Note on a Logarithmic Approximation for Use with Singularity Plots

The analysis or synthesis of band-pass networks using plots of the singularities
\dagger
of their transfer functions[1].[2],[3] can often be shortened and simplified by using a suitable logarithmic transformation of the complex-frequency
s
-plane which permits a simple approximate calculation of the transfer function. This approximation is good for over-all-bandwidth ratios of 2 to 1 or less as compared with a usable bandwidth ratio of 1.2 to 1 for the conventional narrow-band approximation. This transformation is not intended for use with an electrolytic tank, for which better methods have been described in the literature[4],[5],[6] It does not have the power of certain conformal transformations,[8],[9] but is considerably simpler. Unlike the wide-band low-pass to band-pass transformations it is not limited to pole and zero patterns of particular symmetry. This approximation also has interesting properties as an aid in the factoring of network polynomials.