Novel grounded-capacitor active biquads using FiTFNs

MicroelectronicsJournal29 (1998) 123-132 © 1998 Elsevier Science Limited Printed in Great Britain. All rights reserved 0026-2692/98/$19.00

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PH:SO026-2692(97)O0076-1

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Novel groundedcapacitor active biquads using FiTFNs Muhammad Taher Abuelma'atti, Husain Abdullah AI-Zaher and Muhammad Abdullah AI-Qahtani King Fahd University of Petroleum and Minerals, Box 203 Dhahran 31261, Saudi Arabia

Novel current-, voltage- and mixed-mode groundedcapacitor active biquad circuit designs, using the fiveterminal floating nullors, are presented. The proposed circuits enjoy low active and passive sensitivities and independent grounded-element control of the parameters COoand CO0/Q0-© 1998 Published by Elsevier Science Ltd. 1. Introduction

he four-terminal floating nullor (FTFN), shown in Fig. l(a), is a more flexible and T versatile building block than the operational amplifier and the current-conveyor [1-3]. This explains the growing interest in using the FTFN in designing current-amplifiers, voltageto-current converter,s, gyrators, floating immittances [1, 4-6] and,/more recently, in designing current-mode active-RC filters [3, 7-10] and sinusoidal oscillators [11, 12]. In [3] a procedure is described for transforming voltage-mode circuits with operatJional amplifiers to currentmode circuits with FTFNs. Using this procedure, a current-mode lowpass transfer function

is obtained from a Sallen-Key voltage-mode lowpass circuit. In [7] a stable grounded-capacitor first-order allpass filter using an F T F N is presented. In [8] a cascadable current-mode configuration using single F T F N is presented. This configuration can realize second-order lowpass, highpass, bandpass, notch and allpass filters. However, its active sensitivities are large and it requires a floating capacitor. Moreover, these five filters cannot be realized without changing the circuit topology to achieve a specific filter function. In [9] a current-mode configuration using two FTFNs is presented. This configuration enjoys low active and passive sensitivities and can simultaneously realize lowpass/bandpass, or highpass/bandpass or bandpass/notch filters using two grounded capacitors. It can also realize an allpass filter. However, it requires a floating resistor and, more importantly, its parameters c00 and co0/Q0 are interdependent. Thus, it is impossible to adjust any of these two parameters without disturbing the other parameter. In [10] a cascad-

123

M.T. Abuelma'atti et al./Active biquads using FiTFNs

(a) VI_ I [ . . . . . . . . . , Iol y X V2

(b)

V2 = 1/1

- ~ i W 12. . . . . . . . .

,

and

Xe YJ

/03

Io2

Fig. 1. (a) Nullor model of the FTFN. (b) Possibleimplementationof the FTFN. able current-mode configuration using a single FTFN enjoying low active and passive sensitivities is presented. This configuration can realize second-order lowpass, highpass, bandpass, allpass and notch filters using six passive elements at most. These five filters, however, cannot be realized without changing the circuit topology to achieve a specific filter function. Moreover, the parameters COo and CO0/Q0 are interdependent. On the other hand, using the FTFN it is easy to synthesize the minus-type second-generation current-conveyor (CCII-). Thus, the available minus-type CCII-based realizations can be easily implemented using the FTFN. However, it is not possible to synthesize the plus-type CCII using the FTFN. It is, there,fore, proposed here to modify the available FTFN realization, shown in Fig. l(b), by adding a new output terminal as shown in Fig. 2. The resulting building block is the five-terminal floating nullor (FiTFN). The FiTFN is characterized by the following port relations: 11 =/2 = 0

(2)

/02

=

=

I01

(3)

Realization of CCII+ and C C I I - using the FiTFN is straightforward. Thus, the available CCII-based realizations can be easily implemented using the FiTFN. In this paper, the use of the FiTFN in designing current-, voltage- and mixed-mode groundedcapacitor active biquads will be explored. Comparisons with similar realizations available in the literature will be performed and the merits and demerits of the new realizations will be discussed. 2. Proposed current-mode biquad

Figure 3 shows the proposed current-mode biquad circuit. In this circuit the FiTFNs are configured as FTFNs. Using the FiTFN port relations of (1)-(3), routine analysis of the circuit shown in Fig. 3 yields the currenttransfer function expressed by G5 G2C4C6 -$2C1C4I1 + sC4G212 + sC4G213 -G2(G3 + s(C3 + C6))I4

I0-

(4)

D(s)

(1) D

(a)

--P

(b)

w+El::

V2_ I2 i ~ " ~ l l O

V~:

[]

x-I

"-F

x.

yo

W+tW-

Fig. 2. (a) Symbolfor the FiTFN. (b) Possibleimplementation of the FiTFN.

