In our analysis of dc networks, we found it necessary to determine the
algebraic sum of voltages and currents. Since the same will also be true
for ac networks, the question arises, How do we determine the algebraic
sum of two or more voltages (or currents) that are varying sinusoidally?
Although one solution would be to find the algebraic sum on a point-to-point basis, this would be a long and
tedious process in which accuracy would be directly related to the scale
It is the purpose of this chapter to introduce a system of complex
numbers that, when related to the sinusoidal ac waveform, will result
in a technique for finding the algebraic sum of sinusoidal waveforms
that is quick, direct, and accurate. In the following chapters, the technique will be extended to permit the analysis of sinusoidal ac networks
in a manner very similar to that applied to dc networks. The methods
and theorems as described for dc networks can then be applied to sinusoidal ac networks with little difficulty.
Fig. 1: Defining the real and imaginary axes of a
A complex number represents a point in a two-dimensional plane
located with reference to two distinct axes. This point can also determine a radius vector drawn from the origin to the point. The horizontal
axis is called the real axis, while the vertical axis is called the imaginary axis.
Both are labeled in Fig. 1. Every number from zero to
$\infty$ can be represented by some point along the real axis. Prior to the
development of this system of complex numbers, it was believed that any number not on the real axis would not exist-hence the term imaginary for the vertical axis.
In the complex plane, the horizontal or real axis represents all positive numbers to the right of the imaginary axis and all negative numbers to the left of the imaginary axis. All positive imaginary numbers are
represented above the real axis, and all negative imaginary numbers,
below the real axis.
The symbol j (or sometimes i) is used to denote the
Two forms are used to represent a complex number: rectangular
and polar. Each can represent a point in the plane or a radius vector
drawn from the origin to that point.