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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?

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

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  • \$\begingroup\$ electronics.stackexchange.com/questions/321156/… \$\endgroup\$ Commented Feb 8, 2020 at 14:37
  • \$\begingroup\$ 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. \$\endgroup\$ Commented Feb 8, 2020 at 14:51
  • \$\begingroup\$ What are you hoping to achieve with this odd circuit? \$\endgroup\$
    – Andy aka
    Commented Feb 8, 2020 at 14:52
  • \$\begingroup\$ 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. \$\endgroup\$
    – legier
    Commented Feb 8, 2020 at 15:00

1 Answer 1

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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.

enter image description here

The resistance value is calculated to meet a certain quality factor; 5 in this example or a damping ratio of 0.1. The stimulus is a 1-V step. If you run the simulation, you obtain the below transient waveform:

enter image description here

Now extract two consecutive peaks from which you subtract the 1-V offset and calculate the logarithmic decrement as shown in the figure. The below Mathcad sheet shows how measuring the peaks (it works with the valleys too) will give you the damping ratio:

enter image description here

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  • \$\begingroup\$ Yes, you are completely right, but the problem is that because of ADC's measurements there exist small negative/positive feedback which causes that alpha decreases/increases in time and I have no idea how to relate this change to feedback. \$\endgroup\$
    – legier
    Commented Feb 8, 2020 at 17:58

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