HIGHER-ORDER RC-ACTIVE FILTERS

Transcription

1 R752 Philips Res. Repts 26,65-74,1971 HIGHER-ORDER RC-ACTIVE FILTERS by H. W. HANNEMAN Abstract Realization of RC-active filters, as proposed by Sallen and Key, consists of a cascade of suitable second-order building blocks. The number of active elements needed can be reduced if the second-order building block is extended in a canonical way, as is pointed out by Fjällbrant. However, the spread of the dement values and the sensitivity of the transfer function to deviations of the element values determines the order which can be achieved. A method is described to reduce the spread and yet get a low sensitivity (valid up to the fifth order). A method is also presented to realize stiii-higher-order transfer functions. The canonically extended filter configuration is used now as a building block. Design data and examples are included. 1. Introduetion In 1955, Sallen and Key 1) presented a method of designing RC-active filters. A catalogue of second-order active networks was given which enabled the designer to achieve the specified transfer function by cascading the suitable secondorder building blocks. Fjällbrant 2) extended a second-order building block canonically *) up to higher orders. Because of the sensitivity of the transfer function to deviations of the element values and because of the increasing ratios Cmax/Cmln and Rmax/Rmln> the order of the filter which can be attained with a single building block will not be higher than about five. Thus, to achieve still-higher-order transfer functions it is necessary to cascade several building blocks. The (selective) amplification of some building blocks causes a signal level at the output of that building block which is higher than the signal level at the output of the complete filter. This is unadvantageous (e.g. the signal-to-noise ratio decreases) and is th~refore undesirable. In sec. 2.1 the filter configuration given by Fjällbrant is modified and it is shown that more practical filters can be obtained in that way. Design data for low-pass and high-pass filters up to the :fifth order are also presented in this section (Bessel, Butterworth and Chebyshev). In sec. 2.2 attention is paid to the sensitivity of the modulus of the transfer to deviations of the element values. It is shown in sec. 3 that for the realization of RC-active filters with a steep roll-off, the transfer function can be better split up into fourth-order transfer *) Canonical has to be understood here as: realized with the minimum amount of R,C elements.

2 66 H. W. HANNEMAN ratios, which can be obtained as is described in sec. 2. Each building block realizes two pairs of conjugate complex poles in such a way that the pole with a relatively high quality factor is realized together with the pole with a relatively lowquality factor. The merits of this design procedure are discussed in sec. 3.1 and sensitivity considerations in sec An example is described in sec Canonical RC-active filters Like Fjällbrant, we choose as our basic low-pass network configuration the one shown in fig. 1. It is seen that there are, in general, 2n + 1 degrees of freedom for an nth-order filter (n resistances, n capacitances and the gain of the operational amplifier). To determine the transfer function of an nth-order filter within a constant, it is sufficient to have n degrees of freedom. All the resistances will be taken equal to each other, which is attractive from a practical point of view (integration). One element can be given an arbitrary value. Fjällbrant chose an emitter follower for fixed voltage gain of the active element. However, we' have found that a voltage gain larger than unity will decrease the spread of the element values considerably. Further, we found it advantageous to choose two of the capacitances equal to each other (C, = C n - l ). These choices lead to lower capacitance ratios and it is found that an adjustable voltage gain facilitates, if necessary, a precise adjustment of the filter characteristic. The voltage gain then needed lies between 1 and Design procedure The transfer function of the network shown in fig. 1 can be represented as K ---= VIn asps + acp" + a3p3 + a2p2 + alp + 1 The numerical values of the coefficients as... al are determined by the filter characteristic required. For several types of filter characteristics, these data can be found in the literature 3). On the other hand, by analyzing the circuit, Fourth order Fifth order Fig. 1. Filter configuration using an operational amplifier with a voltage gain K.

3 HIGHER-ORDER RC-ACTIVE FILTERS 67 we can find analytical expressions for the coefficients as al- By equating these expressions to the numerical values of the coefficients we obtain a set of non-linear inhomogeneous equations, By way of example the normalized element values (the values of the capacitances) and the voltage gain have been calculated for several commonly used types of filter characteristics (Bessel, Butterworth and Chebyshev), All the resistances are normalized to 1 ohm, The bandwidth is taken as 1 rad/s_ The results are given in table I. The highest value of Cmax/Cmln that appears is less than 35_Comparing the spread of the element values for the fifth-order Butterworth filter we' observe that Fjällbrant obtains an Rmnx/Rmln ratio of about 40 and a CmaxfCmln ratio of about 700 while we get 1 and 10, respectively (though Fjällbrant has added an extra capacitor to reduce the capacitance spread). A network configuration similar to that shown in fig. 1 can be used to achieve a high-pass transfer function, All resistors are replaced by capacitors and all capacitors by resistors, In the high-pass network configuration all capacitances are normalized to 1farad. Table I can also be used: the values of the resistances are equal to the inverse of the corresponding values of the capacitances; the voltage gain remains the same, Band-pass and band-stop filters can be realized by cascading a high-pass and a low-pass filter section, TABLE I Normalized element values for low-pass filters order Cl I C 2 I C 3 I C 4 I K Butterworth 3 _ Bessel Chebyshev dB ripple Chebyshev dB ripple

