# The Basic Concept of Harmonic Balance in EM Fields

When linear EM field systems are excited by a sinusoid, the steady-state response is sinusoidal at the same frequency as the input. While nonlinear EM field systems can exhibit a variety of significant and bizarre behaviors, the systems of interest to designers generally have a periodic steady-state response to a sinusoidal input; the response period is usually equal to that of the input. Since the response is periodic, it is represented as a Fourier series - that is, as a linear combination of sinusoids, whose periods divide the period of the response. If the excited EM field system contains two or more sinusoids that are harmonically unrelated, the system responds in steady-state, with components having the sum and difference frequencies of the input sinusoids and their harmonics. If the response contains an infinite number of sinusoids, then usually all but a few are negligible.

## Definition of Harmonic Balance

As mentioned in section 3.1.1, the waveform can be defined as a summation of trigonometric function, as follows:

where *m* = 2n/T, *P _{ks},P_{kc}&R* are harmonic coefficients, and the pair

*P*= [P

_{k}_{ks},

*P*are the Fourier coefficients of the Fourier series.

_{kc}]& CThe term “harmonic balance” in nonlinear circuit analysis is defined as the set of port voltage waveforms that give the same currents in both linear and nonlinear subcircuits; when that set is found, it must be the solution. This principle actually gave the harmonic balance simulation its name because, through the interconnections, the currents of the linear and non-linear subcircuits have to be balanced at every harmonic frequency [1-5].