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InterviewSolution
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(a) Deriver and expression for the average power consumed in a series LCR circuit connected to a.c., source in which the phase different between the voltage and the current in the circuit is phi. (b) Define the quality factor in an a.c. circuit. Why should the quality factor have high value in receiving circuits ? Name the factors on which it depends. |
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Answer» Solution :(a) Average power in LCR circuit: Let the alternating e.m.f. applied to an LCR circuit is `E = E_(0) sin omega t`..(i) If alternating current developed lags behid the applied, e.m.f. by a phase angle `phi` then `I = I_(0) sin (omega t - phi)`...(ii) Total work done over a complete cycle is `W = int_(0)^(T) EI DT` `= int_(0)^(T) E_(0) sin ometa t. I_(0) sin (omega t - phi) dt = E_(0) I_(0) int_(0)^(T) sin omega t sin (omega t - phi) dt` `= (E_(0)I_(0))/(2) int_(0)^(T) 2 sin omega t sin (omega t - phi) dt` `= (E_(0)I_(0))/(2) int_(0)^(T) " " [cos (omega t + omega t + phi) - cos (omega t + omega t - phi)] dt` `[ :' 2 sin A sin B]` `= [cos (A -B) - cos (A + B)] = (E_(0)I_(0))/(2) int_(0)^(T) [cos phi - cos (2 omega t - phi)] dt` `= (E_(0)I_(0))/(2) [ t cos phi - (sin (2 omega t - phi))/(2 omega)]_(0)^(T)` `= (E_(0) I_(0))/(2) [T cos phi] " " { :' int_(0)^(T) (sin (2 omega t - phi))/(2 omega) = 0}` `= (E_(0)I_(0))/(2).cos phi.T.` `:.` Average power in LCR circuit over a complete cycle is `P = (W)/(T) = (E_(0)I_(0))/(2) cos phi = (E_(0))/(sqrt2).(I_(0))/(sqrt2) cos phi` `:. P = E_(v) I_(v) cos phi` (b) Quality factor in an A.C. : The Q -factor of a resonant LCR circuit is defined as ratio of the VOLTAGE drop across inductor or capacitor to the applied voltage.`:. Q = ("Voltage across L or C")/("applied voltage")` Since, `V_(L) = IX_(L) and V = IR " " :. Q = (IX_(L))/(IR) = (omega_(0)L)/(R)` Hence, at high FREQUENCIES, the Q-factor `(omega_(0)L)/(R)` is quite large. `:.` The voltage drop across the inductor will be quite large as compared to the applied voltage Also, `Q = (1)/(sqrt(LC)).(L)/(R) " " [ :' omega_(0) = (1)/(sqrt(LC))]` `:. Q = (1)/(R) sqrt((L)/(C))` The Q-factor of LCR-series circuit depends on L., C and R |
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