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CHAPTER 5 Sinusoidal Waves. rms Value
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Let us now refer back to the table of values in section 5.3. Now, in that table, replace  with !t and degrees with their equivalent values from the above chart. This gives us the following table of values in terms of radians.
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!t 0 =6 =4 =3 =2 2=3 3=4 5=6  sin !t 0.0000 0.5000 0.7071 0.8660 1.0000 0.8660 0.7071 0.5000 0.0000 cos !t 1.0000 0.8660 0.7071 0.5000 0.0000 0.5000 0.7071 0.8660 1.0000 tan !t 0.0000 0.5774 1.0000 1.7321 1 1.7321 1.0000 0.5774 0.0000 !t 7=6 5=4 4=3 3=2 5=3 7=4 11=6 2 sin !t 0.5000 0.7071 0.8660 1.0000 0.8660 0.7071 0.5000 0.0000 cos !t 0.8660 0.7071 0.5000 0.0000 0.5000 0.7071 0.8660 1.0000 tan !t 0.5774 1.0000 1.7321 1 1.7321 1.0000 0.5774 0.0000
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We have previously (Figs. 81 and 82) sketched the curves of sin , cos , and tan  versus the angle  in degrees. Now, in Fig. 85, we ve used the table immediately above to sketch a couple of cycles of the functions, y A sin !t and y A cos !t, versus the angle !t in radians.
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Fig. 85
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In Fig. 85, the constant A denotes the maximum (peak) value of the sine and cosine functions. The independent variable is time t, in seconds, and, as always, ! 2f , where f is constant frequency in cycles per second (Hz). Also, in the gure, note that for !t 0 the cosine function has the maximum value A, while at the same time the sine function has the value zero. Later in time, however, when !t =2, we see that the value of the cosine has fallen to zero, while the sine has risen to the maximum value of A. Thus, since the sine reaches its peak value at a later time than the cosine, we say that the sine function lags the cosine by =2 radians (90 degrees). This is, of course, the same as saying that the cosine leads the sine by =2 radians (908). As we know, the sinusoidal functions are PERIODIC functions, having a period of 2 radians. Any interval of 2 radians (3608) constitutes ONE CYCLE of a sinusoidal wave. In Fig. 85, for example, we indicate one particular cycle, in the interval from !t 0 to !t 2.
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CHAPTER 5 Sinusoidal Waves. rms Value
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For any given value of !t, the value of a sinusoidal function is repeated over and over for each value of !t 2n, where n is any integer. Thus (corresponding to eqs. 79 and 80) we have that sin !t sin !t 2n 89 cos !t cos !t 2n 90
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where n is any integer. Let us next nd the relationship between the time of one cycle and the frequency f of a sinusoid. The easiest way to do this is to make use of the particular cycle that begins at !t 0 in Fig. 85 (which we ve labeled one cycle in the gure). If we let large T denote the TIME OF ONE CYCLE, then, at the end of this particular cycle, when t T, we see from the gure that !T 2, that is, 2fT 2, from which we get the desired relationship fT 1 91
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in which f is frequency in cycles per second (Hz) and T is the time of one cycle in seconds. We now conclude this section with a discussion of phase shift and phase angle as used in connection with sinusoidal waves. We can begin by pointing out that the term phase, as used in electrical engineering, refers in general to an angular relationship of some kind. As applied to sinusoidal waves, the terms phase shift and phase angle refer to the amount of ANGULAR DISPLACEMENT of such waves; for instance, this might be the angular displacement with respect to the origin of the coordinate axes, or the angular displacement of one wave with respect to another wave of the same frequency. For example, in the preceding discussion of Fig. 85 we noted that the cosine curve leads the sine curve by =2 radians or 908. In phase terminology we could say that the phase angle between the cosine and sine functions is =2 radians, or that the cosine has a phase shift, or is phase shifted, in the amount of =2 radians with respect to the sine. The phase angle between two sinusoidal waves of the same frequency is measured between any two successive, corresponding, points of the two waves. For instance, this can be the angular distance, in radians or degrees, between two successive peak values of the waves, or the angular distance between the points at which the curves are rising in the positive sense as they cross the horizontal axis. This is illustrated in Fig. 86, in which the phase angle, 458, is the angle between two such consecutive crossover points, as shown. In this case, curve A can be said to lead curve B by 45 degrees, because A reaches its peak positive value 458 before B (as mentioned in the discussion of Fig. 85). There is an item of interest in connection with Fig. 86 that should be mentioned. In the gure, the phase angle of 458 (=4 radians) is angular displacement between the two waves themselves, as shown. The point we wish to make, however, is that this
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