Harmonic Balance in EM Fields

Harmonic balance can also be applied to EM field analysis, as the fields that contain the harmonics also satisfy Maxwell’s equations. The harmonics generated in EM fields can be described in the following three ways [10-12]:

  • • When a linear EM object is excited by sources which contain the harmonics, it will exhibit a harmonic field.
  • • When a nonlinear EM object is excited by a sinusoidal signal, it will exhibit harmonic fields.
  • • When both linear and nonlinear EM objects are excited by the sources which contain the harmonics, the result is a complex harmonic field.

One of the most obvious properties of a nonlinear system is the generation of harmonics. For example, we use the following equations to describe the quasi-static EM fields. These can be defined as follows:

A. Nonlinear Magnetic Field

B. Nonlinear Electric Field

where the electric field E, magnetic vector potential A, scalar potential ф on the arbitrary node i in the descretized system, and the source current density Js can be respectively expressed as:

where the vector A0, E0, J0 and scalar ф0 are the DC components, respectively. In practical applications, harmonic к is not infinite. Only a finite number is required in the real system.

The above harmonic approximation can also be expressed by using complex Fourier series with к harmonics:

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