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!

 

[rezeew]

I would suggest taking a step back and reviewing the fundamental concepts of RLC circuits and differential equations. It is important to understand the physical principles and equations involved in order to properly solve the problem.

First, let's review the basics of RLC circuits. RLC circuits consist of a resistor (R), inductor (L), and capacitor (C) connected in series or parallel. These components have different properties that affect the flow of current in the circuit. In a steady state, the current in an RLC circuit is constant and the voltage across each component is also constant.

Now, let's review differential equations. Differential equations are mathematical equations that describe how a system changes over time. In the case of RLC circuits, the differential equation is used to describe the relationship between voltage and current in the circuit.

In order to solve for the stationary current in an RLC circuit, you need to set up and solve the differential equation. In this case, the differential equation is L*i'' + R*i' +1/c*i = u', where i is the current, L is the inductance, R is the resistance, C is the capacitance, and u is the voltage source.

To solve this differential equation, you need to use initial conditions, which in this case are given as i(0+) = 0 and i'(0+) = u/L = 20. These initial conditions represent the current and its derivative at time t=0. Plugging in these initial conditions, you can solve for the constants c1 and c2 in the transient solution c1e-10t + c2e-50t.

Once the capacitor is fully charged, it will block the current and the stationary current will be zero. This is because the capacitor acts as an open circuit for direct current. So, at this point, the current in the circuit is only determined by the voltage source and the resistor.

To solve for the stationary current, you need to set the derivative of the transient solution equal to zero, since there is no change in current. This results in a constant current of 1/3 amps, which is the same as the current calculated using Ohm's law (I=V/R).

In summary, the mistake made in the initial attempt at solving for the stationary current was not properly setting up and solving the differential equation. It is important to understand the underlying principles and equations involved in order to solve problems accurately.