124

i-'

I

Fig. 3. Versatile grounded-capacitor active-biquad using FiTFNs configuredas FTFNs.

Microelectronics Journal, Vol. 2& No. 3

where D(S) =

090 (O2 -}- 5 ~ - 1 -

S2 --

-q- S G5(C3 + C6) G3G5

2C4 GBp -- C3 q- C---~

C4C6

and

C4C6

(9)

+ s2

(5) From eq. (5) the parameters COoand 09o/Q0 can be expressed as

G3G5 0 9 ~ - C4C6

(6)

and 090 G5(C3 -]- C6) = Qo C4C6

--

From eqs. (6) and (7) it can be seen that the parameter 09o can be adjusted by controlling the resistor R3=l/G3 without disturbing the parameter 09o/Q0. Moreover, the parameter 09o/Qo can be adjusted by controlling the capacitor C3 without disturbing the parameter COo-Thus, the proposed circuit enjoys independent groundedelement control of the parameters 090 and 090/Qo. Taking into consideration the FiTFN non-idealities, the port relations of eqs. (2) and (3) can be expressed as

the highpass response can be realized with

and

(ii) the bandpass response can be realized with 11=14=0 and •2=•3;

I02 = 0eI01

(iii) the lowpass response can be realized with I1=0, I2=I3=I4 and 2C4=C3+C6; (iv) the notch response can be realized with I1=I2=I3=I4 and 2C4=C3+C6; (v) the allpass response can be realized with

(12)

where fl=l-g,(]e]<
o¢3f13G3G5

D(S) = (O2 -}- S-'~O0 q- S2 -- °{1fl2C4C6

(13) q- S G5(°{3f13C3 q- alfl2C6) -4- s2 0~1fl2C4C6

11=12=1_3=14, C4=C3+(26, C1=C6 and G2=G5 .

Thus, the circuit of Fig. 3 can realize the lowpass, the highpass, the bandpass, the notch and the allpass responses without changing the circuit topology. From eq. (4) it can also be seen that the highpass gain, the bandpass gain and the lowpass gain are given by G5 C1 G2 C6

(11)

v2 = f l v l

12=13=14=0;

GHp --

(10)

(7)

From eq. (4) it can be seen that: (i)

GLp = 1

(8)

where fli, o~i,i=1-3, are the voltage- and currenttracking errors of the ith FiTFN. Thus, the parameters COoand 09o/Qo can be expressed by 092 -- °~3f13G 3 G 5 0~1fl2C4C 6

(14)

and

090 Qo

--

=

G5( 3/33C3 + 1fl2C6) 0{1fl2C4C6

(15)

125

M.T. Abuelma'atti et al./Active biquads using FiTFNs

~3

From eqs. (14) and (15) it is easy to show that the active and passive sensitivities of the parameters COoand coo/Qo are 1 2

S~(~0 = -So(? = $720 = - S e)0fl3--

s~

: s~,~ = - s ~ , ;

S~o/Qo

-

-

--

:

-s~os = -

sO)o/Q 0 = Se)o/Q 0 f12 c6

-

S~)o/Q o = SCOo/Qo = ScO;O/Qo =_ f13

1

1 1 -[- °~3fl3C3 ad~2C6

-

Vi

1 1 " -~1fl2C6 ~

S~o/Qo -- S¢Oo/Qo ~, = S ~ too 2 =S~=O

(i) Use of grounded capacitors. (ii) Independent grounded-element control of the parameters coo and co0/Q0. (iii) Low active and passive sensitivities. (iv) High impedance outlet

3. Proposed voltage-mode circuit Figure 4 shows the proposed voltage-mode bandpass/lowpass biquad circuit using FiTFNs. In this circuit the FiTFNs are configured as simple unity-gain current and voltage cells. Using the FiTFN non-ideal port relations of

126

LP

Fig. 4. Proposed voltage-mode bandpass/lowpass filter. The FiTFNs are configured as unity-gain voltage and current cells.

