Kerwin, W.J. Passive Signal Processing The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

Transcription

1 Kerwin, W.J. Passive Signal Processing The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 000

2 4 Passive Signal Processing William J. Kerwin University of Arizona 4. Introduction Laplace Transform Transfer Functions 4. Low-Pass Filter Functions Thomson Functions Chebyshev Functions 4.3 Low-Pass Filters Introduction Butterworth Filters Thomson Filters Chebyshev Filters 4.4 Filter Design Scaling Laws and a Design Example Transformation Rules, Passive Circuits 4. Introduction This chapter will include detailed design information for passive RLC filters; including Butterworth, Thomson, and Chebyshev, both singly and doubly terminated. As the filter slope is increased in order to obtain greater rejection of frequencies beyond cut-off, the complexity and cost are increased and the response to a step input is worsened. In particular, the overshoot and the settling time are increased. The element values given are for normalized low pass configurations to 5th order. All higher order doubly-terminated Butterworth filter element values can be obtained using Takahasi s equation, and an example is included. In order to use this information in a practical filter these element values must be scaled. Scaling rules to denormalize in frequency and impedance are given with examples. Since all data is for low-pass filters the transformation rules to change from low-pass to high-pass and to band-pass filters are included with examples. Laplace Transform We will use the Laplace operator, s s + j w. Steady-state impedance is thus Ls and /Cs, respectively, for an inductor (L) and a capacitor (C), and admittance is /Ls and Cs. In steady state s 0 and therefore s j w. Transfer Functions We will consider only lumped, linear, constant, bilateral elements, and we will define the transfer function T(s) as response over excitation. Ts () signal output signal input Ns () Ds () Adapted from Instrumentation and Control: Fundamentals and Applications, edited by Chester L. Nachtigal, pp , copyright 990, John Wiley and Sons, Inc. Reproduced by permission of John Wiley and Sons, Inc. 000 by CRC Press LLC

3 The roots of the numerator polynomial N(s) are the zeros of the system, and the roots of the denominator D(s) are the poles of the system (the points of infinite response). If we substitute s jw into T(s) and separate the result into real and imaginary parts (numerator and denominator) we obtain T( jw ) Then the magnitude of the function, T(jw)ï, is A A + + jb jb (4.) T( j w ) æ A ç è A + B ö + B ø (4.) and the phase T( j w) is B T( j w ) tan tan A B A (4.3) Analysis Although mesh or nodal analysis can always be used, since we will consider only ladder networks we will use a method commonly called linearity, or working your way through. The method starts at the output and assumes either volt or ampere as appropriate and uses Ohm s law and Kirchhoff s current law only. Example 4.. Analysis of the circuit of Fig. 4. for V o Volt I s; V s s s 3 3 I V ( s) s + s ; I I + I 3 V V + I s + s + s + i T() s + ( )( ) + Vo 3 V s + s + s + i 3 3 V 4 /3 I I 3 V i / 3 / V o I I 3 FIGURE 4. Singly terminated 3rd order low pass filter (W, H, F). 000 by CRC Press LLC

4 Example 4. Determine the magnitude and phase of T(s) in Example 4.. T s T s ( ) ( ) 3 s + s + s + s jw ( - ) w w w w + ( ) ( ) 0 - T s 3 w - w tan tan - tan - w w - w - w 3 The values used for the circuit of Fig. 4. were normalized; that is, they are all near unity in ohms, henrys, and farads. These values simplify computation and, as we will see later, can easily be scaled to any desired set of actual element values. In addition, this circuit is low-pass because of the shunt capacitors and the series inductor. By low-pass we mean a circuit that passes the lower frequencies and attenuates higher frequencies. The cut-off frequency is the point at which the magnitude is ( 3 db) of the dc level and is the dividing line between the passband and the stopband. In the above example we see that the magnitude of V o /V i at w 0 (dc) is.00 and that at w rad/s we have T( j w) 6 ( w + ) w rad s (4.4) and therefore this circuit has a cut-off frequency of rad/s. Thus, we see that the normalized element values used here give us a cut-off frequency of rad/s. 4. Low-Pass Filter Functions The most common function in signal processing is the Butterworth. It is a function that has only poles (i.e., no finite zeros) and has the flattest magnitude possible in the passband. This function is also called maximally flat magnitude (MFM). The derivation of this function is illustrated by taking a general all-pole function of third-order with a dc gain of as follows: Ts () 3 as + bs + cs + (4.5) The squared magnitude is T( j w) 3 ( bw ) + ( c w aw ) (4.6) Adapted from Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. 000 by CRC Press LLC

