Numeric Low-Pass Solution for Arbitrary Gain

Figure 3 shows a low-pass filter with selectable gain. The transfer function for this filter is given in Equation (1) where

Figure 3: Low-pass filter with selectable gain.

Backward-solving and substitution of these equations does not result in a simple cubic.

Let M= ? ²+ ? ²

This expression must be solved numerically for C ₁ :

Once C ₁ is found, the other capacitors are

A list of capacitor values for various gain and filter characteristics is given in Tables 1 4 in the appendix.

Table 1: Capacitor values for a Butterworth low-pass filter.
Linear Gain K ₀	C1	C2	C3	R5 (R4=1 ?)
1.5	1.59448	0.529798	1.18383	0.5
2	1.70585	0.676115	0.867076	1
4	2.00002	1.00002	0.500006	3
6	2.20464	1.19602	0.379266	5
8	2.36801	1.34386	0.314254	7
10	2.50608	1.46530	0.272330	9
12	2.62667	1.56975	0.242540	11
14	2.73432	1.66219	0.220033	13
16	2.83194	1.74564	0.202293	15
18	2.92152	1.82202	0.187869	17
20	3.00450	1.89272	0.175858	19

Table 2: Capacitor values for a Chebyshev 0.01 dB low-pass filter.
Linear Gain K ₀	C1	C2	C3	R5 (R4=1 ?)
1.5	1.80347	0.534989	1.31672	0.5
2	1.92172	0.692807	0.954213	1
4	2.22923	1.04493	0.545386	3
6	2.44186	1.25983	0.412965	5
8	2.61181	1.42244	0.341958	7
10	2.75576	1.55614	0.296248	9
12	2.88180	1.67115	0.263796	11
14	2.99457	1.77291	0.239291	13
16	3.09703	1.86472	0.219982	15
18	3.19123	1.94873	0.204285	17
20	3.27863	2.02643	0.191216	19

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Practical Filters and Couplers: A Collection from Applied Microwave & Wireless