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Fig. 166
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" The value of the current I is (from eq. (235)) equal to " I V V 1 !   j XL XC j 1 R !L R 1 1 R R !C 240
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It is important, now, that we somehow get the resonant frequency !0 into eq. (240). This is an interesting exercise in algebraic manipulation, and can be done by rst writing eq. (240) in the form " V I R 1   j!0 L 1 !L 1 R!0 L !C 241
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Equation (241) is, of course, the same as eq. (240), because !0 L=!0 L 1. The next step is not so obvious, but after some study we realize that eq. (241) can also be written in the form " V I R 1   !0 L ! !0 1 J R !0 !  242
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which is true because multiplying by 1=!0 L is the same as multiplying by !0 C (because 1=!0 L !0 C). Next, it is universal practice to represent the ratio of the reactance of the coil at resonance to the circuit resistance by Q ; that is !0 L Q R thus eq. (242) becomes " V I R 1  ! !0 1 jQ !0 !  244 243
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Now, in the following discussion, let it be given that, while the amplitude of the generator voltage will always remain constant at V volts, its frequency ! can be set to any value we might be interested in. Also, let the output voltage of the system be the voltage " drop across the capacitor, VC , as shown in Fig. 166. Then, since " voltage drop across C current reactance of C I j=!C " we have, using the value of I from eq. (244), that for Fig. 166 eq. (244) becomes " VC V j R!C   ! !0 1 jQ !0 !
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245
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Let us now work on the numerator in the above, as follows:          1 !0 !0 1 !0 j j j Q j !R!0 C ! ! R!C R!0 C 1 !0 L, then applied the de nition of eq. (243); in which we made use of the fact that !0 C thus eq. (245) becomes   !0 jQ " VC !   246 ! !0 V 1 jQ !0 ! Equation (246) is said to be in dimensionless form, because it requires only the ratios of like quantities and is thus valid for all systems of measurement. The equation is also said to be in normalized form, because it gives the value of VC relative to the reference voltage V. As previously mentioned, series resonance is of great practical importance because it can be used to select or tune in a signal of any desired frequency while rejecting all others. Thus we are especially interested in the behavior of eq. (246) IN THE IMMEDIATE VICINITY
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OF THE RESONANT FREQUENCY !0 . To aid in the study of eq. (246) in the close vicinity of !0 , let us de ne that ! d 247 !0 that is, d is the ratio of ANY FREQUENCY !, to the RESONANT FREQUENCY !0 . Then eq. (246) becomes " VC V In magnitude, Q j jQ d  1 d jQ d 2 1 1 jQ d d
  " V C  Q   q d !=!0 V  d 2 Q2 d 2 1 2
248
Problem 143 " Letting A jVC =Vj, ll in the following table of values for Fig. 166. (Round nal calculator values o to two decimal places.)
d 0.80 0.85 0.90 0.93 0.95 0.97 0.99 1.00 A, for Q 10 A, for Q 20 d 1.01 1.03 1.05 1.07 1.10 1.15 1.20 A, for Q 10 A, for Q 20
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