 1 l, the input admittance is given by: (1 - a) 1 (1 - a) 1 ac1 Y~ -- sc2r1r--~2 + R + R------~ + R2 C2R2 (7-a) The necessary condition to realize an ideal inductor is given by: ac1 = 1 -t- R2 R2(1 - a) C-'T --R + R1 (7-b) R and a = 1, that is all the Choosing R2 = RI =?- four resistors are taken equal, therefore equation (7-b) simplifies to C1 = 4C2 and in this case L = 2C2R1R2. A gyrator circuit using two CFOAs has been recently reported in [14]. It is well known that it is possible to realize a gyrator circuit using a single conventional op amp [15] or a single CC 1I [16]. A gyrator circuit which employs a single-cfoa is shown in Fig. ll(b). The input admittance of this circuit is given by: sc~- R2j [ 2(1-a)1 -t- RaR2 Y~ = (8-a) sca + ~ - N (a It is seen that the necessary conditions to realize an ideal inductor are given by: a=- a (8- b) 2 R2 = 2R1 (8-c) The magnitude of the realizable inductor is given by: L = CR1R2. (8-d) Practically equation 8(c) must take into consideration the effect of Rx, that is the design equation for R2 should be modified to: R2 = 2(R1 + Rx) (8-e) Fig. 12 shows the simulation results of the magnitude and the phase of the input impedance of the inductor circuit of Fig. ll with a = 1/2, R = 2 Kg2, RI=R2=IKf2, CI=4nFandC2=lnE The simulation results are in close agreement with the theoretical values up to 40 KHz. At higher frequencies however there is a slight increase in the magnitude of L.

11 Applications of the CFOAs K T... Izll 2.OK ~.... lli(v3) 40d T... Pha6e 2oa ~ i p SEL>>I " i Od 0Hz 50KHz IOOKHz 150KHz 200KHz o 180-ip(v3) 1.4K T... Iz~l 1.2K J, I.OK... 1/i(v3) 40d y... Phase J I 9 " " i 20dd I se~>>: i o 180-ip(v3) ~) Fig. 8. The magnitude and phase of Zi of the inductor of Fig. 7. Co) The magnitude and phase of Zi of the inductor of Fig. 7(b).

12 276 Soliman R2 RI R2 C '1 I < vi Yi > Co) Fig. 9. Parallel L-R circuits. Fig. 12(b) shows similar results for the circuit of Fig. ll(b) with a = 1/2, R = 2 Kf2, R 1 = 1.9 Kf2, R2 = 4 Kg2 (with Rx ~ 100f2) and C = 1 ne The simulation results are in close agreement with the theoretical values up to 100 KHz. At higher frequencies however the magnitude of the simulated L is lower than the theoretical expected value FDNR Circuits In some cases it may be desirable to realize an FDNR in series or in parallel with a capacitor. The circuits of Figs. 7 and 9 can be converted to realize FDNR-C circuits using the RC : CR transformation. The realization of an FDNR-R circuit is described next. Fig. 13 realizes an FDNR of magnitude D = C1C2R2/2 in parallel with a resistor of magnitude 2R [17]. The necessary condition for this realization is that, ~ R2 = 2 + ~. cl Fig. 13(b) shows the magnitude and the phase of Yi of the circuit of Fig. 13 which is driven by a voltage source of magnitude 1 V and taking Ca = C2 = 1 nf, R1 = 10 Kf2 and R = R2 = 30Kf2. From the simulations it is found that the resonance frequency at which the phase of Yi changes abruptly from zero to 180 ~ is very close to its theoretical value given by: 27r ~/C1C2RR2 = 5.3 KHz. It is seen that for frequencies above 20 KHz, the circuit behaves as an ideal FDNR of magnitude D = 15 ff.s. Two more circuits realizing an ideal FDNR are given next. Fig. 13 (c) represents the FDNR circuit obtained from that of Fig. ll using the RC : CR transformation. Taking C2 = C1 = C and R 2 = 4R1 it is seen that the circuit realizes an ideal FDNR of magnitude D = 2C2R1. This circuit is of theoretical interest and employs four capacitors. The circuit of Fig. 13(d) employs the minimum number of capacitors namely two. By taking a = 1 and C1 = 2C2, the circuit realizes an ideal FDNR of magnitude D = 2C2R. Fig. 13(e) shows the simulation results of the magnitude and the phase of Yi when the circuit is driven by a unity voltage source and taking Cl = 2 nf, C2 = 1 ne R = 10 Kff2, R1 = 2 K~ and a = 1/2, to realize an FDNR of magnitude D = 20 ff.s. 5. Single-CFOA Filters In this section several new filter configurations using a single-cfoa are given. First the filter sections of order one are considered briefly First Order Filters It is well known that the conventional op amp realizes only a class of noninverting first order transfer functions. With the CFOA however a noninverting first order transfer function of the from given by:

13 Applications of the CFOAs T i/ilv3) 90d ~ Hz 50KHz IOOKHz 150KHZ 200KH~ Q 180-ip(v3 ) 1,0K T... i i o i/i(vl) 90d r... 5od!] : _.... od i i 0Hz 50KHZ IOOKHz 150Kgz 200F/4Z 180-ip(vl) ~) Fig. 10. The magnitude and phase of Zi of the inductor of Fig. 9. (b) The magnitude and phase of Zi of the inductor of Fig. 9(b).

