[SOLVED] Butterworth transfer function transformation to tow-thomas-gm c

Butterworth transfer function transformation to tow thomas gm c

I have derived a 6th order butterworth bandpass filter transfer function with center frequency of 2 MHz and two cutoff frequencies at 1MHz and 3MHz, I am now trying to implement the filter using tow thomas gm c architecture, my transfer function and the bode plot of this transfer function are attached, I am trying to implement the attached tow thomas circuit also using Vo2 and Vi2 as they are the ones that could correspond to my transfer function with 3 cascaded stages to reach the 6th order, however I do not know what I am doing wrong in the transformation, I am comparing each 2nd order bracket from my transfer function and part of the gain multiplied by s with their corresponding coefficients in the tow thomas transfer function -assuming equal 500f capacitance- I get values for the transconductance which are incredibly wrong and their simulation are terribly wrong. I cannot understand what I did wrong in the transformation but it seems there is something I am missing.

Re: Butterworth transfer function transformation to tow thomas gm c

The transfer function doesn't show a Butterworth bandpass, instead it's the combination of 3rd order low- and high passes. It will be preferably implemented by separate circuits.

Re: Butterworth transfer function transformation to tow thomas gm c

sherif96 said:

I have derived a 6th order butterworth bandpass filter transfer function with center frequency of 2 MHz and two cutoff frequencies at 1MHz and 3MHz, I am now trying to implement the filter using tow thomas gm c architecture, my transfer function and the bode plot of this transfer function are attached, I am trying to implement the attached tow thomas circuit also using Vo2 and Vi2 as they are the ones that could correspond to my transfer function with 3 cascaded stages to reach the 6th order, however I do not know what I am doing wrong in the transformation, I am comparing each 2nd order bracket from my transfer function and part of the gain multiplied by s with their corresponding coefficients in the tow thomas transfer function -assuming equal 500f capacitance- I get values for the transconductance which are incredibly wrong and their simulation are terribly wrong. I cannot understand what I did wrong in the transformation but it seems there is something I am missing.

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only 500f? rather small for a gm-C integrator @ 2MHz

Re: Butterworth transfer function transformation to tow thomas gm c

The circuit will be implemented as three separate stages cascaded together, each stage consists of one of the quadratic brackets from the transfer function and a zero with some gain, wouldn't that help?

Re: Butterworth transfer function transformation to tow thomas gm c

FvM said:

The transfer function doesn't show a Butterworth bandpass, instead it's the combination of 3rd order low- and high passes. It will be preferably implemented by separate circuits.

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well let's say I am going to implement them by separate circuits, the problem is I still do not get how to get the parameters in the tow thomas transfer function from my derived transfer function, If i take for example the first 2nd order bracket from the transfer function with a zero and some gain, how can I compare this with the tow thomas transfer function in my hand written paper ?

Re: Butterworth transfer function transformation to tow thomas gm c

SunnySkyguy said:

only 500f? rather small for a gm-C integrator @ 2MHz

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500f is only my starting point, I was waiting to manage to design the tow thomas circuit then I would change the capacitance value based on the response I need but I could not reach that point yet, do you have any idea about how I could compare any of the 2nd order brackets in my transfer with the 2nd order tow thomas transfer function to derive gm values ?

Re: Butterworth transfer function transformation to tow thomas gm c

The transfer function can be implemented by cascading three band pass bi-quads. But the OTA-C circuit has the wrong topology, to create a band pass, the output has to be connected after one integrator.

Re: Butterworth transfer function transformation to tow thomas gm c

FvM said:

The transfer function can be implemented by cascading three band pass bi-quads. But the OTA-C circuit has the wrong topology, to create a band pass, the output has to be connected after one integrator.

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Well this was the only one i could find in a reference " continuous time active filters ". Would you mind attaching an image for the OTA-C with the correct topology and its transfer function?

Re: Butterworth transfer function transformation to tow thomas gm c

You find respective topologies searching for "gm-c bandpass" or "ota-c bandpass". Like this one

Re: Butterworth transfer function transformation to tow thomas gm c

FvM said:

You find respective topologies searching for "gm-c bandpass" or "ota-c bandpass". Like this one

View attachment 145232

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you've been of great help but one last question,where can I find transfer function of such circuit to check the transfer function I derived is there a reference including the transfer function of this exact circuit ?

Re: Butterworth transfer function transformation to tow thomas gm c

FvM said:

I reviewed the first post again and realized that are using the circuit in the vo2/vi2 band pass configuration. The transfer function is similar to the circuit suggested in post #9, all possible C and gm scaling problems apply similarly as well.

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Well if you don't mind this gets us to my first question, How can I compare my transfer function with the general ota-c transfer function, if you may follow my steps and tell me what I am doing wrong.
1- I separate one of the 2nd order brackets from the denominator along with an S from the numerator and some gain forming a resultant partial 2nd order transfer function.
2- I compare this partial transfer function with the general ota-c transfer function as follows in post 1 in my handwritten paper, so for example 1/tao2=1.32987*10^21, similarly for k22 and tao1,
however these steps result in values for the transconductance that are absolutely wrong some numbers reach to *10^-21 and some are in the MEGA range which are obviously wrong, so what am I doing wrong in the previous steps ? Thank you very much for your help you've been great.

Re: Butterworth transfer function transformation to tow thomas gm c

I reviewed the first post again and realized that you are using the circuit in the vo2/vi2 band pass configuration. The transfer function is similar to the circuit suggested in post #9, all possible C and gm scaling problems apply similarly as well.

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I can't reproduce the strange numbers. The pole time constants are in 50 to 150 ns range, Q of the complex pole pair is 1. Here are the separated high pass and low pass transfer function, the two real first order poles can be combined into one biquad, or better implemented as separate first order filters.

