Integrable analogue active filters for implementationin MOS technology

R. Schaumann, Dipl.-lng., Ph.D., Sen. Mem. I.E.E.E., and J.R. Brand, M.S., Mem. I.E.E.E.

Indexing term: Filters

Abstract: A design method for MOS-compatible analogue active filters that use only integrating amplifiers andratios of capacitors is discussed. The circuits retain most of the implementation advantages of sampled-dataswitched-capacitor filters, but are complementary to switched-capacitor filters in providing better and more pre-dictable performance at high frequencies. The effects of circuit parasitics and amplifier nonidealitics are discussedand experimental results are presented to illustrate the performance.

1 Introduction

Among the various methods proposed for implementing mono-lithic analogue active filters [1 -3] , active R filters [4] arereceiving attention because they are suitable for operation oververy wide ranges of frequency (from audio to video frequencies)[5] and because they employ no capacitors other than theones used for internal amplifier compensation. In thesecircuits, amplifiers are used as integrators, and filter par-ameters, such as pole frequency, quality factor and gain, areestablished via resistor ratios [4].

Recently it has been recognised, however, that, in MOStechnology, ratios of capacitors can be implemented moreaccurately, and are less prone to drifts and use less silicon areathan ratios of resistors [6]. This prompted a sudden interest in'switched-capacitor' sampled-data analogue active filters [1,7—9], in which charge transfer on rapidly switched capacitorssimulates current flow in resistors; thus, both RC products andratios of resistors, which set the filter parameters of active RCfilters, are replaced by ratios of capacitors [7].

Although switched-capacitor filters provide excellent resultsat low frequencies [1], their advantages in operation andmonolithic implementation, currently, do not appear trans-ferable to high frequencies because of sampled-data operation,aliasing, and by reason of the amplifiers' frequency-dependentgain which introduces additional poles and phase shifts intothe filter function, resulting in problems similar to thoseexperienced in analogue active RC filters at high frequencies[10]. In particular, for each amplifier used, the degree of thetransfer function increases by at least one over that of theideal function, giving rise to parameter deviations, potentialinstability, and rapidly rising sensitivities with increasing polefrequency [11, 12].

In this paper, a filter design method is outlined, employingthe ideas leading to active R filters and, at the same time,retaining the technological advantages (easy monolithic imple-mentation) of switched-capacitor filters without entailing any

Fig. 1 Block diagram of second-order state variable filter

Paper 1131G, received 15th February 1980The authors are with the Department of Electrical Engineering,University of Minnesota, 123 Church Street SE, Minneapolis, Minn.55455, USA

of the mentioned drawbacks. Some familiarity with active Rdesign [4] suggests that, in principle, the filter behaviourought not to change if each resistor is replaced by a capacitor,thus establishing filter parameters via ratios of capacitors. Inthe following text, this concept is verified and a number ofcircuits for different filter functions are derived; experimentalresults are presented to demonstrate the performance.

2 Circuit configuration

Consider the state-variable-derived topology in Fig. 1 whichrealises

TB = VB/VI = ±oA;1/{(A1A2yl + bA'2

1 + c} (la)

TL = VJV1 = ±al{(£xA%rx +bA~2l + c}

Assume, for simplicity, that the two amplifiers are identical,*with their gain described by the single-pole model with 'excessphase':

Al =A2 = A(s) = — e-« - — fs + a s

= -e~ST(2)

In eqn. 2, GB is the gain-bandwidth product, a the 3 dBfrequency and the excess phase term exp (— ST) is introducedto account for the effect of additional poles and zeros ofA(s) [4]. Assuming further that the frequency of operation,GJ, satisfies CJ > a, one may regard the amplifiers as idealintegratorst in a normalised frequency parameter S = s/GB; Tis defined as T = TGB. Setting, for the time being, T = 0, eqns.1 and 2 indicate that the configuration in Fig. 1 realises thebandpass and lowpass functions

TB = ±aS/(S2 +bS

TL = ±a/(S2 +b'S +

(3a)

(3b)

As in active R circuits, the coefficients a, b, c and the summingfunction in Fig. 1 can be realised via a variety of admittancenetworks such that a, b and c depend only on ratios ofelements. Two specific topologies for second-order filtersections, using only capacitors, are shown in Figs. 2a and b.More complicated passive capacitance networks, analogous tothe active R development in [4], can of course be used. Acareful sensitivity investigation shows, however, that additionalelements would be detrimental to the performance [4, 14]without providing greater design freedom.

