224 Practical MATLAB® Applications for Engineers
The general accepted universal AC wave model is, therefore, a sinusoid function, defi ned
by three variables given as follows:
Vm or Im, the maximum or peak voltage or current
ω the angular frequency
θ the phase angleRecall that the sinusoidal of the form v(t) = Vm cos(ωt + θ) is a periodic function with
period T = 2 π/ω therefore repeats indefi nitely satisfying the time relation f(tx) = f(tx + nT),
for any integer value of n.
The analysis of AC circuits presented in this chapter follows mainly the model devel-
oped by Steinmetz,* which consists in reducing the AC time-dependent circuit into an
equivalent phasor model.
This transformation consists basically in converting the sources and elements (imped-
ances) into phasor representation, by using complex numbers and complex algebra
(Chapter 6 of the book titled Practical MATLAB® Basics for Engineers) to evaluate and represent
currents and voltages. The standard circuit equations, relations, and techniques developed
for DC circuits are equally valid in AC, assuming that the interest is focused on the forced
or steady-state response.
In most cases, the transient response decays rapidly to zero, although the force response
persists indefi nitely making the steady-state solution the solution of extreme practical
importance.
3.2 Objectives
After completing this chapter the reader should be able to
Express mathematically AC currents and voltages
Express AC elements and sources in phasor format
Understand the concepts of
a. Instantaneous value
b. Amplitude or peak value
c. Peak to peak value
d. Periodicity and period T
e. Frequency in cycles/second (f) or radian/second (ω)f. Phase angle in radians and degrees (note that some of these concepts were
introduced in early chapters, in particular Chapter 4 of the book titled Practical
MATLAB® Basics for Engineers and in Chapter 1 of this book)
*^ Charles Proteus Steinmetz (1865–1923), a German–Austrian engineer, worked at the General Electric Corp
(GE) and in the 1890s, he revolutionized the approach, used to analyze AC circuits, by reducing them to simple
algebraic phasor equations, making the process easy and simple avoiding the complications of high level
math. The method developed by Steinmetz was adapted by engineers and scientists around the world and is
commonly referred as the phasor transform method.