Transfer function request

[bitrex]

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:

<https://www.wolframalpha.com/input/?i=transfer+function+s*(s%5E2+%2B+2*0.01*s+%2B+1)>

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.

 

[bitrex]

On 8/16/19 2:31 AM, bitrex wrote:

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:

https://www.wolframalpha.com/input/?i=transfer+function+s*(s%5E2+%2B+2*0.01*s+%2B+1)

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

 

[George Herold]

On Friday, August 16, 2019 at 2:36:23 AM UTC-4, bitrex wrote:

On 8/16/19 2:31 AM, bitrex wrote:
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:

https://www.wolframalpha.com/input/?i=transfer+function+s*(s%5E2+%2B+2*0.01*s+%2B+1)

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

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.

 

[bitrex]

On 8/16/19 8:58 AM, George Herold wrote:

On Friday, August 16, 2019 at 2:36:23 AM UTC-4, bitrex wrote:
On 8/16/19 2:31 AM, bitrex wrote:
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:

https://www.wolframalpha.com/input/?i=transfer+function+s*(s%5E2+%2B+2*0.01*s+%2B+1)

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

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.

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

 

[bitrex]

On 8/16/19 8:58 AM, George Herold wrote:

On Friday, August 16, 2019 at 2:36:23 AM UTC-4, bitrex wrote:
On 8/16/19 2:31 AM, bitrex wrote:
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:

https://www.wolframalpha.com/input/?i=transfer+function+s*(s%5E2+%2B+2*0.01*s+%2B+1)

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

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.

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

<https://imgur.com/a/ndk8Qhy>

 

[Simon S Aysdie]

On Thursday, August 15, 2019 at 11:31:56 PM UTC-7, bitrex wrote:

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:

https://www.wolframalpha.com/input/?i=transfer+function+s*(s%5E2+%2B+2*0.01*s+%2B+1)

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.

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.

 

[Simon S Aysdie]

On Tuesday, August 20, 2019 at 9:46:53 AM UTC-7, Simon S Aysdie wrote:

On Thursday, August 15, 2019 at 11:31:56 PM UTC-7, bitrex wrote:
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:

https://www.wolframalpha.com/input/?i=transfer+function+s*(s%5E2+%2B+2*0.01*s+%2B+1)

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.

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.

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

 

[bitrex]

On 8/20/19 12:46 PM, Simon S Aysdie wrote:

On Thursday, August 15, 2019 at 11:31:56 PM UTC-7, bitrex wrote:
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:

https://www.wolframalpha.com/input/?i=transfer+function+s*(s%5E2+%2B+2*0.01*s+%2B+1)

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.

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)."

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

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.