The circuits of Fig. 2 are sufficiently general to permit

*This is a realistic assumption in any totally monolithic filter imple-mentation or for the use of dual amplifiers on a chip. The extension toAl ¥= A 2 is trivial [4| and yields no significantly different results.•fThe reader is referred to the literature for straightforward extensionsto the case where u> > a is not valid [ 1 3— 15 |.

IEEPROC, Vol. 128, Pt. C, No. 1, FEBRUARY 1981 19

separate specification of gain, pole Q and pole frequency; thetransfer functions realised are

clear that

TB = (ai-a2)S/(S2 +bS + cuc2)

TL = (fli ~a2)c2/(S2 + bS + cnc2)

for Fig. 2a, and

TB = (al-a2)S/{S2 +bS + cnc2)

TL = ~(«i -a2)c2/(S2 +bS + cnc2)

(4a)

(4b)

(5a)

(5b)

for Fig. 2b, with the coefficientsak, b, cxk, c2, k = 1,2, givenby

a, =

a2 =

b =

C5 +CP1)

(6)

C11 = ^ / ( d + C 2 + C 3 + C N 1 )

C12 = CS/(C4 + C 5 +C P 1 )

indicating that all filter parameters are determined by ratios ofcapacitors. The elements Cpt and CNi in eqn. 6 representparasitic capacitors connected from the inverting (N) and non-inverting (P) input terminal of amplifier / to ground (cf.Section 4.3).

Fig. 2 All-capacitor active filter structures

a Realisation of eqns. 4a and bb Realisation of eqns. 5a and b

The circuits in Fig. 2 can realise a second-order lowpass andbandpass of positive or negative gain. Note also, that biquad-ratic transfer functions are available at the inverting inputterminals of amplifier A x in Figs. 2a and b [4]; for a moreversatile and less load-sensitive realisation of such a function,Vj, VB and VL can be fed into a capacitive passive or activesummer (see Fig. 3).

3 Design equations and sensitivity

Because of the similarity of the two filters in Fig. 2 and oftheir performance, the following discussion will consider onlythe circuit of Fig. 2a, described by eqns. 4 and 6. The treat-ment of Fig. 2b is completely analogous. From eqn. 4 it is

20

= cnc2,Sl0/Q = b (7)

where CJ0 is the pole-frequency and Q the pole-quality factor.Thus, with eqn. 6, the design equations are

= bl(cnc2) =

( d +C2+ Cm)/C3 = Q/n0 - 1(8a)(8b)

Further, from eqns. 4a and b, respectively, the midband gainHB and DC gain HL equal

HB = -a2)/b HL = -a2)/cn (9)

Depending on whether the desired gain is positive or negative,eqn. 9, together with eqn. 8, can be solved for the necessarycapacitor ratios.

3.9Mn

22 Mil

Fig. 3 Circuits used for experimental results, with RCA 3140 BiMOSFET-input operational amplifiers (supply: ± 15 V) and an RCA 3600ECMOS inverter (supply: ±6.8 V) for the summer

For example, a bandpass with negative gain results inax = o, i.e. C4 = 0, C5 = °°, so that, from eqns. 8 and 9, thecomplete design equations are

d / C 3 = -HB (HB < 0) (10a)

(CNl + C2)/C3 = Q/£lQ ~1+HB (\Ob)

(CW + Cn)ICe = (Q/a0 -l+HB-Cm/C2)I(SIOQ)- 1(10c)

A simplification of this circuit can be obtained by settingc2 = 1, i.e. C6 = °°, C7 = 0 [16] (cf. also Fig. 3). This savingof two capacitors results in the design equations

(Cm +d)/C3 =

c2/c3 = n0Q

(lla)

(lib)

and is achieved at a cost of not being able to select the gain:— H* — C IC

nB — L»J/V^3.

The design equations (note e.g. eqn. 8b) indicate the needfor potentially very large capacitor ratios, proportional toQ/£l0 = QGB/COQ, if co0 is much less than GB. Note, however,that these filters are aimed at monolithic implementation,using specially designed amplifiers that can be built with GBnot significantly larger than coo- Thus, large capacitor ratioswill be a problem only if applications dictate the choice of

1EEPROC, Vol. 128, Pt. G, No. 1, FEBRUARY 1981

one amplifier with large GB for the realisation of a wide rangeof pole frequencies, (i.e. f20 < 1).

