Replaceability of Circuit Elements and a New Derived Filter†

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Abstract
An identity for a quadratic rational function is here applied to lumped-element linear electric circuits, where expressions involving such functions abound. These include the absolute values of impedances, expressions for the phase angle, natural frequencies of free oscillations, resonance and anti-resonance frequencies and characteristic impedances of wave filters. The arguments of these functions are the elements R, L, C and M, and the result of applying the identity to such functions is that one may replace these elements by other elements of the same kind but having different magnitudes, while the relevant functions remain invariant. For instance, positive elements may replace, in a two-terminal network, fictitious elements with negative, zero or infinite magnitudes, while the absolute value of the impedance remains invariant. In anti-resonant circuits, elements may be replaced by others, thus changing the impedance at anti-resonance, while the anti-resonance frequency is unchanged. In this paper, the basic series and parallel circuits are treated in detail, and several other applications are given. Salient among these is the derivation of a new type of filter section, termed the σ-derived section, which fills an existing gap in image-parameter filter theory. This section is shown to be complementary to the m-derived section in composite band-pass filters, where the joint use of sections of both types is necessary for optimum operation.

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