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Design >> Analog Design >> Amplifier bandwidth for continuous-time filters
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Message started by Berti on Apr 30^th, 2008, 9:30am

Title: Amplifier bandwidth for continuous-time filters
Post by Berti on Apr 30^th, 2008, 9:30am

Hi all,

For a switched-cap. filter the required bandwidth can be calculated from the settling constraints.
It is straight-forward and often appears in literature.
But how is the amplifier bandwidth determined in a continuous-time filter? I have never seen
people doing calculations, but only simulation. How do I calculated the required amplifier bandwidth
for a continuous-time filter (from a set of specifications)?

Regards

Title: Re: Amplifier bandwidth for continuous-time filter
Post by ACWWong on Apr 30^th, 2008, 4:26pm

It depends on the continuous-time filter structure, and most importantly on Q and pole frequency. Analytically I think it you'll find examples if you search for filter sensitivity analyses.
An example is a Tow-Thomas biquad structure, which requires the amplifiers to have UGBW of polefreq*Q^2 (discussed in Laker & Sansen "Design of Analog Integrated Circuits and Systems") for a certain error. An interesting biquad structure which relaxes the this UGBW requirement is Akerbert&Mossberg biquad:
http://ieeexplore.ieee.org/iel5/31/23431/01083785.pdf?arnumber=1083785&htry=4
This paper references some good prior art and does some analyses to show the impact of finite bandwidth on filter Q.

An example of finite OTA bandwidth impact on OTA biquad structures is given at:
http://users.ece.gatech.edu/~phasler/Courses/ECE6414/Unit3/CT_ICfilters.pdf

hope this helps, cheers

aw

Title: Re: Amplifier bandwidth for continuous-time filter
Post by buddypoor on May 1^st, 2008, 3:18am

Hi Berti,

I agree with HCWWong, that the required BW depends on the structuire chosen as well as on the pole parameters (Q resp. pole frequency). However, of course it depends also on the allowed deviation from the nominal response.
More than that, you should not forget the influence of the slew rate, because it makes no sense to look only at the small signal BW.
In addition. the topologies as mentioned by HCWWong (Tow-Thomas resp. Akerberg-Mossberg) are multiple opamp-structures and, therefore, are economical only if their specific advantages (lowpass, highpass and bandpass simultaneously, all of second order) are to be used.
According to my experience and in agreement with nearly all recommendations in filter books, the "best" second order structure from the viewpoint of economical use of the small signal BW is the GIC-based topology. The influence of the limited opamp BW on the filter response will be reduced if both opamps are matched (dual amp chip). The situation is even more complicated since there are other active candidates than opamps, each with their own structure (OTAs, CFAs and Current Conveyors).
In general, your question touches a very important and - at the same time - a very difficult question:
What is the "best" filter topology for a certain application ?
I am involved in filter design since 25 years (and I have written two books in german on filters), but up to now I did not come to a final conclusion. The answer resp. the choice of a specific topology depends on several items (accuracy requirements, filter characteristics, filter order, frequency range, pole data, application conditions and economical as well as operational aspects like power requirements).
But, in any case, I am sure that circuit simulation gives much more information than a simple formula because it can take into account more important opamp nonidealities like input/output impedance, slew rate, load influence, parasitic signal pathes (i. e. through the passive network directly to the opamp output; very important for Sallen-Key-structures).
Sorry, too much came into my mind.......
Regards
Lutz vW

Title: Re: Amplifier bandwidth for continuous-time filter
Post by vivkr on May 14^th, 2008, 5:00am

Hi Lutz,

I would just like to add 1 more point when thinking of filter structures: reliability, and fault-tolerance.
In my opinion, cascaded structures are better at this even though they have poor sensitivity than coupled
structures such as the ladder filter.

If you are making a filter the first time in your life, then it is better to use a cascade of biquads. If something
goes wrong, you can atleast debug it better. A small but nonetheless important point I believe. Also improves
testability in general.

Regards
Vivek

Title: Re: Amplifier bandwidth for continuous-time filter
Post by buddypoor on May 14^th, 2008, 6:35am

Hi Vivek,
I agree, of course, with you that cascade design is more sensitive to tolerances, but easier to design.
(This is another proof of my opinion, that everything in analog design is always a compromize between several conflicting requirements resp. conditions)
More than that, also dynamic range considerations are more critical in cascaded structures.

But let me come back to the original question of this topic:
There is a textbook from M. Herpy and J.C. Berka (in german: Aktive RC-Filter) including a table with formulas for opamp BW recommendations to be applied for various filter topologies of second order*. I am not sure if it is available in english.

* Sallen-Key, Deliyannis, Scultety, Boctor, elliptical, double-T, Fliege (GIC), KHN, Fleischer-Tow, Berka-Herpy.

Regards
L.

Title: Re: Amplifier bandwidth for continuous-time filter
Post by loose-electron on May 14^th, 2008, 3:41pm

As a general rule, the dominant poles of the filter structure needs to be well below the dominant poles of the amplifiers themselves. Any filter will have residual poles in the system which (in the LPF case) are well out of the passband of interest. In a simulation environment you can selctively reduce the BW of the residual poles and see how they affect the gain/phase of the primary poles and their filtering function.

Title: Re: Amplifier bandwidth for continuous-time filter
Post by Berti on May 15^th, 2008, 12:05am

Thanks everybody for the explanations.

In general I was more interested in continuous time loop filters for ΔΣ modulators.
I found that the transfer function is not so important but that distortion and phase-shift
dictate the required amplifier bandwidth.

But anyway, the discussion above was interesting.

Regards

Title: Re: Amplifier bandwidth for continuous-time filter
Post by buddypoor on May 17^th, 2008, 7:59am

Hi Berti,

perhaps you are satisfied already. However - also for somebody else, who is interested in the topic - here are to equations which can be used to select an appropriate opamp:

There are two equations to be considered/evaluated:

1) ft > 20*Qp*fp (opamp transit frequency ft, Qp=pole quality, fp=pole frequency)

2) |Ao(fp)|> 100*Amax (Ao(fp)=opamp open loop gain at fp, Amax=max. filter gain)

Both equations should be fulfilled, that means the most critical of both equations governs and determines the opamp characteristics to be required.

Regards
Buddypoor

Title: Re: Amplifier bandwidth for continuous-time filter
Post by vivkr on May 18^th, 2008, 10:58pm

buddypoor wrote on May 17^th, 2008, 7:59am:

Hi Berti,

There are two equations to be considered/evaluated:

1) ft > 20*Qp*fp (opamp transit frequency ft, Qp=pole quality, fp=pole frequency)

2) |Ao(fp)|> 100*Amax (Ao(fp)=opamp open loop gain at fp, Amax=max. filter gain)

Hi,

Looks like you are assuming some value for some performance parameter implicitly, because of these numbers 20 and 100 in your equations.
Perhaps you should add a note telling what goes into making up those 2 numbers.

Regards
Vivek

Title: Re: Amplifier bandwidth for continuous-time filter
Post by buddypoor on May 18^th, 2008, 11:33pm

Looks like you are assuming some value for some performance parameter implicitly, because of these numbers 20 and 100 in your equations.
Perhaps you should add a note telling what goes into making up those 2 numbers.

Of course, you are completely right. These "magic" numbers require some explanation.
However, both numbers have to be considered as something like a "rule of thumb" . Both numbers have been derived not by exact calculation but by empiric considerations and by measurements (student projects within the last years).

One can expext that the pole frequency as well as the pole-Q error will (most probably) not exceed 5% if these numbers are used as factors in the given equations.
Regards