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



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:



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: