Question
A series LCR circuit is connected to a source having voltage v = v_{m} sin ωt. 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 and (ii) minimum.
Solution
v = v_{m} sin ωt
Let the current in the circuit be led the applied voltage by an angleΦ.
`i= i_m sin(omegat +phi)`
The Kirchhoff’s voltage law gives`L ((di)/dt +Ri +q/C = v)`.
It is given that v = v_{m} sin ωt (applied voltage)
`L(d^2q)/(dt^2) +R(dq)/(dt) +q/C = v_m sinomegat ...... (1)`
On solving the equation, we obtain
`q = q_m sin(omegat + theta)`
`(dp)/(dt) = q_momega cos(omegat +theta)`
`((d^2)q)/(dt^2) = q_momega^2 sin(omegat +theta)`
On substituting these values in equation (1), we obtain
`q_momega[R cos(omegat +theta)+ (X_c X_L)sin(omegat +theta)] = v_msinomegat`
`X_c = 1/(omegaC) X_c = omegaL`
`Z = sqrt(R^2 +(X_c  X_L)^2`
`q_momegaZ[R/Z cos(omegat+theta)+((X_c X_L))/Z sin (omegat+theta)] = v_m sin omegat ........... (2)`
Let `cos phi = R/2` and `(X_c X_L)/Z = sinphi`
This gives
`tan phi = (X_c  X_L)/R`
On substituting this in equation (2), we obtain
`q_momegaZcos (omegat +theta phi) = v_msinomegat`
On comparing the two sides, we obtain
`V_m = q_momegaZ = i+mZ`
`i_m = q_momega`
and `(thetaphi) = pi/2`
`I = (dp)/(dt ) =q_momega cos (omegat+theta)`
`=i_m cos(omegat+theta)`
Or
`i = i_m sin(omegat +theta)`
Where,`i_m = (v_m)/Z = (v_m)/(sqrt(R^2 +(X_c  X_L)^2)`
And
`phi = tan^1((X_c X_L)/R)`
The condition for resonance to occur
`i_m = v_m/sqrt(R^2 +(X_C  X_L)^2)`
For resonance to occur, the value of i_{m} has to be the maximum.
The value of i_{m} will be the maximum when
`X_c = X_L`
`1/(omega C) = omegaL`
`omega^2 = 1/(LC)`
`omega = 1/(sqrtLC)`
`2pif = 1/sqrt(LC)`
`f = 1/(02pisqrt(LC)`
Power factor = cos Φ
Where,`cosphi = R/Z = R/(sqrt(R^2 +(X_c X_L)^2)`
(i) Conditions for maximum power factor (i.e., cos Φ = 1)

X_{C} = X_{L}
Or

R = 0
(ii) Conditions for minimum power factor

When the circuit is purely inductive

When the circuit is purely capacitive