Re: Butterworth transfer function transformation to tow thomas gm c

FvM said:

I reviewed the first post again and realized that you are using the circuit in the vo2/vi2 band pass configuration. The transfer function is similar to the circuit suggested in post #9, all possible C and gm scaling problems apply similarly as well.

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I can't reproduce the strange numbers. The pole time constants are in 50 to 150 ns range, Q of the complex pole pair is 1. Here are the separated high pass and low pass transfer function, the two real first order poles can be combined into one biquad, or better implemented as separate first order filters.

View attachment 145239 View attachment 145238

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Just to make sure I understand correctly, the second order highpass bracket can't be implemented with the tow thomas circuit I attached in post 1, I need to search for a highpass gm c circuit to implement, the lowpass second order can be implemented using Vi1 and Vo2, the two real poles from the high pass and low pass can be implemented with the tow thomas circuit using Vi2 and Vo2, or as two separate first order filters, am I correct ?

Re: Butterworth transfer function transformation to tow thomas gm c

FvM said:

You can implement the poles and zeros as you like. If using three second order band pass blocks (one option), you'll surely split the large numerator coefficient equally among the three blocks. If you make two second order band passes and first order high and low passes, a part is assigned to the high pass. At the end you get regular second and optionally first order filter blocks with characteristic frequencies in the 1 to 3 MHz range.

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a quick question, if I want to implement 3 second order bandpass filter block, what would be their transfer function, I understood how to design the filter using highpass and lowpass blocks, however if I want to implement the filter using the bandpass filter blocks,what would be their transfer function as when I tried to derive the circuit and simulate I got a very weird response. Once again I would like to thank you for all the effort you've been putting to help me, thank you wont 'be enough for what you've done for me.

Re: Butterworth transfer function transformation to tow thomas gm c

You can implement the poles and zeros as you like. If using three second order band pass blocks (one option), you'll surely split the large numerator coefficient equally among the three blocks. If you make two second order band passes and first order high and low passes, a part is assigned to the high pass. At the end you get regular second and optionally first order filter blocks with characteristic frequencies in the 1 to 3 MHz range.

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I see that implementing a Butterworth band pass with 1 and 3 MHz corner frequencies gives a slightly different transfer function with three complex poles. It would be implemented with three second order blocks. Characteristic frequencies and Qs aren't much different from the separate LP/HP implementation, however.

Re: Butterworth transfer function transformation to tow thomas gm c

FvM said:

You can implement the poles and zeros as you like. If using three second order band pass blocks (one option), you'll surely split the large numerator coefficient equally among the three blocks. If you make two second order band passes and first order high and low passes, a part is assigned to the high pass. At the end you get regular second and optionally first order filter blocks with characteristic frequencies in the 1 to 3 MHz range.

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I see that implementing a Butterworth band pass with 1 and 3 MHz corner frequencies gives a slightly different transfer function with three complex poles. It would be implemented with three second order blocks. Characteristic frequencies and Qs aren't much different from the separate LP/HP implementation, however.

View attachment 145257View attachment 145257

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but shouldn't the gain in the numerator be equal to the multiplication of the three coefficients of s in the denominator in the three brackets ?

Re: Butterworth transfer function transformation to tow thomas gm c

but shouldn't the gain in the numerator be equal to the multiplication of the three coefficients of s in the denominator in the three brackets ?

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For a second order band-pass with unity gain, yes. But not generally. I didn't check, but I have no doubts that the tool calculated the transfer function correctly.

Re: Butterworth transfer function transformation to tow thomas gm c

FvM said:

For a second order band-pass with unity gain, yes. But not generally. I didn't check, but I have no doubts that the tool calculated the transfer function correctly.

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one last question I hope, I simulated the bandpass transfer function in post 15, the bode plot in matlab is absolutely fine as attached, however when simulating the tow thomas circuit on cadence the response is not what we hoped for, I have no idea why, attached is a screenshot of the response over cadence on the right the magnitude in dBs on the left the magnitude as absolute, the values of gm was derived as shown in the handwritten paper I have no idea what went wrong, the gain in the numerator was divided equally among the 3 stages then I compared each stage with the tow thomas transfer function with Vi2 and Vo2 and obtained my gm values. on cadence for an ideal circuit I replaced each OTA with a voltage controlled current source for now just to simulate the response and see how will it work out before implementing the non ideal circuit, however the ideal circuit is not yet giving the intended response, so what am I doing wrong if you do not mind checking for the last time?

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The matlab simulation is not so fine after all, the response reaches -3dB at 1 MHz which is fine, however it reaches -3 dB on the other side at around 4MHz not 3 whch is weird

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you can ignore the matlab simulation issue I fixed it already, however cadence simulation is what I do not understand why is the same transfer function after implementation with the tow thomas circuit I get a response which is much different than the one designed as shown in post 18, if you do not mind following the steps I did to obtain my gm values assuming all capacitances equal to 500f F

Re: Butterworth transfer function transformation to tow thomas gm c

There's a typo in one parameter(second G1), an ideal gm-c implementation gives the expected magnitude function however.



I append the Ltspice file for reference.

The output magnitude after the second block shows an uncomfortable gain enhancement relative to input which can result in overload of the second stage. The second stage gain should be probably reduced and the third stage gain respectively increased.

Re: Butterworth transfer function transformation to tow thomas gm c

I was aware that the post #15 transfer function has an incorrect upper cut-off frequency of 4 MHz. Corrected below.



The appended Excel spreadsheet is calculating the gm-c parameters, optionally with gain correction for the 2nd and 3rd stage.