Matrix Methods in the Solution of Ladder Networks

The matrix algebra has a natural application to electric circuit theory, especially to the method of solution by means of substituted variables, such as symmetrical components. The solution of circuit equations by the method of diagonalizing the impedance matrix is developed in the paper, and applied to the solution of the difference equations of ladder networks. An extension to systems of difference equations follows. Diagonalizing is equivalent to the use of substituted currents, voltages and impedances such that there is no mutual coupling between the substituted networks. The substituted currents are then calculated without difficulty and the original circuit currents obtained by a simple transformation. The method has the additional advantage that, since the transformation is independent of the circuit constants, it may be applied to all circuits possessing the same degree of symmetry. Ladder networks may also be solved by regarding them as a series of four-terminal passive networks. The elements of matrix algebra are included for completeness.