State-space synthesis of optimum active biquadsD.T. Nguyen, B.E., Ph.D., M.N.Z.I.E.

Indexing terms: Circuit theory and design, Filters and filtering

Abstract: A state-space technique recently developed for the synthesis of optimum integrator sensitivityactive filters [2, 3] is extended to further optimise the active-/?C biquads. It is found that the optimumsensitivity active biquad, which, having an asymmetric normal system matrix, has optimally distributedsensitivity both to the coefficient of the integrators, as well as to the effect of the finite gain/bandwidthproduct of the op amps. The normal integrator loop thus has two leaky integrators with equal dampingcoefficients and equal gains. The normal loop, however, can provide only the lowpass characteristic at theoutput of either integrator. Any other output characteristic with optimum sensitivity can only be obtainedusing nonzero output feed-forward coefficients ct and c2.

List of symbols

x,x

A, B, C, D

IF(s)

H(s)

G(s)

system

syy1 =

BA(s)

state vector and its time derivativesystem input and outputstandard state-space notations forcoefficientsunit matrixstate vector transfer function (from input tostate variables)system transfer function (from input tooutput)vector transfer function, dual of F(s) intransposed networktransfer function of a leaky (damped) inte-gratorop-amp gain/bandwidth productop-amp open-loop gaindesigned value of undamped (resonant)frequency and quality factor of naturalpoleseffective (implemented) value of co0 and Qo

due to finite gain of op amps

1 Introduction

The state-space technique has been a popular and powerfultool in the analysis of quantisation noise and oscillationphenomena in digital networks over the last decade. The twomost interesting results which concern this paper are theintroduction of the dual function pair F(s) and G(s) and thediscovery of the normal networks, in particular those having anantisymmetric system matrix. The literature on these topics inthe field of digital filters is exhaustive and readers are referredto a representative paper by Jackson et al. [1]. A similartechnique has been applied recently by Snelgrove and Sedra[2, 3] to the synthesis of optimum sensitivity /?C-activebiquads.

F(s) is the state vector gain, i.e. from the input to theoutput of the summing nodes (the integrators in this paper),and G(s) is its counterpart in a conceptually transposed net-work which is obtained by reversing the signal flow direction.F(s) therefore determines the internal signal level and \\F(s) ll2is usually used for (L2 norm) signal scaling to minimise thepossibility of signal clipping at the integrators' outputs. G(s),on the other hand, is the vector gain from the inputs of thenodes to the system output and HG(s)ll2 therefore decidesthe level of the output noise contributed by the integrators.

Snelgrove and Sedra [3] have optimised the biquad sensi-tivity with respect to the integrators' coefficients only. In thispaper, we also optimise the system sensitivity to the effect of

Paper 1965G, first received 4th January and in revised form 16th April1982The author is with the Department of Electrical Engineering, Universityof Auckland, Private Bag, Auckland, New Zealand

the finite gain/bandwidth product of nonideal op amps. Thesystem matrix turns out to be antisymmetric, as in the case ofnormal digital filters. The optimum sensitivity active biquadexpectedly has a perfectly symmetrical circuit topology. Its'normal' integrator loop may have two identical integrators,thus simplifying the circuit design.

As compared to the conventional 2-integrator loop, theabsence of a lossless integrator in the normal loop, however,takes away the advantage of providing both lowpass andbandpass concurrently at the two integrator outputs. Theoptimum sensitivity criterion also requires more complexcalculation for conductances and more components in theinput feed-through and output feed-forward branches.

This paper also extends many aspects of Snelgrove andSedra's technique and presents the design formulas for optimumbiquads in their final form ready for implementation.