14 278 Soliman R2 Two realizations of the first order all-pass transfer function are shown in Figs. 15(b) and 15(c). For both cases the necessary condition is given by: R3 = R4, and in this case the transfer function simplifies to: ( 1 -a) R RI R2 T(s) = -t -R-J-5 9 scr - 1 (11) R4 scr-4-1 The negative sign applies to the circuit of Fig. 15(b) and the positive sign applies to the circuit of Fig. 15(c). The circuit of Fig. 15(b) has the advantage of using grounded capacitor. In both cases the grounded resistor R5 controls the gain factor which can be larger than one, an advantage which does not exist in the conventional op amp first order all-pass circuits Second Order All-Pass Circuits C '1 I < Yi R1 <, < >ar ( 1 -a) R The generalized building block shown in Fig. 15 can also lead to the all-pass second order circuits with real axis poles (Q < 1/2) shown in Figs. 15(d) and 15(e). For the circuit of Fig. 15(d), the transfer function is given by: R5 T(s) = - -- R3 s2c1c2r1r2 - s (\ C2RtR3 R4 C1RI _ C2R2 ) + 1 s2c1c2rir2 + s(c2r1 + C1R1 + C2R2) + 1 (12-a) (b) Fig. 11. Ideal inductor circuits. s+a T(s)=K a>0, K,b>0 (9) s-t-t) is realizable in one of the forms shown in Fig. 14, with the two capacitors grounded. Similar realizations for an inverting first order transfer function using the CFOA can be obtained. Next, the case of a first order all-pass transfer function is considered. Fig. 15 represents a new generalized configuration using a single CFOA and five impedances, its transfer function is given by: z_z -- = (lo) 1,'1 Z1 + Z2 The necessary condition to realize an all-pass response is given by: R--23 = 1 + 2R2 2C1 (12-b) R4 --ffl + c-7 This circuit has an inverting gain factor which can be adjusted to the desirable value which can be larger than one by varying Rs. Of course a notch response with an inverting gain factor (which can be larger than one in magnitude) can also be obtained if: R3 _ R2 + CI R4 R1 ~22 (12-c) Similarly for the circuit of Fig. 15(e), it can be shown that the gain factor equals to & and the condition for R 4 ' all-pass and notch responses are given respectively by: R4 _ 1 -q- 2R1 2C2 (13-a) -kt + c--7 R4 R1 C (13-b) R3 R2 C1

15 Applications of the CFOAs K T... o.s~ / ~... / i/ilvl} 100d T... ', Phase d SEL>>! Od ~... r... ~... ~... n :LT, HZ 10.00:',Hz 20.00KHz 30.00KHz 40.00KHz 50.00KH: = 180-ip(vl) 8.0K T... 1 Izll i" " * : 4.oK ~ 9. --/._~. o... i... i... = i/ilvl) 100d T... Phase,f /,odil I, SEL>> 1 Od~... ~... r..., IKH~ 50.0~z I00.0KBz 150,0KE'IZ 200.0F.H: o 180-ip(vl ) ~) Fig. 12. The magnitude and phase of Zi of the inductor Of Fig. 1 l. (b) The magnitude and phase of Zi of the inductor of Fig. 11 (b).

16 280 Soliman R It? ir CI > > R2 6.0mA T...,.~ i...!... i... ~.o~,i.... OA a i(v3) r... I Phase 2o0a4 10oo~ i S %;LS... ~... ~... r... ~... 0Hz 20KHz 40KHz 60~'tz 80KHz 100KHI 180+ip(v3 ) (b) Fig. 13. A parallel FDNR-R circuit. (b) The magnitude and phase of Yi of the circuit of Fig. 13.

17 Applications of the CFOAs 281 Yi > C ----~ I --tl C1 R1 T < < < > > R2 > (c) C2 Yi > CI- Fig. 13. (c) A grounded R, ideal FDNR circuit. (d) A canonic ideal FDNR circuit.

