GATE EC 2022 Question Paper (06-Feb-2022) (Shift 1)

A p-type semiconductor with zero electric field is under illumination (low level injection) in steady state condition. Excess minority carrier density is zero at x = ±,2 lₙ where lₙ = 10⁻⁴ cm is the diffusion length of electrons. Assume electronic charge, q = −1.6 × 10⁻¹⁹ C. The profiles of photo-generation rate of carriers and the recombination rate of excess minority carriers (R) are shown. Under these conditions, the magnitude of the current density due to the photo-generated electrons at x = +2 lₙ is _________ mA/cm² (rounded off to two decimal places).

Consider the circuit shown with an ideal OPAMP. The output voltage Vₒis __________V (rounded off to two decimal places).

Consider the circuit shown with an ideal long channel nMOSFET (enhancement-mode, the substrate is connected to the source). The transistor is appropriately biased in the saturation region with V${}_{GG}$ and V${}_{DD}$ such that it acts as a linear amplifier. vᵢ is the small-signal ac input voltage. v${}_A$ and v${}_B$ represent the small-signal voltages at nodes A and B, respectively. The value of $\frac{v_A}{v_B}$,is ________ (rounded off to one decimal place).

The block diagram of a closed-loop control system is shown in the figure. R(s), Y(s), and D(s) are the Laplace transforms of the time-domain signals r(t), y(t), and d(t), respectively. Let the error signal be defined as e(t) = r(t) − y(t). Assuming the reference input r(t) = 0 for all t, the steady-state error e(∞), due to a unit step disturbance d(t), is _________ (rounded off to two decimal places).

Consider a channel over which either symbol x${}_A$ or symbol x${}_B$ is transmitted. Let the output of channel Y be the input to a maximum likelihood (ML) detector at the receiver. The conditional probability density functions for 𝑌 given x${}_A$ and x${}_B$ are:

${\mathrm{f}}_{Y|{\mathrm{x}}_{\mathrm{A}}}(\mathrm{y})={\mathrm{e}}^{-(\mathrm{y}+1)}\mathrm{u}(\mathrm{y}+1)$

${\mathrm{f}}_{Y|{\mathrm{x}}_{\mathrm{B}}}(\mathrm{y})={\mathrm{e}}^{(\mathrm{y}-1)}(1-\mathrm{u}(\mathrm{y}-1))$

where u(⋅) is the standard unit step function. The probability of symbol error for this system is _________ (rounded off to two decimal places).