The sensitivity of a filter parameter P to an element x

s; = (dP/p)/(dx/X)can be calculated from eqns. 7 and 9 with eqn. 6. Straight-forward analysis results in

c^jj i . cQ —°GB — 1, JfiH ~

I < 0.5 (; = 1 , 2 , 3 , 6 , 7 , ^ , ^ )

< 1 (/ = 2, 3)

Sc' = 3% = 0 0" = 4,5, i \ )

Possibly of more interest are the sensitivities of £l0, Q and gainto the capacitor ratios:

5"o = 0.5 \S?\ < 1 \S*ain\ < 1

where r represents ax, a2, b, cn, and c2. Thus, the circuitshows an excellent sensitivity behaviour, with very low sen-sitivities to the accurate and stable capacitor ratios r. The onlypotential problem is

signifying that o;0 is proportional to GB and indicating thatGB must be carefully controlled and stabilised [3,4] ifprecision filters are required (cf. Section 4.5).

4 Practical design considerations

In this Section, a number of factors are discussed whichdetract from the idealised circuit behaviour outlined in Section2. In practice, these effects must be accounted for if the designis to perform as expected.

4.1 Amplifier excess phaseTo adequately describe operational amplifier performance atfrequencies high enough to form a significant fraction of GB,the simple 1-pole model must be augmented by an excessphase term* as in eqn. 2. Its effects on the circuit can be

Fig. 4 Bandpass and unity gain notch responses of filter of Fig. 3

Vertical scale, 10 dB/div.Horizontal scales: BP, 200 kHz/div., notch, 20 kHz/div.Measured values: /„ = 1.97 MHz, QR = 19.8,

I VRU**>O )I = 51 mV, HBR = 9 dBNotch depth, —32 dB

t Alternatively, a model with two poles or even with several poles andzeros can be used. As before, the effects modelled by T= GBT will beconsidered identical and tracking for all amplifiers used, a reasonableassumption for fully integrated filters with all amplifiers on the samedie.

assessed by inserting eqn. 2 into eqn. la:

TB(S) =S2 + bS exp (-ST) + c exp ( - 2ST)

(12)

The phase term in the numerator of eqn. 12 adds to the totalphase of TB(S); although it is of no consequence for ourpresent discussion of second-order sections, it does give rise tocharacteristic and large transfer-function distortions in multi-loop feedback filters [17]. Substituting c = Sll, b = no/Qand a=HB£l0/Q, the exponential terms in the denominatorcan, by analogy to active R circuits, be shown [4] to result inthe expressions

(13a)

(13*)

HB = HBR2Q I cosh

I

sin SIRT

2Q

1/2

(13c)

between the designed parameters, co0, Q, HB and the realisedparameters CJR, QR and HBR. Eqns. 13 then provide thedesigner with the predistorted design values £20, Q and HB, tobe used in eqns. 8—11 which, for a given T= GBT, result in£lR, QR and HBR being required. For the frequently satisfiedconditions 4Q2 > 1 and 4QR > 1 > tan Q,RT, eqn. 13 simpli-fies to the approximate equations

fi0 (14«)

(14b)

(14c)

QR =* Q/(l-2QQRT)

HBR ~ HBI{\-2QSIRT)

which indicate a slight shift in pole frequency and seriousenhancement of Q and gain owing to excess amplifier phase— COT. The above development can also be used to show [4]that the circuits will become unstable and oscillate at thefrequency w0 if the designed Q is made larger than Qm&x =1/(2 sin OJ0T).§ From eqn. 14, it is apparent also that, for highQR, the parameters QR and HBR are very sensitive to thevalue T:

Of — 05)

Thus, although excess phase has the very favourable effect ofreducing the size of capacitor ratios required for QR and HBR,this sensitivity result indicates again the need for carefulstabilisation and control of the amplifiers.

4.2 DC bias and feedbackFor practical operation, the simple all-capacitor filters inFig. 2 lack provision for DC input bias currents, and DC feed-back to avoid saturation due to amplifier offset voltages. Theactual implementation of the required DC paths depends onthe technology used to build the filter and on its topology. Inthe circuit of Fig. 3, for example, disregarding the summer, asingle 'resistor' in parallel with C2 would suffice.

