Alternating Current

by Harold Hallikainen

San Luis Obispo, Calif. So far in this series, we've analyzed DC circuits with a few voltage or current sources and a few resistances. Let's start looking at alternating current. After getting a few definitions and derivations out of the way, we'll be able to analyze circuits using the same techniques we've used before, but use complex numbers instead of real numbers. We'll use complex numbers to represent the magnitude and phase of voltages, currents, and impedances.

We'll start with a brief review of trigonometry. Figure 1 shows a right triangle on the x-axis of the x-y plane. Relative to angle a, the three sides can be identified as the adjacent, marked 'x', the opposite, marked 'y', and the hypotenuse, marked 'r'. The adjacent side is marked x, since it is along the x-axis. The opposite side is marked y, since it is in the same direction as the y-axis. Finally, the hypotenuse is marked r, since it can be thought of as the radius of a circle centered at the origin (0,0) as the angle a is varied. Further, a point at the far end of the hypotenuse has coordinates of (x,y) and is r units from the origin.

The three basic trig functions are shown in table 1. The first column gives the English spelling of the function, the second gives the mathematical function notation, and the third column shows how to calculate the function based on the triangle shown in figure 1.

In the triangle of figure 1, we can determine r as the square root of (x²+y²) or about 3.606. This gives makes sin(a)=2/3.606=0.555; cos(a)=3/3.606=0.832; and tan(a)=2/3=0.667. We can use these values and the inverse trig function to determine the angle. For example, the arctangent (tan^-1) of 0.667 is 33.7 degrees. Similarly, the arcsine (sin^-1) of 0.555 and the arccosine (cos^-1) of 0.832 are both about 33.7 degrees.

If we set r = 1, then as a varies from 0 to 360 degrees, it scribes a unit circle, a circle with a radius of 1. This simplifies the sine and cosine as being the y and x values of the point on the unit circle.

Above, we referred to the angle a as being measured in degrees. A degree is 1/360 of the full cycle formed by the angle a before the point on the unit circle returns to the point it started. We generally measure this angle from the positive x-axis. If the point on the unit circle is in quadrant 4 (below the positive x-axis), we can either say the angle is negative, or say it is somewhere between 270 and 360 degrees.

We can also measure angles in radians. A radian is that angle formed when the circle scribed by the increasing angle draws an arc length equal to the radius of the circle. Since we know the circumference of a circle is pi times its diameter, and the diameter is two times the radius, a the circumference of the full circle is 2 pi r. Therefore, the full circle of 360 degrees is equivalent to 2 pi radians. Figure 2 shows how the sin(x) varies as x varies from -2 pi to +2 pi radians.

Note also that the sine varies between -1 and +1. We can multiply this by a constant and end up with a sine wave whose peak amplitude is equal to the constant. Further, we can substitute 360*f*t for a and get a sine wave that progresses with time, assuming the sine function expects an argument (input) in degrees and that f is the frequency in Hz (1/seconds) and t is time in seconds. If our sine function expects an argument in radians, we can use wt (where w is the Greek letter omega) as the argument (the input) to the sine function. Here, w represents the frequency in radians per second.

In figure 3, we've set f to 1,000, and w = 2 pi f . Further, we've multiplied the sine function by 2, giving a peak amplitude of 2. The resulting waveform is a 2 volt peak sine wave. The amplitude (which could represent a voltage or current) can be determined at any instant in time using the function V(t)=2*sin(w*t) where w=2*pi*1000. This is assuming the sine function accepts an argument in radians. If the sine function expects an argument in degrees, the function becomes V(t)=2*sin(360*1000*t) . You might try both of these on your calculator, then compare the results to the instantaneous voltage indicated on an oscilloscope displaying a 2 volt peak 1 KHz sine wave.

Next time, we'll look at other characteristics of the sine wave. These include its average value, RMS value, period and wavelength when it is propogated. Stay tuned!

Harold Hallikainen designs transmitter control and lighting equipment for Dove Systems, a manufacturer serving the broadcast and entertainment industries. He also teaches electronics at Cuesta College and is an avid contra dancer. He can be reached at +1 805 541 0200 (voice), +1 805 541 0201 (fax), harold@hallikainen.com (email), and http://hallikainen.com (World Wide Web, where an archive of these articles is maintained).