Chapter 6, Problem 34P

A manufacturing plant employs a heater and a conveyer belt motor on the same 220 V service line as shown in Fig. P6.34. The voltage across

the heater and motor are given by ${\nu }_{\text{H}}\left(t\right)=66\cos \left(120\pi t\right)\text{V}$ and ${\nu }_{\text{M}}\left(t\right)=180\cos \left(120\pi t+\frac{\pi }{3}\right)\text{V}$, where $t$ is in seconds.

(a) Write down the amplitude, frequency (in Hz), period (in seconds), phase shift (in radians), and time shift in seconds) of the voltage ${\nu }_{\text{M}}\left(t\right)$.

(b) Plot one cycle of the voltage ${\nu }_{\text{M}}\left(t\right)$, and indicate the earliest time after when the voltage is maximum.

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

(d)The total voltage is ${\nu }_{\text{H}}\left(t\right)+{\nu }_{\text{M}}\left(t\right)$ .Write ${\nu }_{\text{T}}\left(t\right)$ in the form ${\nu }_{\text{T}}\left(t\right)=V\cos \left(120\pi t+\theta \right)$ (i.e., find V and $\theta$ ).