On the realizability of non-rational positive real functions



The paper deals with the analysis and synthesis of passive reciprocal one-ports composed of an infinite number of conventional elements (positive R, L. C and ideal transformers), considered as equivalent circuits of physical distributed one-ports. In the generalization from finite to infinite networks, several (generally overlooked) basic difficulties arise, which are discussed and partially clarified. Physically, a prescribed positive real function z(p) is only specified in Rep > 0, and a lossless infinite realization always exists. Since the value of the function in Re p < 0 is then deduced by z(p) + z(–p) = 0, the resistance r(α, ω) = Re z(α + jω) is such that r(0, ω) = 0, but the limit of r(α, ω) for α = +0 may be strictly positive, so that a lossless impedance may have a resistive behaviour in steady-state. The classical Foster and Cauer synthesis procedures may consequently all fail for lossless non-rational impedances, whereas the procedures of Darlington and Bott-Duffin (and sometimes Brune) succeed. Since every point is a transmission zero for an odd function, a cascade synthesis with all zeros at p = 1 always works, and explicit expressions for the element values are obtained. Many examples are treated in detail, and their sometimes pathological behaviour in Re p < 0 is discussed.