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Question #43455

Difficulty: 8/10 | 🟢 Free

The Question

An NPN transistor is operating in the active region within a common-emitter amplifier circuit. The base current is maintained strictly constant at $I_B = 10 \ \mu\text{A}$ . The collector is connected to a DC supply $V_{CC} = 30 \ \text{V}$ through a temperature-dependent load resistor $R_C$ . At a reference temperature $T_0$ , the resistance is $R_C(T_0) = 5 \ \text{k}\Omega$ . The resistance increases with temperature according to $R_C(T) = R_C(T_0)[1 + \alpha(T - T_0)]$ , where the temperature coefficient is $\alpha = 0.02 \ ^\circ\text{C}^{-1}$ .

Due to base-width modulation (the Early effect), the macroscopic current gain $\beta$ is not constant but depends on the collector-emitter voltage $V_{CE}$ as $\beta(V_{CE}) = \beta_0 \left(1 + \frac{V_{CE}}{V_A}\right)$ , where the zero-bias gain is $\beta_0 = 100$ and the Early voltage is $V_A = 150 \ \text{V}$ .

If the operating temperature is raised to $T = T_0 + 50 \ ^\circ\text{C}$ , what is the effective macroscopic current gain $\beta_{eff} = \frac{I_C}{I_B}$ of the transistor in the steady state?

Options

A.
$112.5$
B.
$113.3$
C.
$116.1$
D.
$100.0$

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 Configuration Gains|Drift vs Diffusion Current|Internal Resistance Variation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from ideal circuit theory to semiconductor device physics by incorporating the Early effect (base-width modulation).

The primary trap lies in treating the system sequentially rather than self-consistently.

Students commonly calculate the collector-emitter voltage $V_{CE}$ using the ideal zero-bias collector current ( $I_{C0} = \beta_0 I_B$ ), yielding an incorrect $V_{CE} = 20 \ \text{V}$ , which leads to a false gain of $113.3$ .

In reality, the widening of the depletion region alters the diffusion current gradient in the base, increasing $I_C$ .

This increased $I_C$ causes a larger voltage drop across the temperature-modified load resistor $R_C$ , which in turn reduces $V_{CE}$ .

Recognizing this dynamic feedback loop requires substituting the KVL equation directly into the Early effect transport equation and solving them simultaneously.

Quick Facts About This Question

Question Information

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