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CHAPTER 5 Sinusoidal Waves. rms Value
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Now set  !t in eq. (98), then substitute the right-hand side into eq. (97) in place of sin2 !t. Doing this, eq. (97) becomes p Vp Ip Vp Ip cos 2!t 2 2 99
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Now examine the right-hand side of eq. (99). The average value of the rst term is Vp Ip =2, because it has the same constant value for the entire interval from !t 0 to ! 2. (If an automobile maintains a constant speed of 60 mph over a period of time, the average speed over the period is 60 mph.) Note, however, that the average value of the second term is zero over the same interval from !t 0 to !t 2. To understand why this is true, let us take a moment out to review the basic meaning of average value, as follows. Let y1 ; y2 ; y3 ; . . . ; yn denote n di erent values of a variable y, measured over a certain range of values of whatever independent variable determines the values of y. Note that some of the values of y may be positive in value, and others negative in value. The average value of y, over the particular interval chosen, is then de ned to be equal to the algebraic sum of all the n values, divided by n, that is average value of y Y y1 y2 y3 yn n 100
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Note that if the algebraic sum of the numerator values is zero the average value of y is zero, and this is exactly what happens in the case of the second term on the right-hand side of eq. (99). This is true because, in any number of complete cycles of a sinusoidal wave, there are as many positive values as negative values. Thus we have determined that the average value of eq. (99) (and also of eq. (97)) is equal to Vp Ip =2; that is, letting P denote average power produced in eqs. (99) and (97), we have that P V p Ip V p Ip p p 2 2 2 101
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where the dot means times, and where Vp and Ip are the maximum (peak) values of an applied sinusoidal voltage and the resulting sinusoidal current. Let us now denote the two quantities on the right-hand side of eq. (101) by V and I, without subscripts, thus p 102 V Vp = 2 0:7071Vp p I Ip 2 0:7071Ip 103 The quantities represented by V and I above are called the e ective values or the rms values* of sinusoidal voltages and currents of peak values Vp and Ip . Note that, using this notation, eq. (101) takes the form P VI 104
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* rms stands for root mean square. Noting that mean is the same as average, the origin of rms is as follows. First, instantaneous power, p, is proportional to the square of instantaneous current, i2 , as shown following eq. (96). Thus, over a period of time, AVERAGE POWER, P, is proportional to the average value of i2 , which, for n consecutive samples, let us denote by I 2 ; thus I2
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Thus I, as derived here, is the square root of mean square or rms value of current. Hence rms and e ective have the same meaning, being the value of current or voltage used to calculate average power.
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CHAPTER 5 Sinusoidal Waves. rms Value
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You ll recall that power is expressed in watts in electric circuit calculations (see eq. (15) in Chap. 2). The above equation states that the AVERAGE POWER P produced in a resistance of R ohms is equal to the RMS VOLTAGE TIMES THE RMS CURRENT. If we now apply the foregoing procedure to the equations p v2 =R and p i2 R, we nd that P V 2 =R and P I 2R
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105 106
and, comparing eqs. (106) and (104), we see that I R VI, and thus we have OHM S LAW for the sinusoidal ac case: I V R 107
In eqs. (104) through (106), P is average power in watts, V and I are rms values of voltage and current, and R is load resistance in ohms. It is important to note that eqs. (104) through (107), for the ac circuit, have exactly the same form, and are subject to the same algebraic manipulation, as the equations for dc circuits summarized following eq. (17) in section 2.3. This procedure can be used because, in ac circuit work, we are normally not interested in knowing instantaneous values of power, voltage, and current (given by eqs. (94) and (95) in the sinusoidal case), but only in average power and rms values of voltage and current, which are not functions of time. Problem 70 Are eqs. (104) through (107) basically true for non-sinusoidal periodic waveforms of voltage and current, as well as sinusoidal Problem 71 It is given that ac voltmeters and ammeters are normally calibrated to read rms values. If the meter readings are 120 volts and 8.5 amperes, nd the following values (sinusoidal conditions will always be assumed, unless de nitely stated otherwise): (a) (b) average power input to the circuit, peak power input to the circuit.