Determine voltage - complete response

[Cocoleia]

Homework Statement

I am working on this question from the textbook:

Homework Equations

The Attempt at a Solution

I would use Cramer's method and solve for i2, then get the characteristic equation and find the roots. from there I would attempt to find the natural response. I am not sure how I would find the forced response. I am not even sure that I am on the right track. I had a similar problem on an assignment and they solved using the damping coefficient alpha and resonant frequency omega. I do not know how to do that in this case if it is possible. I didn't want to do all of the calculations and waste my time if I am completely on the wrong track! Any explanations are greatly appreciated.

 

[gneill]

Are you intending to use Laplace Transforms to solve the circuit? I ask because it looks like you've introduced the s variable. Or is that just your approach for finding the characteristic equation? Note that a Laplace Transform approach would yield the complete response, both natural and forced, at the same time.

Still, I have to wonder about the way you're treating the capacitors. Looking at units, you've got capacitance times current times s equaling a voltage. I don't see that being correct (Treat the s variable as having units of 1/seconds).

I think your capacitor impedances should be of the form $\frac{1}{s C}$.

I also note that you've left out the initial charge (voltage) on the first capacitor when you defined v1.

 

[Diegor]

Homework Statement

I am working on this question from the textbook:
View attachment 113157

Homework Equations

The Attempt at a Solution

View attachment 113156

I would use Cramer's method and solve for i2, then get the characteristic equation and find the roots. from there I would attempt to find the natural response. I am not sure how I would find the forced response. I am not even sure that I am on the right track. I had a similar problem on an assignment and they solved using the damping coefficient alpha and resonant frequency omega. I do not know how to do that in this case if it is possible. I didn't want to do all of the calculations and waste my time if I am completely on the wrong track! Any explanations are greatly appreciated.

I only could follow you up to the kvl on right and left side after that is dificult to understand.

You can apply Kvl or any circuit analysis technique and then you need to use the relation of current and voltage over a capacitor that is i(t) = Cdv/dt. After that you will get a differential equation and solve for v1(t).

 

[Diegor]

Homework Statement

I am working on this question from the textbook:
View attachment 113157

Homework Equations

The Attempt at a Solution

View attachment 113156

I would use Cramer's method and solve for i2, then get the characteristic equation and find the roots. from there I would attempt to find the natural response. I am not sure how I would find the forced response. I am not even sure that I am on the right track. I had a similar problem on an assignment and they solved using the damping coefficient alpha and resonant frequency omega. I do not know how to do that in this case if it is possible. I didn't want to do all of the calculations and waste my time if I am completely on the wrong track! Any explanations are greatly appreciated.

Is that s in your equations the s from Laplace transform? If so v(s) = 1/(sC)I(s) not v(s)=sCI(s). I recommend you use R1 R2 C1 C2 instead of the numbers so it is easy to follow equations.

 

[Cocoleia]

Is that s in your equations the s from Laplace transform? If so v(s) = 1/(sC)I(s) not v(s)=sCI(s). I recommend you use R1 R2 C1 C2 instead of the numbers so it is easy to follow equations.

I have no idea if it is Laplace... sometimes the professor puts s = d/dt

 

[gneill]

I have no idea if it is Laplace... sometimes the professor puts s = d/dt

Yes, in what's known as the Laplace Domain, the 's' variable is actually an operator. Multiplying by s is equivalent to differentiation, dividing by s is the equivalent of integration. It's an operator that you can manipulate like a variable in the Laplace Domain. It takes all the blood, sweat, and tears out of working with and solving differential equations

 

[Diegor]

I have no idea if it is Laplace... sometimes the professor puts s = d/dt

Ok l see. It is from Lapalce and problably you teacher is traying to say that when you have derivation on time domain it is equivalent to multiply by s in complex frecuency domain. For example:

i=Cdv/dt is converted to I(s) = CsV(s) in complex frecuency domain.

Anyway you have two ways to solve the problem one using diferential equations like in my first answer or you can use Laplace transform. But i recommend you first solve it using diferential ecuations before jumping to more advanced methods.

 

[Cocoleia]

Anyway you have two ways to solve the problem one using diferential equations like in my first answer or you can use Laplace transform. But i recommend you first solve it using diferential ecuations before jumping to more advanced methods.

For example if I try to find the differential equations for this circuit:

Would it be what is boxed:

 

[Diegor]

The kcl is wrong. I still recommend you to use letters instead of numbers that way is easy to keep track. In the posted pictures I tried to show my point and also how it would look like in frec. domain

. I hope it helps.