Question

ac voltage of frequency ω is applied across a series LCR circuit. Draw the phasor diagram and obtain the impedance of the circuit.

Answer 1

In a series LCR circuit, the current is the same in all three components.

• The voltage across the resistor (V_R) is in phase with the current. 
• The voltage across the inductor (V_L) leads the current by 90 degrees. 
• The voltage across the capacitor (V_C) lags the current by 90 degrees.

Now, the net voltage is given by,

V = `sqrt(V_R^2 + (V_L - V_C)^2)`

= `sqrt((IR)^2 + (IX_L - IX_C)^2)`

= `I sqrt(R^2 + (X_L - X_C)^2)`

The impedance Z is defined as Z = `V/I`. Therefore,

Z = `sqrt(R^2 + (X_L - X_C)^2`    ...`[Because X_L = omegaL], X_C = 1/(omega C)`

= `sqrt(R^2 + (omega L - 1/(omegaC)^2)`