Complex Numbers

While the fundamental signal used in electrical engineering is the sinusoid, it can be expressed mathematically in terms of an even more fundamental signal: the complex exponential. Representing sinusoids in terms of complex exponentials is not a mathematical oddity. Fluency with complex numbers and rational functions of complex variables is a critical skill all engineers master. Understanding information and power system designs and developing new systems all hinge on using complex numbers. In short, they are critical to modern electrical engineering, a realization made over a century ago.

Figure 1 shows that we can locate a complex number in what we call the complex plane. Here,

, the real part, is the

-coordinate and

, the imaginary part, is the

-coordinate.

The complex conjugate of

, written as

, has the same real part as

but an imaginary part of the opposite sign.

Using Cartesian notation, the following properties easily follow.

• If we add two complex numbers, the real part of the result equals the sum of the real parts and the imaginary part equals the sum of the imaginary parts. This property follows from the laws of vector addition.

In this way, the real and imaginary parts remain separate.

• The product of and a real number is an imaginary number: . The product of and an imaginary number is a real number: because . Consequently, multiplying a complex number by rotates the number's position by 90 degrees.

Complex numbers can also be expressed in an alternate form, polar form, which we will find quite useful. Polar form arises arises from the geometric interpretation of complex numbers. The Cartesian form of a complex number can be re-written as

By forming a right triangle having sides

and

, we see that the real and imaginary parts correspond to the cosine and sine of the triangle's base angle. We thus obtain the polar form for complex numbers.

The quantity

is known as the magnitude of the complex number

, and is frequently written as

. The quantity

is the complex number's angle. In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies.

Surprisingly, the polar form of a complex number

can be expressed mathematically as

To show this result, we use Euler's relations that express exponentials with imaginary arguments in terms of trigonometric functions.

The first of these is easily derived from the Taylor's series for the exponential.

Substituting

for

, we find that

because

,

, and

. Grouping separately the real-valued terms and the imaginary-valued ones,

The real-valued terms correspond to the Taylor's series for

, the imaginary ones to

, and Euler's first relation results. The remaining relations are easily derived from the first. We see that multiplying the exponential in Equation by a real constant corresponds to setting the radius of the complex number to the constant.

Adding and subtracting complex numbers expressed in Cartesian form is quite easy: You add (subtract) the real parts and imaginary parts separately.

To multiply two complex numbers in Cartesian form is not quite as easy, but follows directly from following the usual rules of arithmetic.

Note that we are, in a sense, multiplying two vectors to obtain another vector. Complex arithmetic provides a unique way of defining vector multiplication.

Division requires mathematical manipulation. We convert the division problem into a multiplication problem by multiplying both the numerator and denominator by the conjugate of the denominator.

Because the final result is so complicated, it's best to remember how to perform division—multiplying numerator and denominator by the complex conjugate of the denominator—than trying to remember the final result.

The properties of the exponential make calculating the product and ratio of two complex numbers much simpler when the numbers are expressed in polar form

To multiply, the radius equals the product of the radii and the angle the sum of the angles. To divide, the radius equals the ratio of the radii and the angle the difference of the angles. When the original complex numbers are in Cartesian form, it's usually worth translating into polar form, then performing the multiplication or division (especially in the case of the latter). Addition and subtraction of polar forms amounts to converting to Cartesian form, performing the arithmetic operation, and converting back to polar form.