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Question #43903

Difficulty: 8/10 | 🟢 Free

The Question

A circuit consists of a DC voltage source $V_0 = 12 \text{ V}$ , an ideal inductor $L = 2 \text{ H}$ , and a non-linear electronic device $D$ connected in series. The voltage-current characteristic of the device is given by $V_D = k I^2$ (for $I > 0$ ), where $k = 3 \text{ V/A}^2$ . At $t = 0$ , the circuit is closed. Let $t_1$ be the time taken for the current to reach $1 \text{ A}$ . After a long time, the current reaches a steady state. At this operating point, a small AC voltage $v(t) = v_0 \sin(\omega t)$ with angular frequency $\omega = 6 \text{ rad/s}$ is superimposed on the DC source. Let $\phi$ be the phase difference between the small-signal AC current and the AC voltage. Find the value of $t_1$ and the phase angle $\phi$ .

Options

A.
$\frac{1}{6} \ln(3) \text{ s}, \; \frac{\pi}{4}$
B.
$\frac{1}{6} \ln(3) \text{ s}, \; \tan^{-1}(2)$
C.
$\frac{2}{3} \ln(3) \text{ s}, \; \frac{\pi}{4}$
D.
$\frac{2}{3} \ln(3) \text{ s}, \; \tan^{-1}(2)$

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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

Load Line Analysis|Series LCR Impedance Vector|Transient Current in RL Circuit

Why This Question Is Tricky

Common Mistakes

This question tests the ability to analyze non-linear circuits in both transient and small-signal AC steady states.

The primary trap lies in the AC phase calculation: students often incorrectly apply a stationary DC load line by using the static resistance $R_{static} = V_D / I_0 = 6 \; \Omega$ , which yields an incorrect phase angle of $\tan^{-1}(2)$ .

For small AC signals superimposed on a DC bias, the device operates along the tangent of its V-I curve, requiring the dynamic resistance $r_d = dV_D/dI = 12 \; \Omega$ .

Additionally, the transient phase requires setting up and solving a first-order non-linear separable differential equation, testing rigorous application of Kirchhoff's laws beyond standard exponential LCR decays.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12
Concept	Load Line Analysis
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free