EMC concepts explained
EM Waves, Voltage, and Current Waves
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his article presents a concept of a wave together with the wave equations and their solutions. The time-varying EM fields and their propagation in both the time and frequency domains is discussed first. Subsequently, the equations for the voltages and currents on the transmission line are obtained. It is shown that these equations and that their solutions represent voltage and current waves propagating along the line.

1. Concept of a Wave
Consider a function of time t and space z, , with its argument given by
equation
Then,
equation
Eq. (1.2) is valid for any Δz and any Δt. Thus, we could choose any relationship between the two deltas, and the new equation would still be valid. Let’s choose this relationship to be
equation
Then, Eq. (1.2) becomes
equation
or
equation
Therefore, after a time Δt, the function f retains the same value at a point that is Δz = vΔt away from the previous position in space (defined by z), as shown in Figure 1.

This means that any function of the form represents a wave traveling in the positive z direction with a velocity

equation
Similarly, it can be shown that any function of the form f (t ‒ z/v) represents a wave traveling in the negative z direction as the time advances.
2. Uniform Plane EM Wave in Time Domain
The time variations of the magnetic (H) and electric (E) fields give rise to the space variations of the electric and magnetic fields, respectively. This interdependence of the space and time variations gives rise to the electromagnetic wave propagation.
Graph of wave propagating in the positive z direction with a velocity v
Figure 1: Wave propagating in the positive z direction with a velocity v
The two fields are related by Maxwell’s equations (in source-free medium)
equation
equation
where µ, σ, and ε are the permeability, conductivity, and permeability of a medium, respectively.

In general, the electric and magnetic fields have three nonzero components, each of them being a function of all three coordinates and time. That is,

equation
equation
We will focus on a simple and very useful type of wave: uniform plane wave. Uniform plane waves not only serve as a building block in the study of electromagnetic waves but also support the study of wave propagation on transmission lines as we will show.

Under the uniformity in the plane assumption, if the E field points in the +x direction (usual designation) then the Maxwell’s equations show that the H field is pointing in the + y direction, and

equation
equation
This is shown in Figure 2.
Graph of uniform plane EM wave
Figure 2: Uniform plane EM wave
The fields propagate as waves in the positive + z direction. Under the uniformity in the plane assumption, Equations (2.1) for a lossless medium (σ = 0) become
equation
equation
and their general solution, in a lossless medium, is [1],
equation
equation
where
equation
is the intrinsic impedance of a (lossless) medium, and A and B are constants.

Based on the discussion in Section 1, we recognize the functions f and g, as waves propagating in +z and –z directions, respectively, with a velocity of propagation equal to

equation
3. Uniform Plane EM Wave in Frequency Domain
In the previous section, we described the wave equations in a lossless medium for arbitrary time variations. When the time variations are sinusoidal, the wave equations in any (simple) medium become [1]:
equation
equation
where
equation
is the propagation constant of the medium. The general solution of Equations (3.1) is
equation
equation
where
equation
is the intrinsic impedance of the medium. The propagation constant is often expressed in terms of its real and imaginary parts as
equation
where α is the attenuation constant and β = w/v is the phase constant. The complex intrinsic impedance is often expressed in an exponential form as
equation
Then the solution in Equations (3.3) can be written as
equation
equation
Often, the undetermined complex constants can be expressed as
equation
equation
Then, the solutions in Equations (3.7) become
equation
equation
The corresponding time-domain solutions, in a lossless medium, are [1]:
equation
equation
Note that
equation
equation
Thus, the Equations (3.10) represents sinusoidal traveling wave in the +z, and –z direction, respectively!

Figure 3 shows a forward propagating EM wave in a lossless medium.

The wavelength λ is related to the velocity of propagation and frequency by

equation
The phase constant β is related to λ by
equation
From Eq. (3.13) we obtain
equation
We refer to z as a physical length and to z/λ as the electrical length. The time domain solutions in Equations (3.10) can now be written in terms of the electrical length as
equation
equation
Graph of Sinusoidal EM wave in a lossless medium
Figure 3: Sinusoidal EM wave in a lossless medium
The definition of electrical length leads to the concept of the electrically short structures.

Physical object is electrically short if its electrical length z/l ≤ 1/10 or equivalently if its physical length z ≤ l /10.

If the physical object is electrically short, then the lumped-parameter circuit models are an adequate representation of that object. This also means that we can use Kirchhoff’s laws instead of Maxwell’s equations to analyze the circuit models.

4. Voltage and Current Waves along a Transmission Line
In this section, we show that the voltages and current signals propagate as waves along a transmission line.

To obtain the transmission line equations, let’s consider a single segment of a lossless transmission line shown in Figure 4.

The distributed parameters describing the transmission line are: l – inductance per-unit-length (H/m) and c – capacitance per-unit-length (F/m).

Writing Kirchhoff’s voltage law around the outside loop results in
equation
or
equation
Dividing both sides by Δz and taking the limit gives
equation
or
equation
Graph of Single segment of a lossless transmission line
Figure 4: Single segment of a lossless transmission line
Writing Kirchhoff’s current law at the upper node of the capacitor results in
equation
or
equation
Dividing both sides by Δz and taking the limit gives
equation
or
equation
Equations (4.4) and (4.8) constitute a set of first-order coupled transmission line equations. These equations can be decoupled and expressed as [1]:
equation
equation
Compare the Equations (4.9) to the Equations (2.5) describing the EM wave. These two sets of equations have the same mathematical form. This means that the solutions of these two sets will have the same mathematical form! This also means that the voltage and current propagate as waves along the transmission lines!
The general solutions to the transmission-line equations (4.9) are [1]:
equation
equation
ZC is the characteristic impedance of the line
equation
The function V +(t ‒ z/v) represents a forward-traveling voltage wave traveling in the +z direction, while the function V (t + z/v) represents a backward-traveling voltage wave traveling in the -z direction.

Similar statements are valid for the current waves. The total solution consists of the sum of forward-traveling and backward-traveling waves.

The velocity of the wave propagation along the line is given by

equation
References
  1. Bogdan Adamczyk, Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017.
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Dr. Bogdan Adamczyk headshot
Dr. Bogdan Adamczyk is professor and director of the EMC Center at Grand Valley State University (http://www.gvsu.edu/emccenter) where he regularly teaches EMC certificate courses for industry. He is an iNARTE certified EMC Master Design Engineer. Prof. Adamczyk is the author of the textbook “Foundations of Electromagnetic Compatibility with Practical Applications” (Wiley, 2017) and the upcoming textbook “Principles of Electromagnetic Compatibility with Laboratory Exercises” (Wiley 2022). He can be reached at adamczyb@gvsu.edu.