# Understanding Resonant Frequency in a Series RLC Circuit

After spending a couple of hours on the internet searching for answers I'm hoping Stack Exchange can help me.

Recently, using simulation software, I used an oscilloscope to calculate the resonant frequency of the circuit, by measuring the sine wave of the resistor. I used the equation 1/T, with T being the time period between the start of the wave and the peak of the wave. this gave me Wn (or Wo, whatever your preference). My reasoning behind this was that resonance is when Xl = Xc, meaning the circuit it purely resistive so the resistance would be at it's peak.

However

When doing the same tests in the lab irl I was told to measure Wn over the capacitor instead, as Q factor is at it's highest when at at resonance as Q=VC/VS.

Both methods gave the same answer, and I'm sure there is a simple explanation as to why, but I simply cannot get my head around it right now and information online is phrased in a manner that is not helping me to understand.

Which method is better? Are the reasonings for using either method correct?

resonance is when Xl = Xc

No, at series resonance X$_L$ = -X$_C$. This mathematically (and practically) means that those impedances cancel leaving just the series resistance, R at resonance.

My reasoning behind this was that resonance is when Xl = Xc, meaning the circuit it purely resistive so the resistance would be at it's peak.

No, resistance remains exactly the same$^1$ for any frequency.

I'm sure there is a simple explanation as to why, but I simply cannot get my head around it right now

Despite the two reactances cancelling out at resonance they are still their individually and you can still measure a voltage across each one but, as said before, the reactances are oppositely signed and, given that the current is common to all three components, the voltage across C and the voltage across L MUST be equal and opposite at series resonance.

Imagine a 1 ohm resistor in series with two 9V batteries wired in opposite directions. You put that in a black box and try and determine what's in the box: -

simulate this circuit – Schematic created using CircuitLab

All you will ever detect is a 1 ohm resistor.

Which method is better? Are the reasonings for using either method correct?

The reasons are the same for both methods and both methods are equally good.

$^1$ Strictly speaking, as frequency rises so does the resistance of interconnecting leads (due to skin effect) but, for the sake of this answer, that is ignored.

I used the equation 1/T, with T being the time period between the start of the wave and the peak of the wave. this gave me Wn (or Wo, whatever your preference).

You may be misled here - you should use the time to complete one full cycle, then take the reciprocal to find frequency (F) in hertz. If you desire $\omega$ then that is 2$\pi F$.

• One wishes one could give a +2 merely for the fact nobody ever mentions skin effect in these matters. Best I could do is a normal up-vote. Soz. Jan 13, 2016 at 10:31
• @Asmyldof It's always appreciated!! Jan 13, 2016 at 10:46
• @Andyaka Thank's very much, with the exception of the XL=Xc which was a typo, I was really going in the wrong direction with some of that and my assignment would have been pretty incorrect. Life saver. Jan 13, 2016 at 11:30