Results 1 to 4 of 4
Like Tree2Thanks
  • 1 Post By johnsomeone
  • 1 Post By HallsofIvy

Math Help - Setting up a homogenous ODE using Kirchhoff's law of voltage

  1. #1
    Junior Member
    Joined
    Sep 2012
    From
    ohio
    Posts
    68

    Setting up a homogenous ODE using Kirchhoff's law of voltage

    Thank you in advance for your help. I'm having trouble with the setup of this problem. The problem states:

    The loop current I in a series RL circuit with constant voltage E0 satisfies LI' + RI = E0 (where I' is the first derivative of
    I) by Kirchhoff's voltage law. Assume that R and L are constants. Asume that initially there is no current in the circuit.

    a) Find the current as a function of time.
    b) Find the steady state solution.
    c) When will the current be (1-e^-1) times the steady state current?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    Re: Setting up a homogenous ODE using Kirchhoff's law of voltage

    Quote Originally Posted by nivek0078 View Post
    Thank you in advance for your help. I'm having trouble with the setup of this problem. The problem states:

    The loop current I in a series RL circuit with constant voltage E0 satisfies LI' + RI = E0 (where I' is the first derivative of
    I) by Kirchhoff's voltage law. Assume that R and L are constants. Asume that initially there is no current in the circuit.

    a) Find the current as a function of time.
    b) Find the steady state solution.
    c) When will the current be (1-e^-1) times the steady state current?
    a) Solve L\frac{dI}{dt} + RI = E0, with I(0) = 0 (which is the meaning of "initially there is no current in the circuit"), and L, R, and E0 constant. If you don't know how to do that, then your trouble is that you don't understand the basics of differential equations. I'll assume that you do know how to do that, so that you've solved for I(t).

    b) I believe that the "steady state solution" is asking what I(t) is "asymtopically equivalent to" for large time values. If you did (a) and understand what this means, it's not hard to determine.
    Ex: If I(t) were a polynomial in t, then the steady state solution would be the highest power term of the polynomial.
    Ex: If I(t) = 3t^2 -4t +7, then I(t) ~ 3t^2, because -4t +7 are eventually a miniscule fraction of I(t) once t is large.
    Ex: If I(t) = 2t + 5sin(t), then I(t) ~ 2t, since 5sin(t) is a minscule fraction of I(t) when t is large (5sin(t) bounces between -5 and 5).
    Ex: If I(t) = (3t +2)/(t+1), then I(t) ~ 3. That's because I(t) = (3t + 2)/(t+1) = (3(t+1) -3 +2)/(t+1) = (3(t+1)-1)/(t+1) = 3 - (1/(t+1)), and 1/(t+1) goes to 0 as t goes to infinity.
    Ex: Any term going to 0 is discarded from the steady state solution. If I(t) = f(t) + g(t), and limit {t -> infinity) g(t) = 0, then the steady state solution for I(t) is the same as the steady state solution if I(t) has equalled *just* f(t). For the steady state, you can discard terms going to 0.

    c) In part (a), you solved for a function I(t). In part (b), you determined a new function of t, the steady state function of I(t), let's call it SS(t).
    (In other words, I(t) ~ SS(t)). Now part (c) asks "When will the current be (1-e^-1) times the steady state current?". Suppose that specific time(s) is labelled t'.
    Then (c) is asking: "Find t' such that I(t') = (1-e^-1) SS(t')." So simply write down that equation and solve it for t'.
    Note that, in general, there could be more than one solution. Note also that the domain here is t >= 0, so negative solutions must be discarded.
    Last edited by johnsomeone; September 25th 2012 at 12:33 PM.
    Thanks from nivek0078
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2012
    From
    ohio
    Posts
    68

    Re: Setting up a homogenous ODE using Kirchhoff's law of voltage

    Thank you johnsome for the information.

    This is open to anyone! I would like to check if my solution to part A) is correct. I got I(t)= E0 + C, if not what did i do wrong. Is the intergation factor e^t?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,370
    Thanks
    1314

    Re: Setting up a homogenous ODE using Kirchhoff's law of voltage

    Quote Originally Posted by nivek0078 View Post
    Thank you johnsome for the information.

    This is open to anyone! I would like to check if my solution to part A) is correct. I got I(t)= E0 + C, if not what did i do wrong. Is the intergation factor e^t?
    Could you not check it yourself? I(t)= E0+ C is a constant- the general solution to this equation is NOT constant! In particular, if I(t)= E0+ C then I'(t)= 0 so the equation becomes R(E0+ C)= RE0+ RC, NOT E0.
    Thanks from nivek0078
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Voltage Change
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 2nd 2011, 03:10 AM
  2. Phototube and Voltage
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: February 10th 2011, 08:48 AM
  3. find the voltage across
    Posted in the Math Topics Forum
    Replies: 7
    Last Post: October 12th 2010, 12:21 PM
  4. please help me with Kirchhoff's circuit laws
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: August 2nd 2010, 07:51 AM
  5. Approximate the voltage
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 1st 2010, 12:51 AM

Search Tags


/mathhelpforum @mathhelpforum