“If some day they say of me that with my work I have contributed something to the welfare and the happiness of my fellow men, I shall be satis-fied.” — George Westinghouse
The three little pigs. The three branches of government. The holy trinity. The number three is an interesting number. Yes sir, yes sir, three bags full.
Long ago, Nikola Tesla recognized something special about the power of three. He started out working on a solution for an AC induction motor, and in the process, he ended up defining the familiar three-phase system of alternating current that we’ve come to know. We’ve been using three-phase power ever since. Now when you hook up power in most venues, there are three hot phases, a neutral and a ground.
Three-phase power is simply three separate voltage sources produced by a single generator. Three individual phase windings, called armatures, are built around the periphery of the stator, or the stationary part of the generator, and interact with the rotor to create the voltages. In a balanced sys-tem, each of the three voltage sources has the same amplitude and frequency, but they are each separated by 120 degrees.
Why Three-Phase Power and Not Two-Phase or Four-Phase
Tesla found that three-phase power at 60 hertz was ideal for powering motors. Since each of the three current-carrying conductors in a three-phase system is out of phase with one another, each reaches its peak voltage at different times. It has the effect of distributing power transmission evenly throughout a single cycle: As one phase falls off, another kicks in and reaches its peak. That way, one of the three phases is supplying most of the power at any one time. It’s similar to the way a three-piston engine would transfer power more evenly than a single piston would.
Three-phase power has several advantages over single-phase power. First, it’s more efficient for power transmission. Second, it is much easier on the generator, since it produces less resonance and destructive vibrations. Third, three-phase power allows for an option of two different voltages — phase-to-neutral and phase-to-phase.
In North America and many other parts of the world, the phase-to-neutral voltage is 120 volts, and the phase-to-phase voltage is 208 volts. The 120-volts-to-neutral is easy enough to understand. Each phase is an independent voltage source producing 120 volts to ground. If proper grounding techniques are followed, then the neutral and ground on the secondary side of the feeder transformer are bonded, creating a very low impedance path to ground from the neutral at the service transformer. Therefore, the phase voltage on the secondary side is the same whether it’s measured to ground or to the neutral.
The phase-to-phase voltage is somewhat deceptive. You might think that by measuring two hot legs, each of which is 120 volts to ground, that you would come up with 240 volts. And that would be true were they opposite in phase relative to each other. But since they are 120 degrees out of phase with each other, the result is something more than 120 and something less than 240 volts. The — ack! — math behind the calculation of the exact voltage is the very reason that many people are phobic about formulas and equations. It’s very complex, involving imaginary numbers and natural logarithms.
But Wait, Don't Turn the Page
There’s a much simpler way to understand the addition or subtraction of two sine waves that are out of phase with each other. It involves a little math and computation on your part. All it takes is an Excel spreadsheet and a little bit of time. So put away your slide rule, your calculator and your math book, take out a computer and open your Excel spreadsheet.
Now create a series of numbers in the left-hand column from 0 to 360. This represents the phase angle in one complete cycle. In the first cell of the next column, type the following exactly as it appears below: =169.73*(SIN(A1*(3.14/180)))
The peak value of the sine wave is 169.73; SIN is the sine function. A1 is the phase angle, and (3.14/180) converts the phase angle from degrees to radians, which is required by Excel. This will all become clearer to you when you are finished. Now click in that cell and drag the fill handle from the first cell to the 360th cell. It will fill in the same formula in each cell, except it will replace A1 with the appropriate cell to its left. In the end, you will have every value of a 120 VAC sine wave for one complete cycle. You can see it in graphic form by selecting the entire second column and clicking on the Chart Wizard. Then select the line as the Chart Type and click on Finish to create a graph of a typical 120 VAC sine wave.
But we’re not finished yet. Click on the top cell in the second column again and drag the fill handle to the right so that it copies the formula into the top cell in the third column. Now change the value that says B1 to (A1-120), click on the cell again and drag the fill handle from the first cell to the 360th cell. It will fill in the appropriate formula in each of these cells. You have just created another 120 VAC sine wave, except this one is 120 degrees out of phase with the first because you replaced A1 with (A1-120).
Now comes the fun part. In the first cell of the fourth column, type B1-C1. That represents the instantaneous value of the voltage between the two phases at the first point in the sine wave. If you now click in that cell and drag the fill handle from the first to the 360th cell, you will have created a new sine wave showing a complete cycle for the difference between two of the three phases, which is phase-to-phase voltage.
If you then highlight that column and create a graph using the Chart Wizard, you will see the resulting sine wave with the correct peak amplitude. To calculate the RMS value of that sine wave, find the peak and multiply by 0.707. You should get a value of 207.78 volts, or, rounding off, 208 volts.
Of course, if you prefer, you can either use the phasor method or you can use the natural logarithm formula and calculate the peak value the old-fashioned way. I prefer to use the graphical method because it shows you in graphic terms, rather than using imaginary numbers, how two sine waves that are out of phase, add and subtract.
With this method, you can see how two out-of-phase sine waves work together to produce the phase-to-phase voltage. For example, in Europe, the phase-to-neutral voltage is 230 volts. That, of course, is the RMS voltage, so the peak voltage is 325 volts. Try replacing 169.73 with 325 in the formulas above and see what you get.
Math is a great way to model the real world, and it is absolutely vital in engineering calculations. But sometimes you just want to see a picture rather than a bunch of numbers.
Before you pick up your slide rule again, e-mail the author at rcadena@plsn.com.
