noise analysis of second-order analogue active filters

Noise analysis of second-order analogue active

3 E , p E 2 1 2

l Y 1 -

filters

P. Bowrofl K.A. Mezher

Indexing terms: Filters andfiltering, Analoguefilters

Abstract: The output noise of continuous-time active filters is analysed in general terms by directly inserting device noise sources in an equivalent block diagram. This removes some of the restrictions imposed by previous methods of noise-source referral and permits more meaningful analytical optimisation. Design recommendations for minimising noise in a wide range of realisations then emerge. This allows performance com- parisons of active-RC, active-R and OTA-C circuits.

1 Introduction

The application of active filters to the handling of small signals is limited [l, 21 by the inherent noise levels of the constituent components. Operational-amplifier noise sources include flicker, white and burst noise [2, 31. These are dominated by the noise properties of the input stage and are usually [4-61 assumed to be uncorrelated. Under linear time-invariant conditions, the noise sources are regarded as stationary random processes. Passive noise emanates mainly from the thermal effect in the resistors [2]. This can be reduced [l] only at the expense of large capacitance levels, which are inconvenient in integrated circuits [7]. Originally [4], these resistive noise sources were considered separately, but they can be aggregated [ 5 ] into a single, passively based voltage- spectral-density source E,, .

For the noise equivalent circuit of a single amplifier with general negative feedback [SI, as in Fig. 1,

E:,(w) = 4kT Re - [ Y h I

in which k is Boltzmann’s constant, T is the absolute temperature, and Y * ~ ( S ) is the driving-point admittance of the passive section at terminal 2 when K = V, = 0. E,, and I , , are the amplifier noise voltage and current spectral densities, respectively. The [y] and A blocks can then be considered noiseless [SI. The ports marked and V , indicate where the respective input and output signal voltages would appear. Using two-port equivalent generators [SI, the noise performance of second-order single- amplifier filters with ideal amplifiers has been analysed

0 IEE, 1994 Paper 1135G (ElO), first received 19th July 1993 and in revised form 24th January 1994 The authors are with the Department of Electronic & Electrical Engin- eering, University of Bradford, West Yorkshire BD7 lDP, United Kingdom

350

2 Referral o f noise sources

To derive the output noise for multiple-amplifier circuits, it has been considered [ l , 2, 9, lo] necessary to refer equivalent noise generators to locations that are conve- nient for determining individual noise transfer functions. The different source-referral techniques can be explained in terms of the basic single negative-feedback circuit 14, 113 in Fig. 2. This is a constituent stage in several multiple-amplifier arrangements.

In the bidirectional case [2] of Fig. 2a, using Fig. 1 and eqn. 1, the sum of the squared noise voltage spectral densities referred to the input of the mth amplifier is given by

The amplifier noise current spectral density Inam is cus- tomarily [2, 5, 61 neglected compared with Enam. This is justifiable [12] for almost all practical impedance levels associated with both bipolar and MOS realisations. Par- ticularly in the latter, higher resistance values increase the

I E E Proc.-Circuits Devices Syst., Vol. 141, No. 5 , October 1994

relative importance of the passive noise generators rather than the noise current generators. E:, and En-, are the passively derived components at the noninverting and inverting input terminals, respectively. Each of the noise sources is assumed to have a uniform spectral power

- b

- C

Fig. 2 a Bidirectional referral in basic negative-feedback section b Unidirectional referral in basic negative-feedback d o n e Grounded n f e d for open-loop active device

Referral of noise sources

density, which is usually valid except at extra-low frequencies when the frequency dependence of flicker noise becomes significant. Although this was the earliest [2] referral techique, it is unnecessarily cumbersome and, even when corrected [2], is only applicable [l] to a limited number of circuit configurations. On the other hand, the unidirectional method in Fig. 2b exploits the fact that the low output impedance of the operational amplifier allows the output terminal to be considered as short-circuited to ground. As a result, the noise source at the amplifier input is referred only to the circuit input with

(3)

This internal adjustment gives rise [lo] to a simpler and more flexible technique that can be applied to a wider

range of circuits incorporating more complex feedback paths. The original noise source in Figs. 2a and b can equivalently be placed in the noninverting input lead, as shown in Fig. 2c. This represents a grounded referral and is appropriate for open-loop devices, as in the noise analysis [9] of OTA-C filters, occasional ungrounded transconductors necessitating inclusion of floating sources.

