Design ofthird-order notch RC-active filter sections

Design of third-order notch RC-active filter sections M. Biey S. Coco

Indexing

terms: Filters and filtering

Abstract: Two third-order RC-active filter sections with finite transmission zeros, derived from two commonly used single-amplifier biquads, are examined, and a simple iterative procedure for their design is described, which preserves the lowsensitivity performances of the embedded biquads. Next, the third-order notch section proposed by Boctor is considered and a criterion for its lowsensitivity design is given. Finally, the abovementioned sections together with that recently proposed by Rollett are compared from a sensitivity point of view.

R' Cj

Fig. 1

1~

R C

Third-order

notch section proposed by Boctor

Third-order

notch section proposed by Rollett

Introduction

Third-order RC-active filter sections with transmission zeros at finite frequencies have an ideal voltage transfer function which in most practical situations takes the form T(s)

=H

(s

sr[s2 + w;] + Wt)[S2 + (wp/qp)s + w;]

,,,

(1)

with, respectively, r = 0 or r = 1 for the lowpass-notch (LPN) or high pass-notch (HPN) case. These sections are usually realised by cascading a fir:storder RC passive section and a biquad through a buffering amplifier [1]. Such a realisation has the advantages of preserving the sensitivity performances of the employed biquad and of design simplicity. However, it requires one more operational amplifier (OA), increasing the amount of power consumption. To overcome these inconveniences, some authors [2-6] have proposed third-order notch single-amplifier RC-active blocks. This technique saves one OA in the cascade realisation of classical (i.e. elliptic) odd-order lowpass filters and m OAs in the case of special transfer functions with a multiple real pole of multiplicity m [7, 8]. In particular, the solution proposed by Boctor [2] and shown in Fig. 1 offers good sensitivity performances, although some constraints on the equality of RC products and on the accuracy of the ratio of two capacitances constitute a practical disadvantage. In the same reference, explicit design formulas are given that depend on a free parameter, which influences element spread. However, its effect on the overall sensitivity of the section has not been assessed explicitly. Recently, Rollett has proposed a new third-order section [3], shown in Fig. 2, based on a split twin-Tin a Paper 6472G (E10), first received 8th February and in revised form 10th August 1988 The authors are with the Istituto di Elettrotecnica ed Elettronica, Universita di Catania, Catania, Italy 32

y. 1

Fig. 2

negative feedback loop, which simplifies tuning procedures without deteriorating the overall sensitivity. However, to realise the complex pole-pair and the finite transmission zeros, none of the proposed circuits [2-6] makes use of the two popular low-sensitivity single-amplifier building blocks STAR [9] and TTSABB [10, 11] described in the literature. Only an example of the use of TTSABB to construct a third-order section may be found in Reference 12, but details on sensitivity behaviour and design algorithms are not given. Consequently, the circuits mentioned [2-6] look less attractive from the point of view of standardisation. Therefore, it seems of some interest to examine in more detail a realisation of third-order stages that employs the standard biquads [9-11] in a proper context, so as to preserve their low-sensitivity performances. In this paper, we first present an iterative computer procedure for the design of two single-amplifier notch third-order sections, obtained by prefixing a passive firstorder RC network to the above-mentioned STAR and TTSABB biquads. In what follows, these circuits, shown in Figs. 3 and 4 in their LPN version, will be referred to as extended-STAR and extended-TTSABB. Even if the lEE PROCEEDINGS,

Vol. 136, Pt. G, No.1, FEBRUARY

1989

.•.. ----------------------------------~=~

Ra

presented results refer to the LPN case, they may be extended in an obvious way to the HPN case, simply by interchanging the resistor and the capacitor in the RC input section.

Secondly, to reduce the effects of the finite value A of the OA open-loop gain, the biquad is designed by minimizing its gain-sensitivity product [13] (2)

compute

Wz>W R

Fig.3

7

Wz P

1 =

R' z(2Trf,

p

Circuit of extended-STAR

design

C'f!

and H'zH/(2TTf,

biquad

)

for minimum

r

with fz,fp,qpandH Circuit R6 of RSextended- TTSABB

l I

~*c~ Fig.4

I

R7

C3

.

compute new. values of f1 . fz ,fp ,qp and H

analyse the 3rd - order section with real OA

Vo r

are

the N

assigned

Secondly, the attention is focused on the section proposed by Boctor [2], which appears, compared to the others, to have been studied in more detail for a manufacturing environment. In view of a subsequent comparison, its sensitivity is examined and, as result of this analysis, a simple criterion for its low-sensitivity design is gIven. Finally, the three considered blocks (the Boctor section, extended-STAR and extended- TTSABB), designed according to the given guidelines, are compared, from a sensitivity point of view, with the Rollett section, which is attractive in view of an economical design for its interesting tuning features. 2

