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Fig. 128
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First, as you know, the wave of voltage drop across a pure resistance of R ohms is exactly in phase with the wave of current through the resistance, a fact indicated graphically in Fig. 128. Next, in the gure, note that the wave of voltage drop, the counter-emf, across the inductor LEADS the current wave through the inductor by 90 degrees. This is in
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* Ip is the peak value of the steady-state sinusoidal current.
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CHAPTER 8 Reactance and Impedance
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accordance with the basic principles outlined in section 7.4, in which we learned that the voltage induced in an inductor depends NOT upon the AMOUNT of current but only upon the RATE OF CHANGE of the current. In the case of the above gure, note that the RATE OF CHANGE of the current i is MAXIMUM when i 0 and ZERO when i Ip . This is the natural phenomenon that causes the voltage across an inductor of zero resistance to lead the inductor current by 908. Thus, in the sinusoidal steady state, where (by eq. (192)) i Ip sin !t  , the voltage drop across L, vL , at any instant will be of the form vL VL sin !t  908 193 because vL leads Ip sin !t  by 90 degrees. Note that VL (capital V, sub L) denotes the PEAK VOLTAGE DROP ACROSS L. Next, let s consider the VOLTAGE EQUATION for Fig. 127 in the sinusoidal steady state. Such an equation must show that at all instants of time the sum of the voltage drops across R and L is equal to the applied generator voltage. The equation can be arrived at as follows. First, the instantaneous voltage drop across R is vR Ri RIp sin !t  , in which we see that the PEAK VOLTAGE DROP ACROSS R is RIp . Next, VL , the PEAK VOLTAGE DROP ACROSS L, depends, rst of all, upon the magnitude of the MAXIMUM RATE OF CHANGE OF CURRENT, which occurs only at times where i 0. This fact can be seen from examination of Fig. 128, in which it can be seen that the current curve is steepest (maximum amperes per second ), when i 0. Now, if we carefully inspect Fig. 128 we will see that the MAGNITUDE OF THE STEEPNESS of the current at i 0 increases if (1) the PEAK VALUE of current, Ip , is increased, and (2) if the FREQUENCY, !, is increased (for example, note the e ect, on degree of steepness, if the number of current waves in Fig. 128 were doubled). These points, plus the fact that the peak value VL is also proportional to the value of L itself, leads us, correctly, to the conclusion that the PEAK VALUE OF THE VOLTAGE DROP ACROSS L is VL !LIp , and upon substituting this value into eq. (193), and remembering that the peak voltage drop across R is RIp , we have that the SINUSOIDAL STEADY-STATE VOLTAGE EQUATION for Fig. 127 is Vp sin !t RIp sin !t  !LIp sin !t  908 194
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where  phase angle of current with respect to the reference generator voltage. In the above equation the quantity !L is measured in ohms* and is called INDUCTIVE REACTANCE. Inductive reactance is denoted by XL ; thus, XL !L 2fL ohms; thus, by Ohm s law, XL Ip !LIp peak voltage drop across the inductor, as shown in eq. (194). Now, as pointed out in connection with Figs. 93 through 95 in section 5.6, the three rotating voltage components in eq. (194) can be regarded as three stationary vector components, as shown in Fig. 129, where V is now the rms generator voltage, taken as being the reference vector, and I is the rms current vector. Note that RI, the rms voltage drop across the resistance R, is in phase with the current vector I, while !LI, the rms voltage drop across the inductor, leads the current vector I by 90 degrees. In the gure, note that, in accordance with the basic Kirchho voltage law, the vector sum of the two voltage drops RI and !LI is equal to the applied reference voltage V. Or, using the overscore or bar notation to indicate vector quantity (note 4 in the Appendix), the algebraic statement for Fig. 129 is " " " RI !L I 908 V
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