By means of the techniques discussed above, compari- sons of noise performance for different active-RC and, later [lo], active-R and OTA-C configurations have been made. Further insight can be gained by representing the referral processes in terms of equivalent noise block diagrams, which have only been developed [13] for the par- ticularly noisy case of active-R filters. Seen in this perspective, it becomes apparent that any of the referral techniques can be used. Hence, noise sources can be directly inserted in the block diagram of a feedback system [14] of any complexity, with input signals zeroed. Linear systems analysis then suffices to derive the output noise for any configuration. The three-amplifier version is illustrated in Fig. 3, where, for uniformity, positive- feedback notation is depicted. Negative feedback can be provided by the polarities of T,,,, T,,, etc., which are the passive feedback transfer ratios (note that there are no intrinsic feedforward transfer ratios as occur in signal block diagrams). E,,, E,, and E,, are the equivalent input noise spectral densities referred to the inputs of the appropriate amplifiers.

3 Second-order filters

3.1 General analysis In general, the total squared voltage spectral density at the output of the Ith amplifier can be written [lo], by superposition, as

M

El, = c I L I 2 E Z m = 1

(4)

Here, M is the total number of amplifiers. The noise transfer function from the input of the mth

amplifier to the output of the Ith amplifier (in the absence of all other noise sources) is

(5) E#m I En,. Em, '"Emm-,. E.=+, . . 'En,=O

T, ,~~( jo) = ( jw)

Then, the total RMS noise output voltage V,,, over the frequency band o1 to w2 is given [4] by

Fig. 3

IEE Proc.-Circuits Devices Syst., Vol. 141, No. 5, October 1994

Noise block diagram for three-amplifierfilters

351

3.2 Active-RC circuits For the three-amplifier arrangement of Fig. 3, the individual noise transfer ratios to the output of the first amplifier can be derived by block-diagram analysis as

E,,, for I = 1, 2, 3 can then be obtained by substituting in eqn. 4. The single-amplifier [6] and two-amplifier [lS] expressions are merely degenerate cases. Configurations with more than three amplifiers can obviously be analysed by extending the procedure in eqn. 7. However, such arrangements are rarely used as second-order bandpass filters.

Noise is known [16] to be related to sensitivity, and an interesting inter-relationship for the case of the single- amplifier filter is now developed. For a controlled gain K and a passive feedback ratio T b , the gain-sensitivity product [17, 181 of the signal transfer function T can be derived as

K (8) rl0 = A S T - -

A o - l - K T ,

when the amplifier open-loop gain A , + CO. This is the same as the noise transfer function T,, already presented [SI, which is a special case of eqn. 7, i.e.

TATo = T,, (9) To minimise output distortion, it is well known that negative feedback must be increased, implying [15] that I must be large. Eqns. 8 and 9 show that this reduces both r: and T,, . The property of eqn. 9 in the multiple- amplifier case becomes

where 7; is the signal transfer function from the input of the filter. However, eqn. 10 is only true when I = rn, i.e. for the same amplifier.

Although the output noise in eqn. 4 appears to increase with the number of terms (i.e. amplifiers), it is actually more dependent on the nature of This means that circuits whose passive paths have the maximum transfer ratio of unity will tend to have higher noise levels than other circuits. The results of applying the analysis to the well-known two-amplifier and three- amplifier filters of Fig. 4 are presented in Table 1 . The experimental values in Table 1 agree well with both theory and SPICE evaluation. They are measured on an HP 3585A spectrum analyser by setting the marker at the pole frequency on the noise floor with RBW set to 'AUTO' and activating the 'NOISE LVL' function. The reading in V/J(Hz) is then converted into RMS volts by means of a simple transformation [4] developed from eqn. 6. Equal-resistor and equal-capacitor design conditions are used for the nonoptimised composite-amplifier case ( r = 1 , c = 1, 1 = 0.667), and, when optimised [15] as in Appendix 6, r = 2.9, c = 1.47, 1 = 0.0033. The output noise voltages of the GIC-derived circuits are fairly high because of their strong dependence on pass- band gain KO. Minimum noise occurs for design close to the common constrained case, with KO = 2 in both the two- and three-amplifier cases, but the original version [19] of the latter is unstable. The output noise of the composite-amplifier filter circuit [20] is more than halved by optimisation [ 151. As predicted above, configurations containing integrator sections (i.e. the integrator-loop and state-variable arrangements) give the lowest output

Tabla 1 : Output noise voltages for different second-order bandpass filters designed for fo = 1.5 kHz. 0 = 10. K, = lOand R = 10 KR