eqs. (11) and (12), routine analysis of the circuit of Fig. 4 yields the voltage-transfer function expressed by VL p

_ C1C2R2R4 ai~2/h/~a 52 -~- C I ~ "}- Cl°tl°t2fllflgq2a2a3

(16/

and c¢1 VBp = --S Cl/4 Vi s 2 q_ ~ q_ ~lc~2fllfla

(17)

C1C2R2R3

all of which are small. Comparison between the proposed circuit of Fig. 3 and previously published circuits [3, 710] shows that the proposed circuit enjoys the following advantages:

R2

Cl

~/

SC~/QO ---- -Sm°/Q°G5 ---- -- 1

~,

From eqs. (161 and (17) the parameters coo and co0/Q0 can be expressed as

~1~2fllf12

~-

(18)

CIC2R2R3

and coo Qo

1 C1R1

(19)

From eqs. (18) and (19) it can be seen that the lowpass dc gain and the bandpass gain at COoare approximately equal to R3

GLp = ~

(20)

and GBp ~

R1

R4

(21)

From eqs. (20) and (21) it can be seen that the parameter COocan be adjusted by controlling the

Microelectronics Journal, Vol. 2,9, No. 3

resistors R2=l/G2, R3=1/G3and/or the capacitor C2 without disturbing the parameter coo/Qo. Moreover, the parameter COo/Qocan be adjusted by controlling the resistor R1 without disturbing the parameter COo. However, controlling the resistances R1 and/or R3 will disturb the bandpass and/or the lowpass gain. A possible strategy for adjusting the parameters COo,Coo/Qo the lowpass and the bandpass gains is, therefore, as follows: first the resistor R1 is adjusted to control the parameter coo/Qo, then the resistor R 4 is adjusted to control the bandpass gain, the resistor R3 is adjusted to control the lowpass gain and finally the resistor R 2 is adjusted to control the parameter COo. From eqs. (16) and (17) it is easy to show that the active and passive sensitivities of the parameters coo and Q0 are' s °Pl "=

=

_

_SO, O =

--

C2

,82

R2

= -s

C1

1

_ S O , i, R3

= -s

bandpass responses without changing the circuit topology. (v) Low impedance outlets. 4. Proposed mixed-mode circuits

In analog signal processing applications, need may arise for mixed-mode filters with input currents and output voltages [17, 18]. In [18] two currentconveyor-based mixed-mode biquads are presented. Both circuits can simultaneously realize lowpass, highpass and bandpass transimpedance transfer functions. In both circuits the parameters COoand co0/Q0 are interdependent. Thus, while it is possible to adjust the parameter coo without disturbing the parameter co0/Q0, it is not possible to adjust the parameter co0/Q0 without disturbing the parameter coo. The proposed mixed-mode circuits shown in Figs 5 and 6 enjoy the attractive feature of independent control of the parameters COo and co0/Q0. In both circuits each FiTFN is configured as a CCII+.

~

sQo = 'aa2q?Q°~_ sQofll = "'fl2lTQ°= --SQ°R2 = SQ°C1 -- _ s Q o _sQo 1 c2 ~R3 ~ 2 -

and S R1 Q'I= 1

all of which are small.

Routine analysis of the circuit of Fig. 5, using the FiTFN non-ideal port relations, yields the mixed-mode transfer functions expressed as

yl Ii --S~ C4G1 s2C2C4G6 -i--s Pl/32P3=,~2~-----~ CsG1G3 + ~G1G23G3s,

(22) Comparison between the proposed circuit of Fig. 4 and previously published circuits [13-16] shows that the proposed circuit enjoys the following advantages: (i) Use of grounded capacitors, which paves the way for high frequency operation.

,Vj

V2

Ii

(ii) Independent grounded-element control of the parameters coo and co0/Q0. (iii) Low active and passive sensitivities. (iv) Simultaneous realization of lowpass and

Fig. 5. Proposed single-input multiple-output mixedmode lowpass/highpass/bandpass filter. All FiTFNs are configured as C C I I - .

