Capacitor Energy Storage Math Problems

Discussion in 'General Electronics Chat' started by Austin Clark, Aug 30, 2012.

  1. Austin Clark

    Thread Starter Member

    Dec 28, 2011
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    44
    If I try and calculate the energy (J) stored in a capacitor algebraically, I get J = CV^2
    However, I know that in reality J = (CV^2)/2
    I know this because that's the answer I get when I solve it graphically.
    What's causing this contradiction? Correct mathematics will always come out consistently, so what am I doing wrong?
     
  2. Austin Clark

    Thread Starter Member

    Dec 28, 2011
    409
    44
    I think I answered my own question.
    I was confusing instantaneous voltage with average voltage, or something along those lines. I realized that, even though J = QV (where Q is charge), the Voltage across the capacitor isn't constant, so each charge receives a different amount of energy as the capacitor charges.

    Even still, I'd like to see the math worked out, It'd make more sense then. Anyone care to give it a shot?
     
    Last edited: Aug 30, 2012
  3. t_n_k

    AAC Fanatic!

    Mar 6, 2009
    5,448
    782
    If a source is charging a capacitor and at any instant the current is i(t) and the voltage is e(t) then the instantaneous power delivered by the source would be

    p(t)=e(t) \times i(t)

    We also know the relationship between capacitance C, capacitor current i(t) & instantaneous capacitor terminal voltage e(t) is ...

    i(t)=C\frac{de}{dt}

    The energy delivered by the charging source to the capacitor Wc is the integral of the instantaneous power over time or ...

    W_c=\int_{-\infty}^t p(t) dt=\int_{-\infty}^t {e(t) \times i(t) dt}=\int_{-\infty}^t {e(t) \times C\frac{de}{dt} dt}

    It's normally reasonable to permit a change in the limits of integration with the change of variable [from t to e] within the integration. If the capacitor voltage at time t=-∞ is zero and at time t it is E volts, then one may write ...

    W_c=\int_0^E C e(t) de=\frac{1}{2}CE^2 \ Joules
     
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