Chapter 8, Problem 57P

a.

To determine

Prove the expression $\frac{d{v}_C\left(t\right)}{dt}=\left(\frac{{\omega }_0^2}{{\omega }_d}\right)V{e}^{-\alpha t}\sin \left({\omega }_{d}t\right)$.

b.

To determine

Prove the expression $\frac{d{v}_C\left(t\right)}{dt}=0$ when $t=\left(\frac{n\pi }{{\omega }_d}\right),\text{where }n=0,1,2,\dots$.

c.

To determine

Prove the expression $v_C\left(t_n\right)=V-V{\left(-1\right)}^n{e}^{-\frac{\alpha n\pi }{{\omega }_d}}$ when $t_n=\frac{n\pi }{{\omega }_d}$.

d.

To determine

Prove the expression $\alpha =\frac{1}{T_d}\ln \left[\frac{v_C\left(t_1\right)-V}{v_C\left(t_3\right)-V}\right]$ when $T_d=t_3-t_1$.