GATE EE 2023 Question Paper (05-Feb-2023) (Shift 1)

Consider the state-space description of an LTI system with matrices

$A=\left[\begin{array}{cc}0& 1\\ -1& -2\end{array}\right],\mathrm{B}=\left[\begin{array}{c}0\\ 1\end{array}\right],\mathrm{C}=\left[\begin{array}{cc}3& -2\end{array}\right],\mathrm{D}=1$

For the input, sin(𝜔𝑡), 𝜔 > 0, the value of 𝜔 for which the steady-state output of the system will be zero, is ___________ (Round off to the nearest integer)

A three-phase synchronous motor with synchronous impedance of 0.1+j0.3 per unit per phase has a static stability limit of 2.5 per unit. The corresponding excitation voltage in per unit is ___________ (Round off to 2 decimal places).

A three phase 415 V, 50 Hz, 6-pole, 960 RPM, 4 HP squirrel cage induction motor drives a constant torque load at rated speed operating from rated supply and delivering rated output. If the supply voltage and frequency are reduced by 20%, the resultant speed of the motor in RPM (neglecting the stator leakage impedance and rotational losses) is __________ (Round off to the nearest integer).

The period of the discrete-time signal 𝑥[𝑛] described by the equation below is 𝑁 = ___________________ (Round off to the nearest integer).

$\mathrm{x}[\mathrm{n}]=1+3\sin \left(\frac{15\pi }{8}\mathrm{n}+\frac{3\pi }{4}\right)-5\sin \left(\frac{\pi }{3}\mathrm{n}-\frac{\pi }{4}\right)$

The discrete-time Fourier transform of a signal 𝑥[𝑛] is 𝑋(Ω) = (1 + 𝑐𝑜𝑠Ω)𝑒${}^{-\text{j}\Omega }$. Consider that 𝑥${}_{\text{p}}$ [𝑛] is a periodic signal of period N = 5 such that

𝑥${}_{\text{p}}$ [𝑛] = 𝑥[𝑛], for 𝑛 = 0, 1 ,2

= 0, for 𝑛 = 3, 4

Note that ${\mathrm{x}}_{\mathrm{p}}[\mathrm{n}]={\mathrm{\Sigma }}_{\mathrm{k}=0}^{\mathrm{N}-1}{\mathrm{a}}_{\mathrm{k}}{\mathrm{e}}^{\mathrm{j}\frac{2\pi }{\mathrm{N}}\mathrm{kn}}$. The magnitude of the Fourier series coefficient 𝑎₃ is _______________ (Round off to 3 decimal places).