GATE IN 2025 Question Paper (01-Feb-2025) (Shift 2)

Consider an AC bridge shown in the figure with $R=300\Omega$, $R1=1000\Omega$, $R2=500\Omega$, $L=30mH$, and a detector D. At the bridge balance condition, the frequency of the excitation source $V_s$ is _________ kHz (rounded off to two decimal places).

An air filled parallel plate electrostatic actuator is shown in the figure. The area of each capacitor plate is $100\mu m\times 100\mu m$. The distance between the plates $d_0=1\mu m$ when both the capacitor charge and spring restoring force are zero as shown in Figure (a). A linear spring of constant $k=0.01N/m$ is connected to the movable plate. When charge is supplied to the capacitor using a current source, the top plate moves as shown in Figure (b). The magnitude of minimum charge (Q) required to momentarily close the gap between the plates is _________ $\times 10^{-14}C$ (rounded off to two decimal places).
Note: Assume a full range of motion is possible for the top plate and there is no fringe capacitance. The permittivity of free space is ${ϵ}_0=8.85\times 10^{-12}F/m$ and relative permittivity of air (${ϵ}_r$) is 1.

The resistance of a thermistor is measured to be $2.25k\Omega$ at $30°C$ and $1.17k\Omega$ at $60°C$. Its material constant $\beta$ is _________ K (rounded off to two decimal places).

A feedback control system is shown in the figure.

The maximum allowable value of n such that the output $y(t)$, due to any step disturbance signal $d(t)$, becomes zero at steady-state, is _________ (in integer).

The circuit given in the figure is driven by a voltage source $V_s=25\sqrt{2}\angle 30°V$. The system is operating at a frequency of 50 Hz. The transformers are assumed to be ideal. The average power dissipated, in W, in the $50k\Omega$ resistance is _________ (rounded off to two decimal places).