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.
The notion of the square root of originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity could be defined. Euler first used for the imaginary unit but that notation did not take hold until roughly Ampère's time. Ampère used the symbol to denote current (intensité de current). It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. By then, using for current was entrenched and electrical engineers chose for writing complex numbers.
An imaginary number has the form . A complex number, , consists of the ordered pair (,), is the real component and is the imaginary component (the is suppressed because the imaginary component of the pair is always in the second position). The imaginary number equals (,). Note that and are real-valued numbers.
[link] 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.
Some obvious terminology. The real part of the complex number , written as , equals . We consider the real part as a function that works by selecting that component of a complex number not multiplied by . The imaginary part of , , equals : that part of a complex number that is multiplied by . Again, both the real and imaginary parts of a complex number are real-valued.
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.
Use the definition of addition to show that the real and imaginary parts can be expressed as a sum/difference of a complex number and its conjugate. and .
. Similarly,
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.
Convert to polar form.
To convert to polar form, we first locate the number in the complex plane in the fourth quadrant. The distance from the origin to the complex number is the magnitude , which equals . The angle equals or radians ( degrees). The final answer is degrees.
Surprisingly, the polar form of a complex number can be expressed mathematically as
Adding and subtracting complex numbers expressed in Cartesian form is quite easy: You add (subtract) the real parts and imaginary parts separately.
What is the product of a complex number and its conjugate?
. Thus, .
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.
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.
When we solve circuit problems, the crucial quantity, known as a transfer function, will always be expressed as the ratio of polynomials in the variable . What we'll need to understand the circuit's effect is the transfer function in polar form. For instance, suppose the transfer function equals