Bearing in mind that the realisation intended for the filtersdiscussed in this paper is completely monolithic, it stands toreason that the very small bias currents needed for MOSamplifier input stages can be provided by leakage paths on thechip or by suitable means internal to the amplifiers. Of more

§Note that this does not imply that the realised quality factor QR isrestricted to less than {2 sin CJUT} !

IEEPROC, Vol. 128, Pt. G, No. 1, FEBRUARY 1981 21

serious concern is the implementation of 'DC feedback pathsaround a filter block', in the following text simply referred toas 'resistors', that are necessary to keep the filter DC stable.Ideally, in order to not interfere with the circuits' operation,the resistors must be large compared with the impedance ofany parallel capacitors. Note, however, that, say, C = 1 pF at/ = 1 MHz results in l/(a)C)— 160k£2, i.e. an integratedresistor R > 1/(CJC) will be difficult to obtain, and thisproblem becomes worse at lower frequencies.

It is fortunate for this application that the large resistorsdo not need exact, or even repeatable, values from chip tochip as long as one can rely on reasonable parameter trackingon each die. Owing to this fact, there appear to be a number ofmethods for implementing large-valued resistors which arecompatible with monolithic and specifically MOS processingtechniques: using polysilicon layers of 50—2OOk£2/square,becoming standard in, e.g. subthreshold logic circuits, largeresistance values with very small dimensions can be realised.Also, MOS transistors, possibly n-channel/p-channel pairs,suitably biased just above the threshold voltage can beemployed to simulate resistivities of > 100kf2/square.

If, e.g., in lowpass or low-frequency filters, excessivelylarge-valued resistors are needed, one may place in parallelwith each capacitor a large resistor of the type describedabove, chosen such that all RC time constants are approxi-mately equal and the resistance effects cancel. A furtherinteresting possibility, if it can be made compatible withstandard IC processing, lies in the use of a lossy ('leaky')dielectric for the capacitors, so that again all 7?C" productson a wafer are automatically the same.

4.3 Parasitic capacitors

In order to reduce chip size in integrated filter implemen-tations, the values of circuit capacitors should be minimised.Thus, as in switched-capacitor filters, the effect of parasiticcapacitors which are found between each capacitor plate andground or substrate [1] must be considered. A glance at Fig. 2shows that these parasitics appear in one of three locationsthat can be handled without seriously disturbing filter perfor-mance: (a) they are driven by the input voltage or by an ampli-fier output and have therefore no effect as long as they do notbecome too large, (b) they are in parallel with a circuitcapacitor and can be absorbed (e.g. into C5 or C7 in Fig. 2a),(c) they appear connected between an amplifier input terminaland ground and can, along with the amplifier input capaci-tance, be accounted for in the analysis. The 'elements' CPi andCNi in eqn. 6 are used to represent these parasitics.

4.4 Amplifier input and output impedances and finite 3 dBfrequency

In addition to excess phase discussed earlier, major amplifiernonidealities to be considered are finite input and outputimpedances and, especially for MOS amplifiers of relativelylow gain, the finite 3 dB frequency a.

If a in eqn. 2 is not small compared with the frequency ofoperation, it can be shown from eqns. la and 2, with T = 0,that the measured pole Q and pole frequency suffer smalldeviations from the designed values according to

Qm * Q{l-(2Q/ao)A-o1} (16a)

where Qm and £lOm are the measured values, Ao = GB/o is theamplifiers' DC gain, and where higher-order small terms havebeen neglected. In addition, the zero at the origin of the band-pass function, eqn. la, is shifted to —a, resulting in transfer-function deviations at low frequencies. For a more detaileddiscussion the reader is referred to Reference 15.

For MOS amplifiers, the input admittance can be con-sidered purely capacitive, i.e. Yin —sCin, with Cin treated as aparasitic capacitor as discussed in Section 4.3.

Finite amplifier output impedances, here considered purelyresistive for simplicity, can be shown to increase filter order byone for each amplifier used; for example, VB/Vr in Fig. 2 isgiven by a ratio of two fourth-order polynomials. To assess theeffects in detail requires a complete analysis of the circuit inquestion. Since the resulting equations are difficult to in-terpret and of little general use beyond the specific circuitfor which they were derived, they will not be reproduced inthis paper. Suffice it to say that, as a result of finite outputresistances, the dominant pole pair (GJ0, Q) will be shiftedslightly and that, in addition, two distant parasitic poles willbe created and three distant real zeros, one in the right halfplane. There appears to be no danger of instability as long asthe element values, including parasitic capacitors, cause onlyweak loading of the amplifiers and as long as load capacitanceis not excessive.

