Design a six-pole lowpass Chebyshev filter having 0.5 dB passband ripple, bandwidth f0=10kHz, and minimum input impedance (magnitude) of at least 100kΩ. Obtain the voltage transfer function for the filter and plot the gain, phase shift, and group delay versus frequency, for 0.1f0≤f≤10f0.

Question

Design a six-pole lowpass Chebyshev filter having 0.5 dB passband ripple, bandwidth [latex]f_0=10 kHz [/latex], and minimum input impedance (magnitude) of at least [latex] 100 k \Omega [/latex]. Obtain the voltage transfer function for the filter and plot the gain, phase shift, and group delay versus frequency, for [latex] 0.1 f_0 \leq f \leq 10 f_0 [/latex].

Accepted Answer

From Table 19.1, the values of the scale factor [latex] \lambda[/latex] for the three sections are 0.396 , 0.768 , and 1.011. This means, for example, that for the first section [latex] f_{R C}=0.396 f_0 [/latex], where [latex] f_0 [/latex] is the specified passband edge. Thus the R C products for the three sections are:

Table 19.1 Design parameters for VCVS filters: N is the number of poles and [latex] \lambda [/latex] is the scaling factor for Bessel and Chebyshev sections [latex] ^6 [/latex]
N	Butterworth	Bessel		Chebyshev
	k	k	λ	k	λ
2	0.586	0.268	-1.738	0.842	1.231
4	0.152	0.084	1.432	0.582	0.597
	1.235	0.759	1.606	1.66	1.031
6	0.068	0.040	1.607	0.537	0.396
	0.586	0.364	1.692	1.448	0.768
	1.483	1.023	1.908	1.846	1.011
8	0.038	0.024	1.781	0.522	0.297
	0.337	0.213	1.835	1.379	0.599
	0.889	0.593	1.956	1.711	0.861
	1.610	1.184	2.192	1.913	1.006

[latex] \begin{aligned} (R C)_1 &=\frac{1}{2 \pi f_{R C 1}}=\frac{1}{2 \pi \lambda_1 f_0}=\frac{1}{2 \pi(0.396)(10 kHz )} \\ &=40.19 \mu s \\ (R C)_2 &=\frac{1}{2 \pi f_{R C 2}}=\frac{1}{2 \pi \lambda_2 f_0}=\frac{1}{2 \pi(0.768)(10 kHz )} \\ &=20.72 \mu s \\ (R C)_3 &=\frac{1}{2 \pi f_{R C 3}}=\frac{1}{2 \pi \lambda_3 f_0}=\frac{1}{2 \pi(1.011)(10 kHz )} \\ &=15.74 \mu s \end{aligned} [/latex]

We choose [latex] R=100 k \Omega [/latex] for all sections, for which value the input impedance (of the first section) will meet or exceed the specification [latex] \left|Z_{i n} ight| \geq 100 k \Omega [/latex]. The required values for the capacitances are then

[latex] \begin{gathered} C_1=\frac{R C_1}{R}=\frac{40.19 \mu s }{100 k \Omega}=401.906 pF \\ C_2=\frac{R C_2}{R}=\frac{20.72 \mu s }{100 k \Omega}=207.233 pF \\ C_3=\frac{R C_3}{R}=\frac{15.74 \mu s }{100 k \Omega}=157.423 pF \end{gathered} [/latex]

The filter is a cascade of three sections like that in Fig. 19.4(a), with [latex] R=100 k \Omega[/latex] and the capacitance values given above, and with k=0.537 for the first, k=1.448 for the second, and k=1.846 for the third. Again, VCVS sections should generally be cascaded in the order implied by Table 19.1. Figure 19.16 shows graphs of voltage gain, phase shift, and group delay versus frequency and step response versus time for the filter. The voltage gain and the step response are normalized to the dc gain, given by

[latex] \begin{aligned} H_ν(0) &=\left(1+k_1 ight)\left(1+k_2 ight)\left(1+k_3 ight)=10.71 \\ & \Rightarrow A_ν(0)=20.59 dB \end{aligned} [/latex]

The 0.5 dB passband ripple is not visible on the scale used for gain in Fig. 19.16.

Figure 19.18 shows the gain in the passband on a scale that makes the ripple visible.

[latex] { }^6 ext { Adapted from Table } 5.2 ext { in Horowitz and Hill (ibid). } [/latex]

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