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

The expressions of fuel cost of two thermal generating units as a function of the respective power generation 𝑃${}_{\text{G}1}$ and 𝑃${}_{\text{G}2}$ are given as

𝐹₁ (𝑃${}_{\text{G}1}$ ) = 0.1 $\mathrm{a}{\mathrm{P}}_{G1}^2$ + 40 𝑃${}_{\text{G}1}$ + 120 𝑅𝑠/ℎ𝑜𝑢𝑟 0 𝑀𝑊 ≤ 𝑃${}_{\text{G}1}$ ≤ 350 MW

𝐹₂ (𝑃${}_{\text{G}2}$) = 0.2 ${\mathrm{P}}_{G2}^2$ + 30 𝑃${}_{\text{G}2}$ + 100 𝑅𝑠/ℎ𝑜𝑢𝑟 0 𝑀𝑊 ≤ 𝑃${}_{\text{G}2}$ ≤ 300 MW

where a is a constant. For a given value of a, optimal dispatch requires the total load of 290 MW to be shared as 𝑃${}_{\text{G}1}$ = 175 𝑀𝑊 and 𝑃${}_{\text{G}2}$ = 115 𝑀𝑊. With the load remaining unchanged, the value of a is increased by 10% and optimal dispatch is carried out. The changes in 𝑃${}_{\text{G}1}$ and the total cost of generation, F (= F₁ + F₂) in Rs/hour will be as follows

The four stator conductors (A, A', B and B') of a rotating machine are carrying DC currents of the same value, the directions of which are shown in the figure (i). The rotor coils a-a' and b-b' are formed by connecting the back ends of conductors ‘a’ and ‘a'’ and ‘b’ and ‘b'’, respectively, as shown in figure (ii). The e.m.f. induced in coil a-a' and coil b-b' are denoted by E${}_{a-a^{\prime }}$ and E${}_{b-b^{\prime }}$, respectively. If the rotor is rotated at uniform angular speed ω rad/s in the clockwise direction then which of the following correctly describes the E${}_{a-a^{\prime }}$ and E${}_{b-b^{\prime }}$ ?



figure (i): cross-sectional view figure (ii): rotor winding connection diagram

The chopper circuit shown in figure (i) feeds power to a 5 A DC constant current source. The switching frequency of the chopper is 100 kHz. All the components can be assumed to be ideal. The gate signals of switches S₁ and S₂ are shown in figure (ii). Average voltage across the 5 A current source is

In the figure, the vectors u and v are related as: Au = v by a transformation matrix A. The correct choice of A is

One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation 𝜎₀.

The 𝜎₀ is estimated by randomly drawing out 10,000 numbers of samples (𝑥${}_{\text{n}}$). The estimates 𝜎̂₁, 𝜎̂₂ are computed in the following two ways.

${\hat{\sigma }}_1^2=\frac{1}{10000}{\mathrm{\Sigma }}_{\mathrm{n}=1}^{10000}{\mathrm{x}}_{\mathrm{n}}^2$

${\hat{\sigma }}_2^2=\frac{1}{9999}{\mathrm{\Sigma }}_{\mathrm{n}=1}^{10000}{\mathrm{x}}_{\mathrm{n}}^2$