Imagine we are trying to design a "Servo-system". Its a system that takes an input angle and rotates a motor until its pointing at the input angle. The block diagram of this system could be represented like this
The proportional controller outputs tourque based of the difference between the current angle and the desired angle. We will test this system againest step inputs
You might be asking now "What does this has to do with my circuit". The answer would be "They are both second order systems" they both are charactarized by a second order time domain differential equation.
Now defining the variables
\$r\$ : The input reference angle
\$e\$ : The difference between the current motor angle and the required angle
\$T\$ : The proportional controller output that is the motor tourque
\$J\$ : The rotational intertia of the rotating body
\$B\$ : The amount of damping we provide to the system
Imagine we are not providing any kind of damping in our system, once the motor is rotating it will never stop rotating since there is no friction [assuming no frictions at all]
Here is a simulation for the case where this system has no damping value at all
Here is the expected time domain response for this system.
Eventhough the system reached the target value at \$t=T_r\$ and at this point the proportional controller unit output tourque is zero the motor is still rotating at a constant speed since there is no friction to decrease the motor rotational speed and rotational kinetic energy and once the motor is pointing to different angle [more than the desired angle] the proportional controller will drive the motor in the oposite direction and you will have this endless oscillation [NO FRICTION EXISTS]
\$B=0\$ is not a choice then, the system will never be stable if no friction is provided, so how much B should we provide to the system, and how would the system respond to different amounts of damping values.
It turns out that there is a term that charactarizes the system based on the damping value B which is the system damping ratio
Ratio between the system current damping value and the crictical damping value \$B_c\$
We can define \$B_c\$ as the value of the damping that is enough to stop the rotating body at the target value; that is once the motor is pointing to the required angle and the proportional controller output tourque is zero, the rotating body has no energy to rotate anymore.
So if \$\zeta ≥ 1\$ this means you are providing a damping value which is greater than or equal the crictical damping value [NO WAY YOUR SYSTEM WILL OSCILLATE]
The more you increase \$\zeta\$ the more it takes for your system to reach the target value
This is the simulation for our system when \$\zeta=1\$ where the system is circtically damped
And as we increase the damping ratio the time taken to reach the target value is incearsed
If \$\zeta < 1\$ this means that you are still providing damping to your system but this damping value is not high enough to prevent the rotating body from exceding the target value, this is the expected time response for an under damped system
At time \$t=T_r\$ you can see that the rotating body keeps rotating as the case where B=0; But in this case as the rotating body is rotating its energy decreases until it reached a point where its oscillating close to the target value
So its a matter of how much energy is stored and how much damping exists that is sufficent to eat up this amount of stored energy before exceding the target value. The more damping you provide the slower the system approaches the target value. The less damping you provide the more your system oscillates around the target value while its stored energy is eaten up by the damper untill it dies out
If you think this answer is confusing or provides false information or even didnt answer the part you are asking about please let me know in the comments i`ll definetly delete it.