Finding resonance frequency of non ideal inductor

AI Thread Summary
The discussion focuses on finding the resonance frequency of a circuit involving a non-ideal inductor and capacitor. The expression for the output-to-input voltage ratio is derived, but complications arise when determining the resonance frequency due to the non-ideal components. It is suggested that the original problem may be misleading, as the calculations for resonance in a standard RLC circuit should apply here as well. A recommendation is made to revisit textbooks for standard resonance frequency calculations and to simplify the expression into Bode form. The conversation emphasizes the importance of accurately defining parameters and suggests considering the magnitude of the transfer function for further analysis.
Gauss M.D.
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Homework Statement



We have this circuit:

(uin+)---[1/jωC]--[jωL]--[RL]----o------uout+

(resistor of value R connecting the two lines at their o's here)

(uin+)-------------------------o------uout-

Find an expression for the quotient of u(in)/u(out), and then find the resonance frequency.

Homework Equations





The Attempt at a Solution



Uout/Uin = R/(R+RL + jωL + 1/(jωC)) = jωRC/(jωC(R + RL2CL + 1)

Without the added resistance at the inductor, the resonance is 1/√LC but the expression I get when trying to find when this denominator goes to zero is just extremely sticky. Have I done something wrong or is this to be expected?
 
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I'm afraid to say that the title is a bit misleading, since your actual problem is finding the resonance frequency of an ideal circuit for which you have the expression of H(j \omega). The fact that you started with a non-ideal inductor, a non-ideal capacitor or "non-ideal anything" (for that matter) doesn't anymore have a role to play in your equations.

I didn't precisely understand your schematic (can you find a picture of it?), thus I will have to assume that you got the right expression for H(j \omega) (but see below). If so, then you are dealing with a variation of the standard RLC circuit. I strongly suggest you to go back to your textbooks and see how the resonance frequency for RLC was calculated - it is almost exactly the same here! The calculations aren't actually harder, if you carefully deal with them. However, it seems to me you made some mistake in writing H(j \omega) (assuming you got right the first expression). Defining R_{tot} = R+R_L, you should get:
H(j \omega) = \frac{j\omega RC}{1+j\omega R_{tot} C -\omega^2 LC}
Now, it is customary to simplify this expression further by explicitly writing it in Bode form. In other words, you write it as:
H(j \omega)=\frac{j2\zeta' \frac{\omega}{\omega_0}}{1+j2\zeta \frac{\omega}{\omega_0} - \frac{\omega^2}{\omega_0^2}}
where \zeta' , \zeta , \omega_0 are all known. At this point, we are half-way. Now, we must consider: |H(j \omega)| (question: can we, instead, consider |H(j \omega)|^2 to simplify things? If so, why?). To calculate it, I suggest you to use: x=\frac{\omega^2}{\omega_0^2} and do all the math with respect to x. Stated this way, you should be able to conclude it.
 
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