GATE IN 2022 Question Paper (06-Feb-2022) (Shift 2)

A proportional-integral-derivative (PID) controller is employed to stably control a plant with transfer function

$\mathrm{P}(\mathrm{s})=\frac{1}{(\mathrm{s}+1)(\mathrm{s}+2)}$

Now, the proportional gain is increased by a factor of 2, the integral gain is increased by a factor of 3, and the derivative gain is left unchanged. Given that the closed-loop system continues to remain stable with the new gains, the steady-state error in tracking a ramp reference signal ________

A resistor ladder digital-to-analog converter (DAC) receives a digital input that results in the circuit having the state as shown in the figure. For this digital input, the Thevenin voltage, Vₜₕ and Thevenin resistance, Rₜₕ as seen at the output node are ______

The Nyquist plot of a stable open-loop system 𝐺(𝑗𝜔) is plotted in the frequency range 0 ≤ 𝜔 < ∞ as shown. It is found to intersect a unit circle with center at the origin at the point 𝑃 = −0.77 − 0.64𝑗 . The points 𝑄 and 𝑅 lie on 𝐺(𝑗𝜔) and assume values 𝑄 = 14.40 + 0.00𝑗 and 𝑅 = −0.21 + 0.00𝑗. The phase margin (PM) and the gain margin (GM) of the system are _____

In the small signal circuit shown, the enhancement mode n-channel MOSFET is biased in saturation with transconductance g${}_m$. If channel length modulation is ignored, the small signal impedance looking into the node P is given by _____

Consider the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{yln}(\mathrm{y})=0$

If 𝑦(0) = 𝑒, then 𝑦(1) is ________