What's the advantage of building an active filter having more than one stage vs only a single stage?
If you use multi-stage, would you not introduce pole pairs with higher Q factor, and thus producing overshoot and oscillation in your time response, plus you also have to buy more OpAmps, resistors and such?


(two 100V/V Sallen and Key low pass filters)

 

hi cd,
As its a Homework question, please post what you think the advantages are for either type.
We can then advise you.
E

 

What's the advantage of building an active filter having more than one stage vs only a single stage?
If you use multi-stage, would you not introduce pole pairs with higher Q factor, and thus producing overshoot and oscillation in your time response, plus you also have to buy more OpAmps, resistors and such?


View attachment 164603 View attachment 164604 (two 100V/V Sallen and Key low pass filters)

Hello,

I can offer a hint here.
Using a frequency sweep, look at the responses of both ways of doing it, single stage vs double stage, and see if you can spot an advantage.
There are also disadvantages that you may or may not be interested in.

 

If you use multi-stage, would you not introduce pole pairs with higher Q factor, and thus producing overshoot and oscillation in your time response...

Not necessarily. This application note gives a good introduction to active filter design, both single-stage and multi-stage.

...plus you also have to buy more OpAmps, resistors and such?

Yes, that's a trade-off the designer must always consider.

 

If you use multi-stage, would you not introduce pole pairs with higher Q factor, and thus producing overshoot and oscillation in your time response

The Q factor (and thus overshoot and possible oscillation) are determined by the selected values of all the RC components.
More stages can be designed with the same Q factor but it will have a faster rolloff outside the passband.

 

If you try to become familiar with filter SPECIFICATIONS you will notice that there is a passband and a stopband and a transition region between both. Perhaps you can imagine that the width of this transition region (which, of course, depends on the attenuation/damping requirements of the stop band) plays a certain role. And in this context, the complexity (the number of filter elements, active and passive) is important.

 

hi cd,
This is what LTSpice shows for the AC analysis and a 500Hz sqr wave input.
E

 

This is what LTSpice shows for the AC analysis and a 500Hz sqr wave input.

You gains seem off, as I would expect the same low-frequency gain of 40dB for both filters (1 + 1meg/10k).
That's likely because the op amp is not biased fully into its active region.
Try it with a +50mV DC bias value on V2.
(See sim below):

 

hi Carl,
Its just a direct copy of the TS's posted circuit, in order to show him the AC response plots.
I agree it requires some improvement for an actual circuit.
Eric

 

hi Carl,
Its just a direct copy of the TS's posted circuit, in order to show him the AC response plots.
I agree it requires some improvement for an actual circuit.

Yes, but those are not the correct plots for a properly biased op amp.
They need a dual supply or some DC bias on the input source so that they are properly biased in their active region.
When biased at 0V on the input, even rail-rail op amps may not sim properly for AC signals.
Try a +50mV bias on the input and you will see the difference.

 

Hello,

That might indicate an input offset problem. It's very very hard to link two DC coupled circuits together without careful attention to the DC offset when the gains are high. For one stage and a gain of 100, a small offset like 0.001v becomes 0.1v on the output, and then the next stage wants to amplify that by 100 also so that brings the output up to +10v with no input signal. That means either we need extremely low input offset op amps or we need to do some offset adjustment.
Now if the two stages where AC coupled, that would be reduced and might actually be ok with an output offset of only around 0.1 to 0.5v.

 

What's the advantage of building an active filter having more than one stage vs only a
single stage? If you use multi-stage, would you not introduce pole pairs with higher Q
factor, and thus producing overshoot and oscillation in your time response, plus you also
have to buy more OpAmps, resistors and such?

To a point more stages can yield sharper/narrower band BPFs or NOTCH or Band reject, or faster rolloff LPFs or HPFs.
Analog solutions limited by component drift, tolerance. Also certain architectures of digital filters (FIR) are absolutely stable,
unlike some OpAmp/Gain stage filters.

The point about component count, an alternative is to use digital filter, you can achieve higher order filters
in reality vs analog, can easily be made more accurate because they are dependent on a clock which if
chosen correctly will achieve high order filters.

Modern processors have all the necessary HW for filtering, A/D, D/A, DSP engine, DMA. All onchip.

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




But overall you still have to do a end to end noise/tolerance analysis to see whats best for you.


Regards, Dana.[/QUOTE]

 

@confuseddesigner: Perhaps it is helpful to make some general statements about filter techniqies.
For design engineers, it is a rather challenging task to define a certain filter circuit for a specific application.
It is, therefore, not possible to answer your question without additional information about your application (specification, filtering requiremets).
You must be aware that - for lowpass realisations - you have the choice between
(a) Several lowpass approximations (Bessel, Butterworth, Chebyshev,...) corresponding to the required filter order (which is directly related to the hardware complexity);
(b) Cascade structure (series connection of 2nd-order stages) or direct realisations (active realization of passive RLC ladder structures using active L or FDNR techniques);
(c) Various circuit topologies for cascade realisations (Sallen-Key, multi-feedback, integrator stages, ...)
(d) Various strategies for dimensioning of the filter (parts selection).

Example to the last point (dimensioning):
It is common practice to use for Sallen-Key 2nd-order stages three alternatives :
(1) Equal components (two equal R and two equal C) with a positive fixed gain < 3 (depending on the pole Q)
(2) Fixed gain of "2" with two equal gain determining feedback resistors and two equal capacitors;
(3) Fixed unity gain (opamp as a buffer).