All effects discussed in this Section 4 result in coo devi-ations of about the same order of magnitude, whereas shifts inQ are dominated by excess phase, so that predistortion for CJT-effects according to eqns. 13b or 14b is of most importance.

4.5 Temperature drifts

It is apparent from the equations presented earlier that activeC filters, similar to active R and other high-frequency activecircuits, are critically dependent on amplifier performance.Specifically, as shown in eqn. 7, pole frequency is proportionalto GB which may have high initial tolerances and is subject todrifts with aging, temperature and bias voltages. Similar com-ments are valid for r. Thus, in view of eqns. 7 and 15, carefulstabilisation of amplifier parameters is essential.

Solutions to this problem are available in the literature[3-5, 18-20] and will not be repeated here. The moresuccessful ones use phase-locking techniques, whereby anexternal stable reference signal is used to vary DC bias con-ditions or elements in order to keep filter parameters withinspecifications. Even using discrete implementations, thesemethods resulted in parameter control to a fraction of 1%[5, 18, 20]; better performance can be expected for mono-lithic implementations where use can be made of trackingcomponents and where the additional circuitry required addsonly little expense [3].

5 Experiments

Lacking a suitable MOS facility, experimental verification offilter performance was based on a discrete circuit (Fig. 3),although, clearly, the design method outlined in this paper isdirected towards fully integrated filters. The transfer functionsVB/Vj and VL/VI are described by eqn. 4 with eqn. 6 andC4 =C1 = 0, Cs = C6 = °°. Further, VH is a highpass output,resulting in

TH = = a2S2l(S2+bS (17)

and Vo, derived from a buffered summer, gives the biquad-ratic transfer function

_ Vo _ , S2+(b-d2a2)S

which can realise allpass, notch, or highpass filters. The latterone, however, buffered or unbuffered, is more convenientlyimplemented at the terminal VH. In eqn. 18, h = Csi/Cs4,di — CSi/Csi , i = 2, 3 and the gain of the CMOS inverter wasassumed to be infinite. The error introduced by this assump-tion is corrected experimentally by adjusting Cs4. A CMOS

22 IEEPROC, Vol. 128, Pt. G, No. 1, FEBRUAR Y 1981

EFE

;rEr

r.":. :;.,.;:.. ....

."r:.. •

, w w . X.*.,^.. ,->

~ f - • "

...... ..

r.

:.:...:.; ::;... •-:

..

• - • • • • • »

Fig. 5 Lowpass and highpass responses of filter of Fig. 3

Vertical scale, lOdB/div.Horizontal scale, 500 kHz/div.LF: I Vout\ = 430 mV at 100 kHz, HL = 6.9 dBHP:|Kout | = 33 mV at 5 MHz, gain =-15.4dB

inverter was used in the example to illustrate that, evenbuffered, summing functions can be obtained relatively easilywithout requiring a full additional amplifier.

Based on eqns. 4, 11, 14 and 18, a bandpass and a notchfilter were designed for the parameters / 0 = 2 MHz andQR = 15. Using RCA .3140 BiMOS FET-input operationalamplifiers with GB - in ' 3 MHz, r - 17ns, and Cin ^ 4pFand an RCA 3600E CMOS inverter for the summer results inthe capacitors Cx = 3 pF, C2 = 13.7 pF, C3 = 10.2 pF, Csl =8.6 pF, Cs2 =3.5pF, Cs3 = 0 and Cfi4 = 5.5 pF. The experi-mental response, showing good agreement with the predictedbehaviour, is shown in Fig. 4.

Similarly, but now using the exact predistortion formulas,eqns. 13, since Q > 1 is not valid, a lowpass and highpass filterwere designed for the parameters / 3 dB = 1 MHz and QR =l/\/2. The experimental response, obtained with the capacitorvalues Cx = 7.4pF, C2 = 3.3pF, C3 = 15pF and Csi = 0 isshown in Fig. 5; again, the agreement with theory is evident.

