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Question #43459

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The Question

A silicon NPN transistor is used in a common-emitter amplifier circuit with a DC collector supply voltage $V_{CC}$ and a load resistor $R_C$ . The base current $I_B$ is maintained constant. The total collector current is $I_C = \beta I_B + I_{CEO}$ , where $\beta$ is the current gain (assumed temperature-independent) and $I_{CEO}$ is the reverse leakage current. Let $V_{CE}$ be the operating collector-emitter voltage. Based on the kinetic theory of charge carriers, $I_{CEO}$ depends on the junction temperature $T$ as $I_{CEO}(T) = C T^3 \exp\left(-\frac{E_g}{k_B T}\right)$ , where $C$ is a constant, $E_g$ is the bandgap energy, and $k_B$ is the Boltzmann constant. The transistor dissipates heat to the ambient environment with a constant thermal resistance $\theta$ (in $\text{K/W}$ ). Assuming $E_g \gg 3 k_B T$ , which of the following represents the condition for the onset of thermal runaway?

Options

A.
$\theta (2 V_{CE} - V_{CC}) I_{CEO} \frac{E_g}{k_B T^2} \ge 1$
B.
$\theta V_{CE} I_{CEO} \frac{E_g}{k_B T^2} \ge 1$
C.
$\theta (2 V_{CE} - V_{CC}) I_C \frac{E_g}{k_B T^2} \ge 1$
D.
$\theta (V_{CC} - 2 V_{CE}) I_{CEO} \frac{E_g}{k_B T^2} \ge 1$

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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

Transistor Current Equation|Temperature and Kinetic Energy|Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the physical understanding of thermal runaway in semiconductor devices, specifically targeting two common misconceptions.

First, students often incorrectly assume the collector-emitter voltage $V_{CE}$ is constant when differentiating the junction power $P_j = V_{CE} I_C$ , leading to $\frac{dP_j}{dI_C} = V_{CE}$ (Option B).

They fail to account for the non-linear feedback where an increase in $I_C$ increases the voltage drop across the load resistor $R_C$ , thereby reducing $V_{CE}$ .

Second, students might incorrectly apply the exponential temperature dependence derived from kinetic theory to the entire collector current $I_C$ , rather than strictly to the minority carrier leakage current $I_{CEO}$ (Option C).

The correct application requires linking the intrinsic carrier concentration strictly to the leakage component and applying the thermodynamic stability criterion $\frac{dP_j}{dT} \ge \frac{dP_{diss}}{dT}$ .

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