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12-5 Vector representation of sine wave at the start of a cycle (A), at 1 4 cycle (B), at 1 2 cycle (C), and at 3 4 cycle (D).
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The peak amplitude of the wave can be thought of in terms of the length of the vector. In Fig. 12-5, time is represented by the angle counterclockwise from due east, and amplitude is independent of time. This differs from the more common rendition of the sine wave, such as the one in Fig. 12-6. In a sense, whatever force causes the wave is always there, whether there s any instantaneous voltage or not. The wave is created by angular motion (revolution) of this force. This is visually apparent in the rotating-vector model. The reasons for thinking of ac as a vector quantity, having magnitude and direction, will become more clear in the following chapters, as you learn about reactance and impedance. If a wave has a frequency of f Hz, then the vector makes a complete 360-degree revolution every 1/f seconds. The vector goes through 1 degree of phase every 1/(360f) seconds.
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12-6 The four points for the model of Fig. 12-5, shown on a standard amplitude-versus-time graph of a sine wave.
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An angle of 1 radian is about 57.3 degrees. A complete circle is 6.28 radians around. If a wave has a frequency of f Hz, then the vector goes through 1 radian of phase every 1/(57.3f) seconds. The number of radians per second for an ac wave is called the angular frequency. Radians are used mainly by physicists. Engineers and technicians generally use degrees when talking about phase, and Hertz when talking about frequency.
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When two sine waves have the same frequency, they can behave much differently if their cycles begin at different times. Whether or not the phase difference, often called the phase angle and specified in degrees, matters depends on the nature of the circuit. Phase angle can have meaning only when two waves have identical frequencies. If the frequencies differ, even by just a little bit, the relative phase constantly changes, and you can t specify a single number. In the following discussions of phase angle, assume that the two waves always have identical frequencies. 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. (If the amplitudes were the same, you would see only one wave.) The phase difference in this case is 0 degrees. You might say it s any multiple of 360 degrees, too, but engineers and technicians almost never speak of any phase angle of less than 0 or more than 360 degrees.
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12-7 Two sine waves in phase coincidence.
If two sine waves are in phase coincidence, the peak amplitude of the resultant wave, which will also be a sine wave, is equal to the sum of the peak amplitudes of the two composite waves. The phase of the resultant is the same as that of the composite waves.
Phase opposition
When two waves begin exactly 1 2 cycle, or 180 degrees, apart, they are said to be in phase opposition. This is illustrated by the drawing of Fig. 12-8. In this situation, engineers sometimes also 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 degrees. If two sine waves have the same amplitude and are in phase opposition, they will exactly 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 have different amplitudes and are in phase opposition, the peak value of the resultant, which will be a sine wave, is equal to the difference between the peak values of the two composite waves. The phase of the resultant will be the same as the phase of the stronger of the two composite waves. The sine wave has the unique property that, if its phase is shifted by 180 degrees, 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.
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