Before You Begin

Question #43453

Difficulty: 8/10 | 🟢 Free

The Question

A small-signal half-wave rectifier circuit uses a p-n junction diode biased at a constant forward DC voltage $V_0$ . The circuit employs an inductor filter and a negligible load resistance such that the ripple factor $\gamma$ of the output is directly proportional to the diode's dynamic forward resistance $r_d$ . The reverse saturation current of the diode is given by $I_s = C T^3 \exp(-E_g/kT)$ , where $C$ is a constant, $k$ is the Boltzmann constant, and $E_g$ is the bandgap energy. Due to thermal expansion and electron-phonon interactions, the bandgap decreases with absolute temperature $T$ as $E_g(T) = E_{g0} - \alpha T$ , where $E_{g0}$ and $\alpha$ are positive constants. Assuming the diode operates in the region where the forward current is $I \approx I_s \exp(eV_0/kT)$ (with $e$ being the elementary charge), what is the fractional change in the ripple factor, $\frac{\Delta \gamma}{\gamma}$ , for a small increase in temperature $\Delta T$ ?

Options

A.
$-\left( \frac{E_{g0} - eV_0 + 2kT}{kT^2} \right) \Delta T$
B.
$-\left( \frac{E_{g0} - \alpha T - eV_0 + 2kT}{kT^2} \right) \Delta T$
C.
$-\left( \frac{E_{g0} - eV_0 + 3kT}{kT^2} \right) \Delta T$
D.
$-\left( \frac{E_{g0} - \alpha T - eV_0 + 3kT}{kT^2} \right) \Delta T$

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

Rectifier Ripple Factor|Energy Band Gap Temperature Dependency|Approximations (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the integration of solid-state device physics with analog circuit analysis, specifically targeting the temperature dependence of the quiescent operating point.

The primary trap lies in the differentiation of the Boltzmann factor containing the temperature-dependent bandgap $E_g(T) = E_{g0} - \alpha T$ .

Students often mistakenly treat $E_g(T)$ as a static variable during differentiation or directly substitute it back at the end, leading to an erroneous $\alpha T$ term in the final expression (Option B).

They fail to realize that $\exp(\alpha T / kT) = \exp(\alpha/k)$ is a temperature-independent constant that vanishes upon logarithmic differentiation.

A secondary trap involves forgetting the explicit $T$ dependence in the numerator of the dynamic resistance formula ( $r_d = kT/eI$ ), which leads to an incorrect coefficient for the $kT$ term (Option C).

Quick Facts About This Question

Question Information

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