Complex Numbers

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 employed.
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.
Defining the real and imaginary axes of a
complex plane.
Fig. 1: Defining the real and imaginary axes of a complex plane.
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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 imaginary component.
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.
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