Design of extended-STAR and extended- TTSABB

The LPN versions of the extended-STAR and extendedTTSABB are shown, in Figs. 3 and 4, respectively. They are obtained by connecting the RC network formed by R' and C' to the input of the low-sensitivity biquads STAR and TTSABB and have an ideal voltage transfer function given by eqn. 1, with r = O. This arrangement has the advantage of leaving such critical factors as network sensitivity, building-block design, and ease of tuning virtually unchanged. Obviously, the direct connection of R' and C' to the input of the biquads produces loading effects that have to be compensated for. This can be done by modifying the nominal design parameters according to the iterative procedure shown in Fig. 5 and described in the following: First, the designer assigns the nominal parameters f1 = wd2n, fz = wz/2n, fp = wp/2n, qp, and H of the voltage transfer function given by eqn. 1. At the same time also, a preferred value is assigned to the capacitor C'; note that low values of C' tend to overload the connected biquad, whereas high values of C' may lead to unacceptably low values of R', depending on the assigned value off I' and hence a trade-off has to be found. lEE PROCEEDINGS,

Vo/. /36, Pt. G, No. I, FEBRUARY

1989

specifica satisfied

tions

? y

print

component

values

Fig. 5 Block diagram of iterative procedure extended-STAR and extended- TTSABB

proposed

to compute

according to Reference 13 for STAR and to Reference 11 for TTSABB, assuming as design parameters those shown in Fig. 5. Then, the whole circuit is analysed by suitable algebraic routines, and the actual poles, zeros, and multiplicative constant are computed and compared with their nominal values. The analysis may take into account the presence of a nonideal OA and of other parasitics, with the advantage that, at the end of the procedure, the designed circuit is compensated for the considered imper" fections. In this case, spurious poles and zeros are present in the transfer function; however, they can be easily singled out, since they are located at very high frequenCIes. Finally, if actual parameters agree, within an assigned tolerance, with their nominal values, the procedure stops. If that is not the case, new values are computed according to the following equation: Xu+

1)

= XU) + w

. X~)/X(O) (XU)

-

(3)

XU) )

where X denotes any of the design parameters (i.e.f1' fz, fp' qp' H) and X a is the corresponding actual value obtained in the analysis step shown in Fig. 5. Superscripts indicate the iteration number, and X(O) is the nominal value of the considered parameter. 33

The weighting factor W is assumed equal to 0.4 for X

= fl and equal to 1 in all the other cases. Furthermore,

to avoid too large corrections at the first iterations, the ratio X~i)jX(O) is limited within the range 0.8-1.2. Following the outlined algorithm, the convergence is fast for practical values of C. As previously mentioned, the overall sensitivity of the two circuits considered turns out to be very near to that of the embedded biquads. This is confirmed by a comparison of the standard deviation (Ja of the gain G(w), computed by Monte Carlo analysis .. Referring to a practical implementation based on thinfilm or thick-film hybrid technologies, the'specific properties of these processes, such as the tracking accuracy and stability of the RC product, have been taken into account in the statistical analysis. Capacitor and resistor relative variations have been assumed uniformly distributed, with a tolerance of ± 1%, and correlated in such a way as to keep the RC product constant within ±0.1 %. Furthermore, component variations due to nonperfect tracking ha ve been considered less than ± 0.03% (corresponding to ± 5 parts in 106 per degree Celsius, over a 60 deg C range) for resistors and less than ±0.06% for capacitors. The OA has been modelled taking into account its nonzero output impedance Ro, its finite input impedance Ri, open-loop DC-gain Ao and gain-bandwidth product J;. The nominal values of OA parameters and of their respective tolerances are Ro = 150 n ± 20%, Ri = 1 Mn ± 30%, Ao = 100000 ± 40% and J; = 1 MHz ± 20%. Using the described statistical model, for each Monte Carlo analysis 300 samples were generated and analysed, using SPICE as the analysis subprogram. The results shown in Figs. 6 and 7 refer to third-order sections with fl = 3.6 kHz, fz = 3.3 kHz, fp = 3.0 kHz, qp = 20 and H such that G(O) = 0 dB.

