# Easy laplace conversion from s to t-domain gone wrong

1. May 5, 2012

### Twinflower

1. The problem statement, all variables and given/known data

$$I(s)=\frac{2}{s^2 + s}$$

Convert this to the time domain

3. The attempt at a solution

Step 1 - Divide by s
$$I(s)=\frac{\frac{2}{s}}{s + 1}$$

Step 2 - Substract and add 1 to create new fractions
$$I(s)=\frac{\frac{2}{s}-1+1}{s + 1}$$

Step 3 - Split into new fractions
$$I(s)=\frac{\frac{2}{s}}{s + 1} - \frac{1}{s + 1} + \frac{1}{s + 1}$$

Step 4 - Contract the two newbie fractions
$$I(s)=\frac{\frac{2}{s}}{s + 1} - \frac{2}{s + 1}$$

Step 5: Split up the first fraction
$$I(s)=\frac{2}{s} \frac{1}{s + 1} - \frac{2}{s + 1}$$

Step 6 - Convert to time domain
$$i(t)=2e^{-t}-e^{-t}$$

BUT, this is not correct. It should be: $2-2e^{-t}$

What is wrong in my approach here?

2. May 5, 2012

### Ray Vickson

You started with $$I(s) = \frac{2}{s^2+s} = \frac{2}{s(s+1)} = \frac{2}{s}\frac{1}{s+1}$$ and ended up with $$I(s) =\frac{2}{s} \frac{1}{s + 1} - \frac{2}{s + 1}$$ and you see nothing wrong with this? Why did you not just apply partial fractions to
$$I(s) = \frac{2}{s(s+1)}?$$

RGV

3. May 5, 2012

### Staff: Mentor

What's wrong is your first step. Factor the denominator to s(s + 1) and then use partial fraction decomposition to rewrite 2/[s(s + 1)] in the form A/s + B/(s + 1).

4. May 5, 2012

### Twinflower

Thank you Ray.

Can you tell me exactly what step in my first approach which was "illegal"?

5. May 5, 2012

### Ray Vickson

You don't see your claim that
$$-\frac{1}{s+1} + \frac{1}{s+1} = -\frac{2}{s+1}$$ when going from Step3 to step 4? You don't see that the fraction
$$\frac{\frac{2}{s}}{s+1}$$ in the first term in Step 3 is exactly the same as I(s), so you have really done nothing that will be useful?

RGV

6. May 5, 2012

### Twinflower

OK, I'm gonna do this one more time from scratch.
With partial fraction decomposition this time.

$$I(s) = \frac{2}{s^2 + s}$$

Step 1 - Factor the denomenator
$$I(s) = \frac{2}{s(s + 1)}$$

Step 2 - Set up the partial fractions stuff
$$\frac{A}{s}+\frac{B}{s + 1}$$

Step 3 - Merge A and B fractions
$$\frac{A(s+1)+B(s)}{s(s + 1)}$$

Step 4 - Determine A and B
$$(A+B)s = 0$$
$$A = 2$$
Using that A = 2, B must be -2 to fullfill the first equation.

Step 5 - Replace A and B with determined values
$$\frac{2}{s}-\frac{2}{s + 1}$$

Step 6 - Convert to the holy time domain
$$\frac{2}{s} = 2$$
$$\frac{-2}{(s+1} = -2e^-t$$
$$i(t) = 2(1-e^-t)$$

7. May 5, 2012

8. May 5, 2012

### Twinflower

Hey Serena! Long time no see ;)
I managed this one after realizing that I had to utlize the partial fraction-stuff :)

But I still need some help in this other thread of mine about the fifth harmonic of a square wave passing trough a HP filter. And my exam is on monday :/

9. May 5, 2012

### Ray Vickson

This is almost OK now, but it is a bad idea to write things like $$\frac{2}{s} = 2$$ because that is false, and might get you a mark of zero just for writing it. If you mean that the inverse transform of 2/s is 1, why not just say so, or maybe use a better notation, such as $L^{-1}(2/s) = 1$ or $2/s \longrightarrow 1?$ Also, in LaTeX, if you want $e^{-t}$ rather than your $e^-t,$, you need to use a curly bracket. To get $a^{anything},$ put the "anything" between curly brackets, like this: {anything}. If you don't, you will get $a^anything.$

RGV

10. May 5, 2012

### Twinflower

I didnt find the arrow in latex, but in my hand written notation i always use that.

11. May 5, 2012

### Ray Vickson

There are various types of arrows.
$\text{\rightarrow gives} \rightarrow,$ $\text{\longrightarrow gives} \longrightarrow,$ $\text{\Rightarrow gives} \Rightarrow,$ $\text{\Longrightarrow gives} \Longrightarrow.$

12. May 5, 2012

### I like Serena

I like \to ($\to$), which is nice and short to type.

I also like \mathcal{L} $\mathcal{L}^{-1}(\frac 2 s) = 2$, or more formally $\mathcal{L}^{-1}_s[ \frac 2 s ](t) = 2$.

13. May 5, 2012

### Ray Vickson

I, too, like mathcal, but have also seen things like $$\frac{1}{s} \leftrightarrow 1 .$$
However, this uses the somewhat lengthy "\leftrightarrow". Is there a shorter version?

RGV

14. May 5, 2012

### DigitalSwitch

Thread now about latex syntax. I have a feeling I will need to use some Laplace symbols soon so it's great info. Ty