Laplace Transforms of Kirchhoff's Laws

[rdfloyd]

Back again with another problem. These transforms are difficult...

Homework Statement

Had to take a screenshot:
http://i.imgur.com/LM38A.png

The Attempt at a Solution

His instructions are to take the Laplace Transform of each of Kirchhoff's laws. However, when we do, nothing useful comes out of them. I'm completely stuck as to what the problem is asking for.

The question (I think) asks: "Find the voltage across each element as a function of time." But he gives us 3 equations where the voltage is already in respect to time. Can anyone help decode this?

EDIT: Sorry mods. Didn't realize this was Advanced physics. >.<

 

[vela]

From the wording of the question, it looks like this is from a differential equations course. Am I right?

You're looking for an explicit expression of each current as a function of time, like I_L(t) = I₀e^-t/τ cos ωt where you say what the constants τ and ω are equal to in terms of R, L, and C.

You have three unknowns: I_L, I_R, and I_C. You have two equations, e.g., ΔV_C=ΔV_L and ΔV_R=ΔV_L, which will allow you to write two of the currents in terms of the third one. (The third condition, ΔV_C=ΔV_R, automatically follows from the other two, so it's not independent.) You can therefore write I_R+I_L+I_C=0 in terms of just one of the currents and solve for it.

 

[rude man]

He wants an explicit expression for the voltage across R. Not the integro-differential equations he gave you.

 

[rdfloyd]

From the wording of the question, it looks like this is from a differential equations course. Am I right?

You're looking for an explicit expression of each current as a function of time, like I_L(t) = I₀e^-t/τ cos ωt where you say what the constants τ and ω are equal to in terms of R, L, and C.

You have three unknowns: I_L, I_R, and I_C. You have two equations, e.g., ΔV_C=ΔV_L and ΔV_R=ΔV_L, which will allow you to write two of the currents in terms of the third one. (The third condition, ΔV_C=ΔV_R, automatically follows from the other two, so it's not independent.) You can therefore write I_R+I_L+I_C=0 in terms of just one of the currents and solve for it.

It's a math methods course, but we are studying differential equations now. Finished partial diff. eq. on Thursday.

As far as the solution, this is what we did. I was trying to understand exactly what was going on, but then it clicked and I finished it in about half an hour. The answer wasn't very pretty, but it looked right.

He wants an explicit expression for the voltage across R. Not the integro-differential equations he gave you.

But didn't we have to use those to get to Vr?

 

[rude man]

It's a math methods course, but we are studying differential equations now. Finished partial diff. eq. on Thursday.

As far as the solution, this is what we did. I was trying to understand exactly what was going on, but then it clicked and I finished it in about half an hour. The answer wasn't very pretty, but it looked right.



But didn't we have to use those to get to Vr?

Of course. That's why he gave them to you ...
Did he cover transforming differential equations (linear, constant-coefficient ones) to Laplace-transformed algebraic equations, I hope?

 

[rdfloyd]

Did he cover transforming differential equations (linear, constant-coefficient ones) to Laplace-transformed algebraic equations, I hope?

This just clicked for me. He doesn't like to "spoil the surprise" by giving away the whole idea of a concept (forces us to choose whether to use something he taught or not), so it wasn't directly implied.

The whole idea behind Laplace/Fourier Transforms is to take a differential equation, put it in algebraic terms (which I have many more tricks to use), then bring them back to differential terms. It makes so much sense now!

I guess the only question I have now is why don't we do this to every differential equation?

Also, why would we use Fourier Transforms? To my understanding, Fourier Transforms are much more restrictive (it's limits at +/-∞ → 0).

 

[RoshanBBQ]

This just clicked for me. He doesn't like to "spoil the surprise" by giving away the whole idea of a concept (forces us to choose whether to use something he taught or not), so it wasn't directly implied.

The whole idea behind Laplace/Fourier Transforms is to take a differential equation, put it in algebraic terms (which I have many more tricks to use), then bring them back to differential terms. It makes so much sense now!

I guess the only question I have now is why don't we do this to every differential equation?

Also, why would we use Fourier Transforms? To my understanding, Fourier Transforms are much more restrictive (it's limits at +/-∞ → 0).

The Fourier transform has a more intuitive connection with response of a system based on the frequency components of the input function. The system, for example, could be a bunch of connected capacitors, inductors, and resistors, modeled by a differential equation. The input could be a voltage applied to that circuit at a certain point with the output being voltage developed elsewhere. The Fourier transform would give easier insight into the characteristics of the developed voltage relative to the applied voltage if the applied voltage had dominant frequencies -- such as an applied sinusoidal voltage that has only a single frequency at which it interacts with the system.

 

[rude man]

This just clicked for me. He doesn't like to "spoil the surprise" by giving away the whole idea of a concept (forces us to choose whether to use something he taught or not), so it wasn't directly implied.

The whole idea behind Laplace/Fourier Transforms is to take a differential equation, put it in algebraic terms (which I have many more tricks to use), then bring them back to differential terms. It makes so much sense now!

I guess the only question I have now is why don't we do this to every differential equation?

Also, why would we use Fourier Transforms? To my understanding, Fourier Transforms are much more restrictive (it's limits at +/-∞ → 0).

Good question. the answers:
1. Laplace only works for linear, constant-coefficient diff. eq's, basically.
2. You don't learn the 'innards' of diff eq's by looking up the answer in a table of transforms! (There is of course a way to compute the inverse transforms but it involves contour integration in the complex plane, which nobody does if tables are available!)

The Fourier transform is in particular isn't suitable for transient problems, like a step voltage into an R-C network. The Fourier integral is used for single pulses but all you get from that is the frequency spectrum of the output.

 

[rude man]

The Fourier transform has a more intuitive connection with response of a system based on the frequency components of the input function. The system, for example, could be a bunch of connected capacitors, inductors, and resistors, modeled by a differential equation. The input could be a voltage applied to that circuit at a certain point with the output being voltage developed elsewhere. The Fourier transform would give easier insight into the characteristics of the developed voltage relative to the applied voltage if the applied voltage had dominant frequencies -- such as an applied sinusoidal voltage that has only a single frequency at which it interacts with the system.

Pure sinusoidal excitations are handled simply by replacing s → jω.

 

[RoshanBBQ]

Pure sinusoidal excitations are handled simply by replacing s → jω.

Yeah. Then you have turned your Laplace transform into a Fourier transform. The idea is simple: If you deal with analyzing frequency response all the time, you do not want to do the simplifications that follow from letting s be jω. Instead, you just use a Fourier transform whose tables are already in a simplified fashion in terms of ω.

 

[rude man]

Yeah. Then you have turned your Laplace transform into a Fourier transform. The idea is simple: If you deal with analyzing frequency response all the time, you do not want to do the simplifications that follow from letting s be jω. Instead, you just use a Fourier transform whose tables are already in a simplified fashion in terms of ω.

Can't think of anything much simpler than s → jω and computing the magnitude ...

Give me an example of what you mean. Take a simple R-C lo-pass with Laplace 1/(Ts + 1), and tell me what your table of Fourier transforms does for you?