The electrical engineering handbook
Kerwin, W.J. “Passive Signal Processing” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 © 2000 by CRC Press LLC 4 Passive Signal Processing 4.1 Introduction Laplace Transform•Transfer Functions 4.2 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 4.1 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 1/ Cs , respectively, for an inductor ( L ) and a capacitor ( C ), and admittance is 1/ 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 Ns Ds () () () == signal output signal input William J. Kerwin University of Arizona Adapted from Instrumentation and Control: Fundamentals and Applications, edited by Chester L. Nachtigal, pp. 487–497, copyright 1990, John Wiley and Sons, Inc. Reproduced by permission of John Wiley and Sons, Inc. © 2000 by CRC Press LLC 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 = j w into T ( s ) and separate the result into real and imaginary parts (numerator and denominator) we obtain (4.1) Then the magnitude of the function, ÷ T ( j w ) ï , is (4.2) and the phase is (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 1 volt or 1 ampere as appropriate and uses Ohm’s law and Kirchhoff’s current law only. Example 4.1. Analysis of the circuit of Fig. 4.1 for V o = 1 Volt. FIGURE 4.1 Singly terminated 3rd order low pass filter ( W , H, F). Tj AjB AjB ()w= + + 11 22 ΈΈTj AB AB ()w= + + æ è ç ö ø ÷ 1 2 1 2 2 2 2 2 1 2 Tj()w Tj B A B A ( ) tan –tan –– w= 1 1 1 1 2 2 IsV ss s IVsssIII VVIsss Ts V V sss i o i 3 3 2 1 3 2 4 3 2 21 1 2 1 2 3 123 11 32 32 112 221 1 221 ==+ ( ) ( ) =+ = ( ) =+ =+ =+=+++ == +++ ; ; () V i V 1 I 1 I 2 I 3 I 3 V o 1 / 2 4 / 3 3 / 2 1 © 2000 by CRC Press LLC Example 4.2 Determine the magnitude and phase of T(s) in Example 4.1. The values used for the circuit of Fig. 4.1 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 0.707 (–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 1.00 and that at w = 1 rad/s we have (4.4) and therefore this circuit has a cut-off frequency of 1 rad/s. Thus, we see that the normalized element values used here give us a cut-off frequency of 1 rad/s. 4.2 Low-Pass Filter Functions 1 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 1 as follows: (4.5) The squared magnitude is (4.6) 1 Adapted from Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. Ts sss Ts Ts sj ( ) = +++ ( ) = - ( ) +- ( ) = + ( ) =- - - =- - - = -- - 1 221 1 12 2 1 1 0 2 12 2 12 32 2 2 3 2 6 11 3 2 1 3 2 w www w ww w ww w tan tan tan ΈΈTj rads () () . w w w = + = = 1 1 1 0707 6 Ts as bs cs ()= +++ 1 1 32 ΈΈTj bca () (– )(– ) w www 2 22 32 1 1 = + © 2000 by CRC Press LLC or (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 (4.8) We will also impose a normalized cut-off (–3 dB) at w = 1 rad/s; that is, (4.9) Thus, we find a = 1, then b = 2, c = 2 are solutions to the flat magnitude conditions of Eq. 4.8 and our third- order Butterworth function is (4.10) Table 4.1 gives the Butterworth denominator polynomials up to n = 5. In general, for all Butterworth functions the normalized magnitude is (4.11) Note that this is down 3 dB at w = 1 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 12 kHz. Normalizing to a cut-off frequency of 1 rad/s, the –40-dB frequency is thus therefore n = 3.32. Since n must be an integer, a fourth-order filter is required for this specification. ΈΈTj abaccb () (–) (–) w www 2 26 2 4 2 2 1 221 = +++ cb bac 22 20 20–;–== ΈΈTj a () () .w w= = + = 1 2 1 1 0707 Ts sss ()= +++ 1 221 32 ΈΈTj n () () w w = + 1 1 2 TABLE 4.1Butterworth Polynomials Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permis- sion of John Wiley and Sons Limited. s ss sss ssss sssssa + ++ +++ ++++ +++++ 1 21 221 26131 34142 26131 1 32361 52361 52361 32361 2 32 432 5432 . 12 3 4 kHz kHz rad/s= – log40 20 1 4 1 2 = + n © 2000 by CRC Press LLC There is an extremely important difference between the singly terminated (dc gain = 1) 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 1-W terminations), Takahasi has shown that the normalized element values are exactly given by (4.12) for any order n, where k is the L or C element from 1 to n. Example 4.4.Design a normalized (w –3dB =1 rad/s) doubly terminated (i.e., source and load = 1 W) Butter- worth low-pass filter of order 6; that is, n = 6. The element values from Eq. (4.12) are The values repeat for C 4 , L 5 , C 6 so that C 4 = L 3 , L 5 = C 2 , C 6 = L 1 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 func- tions 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 4.2 gives the Thomson denomi- nator 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 1 rad/s. They are given in Table 4.2. 