Inhomogeneous ladder networks, part I

Abstract

SUMMARYIn this paper the theory of orthogonal polynomials is taken as the starting point for developing the theory of semi-infinite inhomogeneous ladder networks of two element kinds. It is shown that the transfer impedance of the ladder network, suitably normalized, is just the Green's function of a self-adjoint difference operator of 'Sturm-Liouville' type. An integral representation for this Green's function is given, and is evaluated in product form. The theory is illustrated with a number of examples based on classical orthogonal polynomials, the use of which enables all the calculations to be carried through explicitly.

Applications of the discrete green's function concept in circuit theory

“…When the findings of the previous section are applied to equations (3) and (4) used in conjunction with (18), then denoting 1 We find that subject to…”

“…When the findings of the previous section are applied to equations (3) and (4) used in conjunction with (18), then denoting 1 We find that subject to The foregoing procedure is applicable to all networks which may be represented as a number of readily identifiable sub-networks in cascade and filters, equalizers, delay lines and coupling networks come to mind. It becomes particularly advantageous if it is necessary to optimize some network characteristics and repeated calculations are required or if the number of sub-networks N is large or both.…”

“…, N + 1 and not inside any network A,). Once the current has been determined then, with reference to (1) and (2), the voltage V, may be deduced with the aid of and/or If the given chain of N networks is not short-circuited at both ends, open-circuit terminal conditions yijl, = y i j J N + = 0 (i, j = 1,2) are appropriate and equation (4) yields the voltage Vn (11 = 1,. . .…”