On the realizability of non-rational positive real functions

✍ Scribed by V. Belevitch

• Publisher: John Wiley and Sons
• Year: 1973
• Tongue: English
• Weight: 863 KB
• Volume: 1
• Category: Article
• ISSN: 0098-9886
• DOI: 10.1002/cta.4490010105

✦ Synopsis

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 oneports. 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 R e p > 0, and a lossless infinite realization always exists. Since the value of the function in R e p < 0 is then deduced by z(p)+z( -p ) = 0, the resistance r(a, w ) = Re z(a+jw) is such that r(0, w ) = 0, but the limit of I(a, w ) for a = +O 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 R e p < 0 is discussed.