CONSTANT-RESISTANCE ALL-PASS OTA-C FILTERS

L. P. CalBba and A. C. M . de Queiroz

Dept. of Electrical Eng., COPPE G EE Universidade Federal do Rio de Janeiro

CP 68504. CEP 21945, Rio de Janeiro. Brazil

Abstract

In th i s paper we present new economical and low- sensitivity all-pass sections suitable f o r the delay equalization of OTA-C continuous-time f i l t e rs . The use of constant-resistance sections allows a delay equalizer t o be inserted between an LC ladder simulation f i l t e r and one of i t s terminations, a s done in passive realiza- tions. An equalizer can be built a s a cascade of 1st and 2nd-order sections, o r directly as a single n th order section. Several canonical s t ruc tures a r e discussed, and the gain and group-delay sensitivities f o r the realiza- t ions a r e compared.

Introduction

Precision analog f i l t e r s a r e usually composed by two sections: a frequency selector and a phase equali- zer. High-order and high-selectivity f i l t e r s must pre- sent low sensitivity t o i t s components. To a t ta in th i s goal in passive realizations, the frequency selector is usually implemented a s a doubly terminated LC ladder, and the phase equalizer i s realized using constant- resistance la t t ice or bridged-T modules. In th i s way, the phase equalizer modules can be directly cascaded between the LC ladder and one of i t s terminations.

The active f i l t e r s with best sensitivity characte- r i s t ics a r e obtained by simulation of passive f i l t e rs . Several continuous-time fully integrated OTA-C f i l t e r s which simulate passive f i l t e r s have been proposed in the recent l i t e ra ture . Most of the research remains res t r ic - ted t o the frequency selector p a r t , in the form of LC doubly terminated ladder simulations [11[21[31. The use of constant-resistance delay equalizers was recently demonstrated t o be practical [41, and superior t o the use of biquad cascade equalizers , in the form of a cascade of second-order sections based on gyra tors L71. In th i s paper, the general case of gyrator-based cons- tant-resistance all-pass OTA-C f i l t e r s i s studied.

Fig. (1) - Basic gyrator-based constant-resistance all- pass section. ( a ) Connected to the output termination. ( b ) Connected to the input termination.

Gyrator-based constant-resistance all-pass f i l t e r s

Passive constant-resistance circuits a r e usually made with la t t ice or bridged-T s t ruc tures , both non- canonical in t h e number of reactive elements. Those s t ruc tures can be simulated with OTA-C circuits, but a simpler solution t o a constant-resistance all-pass fi l- t e r i s obtained if the gyra tor i s allowed a s a circuit element. In OTA-C realizations th i s i s not a problem, since a gyra tor can be easily implemented by two OTAs. In fig. (1) the basic circuit i s shown. An LC admittance YIC(S) i s connected between the two terminals of a

grounded gyra tor . The f igure shows the section in the two possible places in a doubly terminated f i l t e r where i t can be placed.

If the equalizer i s connected t o the output termina- tion (Fig. (la)), the input impedance Zds) of the com- bination is found as:

This resu l t s in ZI(S)=RI if G=l/Rl. In th i s case, if R I is the cor rec t output termination f o r the f i l t e r , the f i l t e r t r a n s f e r function i s changed only by multi- plication by the f a c t o r Vz(s) /Vds) , given by:

YIC(S) is a function in the form N e ( s ) / D n ( s ) o r N d s ) / D e ( s ) , where e and n denote even o r odd polyno- mials. Eq. ( 2 ) can be put in one of the forms below, recognizable a s all-pass functions:

(3a)

The forms (3a) and (3b) a r e non-inverting at infi- nity if the numerator of YIC(S) i s of higher degree than the denominator. If the opposite occurs, the resulting all-pass function is inverting at infinity.

If the equalizer i s connected t o the input termina- tion (Fig. ( lb ) ) , the output impedance of the combina- tion is:

+ Y 1 c ( s )

(4 ) Z g ( s ) = Rg R g R g G ' + YIC(S)

This resu l t s in Zg(s)=Rg if G=l/Rg. If Rg is the cor rec t input termination f o r the f i l t e r , also in this case the overall t r a n s f e r function i s changed only by multiplication by the same f a c t o r Vz(s ) /Vl ( s ) given by eq. ( 2 ) (with Rg in place of R I ) and the f ina l result is the same.

Due t o the constant-resistance property, if a num- ber of sections like these a r e cascaded a t the input o r output sides, the t r a n s f e r function of the result ing f i l t e r i s changed only by the multiplication by the all-pass t r a n s f e r functions of all the sections.

