How to find overshoot and undershoot in an RLC circuit?

[likephysics]

Homework Statement

RLC circuit as shown in the attachment. Input is a pulse, of frequency 1MHz, 5v. Rise/fall time 1ns.
How to find the voltage at the capacitor.
In simulation, I see overshoot and undershoot. But how do I find the overshoot/undershoot amplitude mathematically?
Ringing frequency is f = 1/2*pi*sqrt(LC)

Homework Equations

The Attempt at a Solution

No idea. I tried VL = L di/dt
But not sure how to get dt.

 

[NascentOxygen]

The general approach is to form the second-order differential equation, and solve it. You have a series circuit, with a current i(t). You know the equations for voltage across a resistor, across an inductor, and across a capacitor, all in terms of the current through each, so sum those voltages. This sum equals the applied voltage at all times.

The topic is treated well in most textbooks, and at wiki http://en.wikipedia.org/wiki/RLC_circuit. Possibly your examiner requires nothing more of you than just using the equation the text derives for the overshoot, after plugging in the values for your circuit to determine the damping, ζ.

That 1pF capacitor is very tiny, though not totally unrealistic. Are you sure the value is correct?

Good luck!

 

[likephysics]

The general approach is to form the second-order differential equation, and solve it. You have a series circuit, with a current i(t). You know the equations for voltage across a resistor, across an inductor, and across a capacitor, all in terms of the current through each, so sum those voltages. This sum equals the applied voltage at all times.

The topic is treated well in most textbooks, and at wiki http://en.wikipedia.org/wiki/RLC_circuit. Possibly your examiner requires nothing more of you than just using the equation the text derives for the overshoot, after plugging in the values for your circuit to determine the damping, ζ.

That 1pF capacitor is very tiny, though not totally unrealistic. Are you sure the value is correct?

Good luck!

It's not an assignment problem. I made it up.
The RLC ckt is the equivalent of a scope probe. I am trying to mathematically derive the overshoot amplitude.
Since the input is a pulse, I can't use the standard textbook approach.
I have attached a plot of the voltage across capacitor for rise/fall time of 0.5ns.

 

[NascentOxygen]

Probes are usually carefully designed. I'm skeptical about your equivalent circuit, have you included the resistance of the source in your simulation? Also include the source capacitance. Try 50Ω and 10 pF if you are just making up values.

I wouldn't call that a pulse, your input is a step, repetitively, and that's exactly what the textbooks deal with. If a system rings with a step, it will ring with a delta.

 

[likephysics]

Probes are usually carefully designed. I'm skeptical about your equivalent circuit, have you included the resistance of the source in your simulation? Also include the source capacitance. Try 50Ω and 10 pF if you are just making up values.

The values you mentioned are for passive probes.
I'm trying to play with approximate model of an active probe.
The L and C values are correct. I added the R to dampen the ringing caused from ideal L and C in simulation.

I wouldn't call that a pulse, your input is a step, repetitively, and that's exactly what the textbooks deal with. If a system rings with a step, it will ring with a delta.

Can you point me to resources which will help me find the delta?

 

[NascentOxygen]

Can you point me to resources which will help me find the delta?

Laplace Transforms give you the system response for any input, including Dirac's delta. Look for an electronics textbook introducing Laplace Transforms. Many textbooks will have worked examples. Or search online.

Good luck!

 

[likephysics]

Laplace Transforms give you the system response for any input, including Dirac's delta. Look for an electronics textbook introducing Laplace Transforms. Many textbooks will have worked examples. Or search online.

Good luck!

Found this - http://csserver.evansville.edu/~blandfor/EE210/MatLab/MLEx5.pdf
Plugged in values for time and got voltage close to the one in the plot (page3) for underdamped case.

But in my circuit, the overshoot increases as I make the step input rise time fast.
How do I take this into account?

 

[NascentOxygen]

Found this - http://csserver.evansville.edu/~blandfor/EE210/MatLab/MLEx5.pdf
Plugged in values for time and got voltage close to the one in the plot (page3) for underdamped case.

But in my circuit, the overshoot increases as I make the step input rise time fast.
How do I take this into account?

Have you learned about Laplace Transforms in your studies?

 

[likephysics]

Have you learned about Laplace Transforms in your studies?

Yes, I did learn LT back when I was in school.

 

[NascentOxygen]

In the time domain, to generate a step input with a finite rise-time:
at t=t₀ apply a steep ramp of slope g volts/sec, i.e., v(t)=g·t
at t=t₁ add to this input a ramp of equal but opposite slope, viz., -g volts/sec,

With your system being represented by a linear second order DE, you can use the superposition of responses to more than one input to find the composite output. (I can't help with the mathematics, have forgotten those details.)

Your best hope lies in using Laplace Transforms. Maybe here's your opportunity to revise that topic? I only use LT for sinusoidal responses, but maybe someone else can help you with your piece-wise LT. Good luck!