# Electronic – Discharging a capacitor through a resistor and an LED in series

capacitordiodestheory

Suppose you have a capacitor of capacity \$C\$ and initial voltage \$U_0\$, a resistor \$R\$ and a LED with threshold voltage \$U_S\$ (\$U_0 > U_S\$) in series.

Now I want to calculate duration \$\tau\$ where the led is lighting.

My intuition was that the effect of the LED should be small in this case and I can use just the usual formula for capacitor discharge \$U(t) = U_0 e^{-\frac{t}{RC}}\$. Then I guess that the LED is lighting until the voltage reaches the value \$U_0\$, i.e. I have to solve the equation \$U(\tau) = U_0\$ which leads by elementary algebra to \$\tau = -RC \ln\left(\frac{U_S}{U_0}\right)\$.

However I am not sure if my intuition is correct and how to give reasons for it. So is there a good simple argument, why the approximation above or a similar (correct) approximation is valid?

My second question is about how to derive this (or a similar correct approximation) from first principles.

My idea was to set up a differential equation like follows:

$$C \frac{dU(t)}{dt} = -I(t)$$

And put for \$I(t)\$ the formula for the current through the diode I found on http://en.wikipedia.org/wiki/Diode_modelling#Explicit_solution which involves the Lambert-W-function. However then it get's pretty complicated and I don't know how to solve this differential equation and how to make reasonable approximations (at best with bounds for errors).

PS: I have found this paper: http://www.uncg.edu/phy/hellen/HellenAJPAug03.pdf which discusses the problem in the case when only a diode is present. But it doesn't take the resistor in series into account.

Edit: If I assume approximately that the diode has the voltage \$U_S\$ all the time, after solving the corresponding differential equation, I end up with something like \$U(t) = U_S + (U_0 – U_S) e^{-\frac{t}{RC}}\$ which seems to make no sense because \$U_S\$ is a lower bound (which was actually already in the assumption…). So it would be great if someone could really clarify all the mess here…