Finding resonance frequency of non ideal inductor

[Gauss M.D.]

Homework Statement

We have this circuit:

(u_in+)---[1/jωC]--[jωL]--[R_L]----o------u_out+

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

(u_in+)-------------------------o------u_out-

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

Homework Equations

The Attempt at a Solution

U_out/U_in = R/(R+R_L + jωL + 1/(jωC)) = jωRC/(jωC(R + R_L -ω²CL + 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?

 

[Microgravity]

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.