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 ⑀. LC k n , sin (–) = æ è ç ö ø ÷ 2 21 2 p L C L 1 2 3 2 21 12 05176 2 41 12 14141 2 61 12 19319 == == == sin (–) . sin (–) . sin (–) . p p p H F H TABLE 4.2Thomson Polynomials w –3dB (rad/s) s + 1 1.0000 s 2 + 3s + 3 1.3617 s 3 +6s 2 + 15s +15 1.7557 s 4 + 10s 3 + 45s 2 + 105s + 105 2.1139 s 5 + 15s 4 +105s 3 + 420s 2 + 945s + 945 2.4274 Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. © 2000 by CRC Press LLC (4.13) where A = decibels of ripple; then for a given order n, we define v. (4.14) 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 1 rad/s. The –3-dB frequency for the Chebyshev function is (4.15) The magnitude in the stopband (w > 1 rad/s) for the normalized filter is (4.16) for the singly terminated filter. For equal terminations the above magnitude is multiplied by one-half [1/4 in Eq. (4.16)]. Example 4.5.What order of singly terminated Chebyshev filter having 0.25-dB ripple (A) is required if the magnitude must be –60 dB at 15 kHz and the cut-off frequency (–0.25 dB) is to be 3 kHz? The normalized frequency for a magnitude of –60 dB is Thus, for a ripple of A = 0.25 dB, we have from Eq. (4.13) and solving Eq. (4.16) for n with w = 5 rad/s and ΈT(jw)Έ = –60 dB, we obtain n = 3.93. Therefore we must use n = 4 to meet these specifications. TABLE 4.3Chebyshev Polynomials Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. s ss sss ss ss ss s + +++ ++++ +++´+++ ++ ++ () ()() [] () [] () [] ()( ) sinh sinh sinh sinh sinh sinh . sinh sinh . . sinh sinh . sinh . sinh sinh . n nn nnn nn nn nnn 22 22 2222 22 212 34 075637 085355 184776 014645 061803 090451 [[] () [] ´+ + + . sinh sinh .ss 22 161803 034549nn ⑀= (–) / 10 1 10A v n = æ è ç ö ø ÷ 11 1 sinh – ⑀ w – cosh cosh(/) 3 1 1 dB = é ë ê ê ù û ú ú - ⑀ n ΈΈTj n () cosh(cosh ) – w w 2 22 1 1 1 = +⑀ 15 3 5 kHz kHz rad/s= ⑀==(–). / 10 1 02434 10A © 2000 by CRC Press LLC 4.3 Low-Pass Filters 1 Introduction Normalized element values are given here for both singly and doubly terminated filters. The source and load resistors are normalized to 1 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 = 2, 3, 4, or 5, Fig. 4.2 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 1 rad/s. 1 Adapted from Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited. FIGURE 4.2Singly terminated Butterworth filter element values (in W, H, F). (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) FIGURE 4.3Doubly terminated Butterworth filter element values (in W, H, F). (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) © 2000 by CRC Press LLC Thomson Filters Singly and doubly terminated Thomson filters of order n = 2, 3, 4, 5 are shown in Figs. 4.4 and 4.5. All time delays are 1 s. The cut-off frequencies are given in Table 4.2. Chebyshev Filters The amount of ripple can be specified as desired, so that only a selective sample can be given here. We will use 0.1 dB, 0.25 dB, and 0.5 dB. All cut-off frequencies (end of ripple for the Chebyshev function) are at 1 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 = 2, 3, 4, and 5 and Fig. 4.7 gives the doubly terminated Chebyshev filters for n = 3 and n = 5. 4.4 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 low- pass filter for a specific signal processing application. FIGURE 4.4Singly terminated Thomson filter element values (in W, H, F). (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) FIGURE 4.5Doubly terminated Thomson filter element values (in W, H, F). (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) © 2000 by CRC Press LLC Rule 1. All impedances may be multiplied by any constant without affecting the transfer voltage ratio. Rule 2. 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 2 kHz, that is, –42 dB (dc level is –6 dB). Since 2 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 FIGURE 4.6 Singly terminated Chebyshev filter element values (in W, H, F): (a) 0.1-dB ripple; (b) 0.25-dB ripple; (c) 0.50-dB ripple. (Source: Handbook of Measurement Science, edited by Peter Sydenham, copyright 1982, John Wiley and Sons Limited. Reproduced by permission of John Wiley and Sons Limited.) - + é ë ê ê ù û ú ú 42 1 41 2 dB = 20 log 1 2 n