02

OJ

1J b'"

0

1

00

1J

0.1

2.6

2.7

29

28

frequency. Gain standard deviation

(JG

STAR C' = 47 nF C' = 22 nF

30

3.1

kHz

of extended-ST

Filter parameters:!! = 3.6 kHz;!, = 3.3 kHz;!p = 3.0 kHz; .......

AR qp

= 20

In each Figure, the full-line curve shows the gain standard deviation of the corresponding biquad alone, designed for the same values of fz,fp and qp; it represents the lower achievable limit for each of the considered third-order notch-sections. In the same Figures, the results of the analysis of two third-order realisations, each one involving a different value of C, are also reported. For both the realisations, resistor and capacitor values have been confined in the range 1-100 kn and 1-47 nF, respectively; to reduce resistor spread in the case of extended-STAR, use has been made of pi-to- T transformation, as suggested in Reference 9. 34

29

2.8

frequency,

Fig.7

Gain standard deviation

(JG

30

3.1

kHz

of extended-TTSABB

Filter parameters are as in Fig. 6 .......

TTSABB C' = 47 nF C' ~ 22 nF

values of C, causing lower loading effects, produce lower sensitivities, as shown in Figs. 6 and 7. Several other analyses were performed for a variety of values of the design parameters, without appreciably different results. For completeness, third-order filters realised as a cascade of a first-order RC passive section and a biquad using a buffer, have also been analysed. However, their gain standard deviation turns out to be practically coincident with that of the corresponding biquad and hence has not been reported in Figs. 6 and 7. 3

Sensitivity considerations on the Boctor section

0.0

Fig. 6

2.7

2.6

The third-order section proposed by Boctor and shown in Fig. 1 has an ideal voltage transfer function given by eqn. 1 with r = O. Explicit design formulas are given in Reference 2 for the element values in terms of R and of a free parameter a, which turns out to be not lower than one (see eqn. 26 of Reference 2). The same Figure also shows the negative feedback arrangement, formed by Co and Rb, for the compensation of the unity gain amplifier, as suggested in Reference 2. By choosing

CD

b'"

In general, the loading effect, due to the direct connection of R' and C to the biquad, leads to an increase of the biquad Q during the design phase. Hence, higher

Co Rb

= Ij(2nJ;)

(4)

where J; is the unity-gain bandwidth of the OA, the gain of the amplifier equals one over a wide frequency range; in noncritical applications this compensation may be omitted, by letting Co = Rb = O. The free parameter a influences element spread. Furthermore, from eqn. 27 and the general formulas of Table 2 of Reference 2, it turns out that q~ = qpaJ(1

+ (a -

l)f;jf~)

(5)

where q~ is the pole-Q of the biquad, used to build the third-order section of Fig. 1. Eqn. 5 shows that, for given specifications (i.e. for given qp, fp, fz), q~ is an increasing function of a. Hence, since higher values of the pole-Q are related to higher sensitivities, it is expected that higher values of a lead to more sensitive designs. To confirm this conjecture, several third-order filters have been designed with equal values of fl, fz,fp, qp and H, and different values of the free parameter a. Then, they lEE PROCEEDINGS,

Vol. 136, Pt. G, No.1,

FEBRUARY

1989

have been compared on the basis of the standard deviation erG of the gain G(w), computed by Monte Carlo analysis, using the same component statistics and number of samples as in the preceding section. The curves, shown in Fig. 8, result from the simulation of a filter with the same nominal parameters of the previous examples.

generic quantity Y, WY is defined as follows [14]: n

WY

= i=L1 IS~il

(6)

where S~i is the sensitivity of Y with respect to Xi' The statistical multiparameter sensitivities pIz, ph and pqP, defined as [14]:

0.4 pY aJ

1)

a~02

-

.' ... ;.-.;.,..

'/,,'/

.'/

.. >,

."

"".;-'-

0.0 2.6

2.7

29

2.8 freqency,

Fig. 8 Gain standard deviation values of a Filter parameters as in Fig. 6 a Q

a

(JG

3.0

3.1

kHz of Boctor

section for

J Ctl (S~)2)

(7)

for a generic quantity Y, are reported in columns 4-6 of Table 1. Column 7 of the same Table shows the gainsensitivity product defined in eqn. 2; the last columns contain the capacitor and resistor spread Cmax/Cmin and Rmax/ Rmin, respectively, and the capacitor and resistor count nc and nR. A comprehensive comparison, taking into account the variations of both active and passive elements, may be obtained by considering the statistical results shown in the previous sections. The most favourable of them have been merged in Fig. 9, together with the gain standard

different

= 1.2 = ao = 2.0

Three different values of a have been considered, namely a = 2, a = aO = 1 + (fp/fz)2 = 1.85, and a = 1.2. The case a = ao corresponds to the choice indicated in Reference 2 for low element spread; note that ao has been computed using predistorted values of fp and fz, determined in the design step to compensate for the presence of a nonideal OA. In all the cases, resistor and capacitor values have been confined within the range 1-100 kQ and 1-56 nF, respectively. From Fig. 8 it appears evident that the lower the value of a, the lower is the global filter sensitivity. These considerations suggest a simple criterion for the designer: a should have the lowest possible value consistent with the assigned range of components. 4