I t i s interesting t o observe t h a t the gyrator-based all-pass i s a non-reciprocal network. If a passive LC doubly terminated f i l t e r i s connected with input and output interchanged (with the connections t o the termi- nations changed accordingly), even with constant- resistance reciprocal all-pass networks included, the transfer-function i s not changed (or only the gain chan-

CH2819-1/90/oooO-0488$01.0001991 IEEE

ges if the terminations a r e d i f fe ren t ) . The same can be made t o happen in an OTA-C active simulation. When a gyrator-based all-pass network i s used, however, t h e e f fec t of the reversion in the equalizer i s equivalent to a change in the sign of the gyration conductance G. By eqs. (1) and (4) i t can be observed t h a t the cons- tant-resistance property of the equalizer i s retained when G changes sign, but eq. ( 2 ) shows t h a t the all-pass t ransfer function i s converted into Vz(s)/Vl(s)=l, and the equalizer e f fec t disappears. This charac te r i s t ic res t r ic t s the use of these equalizers t o applications where the bidirectionality property of reciprocal f i l- t e r s is not used, o r f i l t e r s where t h e signal t rave ls in only one direction.

(a)

Y l C *

Fig. ( 2 ) - ( a ) Gyrator-based all-pass. ( b ) Another gyra- tor used to convert the floating Y d s ) into i ts groun- ded dual. (c) OTA-C realization f o r the gyrators. ( d ) Final circuit, with the two gyrators combined and opti- mal sensitivity preservation.

The sensitivity characterist ics a r e also interes- ting. A s the equalizer i s composed only by a gyra tor and reactive elements, i t i s lossless. In the case shown in fig. ( l a ) , the input impedance ZI(S) i s resist ive and equal t o the output load res i s tor RI. The power entering the section in the sinusoidal steady s t a t e i s P= 1 VI( j w ) I 2 / R ~ , and because the gyrator-Yk network can- not dissipate any power, P i s entirely dissipated by RI, and the output voltage magnitude IVz(jw)l must be equal t o lVi(jw)l, maybe with another phase. The same can be concluded by the f a c t t h a t the t r a n s f e r function V z ( s ) / V i ( s ) i s an all-pass function. The interesting f a c t t o observe is t h a t Y d s ) has absolutely no influ- ence in the relation lVz(jw)/Vl(jw)I=l, and so the gain sensitivities of any order t o all the reactive elements composing Y i d s ) a r e equal t o zero at any frequency. When maximum power t r a n s f e r occurs in the equalized fi l- t e r , the f i r s t -order gain sensitivities in relation t o all the lossless elements, the reactive elements of the original f i l t e r and the equalizer gyration conductance a r e also equal t o zero, by the Orchard argument 161. If Y i d s ) i s realized by an OTA-C simulation t h a t respects the rule s ta ted in 151 for optimal sensitivity preserva-

tion (an error in a given OTA transconductance must be equivalent to a change in the impedance level o f the circuit connected to i ts output), the gain sensitivities t o the OTA transconductances used by Y d s ) a r e sums of passive element gain sensitivities, and a r e also null.

Synthesis procedure

The synthesis of an n t h order all-pass section begins with the choice of the gyration conductance G, tha t must be equal t o 1/RI or 1/Rg, depending on where the equalizer i s placed. I t is simpler t o work with the normalized f i l t e r , where i t i s always possible t o use G=l. Y i d s ) can then be obtained f rom one of the identi- f ications shown in eqs. ( 3 ) .

The LC admittance Yids) can be realized by several s t ruc tures . Taking into account t h a t the dual admittance also realizes the desired equalizer (with sign inver- sion), and res t r ic t ing the s t ruc tures t o the canonical Cauer and Foster forms , there a r e eight possible passive realizations f o r a given all-pass section.

An efficient way t o obtain an unbalanced OTA-C simulation i s shown in fig. (2). Yids) i s converted into i t s grounded dual by a gyra tor , and t h e t w o gyra- t o r s a r e combined into a three OTAs circuit . This ap- proach i s t h e best t h a t can be done t o preserve the the sensitivity charac te r i s t ics of the original prototype. The simulation of Yids) cannot be made in the floating position with optimal sensitivity preservation, due t o the coupling between i t s terminals caused by the gyra- t o r . With the conversion t o the grounded dual form, the two OTAs t h a t feed t h e grounded gyra tor in fig. (2c) and the two OTAs forming the grounded gyra tor itself s t i l l don’t sa t i s fy t h e ru le f o r optimal sensitivity preserva- tion. When these f o u r OTAs a r e combined in two, however, the rule i s satisfied (as long as the common-mode t rans- conductance i s negligible), and the circuit i s simpler. The simulation of the dual of YIC(S) can be easily done by the embedding technique [51. Balanced versions can be easily obtained, if necessary [21.

