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Phase voltages of three-phase circuit with Y configuration are equal by absolute value and phase difference between them are \$2\pi/3\$ and \$\theta_A=\theta_B+2\pi/3\$ and \$\theta_A=\theta_C-2\pi/3\$. find complex values of all phase and line voltages if \$u_A=100\cos( 2 \pi ft) V\$ where \$f=50Hz\$.

This is not that hard, I mean, after all, all I have to do is to determine complex voltages. Since I have \$u_A\$ I could easily find \$U_A\$ complex.

\$u_A=50\sqrt{2}\sqrt{2}\cos (2\pi ft) V\$ which means that I already have effective value and phase which is enough for complex representation of this voltage

\$U_A=50\sqrt{2}e^{j0} = 50\sqrt{2}\$ and from this point I could easily find other two complex phase voltages and then I could find line voltages simply by subtracting corresponding phase voltages.
But, the problem is that I don't know what the use of the fact that I know the frequency, I mean there's reason that's the given value, but the way I solved this, it turns out that I don't need it, is there something I did wrong? Why is this frequency given to me as a known value?

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You do not need frequency. When the frequency and time are plugged into the instantaneous equations, the sine waves can be sketched.

Angular velocity: \$2πf = 2π 50Hz = 314 rad/s\$

$$u_A = 100\ cos(314t)\ V$$

This means \$θ_A = 0\$, so \$ θ_B = θ_A\ -\ 2π/3\$ and \$ θ_C = θ_A\ +\ 2π/3\$. So the instantaneous equations for the other two phases are: $$u_B = 100\ cos(314t \ -\ 2π/3)\ V$$ $$u_C = 100\ cos(314t \ +\ 2π/3)\ V$$

with phase voltages as vectors, \$U_A = 100/\sqrt{2}\ \measuredangle 0\ V \$, \$U_B = 100/\sqrt{2}\ \measuredangle -2π/3\ V \$ and \$U_C = 100/\sqrt{2}\ \measuredangle \ 2π/3\ V \$.

In a wye, \$ V_{Line} = \sqrt {3} V_{Phase}\$ and lead phase voltages by 30°.

$$ U_{AB} = U_A - U_B = \sqrt {3}\ 100/\sqrt{2} \ \measuredangle π/6\ V$$ $$ U_{BC} = \sqrt {3}\ 100/\sqrt{2} \ \measuredangle -π/2\ V$$ $$ U_{CA} = \sqrt {3}\ 100/\sqrt{2} \ \measuredangle 5π/6\ V$$

So enter the vectors as complex numbers (\$U_C=100/\sqrt{2}\ e^{j\ 2π/3}\$) into a TI-83 or work out sin and cos components to get real and imagenary components. enter image description here

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