2 State-space formulation

An Mh-order continuous-time linear system can be describedby the familiar state equations, using standard notations:

JC = Ax + Bu (la)

and

y = Cx + Du (Ib)

From eqn. la the vector transfer function from the input u totheNstate variablesxt(i= 1,2,. . .N)is therefore

F(s) = (sI- (2)

and from eqn. Ib, the associated system transfer functionfrom the input u to the output y is

H(s) = C(sI-AYlB (3)

By analogy with eqn. 2, we define also a second vector transferfunction G(s), being the dual of F(s) in a transposed network,i.e. with reversed direction of flow, as:

G(s) = (sI-ATylC (4)

It is customary to assign the state variables xt to the inte-grator outputs. G(s) thus defined is, in fact, the gain from theintegrator inputs to the system output y. When the stateeqns. la and b are used to describe an active biquad usinginverting integrators, the Laplace operator s in eqns. 2, 3and 4 has to be replaced by — s and, in this case,

al2A = ' B = .

a21 a22 J [ b 2

b,

and

280 0143-7089/82/060280 + 05 $01.50/0

C = [c, c2] D = [d]

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

The two state transfer functions are therefore:

b2ax2 — bx(s + a22

F2(s) = X2(s)/U(s) =bxa2X —i

(5a)

(5b)

where A(s) = s2 + s(an +a22) + ana22 — al2a2x, and thesystem transfer function is

H(s) = (6)

The signal flow graph of a state-space biquad using invertingintegrators is shown in Fig. 1. The sensitivity of the systemtransfer function to the network coefficient a,;- running fromthe /th node to the ith node can be expressed in terms of thefunctions F(s) and G(s) as in References 1 and 2:

dH(s)

9a,-,-= Gi(s)Fj(s) (7)

where a,;- may be ati, bt, ct or d. The functions Gi(s) andG2(s) therefore may be found from eqn. 4 or, more con-veniently, from eqn. 7 as:

dH(s) c2a21 —

A(s)

and

G2(s) =bH(s)

bb.cxaX2

A(s)

(8a)

(Sb)

«2 2

Fig. 1 Signal flow graph of state-space active biquad using invertingintegrators

3 Synthesis for optimum coefficient sensitivity biquad

By examining the state equations la and \b, the systemcoefficients a,;- and b{ are seen to have the dimension of s inthe frequency domain. Furthermore, if u, x, y are the systeminput, state and output voltages, respectively, ay and bt willhave the dimension of conductance per unit capacitance, andx will be current per unit capacitance in the time domain.

With the foregoing observation, we rewrite eqn. 6 as:

sy2)-c2$2(l+

(9)

with 7f = l/aihp\ = bilau, ax = ax2/axx and a2 = a2x\a22.

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

In this form, ax, a 2 , /3X, /32 are all dimensionless and only7i and 72 are frequency dependent. Therefore, from theinvariant property of multiparameter sensitivities [3], it canreadily be shown that

which is invariant for a given transfer function to be syn-thesised. This property can also be verified by directsensitivities calculation on H(s) in eqn. 9.

From the calculus of real numbers, the lower bound of theSchoeffler sensitivity measure [4]

i-HW 1

Jy = iV

is attained when

dH(s)

(10)

Again, by direct sensitivities calculation on H(s), or by asimple observation on the interchangeability between yt and7 2 , it can be shown that eqn. 10 is satisfied if

7i = 72

and

= c2p2 = c2b2

(lla)

(lib)

Alternatively, as in Reference 3, we may satisfy eqn. 10 bysetting in eqn. 7

= F2(s)G2(s) (12)

and arrive at the same conditions as in eqns. l l a and b bydirect calculation of eqn. 12 using F(s) in eqns. 5a and b andG(s) in eqns. 8a and b.

It is interesting to note that conditions l l a and b areexactly those required for optimum output noise digitalfilters which have been L2 scaled to avoid overflow at thestates, as stated in eqn. 18 of Reference 1. This is rather to beexpected, since the roundoff noise in digital filters is equiv-lent to the component error in analogue networks. Theoptimum digital filters have output noise being contributedequally by the rounding of coefficients and multiplier productsat all the nodes [1]. The optimum sensitivity active filtersare therefore expected to have output being equally sensitiveto the component error of all the integrators.

