Time Harmonic Fields :

So far, in discussing time varying electromagnetic fields, we have considered arbitrary time dependence. The time dependence of the field quantities depends on the source functions. One of the most important case of time varying electromagnetic field is the time harmonic (sinusoidal or co sinusoidal) time variation where the excitation of the source varies sinusoidally in time with a single frequency. For time-harmonic fields, phasor analysis can be applied to obtain single frequency steady state response. Since Maxwell's equations are linear differential equations, for source functions with arbitrary time dependence, electromagnetic fields can be determined by superposition. Periodic time functions can be expanded into Fourier series of harmonic sinusoidal components while transient non-periodic functions can be expressed as Fourier integrals. Field vectors that vary with space coordinates and are sinusoidal function of time can be represented in terms of vector phasors that depend on the space coordinates but not on time. For time harmonic case, the general time variation is and for a cosine reference, the instantaneous fields can be written as:

                            (5.42)

where is a vector phasor that contain the information on direction, magnitude and phase. The phasors in general are complex quantities. All time harmonic filed components can be written in this manner.