Alternating Current — CBSE Class 12 · 2024 Board Question with Solution

(a)(i) You are given three circuit elements X, Y and Z. They are connected one by one across a given a.c. source. It is found that V and I are in phase for element X, V leads I by

\frac{π}{2}

for element Y, while I leads V by

\frac{π}{2}

for element Z. Identify the elements X, Y and Z. (ii) Establish the expression for the impedance of the circuit when elements X, Y and Z are connected in series to an a.c. source. Show the variation of current in the circuit with the frequency of the applied a.c. source. (iii) In a series LCR circuit, obtain the conditions under which (1) the impedance is minimum and (2) a wattless current flows in the circuit.

Solution

(i) For X, V and I are in phase → X is a resistor. For Y, V leads I by

\frac{π}{2}

→ Y is an inductor. For Z, I leads V by

\frac{π}{2}

→ Z is a capacitor.
(ii) In series the same current flows;

V_{R} = I R

(in phase),

V_{L} = I X_{L}

(leads) and

V_{C} = I X_{C}

(lags). The net source voltage is

V = V_{R}^{2} + (V_{L} - V_{C})^{2} = I Z

, so

Z = R^{2} + (ω L - \frac{1}{ω C})^{2}

The current

I = \frac{V}{Z}

is maximum at the resonant frequency

f_{r} = \frac{1}{2 π L C}

and falls off on either side, giving a sharp peak in the

I

–

f

curve.
(iii) (1) Impedance is minimum,

Z = R

, when

X_{L} = X_{C}

, i.e. at resonance

ω = \frac{1}{L C}

— then current and voltage are in phase. (2) A wattless current flows when

R = 0

(purely reactive circuit), so

ϕ = \frac{π}{2}

and

P_{a v} = V_{r m s} I_{r m s} cos \frac{π}{2} = 0

Z = R^{2} + (ω L - \frac{1}{ω C})^{2}

CBSE Class 12 Board · 2024 · Subjective

CBSE Class 12 · 2024 · Alternating CurrentLong Answer

Solution

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