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(horizontal axis) as seen from the side, graphed as a function of time (vertical axis).
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Figure 12-5 shows the way a rotating vector can be used to represent a sine wave. A vector is a quantity with two independent properties, called magnitude (or amplitude) and direction. At A, the vector points east, and this is assigned the value of 0 , where the wave amplitude is zero and is increasing positively. At B, the vector points north; this is the 90 instant, where the wave has attained its maximum positive amplitude. At C, the vector points west. This is 180 , the instant where the wave has gone back to zero amplitude and is getting more negative. At D, the wave points south. This is 270 , and it represents the maximum negative amplitude. When a full circle (360 ) has been completed, the vector once again points east. The four points in Fig. 12-5 are shown on a sine wave graph in Fig. 12-6. Think of the vector as revolving counterclockwise at a rate that corresponds to one revolution per cycle of the wave. If the wave has a frequency of 1 Hz, the vector goes around at a rate of 1 rps. If the wave has a frequency of
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representation of a sine wave. At A, at the start of the cycle; at B, onefourth of the way through the cycle; at C, halfway through the cycle; at D, threefourths of the way through the cycle.
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vector model of Fig. 12-5, shown in the standard amplitudeversus-time graphical manner.
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100 Hz, the speed of the vector is 100 rps, or a revolution every 0.01 s. If the wave is 1 MHz, then the speed of the vector is 1 million rps (106 rps), and it goes once around every 0.000001 s (10 6 s). The peak amplitude of a pure ac sine wave corresponds to the length of its vector. In Fig. 12-5, time is shown by the angle counterclockwise from due east. Amplitude is independent of time. The vector length never changes, but its direction does.
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Expressions of Phase Difference
The phase difference, also called the phase angle, between two waves can have meaning only when those two waves have identical frequencies. If the frequencies differ, even by just a little bit, the relative phase constantly changes, and it s impossible to specify a value for it. In the following discussions of phase angle, let s assume that the two waves always have identical frequencies.
Phase Coincidence Phase coincidence means that two waves begin at exactly the same moment. They are lined up. This is shown in Fig. 12-7 for two waves having different amplitudes. The phase difference in this
12-7 Two sine waves in
phase coincidence.
Expressions of Phase Difference 193
case is 0 . You could say it s some whole-number multiple of 360 , too but engineers and technicians rarely speak of any phase angle of less than 0 or more than 360 . If two sine waves are in phase coincidence, and if neither wave has dc superimposed, then the resultant is a sine wave with positive or negative peak amplitudes equal to the sum of the positive and negative peak amplitudes of the composite waves. The phase of the resultant is the same as that of the composite waves.
Phase Opposition When two sine waves begin exactly 1 2 cycle, or 180 , apart, they are said to be in phase opposition. This is illustrated by the drawing of Fig. 12-8. In this situation, engineers sometimes say that the waves are out of phase, although this expression is a little nebulous because it could be taken to mean some phase difference other than 180 . If two sine waves have the same amplitudes and are in phase opposition, they cancel each other out. This is because the instantaneous amplitudes of the two waves are equal and opposite at every moment in time. If two sine waves are in phase opposition, and if neither wave has dc superimposed, then the resultant is a sine wave with positive or negative peak amplitudes equal to the difference between the positive and negative peak amplitudes of the composite waves. The phase of the resultant is the same as the phase of the stronger of the two composite waves. Any sine wave without superimposed dc has the unique property that, if its phase is shifted by 180 , the resultant wave is the same as turning the original wave upside down. Not all waveforms have this property. Perfect square waves do, but some rectangular and sawtooth waves don t, and irregular waveforms almost never do. Intermediate Phase Differences Two sine waves can differ in phase by any amount from 0 (phase coincidence), through 90 ( phase quadrature, meaning a difference a quarter of a cycle), 180 (phase opposition), 270 (phase quadrature again), to 360 (phase coincidence again).