Question

(a) Derive an expression for the average power consumed in a series LCR circuit connected to a.c. source in which the phase difference between the voltage and the current in the circuit is ϕ.

(b) Define the quality factor in an a.c. circuit. Why should the quality factor have high value in receiving circuits? Name the factors on which it depends.

Answer 1

(a) In series LCR circuit. 

Voltage, V = V₀sinωt 

Current in circuit, I = I₀sin(ωt + ϕ) Instantaneous Power, P = VI

Average value of cos(2ωt + ϕ) over a complete cycle is zero i.e.,

(b) Quality Factor (Q): In series LCR circuit the ratio of the voltage drop across inductor (or capacitor) to the voltage drop across resistor under resonance condition is called the quality factor

where ω₂ - ω₁ = band width of resonant curve. Smaller is the band width, larger is the quality factor and selectivity (or sharpness of resonance) of the circuit. That is why in receiving circuits, quality factor must be very high. The quality factor depends on the values of resistance, inductance and capacitance of the circuit.