Chapter 6, Problem 22P

A sinusoidal voltage $\nu \left(t\right)=10\sin \left(1000t\right)\text{V}$ is applied to the RLU circuit shown in Fig. P622. The current $i\left(t\right)=0.707\sin \left(1000t+45°\right)$ flowing through the circuit produces voltages across R, L, and C of

${\nu }_{\text{R}}\left(t\right)=7.07\sin \left(1000t+45°\right)\text{V}$ ${\nu }_{\text{L}}\left(t\right)=7.07\sin \left(1000t+135°\right)\text{V}$ ${\nu }_{\text{C}}\left(t\right)=14.14\sin \left(1000t-45°\right)\text{V}$.

1. Write down the amplitude, frequency (in Hz), period (in seconds), phase shift (in radians), and time shill (in m/s) of the current $i\left(t\right)=0.707\sin \left(1000t+45°\right)\text{A}$.

(b) Plot one cycle of the current $i\left(t\right)=0.707\sin \left(1000t+45°\right)\text{A}$ and indicate the earliest time (alter $t=0$ ) when the current is 0.707 A.

(c) Using trigonometric identities, show that ${\nu }_1\left(t\right)={\nu }_{\text{R}}\left(t\right)+{\nu }_{\text{C}}\left(t\right)=15\sin 1000t-5\cos 1000t$.

(d) Write ${\nu }_1\left(t\right)$ obtained in part (c) in the form ${\nu }_1\left(t\right)=M\cos \left(1000t+\theta \right)$ ; in other words, find M and $\theta$.