Converting to a partial fraction.

Question

I'm trying to do an inverse Laplace operation on $I(s)$ shown below but I'm struggling on finding what $A$ & $C$ are on the partial fraction and how to do it. I calculated what $B$ equals by making $s=0$.

$$I(s)=\frac{1}{s^2(R+L)}=\frac{A}{s}+\frac{B}{s^2}+\frac{C}{R+Ls} \\ 1=As(R+Ls)+B(R+Ls)+Cs^2 \\ B=\frac{1}{R}$$

Answer 1

Set $s$ to $-\frac{R}{L}$, eliminating $A$ and $B$ to find $C$ so that $$C\frac{R^2}{L^2}=1\Rightarrow C=\left(\frac{L}{R}\right)^2$$ Note that the coefficient of $s^2$ is zero in your second equation:- $$AL+C=0\Rightarrow A=-\frac{C}{L}=-\frac{L}{R^2}$$

Answer 2

Compute the derivative of both sides at $s = 0$; we have

$$0 = AR + 2 AL s + BL + 2 C s$$

Evaluating at $s = 0$ gives

$$AR + BL = 0 \implies A = -\frac{BL}{R} = -\frac{L}{R^2}$$

To find $C$, perhaps evaluate at $s = 1$ with the previously found values for $A$ and $B$.