A series LCR circuit (R = 30 Ω, X_L = 40 Ω, X_C = 80 Ω) is connected to an AC source of 200 V and 50 Hz. The power dissipated in the circuit is:

Concept:

• In the LCR series circuit,  the resistor of resistance R, capacitor of capacitance C, and inductor of inductance L are connected in series across the A.C voltage.

• Impedance, \(Z=\sqrt{R^2+(X_L-X_C)^2} \)
• At resonance, XL = XC, Z = R, the circuit is purely resistive in nature.
• Phase difference,  \(\phi =tan^{-1}(\frac{X_L-X_C}{R})\)
• The power dissipated in the circuit, P = VrmsIrms cosϕ 
• The power factor is given as, \(cos\phi=\frac{R}{Z}=\frac{R}{\sqrt{R^2+(X_L-X_C)^2}}\)
• The current flowing in the circuit, \(i=\frac{V_{rms}}{Z}=\frac{V_{rms}}{\sqrt{R^2+(X_L-X_C)^2}}\)

Calculation:

Given,

The resistance, R = 30Ω 

The inductive reactant, XL = 40 

The capacitive reactant,  XC = 80 Ω

The AC voltage, V_rms = 200 volt

The angular frequency, ω = 50Hz

The power dissipated in the circuit, P = V_rmsI_rms cosϕ  . . . . . . . . . . .(1)

The current flowing in the circuit, \(i=\frac{V_{rms}}{Z}=\frac{V_{rms}}{\sqrt{R^2+(X_L-X_C)^2}}\)

\(i_{rms}=\frac{200}{\sqrt{(30)^2+(40-80)^2}}\)

i_rms = 4 A

The power factor is given as, \(cos\phi=\frac{R}{Z}=\frac{R}{\sqrt{R^2+(X_L-X_C)^2}}\)

\(cos\phi=\frac{30}{\sqrt{30^2+(80-40)^2}}\)

\(cos\phi=\frac{3}{5}\)

From equation (1), \(P=200\times 4\times \frac{3}{5} = 480 ~W\)