# General second order circuit problem - Find V(t) for t > 0

1. Mar 31, 2007

### VinnyCee

1. The problem statement, all variables and given/known data

For the circuit above, find $v(t)$ for $t\,>\,0$.

Assume that $v\left(0^+\right)\,=\,4\,V$ and $i\left(0^+\right)\,=\,2\,A$.

2. Relevant equations

$$i_c\,=\,C\,\frac{d\,v_c}{dt}$$

3. The attempt at a solution

$$i_1\,=\,C_1\,\frac{d\,V_{C_1}}{dt}$$

$$i_2\,=\,\frac{V_1\,-\,V_2}{2\,\Omega}$$

$$i\,=\,C_2\,\frac{d\,V_{C_2}}{dt}$$

KCL @ $V_1$) $$i_1\,+\,\frac{i}{4}\,-\,i_2\,=\,0$$

$$C_1\,\frac{d\,V_{C_1}}{dt}\,+\,\frac{C_2}{4}\,\frac{d\,V_{C_2}}{dt}\,-\,\frac{V_1\,-\,V_2}{2\,\Omega}\,=\,0$$

$$0.2\,\frac{d\,V_{C_1}}{dt}\,+\,0.25\frac{d\,V_{C_2}}{dt}\,-\,V_1\,+\,V_2\,=\,0$$

KCL @ $V_2$) $$i\,+\,i_2\,=\,0$$

$$C_2\,\frac{d\,V_{C_2}}{dt}\,+\,\frac{V_1\,-\,V_2}{2\,\Omega}\,=\,0$$

$$\frac{d\,V_{C_2}}{dt}\,+\,V_1\,-\,V_2\,=\,0$$

$$V_2\,=\,\frac{d\,V_{C_2}}{dt}\,+\,V_1$$

Now substituting the KCL @ $V_2$ equation into the other equation:

$$0.2\,\frac{d\,V_{C_1}}{dt}\,+\,1.25\,\frac{d\,V_{C_2}}{dt}\,=\,0$$

Here I am stuck. I don't know how to proceed, any hints? I know that I am supposed to get a second order differential equation for the circuit, but where from?

Last edited: Mar 31, 2007
2. Apr 1, 2007

### Ne0

Have you done circuits in the s-domain? That would make this problem much easier to solve for. It would become a simple two-mesh problem in which you could use Cramer's rule to solve for with the constraint equation for the dependant source. And I am pretty sure it is not going to be a second order differential equation since there is only one reactive element per mesh.

3. Apr 3, 2007

### VinnyCee

No, I have not done s-domain yet.

4. Apr 3, 2007

### Staff: Mentor

I think you need to use your substitution to eliminate one of the unknowns, instead of getting an equation that still has both in it. And then you'll need to make an assumption about the form of the solution, differentiate it and plug it back in to solve, and then use the initial conditions for the full form of the final solution. By looking at the circuit, I'd guess the solution is a damped exponential, but it might have other terms....

5. Apr 3, 2007

### SGT

With the reference senses you used, you have

$$i_1\,=\,-C_1\,\frac{d\,V_{C_1}}{dt}$$

$$i\,=\,-C_2\,\frac{d\,V_{C_2}}{dt}$$

and $$i_2 = -i$$

You can use this with

KCL @ $V_2$) $$i\,+\,i_2\,=\,0$$

to eliminate $$i_1$$ and $$i_2$$

6. Apr 3, 2007

### antoker

I'll second to that, s-domain (laplace) will make your life much more easier ;)

7. Apr 3, 2007

### VinnyCee

Okay, I'll try s-domain.

$$i_c\,=\,\frac{V_1\,-\,0}{\frac{10}{s}}\,=\,\frac{V_1\,s}{10}$$

$$i\,=\,\frac{V_1\,-\,0}{\frac{2}{s}\,+2}\,=\,\frac{V_1\,s}{2\,+\,2s}$$

KCL @ $V_1$:

$$-i_c\,-\,\frac{i}{4}\,-\,i\,=\,0$$

$$-i_c\,-\,\frac{5}{4}\,i\,=\,0$$

$$-\frac{V_1\,s}{10}\,-\,\frac{5}{4}\,\frac{V_1\,s}{2\,+\,2s}\,=\,0$$

How do I proceed now?

8. Apr 4, 2007

### SGT

Since you are looking for a zero input response, you must include the initial conditions in your equation.
Or you can use the hint I gave you in my previous post and solve the problem in the time domain.

9. Apr 4, 2007

### VinnyCee

Using your hint in a previous post, I obtained a new KCL @ $V_1$ equation:

$$i_1\,-\,\frac{5}{4}\,i\,=\,0$$

$$-C_1\,\frac{dV_{c1}}{dt}\,-\,\frac{5}{4}\,C_2\,\frac{dV_{c2}}{dt}\,=\,0$$

How do I get initial conditions and how to finally solve?

10. Apr 4, 2007

### SGT

You must have only one variable.
Remember that
$$V_{c1} = V_{c2} - R i = V_{c2} + RC_2\frac{dV_{c2}}{dt}$$
so,

$$\frac{dV_{c1}}{dt} = \frac{dV_{c2}}{dt} + RC_2\frac{d^2V_{c1}}{dt^2}$$
For the initial conditions you must have
$$V_{c2}(0^+)$$ and $$\frac{dV_{c2}}{dt}(0^+)$$

$$V_{c2} = V_{c1} + R i$$
So, $$V_{c2}(0^+) = V_{c1}(0^+) + R i(0^+) = 4 + 2x2 = 8V$$
$$\frac{dV_{c2}}{dt} = -\frac{i}{C_2}$$
So, $$\frac{dV_{c2}}{dt}(0^+) = -\frac{i}(0^+){C_2} = \frac{2}{0.5} = 4V/s$$

Last edited by a moderator: Apr 4, 2007