18 282 Soliman 4.01~A T mA ~ r SEL>>: OA ilv3)... Phase 2ood!~ lood Od ~ , r Onz 20KHz 40KHz 60KHz 80KHz 100KHI D 180+ip{v3} (e) Fig. 13. (e) The magnitude and phase of Yi of the circuit of Fig. 13(d). Vi = Y Vi -- w -Vo 1 a 1-- K b 1 b T I 1 Ka Co) Fig. 14. Two equivalent realizations of the first order noninverting transfer function.

19 Applications of the CFOAs 283 vo Vi 9 R3 R i ra / ~ (b) R3 R3 vi ~ II c > > R5 > :Vo Vi 9 R2 CI -~ RI i O~ --vo >> R5 (c) (d) Vi * R3 i :Vo (e) Fig. 15. A single CFOA generalized configuration. (b), (c) First order all-pass circuits. (d), (e) Second order all-pass and notch circuits. Fig. 16 shows the frequency response of the notch filter of Fig. 15(e) designed for fo = KHz, taking Cl = C2 = 1 nf, R1 = R2 = 10 Kf2, R3 = 10 Kf2,

20 284 Soliman io T... I Gain Db o" -1o" -2o f0=15.813khz r... r... r... r Hz 1.0KHz 10KHz 100KHz 1.0MHz 10MHI. vdb(ul:5) Fig. 16. The frequency response of the notch filter of Fig. 15(e). R4 = 20 Kf2 and R5 = 40 KfL From the simulations the actual fo equals to KI-Iz, which is very close to its theoretical value. In order to have independent control on the gain factor and on the zeros, R5 should be used to control the gain factor and R3 should be used to control the zeros. In this respect the circuit of Fig. 10d is better than that of Fig. 10e since the tuning is achieved by the two grounded resistors R5 and R The Noninverting Single-CFOA Filters The CFOA is a very practical building block in realizing the second order filter circuits based on the positive feedback topology shown in Fig. 17. The Iowpass, highpass and bandpass Sallen-Key filters [ 18] using the CFOA are shown in Figs. 17(b), (c) and (d) respectively, and they have the same equations as in the classical op amp cases. An attractive application of the CFOA in the positive feedback topology is in the realization of a notch filter using a VCVS of gain K < 1 as shown in Fig. 17(e). The transfer function of this noninverting notch circuit is given by: s2 C2 R 2 "Jr- 1 T(s) = K s2c2r 2 +4(1 - K)sCR + 1 The Wo and the Q are given by: COo - CR' (14-a) 1 1 Q (14-b) 4(1 - K) It is seen that the circuit realizes complex poles and can be easily modified to realize lowpass and highpass notch responses. Fig. 18 shows the simulated frequency response of the notch filter of Fig. 17(e) designed for fo = 100 KHz, taking K = 0.95, C = 1 nf, R = Kf2 and R1 = 10 Kf2. It is seen that the theoretical f0 is in close agreement with its simulation value The Inverting Single-CFOA Bandpass Filter The general configuration shown in Fig. 19 has a transfer function given by: V 0 --Z2(Z 3 -~- Z4) -- = (15) V/ Zl (Z3 + Z4) - Z2Z4 This configuration is suitable for realizing a bandpass filter as shown in Fig. 19(b). Its transfer function is

21 Applications of the CFOAs 285 Vi o> vo ~KR > C1 Vi ; RI R2 Rix vo C2 (b) RI Vi CI -- I c: o~ -vo > )KR (c) Fig. 17. The general positive feedback topology using the CFOA. (b) LP filter using the CFOA. (c) HP filter using the CFOA.

22 286 Soliman R2 Vi ~, R1 I C2 R3 vo C C I Vi 9... X~2 o~-----~,vo < > <~ KR1 < (e) Fig. 17. (d) BP filter using the CFOA. (e) Notch filter using the CFOA.

