Noise performance of active-R filters

Noise Performance of Active -R Filters by v. KAPUSTIAN,*

B.B.BHATTACHARYYA

Department of Electrical Montreal, Canada

Engineering,

and

M.N.S.SWAMY

Concordia

University,

A closed-form analytical expression for evaluation of the output noise voltage of active-R filters has been developed. 7’he expression is applicable to any low-pass, band-pass or high-pass two amplifier second order active-R realization. The expression for the noise voltage has been derived in terms of the filter specifications and parameters of noise sources contained within the filter. Thus, it is very convenient for practical calculations. The expression is also useful for obtaining a design that minimizes the magnitude of the output noise. Experimental measurements indicate close agreement with the theroetical analysis. ABSTRACT:

I. Introduction Recently considerable attention has been paid to the design of filter networks by using resistors and operational amplifiers (OA) only (l-4). These designs are known as active-R designs. Since the OA’s and the resistors are easily integrable, such designs are attractive for IC implementation, particularly because of the absence of external capacitors in the resulting networks. Further, these designs considerably widen the frequency range of applications in comparison to the one obtainable from active-RC designs of filter networks. However, the components of active-R filters, namely, the resistors and the OA’s are noisy and hence would contribute noise to the filtered signal. The presence of this noise at the output will restrict the dynamic range of these filters. In addition, in view of this noise, the slew rate of the OA’s will limit the maximum operating frequency obtainable from active-R filters. Thus, it is useful to be able to estimate the amount of noise generated by these filters. An extensive literature search by the authors has failed to reveal any noise analysis of active-R filters. The techniques, that have been used for some specific RC-active filter circuits (57), can, of course, be used for active-R filters. However, the resulting analysis will be complex as, even for a simple realization, 10 or 15 noise sources are Isquired for a complete noise characterization. Consequently, the details are likely to obscure any clear insight into the noise behaviour of active-R filters. The purpose of the present article is to determine a fairly general expression *V. Kapustian was on leave of absence from the Department Devices, Moscow Power Engineering Institute, Moscow, U.S.S.R.

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153

V. Kapustian,

B. B. Bhattacharyya

and M. N. S. Swamy

for the noise voltage generated at the output of active-R filters. Specifically, a relationship will be established that shows clearly the dependence of the output noise voltage on the filter specifications. Further, this relationship will also permit a simple way of minimizing the output noise.

ZZ. Determination

of the Output Noise Voltage

In order to proceed with the analysis, the block schematic of Fig. la is assumed. This configuration for active-R filters is based closely on the one fnst proposed by Schaumann (1,2). All known two-OA realizations of low-pass, band-pass and high-pass active-R filters (l-4) can be shown to be special cases of this configuration. The blocks T,(s) and T*(s) are realized, in practice, as a cascade of an open or closed loop OA and a resistive voltage divider. These blocks can be represented by (1,2): T,(s) = KibiGBi/(s +wli),

i = 1,2,

(1)

where 0 < Ki < 1 is the parameter contributed by the resistive voltage divider, GB, is the gain-bandwidth product of the OA, oli is the’ magnitude of the f%st pole of the closed or open loop OA as the case may be and, finally, bi is a

T;(s)

T,(s)

FIG. 1 .(a) The basic block-schematic

154

of active-R filters. (b) Equivalent circuit for noise calculation.

Noise Performance of Active-R

Filters

parameter related to the dc gain for the inverting closed loop OA; it is unity for the open loop or for the non-inverting closed loop OA. To calculate the output noise voltage, the input has to be assumed to be zero. The output r.m.s. noise voltage E, will then be found by r.m.s. addition of the contributions of the blocks T, and T2 to the output. For this purpose, the noisy block Ti can be represented by a noiseless block Ti with its equivalent noise source eoi(o> in series with its output. Thus, the pertinent equivalent circuit to consider for noise analysis is the one shown in Fig. lb. The output noise spectral densities of block TI and T2 are e&(o) and e&(o), respectively. The total output power spectral density can be written as e;(w) = e&(o)

IToI(i

+ e&(o)

ITo2(i~)12,

(2)

where, as can be shown 1

(s + 4(s + @I*) e,=O = 1 - TIT, = sz + (o,/Q,,)s + 0;’ = T,/(le”,=O

TIT,) =

o. = 2mfo = pole frequency = [oIIw,2+

KlhGh(s +4 sz + (o()/Qo)s +og

of the filter (2),

blb2KIK2GB1GB2]~,

Q. = Q-factor = [~o/(%

of the filter, +

%)I.

