How to Solve a First Order Transient Circuit Using Differential Equations?

[RadiationX]

I'm having serious trouble understanding how to solve this problem using the differential equation method ( I MUST use this method). I provided the answer but my solution attempts are not producing the same result.

Here is the problem. http://img102.imageshack.us/img102/4176/testproblembe8.th.jpg

The first thing I need to do is find the voltage across the capacitor at time $$t_{0^-}$$. By combining the 4k and 6k resistors and using voltage division I see that the voltage across the capacitor for $$t_(0^-)= 8V$$

Now I'm confused here, should I also find the current in the circuit for $$t_(0^-)$$?


Let me assume that I don't need this parameter and then I go on the analyze this circuit for $$t_(0^+)$$

For this circuit all we have is one loop consisting of the capacitor and the 4k and 6k resistors.
Now I can write and equation for the current around this loop:
$$C\frac{dV_c(t)}{dt} + 6ki(t)=0$$

 

[Kenny Lee]

Have you tried using a combination of Kirchoff's voltage and current laws? I tried it and got three equations and three unknowns. But its crazy to solve. I doubt its the right way, but maybe you can give it a shot?

How about Laplace transformations? Find the transfer function for capacitor and output, and then inverse laplace it for the final answer.

 

[piercas]

Hi there,

It seems like you are having some difficulty with understanding how to solve a first order transient circuit using the differential equation method. Don't worry, this is a common problem and I'm here to help.

First, let's start by understanding what a first order transient circuit is. This type of circuit consists of only one energy storage element, in this case, a capacitor. This means that the circuit can be described using a first order differential equation.

Now, let's take a look at the problem you have provided. The first step is to find the voltage across the capacitor at time t_{0^-}. This can be done by using the voltage division rule, as you have correctly done. However, it is important to also find the current in the circuit at this time, as it will be needed for solving the differential equation.

Next, for t_{0^+}, we need to analyze the circuit using the differential equation method. This involves writing an equation for the current in the loop, as you have done. However, it is important to note that the current in the loop will be different from the current at t_{0^-}, as the capacitor will start to charge and the circuit will change.

To solve the differential equation, you will need to use initial conditions, which in this case are the voltage and current at t_{0^-}. You can then use these initial conditions to find the general solution for the circuit, and from there, you can find the specific solution for the given time t_{0^+}.

I hope this helps to clarify the process of solving a first order transient circuit using the differential equation method. If you are still having trouble, I suggest seeking additional resources or consulting with a professor or tutor for further assistance. Good luck!