# Math Help - Formulating Differential equations for RLC circuit with switches.

1. ## Formulating Differential equations for RLC circuit with switches.

Hello,

For Question 2 (attached) I am trying to figure out how to go about answering part a, but I am not confident with the physics so I am unsure how to begin formulating the required equations.

Any help is much appreciated,

Paul

2. ## Re: Formulating Differential equations for RLC circuit with switches.

Switch 1 is closed for a long time so the current in the inductor builds up to its steady state value.

This inductor current (which is easily calculated) is then the initial value for the differential equation involving R2 and L1

For convenience I have assumed that S1 opens and S2 close at time equals zero.

3. ## Re: Formulating Differential equations for RLC circuit with switches.

Thanks, I think I get it.

I have a very similar question:

A fully charged inductor is hooked up to be in series with a resistor and capacitor. How do I figure out the differential equation for the current change over time for the inductor and the voltage change over time for the capacitor? I would be able to figure it out if it was just a resistor and inductor, but I can't find anywhere that explains it it terms of a previously charged inductor.

Thanks

4. ## Re: Formulating Differential equations for RLC circuit with switches.

I'll give you a few pointers.

When your question talks about the capacitor being charged or the inductor carrying current (inductors cannot be "charged") they are giving you a clue about how to work out the initial conditions. YOU DON'T NEED THIS INFORMATION until after you have formulated your differential equation and found the general form of the solution. You then use the initial conditions to determine some constants in the final solution.

You can also FORGET that the question is asking for specific information UNTIL you have solved your differential equation. Once you have a solution for the circuit current (or capacitor charge) then finding the answers to their questions will be easy enough.

FIRST you must formulate the differential equation for an RLC circuit. To start with you don't need any other information from the question. To formulate the DE calculate the voltage across each component in terms of current i or charge q. Add these voltages together and set them equal to the supply voltage (zero in your case).

The last piece of information that you need is that the "charge" on a capacitor is the integral of the current that passes through it. You can formulate your DE in terms of current or capacitor charge, it is YOUR CHOICE.

If you choose current then your DE will have a differential term, the term iR and an integral term.
If you choose charge then your DE will have a second differential term, a differential term an the term q/C.

I hope that this helps.

5. ## Re: Formulating Differential equations for RLC circuit with switches.

Thanks, that makes more sense. Just to clarify the situation I am refering to, attached is the question.

So for the current through the inductor, is the equation I= (I inital)e^(-t/L/R) and the voltage across capacitor V= (V inital)e^(1-e^(-t/RC))?

Then just put in the L,R and C values to get the actual equations?

Thanks so much for your help.

6. ## Re: Formulating Differential equations for RLC circuit with switches.

I'm a bit confused what they are asking for, but the key DE I would be working from is:

$L\frac{di}{dt}+iR+\frac 1C \int i dt = 0$

this is simply the sum of the voltages around the loop.

You also have

$V_C=\frac qC =\frac 1C \int i dt$

$V_{C0}=0$

$I_{L0}=5$

Note that there is also a differential equations page on this site. These electrical problems are so common that you would probably get more responses there.