Question

#8)

A series RCL circuit has a resonant frequency of 1450 Hz. When operating at a frequency other than 1450 Hz, the circuit has a capacitive reactance of 5.0 ? and an inductive reactance of 26.0 ?.

(a) What is the value of L?

(b) What is the value of C?

Answer 1

The vaalue of X_C = 5.0 ? and X_L = 26.0 ? with f = some value iother thatresonance frequency.

In the above slution, it was assumed that these reactancevalues are for fo =1450Hz(resonance frequency);\

Given, X_C = 5.0 ? at some unknown frequencyf.

          X_C = 1/?C = 1/2?f.C = 5.0 ?

       Hence, 1/C = 2?f.x 5.0= 10?.f ---------(1)

Similarly, X_L = ?L = 2?f.L = 26.0?

       Hence, 1/L = 2?f./26 ---------(2)

To eliminate the unknown f, divide (1)/(2)

              L/C = 260/2 = 130 ----------------(3)

From Resonance condition,

            X_L = X_C;   ?².=1/LC

            1/LC  =4?²f_o²       -------------------(4)

From (3) and (4)

         (1/LC ) x  (L/C) =4?2f_o²x130

                 1/C² =4?²f_o²x130

                    1/C= 2?f_ox?130 = 6.282x1450x11.40

                         = 103861 F^-1

Hence capacitance,

              C = 1/103861 = 9.63x10^-6 F

                  = 9.63 ?F

From (3) ,

            L = Cx130 = 9.63x10^-6 x130

               = 0.001252 H =1.252 mH