I need to solve differential Equation

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penrico

Joined: 28 Aug 2001
Posts: 249
Helped: 13
Location: Argentina

16 Jun 2005 21:47 I need to solve differential Equation

I need to solve: (All steps to do it)

A sen(wt) = C δV(t)/δt + 1/L ∫(V(t) δt) + 1/R V(t)

where w ≈ 1/(2 pi sqrt(LC))

Need to get V(t)???

It's the superegenerative initial formula, but I'm looking all the steps to get the sollution.

thanks.

steve10

Joined: 26 Mar 2002
Posts: 177
Helped: 33
Location: Los Angeles (Chinese)

17 Jun 2005 0:06 Re: I need to solve differential Equation

Your question is unclear due to the following:
1. What is "A sen(wt)"?
2. Does 1/R V(t) mean V(t)/R or 1/(R*V(t))?
3. Since you didn't specify boundary or initial conditions, do you mean to seek for general solutions?

nand_gates

Joined: 19 Jul 2004
Posts: 907
Helped: 120

17 Jun 2005 8:53 Re: I need to solve differential Equation

This is simple integro differential equation for LCR series circuit!
Apply KVL and you will get it!

Initial conditions can be assumed to be zero!

penrico

Joined: 28 Aug 2001
Posts: 249
Helped: 13
Location: Argentina

17 Jun 2005 12:56 Re: I need to solve differential Equation

The initial conditions are not zero, the equation has an exitation that is A sin(wt). Its a sinuidal wave.

1/R V(t) mean V(t)/R

I'm looking for general sollution in equations form.

Thanks

Dspnut

Joined: 18 Jul 2002
Posts: 171
Helped: 4

17 Jun 2005 13:50 Re: I need to solve differential Equation

Hi there,

This problem can be simplified as a parallel RLC circuit with a sinusoidal forcing function. To solve this problem you need to find the complete response, v(t) = vn(t) + vf(t), where vn(t) and vf(t) are natural and forced responses, respectively.

The natural response, vn(t), should be in the form of
vn(t) = D*exp(s1*t) + E*exp(s2*t) [second order circuit]
where s1 and s2 are roots of the following characteristic equation
s^2 + (1/(R*C))*s + (1/(L*C)) = 0
the unkonwns, D and E, will be determined later using initial conditions

The forced response, vf(t), should be in the form of
vf(t) = F*sin(w*t) + G*cos(w*t) [the forcing function is 'A*sin(w*t)']
the unknowns, F and G, can be determined by substituting vf(t) in the original differential equation ( this can be done because vf(t) is one of its solutions). Then, you can use the method of undetermined coefficients to find F and G.

HTH

suvendu

Joined: 10 Oct 2004
Posts: 442
Helped: 25

17 Jun 2005 21:10 I need to solve differential Equation

This is 2nd order differential equation for LCR series circuit. so use complementary function and particular integral to solve it.

Dspnut

Joined: 18 Jul 2002
Posts: 171
Helped: 4

20 Jun 2005 7:34 Re: I need to solve differential Equation

Hello friends,

I maybe wrong here but I believe that the above equation is a RLC parallel circuit (not series).

KCL: I(t) = Ic(t) + Il(t) + Ir(t) where
I(t) = A*sin(w*t)
Ic(t) = C*dV(t)/dt
Il(t) = (1/L)*Integrate(V(t))dt + Il(t=0)
Ir(t) = V(t)/R

HTH

steve10

Joined: 26 Mar 2002
Posts: 177
Helped: 33
Location: Los Angeles (Chinese)

20 Jun 2005 9:52 Re: I need to solve differential Equation

Dspnut,
The only thing that is obscure to me is the term Il(t), which is a integral. According to the original poster, Il(t) is a indefinite integral, then the problem can be transformed to a equivalent ordinary differential equation of second order and, therefore, the solution, provided by your previous post, is perfect. If, however, the integral Il(t) is a definite one, then your solution would be problematic. The reason is that, while you can specifify V(0), you are not entitled to specifiy V'(0) as you can get it directly from the equation. In this case, you won't be able to decide the constants "D" and "E" in your previous post, as you only have one condition which is about V(0).

jakjoud

Joined: 06 Feb 2005
Posts: 163
Helped: 9
Location: Marrakech

20 Jun 2005 17:36 I need to solve differential Equation

you can use a derivation with respect of t, then you'll have a DE of second order, use the caracteristique equation: r²+r/(RC)+1/(LC)=0, when getting the solution it's the general one, so you must get a particular solution.

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Dspnut

Joined: 18 Jul 2002
Posts: 171
Helped: 4

22 Jun 2005 9:05 I need to solve differential Equation

Hello friends,
I do agree with steve10 that we need two initial conditions (V(0) and V'(0)). My first impression from reading penrico's post is that both initial conditions are availble. Penrico should be able to clarify this. Smile