# Choosing a capacitance to limit voltage across a switch

Hi everyone. My problem is for an assignment in a network analysis course. I attached the homework page as a pdf.

## Homework Statement

My problem involves an R-L circuit with a switch. There is a 100 V source, and switch that is initially closed, followed by a .25 H inductor and a 200 Ohm resistor in series. After being closed for a while, the switch is opened. The question says "To reduce the voltage that appears across the switch, it is proposed to connect a capacitor as shown"...the capacitor will be connected between the switch and the inductor+resistor and be parallel to them. "Choose a capacitor that will limit the voltage appearing across the open switch contacts to 150 V. Specify the capacitor completely by stating not only its value but also its voltage rating."

## Homework Equations

Not really sure what equations to include that would be helpful. The concept requires knowing all of the necessary equations, as far as I can tell.

## The Attempt at a Solution

I determined that the current across the inductor at 0- and 0+ is 0.5 A. It appears as though the capacitor is not actually in the circuit and then is added at t=0 (since it says it is proposed to connect a capacitor), so I don't know how to approach it since the capacitor will not have any Voltage from t<0 and therefore none for V>0.
So, basically I don't understand how to work it other than perhaps to work backwards plugging in 150 V and then determining somehow what the corresponding capacitance might be, even if you don't have any initial voltage across the capacitor.

#### Attachments

• Hw3.pdf
249.8 KB · Views: 104

## Answers and Replies

gneill
Mentor
The capacitor that is proposed would be a permanent part of the circuit, so it would exist in the circuit before and after the switch is operated.

So is it possible to determine the value of the capacitor without using a computer program like pspice? If the voltage across the open switch is 150 V, would the Voltage as t goes to infinity of the capacitor be -50 V? The original voltage of the capacitor is 100 V, and if the Switch develops a voltage of 150V, doing a KVL of the left loop would mean that the Voltage across the capacitor would need to be -50 V.

gneill
Mentor
With the switch open the energy in the capacitor loop will decay over time because of the resistance. It's an RLC circuit, which will have a natural frequency and damping. So the eventual voltage on the capacitor must be zero.

The question is, is there a simple way to determine the maximum voltage that appears across the capacitor without solving the differential equation or simulating the circuit? I'm thinking about it! NascentOxygen
Staff Emeritus
Science Advisor
I think once one has had a lecture on solving the D.E. for a second-order LCR circuit, knowing the period and exponential envelope, you can probably just plug in the circuit parameters for the solution. Here you are looking for the first downswing to reach -50v.

Did anybody actually figure this out? I have to do the same exact problem. Are we suppose to convert the inductor (jwl) and capacitor (1/jwc)? if so what is the frequency in this circuit?

Did anybody actually figure this out? I have to do the same exact problem. Are we suppose to convert the inductor (jwl) and capacitor (1/jwc)? if so what is the frequency in this circuit?

No, you can't do that. Once you've substituted s=jw in the impedances, you are going after a steady state solution after all the transients have died away. You can think of this in terms of the Laplace transform where setting s=jw is done to find the coefficients of the 1/(s2+w2) term in the partial fraction expansion of the output equation, which corresponds to the steady state sinusoid that results. Or if you prefer to think in phasors, you are assuming the solution takes the form of an everlasting sinusoid ejwt but this solution is not complete as it misses the transient terms. And in both cases you are assuming the excitation is sinusoidal, and in this case it is not.

The solution can be found by replacing the capacitor and inductor by their equivalent circuits including initial conditions. The capacitor can be replaced by an impedance 1/sC in series with its initial voltage as a voltage source v(0)/s. Likewise the inductor can be replaced by an impedance sL and a constant current source i(0)/s in parallel.

Once you've placed those elements in the circuit, solve for the voltage across the capacitor. Keep in mind the total voltage across the capacitor is the voltage across the new 1/sC capacitor *plus* its voltage source v(0)/s. The capacitor is being modelled by two things in series, which comes from the Laplace transform again: i = C dv/dt, I(s) = sC V(s) - Cv(0), V(s) -- the total voltage across the capacitor -- = I(s) / sC + v(0)/s

The result will be some equation with s^2 on the bottom that you should recognize as a damped sinusoid. The maximum value this takes on in the time domain can be read from the coefficients.

Last edited:
rude man
Homework Helper
Gold Member
This is just an RLC circuit, all components in series, with initial conditions on iL and VC. Use Laplace if you have been introduced to it, otherwise classical solution to the 2nd order diff. eq. with both initial conditions given.

I think my networks class must use different variables, what are you saying "s" is? what is 1/sC? I get C is the capacitance.

I've been trying to work on the same problem and I can't figure it out! My class hasn't learned Laplace yet and am struggling to get through it, please help me

I have been working on this problem for a couple hours now and I'm only confusing myself. I read some previous posts, but I'm still lost. Help.

rude man
Homework Helper
Gold Member
I think my networks class must use different variables, what are you saying "s" is? what is 1/sC? I get C is the capacitance.

LIKE I SAID ... if you're not familiar with Laplace transforms (that's what the "s" is all about), use the classical diff. eq. method of solving.

rude man
Homework Helper
Gold Member
I've been trying to work on the same problem and I can't figure it out! My class hasn't learned Laplace yet and am struggling to get through it, please help me

LIKE I SAID ... if you're not familiar with Laplace transforms (that's what the "s" is all about), use the classical diff. eq. method of solving.

rude man
Homework Helper
Gold Member
I have been working on this problem for a couple hours now and I'm only confusing myself. I read some previous posts, but I'm still lost. Help.

Wrire the diff. eq. for your circuit, including the initial conditions on inductor current and capacitor voltage, and solve!