RLC circuit - Differential equation - Stationary current calculation

[DrOnline]

Homework Statement

I'm decent with differential equations for RLC circuits, and I KNOW the stationary current will be zero, but I need help to work it out mathematically, because my maths gets me -2.4A (...)

L: 0.05H
C: 0.04F
R: 3 Ω
U (source) = 1 V

I've got the transient solution: c₁e-^10t + C₂e^-50t

Known:

i(0+) = 0 (coil prevents instant change in current)
i'(0+) = u/L = 20

Once the capacitor is charged up, it will block the current. In other words, the stationary current will be zero.

The Attempt at a Solution

So I could simply declare that, on electrical reasoning, but I want to calculate it. The problem is, my calculation is wrong. Something is wrong.

Been banging my head against this for a while, hope somebody can help me out!

Differential equation: L*i'' + R*i' +1/c*i = u'

i''=0, because i'=20

u'=0, because u=1

I call the stationary current A:

L*0 + R*20 + 1/c A = 0

A=-20*R*C= -2.4

What have I done wrong? If I lack information, please let me know.

 

[vela]

i'' isn't 0 because i'=20. i'=20 only when t=0.

You have two approaches: 1. Solve the differential equation and then take the limit as t goes to infinity; 2. Argue what i'' and i' equal when the system has reached steady state, plug those values into the differential equation, and then solve for the current.

 

[DrOnline]

i'' isn't 0 because i'=20. i'=20 only when t=0.

You have two approaches: 1. Solve the differential equation and then take the limit as t goes to infinity; 2. Argue what i'' and i' equal when the system has reached steady state, plug those values into the differential equation, and then solve for the current.

Hmm.. well going with approach 2, that makes sense. There will be no change of i when the system is stable.

It's real late here, going to take a look at it tomorrow. Thanks! I will ask again if I am still stumped heh.

Edit: And yeah, my assumption about i''= 0 because i'(0)=20 is of course flawed.. It's the Christmas break that made me forget some things!

 

[DrOnline]

Alright, looking at the work again today.

Well it seems to me I already have the starting values I needed!

i(0+)=0
i'(0+)=U/L=20

I don't need i'' for that aspect.

And yeah, if I go with the second approach you outlined, using reason and knowledge of the RLC-circuit, there is no change of i when it is stable, so that means the i''(∞)=0.

Not sure if using infinity in an argument is sound, but it works!