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Question #43327

Difficulty: 8/10 | 🟢 Free

The Question

A circuit consists of a voltage source $V(t) = V_0 + V_m \cos(\omega t)$ , an inductor of inductance $L = 1 \text{ mH}$ , a capacitor of capacitance $C = 10 \text{ \mu F}$ , and a non-linear resistor. The non-linear resistor has a voltage-current relationship given by $V_R = k I_R^2$ (for $I_R > 0$ ), where $k = 2 \text{ V/A}^2$ . The inductor is connected in series with the voltage source. The capacitor and the non-linear resistor are connected in parallel with each other, and this parallel combination is connected in series with the inductor. Given $V_0 = 50 \text{ V}$ and assuming $V_m \ll V_0$ (small-signal regime), find the angular frequency $\omega$ at which the steady-state AC component of the current drawn from the source is exactly in phase with the AC component of the source voltage.

Options

A.
$5000\sqrt{3} \text{ rad/s}$
B.
$0 \text{ rad/s}$
C.
$10000 \text{ rad/s}$
D.
$4000\sqrt{6} \text{ rad/s}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Phasor Rotation Convention|Frequency Sensitivity of Z|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from global impedance to differential impedance in the presence of non-linear elements.

A common trap is incorrectly applying the global (secant) resistance $R = V_0/I_0 = 10 \text{ \Omega}$ to the AC phasor analysis instead of the differential (tangent) resistance $r = dV/dI = 20 \text{ \Omega}$ .

Using $R = 10 \text{ \Omega}$ yields $\omega = 0$ (Option B).

Another trap is ignoring the resistor entirely, yielding the ideal LC resonance $\omega = 1/\sqrt{LC} = 10000 \text{ rad/s}$ (Option C).

In a non-linear element, standard superposition fails, and AC analysis must be treated strictly as a small-signal perturbation around a DC operating point.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Phasor Rotation Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free