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CHAPTER 8 Reactance and Impedance
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at resonance, because ONLY at resonance is it true that XL XC 0. Note that the greater the deviation of the frequency away from the resonant frequency, the greater is the magnitude of (XL XC , thus the greater is the magnitude of the denominator and the less is the magnitude of current. The result is illustrated graphically in Fig. 160, the general form of a plot of eq. (236), where XL !L and XC 1=!C.
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Fig. 160
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Let us now summarize some facts about the basic series circuit of Fig. 159. To begin, let X denote the net reactance in Fig. 159 where, from inspection of eqs. (235) and (236), we see that   1 X XL XC !L !C showing that inductive and capacitive reactances tend to cancel each other out in a series circuit. This is because their voltage drops are 180 degrees out of phase with each other, " " being equal to jXL I and jXC I . Thus, in Fig. 159, if !L is less than 1=!C (below resonance), the generator sees a capacitive circuit, but if !L is greater than 1=!C (above resonance), the generator sees an inductive circuit. Of course, if XL XC (the condition of resonance, ! !0 , the generator sees a pure resistance. These three possible conditions are illustrated in Figs. 161, 162, and 163.
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Fig. 161
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Fig. 163
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A vector diagram for Fig. 159 for the condition of resonance, ! !0 , is given in Fig. 164, where V is generator voltage and I0 is current at resonance I0 V=R . Note that the voltage drops across L and C are equal in magnitude but 1808 out of phase with each other. Note, also, that the magnitudes of the voltage drops across L and C can be MANY TIMES GREATER than the generator voltage V. This is possible because at resonance VL and VC exactly cancel each other out, leaving only the voltage drop RI0 in the circuit (V RI0 :
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CHAPTER 8 Reactance and Impedance
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Now suppose the frequency were to become (for example) GREATER than the reso" nant frequency ! > !0 . In this case the current I would LAG the generator voltage by some angle  (eq. (237)), as shown in Fig. 165. (For convenience, Figs. 164 and 165 are not drawn to the same scale.) A brief discussion follows.
Fig. 165
Since we are no longer at resonance (here we re assuming a frequency above reso" nance), the magnitude of I would now be less than I0 (by eq. (236), and seen in Fig. " 160). The voltage drops VL and VC will still be at right angles to the current vector I and will still be 1808 out of phase with each other, but now VL will be greater than VC , and thus VL and VC will no longer completely cancel each other out, but, instead, a net voltage drop of VL VC will appear between L and C, as shown in Fig. 165. The vector sum of the voltage VL VC and the voltage drop RI across the resistance R must and will be equal to the generator voltage V as shown in the gure. " For frequencies below resonance XC will be greater than XL , and the current I will lead the generator voltage V. The net voltage drop between C and L will be VC VL , and the vector sum of this voltage and the voltage drop RI across the resistance R must again be equal to the generator voltage V.
CHAPTER 8 Reactance and Impedance
Problem 140 Find the resonant frequency of a series circuit in which L 4 microhenrys, C 0:0025 microfarads, and R 0:65 ohm. Problem 141 In a certain series RLC circuit, L 400 microhenrys. Find the value of C if the circuit must resonate at 500 kilohertz (500 kHz). (Answer: 253.3 pF (picofarads)) Problem 142 If, in Fig. 159, L 1 microhenry, C 0:0025 microfarad, R 5 ohms, and if the generator voltage is 20 volts rms, nd the following values: (a) power output of generator at the resonant frequency, (b) voltage drop across C at resonance. If, now, the generator frequency is made equal to 107 rad/sec (all else unchanged), nd (c) (d) (e) magnitude of voltage drop across C, phase angle  of current vector with respect to generator voltage, power output of generator.
Practically speaking, the phenomenon of series resonance is especially important because it can be used to select or tune in a desired signal, while rejecting all others. To investigate this most interesting and useful matter, let us begin with Fig. 166, in which a " generator of reference voltage V V =08 V is applied to a series RLC circuit, as shown.
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