Frequency of a Damped Oscillator (RLC Circuit)

In my last post I mentioned that the frequency of a damped oscillator is not actually the same as of an undamped oscillator.  That makes the assumptions for measuring inductance I used sadly incorrect.  I wanted to briefly run down the math on how to do the calculation correctly:

Frequency of an undamped (perfect) LC oscillator: $\omega_0=\sqrt{\frac{1}{LC}}$

Frequency of a damped oscillator:

Wikipedia page  informs us the formula for the frequency of a tank circuit taking into account the resistance of the coil is:

$\omega = \sqrt{\frac{1}{LC}-(\frac{R}{L})^2}$

I’m too lazy to solve equations since grad school.  So I typed this into my TI-89. I got:

$L=\frac{\sqrt{1-4C^2R^2\omega^2}+1}{2C\omega^2}$

In order to use this we can pulse through the resistor and measure how much voltage is dropped through the tank circuit.  At DC the capacitor is an open, so that will tell us the resistance of the inductor, which should be the only non-negligible part of this.  My R1 is 150 ohms.  I measured one of my 1/4 Watt inductors at 20 ohms.  So, I expect to measure around .6 mV, certainly not more than a volt or so.  That means I’ll do better using the internal voltage reference on the arduino.

I’ll code this up and modify the circuit and we’ll try again in the next post.