# qr code vb.net source Fig. 86 in .NET framework Maker Code 128C in .NET framework Fig. 86

Fig. 86
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CHAPTER 5 Sinusoidal Waves. rms Value
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information alone does not permit us to write the equations of the two waves, even if their peak values are given. This is because the location of the origin of the axes, relative to the waves, is not shown in the gure. To illustrate this, suppose the peak value of A is 10 and the peak value of B is 7, and suppose it is given that curve A passes through the origin (in the manner of the sine wave in Fig. 85). With this information the equations of the two waves can now be written thus (in radians): for A: for B: y 10 sin !t y 7 sin !t =4 92 93
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Equation (93) is the mathematical way of showing that sinusoid B lags sinusoid A by =4 radians (458). Another point to be emphasized is that the curves A and B in Fig. 86 represent two sinusoidal functions having the SAME FREQUENCY. If two sinusoidal functions do not have the same frequency, then no xed phase relationship exists between the two functions, and the term phase shift would have little meaning. This is illustrated in Fig. 87, in which A and B denote curves of two sinusoids having unequal frequencies.
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Fig. 87
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Problem 67 Given that v 100 sin 180 000t8, nd (a) (b) frequency in hertz, value of v at the instant t 0:15 second.
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Problem 68 If ! 533 850 for a certain sine wave, nd the time of one cycle in microseconds. Problem 69 In Fig. 87, which wave, A or B, represents the higher frequency
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Power; rms Value of a Sine Wave of Voltage or Current
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P VI; P V 2 =R; P I 2R
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In section 2.3 we showed that in a dc (direct-current) circuit, the power P in watts, expended in a resistance of R ohms, is given by any of the formulas
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where V is dc voltage and I is dc current.
CHAPTER 5 Sinusoidal Waves. rms Value
Now let s consider the problem of how to calculate power in an ALTERNATING CURRENT (ac) circuit. We at once see that there is a problem here, because the power in an ac circuit is not constant but changes from instant to instant throughout the cycle, because the voltage and current are both continually changing during the cycle. This di culty is resolved by de ning that by power in an ac circuit we will always mean the AVERAGE POWER in the circuit. This de nition leads to what is called the e ective or rms value of an ac voltage or current. The development proceeds as follows. To begin, let us make the following slight change in notation. In eqs. (87) and (88) we used V and I to denote the peak values of sine waves of voltage and current. Let us now, for convenience later on, change that notation and, hereafter, always denote peak values by Vp and Ip , instead of by plain V and I. Upon making this change in notation, eqs. (87) and (88) become v Vp sin !t i Ip sin !t 94 95
In the above equations, v and i denote instantaneous voltage and current at any time t seconds after we start positive time at t 0. It then follows, from the basic considerations used to derive eq. (15) in Chap. 2, that instantaneous power, p, is equal to instantaneous voltage times instantaneous current, that is p vi or, in terms of a load resistance of R ohms, eq. (96) can be written in the forms p v2 =R and p i2 R: 96
But, as already mentioned, we are not interested in instantaneous power; instead, we are interested in nding the AVERAGE POWER obtained over ONE COMPLETE CYCLE of the sine waves of eqs. (94) and (95). (The average power over any one complete cycle is the same for all cycles, and is the average power as long as the waves continue to exist.)* To continue, let us now substitute, into eq. (96), the values of v and i from eqs. (94) and (95). Doing this, we have that the instantaneous power p in an ac circuit is equal to p Vp Ip sin !t 2 Vp Ip sin2 !t 97
in which we ll assume that the peak values, Vp and Ip , will remain constant in any given problem. From eq. (87), the angle !t is in radians. We recall that the function sin !t goes through one complete cycle in the period from !t 0 to !t 2 (Fig. 85). Our problem, therefore, now is to nd the AVERAGE VALUE of eq. (97) when the voltage and current waves, v and i (eqs. (94) and (95)), go through one complete cycle from !t 0 to !t 2. One way this can be done is to make use of the trigonometric identity 1 sin2  1 cos 2 2 98 {
* Power does not accumulate in a circuit, because power is a measure of the RATE at which work is being done. This could be compared to an automobile moving at a constant speed; if the speed is measured for a period of, say, 1 minute, the same speed is measured for all intervals of 1 minute, and is the speed for the entire trip. { See note 6 in Appendix.