“Do not worry about your difficulties in mathematics. I can assure you mine are still greater.” – Albert Einstein
When it comes to math, Albert Einstein would have us believe that he’s no Einstein. Do you suppose he was exaggerating just a tad, or was he serious?
No matter; math is, as scientist Roger Bacon once said, “the gate and the key to the sciences.” But math isn’t all numbers and no fun.
In this space in the November issue of PLSN was an article entitled The Power of Three about poly-phase power. It can be a complex subject, but I tried to present an alternative view of it without the inconvenience of having to do much math. If you read the article and followed the instructions, you should have created an Excel spreadsheet with some charts showing the subtraction of two voltage sine waves and the resulting voltage sine wave. For some of you, this graphic illustration got some synapses firing, as evidenced by some of the e-mail that showed up in my inbox.
“I showed this to my wife,” wrote Daniel Sell, “who teaches geometry and algebra.”
Already I can tell she’s a pretty smart lady, not only because she teaches math, but mostly because she’s not in the entertainment industry. “She helped me diagram it a different way, which actually shows the relationship between the armatures in the generator and the coils in a transformer,” he continued.
He sent me a drawing that sort of looked like this:
It shows a representation of a three-phase power system and each phase drawn as a vector: OA, OB and OC. A vector is a graphical representation of anything with a direction and a strength or magnitude. For example, a hurricane can be drawn as a vector; it has a wind speed and a direction.
A vector looks like an arrow whose length represents its magnitude, and its direction represents, well, its direction. It just so happens that there are 360 degrees in a circle and 360 degrees in single cycle of a sine wave. So a sine wave can be drawn as a vector whose average value is proportional to the length of the vector, and whose direction corresponds to the instantaneous angle of the sine wave.
The direction of the vector is the instantaneous angle of the sine wave because the angle is constantly changing, and a vector is a static representation. So we can use the starting value of the three phases to show the relative phase angle between the three.
In a three-phase electrical system, if we measure across two phases, we’re subtracting the voltages. Why are we subtracting instead of adding the voltages? If we look at the diagram and trace a path along the vectors from B to A through the center O, we can see that we’re traveling in the op-posite direction of OB, but in the positive direction of OA; therefore, it’s the same as subtracting OB from OA.
We can “subtract” the two vectors by superimposing OB so that its head starts at the head of OA. The result, OX, shows the resulting voltage and phase angle.
If the phase vectors OA and OB represent the average, or RMS value of the voltage, then in North America and many other parts of the world, the magnitude would be 120 volts. We can figure out the phase angle and magnitude of OX by drawing a right angle to the line OX from the head of OA. We know that the angle between OA and OB is 120 degrees, so the line we drew to create a right angle now bisects that angle; therefore, it must be 60 degrees. Because the angles in a triangle always add up to 180 degrees, we know that the angle XOA is 30 degrees.
We also know that the length of OA is 120, so we can figure out half of the length of OX by using trigonometry. The cosine of the angle XOA, which is 30 degrees, is by definition the side adjacent to the angle divided by the hypotenuse, which is OA or 120. With our trusty trig calculator, we can find the value of the cosine of 30, which is 0.866; therefore, the length we’re looking for is 0.866 times 120, which is 103.9. Now we double that number to get the magnitude of OX, and we find that the exact value is 207.85 volts.
Isn’t that what we would expect to find as the average value of the voltage for a single-phase 208V system?
We might also have used the peak values instead. We could have used 169.7 volts as the peak value of OA instead of the RMS value of 120 volts. Then we would have found that the magnitude of OX would have been 293.9 volts, which is the peak voltage for a phase-to-phase measurement in our system.
What we’ve done is to build a model of a three-phase power system that graphically illustrates the phase relationships between the three legs of a four-wire plus ground power distribution system. If you are able to follow along, then consider yourself among the math elite of the industry. On the other hand, if you are having trouble grasping the concept, spend some time studying and put some effort into understanding. Just know that you’re not alone. When it comes to math, even Albert Einstein was no Einstein.
Send your Xs and Os to the author at rcadena@plsn.com.
