# How to experimentally measure damping in RLC circuits?

I know there are methods to theoretically determine the damping factor and quality factor in RLC circuits. Equations such as

$$\zeta = \frac{R_s}{2} \sqrt{\frac{C}{L}}$$

$$Q = \frac{1}{2\zeta}$$

Can tell me all about the damping behavior in a certain circuit, however, I would like to find damping experimentally. I have done some research and I know that I can experimentally find some factors which will help formulate a damping graph, but I would like more direct methods.

Is it possible to use an oscilloscope and probe the capacitor to see the actual ac signal?

I am open to using a wide range of methods.

• Examine the output of an MCU, with a hard "O", and you'll see the internal ringing of the silicon "0". In NMOS-on-bulk (Pwell), this will indeed be the silicon substrate ringing. Program the output to a hard "1", and you'll see the VDD ringing, as the MCU executes its program. Commented Sep 12, 2018 at 4:48

Is it possible to use an oscilloscope and probe the capacitor to see the actual ac signal?

Yes it is. The challenge is to do the measurement such that the RLC circuit is not disturbed. Resonance in an RLC circuit is the electric energy travelling back and forth between the inductor and the capacitor. Obviously that energy should only be dissipated ("escaping" the circuit) through the R in the RLC circuit, not through the measurement setup.

It helps if the amount of energy involved in the resonance is larger. So measuring an RLC circuit with C = 1 nF and L = 10 uH (resonates at 10 MHz) will be more difficult than C = 100 nF and L = 1m H (resonates at 100 kHz).

A oscilloscope's probe (use a proper 10:1 probe!) will have a certain input capacitance, usually around 15 - 20 pF. Make sure your circuit's C is significantly larger than that and the probe's capacitance should be no issue.

The probe's 10 Mohm (we're using a 10:1 probe remember) input resistance should be no issue as the R in your RLC circuit is bound to be orders of magnitude lower in value.

Then how do we get some energy into the RLC circuit?

It is possible to use a just use an external power source (a battery would do) and let the power in by using a switch:

simulate this circuit – Schematic created using CircuitLab

Resistor R2 is there to limit the current when SW1 is pressed. It depends on the values used in your RLC circuit what will work best. Feel free to experiment !

I'm assuming that you would be using a digital oscilloscope, then you can trigger it on the starting of the resonance (note that then the voltage will go negative, you can trigger on that).

With Fc the LC resonant frequency, you can :

• Excite the circuit with a square wave of much lower frequency that Fc.

Each edge of the square wave will produce exponentially decaying ringing, ie you will see the transient response, and you will know if it is damped or ringing (underdamped).

Then, you can measure the Q factor by looking at how the ringing decvays. If you solve the circuit's differential equation, you will get an exponentially decaying sine. Measure the decay between one peak and another peak a bit later, also measure the time (or number of periods) and you can compute Q.

• Bode plot

You can use a network analyzer, or in LF you can simply use a soundcard and some freeware audio program to plot the frequency response of your circuit when excited by a source of a known impedance. Then, read this question :

Resonant Frequency from Bode plot

Note: in both cases, make sure you take the output impedance of the signal generator into account.