Construct medium from a transfer function

[dmarek-flex]

Not sure if this is of interest to others, but when working on lumped elements I found that I needed to construct a PoleResidue medium given an admittance function in the Laplace domain, which is most commonly written as a rational expression. This might be useful in other cases where you would like to model a material that is not easily converted to PoleResidue format, but the differential equation governing the polarization current is available.

More precisely, given $\epsilon_\infty$ and an admittance relationship between electric field and a polarization current density

$$ J_p(s) = Y(s)E(s)$$

where

$$Y(s) = \frac{a_0 + a_1 s + \dots + a_M s^M}{b_0 + b_1 s + \dots + b_N s^N}$$

we can find an equivalent complex permittivity (assuming $Y(s)$ is causal, substitute $s = j \omega$)

$$\epsilon(\omega) = \epsilon_\infty - \frac{1}{\epsilon_0 j \omega}Y(j \omega).$$

Finally, we compute the poles and residues of $\frac{1}{\epsilon_0 j \omega}Y(j \omega)$ to convert it into a PoleResidue medium.

At this point I am just curious if this would be useful outside of lumped elements.

[tylerflex]

Just a comment about scipy. otherwise looks good!

[tylerflex]

oh yea, and reminder to add changelog item for it

[caseyflex]

Nice! My one suggestion would be that the name from_transfer_function is a bit ambiguous, I might think that a and b are just the numerator and denominator coeffs of eps(s) unless I read carefully. Maybe from_admittance_coeffs or something to make it clear that there is an additional step before partial fraction decomposition?

Actually, along those lines, maybe we can also split this function into two functions. One from_admittance_coeffs or something, and the other one can just be _real_partial_fraction_decomposition which does partial fraction decomposition using scipy and then takes only one representative of each complex conjugate pair.

[weiliangjin2021]

Nice work!

Actually, along those lines, maybe we can also split this function into two functions. One from_admittance_coeffs or something, and the other one can just be _real_partial_fraction_decomposition which does partial fraction decomposition using scipy and then takes only one representative of each complex conjugate pair.

I also feel it's better to split this function, so that one can apply it directly in other cases when it's not from a transfer function. E.g. it can be applied #1479 and partially (though not the best practice) #1240

[caseyflex]

@weiliangjin2021 @dmarek-flex
In principle the partial fraction decomposition is uniquely defined, so if we start from a transfer function which is stable and passive, we obtain a pole residue model which is stable and passive. I wonder if this is numerically robust here? If we start with a pole on the frequency axis, could we end up finding a pole slightly below it? If we start with Im(eps) = 0, could we end up with Im(eps) < 0?

[dmarek-flex]

@weiliangjin2021 @dmarek-flex In principle the partial fraction decomposition is uniquely defined, so if we start from a transfer function which is stable and passive, we obtain a pole residue model which is stable and passive. I wonder if this is numerically robust here? If we start with a pole on the frequency axis, could we end up finding a pole slightly below it? If we start with Im(eps) = 0, could we end up with Im(eps) < 0?

I assume that is a possibility, I was going to try and give a better answer, but could not find one! Instead I at least added more tests and validation to this convenience function to ensure that the supplied admittance transfer function is realizable from passive components. But it doesn't guarantee that the final computed PoleResidue medium will be passive over all frequencies if there is some kind of numerical error when computing residues and poles.

[caseyflex]

@weiliangjin2021 @dmarek-flex In principle the partial fraction decomposition is uniquely defined, so if we start from a transfer function which is stable and passive, we obtain a pole residue model which is stable and passive. I wonder if this is numerically robust here? If we start with a pole on the frequency axis, could we end up finding a pole slightly below it? If we start with Im(eps) = 0, could we end up with Im(eps) < 0?

I assume that is a possibility, I was going to try and give a better answer, but could not find one! Instead I at least added more tests and validation to this convenience function to ensure that the supplied admittance transfer function is realizable from passive components. But it doesn't guarantee that the final computed PoleResidue medium will be passive over all frequencies if there is some kind of numerical error when computing residues and poles.

Maybe you can check PoleResidue._imag_eps_extrema_with_samples to warn the user if the resulting model is not passive?

[dmarek-flex]

@caseyflex Good idea, I added the test you suggested and at very high frequencies one of the admittance functions I was testing does show slight gain, although it is very small ~-1e-22. In fact, in this case I think it is numerical error when computing the complex permittivity that is causing a small negative part to appear. So I added a warning to the log when a gain term larger than machine precision is detected.

[caseyflex]

Sounds good, thanks!

…

On Fri, Sep 27, 2024 at 4:41 PM dmarek-flex ***@***.***> wrote: @caseyflex <https://github.com/caseyflex> Good idea, I added the test you suggested and at very high frequencies one of the admittance functions I was testing does show slight gain, although it is very small ~-1e-22. In fact, in this case I think it is numerical error when computing the complex permittivity that is causing a small negative part to appear. So I added a warning to the log when a gain term larger than machine precision is detected. — Reply to this email directly, view it on GitHub <#1974 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/A3KLECP5HP7DG2CJD2TGU3TZYV4D5AVCNFSM6AAAAABOWZZSOWVHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZDGNZZGU3TGMJSGU> . You are receiving this because you were mentioned.Message ID: ***@***.***>

[dmarek-flex]

I'm gonna merge this unless you have any final comments @tylerflex

[tylerflex]

go for it!