This tutorial is to get you familiarized with complex numbers and is in no way intended to replace appendix A in our text "Basic Electrical Circuit Theory" by Mayergoyz and Lawson. Instead, this tutorial will give you a guide line on how to deal with complex numbers and such concepts as converting a complex number from its algebraic or Cartesian form to its exponential-polar form and vice versa, plotting a complex number on the complex plane, arithmetic with complex numbers, finding the magnitude and polar angle of a complex number, and converting sinusoidal quantity into a phasor. Think of this tutorial as a tool that you are going to add to your tool box of electrical engineering knowledge; as the semester goes on, you will be adding different tools into your tool box, which will help you become a good engineer.

Complex numbers become very important in electrical engineering, because they help make many complex/difficult problems very simple. As you get familiar with complex numbers, more specifically phasors, you will see that a complex circuit with n number of branches will result in a system of n linear differential equations. Sometime solving the system of linear differential equations may be simple, but when you start to use sinusoidal quantities, such as in the case of ac steady state, the system of differential equations gets to be very messy. By reducing the differential equations for circuit variables into algebraic equation for phasors, a complex problem is reduced to simple high school algebra.

A complex number is formed from two parts, a real part and an imaginary part. The basis for complex variables is the following equation: j = Ö -1. Using the previous equation for the definition of the number j, we can represent a complex number in two different forms and plot it on the complex plane where the vertical axis is the imaginary axis and the horizontal axis is the real axis.

Algebraic or Cartesian Form: z = x + jy

Exponential-Polar Form: z = |z|e^jq where |z| = Ö x² + y² and q = tan^-1 (y/x)

|z| is the magnitude of z and q is the polar angle of z.

Knowing a complex number in algebraic form you can convert it into exponential-polar form and vice versa. Remember that when plotting a complex number the correct quadrant is important, because two different complex numbers may have the same polar angle but are in different quadrants. For example, the complex numbers -4 - j3 and 4 + j3 have the same polar angle, 36.86 degrees, but one lays in the third quadrant and one lays in the first quadrant. When plotting complex numbers use the algebraic form.

When doing arithmetic with complex numbers follow these rules:

1. For adding and subtracting two complex numbers, use the algebraic form to represent the two numbers and then do the arithmetic.

Example 1: (a + jb) + (c - jd) = (a + c) + j(b - d)

Example 2: (a + jb) - (c + jd) = (a + c) - j(b + d)

1. For multiplying and dividing two complex numbers, use the exponential-polar form to represent the two numbers and then do the arithmetic.

Example 3: ae^jq * be^jf = (|a| * |b|)e^{j(q + f )}

Example 4: ae^jq / be^jf = (|a| / |b|)e^{j(q - f )}

The examples on the next three pages(Page 1, Page 2, Page 3) will show you how to convert complex numbers between the two forms, how to plot a complex number, how to find the magnitude and polar-angle of a complex number, and how to convert sinusoidal quantity into a phasor such as sinusoidal voltages and currents. Some of these examples use the problems in appendix A of our book.