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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 $f_c = \frac{1}{2\pi}\sqrt{\frac{1}{LC}}$.

Circuit schematic

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):

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

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

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

\begin{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 \end{equation}

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):

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

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: Experimental data

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