Circuit V ~ J J ~ Q V", I PV

Theoretical SPICE Experimental

Non-optimised 55.5 55.1 54.5 composite amplifier

15.8 16.4 16.7

Two-amplifier 54.3 55.0 GIC derived

Integrator loop (1 + !$)€:, + [ 1 + il + y)2] !$ + E:#* + + 11.2 11.3 11.3

State variable E:az + 10.2 10.3 10.3 2

2 E 5 + E:=, +

Active- R 460.0 - U31

500.0 31

352

E,,=J(.UtTR) = 12.87 nV/J(Hz) and measured Ens, z z Ens, = 19.9 nV/J(Hz) forpA741 amplifiers

I E E Proc.-Circuits Devices Syst., Vol. 141, No. 5, October 1994

noise. For such active-RC networks with three equal amplifiers and high-Q design, the absolute minimum output noise can be obtained from relevant expressions in Table 1 as

3.3 Active - R circuits The higher-frequency design of the bandpass active-R circuit [13] gives the expression and values listed in Table 1. Here, K~ and K~ are factors determined by

resistor ratios [13], and W, is the radian gain-bandwidth product of the amplifiers used. Apart from the obvious resistive thermal noise, the dependence of the design on U, is seen to exacerbate the output noise considerably.

3.4 OTA-C circuits The analysis can also be applied to continuous-time filters using canonical [lo] or noncanonical [9, 213 OTA-C circuits. These offer the important advantages of integrability, tunability and higher-frequency operation, but present dynamic-range problems [9, 10,21-231.

4 a

R

I C n

C

1 + KO-Q 7 i - R

R

d

I I

f e

Fig. 4 a General composite-amplifier MMNFB circuit b Bhattacharyya-MikhaeI-Antoniou two-amplifier GICD circuit c Tow-Thomas integrator-loop circuit d Kenvin-Huelsman state-variable circuit e Mikhael-Bhattacharyya thra-amplifier GICD circuit

IEE Proc.-Circuits Devices Syst, Vol. 141, No. 5, October 1994

Noise equivalent circuits for various active-RC bandpassfilters

353

For the canonical case, the general noise block diagram [lo] is shown in Fig. 5, the noise outputs being

and

where, substituting s = j w ,

Fig. 5 Noise block diagram for canonical second-order OTA-Cfilters

The equivalent noise block diagrams for some noncanonical circuits with M ( > 2 ) devices are shown in Fig. 6. After applying eqn. 7 to these, eqns. 4 and 6 give the output noise voltage. In most cases, assuming identical devices, the minimum level of output noise voltage can be expressed in the form

For a given E,, , this is apparently lower than in eqn. 11, as expected from the topological simplicity. However,

E

a

this is offset by the higher design w, for OTA-C filters. Eqn. 13 indicates that, for high Q, the noise level is inde- pendent of the number ( M ) of OTAs used. This is con- trary to the conclusions of earlier analysis [SI, in which variables were not fixed, and results were not confirmed by experiment. Table 2 compares the theoretical and measured noise at the bandpass output of some canonical and noncanonical filter circuits.

Noncanonical OTA-C circuits, in which two scaling amplifiers are added to the dominant loop, are not covered by eqn. 13. In such cases (e.g. Fig. 6b and c), the ( M - 1) coefficient is redistributed to increase the Q term at the expense of the 1/Q term, so that eqn. 13 is replaced by

Table 2: Numerical values of output noise voltages for OTA-C second-order bandwss filter circuits

Circuit v,, I IJV

Theoretical Experimental

Canonical 2-OTA 9.46 8.03 grounded-C 2-OTA 9.46 8.10 floating-C

Non- Unscaled 3-OTA 9.54 8.10 canonical 4-OTA 9.60 8.1 5

5-OTA 9.67 8.20 6-OTA 9.74 8.20

Scaled 6-OTA 16.37 14.40 8-OTA 16.50 14.80

Designed for f , = 140.0 kHz, 0 = 8.2, I,,, = 500.0 PA, assuming typical g,, =g,, = 9.6 mS, and = ,FnS2 = 7 nV/J( Hz) for CA3280 devices

r

E

Fig. 6 a 5-OTA circuit (from Fig. 26 in Reference 9) b 6-OTA circuit (from Fig. 2d in Reference 9) e 8-OTA circuit (from Fig. 2e in Reference 9)

354

Equiualent noise block diagrams for noncanonical OTA-Cfilters

IEE Proc.-Circuits Deuices Syst., Vol. 141, No. 5, October I994

This necessarily exacerbates the overall output noise, as can be seen for the second 6-OTA and 8-OTA entries in Table 2.

As MOS devices [22] can have high noise (E,,== 50 nV/J(Hz) compared with 7 nV/,,/(Hz) for the bipolar CA 3280), the noise levels for some recent integrated continuous-time filters [23] may present a problem.

