Oscar

CHAPTER

7
Review of Basic Laws of Electromagnetism

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An exploded view of a power transformer, highlighting its magnetic core and cast windings. (Courtesy of Square D Company)
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Review of Basic Laws of Electromagnetism

2.1

Introduction
We can best understand and predict the behavior and characteristics of an electric machine if we understand not only its physicalconstruction but also the role of the magnetic field in that machine because almost all practical energy conversion devices use magnetic field as a medium. The magnetic field may be set up by a winding (coil) or a permanent magnet. If the magnetic field is produced by a winding, it can be either of constant magnitude (dc) or a function of time (ac). In transformers, the ac magnetic field helps totransfer energy from the primary (input) side to the secondary (output) side. No electrical connection is necessary between the two sides of the transformer. The energy transfer process is based upon the principle of induction. However, in dc machines and synchronous machines it is the uniform (constant) magnetic field that facilitates the conversion of electric energy to mechanical energy (motoraction) or mechanical energy to electrical energy (generator action). In fact, we would discover that both motor and generator actions exist simultaneously in every electromechanical energy conversion device. In other words, one action cannot exist without the other. It is therefore evident that the study of electric machines requires a basic understanding of electromagnetic fields. The purpose of thischapter is to review the basic laws of electromagnetism, with which the reader is expected to have some familiarity. A detailed discussion of these laws can be found in any book on electromagnetic field theory.

2.2

Maxwell’s Equations
The fundamental theory of electromagnetic fields is based on the four Maxwell’s equations. These equations are, in fact, generalizations of laws based uponexperiments. Our aim in this chapter is not to trace the history of these experiments but to present them in the form that is most useful from the application point of view. The four Maxwell’s equations are

vG=o
4

or or

V*D=p

f f

ii*~=o
4 -

D * d s = I pdv

Maxwell’s Equations

65

A

- L -

where E is the electric field intensity in volts/meter (V/m) H is themagnetic field intensity in amperedmeter (A/m) B is the magnetic flux density in teslas (T) or webers/meter2 (Wb/m2) D is the electric flux density in coulombs/meter2 (C/m2) J is the volume current density in amperes/meter2 (A/m2) p is the volume charge density in coulombs/mete6‘ (C/m3). In Eqs. (2.la) and (2.lb) s is the open surface bounded by a closed contour c. However, in Eqs. (2.1~) (2.ld) u is thevolume bounded by a closed surface s. and The above equations are tied together by the law of conservation of charge, which is also known as the equation of continuity. That is,
A A
- L -

In addition to Maxwell’s equations and the equation of continuity, we must include the Lorentz force equation

A

which defines the force experienced by a charge q moving with a velocity through anelectric field Eand a magnetic field We must mention here that x a n d 3 are the fundamental fields and D and B are the derived fields. The derived fields are related to the fundamental fields through the following constitutive relations
2 -

4

D
A

= E E = E , E ~E
4
2

(2.4a) (2.4b)

B = p H = prpo H

where E is the permittivity, p is the permeability, E, is the dielectric constant,and pr is the relative permeability of the medium. e0 and p0 are the permittivity and the permeability of free space. In the International System (SI) of units,

e0 = 8.854 X

10-9 = -farad/meter (F/m) 36n

(2.5a) (2.5b)

k0 = 4 n X

henry/meter (H/m)

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Review of Basic Laws of Electromagnerisrn

Induced Electromotive Force
Equation (2.la) is a special case of Faraday’s law...