( d ) L 2

>-Idv2 $ C l c 3 + + L 4 v 1

Fig. ( 3 ) - Realizations f o r a 4th-order all-pass cons- tant-resistance section with Y d s ) realized by Cauer expansions. Cases ( a ) and (b) are non-inverting, and cases ( c ) and ( d ) inverting.

Figs. ( 3 ) and (4) show several possible realiza- t ions f o r a 4th-order constant-resistance all-pass ob-

489

tained by th i s approach, all in the most economical ver- sion. The ones in fig. (3) a r e derived f rom the Cauer expansions of Y i d s ) o r i t s dual, and the ones in fig. (4) f rom Foster expansions. With the exceptions of cases (3a) and (4b), where Ci was le f t at i t s original posi- tion f o r economy, all the s t ruc tures present optimal preservation of the sensitivity characterist ics of the passive prototypes. There i s no practical s t ruc ture t o the OTA-C simulation when Y i d s ) i s in the form shown in fig. (4d), with optimal sensitivity preservation, becau- s e f o r th i s the ser ies LC tanks in parallel must at f i r s t be dualized into the form (4a) , and again dualized f o r the OTA-C simulation. The final s t ruc ture would be similar t o (4a) with two more OTAs forming a gyrator. The same problem would occur with the form (4b) f o r higher orders.

11 1 I I i r T i r - + - + - + - \ + - //+ -!\ -/ ( a

i t s adjacent termination. The other s t ruc tures can also be made t o present all the capacitors grounded, a t the expense of more OTAs.

L

An nth-order equalizer can be realized as a cascade of independent f i r s t and second-order sections. Fig. ( 5 ) shows two possible 4th-order realizations, one a s a cascade of economical non-inverting sections (5a) , and another as a cascade of inverting sections, with all the capacitors grounded and optimal sensitivity preservation (5b). The problems of th i s approach a r e the higher num- ber of OTAs required, and in the case (5a) the subopti- mal sensitivity charac te r i s t ics and the lacking of a way t o absorb the bottom-plate capacitance of one of the floating capacitors.

Yl ’ c -

( b )

( C )

j L 2 2 L 4 r ,Tc1Tc3 Y IC

( d )

Fig. (4) - Realizations f o r a 4th-order all-pass cons- tant-resistance section with Y i d s ) realized by Foster expansions. Cases ( a ) and ( b ) are non-inverting, and case ( c ) inverting. The simulation in case ( d ) i s not practical.

The most economical solution i s the one in fig. (4b). I t presents floating capacitors, but each one presents at one side a grounded capacitor (one of them f rom the frequency selector f i l t e r ) t h a t can absorb the bottom-plate parasit ic capacitance. Similar si tuations occur in cases (3a), (3d) and ( 4 ~ ) . Due t o the bottom- plate capacitances, the form (3b) may be impractical. The most interesting s t ruc ture i s (3~1, t h a t presents all the capacitors grounded. The complete f i l t e r can present capacitors t o ground at all the nodes, with the exception of the connection of the delay equalizer with

I C2TT

I C4TT

Fig. (5) - Realization of a 4th-order equalizer by a cascade of two 2nd-order sections. ( a ) With non- inverting sections. ( b ) With Inverting sections.

Example

A s an example, a 4th-order normalized Chebyschev low-pass f i l t e r with 0.25 dB ripple and passband edge a t 1 rad/s, equalized by a 4th-order all-pass is used. The optimized all-pass function required i s shown in eq. ( 5 ) [SI, and the s t ruc ture f o r the complete f i l t e r i s shown on fig. ( 6 ) .

s4-1. 895s3+1. 97275s2-1. 08447s+O. 252504

s4+1, 895s3+1, 97275s2+1. 08447s+O. 252504 (51 V O ( S ) - ~-

’(’)

Fig. ( 6 ) - Passive f i l t e r f o r the example (a) , and i ts OTA-C simulation (b).

490

Table I lists the component values obtained by the application of eqs. (3a) or (3b) t o (51, with G=l. Cases 3a t o 4c correspond t o the networks in figs. (3) and (4). Cases Sa and 5b correspond t o the cascade realiza- tions in fig. (5).

Table I - Component va lues f o r the c i r c u i t s i n f i g s . ( 3 ) , (4), and (5) f o r the rea l iza t ion o f t h e t r a n s f e r func t ion (5). Gm=I f o r a l l the OTAs.