4 Normal active biquad using inverting op amps

Implementation of the optimum sensitivity biquad usinginverting op amps may now be commenced. Fig. 2 shows ageneral damped integrator using nonideal op amps with finite

•out

Fig. 2 Components in leaky {damped) integrator using nonidealop amp

281

gain/bandwidth product B. An obvious effect of the op-ampfinite gain is the absence of a virtual ground at the invertinginput terminal. At moderate frequencies, the op-amp open-loop gain can be approximated as:

A(s) = Bis u/B < 1

and the transfer function of the ith integrator can be shownto be:

Us) = - (13)

in which |a,-| = Gi/GQi is the integrator gain and yt = CilGQi

is the integrator damping constant. The use of the lettersla,| and 7,- here is deliberate, since these are, in fact, theelectrical analogue of a,- and yt in eqn. 9.

The transfer function of a nonideal op-amp inverter can beconveniently obtained by letting |a, | = 1 and 7,- = 0, that is:

1

1 +(14)

At frequencies well below the op-amp GB product, we canignore the s2 term in eqn. 13, as a rough approximation. Theeffect of finite Bt is therefore roughly equivalent to anincrease in the capacitance AC,- = (G,- + GQ,)/Z?,-, with thevirtual ground being assumed unaffected at the invertinginput. A more accurate analysis will be presented in Section 5following. With this approximation, the state-space analysis inthe preceding Sections can be used to synthesise an RC-activebiquad by assigning the state xt to the output voltage of theith integrator and the variable xt to the current per unitcapacitance flowing in the zth integrator's capacitor Q. Thesystem coefficients a^ and &,• are then the conductances perunit capacitance of C,-. Thus:

an = GQilCi = 1/7,- i = 1,2

\an\ = G1/C1 = l<*i

and

= G2/C2 = \oc2\a22

The effect of finite B{ is roughly to modify yt to

+ \at\= 7«

Bt

It is therefore obvious that, in order to maintain the optimumsensitivity condition in eqn. 1 la, we must have

l«il = local

that is,

\a\7\ = \a21\

assuming B1 = B2 = B.The system matrix then has the form

(15)

A =K ± X,

+ X,- Xr

i.e. a normal matrix, where X1>2 = Xr ±/X,- are the eigenvaluesof A which are also the poles of//(s). Ani?C-active integratorloop having a normal system matrix is termed a normal loop.

The 2-integrator normal loopThe typical denominator of a biquad transfer function is:

therefore

« 1 1 = <*22 =

land

or

lRQl = UC2RQj

l/2

C2R= wo(l - l/4Go)2\l/2

2 ^ 2

Two possible topologies for an optimum sensitivity activebiquad are shown in Figs. 3A and B, for which:

A =±a>o(l-l/4<2o2)

2N1/2

l/2

The normal loop biquad has been reported recently by Mayand Mehdi [6]. Their study, however, involves only thebandpass filter for the case ax2 > 0 and, most important ofall, their filter is not optimal since b^cx =£ b2c2.

5 Effect of op-amp finite GB product

A more accurate analysis of the effect of the op-amp finite GBproduct is presented in this Section. The condition in eqn. 15is established by minimisation of the effect of the finite valueof B on co0 and Qo.

From eqns. 13 and 14, the denominator of the transfer

Fig. 3A Normal integrator loop active biquad having a X2 < 0

Output forward connection not shown

Fig. 3B Normal integrator loop active biquad having a l2 > 0

Output forward connection shown for bandpass case, i.e. d = 0

282 IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

function of the optimum sensitivity biquads in Fig. 3 is:

A(s) =

l+S 7272 Is

in which yx = y2 =2Q0lco0 and ot1a2 = 1-4Q%. Byassuming that all op amps are matched, that is Bx = B2 =B3 =B and by neglecting the terms containing B~2 and B~3,it can be shown that:

