# Complex Numbers and AC

Earlier, we learned that when a capacitor is connected across a DC source, current flows only until the capacitor is fully charged, at which time the current flow stops. We also learned that when an inductor is connected across a DC source, the current starts out small, but continues to increase until the magnetic field surrounding the inductor reaches its maximum value, at which time the current flow is constant and very large and the voltage across the inductor is almost 0. (the inductor looks almost like a short circuit, with only the parasitic resistance of its wires limiting the current flow).

When capacitors and inductors are connected across an AC source, their behavior is very different. The inductor is not a short circuit - the current through the inductor is much smaller, and decreases as the frequency of the AC voltage increases. On the other hand, the capacitor does not look like an open circuit - the AC current through a capacitor increases as the frequency of the AC source increases. These devices act somewhat like resistors, whose resistance is dependent on frequency. However, unlike a resistor, neither the capacitor nor the inductor consume any power. This is very unusual! Current is drawn, but no power is consumed. These components store energy temporarily in either an electric field (capacitor) or magnetic field (inductor) and return it to the circuit when the polarity of the AC voltage changes. This energy storage property is called reactance, and its symbol is X.

The concept of reactance leads us into some new mathematics. Resistance may be positive or negative. A resistor with a positive resistance consumes power and a resistor with a negative resistance (there really is such a thing) supplies power. We need a different kind of number to describe reactance, something that is neither positive or negative. Fortunately there are such numbers - they are called imaginary numbers, and in spite of their name, they are very real.

Imaginary numbers were created by mathematicians to solve equations that could not be solved with conventional "real" numbers. Let us look at an example. The equation x2 -1 = 0 has two answers - x = -1 and x = 1. Suppose the minus sign is changed to a plus sign, to give x2 +1 = 0. What is the solution to this equation? There is no positive or negative number that works.  This may become more obvious, if we rewrite the equation as x2 = -1. Technically, the solution is the square root of negative one, but what is that? For hundreds of years, mathematicians simply said that equations like x2 = -1 have no solution. In the sixteenth century, mathematicians began to wonder if the solutions to equations like x2 = -1 were actually an entirely new class of number. They called these numbers "imaginary numbers" because they were solutions to math problems that did not seem to have any application in the everyday world (of the sixteenth century). By the end of the nineteenth century, it was clear that imaginary numbers were essential to solving problems in the new field of electrical engineering, and imaginary numbers became very "real" and very important indeed.

What does an imaginary number look like? Consider the solution of the following equation:

Figure C-1: Solution of an equation resulting in an imaginary number

We have defined the square root of negative one to be "j". An real number can be made imaginary by placing +j or -j in front of it. All the rules of arithmetic apply to imaginary numbers, although the results can be surprising. Here are some examples:

j + j2 = j3

j7 - j2 = j5

j6*j6 = -36

-j6*j6 = +36

It is interesting to note that when two imaginary numbers are multiplied together, the result is real. This relationship between real everyday numbers and imaginary numbers suggests that they are part of something more general. That more general thing is a complex number. A complex number has a real and an imaginary part. Here is an example:

5-j3

Imaginary numbers and complex numbers are very important in AC analysis, because reactance is an imaginary number. Inductive reactance is a positive imaginary number, jX and capacitive reactance is a negative imaginary number, -jX.

 Previous Page Index Next Page