1.7 Complex EM Properties

For many geophysical applications we neglect the permittivity (e or K_e) term and assume the magnetic permeability is that of free space, i.e., K_m = 1. This cannot be done for ground penetrating radar, GPR. In this case we must treat the total current density, J_T as

Here, J_C is the same current density, J defined earlier, however, in the context here it is usually referred to as the conduction current density to distinguish it from the displacement current density, ∂D/∂t.

By assuming that the time dependency of all fields is e^+iwt we see that

JT = JC + iwD	(1.7.2)
= (s + iwe)E.	(1.7.3)

(Using the exponential, e^+iwt to express a sinusoidal variation with time is discussed in Appendix A). Equation 1.7.3 allows a generalized complex expression of Ohm's law (equation 1.5.1). This uses the definition of complex conductivity, s* as

and yields

where s' = s and s'' = we are the real and imaginary parts of the complex conductivity, s* respectively. A convenient way to express the relative amount of conduction current versus displacement current is the loss tangent, tan d.

tan d = s / we = conduction current density displacement current density

(1.7.6)

Another way to express J_T is to modify the J_T equation 1.7.3 above as

This leads to the definition of complex electric permittivity, e* (von Hippel, 1954) as

where e' and e'' are the real and imaginary parts of the complex permittivity, e*, respectively. This allows the alternate expression of the loss tangent to be

The complex dielectric constant, K* follows as

Davis and Annan (1989) prefer to separate the imaginary part of the dielectric constant into two parts: 1) a dc component of conductivity s_dc and 2) a high frequency contribution, K''. Then

The relations above can be represented in the complex plane. Figure 1.9 a, b, and c contain such examples.

a)	b)	c)

Appendix A

Exponential Notation of Time

The widespread use of complex notation in electromagnetics (and in other branches of physics) and Euler’s formula leads to an effective shorthand for expressing sinusoidal variations of time. Using Euler’s formula we can write

which leads to

and

where Im and Re refer to the imaginary and real parts, respectively.

These “sinusoidal” time variations are often simply expressed by saying that the time dependency is e^+iwt (or e^-iwt ). It does not mean that the time dependency is complex. The resulting expression can be returned to a real one by extracting the real or imaginary part in the end. This is acceptable as long as various manipulations involve only addition, subtraction, differentiation, or integration where the real and imaginary parts do not mix. However, usually the final expression is simply left in exponential form.

Using the exponential form is preferred because the algebra of complex exponentials is easier to work with compared to sines and cosines. For example, the differentiation operation d/dt is just a multiplication by +iwt. That is,

To illustrate the use of complex exponential notation, again consider the earlier equation 1.2.6, i.e.,

with

Equation A.5 becomes

after the integration

Now, upon taking the imaginary part of the expression for V after inserting

we have

which is the identical to the expression we derived earlier (equation 1.2.10) using
I = I_o sin wt.