Circuit Analysis - Laplace notation

In summary: If you get stuck, post what you have and I'll try to assist.Ok I have two equations, here is the first:(2s+4I1+4I2)=(2s+4I1)And here is the second:(4s+8I2+4I1+20I1+4I2)=(4s+4I1+4I1+4I2+10I1)I set the Laplace expressions for the currents equal to the current expressions in the first set of equations you posted and came up with a different result in both cases (i.e., I didn't get what you have above).I've got to leave for a while; I'll be back
  • #1
anol1258
26
0

Homework Statement



The circuit below uses ideal components and disguises itself as a second order system when in fact it is really two first order systems. Prior to t=0 switch S is open. Then suddenly at t=0 switch S is closed. Find the peak output voltage [itex]V_{0max}[/itex], and the time taken for this peak to occur.
Hint: Use KVL loop equations formulated in Laplace notation.

circuit:

10.jpg


Homework Equations



V=IR

The Attempt at a Solution



Im stuck on most of it. I know how to take the inverse Laplace, but forming the right equations is getting me stuck.
 
Last edited:
Physics news on Phys.org
  • #2
anol1258 said:

Homework Statement



The circuit below uses ideal components and disguises itself as a second order system when in fact it is really two first order systems. Prior to t=0 switch S is open. Then suddenly at t=0 switch S is closed. Find the peak output voltage [itex]V_{0max}[/itex], and the time taken for this peak to occur.
Hint: Use KVL loop equations formulated in Laplace notation.

circuit:
[Broken]

Homework Equations



V=IR

The Attempt at a Solution



Im stuck on most of it. I know how to take the inverse Laplace, but forming the right equations is getting me stuck.

I can't see your image. By the URL, it looks like it's on a yahoo mail server. Can you upload a copy to the PF server and post it as an attachment? Use the paperclip icon in the edit frame.

By the way, Welcome to PF!
 
Last edited by a moderator:
  • #3
What do you know about expressing circuit elements in Laplace notation? What have you attempted so far?
 
  • #4
I know when you differentiate you multiply by s, and when you integrate you divide by s in the frequency domain.
 
  • #5
Here is what I have so far:

KVL on primary side and secondary side
final10.jpg


I am doing this right?
 
  • #6
anol1258 said:
Here is what I have so far:

KVL on primary side and secondary side
View attachment 58454

I am doing this right?

You've got the right idea, but the mutual inductance terms should have an 's', and beware of the signs you assign to them; check the current directions with respect to the dots.

Current flowing into the dot in one loop will cause a current to flow out of the dot on the other. In terms of the controlled voltage source that is used to represent the mutual inductance, the voltage source should have a polarity that would want to drive the current in the same direction.
 
  • #7
Is this correct now?
KVLloop.jpg


Would I now put these equations in matrix form and solve for either I1 or I2?
 
  • #8
anol1258 said:
Is this correct now?
View attachment 58478

Would I now put these equations in matrix form and solve for either I1 or I2?

Sure. Looks like you'll need I2 in order to find Vout.

Since there are only two equations in two unknowns you could always resort to substitution and solving by hand. Whatever you're comfortable with.
 
  • #9
Ok. Once I find I2 though how can I solve for Vout. This isn't a simple case where I can use Ohm's Law with the R2.
 
  • #10
anol1258 said:
Ok. Once I find I2 though how can I solve for Vout. This isn't a simple case where I can use Ohm's Law with the R2.

Sure it is. I2 flows through R2. Vout is across R2.

Of course, you'll need to take the inverse Laplace transform to I2 first...
 
  • #11
But there isn't another node on the very bottom right part of the circuit. From looking at the diagram are you sure they are asking Vout across R2?
 
  • #12
anol1258 said:
But there isn't another node on the very bottom right part of the circuit. From looking at the diagram are you sure they are asking Vout across R2?


R2 is situated between the node labeled Vout and the reference node at the bottom (labeled with the ground symbol). So yes, I'm sure.
 
  • #13
Thank you gneill

I'm having a hard time putting my current in laplace form so that I can do some inverse laplace on it. Here is the equation for the current:
10.1.jpg
 
  • #14
anol1258 said:
Thank you gneill

I'm having a hard time putting my current in laplace form so that I can do some inverse laplace on it. Here is the equation for the current:
View attachment 58515

Can you show how you arrived at this expression for I2 from your equations in post #7? The result doesn't look quite right to me.
 
  • #15
I did Cramer's rule. I thought it would work for this problem, maybe not.
Here's cramers rule:
cramers.jpg
 
  • #16
yeah I messed up should be:

Laplace.jpg
 
  • #17
How can i do this laplace. Its killin me
 
  • #18
anol1258 said:
How can i do this laplace. Its killin me

I don't think that your Laplace equation for I2 is quite right yet. Can you show me your two KVL loop equations? As I mentioned previously, pay careful attention to the signs of the mutual inductance terms. They are directly dependent upon your choice of loop current direction and the location of the coupling dots.

So, what directions have you chosen for the loop currents? Do they flow into or out of the dots on the coupled inductors? How does this affect the polarity of the induced-voltage source in the other loop?

As for dealing with the Laplace inversion, if you have a quadratic in the denominator you want to reduce the coefficient of the s2 term to unity (1), then factor the quadratic by finding its roots (say, ##\alpha_1## and ##\alpha_2##. Write the quadratic as ##(s - \alpha_1)(s - \alpha_2)##. That should get you going.
 

1. What is Laplace notation in circuit analysis?

Laplace notation is a mathematical technique used in circuit analysis to simplify complex differential equations into algebraic equations. It involves transforming time-domain functions into the s-domain, where s is a complex variable. This allows for easier analysis and calculation of circuit parameters.

2. How is Laplace notation used in circuit analysis?

Laplace notation is used to solve differential equations that describe the behavior of electrical circuits. By converting these equations into the s-domain, it becomes easier to manipulate and solve them using algebraic equations. This helps to determine important circuit parameters such as voltage, current, and impedance.

3. What are the advantages of using Laplace notation in circuit analysis?

One of the main advantages of using Laplace notation is its ability to simplify complex circuits and make them easier to analyze. It also allows for the use of algebraic equations, which are more straightforward to solve than differential equations. Additionally, Laplace notation can be used to analyze circuits with different types of input signals, such as sinusoidal, step, or impulse signals.

4. Are there any limitations to using Laplace notation in circuit analysis?

While Laplace notation is a powerful tool in circuit analysis, it does have some limitations. It can only be applied to linear circuits, meaning that the components in the circuit must have a linear relationship between voltage and current. It also assumes that the circuit is in a steady-state, meaning that the behavior of the circuit does not change over time.

5. How does Laplace notation compare to other methods of circuit analysis?

Laplace notation is one of the most commonly used methods of circuit analysis, along with other techniques such as Kirchoff's laws and Thevenin's theorem. Compared to these methods, Laplace notation offers a more systematic approach that can handle a wider range of input signals. It also allows for the use of complex numbers, making it easier to analyze circuits with reactive components.

Similar threads

  • Electrical Engineering
Replies
3
Views
894
  • Introductory Physics Homework Help
Replies
6
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
3
Views
1K
  • Electrical Engineering
Replies
3
Views
691
  • Introductory Physics Homework Help
Replies
9
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
2K
  • Introductory Physics Homework Help
Replies
20
Views
3K
  • Introductory Physics Homework Help
Replies
12
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
1K
Back
Top