AC Electromagnetics


A special case in linear electromagnetics occurs when the field is excited by an AC source. In fact this sort of analysis is useful for periodic fields, through resolution to frequency components by Fourier analysis.

Working from Maxwell's equations for linear, isotropic non-dispersive materials :

H
t
 = - 1
m
 Ñ×E
(1)

E
t
 = - 1
e
 Ñ×H- s
e
 E
(2)

Ñ. H = 0
(3)

Ñ. E = r
e
(4)

All the quantities are assumed to be sinusoidal. Every quantity can be expressed in terms of its magnitude and relative phase. A complex or phasor quantity is used to represent each quantity in this way.

For example a quantity A, which may be a function of space and time, is written as A¢ei wt. A is now a complex quantity that is not now a function of time but represents a sinusoidal variation with time, it signifies a magnitude (its modulus) and phase delay (its argument).

Substituting the sinusoidal terms into the equations above results in the following

j wH¢ = - 1
m
 Ñ×E¢
(5)

j wE¢ = - 1
e
 Ñ×H¢- s
e
 E
(6)

Ñ. H¢ = 0
(7)

Ñ. E¢ = r
e
(8)

Further simplifications can be made by regarding equations (5) and (6) as simultaneous equations. Taking the curl of both sides of equation (5) gives

j wÑ×H¢ = - 1
m
 Ñ×Ñ×E¢ .
(9)
Substituting Ñ×E¢ from (6) gives

j w( j weE¢ + sE¢ ) = 1
m
 Ñ×Ñ×E¢
(10)
or

smE¢- w2 emE¢ = Ñ×Ñ×E¢  .
(11)

Using the general identity Ñ×Ñ×A = Ñ(Ñ.A) - Ñ2 A,

Ñ×Ñ×E¢ = Ñ(Ñ.E¢) - Ñ2 E¢ .
[Note that Ñ2 A = Ñ(Ñ. A).]

Substituting the result of equation (8) gives

Ñ×Ñ×E¢ = Ñ r
e
 - Ñ2 E¢
which, when substituted into equation (11) gives the following

smE¢- w2 emE¢ = Ñ r
e
 - Ñ2 E¢  .
(12)

A similar development, working from equation (6) gives

-w2 meH¢ = Ñ2 H¢+ j wmsH¢
(13)

Source-free Region

In a source-free region, equations (12) and (13) become

Ñ2 E + w2 emE¢ = 0 
(14)
and

Ñ2 H¢+w2 meH¢ = 0 .
(15)
By putting k = wÖ{em} the equation become

Ñ2 E + k2 E¢ = 0 ,
(16)
and

Ñ2 H¢+k2 H¢ = 0 ;
(17)
the components of E and H all satisfy the Helmholtz equation.

Numerical Solution

Clearly the key to finding sinusoidal (or periodic) solutions to electromagnetic problems is through the solution of the Helmholtz equation. The Helmholtz equation can be solved analytically only for simple domains. In general it is necessary to find numerical solution.

The finite element method (FEM ), and finite difference method (FDM) are traditional methods of solution. In these methods the a mesh of the electromagnetic domain is required. The boundary element method (BEM) is a very powerful method of solution, particularly with exterior problems since a mesh of the bounding surfaces of the domain alone is required.