GATE Technical: GATE 2022 Electrical Engineering (EE) questions with solutions

Q1. For an ideal MOSFET biased in saturation, the magnitude of the small signal current gain for a common drain amplifier is

1. 0
2. 1
3. 100
4. infinite

Answer: infinite

For an ideal MOSFET the gate (input) current is zero, so the small-signal input current of a common-drain (source-follower) stage is essentially zero while the output current is finite. The current gain (output/input) is therefore infinite.

Q2. The most commonly used relay, for the protection of an alternator against loss of excitation, is

1. offset Mho relay.
2. over current relay.
3. differential relay.
4. Buchholz relay.

Answer: offset Mho relay.

The offset Mho relay is specifically designed to detect loss of excitation in alternators by monitoring the impedance of the generator, making it the most effective choice for this protection scenario.

Q3. The geometric mean radius of a conductor, having four equal strands with each strand of radius ‘r’, as shown in the figure below, is

1. 4 r
2. 1.414 r
3. 2 r
4. 1.723 r

Answer: 1.723 r

The geometric mean radius for a conductor with multiple strands is calculated using the formula that accounts for the arrangement of the strands. In this case, with four equal strands, the geometric mean radius is determined to be approximately 1.723 times the radius of a single strand.

Q4. The Bode magnitude plot of a first order stable system is constant with frequency. The asymptotic value of the high frequency phase, for the system, is −180°. This system has

1. one LHP pole and one RHP zero at the same frequency.
2. one LHP pole and one LHP zero at the same frequency.
3. two LHP poles and one RHP zero.
4. two RHP poles and one LHP zero.

Answer: one LHP pole and one RHP zero at the same frequency.

The system exhibits a constant magnitude in the Bode plot, indicating a first-order behavior, while the high-frequency phase of -180° suggests the presence of a zero that cancels out the effect of a pole, resulting in one left-half plane (LHP) pole and one right-half plane (RHP) zero at the same frequency.

Q5. In the circuit shown below, a three-phase star-connected unbalanced load is connected to a balanced three-phase supply of 100√3 V with phase sequence ABC. The star connected load has Z_A = 10 Ω and Z_B = 20∠60° Ω. The value of Z_C in Ω, for which the voltage difference across the nodes n and n' is zero, is

1. 20∠−30°
2. 20∠30°
3. 20∠−60°
4. 20∠60°

Answer: 20∠30°

The correct option is 20∠30° because for the voltage difference across nodes n and n' to be zero in a star-connected load, the sum of the phase currents must equal zero. By calculating the phase currents for the given impedances and ensuring they balance out, we find that Z_C must be 20∠30° to achieve this condition.

Q6. A charger supplies 100 W at 20 V for charging the battery of a laptop. The power devices, used in the converter inside the charger, operate at a switching frequency of 200 kHz. Which power device is best suited for this purpose?

1. IGBT
2. Thyristor
3. MOSFET
4. BJT

Answer: MOSFET

MOSFETs are ideal for high-frequency applications like this one due to their fast switching capabilities and low on-resistance, making them efficient for power conversion at 200 kHz.

Q7. A long conducting cylinder having a radius ‘b’ is placed along the z axis. The current density is J = Jₐ r³ ẑ for the region r < b where r is the distance in the radial direction. The magnetic field intensity (H) for the region inside the conductor (i.e. for r < b) is

1. Jₐ/4 r⁴
2. Jₐ/3 r³
3. Jₐ/5 r⁴
4. Jₐ r³

Answer: Jₐ/5 r⁴

Enclosed current I(r) = integral_0^r (Ja r'^3)(2*pi*r') dr' = 2*pi*Ja*r^5/5. Then H = I/(2*pi*r) = Ja*r^4/5. Stored answer (Ja/4 r^4) is wrong; the correct one is Ja r^4 /5.

Q8. The type of single-phase induction motor, expected to have the maximum power factor during steady state running condition, is

1. split phase (resistance start).
2. shaded pole.
3. capacitor start.
4. capacitor start, capacitor run.

Answer: capacitor start, capacitor run.

The capacitor start, capacitor run motor is designed to improve both starting torque and efficiency, resulting in a higher power factor during steady state operation compared to other types of single-phase induction motors.

