# How to solve this problem using transient time-domain analysis?

Find V0(t) in this circuit assuming zero initial conditions.

I solved it (refer image above) using frequency-domain analysis using Laplace and found the answer, but I am not able to get the same answer using time-domain analysis. How to solve it in thr time domain using transient analysis?

• Simulators use transient analysis and would easily solve this problem. Maybe have a quick word with Alfred? Commented Sep 12, 2022 at 13:57
• @Andyaka Alfred's asking, which one? ;P Commented Sep 12, 2022 at 14:02
• Tell Alfred that I use micro-cap 12: zip file or link. Commented Sep 12, 2022 at 15:53

Using nodal analysis at the $$\\small V_x\$$ node: $$\\Sigma \small currents\:away\:from \:node =0 \$$

$$\\large\frac{(V_x -1)}{1}\: +\large\frac{1}{3}\frac{dV_x}{dt}+\large\frac{1}{5}\left(\small V_x +\large\frac{dV_x}{dt}+\frac{1}{3}\frac{d^2V_x}{dt^2}\right)\small=0\$$

Simplifying,

$$\\large\frac{d^2 V_x}{dt^2}\small+8\large\frac{dV_x}{dt}\small +18V_x=15 \$$

Auxiliary equation:

$$\\small m^2 +8m +18=0\$$

$$\\small m=-4\pm j\sqrt{2}\$$

Complementary function:

$$\\small V_{xcf}(t)=e^{-4t}(A cos\sqrt{2}t+Bsin\sqrt{2}t) \$$

Supplement, in response to comment:

$$\\small i_L=\large \frac{1}{5}\small (V_x-V_0)=\large \frac{1}{5}\small \left(V_x-\large\frac{d}{dt}\small i_L\right) =\large \frac{1}{5}\small \left(V_x-\large\frac{d}{dt}\small(i-i_C)\right) =\large \frac{1}{5}\small \left(V_x-\large\frac{di}{dt}+ \frac{di_C}{dt}\right)\$$

where $$\ i\$$ is the source current, $$\i_L \$$ is the inductor current, and $$\\small i_C\$$ is the capacitor current.

Thus,

$$\ i_L=\large \frac{1}{5}\small \left(V_x+\large\frac{dV_x}{dt}+\frac{1}{3} \frac{d^2V_x}{dt^2}\right)\$$

Also, the other two currents are,

$$\\small i=1-V_x \$$, and $$\ i_C=\large\frac{1}{3}\frac{dV_x}{dt}\$$

Finally, node current balance,

$$\i+i_C +i_L=0\$$

gives,

$$\\large\frac{d^2 V_x}{dt^2}\small+8\large\frac{dV_x}{dt}\small +18V_x=15 \$$

• Could you care to explain how you got 1/5(Vx+dVx/dt...) that equation? Commented Sep 14, 2022 at 10:26
• See supplement.
– Chu
Commented Sep 14, 2022 at 13:33

You've already used Kirchhoff for the Laplace domain, why not use it for the time domain, too? I'll use the loop:

\begin{align} &\begin{cases} 1\cdot i_1(t)+\dfrac{1}{\frac13}\int{[i_1(t)-i_2(t)]\mathrm{d}t}&=\theta(t) \\ \dfrac{1}{\frac13}\int{[i_1(t)-i_2(t)]\mathrm{d}t}&=5\cdot i_2(t)+1\cdot \dfrac{\mathrm{d}i_2(t)}{\mathrm{d}t} \end{cases} \\ \Rightarrow \\ &\begin{cases} 1\cdot \dfrac{\mathrm{d}i_1(t)}{\mathrm{d}t}+\dfrac{i_1-i_2}{\frac13}&=\delta(t) \\ \dfrac{i_1-i_2}{\frac13}&=5\cdot \dfrac{\mathrm{d}i_2(t)}{\mathrm{d}t}+1\cdot \dfrac{\mathrm{d}^2i_2(t)}{\mathrm{d}^2t} \end{cases} \tag{1} \\ \Rightarrow \\ i_2(t)&=\dfrac16-\dfrac13\mathrm{e}^{-4t}\left(\sqrt2\sin(\sqrt2 t)+\cos(\sqrt2 t)\right) \tag{2} \end{align}

$$\i_1(t)\$$ can be discarded. Since the voltage across the inductor is the derivative of the current through it:

$$\dfrac{\mathrm{d}i_2(t)}{\mathrm{d}t}=\dfrac{3}{\sqrt2}\mathrm{e}^{-4t}\sin(\sqrt2 t) \tag{3}$$