4 Conclusions

The utility of the equivalent noise block diagram in sim- plifying the analysis procedure by direct insertion of noise sources has been demonstrated. By removing the restrictions of previous techniques [l, 21, this has allowed derivation of the output noise for general multiple- amplifier second-order filter circuits. Succinct expressions for minimal noise have been developed, allowing com- parison of active-RC, active-R and OTA-C configurations.

The output noise of active networks appears to depend upon the nature of the constituent circuit stages in preference to the topological arrangement of feedback paths in the associated signal-flow graph. The simpler the structure of each individual stage, the lower the noise. Lossless inverting integrators (RC or gJC) contribute least, followed by lossy integrators. Inverters, summers and noninverting integrators contribute more, as do first- order allpass stages. Filter circuits that are not based on first-order sections have markedly more noise. Examples are the MNFB circuit (with second-order RC feedback path) and the GIC-derived circuit (with complex inter- connections of active and passive components). For the important OTA-C circuits, the general noise behaviour has been described, and the dependence of output noise on Q and on the number of amplifiers has been clarified.

For precision instrumentation circuits operating at extremely low frequencies and for MOS designs at voice frequencies, flicker noise [3, 121 can be significant. It is then necessary to represent the frequency dependence of the device noise spectral density as linearly filtered stationary noise: E,&) = (1 + d/s)E,, where d is a real constant determined by the semiconductor used. The integral of eqn. 6 then readily extends from a single function [4] of w to three standard functional components, which are tabulated [4] and have already been evaluated individ- ually. Experimental confirmation would necessitate special measurement facilities. However, the experimental results presented in Tables 1 and 2 confirm that the underlying theory is valid in the majority of practical situations. In higher-frequency communication circuits, the dependence of output noise on the square root of frequency can contribute to a restriction of the dynamic range [lS].

Identification of filter arrangements with low output noise permits design with lower-cost devices. The technique described here can also be extended [24] to derive and evaluate noise levels in high-order active filters.

5 References

1 WEINRICHTER, H., and NOSSEK, J.A.: ‘Noise analysis of active-

2 BRUTON. L.T.. TROFIMENKOFF. F.N.. and TRELEAVEN. RC filters’. IEEE Proc. ISCAS, 1976, pp. 344-347

D.H.: ‘Noise pekormance of low-sensitivity active filters’, IEEE J: Solid-state Circuits, 1973,8, pp. 85-91 (corrections, p. 290)

3 HSU, S.T., and WHITTIER, R.J.: ‘Physical model for burst noise in semiconductor devices’, Solid-State Electron., 1970, 13, pp. 1055- 1071

4 TROFIMENKOFF, F.N., TRELEAVEN, D.H., and BRUTON,

IEE Proc.-Circuits Devices Syst., Vol. 141, No. 5, October 1994

L.T.: ‘Noise performance of RC-active quadratic filter section’, IEEE Trans., 1973, CT-20, pp. 524-532

5 BACHLER, H.J., and GUGGENBUHL, W.: ‘Noise and sensitivity optimisation of a single-amplifier biquad’, IEEE Trans., 1979, CAS-26, pp. 30-36

6 BOWRON, P., and MEZHER, K.A.: ‘Noise and sensitivity optimisation in the design of second-order single-amplifier filters’, Int. J. Circuit Theory Appl., 1991,19, pp. 389-402

7 KHOURY, J.M., and TSIVIDIS, Y.P.: ‘Analysis and compensation of high-frequency effects in integrated MOSFET-C continuous-time filters’, IEEE Trans., 1987, CAS-34, pp. 862-875

8 HUBER, W.R.: ‘Two-port equivalent noise generators’, Proc. IEEE (Lett.), 1 9 7 0 , q pp. 807-809

9 BRAMBILLA, A., ESPINOSA, G., MONTECCHI, F., and SANCHEZ-SINENCIO, E.: ‘Noise optimization in operational transconductance amplifier filters’. IEEE Proc. ISCAS, 1989, 1, pp. 118-121

10 BOWRON, P., and MEZHER, K.A.: ‘Noise analysis of continuous- time active filters’. Proc. IEEE ISCAS, 1990,2, pp. 1181-1 184

11 ZURADA, J., and BIALKO, M.: ‘Noise and dynamic range of active filters with operational amplifiers’, IEEE Trans., 1975, CAS-22, pp. 805-809

12 DALE, D.P.E.: The LinCMOS design manual’ (Texas Instruments, 1985). pp. 3.15-3.20

13 KAPUSTIAN, V., BHATTACHARYYA, B.B., and SWAMY, M.N.S.: ‘Noise performance of active-R filters’, J. Franklin Inst., 1979,308, pp. 153-162