4 68 H. W. HANNEMAN 2.2. Sensitivity Several authors have formulated the sensitivity of second-order transfer functions in an explicit form 4.5.6). It is pointless to do so for higher-order transfer functions, for the sophisticated formulas which describe the sensitivity do not offer a clear insight. As a measure of the sensitivity, we have therefore calculated the change of the modulus of the transfer ratio for deviations of the values of each element. This was done for each order. As expected, the sensitivity increases with the steepness of the filter characteristic (see also table II). In general, the sensitivity reaches a maximum at about the cut-off frequency, irrespective of which element is varied. Some typical curves are given in fig. 2: the proportional change ofthe modulus of a fourth-order transfer ratio (realizing a Chebyshev characteristic with 0'5-dB ripple) is shown as a function of the normalized frequency for a change of 1% in the capacitors and the voltage gain. The curves for other types of filters are similar: In table II the value of TABLE II Maximum deviation of transfer function (in db) order Butterworth Bessel Chebyshev Chebyshev 0 5 db-ripple I O-dB ripple '0' I Change of I V~utl(%) Vm 7' Fig. 2. The proportional change of the modulus of a fourth-order transfer function as a function of the frequency for a change of 1% in Cl (a), C 2 (b): C 3 (c) and (1 - K) (d).

5 HIGHER-ORDER RC-ACTIVE FILTERS 69 the maximum deviation of the transfer function (in db) is given for a 1% change in the element that causes the largest deviation. Deviations of the filter characteristic from the prescribed behaviour, due to the inaccuracies of the element values, can, to a large extent, be removed by a minor adjustment. of the voltage gain of the operational amplifier. 3. Higher-order RC-active filters Realizing RC-active filters of an order higher than five in the way described in the previous section is not readily possible because of the rapidly increasing sensitivity. Thus, we are obliged to cascade several building blocks. If we wish to cascade building blocks we face the following problem. Let G(p) be the transfer ratio of a second-order low-pass building block. Then V. K K' G(p) = ~ = Vin (pjwo)2 + (2a/wo) (p/wo) + 1 (p + a +jw) (p + a-jw)' where It is seen that at about the frequency Wo the input signal will be amplified Q = wo/2a times at w = Wo, if Q is the figure of merit of the poles realized with the circuit (see fig. 3). This means that the maximum input signal which can be handled is determined by that amplification. In general, this second-order building block is one of a chain of second-order building blocks of which the next one amplifies Q' times at w = wo'. If wo' is close to Wo - and this will be the case if we want to design filters with a steep roll-off - the maximum allowable signallevel will be exceeded. Thus, the input signal has to be diminished, etc. That is why the input signalof the complete filter has to be low. Fig. 3. The frequency response of a second-order building block. r

6 70 H. W. HANNEMAN The noise level is quite high because of the relatively high number of active elements (n/2) required. Some progress can be made by choosing the sequence of the building blocks as n, 1, (n - 1), 2..., etc. (1 corresponds to the building block realizing the poles with the highest quality factor, n to the one with the lowest quality factor) Design procedure Let G(p) be the higher-order transfer ratio. Split G(p) up into as much fourthorder sections as possible. Thus, where N I (P) N 2 (P) N n -1(p) D I (P)' D 2 (p)'..., G n - I (p)' are ail fourth-order transfer ratios and is 'a third-, second- or first-order section or is a constant. It is supposed that either all Nlp) are constants (low-pass filter) or all Nlp) are of the form pk. with k = 4, 3, 2 or 1 for DI(P) being of the fourth, third, second or first order,! respectively (high-pass filter). Each fourth-order building block realizes two pairs of conjugate complex poles. We now design the fourth-order building block so that the pole with the highest quality factor is realized together with the pole with the lowest quality factor. The next fourth-order building block (if any) realizes the pole with the highest-but-one quality factor together with the lowest-but-one quality factor, etc., see fig. 4. All these buildings blocks can be designed as described in sec. 2. The advantages are threefold. First, let the input signalof the first fourthorder building block be 0 db. Then the output signallevel will not exceed 0 db at any frequency. The output signalof the first section is applied to the second. Again, it can be proved that the output signalof the second stage will not exceed! o db at any frequency if the voltage gains of the active elements are assumed to be 1. Second, the number of active elements needed, compared with Sallen's and Key's solution, is reduced from n/2 to n/4 or n/4 + 1, which implies that the filters can be produced more cheaply and made' less noisy and can have a lower power dissipation.

