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I have 5th order filter circuit as shown below. But I am trying to find a mathematical expression to calculate the values of inductances and capacitors. Using Wolfram-Mathematica, I found Iout/Vin transfer function. The program can give the zeros but it is not possible to get poles.

Could anyone help to me that how I can get the formula for poles? I am just trying to find which L5=L6 value can give the stabilization of the circuit. When I put the L5=L6=10uH, I can see the oscillation in the circuit.

C3-C4--->C1,
L5-L6--->L,
Iout---->I_R3

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    \$\begingroup\$ Why not simulate it. You should get exceptionally accurate results and, you can add parasitic components too to make it more representative of the real world implementation. \$\endgroup\$
    – Andy aka
    Jun 6, 2023 at 19:34
  • \$\begingroup\$ @Andy aka Yes, I simulated as well, I have a bode plot and time domain result. But I need a math expression about the circuit. \$\endgroup\$ Jun 6, 2023 at 19:38
  • \$\begingroup\$ Which branch is Iout? \$\endgroup\$ Jun 6, 2023 at 19:51
  • \$\begingroup\$ @Tim Williams Sorry, I forgot to write it. I_R3 is Iout \$\endgroup\$ Jun 6, 2023 at 19:54
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    \$\begingroup\$ Why not draw L6, L7, B5, B6 as a coupled inductor? \$\endgroup\$ Jun 6, 2023 at 19:58

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The program can give the zeros but it is not possible to get poles.

But.. it gave you the poles - in terms of roots of a 5th order polynomial! Root objects are used because they have better symbolic manipulation properties and numerically evaluate with lower errors than the radical form. And there are no closed form solutions for 5th order polynomial roots, so you can't "see" them, but you can still evaluate them.

It is not possible to find root if your equation is higher than 3rd order.

It is possible to find the roots numerically, or to find series expansions for them. It is also possible to solve root location inequalities, so you can force the poles to be where you want them to be, and Mathematica will hopefully give you an optimization solution.

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