Laplace and Equating Coefficients

**Hime** · May 20th 2012, 11:22 AM

I have a couple of questions I'm hoping someone can help me with. I'm using partial fractions to transfer a couple of equation from F(s) to f(t).

First question is:

F(s) = 4/s^2(s+1)(s+2)

By equating coefficients this becomes: 4/s^2(s+1)(s+2) = A/s + B/s^2 + C/(s+1) D/(s+2)

By multiplying by the numerator the equation becomes 4= As(s+1)(s+2) + B(s+1)(s+2) + Cs^2(s+2) + Ds^2(s+1)

Getting B,C & D is enough enough by substituting s=0,-1&-2. The answer I have for getting A I just don't understand, it says equate coefficients of s so that:

0=2A + 3B

I just don't see where this comes from.

Second question is to do with a closed loop control system. Using a standard control loop equation I have G(s)/1+ G(s).H(s)

This then becomes: (s/s+2) / [1+ 4.(1/s+2)] I have the answer as 2/s+10. I'm useless at working out fractions like this so any help would be appreciated.

I'm sure it's really basic stuff but I'm going number blind at this point. Thanks for your help.

**BAdhi** · May 20th 2012, 07:42 PM

Originally Posted by Hime

By multiplying by the numerator the equation becomes 4= As(s+1)(s+2) + B(s+1)(s+2) + Cs^2(s+2) + Ds^2(s+1)

Getting B,C & D is enough enough by substituting s=0,-1&-2. The answer I have for getting A I just don't understand, it says equate coefficients of s so that:

0=2A + 3B

I just don't see where this comes from.

first you've mentioned that you get,

$4=As(s+1)(s+2)+B(s+1)(s+2)+Cs^2(s+2)+Ds^2(s+1)$

by expanding you'll get,

$4=A(s^3+3s^2+2s)+B(s^2+3s+2)+C(s^3+2s^2)+D(s^3+s^2 )$

not lets make some adjustments,

$4=(A+C+D)s^3+(3A+B+2C+D)s^2+(2A+3B)s+2B$

now equate the coefficient of s you'll get,

$0=2A+3B$

just like that you can also equate coefficients of $s^3$ , $s^2$ ,... This is another way of finding constants on partial fractions (of course there are much easier methods)

**Hime** · May 21st 2012, 04:37 AM

Thanks a bunch mate, most appreciated.

Do you have any idea how to solve the second question?

**Prove It** · May 21st 2012, 05:04 AM

Originally Posted by Hime

I have a couple of questions I'm hoping someone can help me with. I'm using partial fractions to transfer a couple of equation from F(s) to f(t).

First question is:

F(s) = 4/s^2(s+1)(s+2)

By equating coefficients this becomes: 4/s^2(s+1)(s+2) = A/s + B/s^2 + C/(s+1) D/(s+2)

By multiplying by the numerator the equation becomes 4= As(s+1)(s+2) + B(s+1)(s+2) + Cs^2(s+2) + Ds^2(s+1)

Getting B,C & D is enough enough by substituting s=0,-1&-2. The answer I have for getting A I just don't understand, it says equate coefficients of s so that:

0=2A + 3B

I just don't see where this comes from.

Second question is to do with a closed loop control system. Using a standard control loop equation I have G(s)/1+ G(s).H(s)

This then becomes: (s/s+2) / [1+ 4.(1/s+2)] I have the answer as 2/s+10. I'm useless at working out fractions like this so any help would be appreciated.

I'm sure it's really basic stuff but I'm going number blind at this point. Thanks for your help.

Equating coefficients is the long and complicated method. It's easier to note that this equation: $\displaystyle \begin{align*} 4 \equiv A\,s(s + 1)(s + 2) + B(s + 1)(s + 2) + C\,s^2(s + 2) + D\,s^2(s + 1) \end{align*}$ holds true no matter the value of $\displaystyle \begin{align*}x \end{align*}$ , so by using some inspired guesswork, we can see that if we let $\displaystyle \begin{align*} s = 0 \end{align*}$ we have

$\displaystyle \begin{align*} A(0)(0 + 1)(0 + 2) + B(0 + 1)(0 + 2) + C(0)^2(0 + 2) + D(0)^2(0 + 1) &= 4 \\ 2B &= 4 \\ B &= 2 \end{align*}$

By letting $\displaystyle \begin{align*} s = -1 \end{align*}$ we have

$\displaystyle \begin{align*} A(-1)(-1 + 1)(-1 + 2) + B(-1 + 1)(-1 + 2) + C(-1)^2(-1 + 2) + D(-1)^2(-1 + 1) &= 4 \\ C &= 4 \end{align*}$

By letting $\displaystyle \begin{align*} s = -2 \end{align*}$ we have

$\displaystyle \begin{align*} A(-2)(-2 + 1)(-2 + 2) + B(-2 + 1)(-2 + 2) + C(-2)^2(-2 + 2) + D(-2)^2(-2 + 1) &= 4 \\ -4D &= 4 \\ D &= -1 \end{align*}$

By letting $\displaystyle \begin{align*} B = 2, C = 4, D = -1 \end{align*}$ and $\displaystyle \begin{align*} s = 1 \end{align*}$ we have

$\displaystyle \begin{align*} A(1)(1 + 1)(1 + 2) + 2(1 + 1)(1 + 2) + 4(1)^2(1 + 2) - 1(1)^2(1 + 1) &= 4 \\ 6A + 12 + 12 - 2 &= 4 \\ 6A + 22 &= 4 \\ 6A &= -18 \\ A &= -3 \end{align*}$

Thread: Laplace and Equating Coefficients

LinkBack

Thread Tools

Display

Laplace and Equating Coefficients

Re: Laplace and Equating Coefficients

Re: Laplace and Equating Coefficients

Re: Laplace and Equating Coefficients

Similar Math Help Forum Discussions

The act of equating

Equating two differential equations

Equating two SI units of time

Equating Coefficients of Polynomials Proof

Laplace transform DE with non-constant coefficients

Search Tags