Chapter 6, Problem 21P

A series RL circuit is subjected to a sinusoidal voltage of frequency $120\pi \text{rad/s}$, as shown in Fig. P6.20. The current $i\left(t\right)=500\sin \left(20\pi t\right)$ mA is flowing through the circuit. The voltage across the resistor and inductor are given by ${\nu }_{\text{R}}\left(t\right)=0.5\sin \left(20\pi t\right)$ and ${\nu }_{\text{L}}\left(t\right)=0.1\sin \left(20\pi t+\frac{\pi }{2}\right)$, where $t$ is in seconds.

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

(b) Plot one cycle of the voltage ${\nu }_{\text{L}}\left(t\right)$, and indicate the earliest time after $t=0$ when the voltage is minimum.

(c) The total voltage across the circuit is given by $\nu \left(t\right)={\nu }_{\text{R}}\left(t\right)+{\nu }_{\text{L}}\left(t\right)$ .Write $\nu \left(t\right)$ in the form $\nu \left(t\right)=M\cos \left(20\pi t+\theta \right)$, in other words, find M and $\theta$.