So I am trying to understand how to calculate the value of an inductor in a LC circuit with a known value capacitor and I would like to know if one or both of my current systems are flawed.

The method to my madness currently is 1/ get the tank (3rd pic) to resonate with a pulse and then count the frequency of the oscillation (1st pic) or, 2/ vary the frequency of a sine wave till I hit the highest amplitude of the tank and measure the frequency (2nd pic) which then gives me enough to calculate the inductor value.

Edited because I can't load pictures in order...

Hi there,

[Note this has been edited for content]

Actually the two methods will yield two different values for the resonate frequency. That might sound strange but that's life.

The difference may not be that great, but in general that's the way it is.

Also, to actually have resonance in the first place the values of the external resistance and the inductor series resistance have to be of certain values or else it's just a damped response. That does not mean you cannot calculate the inductance that way though, but the values will be different. How much different depends on the resistances and the L and C values.

The nice thing is, the pulse (stepped) wave technique can show if there is resonance by observing the wave as it looks like you did. If there is no apparent sinusoidal part then it's not resonating.

The test circuit would be comprised of an inductor L with some series resistance RL (present in all real inductors we normally see), and the generator series resistance we can call Rs, and we can also add more resistance to Rs by placing a resistor in series with Rs, but then the Rs in the formulas below is the total series resistance not just the resistance of the generator.

Ok, so in the time domain with a stepped input wave the 'resonant' frequency w (if there is a real one) will measure:

w=sqrt(-Rs^2*C^2*RL^2+2*Rs*C*L*RL-L^2+4*Rs^2*C*L)/(2*Rs*C*L)

and that part under the radical must be greater than zero or there is no real resonance.

In the frequency domain (frequency sweep) the 'resonant' frequency w (if there is one) will be:

w=sqrt(sqrt(2*Rs^2*C*L*RL^2+2*Rs*L^2*RL+Rs^2*L^2)/(Rs*C*L^2)-RL^2/L^2)

and also this expression illustrates the departure from the ideal LC resonance which is w=sqrt(1/(L*C)).

Note that I put the word 'resonant' in quotes. That is because there are three different forms of resonance, so in general I like to call this the 'resonant peak' and the 'resonant peak frequency' we often refer to as w0. That's the frequency where we observe the maximum peak when we apply a variable frequency sine wave for testing.

Note that often the two w's above are not that far apart so an estimate either way MAY get you there. There could be times when they do not match well though so beware of that.

The nice thing about the pulse method is it seems to show if the circuit can resonate to begin with.

Also note that you can also just use the voltage divider formula to calculate the inductance using:

VL=Vs*w*L/sqrt(w^2*L^2+Rs^2)

where

VL is the voltage across the inductor,

Vs is the source sine wave voltage,

L is the inductance,

Rs is the total resistance of the generator plus any added series resistance for the test.

You will note that there is no capacitor in this circuit so it simplifies a little.

With that expression you solve for L.

Note however that in that expression there is no RL. If we do include RL the expression becomes more complicated:

V=Vs*sqrt(RL^2+w^2*L^2)/sqrt((RL+Rs)^2+w^2*L^2)

and to solve for L we also have to solve for RL (unless we rely on a low frequency or DC measurement). To solve for both we would also have to measure the phase and include that into a more advanced formula.

If you feel like leaving the cap in the circuit but still employ the voltage divider method, you can use:

V=Vs*sqrt(w^2*C^2*RL^2+(1-w^2*C*L)^2)/sqrt((w*C*RL+Rs*w*C)^2+(1-w^2*C*L)^2)

but of course that just complicates things.

Hey, let us know how you make out