Parallel LCR resonant frequency derivation for larger resistances

[QuarterWit]

[SOLVED] Parallel LCR resonant frequency derivation for larger resistances

There have been similar questions here before which I have seen, but I couldn't find an answer with the detail I require.

Homework Statement

I would like to derive an expression for the resonant frequency of a parallel LCR circuit with a large (not negligible) resistance.

I have seen a https://www.physicsforums.com/showthread.php?t=121069" of a solution (the same as mine but with the "4" omitted) but it doesn't fit my experimental data by a long shot, though my data could be wrong. The solution I came up with is a lot closer to my data but that doesn't necessarily mean it's right, could someone check through and explain if/where I've gone wrong
http://gray.slightlybeanish.com/Other/LCR.jpg

Homework Equations

The Attempt at a Solution

http://gray.slightlybeanish.com/Other/LCRDerivation.jpg

I mostly suspect the first step is wrong, I was trying to apply Kirchoff's law

 

[turin]

... an expression for the resonant frequency of ...
http://gray.slightlybeanish.com/Other/LCR.jpg
...
http://gray.slightlybeanish.com/Other/LCRDerivation.jpg

I[/URL] agree with this. You just have to make sure that 1/LC > R²/4L² (otherwise the system is overdamped).

 

[FedEx]

I mostly suspect the first step is wrong, I was trying to apply Kirchoff's law

Kirchoof is never wrong. What you have done is right. If you want to get the solution of the equation by the help of complex numbers. However that is unnecessary

 

[QuarterWit]

Thanks for replying so quickly, I'm glad you agree.
Fedex: I'm not sure whether you mean your 3rd sentence to be connected to the 2nd or the 4th. Anyway, I have used complex numbers, which is why the arguments under the square root changed signs.

As turin said, I'd assumed 1/LC > R²/4L². Which would give a complex solution to the quadratic, leading to an oscillating solution to the differential equation, with the angular frequency stated.

For any other readers, this can also be solved by calculating the impedance directly and finding when it's a maximum, I was just looking for a short derivation.

 

[FedEx]

Fedex: I'm not sure whether you mean your 3rd sentence to be connected to the 2nd or the 4th. Anyway, I have used complex numbers, which is why the arguments under the square root changed signs.

My mistake. What i wanted to say is that we can solve the second order differential equation with the help of complex numbers. But that is quite long.

 

[QuarterWit]

Ah ok, that is what I did, I wasn't aware there was another way to do it, I just omitted it from the working because it's the same for any question.

The method is shown http://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx" for those wondering how I suddenly leapt to the frequency from the characteristic equation. I just realized that if the system is to oscillate, the roots will be complex, so the trial solution will have a decay term e^-R/2L multiplied by (Acosωt + Bsinωt)
(ω as above)

 

[turin]

For any other readers, this can also be solved by calculating the impedance directly and finding when it's a maximum, I was just looking for a short derivation.

I thought that was the short derivation. That's how I did it. I just found the poles of the transfer function in extended Fourier space. It is just a few lines of algebra (depending on how much detail you want to show).