Periodic Functions (1944)

[Music] Founding demonstration proof that the unseen forces of the universe can be made to respond to the will of man. And now I cover the queen's favorite. I concentrate pristo but that is nothing. I will now demonstrate the superiority of mind over matter. Mere science can be bent to the will of man. I will now produce an electric generator. Presto Chang. And over here, another generator. Presto Chang. Now I will bring them both to life. First one here and the second here. And here I will produce a voltmeter presto. For the benefit of skeptics, I will show you the voltage they produce. This one 60 and this one 60. I will now connect both generators to the voltmeter. And of course, you will expect to read 130 volts. >> You're wrong. Huh? You mean 120 volts? >> So, it's 120 volts. I'm a 10 volt liar. I will now connect both generators. I now concentrate on the volt meter. It cannot read 120 now because I wish it to read 85. What a phony. You're a hell of a seer. What makes you think you can get away with stuff like that? That isn't magic. That's a simple scientific fact. Didn't you ever hear of a periodic function? >> What's that? >> Here, let me show you a thing or two. Where's your crystal ball? >> Simple. Very simple. Presto Chang. Very good. Very good. >> Just one moment, please. Where is your membership card in the magician's union? >> I don't need one. This stuff isn't magic. Now, hold on to your turban, Swami. We're going for a ride into the strange world of fact. But first, I need a paper and a pencil. >> Here, let me do that. Is one of my best tricks. How about black paper? Okay. And a pencil that makes white lines. Fine. Now let me show you what a periodic function is. First a point. Next a vector. Now using the vector as a radius draw a circle. Here we are back at the starting point which we will call zero. Now we'll turn the vector and form an angle. say one of 30°. Next, we draw a line from the point of the vector perpendicular to the zero line. Notice that this perpendicular line changes in length as the angle of the vector changes. Since the length of the line depends upon the size of the angle, we can say that its length is a function of the angle. If we call the angle at the center of the circle A, then the sign of the angle A is equal to B C / A. We will call the length of the vector AB 1. Then we can say that this line BC is the sign of the angle. Any sign measured in the upper half of the circle is positive. Any sign measured in the lower half is negative. As the vector makes a complete revolution, the sign increases, then decreases to zero, then begins to increase in a negative direction, and finally decreases to zero again. All this in one revolution. If the vector makes another turn, the same values of the sign are repeated. A function which repeats itself again and again as the sign does is called a periodic function. like the waves of the sea that occur time and time again. Each one is pretty much like the other. So even waves of the sea are periodic. And now that we know what periodic functions are, let me ask you, what are they good for? I'll show you by going back to our rotating vector that functions which repeat themselves time after time can be diagrammed or plotted to show their values throughout these periods of time. To simplify the job of plotting, we'll attach a rider to the vector. When the vector turns, the rider moves up and down. If we place a scale alongside the arm of the rider, we can use the scale to read the sign of any angle. The center of the scale is marked zero. Measurements above the zero are positive, those below negative. When the vector is at zero, the sign is zero as shown on the scale. When the vector is at 90°, the sign is + 1. When the vector is at 180°, the sign is again zero. And when the vector is at 270, the sign is minus one. With this scale, we can read off the sign of any angle. Let's turn the vector to 30°. And here's the sign of 30°.5. We can read the values of the signs of other angles just as well. The sign of 60° is about 87. Now let's put a pencil on the arm. Then some graph paper under the pencil. The graph paper is scaled off in degrees just as the circle made by the rotating vector. Now we can plot the values of the signs of the angles on paper. In order to do this, we must move the paper at a constant speed. In this way, time is measured along the horizontal axis. Now, let's continue. We'll turn the vector to 30° and stop a moment. If we take the sign of the angle and move it up to the graph, it fits. We can do the same with 60°. and 90° and so on all the way around. We see that the shape of the wave is determined by the signs of the angles. Why we call it a sinewave? >> Well, what's this sign business got to do with me? A peaceful man trying to make an honest dollar. >> It's got something to do with your generator. Let's take a look at a wide open one instead of a closed up secret generator like yours. Notice that the rotor of the generator turns like our vector. Now we need an oscilloscope. >> Osello who? Oscilloscope. This instrument will show the waves made by the generator. We'll connect the generator to it and then start up the generator and we get a sine wave. How come? Take a look in your crystal ball again. The lines of force run from one field magnet to the other so that when the rotor is turning through this part of the field, it's cutting across only a few lines and producing a small voltage. But when it reaches this part of the field, it cuts across many lines and produces a higher voltage. This scale will show the variations in voltage as the generator makes one revolution. We can plot these variations by marking them on a moving paper. and they turn out to be a sine wave. As you know, all generators are not alike. They may differ in the voltages they produce. Let's compare this generator with a larger one. If we turn both rotors at the same speed, we get a difference in voltage which is shown by a difference in amplitude of the sine waves. Generators may also differ in frequency. Here are two generators that produce the same voltage, but one must turn twice as fast as the other to do it. Each complete wave is called a cycle. Since the lower generator completes two cycles while the upper generator completes only one cycle, we say that the frequency of the lower generator is twice that of the upper one. Now there's a third difference between generators and that's the one you didn't reckon with Swami. You had two generators producing the same voltage and having the same frequency. If we run the rotors of these two generators in phase that is if we start them at the same place and run them at the same speed we will get identical sine waves. We can add the voltages of these two generators. First, let's put in on the lower wave the signs that determine its shape. To get the combined voltages of the two generators, we'll add the signs of the lower wave to the top wave. We get a resultant that is twice as high as either one. So if this wave represents 60 volts and this one another 60 volt then the resultant must represent 120 volts because it's twice as high. But if we start one rotor 180° ahead of the other and run them at the same speed, we get waves like this. Now look what happens when we add them. The two waves each representing 60 volts cancel each other out and the result is no voltage at all. When one rotor is ahead of the other, the angle between them is called the phase angle. In this case, the phase angle is 90°. If the rotors are always 90° apart as they turn, the sine waves will be 90° out of phase. Now when we add these waves, we get a resultant like this. So if this wave represents 60 volts and this one represents 60 volts, the resultant isn't twice as high. It actually represents 85 volts. And that's what occurred with you, my friend. One of your generators was 90° out of phase with the other. So that when you added them, you didn't get 120 volts, you got 85 volts. And your mental powers had nothing to do with it. >> It is a good trick, but I can't use it. >> Well, why can't you use it? >> Because it ain't mysterious.