Foundations of Electromagnetic Compatibility with Practical Applications

6Complex Numbers and Phasors

6.1 Definitions and Forms

A complex number z is a number that can be expressed as

(6.1)

where x and y are real numbers and

(6.2)

x is called a real part of z, and we write x = Re(z).

y is called an imaginary part of z, and we write y = Im(z).

j is called an imaginary unit, sometimes expressed as .

As we will soon see, there are several equivalent representations of complex numbers. The representation in Eq. (6.1) is called the rectangular form.

Example 6.1 Rectangular form of a complex number

When dealing with complex numbers, it is often expedient to represent them graphically. We can easily do that in a complex plane using a Cartesian coordinate system, where the x axis is the real axis and the y axis is the imaginary axis, as shown in Figure 6.1.

Figure 6.1 Complex plane.

Example 6.2 Complex plane

Let . Determine the location of this number in the complex plane.

Solution: ...