# Circuit analysis problem - Laplace

Discussion in 'Homework Help' started by ineedmunchies, Apr 3, 2008.

1. ### ineedmunchies Thread Starter New Member

Apr 3, 2008
1
0
The question is as shown in the first picture. Question1.jpg

It asks that initial condition current generators are used, what I believe models the circuit for t>0 is shown in the second picture with the switch open, and a current generator added. (I am not sure if this is correct.)

I then used KCL to write equations at node 1 and node 2.

Node 1:
- =+

Node 2:
+=-()

These can then be rearraged to give

=(1+)-()

and

=()+(1+)

Which I then put into a matrix and solved for and

Giving
= -
which can be simplified to
= -

and
= ()+

(*Note the 4/s is not part of the denominator, I couldn't get the brackets to work properly.)

Which can be simplified to
= -

Then convert these back to the time domain to give:
(t) = 3-4

and (t) = 4-3

Can anyone tell if there is a mistake here or not?
I don't feel confident this is the correct answer. I think the 5 ohm resistor and 5A current should effect the circuit somehow but do not know how to encorporate it into my equations.

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2. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
I think your algebra is off in your solutions for V1 and V2. Starting with your original equations for nodes 1 and 2, I get:
V1 = (3s + 7/2)/(s(s + 1))
V2 = (4s + 7/2)/(s(s + 1))

I've checked, and these values satisfy your original equations, so I'm pretty confident about them.
To do the inverse Laplace transforms, you'll need to break each of these apart into a sum of two rational expressions. The denominators will be s and s + 1, not s and s + 2 that you show.
Mark

3. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
In case the bit about breaking up the expressions for V1 and V2 is not clear enough, you'll need to do a partial fraction decomposition for each of them.

For V1, you need to solve for A and B in this equation:
(3s + 7/2)/(s(s+1)) = A/s + B/(s + 1)

When you do the inverse Laplace transform, you'll end up with v1(t) = A + B*e^(-t)

Do the same thing to find v2(t).