MHB Laplace Transform Help: Solving for Unknown Variables in Electrical Circuits

[Sophie1]

Evening All

I have had a go at a laplace transform and got stuck.

$$\frac{d^2v}{dt^2}+\frac R L \d v t+\frac 1{LC}v=\frac 1{LC}V_0$$

$$R=12 \Omega, L=0.16H, C=10^{-4}F, V_0=6V, v(0)=0, v'(0)=0$$

so subbing these in i get
$$\mathscr L \left[ \frac {d^2v}{dt^2}+75\d v t+62500 v \right]=\mathscr L[375000]$$

$$S^2X-Sx(0)-x'(0)+75(SX-x(0))+62500v=\frac{375000}{S}$$

subbing $v(0)=0, x'(0)=0$

$$S^2X+75SX+62500X=\frac{375000}{S}$$

$$X(S^2+75S+62500)=\frac{375000}{S}$$

$$X= \frac{375000}{S(S^2+75S+62500)}$$

not I'm stuck as i can't an inverse form to change it back into.

can someone help?

 

[I like Serena]

Evening All

I have had a go at a laplace transform and got stuck.

$$\frac{d^2v}{dt^2}+\frac R L \d v t+\frac 1{LC}v=\frac 1{LC}V_0$$
$$R=12 \Omega, L=0.16H, C=10^{-4}F, V_0=6V, v(0)=0, v'(0)=0$$
(snip)
$$X= \frac{375000}{S(S^2+75S+62500)}$$

not I'm stuck as i can't an inverse form to change it back into.

Hi Sophie! Welcome to MHB! ;)

Suppose we could write $X$ as:
$$X = \frac A S + \frac {BS + C}{(s+\alpha)^2 + \omega^2}$$
for some yet to be determined $A,B,C,\alpha,\omega$.
Could you then find the inverse transform? (Wondering)

 

[Sophie1]

could you give me a pointer of where to begin. I have not used or even see α,ω. With respect to laplace transform.

 

[I like Serena]

could you give me a pointer of where to begin. I have not used or even see α,ω. With respect to laplace transform.

From wiki:
$$\mathcal L^{-1}\left[\frac{s+\alpha}{(s+\alpha)^2+\omega^2}\right] = e^{-\alpha t} \cos(\omega t) \cdot u(t)$$

So if we can find such constants $\alpha,\omega$ we can find the inverse transform.
And we have that:
$$S^2+75S+62500 = (S+\alpha)^2 + \omega^2$$
so we just need to figure out what these $\alpha,\omega$ are... (Thinking)

 

[Sophie1]

Would you be able to confirm that my working out upto the point i asked is correct as I'm not finding the answer. I will carry on trying.

 

[I like Serena]

Would you be able to confirm that my working out upto the point i asked is correct as I'm not finding the answer. I will carry on trying.

Your working up to:
$$X= \frac{375000}{S(S^2+75S+62500)}$$
is correct (disregarding some inconsistencies in $v$, $V$, and $x$ in the intermediate steps).
(Nod)