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Question #43458

Difficulty: 8/10 | 🟢 Free

The Question

An NPN transistor operates in the active region with a high-frequency AC base current $I_B(t) = I_0 \sin(\omega t)$ . The time-varying voltage across the collector-base junction is given by $V_{CB}(t) = V_0 \sin(\omega t)$ . The collector-base depletion region acts as an ideal parallel-plate capacitor with cross-sectional area $A$ , width $d$ , and uniform permittivity $\epsilon$ . The transistor has a constant DC current gain $\beta$ . According to the generalized Ampere-Maxwell law, the total collector current must account for both conduction and displacement currents. Assuming the emitter-base junction capacitance is negligible, which of the following expressions gives the peak value of the total emitter current $I_{E,max}$ ?

Options

A.
$\sqrt{ (\beta + 1)^2 I_0^2 + \left( \frac{\epsilon A V_0 \omega}{d} \right)^2 }$
B.
$I_0 + \sqrt{ \beta^2 I_0^2 + \left( \frac{\epsilon A V_0 \omega}{d} \right)^2 }$
C.
$(\beta + 1)I_0 + \frac{\epsilon A V_0 \omega}{d}$
D.
$\sqrt{ \beta^2 I_0^2 + \left( \frac{\epsilon A V_0 \omega}{d} \right)^2 }$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Transistor Current Equation|Ampere’s Law Displacement Current|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of semiconductor device physics with Maxwell's equations and AC circuit analysis.

The primary trap exploits the assumption that a transistor operates purely under steady-state KCL rules ( $I_C = \beta I_B$ ).

At high frequencies, the time-varying electric field in the collector-base depletion region necessitates the inclusion of displacement current (Ampere-Maxwell Law).

A secondary trap tests phasor addition: students often incorrectly calculate the peak collector current first and then linearly add the peak base current (Option B), violating the superposition principle for time-varying signals.

The correct approach requires summing the instantaneous time-domain signals before extracting the peak amplitude via orthogonal phasor addition.

Quick Facts About This Question

Question Information

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