GATE EC 2023 Question Paper (05-Feb-2023) (Shift 2)

A closed loop system is shown in the figure where k > 0 and α > 0. The steady state error due to a ramp input (R(s) = α/s²) is given by

In the following block diagram, R(s) and D(s) are two inputs. The output Y(s) is expressed as Y(s) = G₁(s)R(s) + G₂(s)D(s).

G₁(s) and G₂(s) are given by

The state equation of a second order system is

𝒙̇ (𝑡) = A𝒙(𝑡), 𝒙(0) is the initial condition.

Suppose λ₁ and λ₂ are two distinct eigenvalues of A and 𝒗₁ and 𝒗₂ are the corresponding eigenvectors. For constants 𝛼₁ and 𝛼₂, the solution, 𝒙(𝑡), of the state equation is

The switch S₁ was closed and S₂ was open for a long time. At 𝑡 = 0, switch S₁ is opened and S₂ is closed, simultaneously. The value of i${}_c$ (0⁺), in amperes, is

Let a frequency modulated (FM) signal

$\mathrm{x}(\mathrm{t})=\mathrm{A}\cos ({\omega }_{\mathrm{c}}\mathrm{t}+{\mathrm{k}}_{\mathrm{f}}{\int }_{-\infty }^{\mathrm{t}}\mathrm{m}(\lambda )\mathrm{d}\lambda )$, where 𝑚(𝑡) is a message signal of bandwidth W. It is passed through a non-linear system with output 𝑦(𝑡) = 2𝑥(𝑡) + 5(𝑥(𝑡))² . Let 𝐵${}_{\text{T}}$ denote the FM bandwidth. The minimum value of 𝜔${}_{\text{c}}$ required to recover 𝑥(𝑡) from 𝑦(𝑡) is