# Crossover filter gives weird results

I was analyzing results of an experiment on a crossover filter and I was trying to take into consideration the parasitic resistance of the inductor L (called $$R_L$$).

I know the theoretical crossover frequency under the assumption that $$R_1 = R_2$$ and $$R_L = R_3 = 0 \Omega$$ is $$f_c = \frac{1}{2\pi}\sqrt{\frac{1}{LC}}$$.

Dropping this assumpions I get in the phasor domain that (letting $$v_t$$ be the tweeter output signal, $$v_w$$ the woofer one and $$v_{fgen}$$ the voltage of the function generator. I used tilde to denote fasors):

$$$$|\tilde{v_t}| = \frac{R_2|\tilde{v}_{fgen}|}{\sqrt{(R_2 + R_3)^2 + (\omega C)^{-2}}}$$$$

$$$$|\tilde{v}_w| = \frac{ R_1 |\tilde{v}_{fgen}|}{\sqrt{(R_1 + R_L) ^ 2 + (\omega L)^2}}$$$$

So by requiring $$|\tilde{v_t}| = |\tilde{v}_w|$$ I get the equation:

$$$$(R_2 L)^2 (2 \pi f_c) ^ 4 + \left(R_2^2(R_1 + R_L)^2 - R_1^2(R_2 + R_3)^2\right)(2 \pi f_c) ^ 2 -\left(\frac{R_1}{C}\right)^2 = 0$$$$

Now taking the already stated assumptions the term in front of the squared term vanishes and I get the expected result of $$f_c = \frac{1}{2\pi}\sqrt{\frac{1}{LC}}$$. And the addition of $$R_3$$ in the tweeter branch is actually required to make the term $$\left(R_2^2(R_1 + R_L)^2 - R_1^2(R_2 + R_3)^2\right)$$ vanish. Let's call this goddamned term $$B = \left(R_2^2(R_1 + R_L)^2 - R_1^2(R_2 + R_3)^2\right)$$.

Turns out that building the circuit using two equal nominal value resistors for $$R_1$$ and $$R_2$$ and some close value to $$R_L$$ for $$R_3$$, I get a frequency that is pretty close to the crossover frequency given by $$\frac{1}{2\pi}\sqrt{\frac{1}{LC}}$$, which seemes logical.

At this point I though. Oh $$B$$ is small let's do perturbative series and find first order correction terms. And I got the formula (for the angular frequencies):

$$$$\omega_{c,corrected} = \omega_{c,0} - \frac{B}{4(R_2 L)^2 \omega_{c,0}}$$$$

Which actually gives a correction of ~30 Hz, which could be correct.

The problem started when I calculated $$B$$. I thought that $$B$$ should be a small number, but calculating it it turns out to be of the order of $$7 \times 10^9 \Omega^4$$. The values measured of the components is $$R_1 = (997 \pm 1)\Omega$$, $$R_2 = (999\pm 1)\Omega$$, $$R_3 = (200.8\pm 0.1)\Omega$$, $$R_L = (203.5\pm 0.2)\Omega$$, $$C = (44.0 \pm 0.4)nF$$, $$L = (47.6 \pm 0.5)mH$$.

But experimental data shows that the zeroth order solution (aka $$\frac{1}{2\pi}\sqrt{\frac{1}{LC}}$$) is almost right (the correction of those 30ish Hz would be okay, but the perturbative approach is wrong because we can't assume a small $$B$$).

Also the exact solution to the equation goes against experimental evidence. What am I doing wrong?

Experimental data: