The following series $\text{RLC}$ circuit with zero initial conditions is excited by a unit impulse function $\delta(t)$.
For $t>0$, the voltage across the resistor is
- $\frac{1}{\sqrt{3}}\left(e^{-\frac{\sqrt{3}}{2} t}-e^{-\frac{1}{2} t}\right)$
- $e^{-\frac{1}{2} t}\left[\cos \left(\frac{\sqrt{3}\; t}{2}\right)-\frac{1}{\sqrt{3}} \sin \left(\frac{\sqrt{3}\; t}{2}\right)\right]$
- $\frac{2}{\sqrt{3}} e^{-\frac{1}{2} t} \sin \left(\frac{\sqrt{3}\; t}{2}\right)$
- $\frac{2}{\sqrt{3}} e^{-\frac{1}{2} t} \cos \left(\frac{\sqrt{3} \;t}{2}\right)$