Ideally, 5 = °°and

Since ax a2 is a constant, \oci\+ |a21 will be minimised if

l«il = l<*2| = 2 g o ( l - l/4(2o)1/2 - 2 g o - l / 4 g o

as already predicted in eqn. 15.Therefore, the minimised effect of the finite GB product is

to decrease the value co0 to

co0 = w 0B BQQ 4BQZ

-1/2

—ir i + ; r Go >

and to enhance the value of Qo to

BQ0 SBQ0

- l

or

(2o(^o) - <2o l- l

go > (17)

Eqns. 16 and 17 turn out to be approximately the same aseqns. 8.46 and 8.47 of Reference 4 for a conventional2-integrator loop biquad, which assumes co0 = \ICXRX =I/C2R2 m order to minimise the sensitivity of Q (alone) to

-the finite op amp gain [5]. Most conventional biquads have,in fact, a state-space realisation with

A =0

and B =0

in which the arbitrary constant kx is chosen to be CJ0 , i.e.a 12 = — 021 and a22 = 0 . When *;, = 1, we have the 'direct-form' state-space biquad.

6 The lowpass and bandpass optimum active biquads

6.1 Lowpass biquadA lowpass biquad with gain K has the transfer function

LP(s) =

Qo

and may be synthesised in the state space with the statevariable x2 assigned to the lowpass output, i.e. LP(s) = c2F2(s).This is achieved by simply setting b2 = 0, ct = 0 and d = 0 ineqns. 5 and 8, so that

A(s) A(s)

and

G2(s) =

Note that F1(s)Gl(s) = F2(s)G2(s) if an = a22 =u>0/2Q0,since the chosen values of b\, cx above imply that conditionb\Cx = b2c2 is satisfied for all values of bx and c2.

Therefore, in Fig. 3A, the output of the second integratorhas a lowpass characteristic if c2 = 1 and

1/C^Q! = l/C2RQ2 = o>0/2Q0

l/CxR1 = l/C2R2 = w o ( l ^ 1 / 2

R02 = 00

6.2 Bandpass biquadIt is possible to obtain a bandpass characteristic

BP(s) =

at the output of either integrator in the normal loop, e.g. atthe state JCX , so that F1 (s) = BP(s) by making

bla22 = (18)

However, eqn. 18 implies that b^O and b2^0 and thecondition brCi =b2c2 cannot be satisfied unless cx =£0 andc2

:£0. Therefore, F1(s) as obtained is not an optimumsensitivity bandpass function. Also it can be verified that, inthis case,F1 (s)Gx{s)^F2(s)G2(s).

To obtain an optimum sensitivity frequency response otherthan lowpass, the biquad must have nonzero feedforwardcoefficients cx and c2, feeding a summing op amp. Forexample, BP(s) can be obtained from H(s) in eqn. 6 by making

2clblan = (cYb2 —c2bx)ax2

and

d = 0

i.e.

v-1/2

Noting that cx and c2 are dimensionless and that bx and b2

have dimensions of frequency, a possible solution for Fig. 3Bis:

= koj0(2Q0 - b2 = kuo(2Qo\l/2

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 283

Filtertype

LowpassBandpassHighpassAll pass

d

00K1

bl

0o (2Q0

o (2Q0

o <2Q0

Table 1: Coefficient values for Fig. 3B '

kio0 (2Q0 + 1 )" 1 / 2

- 1 ) l / 2 koo0 (2Q0 + 1 ) U 2

- 1 ) 1 ' 1 * w o ( 2 Q o + 1 ) " 1 ' 2

- 1 ) 1 / 2 /rto0(2Q0 + 1 ) I / 2

(a I2 > 0)

(K2QJk)(2Q0

(K/k2Q0)(2Q0

{K/k2Q0){2Q0

(1/*Q0)(2Q0 -

- 1 ) " 1 / 2

- 1 ) " 1 / 2

c2

0(K/k2Q0)i2Q0 + 1)"1'2

{K/k2Q0)(2Q0 + 1)1/2

(1/*Q0)(2Q0 + 1)"1'2

and

c, = (2<2o - v-1/2 C2 = (2Go\-l/2

where A: is an arbitrary constant.

