(a) Derive an expression for the average power consumed in a series LCR circuit connected to an a.c. source in which the phase difference between the voltage and the current is $\phi$. (b) Define the quality factor of an a.c. circuit. Why should the quality factor have a high value in receiving circuits? Name the factors on which it depends.

(a) $P_{av}=V_{rms}I_{rms}\cos\phi$; (b) $Q=\frac{1}{R}\sqrt{\frac{L}{C}}$; a high $Q$ gives sharp tuning (good selectivity); it depends on $L$, $C$ and $R$.

CBSE Class 12 · 2009 · Alternating CurrentLong Answer

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

ϕ

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

Solution

(a) Let

V = V_{0} sin ω t

and

I = I_{0} sin (ω t - ϕ)

. The instantaneous power is

P = V_{0} I_{0} sin ω t sin (ω t - ϕ)

. Using

2 sin A sin B = cos (A - B) - cos (A + B)

P = \frac{V _{0} I _{0}}{2} [cos ϕ - cos (2 ω t - ϕ)]

Averaging over a cycle,

⟨ cos (2 ω t - ϕ)⟩ = 0

, so

P_{a v} = \frac{V _{0} I _{0}}{2} cos ϕ = \frac{V _{0}}{2} \cdot \frac{I _{0}}{2} cos ϕ = V_{r m s} I_{r m s} cos ϕ

(b) The quality factor measures the sharpness of resonance:

Q = \frac{ω _{0} L}{R} = \frac{1}{ω _{0} C R} = \frac{1}{R} \frac{L}{C}

A receiving circuit should have a high

Q

so that resonance is sharp and the set responds strongly to the desired frequency while rejecting nearby ones (good selectivity).

Q

depends on

L

C

and

R

P_{a v} = V_{r m s} I_{r m s} cos ϕ, Q = \frac{1}{R} \frac{L}{C}

CBSE Class 12 Board · 2009 · Subjective

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CBSE Class 12 · 2009 · Alternating CurrentLong Answer

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