Then the total r.m.s. output voltage E, is given by

Thus, to complete the analysis, the value of e&(o) and e&(o) have to be determined. The most general form that either the block TX or T, can assume is of the

e.,

FIG. 2. The electrical schematic of a block of active-R filter. Vol. 308, No. 2, August 1979 Printed in Northern Ireland

155

V. Kapustian,

B. B. Bhattacharyya

and M. N. S. Swamy

form shown in Fig. 2. As shown, the block Y& consists of an OA, feedback resistor Rif, resistors RT’s and Rt’s connected to the inverting and noninverting input terminals respectively. The output noise of the circuit of Fig. 2 will be due to Johnson noise and resistors and due to the noise contributed by the OA. Each of the noisy resistors, Ri can be assumed to be a noiseless resistor of value Ri in series with tin equivalent noise source with power spectral density given by E &(o) = 4KTRi, where constant = 1.38 x 1 OP23JK- ’ ,

K = Boltzmann’s

T = absolute temperature, Ri = resistance,

K,

0.

The equivalent circuit of a noisy OA is shown in Fig. 3. The OA shown is regarded as noise free. The values of the noise voltage E and noise currents i, and i, may be obtained from manufacturer’s specifications or measurement. Since the operating frequency of active-R filters are much higher than 1 kHz, all background sources of noise associated with an OA, such as its Flickernoise, can be neglected. Under this situation, the spectral density for the equivalent noise sources E, il and i2 can be assumed to be constant in the operating frequency range. Their typlcai values, for 741 PA, are (8) E = 20 nV Hz-f, i, = i, = i = 0.2 pA Hz-i. Assuming all the above mentioned noise sources to be uncorrelated, the total output voltage spectral density eoi of the circuit of Fig. 2 can be obtained by an r.m.s. addition of the contribution to output due to each of the noise sources. This has been done in Appendix A. Using Appendix A, we have the output voltage spectral density e,,i for the ith block as e%(w)

=

eZ(w>

+

G,(o),

(4)

where e,% and e,% are the contributions to the output spectral density by the OA and the resistors contained in the ith block. Also, e&(W) = 4K7’ (5)

FIG. 3. OA’s noise model.

156

Noise Performance of Active-R g,, + 2

gz =sum

of all conductances

connected

to the inverting

Filters

terminal

k=l

of the OA. z,

g&=sum

of

all

the

conductances

connected

to

the

non-inverting

terminal of the OA. @ik

=

GBigifl(gif

+

C

gik)

(6) In view of the fact that the same type of OA’s are used in most applications, and also the fact, that the two OA’s are fabricated on the same chip, at the same time and using the same processing steps, it is reasonable to assume that they all possess similar characteristics. Thus let GB, = GB, = GB, E,+ = .sA, = . . it is likely that the sum CA, lA, = lA, = iA. Also in practice,

gif+C gik+I g:k 1

1

will be less than 50 kR. Under this situation, cA >>iA [

1 &+C gZ+I SL 1

1

for typical values of cA and iA, as mentioned before. With these assumptions, GB e:,(m) = E: ___ jo+wli

I

2

I

.

(7)

Substituting (5) and (7) into (4), we finally get the output spectral density for the ith block as

(84 =Ai where Ai is determined Substituting

GB * ~ 10+6.Qi ’

I

I

@b)

from Eqs. (8)

now (8) into (2), and assuming Q. 2 3, we can show that (9)

where

Vol. 308, No. 2, August 1979 Printed in Northern Ireland

157

V. Kapustian,

B. B. Bhattacharyya

and M. N. S. Swamy

Using (3) and (9) we now have m

e:(o) dw =

6

df,

It has been shown in (7) that

Jw ITBp(j~)lzdf = lalTLp(j~)12 df =(d2)foQo. 0

From (lo),

then we obtain E;=--

nGB2 2

Equation

(lob)

0

(11) can be rewritten

fo

1 .

(11)

in the form

(12) Equation (12) can be used to calculate the output r.m.s. noise voltage of any second order two-OA active-R realizations of low-pass, band-pass and highpass filters. The first component of (12) is due to OA’s noise and second component is due to the noise of the resistors. It should be noted that the output noise is directly proportional to the Q-factor of the filter. A careful examination of either Eq. (11) or Eq. (12) shows that the noise contribution of the second block of the filter is proportional to the ratio (KJK,). Thus, in order to minimize the output noise, the circuit should be designed in such a way that K, is as much larger as possible than K1. III. Experimental

Verifications

In order to verify the results df the theoretical analysis, a band-pass filter using the circuit of Mitra-Aatre (3) was built and tested. The circuit is shown in Fig. 4. For convenience in understanding, the blocks T, and T2 are indicated in this figure. The OA’s used were MS1741cp. The values of E and GB for a supply voltage of +15 V were measured to be GB = 2~ X lo6 rad s-l, E = 19 nV Hz-f. Two types of experiments were conducted. First, the dependence of the output r.m.s. noise voltage E. on o. was investigated. The resonant frequency o. was varied by the trim-pot R,. The results are shown in Fig. 5. The second experiment was on the dependence of E, on the Q-factor. The Q-factor was varied by the trim-pot R,, while the value of w0 was maintained constant at

Noise Performance of Active-R

Filters

__--_---FIG. 4. Mitra-Aatre

active-R filter.

