Chapter 6, Problem 38P

In the three-phase circuit shown in Fig.P.36. ${\nu }_{ca}\left(t\right)={\nu }_{cn}\left(t\right)-{\nu }_{an}\left(t\right)$, where ${\nu }_{cn}\left(t\right)=120\sin \left(120\pi t-120°\right)\text{V}$ and ${\nu }_{an}\left(t\right)=120\cos \left(120\pi t-90°\right)\text{V}$ are the line-to-neutral voltages and ${\nu }_{ca}$ the line-to-line voltage of the three- phase system. (a) Write down the amplitude, frequency (in Hz), period, phase angle (in degrees), phase shift (in seconds) of the voltage ${\nu }_{an}\left(t\right)$.

(b) Plot one cycle of the line-to-neutral voltage ${\nu }_{an}\left(t\right)$, and indicate the earliest time after when the voltage reaches 60 V.

(c) Write the line-to-line voltage ${\nu }_{ca}\left(t\right)$ in the form ${\nu }_{ca}\left(t\right)=A\sin \left(120\pi t\right)+B\cos \left(120\pi t\right)$ (i.e., Find A and B).

(d)Write ${\nu }_{ca}\left(t\right)$ in the form ${\nu }_{ca}\left(t\right)=V\cos \left(120\pi t+\theta \right)$ (i.e., find V and $\theta$ ).