As an additional example, consider the realisation of

T(S) =

-K(S2 +3.3551?2)(S2 + 0.6650?S + 1.051Sq2)(S2 + 2.0905^S + 1.3544?2)

(19)

a fourth-order maximally flat lowpass with a minimum stop-band attenuation of 30 dB. A" is an arbitrary gain constant,S = s/GB, and q = co3liE/GB. The realisation can be accomp-lished by cascading a second-order lowpass and a second-ordernotch section, with the latter requiring a separate summer asin Fig. 3. Alternatively, the function may be implemented asshown in Fig. 6 by utilising the unused amplifier input ter-minals (grounded in Fig. 3). This circuit is easy to tune, allowsindependent notch control and saves one amplifier; it realisesthe function

out

Vi (S \V)(20)

where the superscript refers to the Section number and oc isdefined in Fig. 6. For / 3 d B = 455 kHz, and using eqns. 11 and13, a comparison of eqns. 19 and 20 results in the elementvalues C u = 3 1 . 3 p F , C 2 i=0.9pF, C 3 i=4.0pF, C12 =14.3 pF, C22 = 0.8 pF, C32 = 8.7 pF, aC = 8.0 pF, (1 - a)C =6.8 pF. Three resistors of value R = 4.7 M£2 were used for DCstabilisation and bias, as shown in Fig. 6. The measured per-formance, after trimming aC to 22 pF to correct the notchfrequency, is reproduced in Fig. 7. The Figure shows clearlythe low-frequency attenuation caused by the finite resistors R;

note that, for the element values used, the critical RC productis approximately (50 kHz)"1!

The notch depth, both in Fig. 7 and in Fig. 4, was restrictedto — 33 dB. The reason for this behaviour can be found inamplifier excess phase (see eqn. 2 and Section 4.1) whichchanges the numerator factor of eqn. 20 to (S2e2ST + c\2)/oc),thereby shifting the zero away from the imaginary axis by anamount determined by the amplifier excess phase T — GBT.

To illustrate and support the applicability of the designmethod to monolithic fabrication, all capacitors in the experi-ments were chosen to be in the low picofarad range. It is notedthat theory and measurements in all cases agree surprisinglywell, in spite of the fact that the chosen capacitor values arereally 'too small' to dominate the unavoidable parasitic effectsencountered in the discrete circuits used in the experiments.

F ig. 6 Experimen tal circuit for fourth-order lo wpass filter

6 Discussion and conclusion

A design method for analogue active filers has been introducedwhich, in addition to integrating amplifiers, uses only ratios ofsmall capacitors for the synthesis of a variety of filterfunctions. The procedure is particularly well suited for high-frequency active filters and is analogous to the well-knownactive R design method [4]. Thus, the label 'active C" filtersappears appropriate for these circuits which are promising forthe implementation of fully monolithic analogue active filters,especially in MOS technology, using MOS amplifiers and ratiosof MOS capacitors. The few 'resistors' necessary for biasingand DC stabilisation can be realised in a manner compatiblewith MOS technology.

Compared to active R filters, active C circuits possess anumber of advantages: Ratios of capacitors use less Si areathan resistor ratios, they are also more accurate and less proneto drift due to temperature, voltage level, and aging [6]. Also,whereas resistors in an active R design must be kept relativelylarge to prevent overloading the amplifiers, capacitors in activeC filters can be made arbitrarily small, down to values whereparasitics become significant; in carefully designed monolithicrealisations, capacitors of fractions of picofarads are feasible[6]. Circuit-internal amplifier loads are therefore negligibleuntil very high-frequency operation (say/— 10MHz). Further-more, observe that parasitic capacitors, which may causesevere deviations in active R filters [21], can readily beaccounted for by absorbing them into the existing capacitors.The only disadvantage compared to active R filters lies in thespecial care required for achieving good low-frequency perfor-mance in lowpass applications (see Section 4.2 and Fig. 7).

The price paid for the advantages of active C filters is the

IEEPROC, Vol. 128, Pt. G, No. 1, FEBRUARY 1981 23

r

—-j

Fig. 7 Response of circuit in Fig. 6

Vertical scale, lOdB/div.Horizontal scale, 200 kHz/div.Measured values: /3dB = 450 kHz

fnotch — 8 4 6 kHz> notch depth: — 33 dB\vout\ = lOOmV a t / = 200kHz

unavoidable dependence of some filter parameters on the(normally) poorly controlled and drift-prone parameter GB.Bearing in mind that it appears impossible to design a high-frequency active filter whose performance is independent ofamplifier parameters, and noting that GB is stabilised relativelyeasily [4], particularly in IC form [3], this is considered asmall price to pay. It is noted also, that this stabilisation willusually not be needed in one useful and highly importantapplication requiring analogue, and preferably integrablecircuits — bandlimiting filters in sampled-data or digital signalprocessing systems to avoid aliasing problems. For thesesystems, highly accurate response parameters of the (usuallyrequired) lowpass filters are unnecessary, so that the GB or rstabilisation can be avoided.

