Chapter 6, Problem 18P

A sinusoidal current $i\left(t\right)=0.1\sin \left(100t\right)$ amps is applied to the RC circuit shown in Fig. P6.18. The voltage across the resistor and capacitor are given by

$\begin{array}{l}{\nu }_{\text{R}}\left(t\right)=20\sin \left(100t\right)\text{V}\\ {\nu }_{\text{C}}\left(t\right)=-20\cos \left(100t\right)\text{V}\end{array}$ where $t$ is in seconds.

(a) The voltage applied to the circuit is given by $\nu \left(t\right)={\nu }_{\text{R}}\left(t\right)+{\nu }_C\left(t\right)$. Write $\nu \left(t\right)$ in the form $\nu \left(t\right)=M\sin \left(100t+\theta \right)$ ; in other words, find $\text{M}$ and $\theta$.

(b) Suppose now that $\nu \left(t\right)=28.28\sin \left(100t-\frac{\pi }{4}\right)$ volts. Write down the amplitude, frequency (in Hz), period (in seconds), phase angle (in degrees), and time shift (in seconds) of the voltage $\nu \left(t\right)$.

(c) Plot one cycle of the voltage $\nu \left(t\right)=28.28\sin \left(100t-\frac{\pi }{4}\right)$, and indicate the earliest time (after $t=0$ ) when voltage is 28.28 V.