Kerwin, W.J."Passive Signal Processing The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Kerwin, W.J. “Passive Signal Processing” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
4 Passive Signal Processing aplace Transform. Transfer Functions 4.2 Low-Pass Filter Functions Thomson Functions. Chebyshev Functions 4.3 Low-Pass Filters Introduction. Butterworth Filters. Thomson Filters Chebyshe William J. Kerwin 4.4 Filter Design aling 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 impedanc 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 example Laplace Transform We will use the Laplace operator, s=0+ jo. Steady-state impedance is thus Ls and 1/Cs, respectively, for an inductor (L)and a capacitor(O), and admittance is 1/Ls and Cs. In steady state o =0 and therefore s= ja. Transfer functions We will consider only lumped, linear, constant, bilateral elements, and we will define the transfer function T() T(s) N(s) signal input D(s) 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. c 2000 by CRC Press LLC
© 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 + jw. 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 = jw. Transfer Functions We will consider only lumped, linear, constant, bilateral elements, and we will define the transfer function T(s) as response over excitation. T s N s D s ( ) ( ) ( ) = = 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
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= jo into T(s) and separate the result into real and imaginary parts(numerator and denominator)we obtain (o)=At (4.1) A2+ jB, Then the magnitude of the function, IrGo)I,is Bi A2+B2 and the phase T(o) is TGo)=ta (4.3) A A, 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.=1 Volt. 3=s;V1=1+()(s) 12=V(S=2s+s; I=I2+I3 V=V+,=s3 T(s)== 32 FIGURE 4.1 Singly terminated 3rd order low pass filter(Q H, F) e 2000 by CRC Press LLC
© 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 = jw 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(jw)Ô, 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 Vo = 1 Volt. FIGURE 4.1 Singly terminated 3rd order low pass filter (W, H, F). T j A jB A jB ( w = ) + + 1 1 2 2 *T j * A B A B ( w = ) + + Ê Ë Á ˆ ¯ ˜ 1 2 1 2 2 2 2 2 1 2 T(jw) T j B A B A ( ) tan – tan – – w = 1 1 1 1 2 2 I s V s s s I V s s s I I I V V I sss T s V V sss i o i 3 3 2 1 3 2 4 3 2 2 1 1 2 1 2 3 1 2 3 1 1 3 2 3 2 1 1 2 2 2 1 1 2 2 1 = = + ( ) ( ) = + = ( ) = + = + = + = + + + = = + + + ; ; ( ) Vi V1 I 1 I 2 I 3 I 3 Vo 1/ 2 4/ 3 3/ 2 1
Example 4.2 Determine the magnitude and phase of T(s)in Example 4.1 s3+2s2+2s+1= (s)= 20-]+(20-0 T(s =tan0-tan-120-03 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 dividin line between the passband and the stopband. In the above example we see that the magnitude of V/v,at o o(dc)is 1.00 and that at o= 1 rad/s we have TGo) =0.707 (44) (O°+1) 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 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 follow The squared magnitude TGo)P (4.6) '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. e 2000 by CRC Press LLC
© 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 Vo /Vi 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 Functions1 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. T s sss T s T s s j ( ) = + + + ( ) = ( - ) + - ( ) = + ( ) = - - - = - - - = - - - 1 2 2 1 1 1 2 2 1 1 0 2 1 2 2 1 2 3 2 2 2 3 2 6 1 1 3 2 1 3 2 w w w w w w w w w w w tan tan tan *T j * rad s ( ) ( ) w . w w = + = = 1 1 1 0 707 6 T s as bs cs ( ) = + + + 1 1 3 2 *T j * b c a ( ) ( – ) ( – ) w w w w 2 2 2 3 2 1 1 = +
T()P (4.7) a2o°+(b2-2ac)o04+(c2-2b)o02+ MFM requires that the coefficients of the numerator and the denominator match term by term (or be in the excep Therefore 2-2b=0;b2-2ac=0 (4.8) We will also impose a normalized cut-off(-3 dB)at o= 1 rad/s; that is =0.70 (49) 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) s3+2s2+2s+1 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 o=l rad/s for all n This may, of course, be multiplied by any constant less than one for circuits whose dc gain is deliberately Example 4.3. A low-pass Butterworth filter is required whose TABLE 4.1 Butterworth Polynomials 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 I rad/s, the-40-dB frequency is 12 kHZ s4+2.6131s3+3.414252+2.6131s+1 3 kHz Source: Handbook of Measurem edited by Peter Sydenham, copyrigh wiley and Sons Limited. Reproduced sion of John Wiley and Sons Limited. therefore n=3.32. Since n must be an integer, a fourth-order filter is required for this specification. e 2000 by CRC Press LLC
© 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 thirdorder 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. *T j * a b ac c b ( ) ( – ) ( – ) w w w w 2 2 6 2 4 2 2 1 2 2 1 = + + + c b b ac 2 2 – 2 0 = = ; – 2 0 *T j * a ( ) ( ) w . w = = + 1 = 2 1 1 0 707 T s s s s ( ) = + + + 1 2 2 1 3 2 *T j * n ( ) ( ) w w = + 1 1 2 TABLE 4.1 Butterworth 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 s s s s s s s s s s s s s s a + + + + + + + + + + + + + + + 1 2 1 2 2 1 2 6131 3 4142 2 6131 1 3 2361 5 2361 5 2361 3 2361 2 3 2 4 3 2 5 4 3 2 . . . . . . . 12 3 4 kHz kHz = rad/s – 40 20 log 1 4 1 2 = + n