=

Comparative examples

To compare the sensitivity performances of the three considered blocks, the most significative parameters, such as passive multiparameter sensitivities, gain-sensitivity product, component spread and count, have been computed and collected in Table 1 for each of the previously examined circuits, and for the Rollett section, designed according to the equations reported in Reference 3. The worst-case multiparameter sensitivities WIz, Wh and wqp with respect to the passive components are shown in the first three columns of Table 1. For each 785-6 6nR 21.1 111 321.3 0.97 0.96 67.9 2.21 0.80 0.84 27.1 72.1 24.3 17.1 rR104.0 103.0 498.5 510.0 81.1 5-6 0.94 2.15 23.9 0.74 0.85 48.1 40.9 2.00 1.07 8.87 102.0 70.6 38.6 2.00 F" 2.02 2.63 wap W'p Fap F'p Rmin Cmin maxi 2.04 Table 1: Multiparameter nc C' = 47 nF Rollett Extended spread and count for the four blocks considered

sensitivities, Cmax/

aJ

1)

a"

0 1

0.0 26

2.7

2.8

29

3.0

3.1

frequency, kHz

Fig.9

Gain standard deviation extended- TTSABB Boctor section extended-STAR Rollett section

(JG

of the four blocks considered

deviation of the Rollett section. From this Figure, it is evident that the four blocks present only slight sensitivity differences, and hence other features, such as ease of tuning and/or the possibility of using standard buildingblocks, should be considered by the designer in his choice. To have an idea of how the considered blocks behave as the design frequency is increased, the notch depth of each of them has been examined at higher frequencies. Fig. 10 reports typical upper-envelopes of the gain in the proximity of the notch frequency, obtained by Monte Carlo simulation, with the same statistics used throughgain-sensitivity

product,

component

35

out the paper of a filter with!,. = 14.4 kHz,fz = 13.2 kHz, qp = 20.0. Finally, to compare the four considered blocks in a practical situation, a sixth-order lowpass trasfer function

fp = 12.0 kHz and

have slightly lower values of a G and an particularly in the case of extended-STAR and extended-TTSABB. On the other hand, the Rollett section offers some advantage

-.-'-'-...

-10

------

0.05 aJ

D t9

b aJ

D

c

0.0

o

2.8

(J1

2.9

30

3.1

3.2

frequency.

Fig.11

Gain standard deviation

rJG

3.3

3.2

3.5

k Hz

of the four considered sixth-order

filters Building-blocks employed: extended- TTSABB Boctor section extended-STAR Rollett section

-50 13.0

13.1

13.2

frequency,

13.3

134

kHz

Fig. 10

Upper envelopes of the gain in the proximity of the notchfrequency for the four blocks considered extended- TTSABB Boctor err Boctor section Nominal ripple = 0.053 dB Jir extended-STAR Rollett section

with a double real pole [7], suitable for PCM applications, has been realised by four different filters, employing, respectively, the three sections considered in this paper and designed according to the given guidelines, and the section proposed by Rollett [3]. The considered transfer function has a passband attenuation lower than 0.053 dB in the range 0-3400 Hz, and a stopband attenuation greater than 30 dB for frequencies greater than 4600 Hz. Its poles and zeros are reported in Table 2; the Table 2: Poles and zeros of the transfer function considered in the example 1.285093 6.382767 Poles real 1.706413 1.265937 Q-factor Zeros frequency

12961 99876

Normalisation

frequency = 3407 Hz

multiplicative constant is such that the gain equals 0 dB at zero frequency; the normalisation frequency, equal to 3407 Hz, has been chosen to maximise passband and stopband margins in a statistical way. The results of a Monte Carlo simulation, based on the same component statistics used in the previous examples, are shown in Fig. 11. Furthermore, since for the filter designer it is the passband ripple r (i.e. the difference between maximum and minimum attenuation in the passband) that is important, it has been computed for each sample filter, and then its mean value /l, and its standard deviation a, have been evaluated and reported in Table 3. From Figs. 9 and 11 and Table 3, it turns out that the three filters built with the blocks considered in this paper 36

Table 3: Mean value It, and standard deviation 0". in dB of the passbandripple for the four filters considered 0.030 0.096STAR ExtendedExtended0.024 0.087 0 0.035 0.108 .028 .086 T Rollett TSABB

in the tuning procedure and in the achievable notch depth (Fig. 10), at the expense of a very moderate increase in its overall sensitivity.