127

M. 7. Abuelma'atti et al./Active biquads using FiTFNs

dance and the bandpass transimpedance are approximately given by

,V2

1 ~V3

(27)

Z L P ,--v G5

1

----~-_~_Y5

ZH p¢v

(28)

G6

and C4

ZBp z~ - -

(29)

CsG3

Fig. 6. Proposedmixed-modebiquad circuit. All FiTFNs are configuredas CCII+. where fli, o{i, i = 1-3, are the voltage- and currenttracking errors of the ith FiTFN,

V2 I; /31/32G1 G3 0~1~2

S2C2 C4 G6 + s 31J~2/~3 C5 G1 G3 _1_/~1flaJ~3G1 G3 G5 R1{120{3 ~10~2~3 (23)

and

From eqs. (25) and (26) it is easy to show that the active and passive sensitivities of the parameters coo and Qo are S0? = S~2~O0= S~3C00:

V3 Ii -- 52C2C4G6

-S~i' :

-Sfl); I :

-Sfl); ) = - ~

1

1

52C2C4

s~°°=~2 s°~°=< s~: = _s~o = -s~'; = - s ~'o~= - 5

+ s ~~1~20c 3 3 C s G t G 3 + ~&~3 ~10c20c3G 1 G 3 G 5

(24) From eqs. (22)-(24), the parameters coo and COo/ Qo can be expressed as 692 = fll/~2fl3 G1 G 3 G 5

(25)

O~10~20~3C2C4C6

and COO= f11~2~3CsG1G3 Qo oqo~20~3C2C4G6

s~o = s~o = s~o = _soo

131 =

sQ 0 : C2

(26)

sQ 0 : C4

sQ 0 : G5

sQ 0 : G6

- s Q°

132 =

-sQ"

1

113 = ~

_ s Q o _~ _ s Q o 1 G1 G3 ~ 5

and SQ°c5 :

From eqs. (22)-(24), it can also be seen that the lowpass transimpedance, the highpass transimpe-

128

From eqs. (25) and (26) it can be seen that the parameter coo can be adjusted by controlling the resistor Rs=I/G5 without disturbing the parameter Coo/Qo. Moreover, the parameter Coo/Qo can be adjusted by controlling the capacitor C5 without disturbing the parameter COo.Thus, the mixed-mode circuit of Fig. 6 enjoys the attractive feature of independent control of the parameters coo and Coo/Q0.

-- 1, S~ = 0

all of which are small. Comparison between the proposed circuit of Fig. 5 and previously published circuits [18]

Microelectronics Journal, Vol. 29, No. 3

shows that the proposed circuit enjoys the following advantages:

D(s) = s2 .--}-S Qo q- CO2 = s2 --}-S c-~-~g4G6.-I- GIG3c2c7

(i) All the passive elements are grounded.

(36)

(ii) Independent grounded-element control of the parameters COoand co0/Qo. (iii) Low active and passive sensitivities. (iv) Simultaneous re;dization of lowpass, highpass and bandpass nfixed-mode transfer functions without changing the circuit topology.

Equations (33)-(35) represent the transimpe= dances of lowpass, notch and bandpass transfer functions, respectively. From eq. (36), the parameters coo and Coo/Qo can be expressed as O32

Routine analysis of the circuit of Fig. 6, using the ideal port relations of the FiTFN, yields the mixed-mode transfer functions expressed as

_

_

--

=

V2

Ii

-YlY4 Y2Y5YT+Y2Y4Y6+YlY3Y5

(30)

YlY3+Y2Y7 Y2Y5Y7+Y2Y4Y6+YlY3Y5

(31)

and

(20

--Y2Y4

Ii

Y2Y5Y7+ y2Y4Y6 + ylY3Y5

(37)

G4 G6 CvG5

(38)

From eqs. (33)-(35) it can be seen that the lowpass dc transimpedance, the bandpass transimpedance at COoand the notch transimpedance are given by ZLp -

V3

C2C7

and COO

71

GIG3

(32)

1 G4

G5 G3

(39)

1

Now if we choose

Zu~, = G66

Yl = G1, Y2 = sC2, Y3 = G3, Y4 = G4, Y5 = Gs,

and

Y6 = G6, Y7 = sC7

1

Znotch = 7

then eqs. (30)-(32) reduce to V1

--

Ii

=

V2 Ii

1 GIG4/C2C 7 G5 D(s) 1 S2 -it-G1G3/C2C7 G5 D(s)

t_,5

(33)

(34)

and V3

Ii where

1 sG4/Cv G5 D(s)

(35)

(40)

(41)