23 Applications of the CFOAs i -15 i r ~ r r Hz 1.0KHz 10gHz IOOKHz 1.0MHz IOMH~ a vdb(ulz5) Fig. 18. The frequency response of the notch filter of Fig. 17(e). given by: T(s) = -sc1r2 s2cic2r1r2 d-s(c1r1 -]- C2R2 - KC1R2) -}- 1 (16-a) Taking: K = 1, C1 = C2 = C, the design equations are given by: 1 Q R I and Rz= (16-b) cooc Q woc In this case the gain at Wo is given by: T(jwo) : - Q2 (16-c) An alternative design however is to take K = 1/2 and C2 = C1/2. The design equations are given by: In this case 1 2Q R1---- and R2=-- (17-a) w0c1 Q cooc1 T(jwo) = -2Q 2 (17-b) Fig. 19(c) shows the simulation results using the first design with C1 = C2 = 1 nera = 4 Kf2 and R2 = 100 Kf2. It is seen that Afo/fo = -0.65%. Fig. 19(d) shows the simulation results using the second design with C1 = 2 nf, C2 = 1 nf, R1 = 1 Kf2, R2 = 50 Kf2, R = 20 Kf2 and K = 1/2. In this case it is found that Afo/fo = -0.41%. 6. The Two-CFOA Filters In this section four circuits realizing ban@ass and lowpass voltage responses at the two-cfoa outputs are proposed. All circuits have the attractive advantage of using grounded capacitors, and having very low 090 and Q sensitivities to all circuit components The Inverting BP-Inverting LP Filter Fig. 20 represents the inverting bandpass inverting lowpass filter circuit. The voltage transfer functions at the two-cfoa outputs are given by: where 1 VOl ~R Vo2 C1C2RzR -- D(s---~' Vi = D(s) (18) s 1 (19) D(s) = s 2 + ~ + C1C2RzR3 The 090 and the Q of the filter are given by:

24 288 Soliman v~ 9 ovo Vi r RI CI t o> ~ ~-~C2 (l-k) R (b) Fig. 19. An inverting general configuration using a single CFOA. (b) An inverting BP filter using a single CFOA.

25 Applications of the CFOAs 289 \ ain Db 9 i.... O~ fd~7. 06Khz d ~ -280d -320d >>I \ , ~ , r... --~ Hz ~.0KHz 10KHz 100EHz 1.0MHz 10MH: [] n vp(ul:5) [] 9 vdb(ul:s) F r e q u e n c y (c) -80d ~ 2 40 T odq -160d q -2ood~ -24od -28od~ >>1-320d J ~... ~...,... loohz 1. OKItZ IOKItZ 100KHZ 1 9 O~z 1O14111 [] * ",p(ul:s) [], vdb(ul:5) FreqaaRne'/ (d) Fig. 19. (c) The magnitude and phase of the bandpass filter of Fig. 19(b) with K = 1. (d) The magnitude and phase of the bandpass filter of Fig. 19(b) with K = 1/2.

26 0 ~~ o~ t~ o --7 e~ o 7 0 ~t o o~

27 Applications of the CFOAs 291 Od ~ Gain Db -SOd~ II... i zoodq... i 7!ii \...,... -isoa q -200d q -2soa q = GOd >>1 J ~ ~ , Hz 1 9 0KHz 10KHz 100KHz i. 0MH~ [] D vp(ul:5) [] 9 vdb(ul:5) Fig. 21. The magnitude and phase of the bandpass filter of Fig. 20(d). [ 1 / C 1 COO -- ~/C1C2R2R3, Q = RI~/ C2R2R3 R1 The bandpass gain at COo = T(jCOo) = --- R R3 The lowpass DC gain ---- T(0) = --- R (20) (21-a) (21-b) It is seen that the Wo and the Q sensitivities to all circuit components are very low (< 1). For a specified COo and Q, there are many posible choices for the element values of the filter. Taking C1 = C2 = C, R2 = R3, the design equations are given by: R] = Q cooc (22-a) R2 = 1 R3-- co0c (22-b) The grounded resistor R 1 controls Q without affecting COo. The resistor R controls the gain, and for a specified T(jCOo), or T(0), the design equation of R is given by: R1 R3 R= - - orr=-- (23) IT(jCOo) l IT(0) I 6.2. The Noninverting BP-Noninverting LP Filters The circuit of Fig 20(b) has the advantage of having an infinite input impedance. The transfer functions of this circuit are given by: s ~ 0+-~) Vo~ _ C7R3 (1 + -~) and Vo2 = ClC2R2R3 Vi D(s) Vi D(s) (24) where D(s) is the same as given by equation (19), and of course wo and the Q are the same as given by equation (20). The design equations can be taken as given by (22). The grounded resistor R controls the gain of the filter as seen from the following equations: R1 R1 T ( j coo) = -~3-6 --~ (25-a) T(O) = i + - R3 - R (25-b) If the magnitude of the gain is not one of the specified parameters, then R can be taken as open circuit resulting in a unity mc gain and a bandpass gain at coo equals to a. It is worth noting that the filter based on the two CFOAs gyrator circuit [14] has one of the capacitors floating.

