FUNDAMENTALS OF

FIELDS AND WAVES

All electrical systems, at the most fundamental level, obey

Maxwell's equations and the postulates of electromagnetics.

Under certain circumstances, approximations can be made that

allow simpler methods of analysis, such as circuit theory, to be

employed. However, the problems associated with EMC usually

involve departures from these approximations. Therefore, a

review of fundamental concepts is the logical starting point for a

proper understanding of electromagnetic compatibility.

Fundamental quantities and electrical dimensions

speed of light, permittivity and permeability in free space

The speed of light in free space has been measured through very precise experiments,

and extremely accurate values are known (299,792,458 m/s). For most purposes,

however, the approximation

is sufficiently accurate. In the International System of units (SI units), the constant

known as the permeability of free space is defined to be

𝜇0 = 4 × 10−7 𝐻/𝑚.

The constant permittivity of free space is then derived through the relationship

and is usually expressed in SI units as

Both the permittivity and permeability of free space have been repeatedly verified through

experiment.

wavelength in lossless media

Wavelength is defined as the distance between adjacent equiphase

points on a wave. For an electromagnetic wave propagating in a

lossless medium, this is given by

The free space wavelengths associated with waves of various

frequencies are shown below:

relationship between physical and electrical dimensions

The size of an electrical circuit or circuit component, as compared to a

wavelength, determines, to a certain extent, the manner in which it interacts with

EM fields. For instance, in order for an antenna to effectively receive and transmit

signals at a certain frequency, it must be a significant fraction of a wavelength long

at that frequency.

Likewise, other types of electrical components may emit or receive interference-

causing signals if they are large compared to a wavelength. In addition, the

electrical characteristics of a circuit component are often very different when it is

electrically large (i.e. the frequency of operation is high) than when it is

electrically small (the frequency of operation is low).

Kirchoff's voltage and current laws are only valid if the circuit elements under

consideration are small compared to a wavelength. If the components under

consideration are electrically large, then Maxwell's equations must be applied in

order to analyze device behavior.

The electrical dimensions of a device or circuit are determined by comparing

physical dimensions to wavelength. A device with length l has electrical

dimensions (in wavelengths)

The electrical dimensions of a circuit are determined by first calculating the

wavelength at the highest frequency of interest, and then determining de.

Devices or circuits are considered to be electrically small if the largest

dimension is much smaller than a wavelength (kde«1).

Typically structures which are less than one tenth of a wavelength long are

considered to be electrically small. It must be remembered that the electrical

dimensions are dependent upon the material in which EM waves propagate.

A device may be electrically much larger when it is embedded in a printed

circuit board than it is when surrounded by air. Likewise, a capacitor which

contains a high permittivity dielectric is electrically larger than a similar

capacitor whose plates are separated by air.

Voltage gain and current gain are also sometimes

represented in terms of decibels. If the input and output

powers of an amplifier are dissipated by two equal

resistances then the power gain in decibels is

It is important to note that decibels represent the ratio of two

quantities, or more precisely, the value of one quantity as

referenced to some base quantity. The units dB describe

power in watts referenced to 1 watt

while the units dBm describe power in watts referenced to 1 milliwatt

In each of these cases, a positive value of dB or dBm indicates that the

power in watts is greater than the reference quantity, while a negative

value of dB or dBm indicates that the power in watts is less than the

reference quantity. Other quantities are sometimes expressed in

decibels, including

BASIC CONCEPTS AND RELATIONS OF ELECTROMAGNETIC FIELDS

AND WAVES ARE MAXWELL’S EQUATIONS

Maxwell`s equations

The constitutive relations for a linear and isotropic medium are given as follows;

Classifications of Media

Classification of a medium is governed by the constitutive relationships and the

parameters appropriate for the medium. The commonly used definitions for a medium

are discussed as follows:

1. If D, B , J vary linearly with E, H, E ,respectively, then 𝜀, 𝜇 𝑎𝑛𝑑 𝜎 are independent

of the field amplitudes. Under these conditions the medium is called linear otherwise it

is nonlinear.

2. If 𝜀, 𝜇 𝑎𝑛𝑑 𝜎 do not depend on the spatial coordinates, the medium is homogeneous;

otherwise, it is inhomogeneous.

3. If D is parallel to E, B is parallel to H, and J is parallel to E, then the medium is

isotropic; otherwise, it is anisotropic.

A medium can be linear, isotropic, and homogeneous, and its characteristic

𝜀, 𝜇 𝑎𝑛𝑑 𝜎 are constant in time and space. Such a medium is called a simple medium

when it is lossless, meaning when it has 𝜎 = 0. An example of a simple medium is the

free space.

We now classify linear, isotropic, and homogeneous media in the following

manner:

1 . Perfect Conductor: It has , 𝜀 = 𝜀0, 𝜇 = 𝜇0 𝑎𝑛𝑑 𝜎 = ∞. Perfect

conduction is not realistic, but the assumption of infinite conductivity

simplifies the theoretical analysis.

2. Conductor: If the conductivity 𝜎 is large but not infinite, then the material

is referred to as a conductor (i.e., most metals). It should be noted that under

steady state conditions, a conducting medium (including the case 𝜎 = ∞)

cannot sustain a free volume density of charge; in fact, for the general media

under consideration, the volume density of charge is zero if 𝜎 > 0.

3. Dielectric Medium: Materials with 𝜀0 > 1, 𝜇 = 𝜇0 𝑎𝑛𝑑 𝜎 = 0 are called

perfect dielectric, and materials with small values of 𝜎 are called lossy

dielectric media.

4. Magnetic Medium: Materials for which , 𝜇 ≅ 𝜇0are called nonmagnetic;

otherwise, they are called magnetic. The medium is lossless if

𝜎 = 0 and lossy if 𝜎 > 0.

Energy Flow and Poynting’s Theorem

The Poynting theorem provides a representation for energy

flow in a time-varying electromagnetic field.

HARMONICALLY OSCILLATING FIELDS

THE WAVE EQUATIONUsing his original algebraic equations in free space, Maxwell showed

that the two transverse components of time dependent electric and

magnetic fields together form an electromagnetic wave propagating in

the longitudinal (or z - ) direction at the velocity of light in free space. In

this section we describe how the wave equation arises from the modern

vector form of Maxwell’s equations given earlier.

The one-dimensional version of the scalar

wave, equations as

The general solution is

Time Harmonic Case

The complete expression for the forward traveling wave

can now be written as

The real time dependent form for the forward traveling (+z direction) wave

General Representation of TEM Waves

In many situations we need to consider a uniform plane or TEM wave propagating

in an arbitrary direction in a homogeneous, lossless, and isotropic medium.

Example 3.3

Determine the fields and power flow for a Ƹ𝑧 polarized TEM wave

propagating in the y-direction.

References

1. Cheng, D., Field and Wave Electromagnetics, Addison-Wesley Publishing Company, second

edition, 1989.

2. Ramo, Whinnery, and Van Duzer, Fields and Waves in Communication Electronics, John

Wiley & Sons, second edition, 1984.

3.

Paul, C., Introduction to Electromagnetic Compatibility, John Wiley & Sons, 1992.

4.

Irwin, J. D., Basic Engineering Circuit Analysis, Macmillan Publishing Company, third

edition, 1989.

5. D. Nyquist and E. Rothwell, class notes for EE 305, EE 306, and EE 435, Michigan State

University, Dept. of Electrical Engineering.

6. https://www.egr.msu.edu/emrg/electromagnetic-compatibility-emc-course-notes