Theorems regarding the driving-point impedance of two-mesh circuits

The necessary and sufficient conditions that a driving-point impedance be realizable by means of a two-mesh circuit consisting of resistances, capacities, and inductances are stated in terms of the four roots and four poles (including the poles at zero and infinity) of the impedance. The roots and the poles are the time coefficients for the free oscillations of the circuit with the driving branch closed and opened, respectively. For assigned values of the roots, the poles are restricted to a certain domain, which is illustrated by figures for several typical cases; the case of real poles which are not continuously transformable into complex poles is of special interest. All driving-point impedances satisfying the general conditions can be realized by any one of eleven networks, each consisting of two resistances, two capacities, and two self-inductances with mutual inductance between them; these are the only networks without superfluous elements by which the entire range of possible impedances can be realized; the three remaining networks of this type give special cases only. For each of these eleven networks, formulas are given for the calculation of the values of the elements from the assigned values of the roots and poles.