From eqs. (37) and (38) it can be seen that the parameter COocan be adjusted by controlling the grounded resistors RI=I/G1 and/or R3=l/G3 without disturbing the parameter COo/Q0. Moreover, the parameter CO0/Q0 can be adjusted by controlling the grounded resistors R4=1/G4 and/or Rs=1/G5 and/or R6=l/G6 without disturbing the parameter COo. Thus, the proposed circuit enjoys independent groundedelement control of the parameters coo and COo/ Q0. However, controlling the parameter co0/Qo may disturb the bandpass, lowpass and notch

129

M. 7". Abuelma'atti

et

al./Active biquads using FiTFNs

transimpedances. Therefore, a possible strategy for adjusting the parameters (00, (00/Qo and the transimpedances is as follows: first, the resistors R s = l / G 5 and R 6 = l / G 6 are adjusted to control the notch and bandpass transimpedances, respectively, then the resistor R4=l/G4 is adjusted to control the parameter (00/Qo, and the resistor R3=l/G3 is adjusted to control the lowpass transimpedance. Finally, the resistor R1=1/G1 is adjusted to control the parameter (0o-

all of which are small.

Taking into consideration the non-idealities of the FiTFNs, eq. (36) becomes

(i) All the passive elements are grounded.

(00

G 4G6 = s 2 -1- s ~-']-_., o (o 2 -- 52 ~- s 0{1~2#1fl 2-C7G5

O(s)

0~5

R R G1G3 0(10~20~30(4#1 P 2 P 4

(42) From eq. (42), the parameters (00 and (00/Q0 can be expressed as (02 =

1

G1 G3

0(10~20~30{4#1f12f14

C2 C7

(43)

and (0o

--

Qo

°~5

=

G4G6

(44)

0~10~2flif12 CvGs

From eqs. (43) and (44) it is easy to show that the active and passive sensitivities of the parameters COoand co0/Q0 are S~; 1, ~--- $7;1 ~- S (00 =~3

S~O);, ~-- S('OOfll= S(OOfl2= S0)(1#4

1

2 S~',, C2 = S°" C7

~-

_ S~',, G1 = _ S~°O G3

So,,,/Q,, _ S;;,/Q,, = - S ~ f Q'' _

_

--

130

_ Se)o / Q, , &

So)o/Q o _ C7 --

SCOo/Q o G5

_Stoo/Qo ~-

G4

So),,/Q, ~ ~-- - -

G6

1 ~

--

Comparison between the proposed circuit of Fig. 6 and previously published circuits [17, 18] shows that the proposed circuit enjoys the following advantages:

(ii) Independent grounded-element control of the parameters (o0 and (00/Qo. (iii) Low active and passive sensitivities.

1

+

and

=

-

1

_

_

1 2

_S',,/Qo

(iv) Simultaneous realization of lowpass, bandpass and notch mixed-mode transfer functions without changing the circuit topology. 5. Simulation results

To verify the theoretical analysis, the proposed circuits were used to realize a variety of current-, voltage- and mixed-mode transfer functions. The required FiTFNs were obtained by modifying the available FTFN realizations by adding an additional output. Although there are several ways to simuhte the FTFNs required [1, 2, 4, 5, 11, 19], the SPICE simulation results were obtained using the realization of the FiTFN shown in Fig. 2. The FiTFN consists of a 741 operational amplifier and the power-supply current-sensing technique proposed in [20] and realized using the QN2907 PNP and QN2222 N P N transistors. The simulation results obtained from the current-mode biquad of Fig. 3 are shown in Fig. 7, which agrees very well with the presented theory. 6. Conclusion

In this paper the FiTFN has been proposed. The FiTFN is a more versatile device and can be

Microelectronics Journal, Vol. 29, No. 3

easily configured to realize unity-gain current and voltage cell and plus- and minus-type firstand second-generation current-conveyors ( C C I + , C C I I + ) . Thus, in addition to the FiTFN-based realizations, all the available unity-gain cell and current-conveyor-based realizations can be implemented using the proposed FiTFN.

(a) f

~

-8

-12 --16 I

I

2K

5K

I

I

I

I

I

10K 20K 50K 100K 200K F r e q u e n c y in (Hz)

I 500K

(b) _1[

N e w current-, voltage- and mixed-mode circuit designs using the FiTFN or the FiTFN configured as a CCII+ or a unity-gain cell have been presented. All the proposed circuits enjoy the attractive features of." (i) Use o f grounded capacitors. (ii) Independent grounded-element control o f the parameters COoand CO0/Q0. (iii) Low active and passive sensitivities.

