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A series LCR circuit is connected to an a.c. source having voltage V=V_(m) sin omega t Derive the expression for the instantaneous current I and its phase relationship to the applied voltage. Obtain the condition for resonance to occur. Define 'power factor'. State the conditions under which it is (i) maximum, (ii) minimum. |
Answer» Solution : Let as shown in Fig. 7.42 an alternating voltage`V = V_(m) sin omega t`be applied across a series combination of an inductor L, capacitor C and resistance R. As all components are in series, same current I flows through them. Let `vecV_(L), vecV_(C)` and `vecV_(R)` represent the instantaneous across L,C and R respectively, where (i) `vecV_(L) = IX_(L)` and leads the current in phase by `pi/2`. (ii) `vecV_( c) = IX_( c)` and lags behind the current by `pi/2`, and (iii) `vecV_(R) = IR` in same phase as the current I. The VOLTAGES are shown in phasor diagram in Fig. 7.43. Since `vecV_(L)` and `vecV_( c)`are in mutually opposite phase, they can be combined into a single phasor having magnitude `(V_(L)-V_(C))`and leading the current by `pi/2` Resultant of `V_(R)` and `(V_(L)-V_(C))`gives the total voltage, which is equal to the voltage of a.c. source. Thus, `V = sqrt(V_(R)^(2) + (V_(L)-V_( C))^(2)) = sqrt((IR)^(2) + I(X_(L)-X_(C))^(2))` `=I sqrt(R^(2) + I(X_(L) -X_(C))^(2))` `=I sqrt(R^(2) + (X_(L)-X_(C))^(2)) = IZ`  where Z is known as the impedance of given LCR series circuit. Hence, impedance `Z = sqrt(R^(2) + (X_(L)-X_(C))^(2))` `= sqrt(R^(2) + (L omega -1/(C omega)^(2))` Moreover, the voltage leads the current (or current lags behind the voltage) by a phase angle `phi`such that `tan phi =(V_(L)-V_(C))/V_(R) =(X_(L)-X_(C))/R = (Lomega-1/(Comega))/R` Thus, current in the circuit is given by `I = V/Z sin(omega t - phi)`, where `Z= sqrt(R^(2) + (X_(L)-X_(C))^(2))` and `phi = tan^(-1) (X_(L)-X_(C))/R`. The current amplitude `I_(m) = V_(m)/Z` Condition for Resonance and Resonant Frequency : Impedance of the given a.c. circuit will be MINIMUM and the current maximum if `X_(L) = X_( C)`because then Z = R and `I =V/R`. It is called resonance condition. For resonance to happen angular frequency of a.c. should be `omega_(0)`), so that `X_(L) = (L omega_(0)) = X_(C) = (1/(C omega_(0))) rArr omega_(0) = 1/sqrt(LC)` Power factor of an a.c. circuit is the cosine of the angle (cos ) by which the current lags or leads the a.c. voltage. Power factor, `cos phi = R/sqrt(R^(2) + (Lomega - 1/(C omega)^(2))) = R/sqrt(R^(2) + X^(2))` (i) The power factor is maximum, having a value one, when either the circuit is purely resistive circuit or when `X_(L) = X_(C)`so that Z=R. (ii) The power factor is minimum, having a value zero, when no resister R is PRESENT in the circuit and impedance is purely reactive impedance |
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