GATE 2024 EC Q.52: solution and answer

GATE 2024 · ECQ.52 · 2 marks · MSQ

Which of the following statements is/are true for a BJT with respect to its DC current gain $β$ ?

(A)
Under high-level injection condition in forward active mode, $β$ will decrease with increase in the magnitude of collector current.
(B)
Under low-level injection condition in forward active mode, where the current at the emitter-base junction is dominated by recombination-generation process, $β$ will decrease with increase in the magnitude of collector current.
(C)
$β$ will be lower when the BJT is in saturation region compared to when it is in active region.
(D)
A higher value of $β$ will lead to a lower value of the collector-to-emitter breakdown voltage.

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Answer and solution

ANSWER

Statements (A), (C) and (D) are true and (B) is false, so the answer is A, C, D.

Official answer: A, C, DSource: Official answer key, GATE 2024 EC (IISc Bengaluru)

Recall the general shape of the $β$ vs $I_{C}$ curve for a BJT in forward active mode: $β$ is low at very small $I_{C}$ (base current dominated by recombination), rises to a peak at moderate $I_{C}$ , and falls again at large $I_{C}$ (high-level injection).
Option A (high-level injection): when the injected minority-carrier density at the base edge of the emitter-base junction becomes comparable to the base doping $N_{A}$ , the effective Gummel number of the base increases with current, so the base current $I_{B}$ grows faster than the ideal $I_{B} \propto e^{V_{B E} / V_{T}}$ law (the Webster effect). Since $β = I_{C} / I_{B}$ , this makes $β$ decrease as $∣ I_{C} ∣$ increases. $\Rightarrow$ (A) is TRUE.
Option B (low-level injection, recombination-dominated $I_{B}$ ): here the base current is $I_{B, rec} \propto e^{V_{B E} /2 V_{T}}$ (non-ideality factor $n = 2$ , from SRH recombination in the emitter-base depletion region), while the ideal diffusion component of $I_{C} \propto e^{V_{B E} / V_{T}}$ (ideality factor $n = 1$ ).
As $V_{B E}$ (and hence $I_{C}$ ) increases from very small values, $I_{C}$ (slope $\propto 1/ V_{T}$ in $ln I$ vs $V_{B E}$ ) rises faster than this recombination-dominated $I_{B}$ (slope $\propto 1/2 V_{T}$ ). Therefore $β = I_{C} / I_{B}$ increases with increasing $I_{C}$ in this regime, not decreases. $\Rightarrow$ (B) is FALSE.
Option C (saturation vs active): in saturation both the emitter-base and collector-base junctions are forward biased, so the base current picks up an extra component from hole injection back into the collector (reverse injection across the collector-base junction) that does not appear in $I_{C}$ . This makes $I_{B}$ rise much faster relative to $I_{C}$ than in the active region, so the forced current gain $β_{forced} = I_{C} / I_{B}$ in saturation is always lower than the active-region $β$ . $\Rightarrow$ (C) is TRUE.
Option D (breakdown voltage relation): in the common-emitter (open-base) configuration, breakdown occurs when the avalanche multiplication factor $M (V_{C E})$ satisfies $M (V_{C E}) \cdot β \to 1$ (each collected carrier must, on average, generate a fresh base current carrier to sustain the loop), giving $B V_{C E O} = \frac{B V _{C B O}}{( β ) ^{1/ n}}$ , where $n$ (typically $3$ to $6$ for silicon) comes from the empirical avalanche multiplication law $M (V) = [1 - (V / B V_{C B O})^{n}]^{- 1}$ .
Since $B V_{C E O}$ scales as $β^{- 1/ n}$ , a larger $β$ directly forces a smaller $B V_{C E O}$ . $\Rightarrow$ (D) is TRUE.
Collecting the verdicts: A, C, D are true statements about $β$ ; B is false.

KEY IDEA

The $β$ vs $I_{C}$ curve of a BJT rises then falls: it is low at very small currents (base current dominated by $n = 2$ recombination, so $β$ actually increases with $I_{C}$ there), peaks at moderate currents, and falls at high currents due to high-level injection (Webster effect). Separately, $β$ drops in saturation (extra reverse base-current injection) and a larger $β$ forces a lower $B V_{C E O}$ through $B V_{C E O} = B V_{C B O} / β^{1/ n}$ .

WATCH OUT FOR

Assuming $β$ always decreases with increasing $I_{C}$ everywhere on the curve (over-generalizing the high-injection roll-off in option A to the low-injection, recombination-dominated regime in option B), which leads students to wrongly mark B as true as well.

From the GATE 2024 EC paper. Every question of this paper · Download the full paper · All 3 BJT questions, with notes