28 292 Soliman Vi -~ R *Vol,r, Vo2 RI -~CI % Vi 9 Ca) *'col ;vo2 (b) Fig. 22. Inverting BP- inverting LP filter using three CFOAs. (b) Noninverting BP- noninverting LP filter using three CFOAs. Another noninverting BP-noninverting LP filter is shown in Fig. 20(c). The circuit has the same w0 and Q as given by equation (20). The gain at COo and the DC gain are given by: gl T(jwo) = (26-a) R3 T(0) = 1 + R3 (26-b) R1 For a specified 090, Q, T(jcoo) or (T(0)) and taking C1 = C2 = C, the design equations are given by: R1 = R or R2 = Q cooc (27-a) R1 R3 = RI(T(0) - 1) T(jwo) - 1 (27-b) 1 (woc)2r3 (27-c) For a specified w0 and Q only, the design equations can be taken as in equation (22), and in this case 1 T(jwo) = Q + 1 and T(O) = 1 +-~ The Inverting BP-Noninverting LP Filter Fig. 20(d) represents another BP-LP filter using the same number of circuit components and employs an inverting integrator in the second stage instead of a noninverting one as in the previous three cases. The transfer functions in this case are given by: 1 Yo1 csr3 V02 C,C2R2R3 (28) Vi D(s) ' Vi D(s) ' where D(s) is given by equation (19). The design equations can be taken as given by equation (22). For

29 Applications of the CFOAs 293 P h a e Od r... G a i n D b 20~ / / \ \ \ / >> i , ~ r h 1.0Kh 10Kh 100Kh 1.0Mh lomb [1 D vp(ulzs) [2 9 vdb(ul:5)?requency (c) G a s n r... 7 / J // \\ -2 o ; , ~ ~ , h l*0kh 10Kh 100Kh 1.0Mh 10Mh [i o vp(ul;5) [2 9 vdb(ul~5) (d) Fig. 22. (c) The magnitude and phase of the bandpass filter of Fig. 22. (d) The magnitude and phase of the bandpass filter of Fig. 22(b).

30 294 Soliman Ri R3! ~ ~ ~ ' ~ kvo~ Vi 9 RI C 2 'VV~ (6) Fig. 23. Inverting HP-BP-LP filter using three CFOAs. (b) Noninverting HP-BP-LP filter using three CFOAs. Vi Ri > Ii ~ BP~ V~ Fig. 24. Inverting HP-BP-LP filter using five CFOAs.

31 Applications of the CFOAs 295 4oT... n -10Od 20 -o" 1 1 / /' / '\ -20-4o~ I -,og2. looh 1.0Kh 10Kh lookh 1.OMh lohh [i a Vp(U2~5) [2 9 vdb(u2~5) (b) Fig. 24. (b) The magnitude and phase of the bandpass filter of Fig. 24. Vi Vol (c) Fig. 24. (c) Noninverting HP-BP-LP filter using five CFOAs.

32 296 Soliman this circuit the DC gain equals to unity, and T(jwo) = Q. Fig. 21 shows the magnitude and phase simulation results of this bandpass filter designed for f0 = KHz and Q = 40, taking C~ = Cz = 1 nf, R1 = 200 Kf2 and R2 : R3 = 5 Kf2. 7. The Multiple-CFOA Filters In this section several novel multiple-cfoa second order filters are introduced The Bandpass-Lowpass Filters Fig. 22 represents an inverting BP-inverting LP filter which is generated from the circuit of Fig. 20 using a third CFOA acting as a voltage to current converter feeding-back the current ~ to terminal X of the first CFOA. The circuit has the same equations as that of Fig. 20, it has the advantage however of using a grounded resistor R3. This circuit is suitable for current excitation and in this case it will be classified as a mixed mode (current excitation and voltage responses) bandpass-lowpass tilter. Fig. 22(b) represents a very attractive noninverting BP-noninverting LP filter with infinite input impedance and with all resistors and capacitors being grounded. The circuit is generated from that of Fig. 20(b) using a third CFOA to act as a voltage to current converter. The equations for this circuit however are different from those of the circuit of Fig. 20(b), and are given by: s 1 VO1 -- CI"~ V02 - C1C2R2R Vi D(s) and ~ D(s) (29) where D(s) is the same as given by equation (19). It is seen that the circuit has the same magnitudes of T(0) and T(jcOo) as the circuits of Figs. 20 and 22, thus the design equations (22) and (23) apply also to this circuit. Similarly the circuit of Fig. 20(c) may be modified using two more CFOAs to provide the necessary feedback currents, resulting in an alternative noninverting BP-noninverting LP filter with grounded elements and using four CFOAs. Fig. 22(c) shows the magnitude and phase responses of the bandpass filter of Fig. 22 with C1 = C2 = 0.2 nf, R1 ~ Kf2 and R = R2 = R3 : 10 Kf2. Fig. 22(d) shows similar results for the bandpass circuit of Fig. 22(b) designed with the same circuit val- ues. It should be noted that although the magnitude response is identical to that of Fig. 22(c), the phase response deviates at frequencies above 1 MHz due to the capacitance Cx of the first CFOA The Inverting HP-BP-LP Filter The filter circuit shown in Fig. 23 realizes an inverting highpass, inverting bandpass and inverting lowpass response at the three-cfoa outputs. The transfer functions are given by: RS2 R Vol ~ Vo2 C,R~R, 's Vi D(s) ' Vi D(s) R and Vo3 _ CIC2R1R2Ri (30) Vi D(s) ' where R D(s) = s 2 + ci~s/~ik4 + C1C2R1R2R3 (31) From the above equation the coo and the Q of the filter are given by: R and Q = R4 (32) coo = CtC2R1R2R3 C2R2R3R For a specified wo and Q the design equations may be taken as: choose C 1 = C2 ~-- C, Ra = R2, R3 = R (33) Thus, 1 R1 = R2- ooc' R4 = QR3 (34) The resistor R4 controls Q without affecting COo. Ri controls the magnitude of the gain without affecting coo or Q and for the chosen design it can be easily seen that the magnitude of the gain at coo at any of the three outputs-- ~.R The Noninverting HP-BP-LP Filter Fig. 23(b) represents an infinite input impedance three- CFOA noninverting highpass, bandpass and lowpass R

