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Three terminals X, Y and Z carry the instantaneous potentials Vx = V0*sin(wt), Vy = V0*sin(wt + 2*pi/3) and Vz = V0*sin(wt + 4*pi/3). An ideal voltmeter reads the rms value of the potential difference across it. It is first placed across X and Y, then across Y and Z. Which statement(s) about its reading(s) is/are correct?

1. VrmsXY = V0
2. VrmsYZ = V0*sqrt(1/2)
3. Independent of the choice of the two terminals
4. VrmsXY = V0*sqrt(3/2)

Correct answer: VrmsXY = V0*sqrt(3/2)

Solution

The phase difference between any two of the symmetric phasors is 120 deg. The difference of two phasors of amplitude V0 separated by 120 deg has amplitude V0*sqrt(1² + 1² - 2*cos(120 deg)) = V0*sqrt(3). Its rms value is V0*sqrt(3)/sqrt(2) = V0*sqrt(3/2). By symmetry this is the same for XY and YZ, so it is independent of the chosen pair, but the listed value VrmsYZ = V0*sqrt(1/2) is wrong.

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