GATE 2024 IN Q.47: solution and answer

GATE 2024 · INQ.47 · 2 marks · MCQ

In the figure shown, assume

$α$ is the phase angle between the load current and the load voltage
$β$ is the phase angle by which pressure coil current lags the pressure coil voltage of the wattmeter
$γ$ is the phase angle between currents in the pressure coil and the current coil of the wattmeter
$δ$ is the phase angle of the voltage transformer
$θ$ is the phase angle of the current transformer

When the load has a lagging phase angle of $α$ , which one of the following options is correct?

GATE 2024 IN question 47: A single-phase supply (vertical double-headed arrow labelled 'Single Phase Supply') on the left, with a top line and a… — Figure from the GATE 2024 IN paper. It is part of the question.

(A)
$α = - γ \pm δ \pm θ - β$
(B)
$α = - γ \pm δ \pm θ + β$
(C)
$α = γ \pm δ \pm θ + β$
(D)
$α = γ \pm δ \pm θ - β$

Answer and solution

ANSWER

Option (C): $α = γ \pm δ \pm θ + β$ .

Official answer: CSource: Official answer key, GATE 2024 IN (IISc Bengaluru)

Take the load voltage $V$ as the reference phasor, at angle $0$ . The load is lagging, so the load current $I$ sits at angle $- α$ (it lags $V$ by $α$ ).
Voltage (potential) transformer: the pressure-coil circuit, and the voltmeter, are connected across the PT secondary. The reversed secondary voltage $V_{s}$ differs in phase from $V$ by the PT phase angle $δ$ , and it can lead or lag. Let $V_{s}$ lead $V$ by $δ^{'}$ , where $δ^{'} = \pm δ$ (positive when it leads). Then $V_{s}$ is at angle $δ^{'}$ .
Current transformer: the current coil and the ammeter carry the CT secondary current. The reversed secondary current $I_{s}$ differs in phase from $I$ by the CT phase angle $θ$ , and it can also lead or lag. Let $I_{s}$ lead $I$ by $θ^{'}$ , where $θ^{'} = \pm θ$ . Then $I_{s}$ is at angle $- α + θ^{'}$ .
Inside the wattmeter: the pressure coil has some inductance, so its current $I_{p}$ lags its applied voltage $V_{s}$ by $β$ . That puts $I_{p}$ at angle $δ^{'} - β$ .
With a lagging load, the current-coil current $I_{s}$ lags the pressure-coil current $I_{p}$ by $γ$ , the angle between the two coil currents. So $I_{s}$ is also at angle $δ^{'} - β - γ$ .
Set the two expressions for the angle of $I_{s}$ equal: $- α + θ^{'} = δ^{'} - β - γ \Rightarrow α = γ + β + θ^{'} - δ^{'}$
The signs of $θ^{'} = \pm θ$ and $- δ^{'} = \mp δ$ depend only on whether each transformer's secondary quantity leads or lags, and the two are independent. So $α = γ \pm δ \pm θ + β$ This is option (C).
Check: with an ideal PT and CT ( $δ = θ = 0$ ) and a purely resistive pressure coil ( $β = 0$ ), $I_{p}$ is in phase with $V$ and $I_{s}$ is in phase with $I$ . The coil currents are then exactly $α$ apart, and the formula correctly gives $α = γ$ .

KEY IDEA

Draw every phasor relative to the load voltage. The CT and PT each rotate their secondary quantity by their own phase angle, which can go either way, hence the $\pm$ . The pressure-coil inductance makes $I_{p}$ lag its voltage by $β$ . So the angle between the coil currents, $γ$ , is the load angle $α$ reduced by $β$ and shifted by the two transformer angles: $α = γ + β \pm δ \pm θ$ .

WATCH OUT FOR

Getting the sign of $β$ wrong, which leads to option (D). Because the pressure-coil current lags its voltage by $β$ , it swings toward the lagging load current, so the gap between the coil currents is smaller than the load angle: $γ = α - β$ (ignoring transformer errors), not $α + β$ . Options (A) and (B) also flip the sign of $γ$ . That would make the coil currents close to $- α$ apart for a lagging load, which is impossible.

From the GATE 2024 IN paper. Every question of this paper · Download the full paper · All 3 Instrument transformers questions, with notes