5 or T( j w) 6 4 a w + ( b ac) w + ( c b) w + (4.7) MFM requires that the coefficients of the numerator and the denominator match term by term (or be in the same ratio) except for the highest power. Therefore c b 0 ; b ac 0 (4.8) We will also impose a normalized cut-off ( 3 db) at w rad/s; that is, T( j w) w ( a + ) (4.9) Thus, we find a, then b, c are solutions to the flat magnitude conditions of Eq. 4.8 and our thirdorder Butterworth function is Ts () 3 s + s + s + (4.0) Table 4. gives the Butterworth denominator polynomials up to n 5. In general, for all Butterworth functions the normalized magnitude is T( j w) n ( w + ) (4.) Note that this is down 3 db at w rad/s for all n. This may, of course, be multiplied by any constant less than one for circuits whose dc gain is deliberately set to be less than one. Example 4.3. A low-pass Butterworth filter is required whose cut-off frequency ( 3 db) is 3 khz and in which the response must be down 40 db at khz. Normalizing to a cut-off frequency of rad/s, the 40-dB frequency is thus khz 3 khz 40 0 log 4 rad/s n 4 + TABLE 4. Butterworth Polynomials Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. therefore n 3.3. Since n must be an integer, a fourth-order filter is required for this specification. s s + + s + 3 s + s + s s s s s s s s s s + a 000 by CRC Press LLC

6 There is an extremely important difference between the singly terminated (dc gain ) and the doubly terminated filters (dc gain 0.5). As was shown by John Orchard, the sensitivity in the passband (ideally at maximum output) to all L, C components in an L, C filter with equal terminations is zero. This is true regardless of the circuit. This, of course, means component tolerances and temperature coefficients are of much less importance in the equally terminated case. For this type of Butterworth low-pass filter (normalized to equal -W terminations), Takahasi has shown that the normalized element values are exactly given by æ k L, C sin ( ç ) p ö è n ø (4.) for any order n, where k is the L or C element from to n. Example 4.4. Design a normalized (w 3dB rad/s) doubly terminated (i.e., source and load W) Butterworth low-pass filter of order 6; that is, n 6. The element values from Eq. (4.) are L C L 3 sin ( ) p H 4 sin ( ) p 44. F 6 sin ( ) p 939. H The values repeat for C 4, L 5, C 6 so that C 4 L 3, L 5 C, C 6 L Thomson Functions The Thomson function is one in which the time delay of the network is made maximally flat. This implies a linear phase characteristic since the steady-state time delay is the negative of the derivative of the phase. This function has excellent time domain characteristics and is used wherever excellent step response is required. These functions have very little overshoot to a step input and have far superior settling times compared to the Butterworth functions. The slope near cut-off is more gradual than the Butterworth. Table 4. gives the Thomson denominator polynomials. The numerator is a constant equal to the dc gain of the circuit multiplied by the denominator constant. The cut-off frequencies are not all rad/s. They are given in Table 4.. TABLE 4. Thomson Polynomials w 3dB (rad/s) s s + 3s s 3 +6s + 5s s 4 + 0s s + 05s s 5 + 5s 4 +05s s + 945s Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. Chebyshev Functions A second function defined in terms of magnitude, the Chebyshev, has an equal ripple character within the passband. The ripple is determined by. 000 by CRC Press LLC

7 TABLE 4.3 Chebyshev Polynomials s s + sinh n ( ) + sinh n s + sinh n + ( s + sinh n) [ s + ( sinh n) s + sinh n + 34] [ s + ( sinh n) s + sinh n ] [ s + ( sinh n) s + sinh n ] [ ] [ s + (. sinh ) s + sinh +. ] 6803 n n ( ) ( ) s + sinh n s sinh n s + sinh n Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. A / 0 ( 0 ) (4.3) where A decibels of ripple; then for a given order n, we define v. v æ ö sinh ç n è ø (4.4) Table 4.3 gives denominator polynomials for the Chebyshev functions. In all cases, the cut-off frequency (defined as the end of the ripple) is rad/s. The 3-dB frequency for the Chebyshev function is w cosh cosh ( / ) 3dB é - ù ê ú ëê n ûú (4.5) The magnitude in the stopband (w > rad/s) for the normalized filter is T( j w) + cosh ( n cosh w) (4.6) for the singly terminated filter. For equal terminations the above magnitude is multiplied by one-half [/4 in Eq. (4.6)]. Example 4.5. What order of singly terminated Chebyshev filter having 0.5-dB ripple (A) is required if the magnitude must be 60 db at 5 khz and the cut-off frequency ( 0.5 db) is to be 3 khz? The normalized frequency for a magnitude of 60 db is 5 khz 3 khz 5 rad/s Thus, for a ripple of A 0.5 db, we have from Eq. (4.3) A / 0 ( 0 ) and solving Eq. (4.6) for n with w 5 rad/s and T(jw) 60 db, we obtain n Therefore we must use n 4 to meet these specifications. 000 by CRC Press LLC