33 Applications of the CFOAs 297 r... O~ ",. C=0-2o -,' -40 ~... u. vdb(5)... r... Od ~: : : ~ o.' :."~ \ 1 \\ '\ SEL>> i ~ ~... [... ~... ~.. t... IOOHz i 9 0KHz IOKHz 100KHz I. 0MHz 10MHz 100MH o * vp(5) Fig. 25. The magnitude and phase characteristics of the uncompensated and the compensated noninverting VCVS. R1 = 10 K~, R2 = 20 K~. r... " Gain Db " = ".. O" ~ o\ " " ".... c=0 ~ ' ~. ~... i i i i i i',\'k!i.;... i i... ~..>... i o ~ vp(s} Frec~uency (b) Fig. 25. (b) RI = 2 K~2, R2 = 4 K~.

34 m Soliman 40 T... GaLn Db 0, C=55pf o~, SZL>>I -40 ~ : o * vdb(5) i Phase.... c=55pf o * vp(5) (c) Fig. 25. (c) R1 = 2 Kf2, Rz = 20 Kf2. filter. The transfer functions are given by: R(R3+R4) ~2 VO 2 R(R~+R4) S go1 R3R4 ~ CIR1R3R4 Vi D(s) ' Vi D(s) and R(R~-FR4) go3 C1C2R1R2R3R4 - (35) Vi D(s) where D(s) is the same as given by equation (31). The design equations are the same as given by equations (33) and (34). For this design the gain at o~0 at any of the filter three outputs is given by Q The Inverting Universal Filter Fig. 24 represents a five-cfoa inverting universal filter. The circuit has the same equations as that of Fig. 23. This circuit is also suitable to be driven by a current signal and as such it can be considered as a mixed mode inverting filter. Of course the circuit can also serve as a current mode universal filter by taking the currents in Rb R2 and R3 as the HP, BP and LP output currents respectively. It is worth noting that if a generalized second order response is required, the combinations of the three output currents may be taken to a current summer circuit. Fig. 24(b) shows the magnitude and phase of the bandpass circuit of Fig. 24 designed for Q = 50 and f0 = KHz, taking C1 = C2 = 1 nf, R1 = R2 = R3 = Ri = R = 5 Kf2 and R 4 = 250 Kg The Noninverting Universal Filter The modified version of the filter circuit of Fig. 23(b) using two more CFOAs to realize voltage to current converters is given in Fig. 24(c). The numerators of the equations for this filter circuit however are different from those of the circuit of Fig. 23(b) and are the same as those of the circuits of Figs. 23 and 24 except for the polarities which are all positive in this case. This circuit is considered to be a very attractive voltage mode universal filter, with infinite input impedance, very low output impedances [19], and all the four resistors and the two capacitors being grounded. 8. Limitations and Compensation Methods Like the conventional op amp, the CFOA has frequency limitations caused by Rx, Cx, Rz and Cz.

35 Applications of the CFOAs 299 r... 0~ Gain Db -20 q... \ -,o~--?;~);~ dr ~ \ \ 1SOd ~ ~',.\ 9... \\... \ \ ', 9. - C=O 9 \\ 9 \ I \ \... r... r... ~... ~ r... loohz 1.0KHz lok~z lookhz 1.0MHz 10MHz 100MHZ o vp(5) [ o" lp.~... " ~176 ~ i... i -20 i... \\ o x ~, \ \,, D o vdb(5)... 'i. " i. i\\ ~\ i... " ~ nod ~... i I z vpi5) i, OKHZ 10Fddz 100KHz i. 0/4Hz IOMHz loomhz i. 0GHz a o (b) Fig. 26. The magnitude and phase characteristics of the uncompensated and the compensated inverting VCVS. R1 = 10 K~2, R2 = 20 K~2. (b) RI = 2 K~2, R2 = 4 K~2.