-31 -41

I

I

2K

5K

(c)

I

I

I

I

I

10K 20K 50K 100K 200K F r e q u e n c y in (Hz)

I 500K

References

15 5 =

It is expected that the FiTFN will be a useful device for analog signal processing circuit designs.

-5

-15 -25 I

I

I

I

I

2K

5K

I0K

20K

50K

I

I

100K 200K

I 500K

Frequency in (Hz) Fig. 7. Simulated results obtained from the circuit of Fig. 3 with: (a) notch: CwC3=C4=C6=5OOpF, R2=160K, R3=6.4K, Rs=160K; (b) LPF: CwC3=500pF, C6=lnF, R2=10K, R3=IOK, R3=10K, Rs=5K; (c) BPF: C1=500 pF, C3=C6=200pF, C4=2nF, R2=R3 =10 K, Rs=100K.

[1] Nordholt, E.H. Extending op amp capabilities by using a current-source power supply, IEEE Trans. Circuits Syst., CAS-29 (1982) 411-414. [2] Huijsing,J.H. and De Korte, J. Monolithic nullor--a universal active network element, IEEEJ. Solid-State Circuits, SC-12 (1977) 59-64. [3] Higashimura, M. Ikealisation of current-mode transfer function using four-terminal floating nullor, Electron. Lett., 27 (1991) 170-171. [4] Huijsing, J.H. Operational floating amplifier, IEE Proc., G-137 (1990) 131-136. [5] Senani, R. A novel application of four-terminal floating nullors, Proc. IEEE, 75 (1987) 1544-1546. [6] Senani,R. Generation of new two-amplifiersynthetic floating inductors, Electron. Lett., 23 (1987) 12021203. [7] Higashimura, M. Current-mode allpass filter using FTFN with grounded capacitor, Electron. Left., 27 (1991) 1182-1183. [8] Liu, S.-I. Cascadablecurrent-mode filters using single FTFN, Electron. Lett., 31 (1995) 1965-1966.

131

M.T. Abuelma'atti et al./Active biquads using FiTFNs

[9] Liu, S.-I. and Lee, J.-L. Insensitive current/voltagemode filters using FTFNs, Electron. Lett., 32 (1996) 1079-1080. [10] Abuelma'atti, M.T. Cacadable current-mode filters using single FTFN, Electron. Lett., 32 (1996) 14571458. [11] Senani, R. On equivalent forms of single op-amp sinusoidal RC oscillators, IEEE Trans. Circuits Syst. 1: Fundam. Theory Applic., 41 (1994) 617624. [12] Liu, S.-I. and Liao, Y.-H. Current-mode quadrature sinusoidal oscillator using single FTFN, Int. J. Electron., 81 (1996) 171-175. [13] Zele, R.H., Allstot, D.J. and Fiez, T.S. Fully balanced CMOS current-mode circuits, IEEE J. Solid-State Circuits, 28 (1993) 569-575. [14] Ramirez Angulo, R. and Sanchez-Sinencio, E. Two approaches for current-mode filters using voltage follower and transconductance multipliers building

132

[15] [16] [17]

[18] [19] [201

blocks, IEEE Int. Syrup. on Circuits and Systems, Vol. 5, 1994, pp. 669-672. Tsividis, Y. and Papananos, Y. Continuous time filters using buffers with gain lower than unity, Electron. Lett., 30 (1994) 629-630. Celma, S., Sabadell, J. and Martinez, P. Universal filter using unity-gain cells, Electron. Lett., 31 (1995) 1817-1818. Ramirez-Angulo, J., Robinson, M. and SanchezSinencio, E. Current-mode continuous time filters, IEEE Trans. Circuits Syst. I: Fundam. Theory Applic., 39 (1992) 337-341. Soliman, A.M. Mixed-mode biquad circuits, MicroelectronicsJ., 27 (1996) 591-594. Stevenson, J.K. Two-way circuits with inverse transmission properties, Electron. Lett., 20 (1984) 965-967. Brinson, M.E. and Faulkner, D.J. SPICE macromodel for operational amplifier power supply current sensing, Electron. Lett., 30 (1994) 1911-1912.