8 4.3 Low-Pass Filters Introduction Normalized element values are given here for both singly and doubly terminated filters. The source and load resistors are normalized to W. Scaling rules will be given in Section 4.4 that will allow these values to be modified to any specified impedance value and to any cut-off frequency desired. In addition, we will cover the transformation of these low-pass filters to high-pass or bandpass filters. Butterworth Filters For n, 3, 4, or 5, Fig. 4. gives the element values for the singly terminated filters and Fig. 4.3 gives the element values for the doubly terminated filters. All cut-off frequencies ( 3 db) are rad/s. FIGURE 4. Singly terminated Butterworth filter element values (in W, H, F). (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) FIGURE 4.3 Doubly terminated Butterworth filter element values (in W, H, F). (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) Adapted from Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. 000 by CRC Press LLC

9 FIGURE 4.4 Singly terminated Thomson filter element values (in W, H, F). (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) FIGURE 4.5 Doubly terminated Thomson filter element values (in W, H, F). (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) Thomson Filters Singly and doubly terminated Thomson filters of order n, 3, 4, 5 are shown in Figs. 4.4 and 4.5. All time delays are s. The cut-off frequencies are given in Table 4.. Chebyshev Filters The amount of ripple can be specified as desired, so that only a selective sample can be given here. We will use 0. db, 0.5 db, and 0.5 db. All cut-off frequencies (end of ripple for the Chebyshev function) are at rad/s. Since the maximum power transfer condition precludes the existence of an equally terminated even-order filter, only odd orders are given for the doubly terminated case. Figure 4.6 gives the singly terminated Chebyshev filters for n, 3, 4, and 5 and Fig. 4.7 gives the doubly terminated Chebyshev filters for n 3 and n Filter Design We now consider the steps necessary to convert normalized filters into actual filters by scaling both in frequency and in impedance. In addition, we will cover the transformation laws that convert low-pass filters to high-pass filters and low-pass to bandpass filters. Scaling Laws and a Design Example Since all data previously given are for normalized filters, it is necessary to use the scaling rules to design a lowpass filter for a specific signal processing application. 000 by CRC Press LLC

10 FIGURE 4.6 Singly terminated Chebyshev filter element values (in W, H, F): (a) 0.-dB ripple; (b) 0.5-dB ripple; (c) 0.50-dB ripple. (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 98, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) Rule. Rule. All impedances may be multiplied by any constant without affecting the transfer voltage ratio. To modify the cut-off frequency, divide all inductors and capacitors by the ratio of the desired frequency to the normalized frequency. Example 4.6. Design a low-pass filter of MFMtype (Butterworth) to operate from a 600-W source into a 600-W load, with a cut-off frequency of 500 Hz. The filter must be at least 36 db below the dc level at khz, that is, 4 db (dc level is 6 db). Since khz is four times 500 Hz, it corresponds to w 4 rad/s in the normalized filter. Thus at w 4 rad/s we have - 4 db 0 log é ê ëê ù ú n 4 + ûú 000 by CRC Press LLC

11 Ripple (db) C L C Ripple (db) C L C L C FIGURE 4.7 Doubly terminated Chebyshev filter element values (in W, H, F). FIGURE 4.8 Third-order Butterworth low-pass filter: (a) normalized (in W, H, F); (b) scaled (in W, H, mf). therefore, n.99, so n 3 must be chosen. The / is present because this is a doubly terminated (equal values) filter so that the dc gain is /. Thus a third-order, doubly terminated Butterworth filter is required. From Fig. 4.3 we obtain the normalized network shown in Fig. 4.8(a). The impedance scaling factor is 600/ 600 and the frequency scaling factor is p500/ p500: that is, the ratio of the desired radian cut-off frequency to the normalized cut-off frequency ( rad/s). Note that the impedance scaling factor increases the size of the resistors and inductors, but reduces the size of the capacitors. The result is shown in Fig. 4.8(b). Transformation Rules, Passive Circuits All information given so far applies only to low-pass filters, yet we frequently need high-pass or bandpass filters in signal processing. 000 by CRC Press LLC