36 300 Soliman _.o~~... : : :... ~ i... ~ \!:... : :\... ii -40 ~... ~... s O vdb(5) 180d ~.ooj i l~od~... i... i... i... c-o \ Hz 1 9 0Kttz 10KHZ 1001r 1 9 0Y~z 10~z 100/fflz 1.0GH=. * vp(5) (c) Fig. 26. (c) R1 = 2 Kf2, R2 = 20 K~2. 40~... n 20; / / -0 / I / -20" >>I...,... ~... ~...,... looh 1,OKh lokh lookh 1.OMh lomh [1. vp{u2:5] [2 9 vdb(u2:5) Fig. 27. The magnitude and phase characteristics of the compensated bandpass filter of Fig. 24.

37 Applications of the CFOAs 301 Although the effect of Rx can be minimized in most cases by taking the resistor connected to port X much larger than Rx, there are few circuits which require compensation for the effect of Rx, examples are the inductor circuits of Figs. 9(b) and ll(b). It has been demonstrated in Sections 4.2 and 4.3, that the compensation for the effect of Rx can be easily achieved, leading to simulation results that are very close to the theoretical expected ones. The effect of Cz on limiting the high frequency range of operation and passive compensation methods are discussed next. From the simulation results, it is seen that all the circuits with a resistor connected between port Z and ground are frequency limited by the pole which results due to the parasitic capacitance Cz at port Z. Well known passive compensation methods [20-21] can be applied to this class of circuits in order to extend the frequency range of the circuit. As an example, Wilson's passive compensation method [20] can be applied to the noninverting VCVS shown in Fig. 2(b) by adding a capacitor C of magnitude given by C = (Rz/R1)Cz, in parallel with R1, this will result in a pole-zero cancellation. Fig. 25 shows the magnitude and phase of a noninverting VCVS of gain 2, realized by taking RI = 10 Kf2, R2 = 20 K~2 and C = 11 PE It is seen that f3db has been extended from 1.5 MHz for the uncompensated VCVS to about 44.9 MHz. Fig. 25(b) shows similar simulations for the VCVS with same gain of 2, by taking R1 = 2 Kf2 and R2 = 4 K~2. It should be noted that although the uncompensated f3ab for this VCVS which is about 5 times larger than that of the previous case (since f3db = 1/2Jr R2Cz), the f3db of the compensated VCVS is almost the same as in Fig. 25. This is due to the fact that bandwidth of the compensated amplifier is limited by C, Rx, Cx and the voltage follower action from port Z to the output port of the CFOA. Fig. 25(c) shows similar simulations for the noninverting VCVS with gain of 10, realized with R~ = 2 Kf2 and R2 = 20 Kf2 For the inverting VCVS shown in Fig. 2(c), passive compensation is achieved using a single capacitor C, in parallel with R1 in the same way as was done for the conventional op amp VCVS circuits [21]. For pole-zero cancellation, the magnitude of C should also be taken equal to KC z, where K = R2/R1. Fig. 26, (b) show the magnitude and phase for the case K = 2 taking Ra = 10 Kf2, R2 = 20 Kf2 and R1 = 2 Kf2, R2 = 4 Kf2 respectively. From the simulation results it is seen that f3db is approximately the same in both cases and is given by 48.4 MHz which is higher than that of the noninverting VCVS of the same gain. This is due to the fact that Cx has no effect on the frequency response since port Y is grounded. Fig. 26(c) shows similar simulations for the inverting VCVS with gain of 10, realized with R1 = 2 Kf2 and R2 = 20 Kf2. The same method of compensation can be applied to some of the filter circuits reported in this paper. As an example consider the filter of Fig. 24. Improvement in both the phase and the magnitude response can be achieved as shown in Fig. 27 by adding a capacitor C of magnitude 5.5 pf in parallel with Ri. The effect of Rz (typically 2.2 Mr2) on the circuits which employ a capacitor C connected between port Z and ground is observed at very low frequencies, since the pole produced by Rz and C is at 72 Hz (assuming C = lnf). Of course this effect is observed on the phase of the integrators at low frequencies. For filter applications however which are intended for frequencies >> 100 Hz, the effect of Rz can be minimized by using capacitors in the nano-farads range. 9. Conclusions The versatility of the current feedback op amp (CFOA) with an available Z terminal [2] in realizing analog circuits is demonstrated by numerous applications. These applications include the realization of voltage amplitiers, voltage integrators, inductors, FDNRs and filters. One of the major objectives of the paper is to give an overview of the second order filter circuits realized using the CFOAs. PSPICE simulations indicating the frequency limitations of some of the reported circuits using the AD 844-CFOA are given. Passive compensation methods for the noninverting and the inverting VCVS structures have been considered in this paper. Although the direct compensation methods can be applied to some of the circuits reported in this paper, other circuits may require special methods of compensation. It is not the intention of this paper however to concentrate on the compensation of filter circuits.

