5 replies to this topic

[Melchoire]

Working on solving a circuit problem.

 

My problem boils down to applying the reverse Laplace transform on a certain function. I can't seem to manipulate it to a form that easily reversable.

 

Here's the s-domain function:

 

 

The function on the left is a sine with frequency w. The one on the right is an exponential. But not sure what to do about their product.

 

Any ideas?

[ShadowLink64]

Inverse Laplace transforms follow the principle of superposition, so I think you'll have to do partial fraction decomposition. You have two distinct factors on the bottom, and so you want to split the problem where you have a sum of:

 

w/((s^2+w^2)(s+R/L)) = (As+B)/(s^2 + w^2) + C/(s + R/L)

 

Or something like that. Then just find what A,B, and C are by matching coefficients with the correct powers of s, and take the inverse Laplace transform of each term.

 

Hopefully this works out.

[Melchoire]

Inverse Laplace transforms follow the principle of superposition, so I think you'll have to do partial fraction decomposition. You have two distinct factors on the bottom, and so you want to split the problem where you have a sum of:

 

w/((s^2+w^2)(s+R/L)) = (As+B)/(s^2 + w^2) + C/(s + R/L)

 

Or something like that. Then just find what A,B, and C are by matching coefficients with the correct powers of s, and take the inverse Laplace transform of each term.

 

Hopefully this works out.

 

Doesn't work out for me, I'm left with:

 

w = As(s² + R/L) + B(s + R/L) + C(s² + w²)

 

Which means A, B, and C are 0?

[ShadowLink64]

 

Doesn't work out for me, I'm left with:

 

w = As(s² + R/L) + B(s + R/L) + C(s² + w²)

 

Which means A, B, and C are 0?

No. You should have:

w = As(s + R/L) + B(s + R/L) + C(s² + w²)

 

w = As² + As(R/L) + Bs + B(R/L) + Cs² + Cw²

 

A + C = 0 to balance the s² terms.

A(R/L) + B = 0 to balance the s terms.

B(R/L) + Cw^2 = w to balance the zeroth order terms.

 

I can tell you right now that A, B, and C are not all zero.

[Melchoire]

No. You should have:

w = As(s + R/L) + B(s + R/L) + C(s² + w²)

 

w = As² + As(R/L) + Bs + B(R/L) + Cs² + Cw²

 

A + C = 0 to balance the s² terms.

A(R/L) + B = 0 to balance the s terms.

B(R/L) + Cw^2 = w to balance the zeroth order terms.

 

I can tell you right now that A, B, and C are not all zero.

 

You are correct, I'd forgotten how to do these decompositions.

 

In any case I solved the 3 systems and ended up with some gnarly looking expressions. But can't argue with the math right? Thanks for your help =)

[Melchoire]

So after I went with this path my final answer didn't work out.

 

It turns out the way you set up the partial fraction expansion is like so:

 

F(s) = A/(s + R/L) + B/(s - jw) + B*/(s + jw)

 

With this approach you end up with complex values for B and B*