7 HIGHER-ORDER RC-ACTIVE FILTERS 71 Im(p) t -Re(p) o Fig. 4. The poles of a transfer function of the order 211.The poles 1, 1*, 11,11* and 2, 2*, (11-1), (11-1)*, and so on, are each realized with a single fourth-order building block. In the third place, it will be indicated in sec. 3.2 that the sensitivity of the modulus of the transfer function to deviations of the element values is reduced or remains about the same. Note. If G(p) is an odd-order transfer function, the pole with the lowest quality factor is on the negative real axis. Thus, it is necessary to realize the highestquality-factor pole together with the pole with the lowest-but-one quality factor to get a fourth-order building block Sensitivity considerations As we have found in the previous section the sensitivity of the modulus of he transfer ratio increases with the order of the transfer function and with the steepness of the filter characteristic prescribed by that transfer function. Now let us compare the sensitivity of a second-order building block realizing a pole with a high quality factor, say Q, with the sensitivity of a fourth-order building block realizing a pole with the same quality factor Q and a pole with a relatively low quality factor. From the viewpoint of sensitivity it is significant that the order ofthe fourth-order building block is higher, but the steepness ofthe filter characteristic is lower than for the second-order building block. Thus, the sensitivity of the fourth-order transfer function will not necessarily be higher than that of the second-order transfer function; see fig. 5.

8 72 H. W. HANNEMAN I~~nutl (db) , Freq. Fig. 5. The frequency response of (a) a second-order and (b) a fourth-order building block Example Consider the transfer function of a tenth-order Chebyshev high-pass filter with a ripple of 0 5 db in the pass band. Figure 6 shows the network configuration and the normalized element values if the transfer function is realized in conformity with the method described in sec. 3.1 (note the low spread ofthe element values). The sensitivity of a second-order circuit realizing the pole with the highest quality factor (Q is about 18) is now compared with the sensitivity of a fourth-order circuit which realizes the same pole and the pole with the lowest quality factor (Q ~ 1). In fig. 5 the transfer function vs the frequency is shown for both network configurations. The sensitivity as a function of the frequency is shown in fig. 7. In this case the sensitivity of the fourth-order section is even smaller than that of the second-order section. A comparison of the sensitivity of the second-order circuit which realizes the highest-but-one pole, with the sensitivity of the fourth-order design which realizes the highest-butone and the lowest-but-one pole shows that these are about the same. Fig. 6. A tenth-order high-pass filter with a Chebyshev characteristic (O'S-dB ripple).

9 HIGHER-ORDER RC-ACTIVE FILTERS S(%} 40 r 20 o ;JO Freq. a} 30 S(%} Î b} Fig. 7. a. The proportional change S of the modulus of a second-order transfer function as a function of the frequency for a change of I % in Cl (a), C 2 (b) and (I - K)(c). b. The proportional change S of the modulus of a fourth-order transfer function as a function of the frequency for a change of I % in Cl (a), C 2 (b), C 3 (c) and (1 - K) (d). 4. Conclusion A practical and efficient method to design RC-active filters has been described. Up to the fifth order a canonical extension of the network configuration given by SalIen and Key is employed. For still higher orders a cascade of these circuits is used, pairing the poles in a particular way. An extension of the method to cover the realization of transfer functions containing zeros on the imaginary axis is at present under investigation.

10 74 H. W. HANNEMAN Acknowledgement I wish to thank my colleagues Ir D. Blom and Ir J. O. Voorman for valuable suggestions, especially concerning the first part of this paper. Eindhoven, August 1970 REFERENCES 1). R. P. Sallen and E. L. Key, IRE Trans. Circuit Theory CT-2, 74-85, ) T. Fjällbrant, Ericsson Technics 23, , ) E. Christian and E. Eisenmann, Filter design tables and graphs, Wiley, New York, ) P. J. McVey, lee Proc. 112, , S) A. G. J. Holt, A. S. Reid and J. N. T'o r ry, The Radio and Electr. Engr. 37, , ) W. Saraga, Electronics Letters 3, , 1967.