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I'm in university studying 'Electrical Engineering'. We've to model a speaker, the model we came up with has the following formula representation:

$$\left|\underline{\text{Z}}_\text{in}\left(\omega\right)\right|=\left|\text{R}_1+\text{j}\omega\text{L}_1+\frac{1}{\frac{1}{\text{R}_1}+\frac{1}{\text{j}\omega\text{L}_2}+\frac{1}{\left(\frac{1}{\text{j}\omega\text{C}}\right)}}\right|$$

Where are variables are real and $\ge0$, and $\text{j}^2=-1$.

The resonancefrequency can be found using, two methods:

  1. $$\frac{\partial\left|\underline{\text{Z}}_\text{in}\left(\omega\right)\right|}{\partial\omega}=0\space\Longleftrightarrow\space\omega=\omega_\text{res}=\dots$$
  2. $$\Im\left[\underline{\text{Z}}_\text{in}\left(\omega\right)\right]=0\space\Longleftrightarrow\space\omega=\omega_\text{res}=\dots$$

Question: What can we say about $\underline{\text{Z}}_\text{in}\left(\omega_\text{res}\right)$ and $\left|\underline{\text{Z}}_\text{in}\left(\omega_\text{res}\right)\right|$, I mean: what will the closed form representation of $\underline{\text{Z}}_\text{in}\left(\omega\right)$ and $\left|\underline{\text{Z}}_\text{in}\left(\omega\right)\right|$ be at the resonant frequency?

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    Are you asking for a closed form for the impedance at the resonant frequency? – 2017-01-14
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    @Dr.MV Yes I do, you understood that the right way! – 2017-01-14
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    I would advise that you pose your question more clearly. – 2017-01-14
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    @Dr.MV Done, thanks for the hint! – 2017-01-14
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    You have two methods of finding the resonant frequency. What stops you from proceeding? – 2017-01-14
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    @Dr.MV I'm not able to use one of the methods the straight way, because it becomes very nasty (big formulas turn up very quickly). – 2017-01-14
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    And have you tried to other? – 2017-01-14
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    @Dr.MV Sure, it gives the same big nasty formulas – 2017-01-14
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    Well, perhaps the problem is simply not clean. And why do you think it would or should be? – 2017-01-14
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    @Dr.MV Sure, it gives the same big nasty formulas, I think it because I just add a coil into the model if I do that not the resonancefrequency is just $\frac{1}{\sqrt{\text{CL}}}$ and now I just add one coil and it becomes so that nasty – 2017-01-14

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