# 7.1 Time-Harmonic Fields

Time-varying electric and magnetic fields (E, D, B, and H) and their sources (the charge density ρ v and current density J) generally depend on the spatial coordinates (x,y,z) and the time variable t. However, if their time variation is sinusoidal with angular frequency ω, then these quantities can be represented by a phasor that depends on (x,y,z) only. The vector phasor E˜(x,y,z) and the instantaneous field E(x,y,z;t) it describes are related as

E(x,y,z;t)=R e E˜(x,y,z)e j ω t.(7.1)

Similar definitions apply to D, B, and H, as well as to ρ v and J. For a linear, isotropic, and homogeneous medium with electrical permittivity ε, magnetic permeability μ, and conductivity σ, Maxwell’s equations (6.1) to (6.4) assume the following form in the phasor domain.

To derive these equations, we used D=ε E and B=μ H as well as the fact that, for time-harmonic quantities, differentiation in the time domain corresponds to multiplication by j ω in the phasor domain. These equations are the starting point for the subject matter treated in this chapter.

# 7.1.1 Complex Permittivity

In a medium with conductivity σ, the conduction current density J˜ is related to E˜ by J˜=σ E˜. Assuming no other current flows in the medium, Eq. (7.2d) may be written as

∇×H˜=J˜+j ω ε E˜=(σ+j ω ε)E˜=j ω ε−j σ ω E˜.(7.3)

By defining the complex permittivity ε c as

Eq. (7.3) can be rewritten as

∇×H˜=j ω ε c E˜.(7.5)

In a source-free medium, ρ v=0. Hence, Maxwell’s equations become

The complex permittivity ε c given by Eq. (7.4) is often written in terms of a real part ε′ and an imaginary part ε′′. Thus,

ε c=ε−j σ ω=ε′−j ε′′,(7.7)

with

ε′=ε,(7.8a)
ε′′=σ ω.(7.8b)

For a lossless medium with σ=0, it follows that ε′′=0 and ε c=ε′=ε.

# 7.1.2 Wave Equations

Next, we derive wave equations for E˜ and H˜ and then solve them to obtain explicit expressions for E˜ and H˜ as a function of the spatial variables (x,y,z). To that end, we start by taking the curl of both sides of Eq. (7.6b) to get

∇×(∇×E˜)=−j ω μ(∇×H˜).(7.9)

Upon substituting Eq. (7.6d) into Eq. (7.9), we obtain

∇×(∇×E˜)=−j ω μ(j ω ε c E˜)=ω 2 μ ε c E˜.(7.10)

From Eq. (3.113), we know that the curl of the curl of E˜ is

∇×(∇×E˜)=∇(∇⋅E˜)−∇2 E˜,(7.11)

where ∇2 E˜ is the Laplacian of E˜, which in Cartesian coordinates is given by

∇2 E˜=∂2∂x 2+∂2∂y 2+∂2∂z 2 E˜.(7.12)

In view of Eq. (7.6a), the use of Eq. (7.11) in Eq. (7.10) gives

∇2 E˜+ω 2 μ ε c E˜=0,(7.13)

which is known as the homogeneous wave equation for E˜. By defining the propagation constant γ as

γ 2=−ω 2 μ ε c,(7.14)

Eq. (7.13) can be written as

To derive Eq. (7.15), we took the curl of both sides of Eq. (7.6b) and then we used Eq. (7.6d) to eliminate H˜ and obtain an equation in E˜ only. If we reverse the process, that is, if we start by taking the curl of both sides of Eq. (7.6d) and then use Eq. (7.6b) to eliminate E˜, we obtain a wave equation for H˜:

Since the wave equations for E˜ and H˜ are of the same form, so are their solutions.