7 Summary and conclusion

The paper has presented a state-space synthesis for .KC-activebiquads and has arrived at a circuit topology which gives theactive biquad an optimum sensitivity both to the integrators'component errors and to the effect of the finite gain of opamps. The optimum sensitivity criterion requires (i) that thebiquad has a normal system matrix and (ii) that the productof the input feed through and output feed forward is the samein both state branches, i.e. bxcx =b2c2 The normal integratorloop is, therefore, perfectly symmetrical. If the integratorcapacitors are chosen to be the same, the two integrators inthe normal loop are in fact identical.

The condition that blcl=b2c2 is very restrictive. Toobtain a certain frequency response, bx and b2 cannot be zero,and hence cx and c2 cannot be zero. This leads to the resultthat the normal integrator loop can provide only the lowpasscharacteristic at either of its integrators, whereas the con-ventional biquad loop can provide both lowpass and bandpass.

In general, the input feedthrough resistors Rol, Ro2 arecalculated from the values of bx, b2 as below:

ROi = 1/bfC i = 1,2

The state feed-forward coefficients cx and c2 (dimensionless)

D. Thong Nguyen was born in Ha-Tinh,Vietnam, on 1st January, 1941. Hereceived his B.E. degree in electrical engin-eering with lst-class honours in 1965from the University of Canterbury, NewZealand, and his Ph.D. degree in antennatheory in 1969 from the University ofAuckland, New Zealand. Dr. Nguyenjoined the teaching staff at the NationalInstitute of Polytechnique, Saigon, Viet-nam, in 1969, and became Associate

Professor in 1974. In 1975, he was appointed senior lecturer inthe Department of Electrical Engineering, University of Auck-land. His current research interests are in the fields of digitalsystems, data transmission, bandwidth compression and speechresearch, digital and analogue filters.

represent the weightings of the integrators' output voltages tobe added to the input voltage having weighting d in a summingop amp, i.e.

y = c2x2 +du

By equating the terms of H(s) in eqn. 6 with correspondingterms in a general 2nd-order transfer function of gain K andtaking into account the optimality requirements in eqns. 1 \aand b, the values of bt and ct are calculated and are given inTable 1 for various types of filter.

For the case a12 < 0 in Fig. 3A, we interchange bx and b2,Cj andc2 in Table 1.

The constant k is arbitrary and may be used to control thecomponent spread or as a scaling factor to avoid clipping ofthe integrators' outputs.

8 References

1 JACKSON, L.B., LINDGREN, A.G., and KIM, Y.: 'Optimalsynthesis of second-order state-space structures for digital filters',IEEE Trans., 1979,CAS-26,pp. 149-153

2 SNELGROVE, W.M., and SEDRA, A.S.: 'A novel synthesis methodfor state-space active networks'. Proceedings of 1980 Midwestsymposium on circuits and systems, Toledo, Ohio

3 SNELGROVE, W.M., and SEDRA, A.S.: 'State-variable biquadswith optimum integrator sensitivities', IEE Proc. G, Electron. Circ.& Syst., 1981,128, (4), pp. 173-175

4 BRUTON, L.T.: 'RC-active circuits, theory and design' (Prentice-Hall,NJ,1980),Chap.8

5 MUIR, A.J.L., and ROBINSON, A.E.: 'Design of active RC filtersusing operational amplifiers', Syst. Technol, 1968, (4), pp. 18-30

6 MAY, E.J.P., and MEHDE, H.H.: 'Low-noise state-space realisationof bandpass active filter', Electron. Lett., 1981, 17, (17), pp.615-617

284 IEE PROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982