# Matrix Solution to Multi-Branch Networks

## Abstract

For simple circuits we should be able to derive a direct relation between input and output as a function of frequency; this is we call the transfer function. For more complicated circuits, with multi-branch it becomes exceedingly difficult to find this direct relation. Instead we have to resort to matrix solution methods, where the network is assigned node voltages and branch currents, and KVL/KCL solved concurrently. This leads to a system of equations with a number of unknowns. The system, still in the frequency domain, can now be solved most often using matrix techniques, such as direct inversion or numerical techniques. The end result is each node voltage and each branch current figured in the frequency domain. From there we can go back to the time domain by simple inversion. We illustrate this flow on a few examples, ending with a system of size 3 × 3 which still can be inverted analytically. For larger systems it is most efficient to write a program or pass the matrix to a mathematical package to solve for it. Lastly the Problems section also sheds light on self and mutual impedance.