Note on a logarithmic approximation for use with singularity plots

THE ANALYSIS or synthesis of bandpass networks using plots of the singularities of their transfer functions 1‾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‾6 It does not have the power of certain conformai transformations7, 8 but is considerably simpler. Unlike the wide-band low-pass to bandpass transformation9 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, as will be shown.