Please note that in your design the opamps are wired for a postive gain of "101", which is rather "uncommon" for a series connection of Sallen-Key stages because - as already mentioned by MrAl - the unwanted DC offset voltage would be remarkably amplified. This will be not a problem for a design with gain values as low as "1" or "2". More than that, the Q-values of Sallen-Key stages are very sensitive to gain tolerances (tolerances of the gain determining feedback resistors). This is the most important aspect for selecting design alternatives with gain values "1" or "2".

 

What's the advantage of building an active filter having more than one stage vs only a single stage?
If you use multi-stage, would you not introduce pole pairs with higher Q factor, and thus producing overshoot and oscillation in your time response, plus you also have to buy more OpAmps, resistors and such?


View attachment 164603 View attachment 164604 (two 100V/V Sallen and Key low pass filters)

Hello again,

If we look at the ratio of the gain of the peak to the gain at w=0 in the frequency domain then we see for two stages we get the square of the ratio that we get for one stage. That of course means that the 'overshoot' increases for two stages.
However, if the gain is adjusted differently we have all real poles so there is no overshoot no matter how many stages we add.
It looks like the value range for complex poles is, adjusting R4 alone and using the other first op amp stage values, approximately 900k to 1250k. This suggests the other values might be adjusted instead so that we can get less DC gain and still get complex poles.

 

If you use multi-stage, would you not introduce pole pairs with higher Q factor, and thus producing overshoot and oscillation in your time response,
View attachment 164603 View attachment 164604 (two 100V/V Sallen and Key low pass filters)

The properties of a filter in the frequency domain as well the time domain (peaking, overshoot,...) is independent on the selected and realized design. Hence, it plays no role if you select a cascade structure or any other topology. This choice depends on other requirements (amount of parts, accuracy, sensitivity to tolerances, frequency range,...)

Summary: To make it clear - sometimes resp. very often you have no other choice than to use 2 or 3 stages (of 2nd order) in cascade (provided you have decided to use the cascade and not the "direct" approach). But this depends solely on the required filter order!!

 

The properties of a filter in the frequency domain as well the time domain (peaking, overshoot,...) is independent on the selected and realized design. Hence, it plays no role if you select a cascade structure or any other topology. This choice depends on other requirements (amount of parts, accuracy, sensitivity to tolerances, frequency range,...)

Hi,

Maybe you could elaborate a little there.

When we connect two of the same in tandem, the response is squared, and thus the ratio of peak to some other place will increase.
For example, with the given values the gain could be 100 at w=0 and 130 at the peak, and so the ratio is 1.3. Squaring this, we get 1.69, so the ratio went from 1.3 to about 1.7 which means more of a peak relative to the other values. If we added a third stage, this ratio would go up to about 2.2, which is nearly twice what the single stage offers.
In general, a function that is squared has steeper slopes than one which is not.

 

Well - I must admit that in my contribution I have suppressed the following (in italics) :
The properties of a filter in the frequency domain as well the time domain (peaking, overshoot,...) for a given specification is independent on the selected and realized design

Of course, I have assumed that somebody who is designing a filter will follow a certain specification (filter requirements).
And in this case, all realization alternatives will have (under ideal conditions) the same function and the same properties.

Of course, it is obvious that a single stage will have other properties than a a cascade of two such stages. But that is not the way we are designing filters, I think.

 

Well - I must admit that in my contribution I have suppressed the following (in italics) :
The properties of a filter in the frequency domain as well the time domain (peaking, overshoot,...) for a given specification is independent on the selected and realized design

Of course, I have assumed that somebody who is designing a filter will follow a certain specification (filter requirements).
And in this case, all realization alternatives will have (under ideal conditions) the same function and the same properties.

Of course, it is obvious that a single stage will have other properties than a a cascade of two such stages. But that is not the way we are designing filters, I think.

Hello again,

Thanks for talking about this more.

Well, if we want an analog design and we want something like a sharp 4th order response, what else could we do?
I would think we would need two op amp stages, but you can certainly suggest otherwise if you would like too and im sure that would be interesting.

 

Well, if we want an analog design and we want something like a sharp 4th order response, what else could we do?
I would think we would need two op amp stages

Certainly the usual way to do a 4-pole active filter with op amps is to use two 2-pole filters in series.

 

Well, if we want an analog design and we want something like a sharp 4th order response, what else could we do?
I would think we would need two op amp stages, but you can certainly suggest otherwise if you would like too and im sure that would be interesting.

Hello again,
because you have indicated your interest: As outlined in my answer #13, we always have two different alternatives for a 4th-order lowpass (example):
(a) cascade design (based on a series of 2nd.order stages)
(b) "direct" design (mostly based on active realisation of passive RLC topologies).

Al always in our live, the two alternatives have some advantages and some disadvantages.
* Alternative (a) is easy to calculate and to fine tune - that means: Easy to design. But it is rather sensitive to parts tolerances
* Alternative (b) has opposite properties (involved calculations; better: Tabulated normalized parts values) but it is much less sensible to parts tolerances.

Examples for (b): FDNR- technique, Leapfrog topology, FLF (Follow-the Leader-Feedback), PRB (Primary-Resonator-Block).

Comment 1: These methods (in particular: Leapfrog) are the starting point for inegrated Switched-Capacitor (SC) realizations.

Comment 2: The filter design programm from NuhertzTechnologies (www.filter-solutions.com) allows - amomg many other design alternatives - also a design based on "leaprog"-structures.