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Transfer function request

Is there a filter topology to synthesize the following transfer function (with omega normalized to 1):

s*(s^2 + 2*zeta*s + 1)

A zero at the origin, and a complex conjugate pair of zeros.

Bode plot here for a Q of 50:

Or conversely the "inverse", complex conjugate zeroes, pole at the origin so the response starts out like an integrator, and another pole at some point to cancel out the response rising again after the notch at resonance. don't know how to express that mathematically offhand, though.

Reply to
bitrex
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Obviously for this one the gain can't keep rising forever but that's OK.

Reply to
bitrex

Ahh not sure, but isn't there something about having the poles on the left side of the plane... or it's an oscillator.

George H.

Reply to
George Herold

Actually I bungled my requirements anyway, what I need is the input impedance function to the network to look like that, not the transfer function.

In that case with passive components at least I think what I'm looking for is just a RCLC circuit in both cases, RC high-pass or low pass and then a series LC band-reject in parallel with the R or C, depending from the output to ground

Reply to
bitrex

this elegant passive structure should work for the second type I mentioned. at low frequency the input impedance is just R1 + R2. then it dips due to the complex-conjugate zeroes. Then rises again due to CC poles. Then dips out to high frequency due to the single pole.

Thanx to Wolfram Alpha/Mathematica for help with the analysis

Reply to
bitrex

You mentioned you meant this as a "driving point immittance." What you gave is not a rational function: s*(s^2 + 2*zeta*s + 1)

So, automatically, the answer is "no, it can't be synthesized (realized)."

For a driving point immittance to be realizable requires

  1. F(s) is real if s is real.
  2. Real(F(s)) >= 0 if Real(s) >= 0

These are the Brune conditions. (Otto Brune, 1931)

F(s) is the immittance function in s, whether impedance or admittance.

Some other interesting rules are derived if it is something like a lossless one-port or a doubly terminated ladder.

Reply to
Simon S Aysdie

I should have said that it has to be a (positive real) rational function. 1 & 2 ensure the rational function is "positive real."

Reply to
Simon S Aysdie

Yep. Infinite gain at infinite frequency is indeed a tall order

Reply to
bitrex

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