Equivalent Ladder Networks by the Use of Signal Flow Graphs

Signal flow graphs of ladder networks have properties that make them convenient for determining impedances and transfer ratios. Because of the symmetry of these flow graphs it is possible to recognize equivalent flow graphs, and hence equivalent networks, with respect to some desired characteristic. In particular, the output-input voltage ratio is the characteristic that is used as the basis for equivalence. Evaluation of elements in the equivalent circuits results from relating coefficients in the transfer voltage ratio to the element values. Using 4-branch ladder networks, examples are given of distributing resistance, determining when networks must use active elements for certain transfer functions, and finding the number of equivalences that exist. A particular equivalence is derived between a bridged-T and ladder, and between a lattice and ladder. In each case, three of the branches of the ladder are the same as in the bridged-T or lattice, while the fourth branch of the ladder is a function of all the impedances. These equivalences are derived by recognizing the flow graph configuration for a ladder within the flow graph of each of the other circuits and then reducing these flow graphs to the one for the ladder.