Circle diagrams of impedance or admittance for four-terminal networks

Abstract

The paper offers a theory of circle diagrams of impedance or admittance representing the relations between terminal impedances and input impedances for the general passive four-terminal network, including the cases of asymmetry and dissipation, so formulated that common transmission-line diagrams become special cases.After a review of earlier work on the subject, the relations between terminal and input impedances are expressed in terms of the iterative quantities of the network in a circle diagram. It is shown that this diagram is actually a graphical representation of the complex hyperbolic tangent function. Special circle diagrams are discussed corresponding to a non-dissipative network in the stop-band or pass-band, or at the cut-off frequency. These diagrams are studied by the application of projective geometry.An inversion of the ordinary circle diagram leads to the Smith diagram, which is described for both real and complex iterative impedances. It is pointed out that for a transmission line of less than ¼λ the error resulting from considering Z a pure resistance is of the same order of magnitude as that which would result from neglecting the damping.