In summary, the theory and experiments presented demon-strate that the active C design procedure should prove attractivefor the implementation of monolithic high- and low-frequencyanalogue active filters. Extensions of this technique from thesecond-order sections discussed in this paper to high-orderfilters via simulated ladders or multiple feedback topologies, asin References 22 and 23, respectively, are evident andanalogous to methods reported in the literature [17, 24].There are numerous potential applications of this design pro-cedure, starting from integrable, technology-compatible, band-limiting filters for sampled-data or digital signal processingapplications to high-precision filtering needs in low- and high-frequency analogue communications.

7 Acknowledgment

This work was supported by NSF Grant ENG 76-11218-A01.

8 References

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2 MOULDING, K.W., and WILSON, G.A.: 'A fully integrated fivegyrator filter at video frequencies', IEEE J. Solid-State Circuits,1978, SC-13, pp. 303-307

3 TAN, K.-S., and GRAY, P.R.: 'Fully integrated analog filter usingbipolar-JFET technology', ibid., pp. 814-821

4 BRAND, J.R., and SCHAUMANN, R.: 'Active R filters: review oftheory and practice', IEE J. Electron. Circuits & Syst., 1978, 2, (4)pp. 89-101

5 BRAND, J.R., and SCHAUMANN, R.: 'A temperature-compensatedhigh-frequency, tunable active R filter using programmableoperational amplifiers'. Proceedings 18th Midwest symposium oncircuits and systems, 1975, pp. 603-607

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7 FRIED, D.L.: 'Analog sample-data filters', ibid., 1972, SC-7, pp.302-304

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13 BERMAN, B.D., and NEWCOMB, R.W.: 'Transistor-resistor syn-thesis of voltage transfer functions', IEEE Trans., 1973, CT-20, pp.591-593

14 SCHAUMANN, R.: 'On the design of active filters using onlyresistors and voltage amplifiers', AEU, 1978, 30, pp. 245-252

15 SERIKI, O.A., INDRAJO, G., and NEWCOMB, R.W.: 'High-frequency extended CMOS active R filters'.. Proceedings 21stMidwest symposium on circuits and systems, 1978, pp. 174-178

16 SCHAUMANN, R., and BRAND, J.R.: 'Design method for mono-lithic analogue filters', Electron. Lett., 1978, 14, pp. 710-711

17 SCHAUMANN, R., and FARRELL, S.R.: 'The effect of excessphase and its compensation in the design of high-order multiplefeedback active filters'. Proceedings 21st Midwest symposiumon circuits and systems, 1978, pp. 178-183

18 BRAND, J.R., SCHAUMANN, R., and SKEI, E.M.: 'Temperature-stabilized active/? bandpass filters'. Ibid., 1977, pp. 295-300

19 RAO, K.R., SETHURAMAN, V., and NEELAKANTAN, P.K.: Anovel 'follow-the-master' filter', Proc. IEEE, 1977, 65, pp. 1725-1726

20 BRAND, J.R., and SCHAUMANN, R.: 'Comments on a novel'follow the master filter' ', ibid., 1978, 66, pp. 590-592

21 CHOI, T.C., and SCHAUMANN, R.: The effect of parasiticcapacitors in high frequency active filters'. Proceedings 20thMidwest symposium on circuits and systems, 1977, pp. 306-310

22 SODERSTRAND, M.A.: 'Active R ladders: high-frequency high-order active R filters without external capacitors', IEEE Trans.,1978,CAS-25,pp. 1032-1038

23 LAKER, K.R., SCHAUMANN, R., and GHAUSI, M.S.: 'Multipleloop feedback topologies for the design of low sensitivity activefilters', ibid., 1979, CAS-26, pp. 1-21

24 LAKER, K.R., SCHAUMANN, R., and BRAND, J.R.: 'Multipleloop feedback active R filters', Proc. IEEE ISCAS, 1976, 64, pp.279-282

24 IEE PROC, Vol. 128, Pt. G, No. 1, FEBRUARY 1981