PROBLEM:
I have been trying to measure and explain the oscillation frequency of an LC circuit I have been working on. The inductor that I’m using is made from conductive polymer so it has a big 200Ω resistance and my measured inductance is much lower than what I expect.

WHAT I KNOW/TRIED:
Because of such high resistance, obviously the ω=\(\sqrt{1/LC}\) won’t work here and my measured resonant frequency shifted down significantly. I found a tutorial on All About Circuits that talks about the idea of anitresonance. I was able to derive the oscillation frequency by calculating the total Z and setting the imaginary part to zero:

and thought maybe the (R/L)^2 is the antiresoance frequency that lowers my measured frequency. Using the equation above and putting values for L,R,C would give me an imaginary numbers. Why is that? I have verified the equation in 2 or 3 textbooks. I am not sure how to justify an imaginary frequency.
I have been trying to learn more about antiresonance but my limited online source don’t seem to explain it properly/mathematically. Also, Sometimes they call it parallel resonance which confuses me more: a simple LC circuit is parallel and the resistance is in resistance. SO is the ω=√1/LC also a parallel resonance(aka: antiresonance)??. I am just very confused about the whole thing.
HOW TO HELP
-Am I doing something wrong?
-Can someone please introduce me a good source ( preferably a textbook) that explains the idea of antiresoannce.
-Is there a way to calculate the resonance correctly?

Thank you.

 

Any guess what the L is? You control C, right?

 

Hi MikeML,
Thank you for your reply. Yes, I do control the C. I have done a COMSOL simulation of the coil and the inductance is in the nH range because of the low conductivity of the conductive polymer that I am using (~200 nH but this is just an approximation).

I have also repeated the experiment with a known inductor (19μH) that I bought and put a 200Ω series resistor (with my C=1nF) and what I measure does not agree with my calculations from any of the above equations. In fact, I get an imaginary frequency from the second equation. When I remove the resistor the value is an exact match with what I get from ω=1/√LC.

 

Consider how the equation

\(\omega_o=\sqrt{\frac{1}{L C} - \( \frac{R}{L} \)^2}\)

is derived.

The derivation is based on the assumption that at some frequency the complex imaginary part of the effective circuit impedance is zero. The circuit appears purely resistive at some frequency. If that condition cannot be met at any frequency then the equation has no real solution. Clearly if

\( \( \frac{R}{L} \)^2 \ge \ \frac{1}{L C}\)

then the condition cannot be met.

 

Using the equation above and putting values for L,R,C would give me an imaginary numbers. Why is that?

You get imaginary results because R is too large for the natural response of the circuit to be oscillatory. Your circuit is overdamped.

 

Maybe that this Wiki will give you some ideas...

Says it is good for measuring low Q inductors...

 

Try a series resonant circuit.

An Agilent Vector Impedance meter could measure this in a heartbeat...

 

I have also repeated the experiment with a known inductor (19μH) that I bought and put a 200Ω series resistor (with my C=1nF) and what I measure does not agree with my calculations from any of the above equations.

What did you measure? How did you measure it (what instrument did you use to make the measurement)?

What was the result of the measurement, and what was the result of the calculation?