20 kHz. The results are shown in Fig. 6. The predicted values were obtained from E!q. (12). Good agreement is found to exist between the predicted and measured values in both the experiments. The slight difference observed is probably due to the inaccuracy in the measurement of the OA parameters.

IV. Conclusion An analytical expression for the calculation of output noise voltage of a general configuration of active-R filters has been obtained. The expression is

IO

20

30 f0,

FIG.

5.

40

50

60

kHz

Dependence of the output r.m.s. noise voltage of fc(R4-VW).

Vol. 308, No. 2, August 1979 Printed in Northern Ireland

159

V. Kapustian, B. B. Bhattacharyya

and M. N. S. Swamy

0.9 -

0.9 -

0.7z 0.6-

A - Experiment

G

-

- Predicted

05-

04-

0.3<=I IO

I

I

I

20

30

40

0 FIG.

6. Dependence

of the output r.m.s. noise voltage on Q(f,, = 29.75 kHz).

valid for any two-OA second order realizations of low-pass, band-pass and high-pass filters. The form of the expression is particularly convenient for carrying out practical calculations, since it is obtained in terms of the parameters of the filter and the noise sources contained within the filter. Experimental measurements made on a band-pass active-R filter indicated good agreement with the theoretical predictions.

References active-R filters”, proc. of 1975 (1) R. Schaumann, “A note on state-variable-type Midwest Symposium on Circuits and Systems, pp. 598-602, Montreal. Canada, Aug. 1975. (2) R. Schaumann, “Low-sensitivity high-frequency tunable active filter without external capacitors”, IEEE Trans. Circuits and Systems, Vol. CAS-22, pp. 39-44, Jan. 1975. (3) A. K. Mitra and V. K. Aatre, “Low-sensitivity high frequency active-R filter”, IEEE Trans. Circuits and Systems, Vol. CAS-23, pp. 670-676, NOV. 1976. (4) H. K. Kim and J. B. Ra, “An active biquadratic building block without external capacitors”, IEEE Trans. Circuits and Systems, Vol. CAS-24, pp.689-694, Dec. 1977. (5) L. T. Bruton, F. N. Trofimenkoff and D. H. Treleaven”, Noise performance of low-sensitivity active filters”, IEEE J. Solid-State Circuits, Two-Integrator Loop, Vol. SC-8, pp. 85-91, Feb. 1973. (6) H. Weinrichter and T. A. Nossek, “Noise analysis of active-RC filters”, Proc. 1976 IEEE ht. Symp. Circuit and Systems, pp. 344-347, F.R. Germany, April 1976. (7) F. N. Trofimenkoff, D. H. Treleaven and L. T. Bruton, “Noise performance of RC-active quadratic filter sections”, IEEE Trans. Circuit Theory, Vol. CT-20, pp. 524-532, Sept. 1973.

160

Noise Performance of Active-R

Filters

(8) L. Smith and D. H. Sheingold, “Noise and operational amplifier circuits”, Analog Dialogie, A Journal for the exchange of analog techniology. Published by Analog Devices, Inc., Vol. 3, March 1969.

Appendix

A

To calculate the output noise spectral density eoi of Fig. 2, the signals e, and e, can be assumed shorted. They only show the probable signal injection point from one of the blocks to the other. To simplify analysis the subscript i also will be dropped. Then, replacing the noisy resistors and the OA by their noise equivalents, we have the circuit of Fig. 7a to consider. In Fig. 7a, the resistors and the OA shown are now noise free. The circuit contains, in general, (4 + n + m) noise sources. The output voltage spectral density can be found by summation of the contributions of all the noise sources. For simplicity, lirst the output noise due to the resistors was calculated. This was done by short circuiting the voltage sourcce E and open circuiting current sources i, and il. The result is the voltage spectral density eOR(W) given by 1 -.-!-x, g;+pr +c;=,g;

GB

2

II jw+o,

I’

-

FIG.

7.(a)

Equivalent

Vol. 308, No. 2, August 1979 Printed in Northern Ireland

(A. 1)

e.

circuit for calculating output noise of T, or T2. (b) Noise equivalent circuit of the block T.

161

V. Kapustian, B. B. Bhattacharyya

and M. N. S. Swamy

where

g;=l/R,, g;=l/R;, g,=4 Rf o1

=

GB----

gf

gf+cgi

GB = gain-bandwidth product of the OA. To calculate the contribution to the output by the noise sources of the OA, then current sources i, and i, can be converted into equivalent voltage sources. Analysis then yields the voltage spectral density eoA given by

It is reasonable

to assume (4,s)

that i, = i2. Then

(A.21 Assuming the noise sources to be uncorrelated the total noise voltage spectral density e, is given by e&0) = e&(w) f&,(o)

(A.3)

where eoR and eoA are given by (A.l) and (A.2). The above analysis is valid for either T, or Tz with appropriate values for the set of resistors R, and R; and R,. With the value of the output spectral density e, calculated as above the noisy block T can be represented as a noiseless block T with an equivalent voltage e, connected in series with its output as shown in Fig. 7b.

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