12 Low-Pass to High-Pass Transformation To transform a low-pass filter to high-pass, we first scale it to a cut-off frequency of rad/s if it is not already at rad/s. This allows a simple frequency rotation about rad/s of s /s. All L s become C s, all C s become L s, and all values reciprocate. The cut-off frequency does not change. Example 4.7. Design a third-order, high-pass Butterworth filter to operate from a 600-W source to a 600-W load with a cut-off frequency of 500 Hz. Starting with the normalized third-order low-pass filter of Fig. 4.3 for which w 3 rad/s, we reciprocate all elements and all values to obtain the filter shown in Fig. 4.9(a) for which w 3 rad/s. Now we apply the scaling rules to raise all impedances to 600 W and the radian cut-off frequency to p500 rad/s as shown in Fig. 4.9(b). Low-Pass to Bandpass Transformation FIGURE 4.9 Third-order Butterworth high-pass filter: (a) normalized (in W, H, F); (b) scaled (in W, H, mf). To transform a low-pass filter to a bandpass filter we must first scale the low-pass filter so that the cut-off frequency is equal to the bandwidth of the normalized bandpass filter. The normalized center frequency of the bandpass filter is w 0 rad/s. Then we apply the transformation s s + /s. For an inductor For a capacitor æ Z Ls transforms to Z Lçs + è æ Y Cs transforms to Y C çs + è ö s ø ö s ø The first step is then to determine the Q of the bandpass filter where Q f w B 0 0 B r (f 0 is the center frequency in Hz and B is the 3-dB bandwidth in Hz). Now we scale the low-pass filter to a cut-off frequency of /Q rad/s, then series tune every inductor, L, with a capacitor of value /L and parallel tune every capacitor, C, with an inductor of value /C. Example 4.8. Design a bandpass filter centered at 00 khz having a 3-dB bandwidth of 0 khz starting with a third-order Butterworth low-pass filter. The source and load resistors are each to be 600 W. The Q required is Q 00 khz 0, or 0. 0 khz Q Scaling the normalized third-order low-pass filter of Fig. 4.0(a) to w 3dB /Q 0. rad/s, we obtain the filter of Fig. 4.0(b). 000 by CRC Press LLC

13 FIGURE 4.0 Third-order Butterworth low-pass filter: (a) normalized (in W, H, F); (b) scaled in (in W, H, F). FIGURE 4. Sixth-order Butterworth bandpass filter (Q 0): (a) normalized, w 0 rad/s (in W, H, F); (b) scaled. Now converting to bandpass with w 0 rad/s, we obtain the normalized bandpass filter of Fig. 4.(a). Next, scaling to an impedance of 600 W and to a center frequency of f 0 00 khz (w 0 p00 k rad/s), we obtain the filter of Fig. 4.(b). Defining Terms Bandpass filter: A filter whose passband extends from a finite lower cut-off frequency to a finite upper cutoff frequency. Equal ripple: A frequency response function whose magnitude has equal maxima and equal minima in the passband. Frequency scaling: The process of modifying a filter to change from a normalized set of element values to other usually more practical values by dividing all L, C elements by a constant equal to the ratio of the scaled (cut-off) frequency desired to the normalized cut-off frequency. High-pass filter: A filter whose band extends from some finite cut-off frequency to infinity. Impedance scaling: Modifying a filter circuit to change from a normalized set of element values to other usually more practical element values by multiplying all impedances by a constant equal to the ratio of the desired (scaled) impedance to the normalized impedance. Low-pass filter: A filter whose passband extends from dc to some finite cut-off frequency. 000 by CRC Press LLC

14 Maximally flat magnitude (MFM) filter: A filter having a magnitude that is as flat as possible versus frequency while maintaining a monotonic characteristic. Passband: A frequency region of signal transmission usually within 3 db of the maximum transmission. Stopband: The frequency response region in which the signal is attenuated, usually by more than 3 db from the maximum transmission. Transfer function: The Laplace transform of the response (output voltage) divided by the Laplace transform of the excitation (input voltage). Transformation: The modification of a low-pass filter to convert it to an equivalent high-pass or bandpass filter. Related Topics 6. Definitions and Properties 0.3 The Ideal Linear-Phase Low-Pass Filter 0.6 Butterworth Filters 0.7 Chebyshev Filters References A. Budak, Passive and Active Network Analysis and Synthesis, Boston: Houghton Mifflin, 974. C. Nachtigal, Ed., Instrumentation and Control: Fundamentals and Applications, New York: John Wiley, 990. H.-J. Orchard, Inductorless filters, Electron. Lett., vol., pp. 4 5, 966. P. Sydenham, Ed., Handbook of Measurement Science, Chichester, U.K.: John Wiley, 98. W. E. Thomson, Maximally flat delay networks, IRE Transactions, vol. CT-6, p. 35, 959. L. Weinberg, Network Analysis and Synthesis, New York: McGraw-Hill, 96. L. Weinberg and P. Slepian, Takahasi s results on Tchebycheff and Butterworth ladder networks, IRE Transactions, Professional Group on Circuit Theory, vol. CT-7, no., pp. 88 0, by CRC Press LLC