A Graphical Contribution to the Analysis and Synthesis of Electrical Networks

The contribution of the poles and zeros of a network function to the steady-state characteristics at real frequencies is considered by means of vectors, and the connection between attenuation and phase is thereby accentuated. Bode´s relation between attenuation and phase in minimum-phase networks is derived by considering the contributions from individual singularities and using a known definite integral in a real variable. Details are derived for the construction of a series of templates, useful for finding pole-zero positions which yield certain attenuation, phase and delay characteristics, and vice versa. These are attenuation and phase templates for individual singularities on a linear frequency scale, and delay and phase templates on a logarithmic frequency scale. Apart from uses in the usual analysis and synthesis problems, the templates provide a particularly easy means, for simple networks, of obtaining the phase and delay characteristics corresponding to a given attenuation characteristic of a minimum-phase network, and vice versa, and of obtaining the transient response corresponding to given steady-state characteristics. The paper concludes with illustrations of the procedure for estimating the phase-angle characteristic from a given impedance-magnitude characteristic of a driving-point impedance; for estimating the phase and delay characteristics corresponding to the given attenuation characteristic of a minimum-phase wide-band amplifier; for estimating the response of the amplifier to a step-function drive; and for the design of a first-degree delay equalizer.