Question

An ac source of voltage v = v_m sin ωt is connected to a series combination of LCR circuit. Draw the phasor diagram. Using it obtain an expression for the impedance of the circuit and the phase difference between applied voltage and the current.

Answer 1

From the definition of impedance:

For inductance (V_L) = I X_L = IωL

For capacitor: (V_c) = I X_C = `I/(omega C)`

For resistor (V_R) = IR

From the phasor diagram,

`V_S^2 = V_R^2 + (V_L - V_C)^2`

(IZ)2 = `(IR)^2 + (IomegaL - I/(omegaC))^2`

Z = `sqrt(R^2 + (omegaL - 1/(omega C^2))`

As resonance, impedance is minimum:

For V_C > V_L, the circuit is capacitive.

For V_L > V_C, the circuit is inductive.

I = `V/Z sin(omega t - phi)`

and, θ = `tan^-1((X_L - X_C)/R)`