5

Conclusions

An iterative computer procedure has been proposed for the design of two third-order notch sections employing two widely used single-amplifier biquads, namely STAR [9] and TTSABB [10, 11]. A statistical comparison, made on the basis of the gain standard deviation aG computed by Monte Carlo analysis, shows that the two considered third-order sections, when designed according to the outlined procedure, have an overall sensitivity very near to that obtainable by connecting a first-order RCactive section to the input of the considered biquad. The sensitivity of the section proposed by Boctor has been examined, and the influence of the design parameter a on its overall sensitivity has been investigated. Hence, a simple criterion for the choice of a for low-sensitivity designs has been given. Finally, the sensitivities of the three sections considered and of the one recently proposed by Rollett [3] have been compared. From the examples considered it turns out that the practical advantages offered by the Rollett section imply only a little decay in sensitivity performances. Moreover, it is worth noting that better results in this sense could probably be obtained by adding some amount of positive resistive feedback through the noninverting input terminal of the OA. On the other hand, in applications where a high degree of standardisation is required, extended-STAR and extended- TTSABB seem to be valid alternatives to the other existing third-order sections, at the cost of slightly more complex tuning procedures. lEE PROCEEDINGS,

Vo/. 136, Pt. G, No. I, FEBRUARY

1989

r

6

Acknowledgments

r

!

The authors are grateful for the computer facilities, freely offered by CSI-Piemonte, Torino, Italy, and partial financial support from C.N.R., Roma, Italy. Part of this work has been presented at the 1984 International Symposium on Circuits and Systems, Montreal, Canada, May 7-10, 1984. 7

References

1 WAIT, lV., HUELSMAN, L.P., and KORN, G.A.: 'Introduction to operational amplifier: theory and applications' (McGraw-Hill, New York,1975) 2 BOCTOR, SA: 'Design of a third-order single amplifier filter', IEEE Trans., 1975, CAS-22, pp. 329-334 3 ROLLETT, 1.M.: 'Easily trimmed third-order RC-active-filter section with finite transmission zeros', Electron. Lett., 1983, 19, pp. 677-678 4 QUACH, L.T.: 'Design of single amplifier filters with finite transmission zeros', Trans. IECE Jpn., 1982, E6S, pp. 9-15 5 ISHIBASHI, Y.: 'A single amplifier 3rd-order lowpass notch filter', Trans. IECE Jpn, 1980, J63-A, pp. 227-229

lEE PROCEEDINGS,

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1989

6 KUMAZAWA, M., and YANAGISAWA, T.: 'Video frequency active filters using balanced type NICs', Trans. IECE Jpn, 1985, J68-C, pp. 240--247 7 BIEY, M., and PREMOLI, A.: 'Multiple real-pole equal-ripple rational (MRPER) approximation for lowpass RC-active filters', Int. J. Circuit Theory Appl., 1980,8, pp. 219-228 8 BIEY, M., and PREMOLI, A.: 'Lowpass filters by cascading low-Q 3rd-order blocks', lEE Proc. G, Electron. Circuits & Syst., 1982, 129, (1), pp. 26--27 9 FRIEND, 1.J., HARRIS, CA., and HILBERMAN, D.: 'STAR: an . active biquadratic filter section', IEEE Trans., 1975, CAS-22, pp. 115-121 10 MOSCHYTZ, G.S.: 'A universallow-Q active-filter building block suitable for hybrid-integrated circuit implementation', IEEE Trans., 1973, CT-20, pp. 37--47 11 MOSCHYTZ, G.S.: 'Linear Integrated Networks: design' (Van Nostrand Reinhold, New York, 1975) 12 FRIEDENS ON, R.A., DANIELS, R.W., DOW, R.J., and McDONALD, P.H.: 'RC active filters for the D3 channel bank', Bell Syst. Tech. J., 1975,54, pp. 507-529 13 MOSCHYTZ, G.S., and HORN, P.: 'Active filter design handbook' (Wiley, New York, 1981) 14 BRUTON, L.T.: 'RC-active circuits, theory and design' (PrenticeHall, Englewood Cliffs, N.J., 1980)

37