# What is the theory behind LC circuit's damping ratio value with a little positive/nagtive feedback?

I need to measure damping ratio of LC circuit attached to microcontroller. I want to achieve it by measuring voltage of the highest/lowest point of consecutive oscillations by internal ADC.

ADC measures voltage by attaching for a short period of time additional discharged capacitor (typ. 5-8pF) and then by succesive approximation unit it outputs digital value.

But attachment of such capacitor acts as negative or positve feedback during measurements.

In such circuit damping ratio between any two consecutive amplitudes should be constant, but with this little feedback the smaller the amplitude the lower/higher damping ratio is (depending on feedback sign). I tried to find out some theory behind this and get formula describing damping ratio's change but with no success. Can any one give me a formula to calculate it?

P.S. If it is nessecery I can bring some data to shows this behaviour.

• electronics.stackexchange.com/questions/321156/… Commented Feb 8, 2020 at 14:37
• Your question is inconsistent with model. Shown is a lossless Cct with a freq. shift. Both undamped. yet this is easily analyzed Search LC oscillator. Commented Feb 8, 2020 at 14:51
• What are you hoping to achieve with this odd circuit? Commented Feb 8, 2020 at 14:52
• I did not place resistor there, due to non ideal coil and capacitor in my circuit. Coil's resistance is about 2 ohms. Main goal is to frequently short lower pin to GND just to load the C capacitor, release it and measure the damping ratio. It is build for detecting metal object in front of the coil. Commented Feb 8, 2020 at 15:00

The damping ratio $$\\zeta\$$ (zeta) of a $$\RLC\$$ filter can be evaluated using the logarithmic decrement $$\\delta\$$ (delta): $$\\delta=ln(\alpha)=\frac{2\pi}{\sqrt{4Q^2-1}}\$$ from which you extract $$\Q\$$ or $$\\zeta\$$ as detailed in the book I published on loop control: $$\\zeta = \frac{1}{\sqrt{(\frac{2\pi}{\delta})^2+1}}\$$.
Let's see a practical application with the below $$\RLC\$$ circuit.