14 BOWRON, P., and BUTLAND, J.: ‘A frequencydomain evaluation of multiple-nonlinearity systems exhibiting jumpresonance responses’, IEEE Trans., 1984, CAS-31, pp. 175-181

15 BOWRON, P., MEZHER, K.A., and MUHIEDDINE, A.A.: ‘Dynamic-range prediction of continuous-time active filters’. Proc. European Cod. Circuit theory and design, 1991, HI, pp. 921-929

16 LEE, M.R., and MACROW, S.: ‘Noise characteristics of cascaded second-order active filters’, Elecrron. Lett., 1976, 12, pp. 301-302

17 SEDRA, A.S., and BRACKETT, P.O.: ‘Filter theory and design: active and passive’(Pitman, 1978), pp. 43&447

18 MOSCHYTZ, G.S.: ‘Gain-sensitivity product - a figure of merit for hybrid integrated filters using single operational amplifiers’, IEEE J. Solid-State Circuits, 1971, SC4, pp. 103-110

19 MIKHAEL, W.B., and BHATTACHARYYA, B.B.: ‘A practical design for insensitive RC-active filters’, IEEE Trans., 1975, CAS-22, pp. 407-415

20 SCHAUMANN, R.: ‘Two-amplifier active-RC biquads with mini- m i d dependence on op. amp. parameters’, IEEE Trans., 1987, CAS-34, pp. 431-433

21 SANCHEZ-SINENCIO, E., GEIGER, R.L., and NEVAREZ- LOZANO, H.: ‘Generation of continuous-time two integrator-loop OTA filter structures’, IEEE Trans., 1988, CAS-35, pp. 936-945

22 NEDUNGADI, A., and VISWANATHAN, T.R.: ‘Design of linear CMOS transconductance elements’, JEEE Trans., 1984, CAS-31, pp. 891-894

23 PARK, C.-S., and SCHAUMANN, R.: ‘High-frequency fully-tuned CMOS transductance-C filter’. IEEE Proc. ISCAS, 1988, 111, pp. 2161-2164

24 BOWRON, P., and MEZHER, K.A.: ‘Noise analysis for high-order active filters’. Twelfth IEE Colloq. Digital and analogue filters and filtering systems, 6th November 1982, pp. 8/1-8/5

6

The output noise voltage of the second-order filter in Fig. 4a is given by Table 1 in terms of various circuit design parameters. The latter are defined and optimised below.

6.1 Definitions The primary variables qpr q?, K , R , and 7 are first defined. If the passive section In Fig. 4a has [6] feedback transfer function

Appendix: Derivation o f parameters for composite-ampli f ier f i l ter circuit

in which, almost always, wp = CO, = w, , then the quality factor of the denominator (i.e. the selectivity of the

355

passive network) is

rc 1 + r2 + c2 (16)

and the quality factor of the numerator (i.e. of the zeros) is

q p =

rc

The gain of the controlled source in Fig. 4a is

Rb 1 + r 2 + c 2 I C K = l + - = --- R* r2 Q r

where Q is the active selectivity.

noise resistor at the noninverting input of amplifier A , is When I = 1 and m = to [20], the equivalent thermal

(19) The component ratio a is defined along with 1, m, r and c in Fig. 4a. The ratio of the gain-defining resistance level to the internal composite-amplifier resistance R A is denoted [lS] by

Re = RJ/&/ / ( l + @ R A

where

Y = R d R A (21) Ratio a merely controls KO and does not affect the noise analysis.

6.2 Optimisation The output noise voltage can be minimised [6, 151 by optimising the passive parameters r and c. Differentiating the nonoptimised expression for V, in Table 1 with respect to r and equating to zero gives an eighth-order equation. This can be solved numerically for a fixed value of c. In the common practical case [ l S , 201 when q 4 1, an analytical expression can be derived [6] for the optimal resistor ratio

Here, the active noise sources normalised with respect to the thermal noise are

v1 = E&,/4kTR and v2 = E&/4kTR

and

I = R J R

Optimising with respect to c for a fixed r gives another eighth-order equation whose optimal solution, for q 4 1, is

= -i J[i2 + i(1 + r2)] (23) in which

v1 + I. + (1 + vl)r2 3(v1 + 2 + r2) i=

Mutual substitution of eqns. 22 and 23 enables [6] ulti- mate noise-optimisation conditions to be evaluated for a specified design. When r and c are optimised, the other parameters are denoted as cj,, &, and I?.

356 IEE Proc.-Circuits Devices Syst., Vol. 141, No. 5 , October I994

noise analysis of second-order analogue active filters

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