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IN ALTERNATING CURRENT, EACH 360 CYCLE IS EXACTLY THE SAME AS EVERY OTHER. IN EVERY CYCLE, the waveform of the previous cycle is repeated. In this chapter, you ll learn about the most common type of ac waveform: the sine wave.
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An ac sine wave has a characteristic shape, as shown in Fig. 12-1. This is the way the graph of the function y = sin x looks on an (x,y) coordinate plane. (The abbreviation sin stands for sine in trigonometry.) Suppose that the peak voltage is 1 V, as shown. Further imagine that the period is 1 s, so the frequency is 1 Hz. Let the wave begin at time t = 0. Then each cycle begins every time the value of t is a whole number. At every such instant, the voltage is zero and positive-going.
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12-1 A sine wave with a period of 1 second. It thus has a frequency of 1 Hz.
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If you freeze time at, say, t = 446.00, the voltage is zero. Looking at the diagram, you can see that the voltage will also be zero every so-many-and-a-half seconds, so it will be zero at t = 446.5. But instead of getting more positive at these instants, the voltage will be negative-going. If you freeze time at so-many-and-a-quarter seconds, say t = 446.25, the voltage will be +1 V. The wave will be exactly at its positive peak. If you stop time at so-many-and-three-quarter seconds, say t = 446.75, the voltage will be exactly at its negative peak, 1 V. At intermediate times, say, somany-and-three-tenths seconds, the voltage will have intermediate values.
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Figure 12-1 shows that there are times the voltage is increasing, and times it is decreasing. Increasing, in this context, means getting more positive, and decreasing means getting more negative. The most rapid increase in voltage occurs when t = 0.0 and t = 1.0. The most rapid decrease takes place when t = 0.5. When t = 0.25, and also when t = 0.75, the instantaneous voltage neither increases nor decreases. But this condition exists only for a vanishingly small moment, a single point in time. Suppose n is some whole number. Then the situation at t = n.25 is the same as it is for t = 0.25; also, for t = n.75, things are the same as they are when t = 0.75. The single cycle shown in Fig. 12-1 represents every possible condition of the ac sine wave having a frequency of 1 Hz and a peak value of 1 V. The whole wave recurs, over and over, for as long as the ac continues to flow in the circuit. Now imagine that you want to observe the instantaneous rate of change in the voltage of the wave in Fig. 12-1, as a function of time. A graph of this turns out to be a sine wave, too but it is displaced to the left of the original wave by 1 4 of a cycle. If you plot the instantaneous rate of change of a sine wave against time (Fig. 12-2), you get the derivative of the waveform. The derivative of a sine wave is a cosine wave. This wave has the same shape as the sine wave, but the phase is different by 1 4 of a cycle.
12-2 A sine wave representing the rate of change in the instantaneous
voltage of the wave shown in Fig. 12-1.
190 Phase
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An ac sine wave represents the most efficient possible way that an electrical quantity can alternate. It has only one frequency component. All the wave energy is concentrated into this smoothly seesawing variation. It is like a pure musical note.
Circular Motion Suppose that you swing a ball around and around at the end of a string, at a rate of one revolution per second (1 rps). The ball describes a circle in space (Fig. 12-3A). If a friend stands some distance away, with his or her eyes in the plane of the ball s path, your friend sees the ball oscillating back and forth (Fig. 12-3B) with a frequency of 1 Hz. That is one complete cycle per second, because you swing the ball around at 1 rps. If you graph the position of the ball, as seen by your friend, with respect to time, the result is a sine wave (Fig. 12-4). This wave has the same fundamental shape as all sine waves. Some sine waves are taller than others, and some are stretched out horizontally more than others. But the general waveform is the same in every case. By multiplying or dividing the amplitude and the wavelength of any sine wave, it can be made to fit exactly along the curve of any other sine wave. The standard sine wave is the function y = sin x in the coordinate plane. You might whirl the ball around faster or slower than 1 rps. The string might be made longer or shorter. This would alter the height and/or the frequency of the sine wave graphed in Fig. 12-4. But the sine wave can always be reduced to the equivalent of constant, smooth motion in a circular orbit. This is known as the circular motion model of a sine wave. Rotating Vectors Back in 9, degrees of phase were discussed. If you wondered then why phase is spoken of in terms of angular measure, the reason should be clearer now. A circle has 360 . A sine wave can be represented as circular motion. Points along a sine wave thus correspond to angles, or positions, around a circle.
12-3 Swinging ball and string as seen from above (A) and from the side (B).
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