L,C1 ,528 4.29 .528 4.29 3 . 5 3 ,528 ,528 3 . 2 0 . 9 3 5 L , C 2 1.35 1.41 1.35 1.41 2 .06 4.29 4.29 , 9 3 5 3 . 2 0 L,C3 1.89 ,775 1.89 ,775 ,768 ,885 ,885 1 , 0 9 1 . 2 1 L,C4 2.94 ,843 2.94 ,843 ,709 1.98 1.98 1.2 1 1 . 0 9

In figs. (7) and (8) , The sensitivity charac te r i s - t i cs of the realizations a r e compared. The comparison cr i te r ia used a r e the s ta t i s t ica l deviation of the gain and of the group-delay in the passband, considering 5% uncorrelated variabilities f o r all the circuit parame- t e r s (capacitances and transconductances) of the comple- t e f i l t e r . The e r r o r s in the unequalized f i l t e r a r e also shown as a reference. The deviations were computed f rom the sensitivities of the t r a n s f e r function T(jw) t o the circuit parameters xi by the use of eqs. (6) and (71, f o r the e r r o r in gain and group-delay, respectively. V i a r e the variabilities, s e t at 0.05.

0.90 I \

F

0.00 rrd,r 1.10 0.40

Fig. (7) - Gain s ta t i s t ica l deviations f o r the equal i - z e r s of the example , represented by cases 3a (identical t o 4b), 36 (identical t o 4a), 3c (identical t o 3d and 4c), 5a, 5b and F, the unequalized f i l t e r .

The be t te r gain sensitivity characterist ics of the 4th-order sections, and the be t te r group-delay sensiti- vity characterist ics of the cascade realizations can be observed. The higher gain e r r o r s f o r the cascade reali- zations a r e mainly due t o the la rger number of OTAs controlling the DC gain of the f i l t e r , but in case 5b the e r r o r due t o the imperfect preservation of the pro- totype f i l t e r sensitivities i s significant. These re - sults appear t o be typical f o r similar realizations of other f i l t e r approximations. I t i s interesting t o note tha t the group-delay deviations f o r the cascade form a r e similar t o what i s obtained with o ther forms of cascade realizations f o r the equalizer, as the biquad cascade,

49 1

tha t a r e much worse in te rms of gain sensitivity. The losslessness of the constant-resistance realizations ensures only low gain sensitivities. Nothing can be guaranteed about phase or group-delay sensitivities.

F I

Fig. (8 ) - Group-delay s ta t i s t ica l devia t ions f o r the e q u a l i z e r s of the example . I n t h i s case , a l l the c i r - c u i t s p r e s e n t d i f f e r e n t charac te r i s t ics . F i s the une- qua l ized f i l t e r .

Conclusions

The constant-resistance gyrator-based all-pass i s a convenient circuit f o r realization by OTA-C structures. The networks with lower gain sensitivities, and the simplest ones, a r e obtained when the all-pass network i s realized as a single nth-order section, but somewhat be t te r group-delay sensitivity characterist ics, and, a t least in t h e example shown, a smaller spread in capaci- t o r values, a r e obtained when the equalizer i s realized a s a cascade of low-order sections.

References

A. C. M. de Queiroz, L. P. Cal6ba, and E. Sanchez- Sinencio, "Signal f low graph OTA-C integrated fi l- ters", Proc. 1988 IEEE ISCAS, Espoo, Finland, pp. 2165-2168, June 1988. M. A. Tan and R. Schaumann, "Simulating general- parameter LC-ladder f i l t e r s f o r monolithic realiza- t ions with only transconductance elements and grounded capacitors", IEEE Trans. Circuits and Systems, vol. 36, no. 2, pp. 299-307, February 1989. F. Krummenacher and G. Van Ruymbeke, "Integrated selectivity f o r narrow-band FM IF systems", IEEE J. Solid-state Circuits, vol. 25, no. 3, pp. 757-760, June 1990. V. Gopinathan and Y. Tsividis, "Design considera- t ions f o r integrated continuous-time video fi l- ters", Proc. 1990 IEEE ISCAS, New Orleans, USA, pp. 1177-1180, May 1990. L. P. CalBba and A. C. M. de Queiroz, "OTA-C simu- lation of passive f i l t e r s via embedding", Proc. --- 1989 IEEE ISCAS, Portland, USA, pp. 1083-1086, May 1989. H. J. Orchard, "Inductorless fi l ters", Electronics Let te rs , vol. 2, pp. 224-225, June 1966. L. T. Bruton, "RC-Active Circuits, Theory and De- w, Prentice-Hall, 1980, pg. 502. G. Daryanani, "Principles of Active Network Synthesis and Design", John Wyey and Sons, 1976, pg. 123.