38 302 Soliman Acknowledgements The author would like to thank the reviewers for their useful comments. The author would like also to thank his graduate student A.S. Elwakil for his assistance with the PSPICE simulations included in this paper. 19. A. M. Soliman, "Current conveyors steer universal filter." IEEE Circuits and Devices Magazine. 11, pp , March G. Wilson, "Compensation of some operational amplifier based RC active networks?' IEEE Trans. Orcuits and Systems. CAS-23, pp , July A.M. Soliman and M. Ismail, "Passive compensation of Op Amp VCVS and weighted summer building blocks?' 1EEE Trans. Orcuits and Systems. CAS-26, pp , Oct. 197~, References 1. S. Evans, Current Feedback Op Amp Applications Circuit Guide. Complinear Corporation, Fort Collins, CO., 1988, pp Analog Devices, Linear Products Data Book. Norwood, MA., E. Bruun, "A dual current feedback op amp in CMOS technology." Analog Integrated Orcuits and Signal Processing. 5, pp , C. Toumazou, J. Lidgey and A. Payne, "Emerging Techniques For High BJT Amplifier Design: A Current Mode Perspective" in First Intentional Conference on Electronics Circuits and Systems, Cairo, A. Fabre, "Insensitive voltage mode and current mode filters from commercially available transimpedance op amps?' lee Proceedings-G, 140, pp , A. S. Sedra and K. C. Smith, "A second generation current conveyor and its applications?' IEEE Trans. on Circuit Theory. CT-17, pp , B. Wilson, "Universal conveyor instrumentation amplifier." Electronics Letters. 25, pp , B. Wilson, "Recent developments in current conveyor and current mode circuits?' IEEProceedings-G. 137, pp , A. J. Prescott, "Loss compensated active gyrator using differential input operational amplifier." Electronics Letters. 2, pp , A.C. Caggiano, "Operational amplifier simulates inductance?' Electronics. 41, p. 99, R.L. Ford and F. E. J. Girling, "Active filters and oscillators." Electronics Letters, 2, p. 52, A.M. Soliman, "Ford-Girling equivalent circuit using CC II?' Electronics Letters. 14, pp , A.M. Soliman and S. S. Awad, 'A tunable active inductance using a single operational amplifier." AEU (Electronics and Communications). 32, pp , A. Fabre, "A gyrator implementation from commercially available transimpedance operational amplifiers?' Electronics Letters. 28, pp , H.J. Orchard and A. N. Wilson, "New active-gyrator circuit?' Electronics Letters. 10, pp , A.M. Soliman, "New active-gyrator circuit using a single current conveyor." Proceedings IEEE. 66, pp , A.M. Soliman and S. S. Awad, "Canonical high selectivity parallel resonator using a single operational amplifier and its applications in filters?' lee Electronic Circuits and Systems. 1, pp , July A. Budak, Passive and Active Network Analysis and Synthesis. Houghton Mifflin, Ahmed M. Soliman was born in Cairo Egypt, on November 22, He received the B.S. degree from Cairo University, Egypt, in 1964, and the M.S. and Ph.D. degrees from the University of Pittsburgh, PA., U.S.A. in 1967 and 1970, respectively, all in electrical engineering. He is currently Professor and Head Electronics Group, Electronics and Communications Engineering Department, Cairo University, Egypt. Dr. Soliman served as Professor and Chairman of the Electrical Engineering Department, United Arab Emirates University ( ), and as the Associate Dean of Engineering at the same University ( ). He has held visiting academic appointments at the American University of Cairo ( ), Florida Atlantic University, FL. ( ) and San Francisco State University, CA. ( ). Dr. Soliman served also as Associate Professor of Electrical Engineering at Florida Atlantic University, U.S.A. ( ). He was a visiting scholar at the Technical University of Wien, Austria (Summer 1987) and at Bochum University, Germany (Summer 1985). He was a Research Analyst at the Central Research, Rockwell Manufacturing Company, Pittsburgh, PA., U.S.A. (Summer 1970). Dr. Soliman received the First Class Science Medal from the President of Egypt in 1977, for his services to the field of Engineering and Engineering Education.