Question 66: AC fundamentals

Question

The instantaneous voltages at three terminals marked $$X,Y$$ and $$Z$$  are given by 
$${V_x} = {V_0}\,\sin \,\omega t,$$ 
$${V_Y} = {V_0}\,\sin $$ $$\left( {\omega t + {{2\pi } \over 3}} \right)$$  
and $$Vz = {V_0}\sin \left( {\omega t + {{4\pi } \over 3}} \right)$$ 
An ideal voltmeter is configured to read $$rms$$ value of the potential difference between its terminals. It is connected between points $$X$$ and $$Y$$ and then between $$Y$$ and $$Z.$$ The reading(s) of the voltmeter will be

Accepted Answer

The potential difference between point X and Y is given by
V
XY
= V
X
$$-$$ V
Y
= V
0
sin($$\omega$$t) $$-$$ V
0
sin($$\omega$$t + $${{2\pi } \over 3}$$)
= 2V
0
cos($$\omega$$t + $${{\pi } \over 3}$$) sin ($$-$$$${{\pi } \over 3}$$)
= $$-$$$$\sqrt 3 $$V
0
cos($$\omega$$t + $${\pi  \over 3}$$) = $$\sqrt 3 $$V
0
sin ($$\omega$$t $$-$$ $${\pi  \over 6}$$).
Similarly, the potential difference between point Y and Z is
V
YZ
= V
Y
$$-$$ V
Z
= V
0
sin($$\omega$$t + $${2\pi  \over 3}$$) $$-$$ V
0
sin($$\omega$$t + $${4\pi  \over 3}$$)
= 2V
0
cos($$\omega$$t + $$\pi$$) sin ($$-$$$${\pi  \over 3}$$)
= $$\sqrt3$$V
0
cos($$\omega$$t) = $$\sqrt3$$V
0
sin($$\omega$$t + $${\pi  \over 2}$$),
and the potential difference between point Z and X is
V
ZX
= V
Z
$$-$$ V
X
= V
0
sin($$\omega$$t + $${4\pi  \over 3}$$) $$-$$ V
0
sin($$\omega$$t)
= $$\sqrt3$$V
0
cos($$\omega$$ + $${2\pi  \over 3}$$) = $$\sqrt3$$V
0
sin($$\omega$$t + $${7\pi  \over 6}$$).
The rms value of the potential V = V
0
sin($$\omega$$t + $$\phi$$) is given by V
rms
= V
0
/ $$\sqrt2$$. Thus, rms values of the potentials are $$V_{XY}^{rms} = V_{YZ}^{rms} = V_{ZX}^{rms} = {V_0}\sqrt {3/2} $$. Hence, the reading of the voltmeter is independent of the two terminal i.e., reading is same whether it is connected across X-Y or Y-Z or Z-X.

Answer

$$V_{xy}^{rms} = {V_0}\sqrt {{3 \over 2}} $$

Answer

$$V_{YZ}^{rms} = {V_0}\sqrt {{1 \over 2}} $$

Answer

$$V_{XY}^{rms} = {V_0}$$

Answer

Independent of the choice of the two terminals

AC fundamentals

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