Q9. The output impedance of a non-ideal operational amplifier is denoted by Z_out. The variation in the magnitude of Z_out with increasing frequency, f, in the circuit shown below, is best represented by

1. (A) log(|Z_out|) versus log(f) is a horizontal line
2. (B) log(|Z_out|) versus log(f) is a straight line with positive slope
3. (C) log(|Z_out|) versus log(f) is low at first, then increases to a higher constant value
4. (D)

Answer: (C) log(|Z_out|) versus log(f) is low at first, then increases to a higher constant value

The correct option indicates that as frequency increases, the output impedance initially remains low due to the operational amplifier's feedback mechanisms, but eventually rises to a higher constant value as the effects of parasitic capacitances and other frequency-dependent factors become significant.

Q10. An LTI system is shown in the figure where G(s) = 100/(s² + 0.1s + 10). The steady state output of the system, to the input r(t), is given as y(t) = a + b sin(10t + θ). The values of ‘a’ and ‘b’ will be

1. a = 1, b = 10
2. a = 10, b = 1
3. a = 1, b = 100
4. a = 100, b = 1

Answer: a = 1, b = 10

The steady state output of an LTI system to a sinusoidal input can be determined using the system's frequency response. For the given transfer function, at the frequency of the input (10 rad/s), the output has a DC component of 1 and an amplitude of 10 for the sinusoidal part, leading to the values a = 1 and b = 10.

Q11. The fuel cost functions in rupees/hour for two 600 MW thermal power plants are given by Plant 1: C1 = 350 + 6P1 + 0.004P1² Plant 2: C2 = 450 + aP2 + 0.003P2² where P1 and P2 are power generated by plant 1 and plant 2, respectively, in MW and a is constant. The incremental cost of power (λ) is 8 rupees per MW. The two thermal power plants together meet a total power demand of 550 MW. The optimal generation of plant 1 and plant 2 in MW, respectively, are

1. 200, 350
2. 250, 300
3. 325, 225
4. 350, 200

Answer: 250, 300

Plant 1: dC1/dP1 = 6 + 0.008 P1 = 8 gives P1 = 250 MW. With total demand 550 MW, P2 = 300 MW (consistent with a + 0.006(300) = 8). The optimal dispatch is 250 MW and 300 MW, option (250, 350) being wrong; correct is (250, 300).

Q12. If the magnetic field intensity (H) in a conducting region is given by the expression, H = x² î + x² y² ĵ + x² y² z² k̂ A/m. The magnitude of the current density, in A/m², at x = 1 m, y = 2 m and z = 1 m, is

1. 8
2. 12
3. 16
4. 20

Answer: 12

J=curl H gives Jx=2x^2 y z^2, Jy=-2x y^2 z^2, Jz=2x y^2. At (1,2,1): J=(4,-8,8), so |J|=sqrt(16+64+64)=12 A/m^2.

Q13. Let a causal LTI system be governed by the following differential equation y(t) + 1/4 dy/dt = 2x(t), where x(t) and y(t) are the input and output respectively. Its impulse response is

1. 2e^(-t/4)u(t)
2. 2e^(-4t)u(t)
3. 8e^(-t/4)u(t)
4. 8e^(-4t)u(t)

Answer: 8e^(-4t)u(t)

The equation y + (1/4)dy/dt = 2x becomes dy/dt + 4y = 8x. The transfer function is 8/(s+4), so the impulse response is 8e^(-4t)u(t). This is option 3, not the stored option 0.

Q14. As shown in the figure below, two concentric conducting spherical shells, centered at r = 0 and having radii r = c and r = d are maintained at potentials such that the potential V(r) at r = c is V1 and V(r) at r = d is V2. Assume that V(r) depends only on r, where r is the radial distance. The expression for V(r) in the region between r = c and r = d is

1. V(r) = cd(V2 − V1)/((d − c)r) − (V1 c + V2 d − 2V1 d)/(d − c)
2. V(r) = cd(V1 − V2)/((d − c)r) + (V2 d − V1 c)/(d − c)
3. V(r) = cd(V1 − V2)/((d − c)r) + (V1 c − V2 c)/(d − c)
4. V(r) = cd(V2 − V1)/((d − c)r) + (V2 c − V1 c)/(d − c)

Answer: V(r) = cd(V1 − V2)/((d − c)r) + (V2 d − V1 c)/(d − c)

The correct option is derived from the principles of electrostatics, where the potential difference between two points in a radial electric field is expressed in terms of the distances and potentials at the boundaries. This option correctly accounts for the linear variation of potential in the region between the two spherical shells, ensuring that it satisfies the boundary conditions set by the potentials at r = c and r = d.