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9-7 At A, pure 60-Hz sine wave on spectrum analyzer. At B, 60-Hz wave containing harmonics.
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172 Alternating current basics
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Engineers break the ac cycle down into small parts for analysis and reference. One complete cycle can be likened to a single revolution around a circle.
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One method of specifying the phase of an ac cycle is to divide it into 360 equal parts, called degrees or degrees of phase. The value 0 degrees is assigned to the point in the cycle where the magnitude is 0 and positive-going. The same point on the next cycle is given the value 360 degrees. Then halfway through the cycle is 180 degrees; a quarter cycle is 90 degrees, and so on. This is illustrated in Fig. 9-9.
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9-9 A cycle is divided into 360 degrees.
Radians
The other method of specifying phase is to divide the cycle into 6.28 equal parts. This is approximately the number of radii of a circle that can be laid end-to-end around the circumference. A radian of phase is equal to about 57.3 degrees. This unit of phase is something you won t often be needing to use, because it s more common among physicists than among engineers. Sometimes, the frequency of an ac wave is measured in radians per second, rather than in hertz (cycles per second). Because there are about 6.28 radians in a complete cycle of 360 degrees, the angular frequency of a wave, in radians per second, is equal to about 6.28 times the frequency in hertz.
Amplitude of alternating current 173
Phase difference
Two ac waves might have exactly the same frequency, but they can still have different effects because they are out of sync with each other. This is especially true when ac waves are added together to produce a third, or composite, signal. If two ac waves have the same frequency and the same magnitude, but differ in phase by 180 degrees (a half cycle), they will cancel each other out, and the net signal will be zero. If the two waves are in phase, the resulting signal will have the same frequency, but twice the amplitude of either signal alone. If two ac waves have the same frequency but different magnitudes, and differ in phase by 180 degrees, the resulting composite signal will have the same frequency as the originals, and a magnitude equal to the difference between the two. If two such waves are exactly in phase, the composite will have the same frequency as the originals, and a magnitude equal to the sum of the two. If the waves have the same frequency but differ in phase by some odd amount such as 75 degrees or 310 degrees, the resulting signal will have the same frequency, but will not have the same waveshape as either of the original signals. The variety of such cases is infinite. Household utility current, as you get it from wall outlets, consists of a 60-Hz sine wave with just one phase component. But the energy is transmitted over long distances in three phases, each differing by 120 degrees or 1/3 cycle. This is what is meant by three-phase ac. Each of the three ac waves carries 1/3 of the total power in a utility transmission line.
Amplitude of alternating current
Amplitude is sometimes called magnitude, level, or intensity. Depending on the quantity being measured, the magnitude of an ac wave might be given in amperes (for current), volts (for voltage), or watts (for power).
Instantaneous amplitude
The instantaneous amplitude of an ac wave is the amplitude at some precise moment in time. This constantly changes. The manner in which it varies depends on the waveform. You have already seen renditions of common ac waveforms in this chapter. Instantaneous amplitudes are represented by individual points on the wave curves.
Peak amplitude
The peak amplitude of an ac wave is the maximum extent, either positive or negative, that the instantaneous amplitude attains. In many waves, the positive and negative peak amplitudes are the same. But sometimes they differ. Figure 9-9 is an example of a wave in which the positive peak amplitude is the same as the negative peak amplitude. Figure 9-10 is an illustration of a wave that has different positive and negative peak amplitudes.
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