# Why we use a 2nd order filter to build a higher order filter?

Why would we use 2nd order filters to build a higher order filter, like a Sallen & Key and others?

Is the oscillation problem in higher order filters?

• Please capitalise your title and post properly for legibility and credibility. – Transistor Feb 6 '18 at 21:06
• Because the mathematics gets pretty gnarly by the time you get to 3rd order. If you are a mathematics whiz and like to explore these things, then you can build a higher order filter directly. It's just that you introduce more concepts into the process. With 2nd order, you have a very nice and simple idea of "damping" or Q. With higher order, that gets muddied into more complex behaviors that are harder for most normal humans. Also, there are lots and lots of tables for chaining 2nd order stuff to make any kind of resulting filters. All the hard work is taken care of for you; neatly factored. – jonk Feb 6 '18 at 21:06
• One of the better texts, because it is targeted at explaining some of the reasons behind the Sallen & Key approach, is Technical Report 50, "A Practical Method of Designing RC Active Filters" by the infamous R P Sallen and E L Key, dated 6 May 1954. This is a MUCH better resource than their 1955 publication. TR-50 was held secret for a while, but has been "unclassified." However, it is not published. So you have to write MIT directly in order to obtain a copy of it. They hold the ONLY edition of it and I think the military here has to approve each request. (But I got one. So anyone may.) – jonk Feb 6 '18 at 21:17
• I had to edit. It was too painful to look at. VTC because this is mostly a research problem. – Sparky256 Feb 6 '18 at 22:33
• @Sparky256 Thanks for the edit! I almost feel like writing an answer, now. :) – jonk Feb 6 '18 at 23:11

(1) Up to now - all responses were related to a filter design strategy which is called "Cascade design". This method uses filter design design tables to realize higher order filters by cascading 1st and 2nd order active stages.

(2) However, there is another strategy called "direct synthesis". Using this approach you can transfer a passive RLC reference ladder network of any order into an active circuit (active L-simulation, FDNR-technique). For the passive reference structure, tabulated parts values are available.

(3) Another method of direct filter synthesis is based on multi-feedback-topologies for basic active stages like integators (Leapfrog structure, Follow-the-Leader FLF, Primary Resonator Block PRB).

Comment 1: It was shown that all realizations based on those "direct methods" have much better passive sensitivity figures than the "cascade filters" (passive sensitivity is the sensitivity of the whole filter circuit upon tolerances of the passive parts R and C)

Comment 2: Regarding oscillation: No, there is not a specific problem for higher order filters. This is true for the cascade approach as well as the direct filter synthesis.

The tendency to oscillation does not depend on the order of the filter but to the pole location only. For high-Q filters the poles must be placed rather close to the Im-axis of the s-plane. In this case, there could be a problem if - due to unwanted influences (parts tolerances) - one of the pole pairs is shifted towars the Im-axis.

However, it is to be noted that the pole location of the various 2nd-order blocks does depend also on the filter order. For example, for a 10th-order Chebyshev lowpass (ripple 1 dB) the maximum pole-Q is Q=22.3. This is equivalent to an angle alpha=88.7 deg (the imag. axis is at 90 deg). The angle alpha is between the real axis and the vector to the pole location in the left part of the s-plane.

You have a higher rolloff with a sallen key than with a normal 1-pole filter. Sometimes the signal in the bandpass and the noise are too close toghether and you need a high roll off rate to isolate them. (like if you had a 100Hz signal and wanted to bury 60Hz out of your data, since they are spaced close toghether if you got a 40dB 60Hz noise, you'd need a really high filter to isolate it from your 100Hz signal). Because there are two poles you can also build bandpass filters and notch filters.

Yes there is a problem with oscillation, especially when filter poles are spaced close together. In that case it will be better to either space the poles slightly apart and use component values that will give a better filter response.

• This answer does not address the OPs question about 'oscillation'. – Sparky256 Feb 6 '18 at 23:21

The answer is that we aren't building the filters from second order circuits. We are building them using active circuits that simulate inductors. It just so happens that in all these circuits we can easily incorporate a capacitive element as well, so they can be turned into second order blocks which generally have superior filtering characteristics.

In the real world, inductors are annoying to use. They are large, prone to picking up interference, mechanically delicate, difficult to make accurate, and relatively expensive. In comparison, capacitors are cheap, stable and robust. So early on in the history of analogue electronics, designers created circuits that would simulate an inductive impedance using only capacitors. All the various active filter topologies are fundamentally just negative impedance converters. You can even adapt them to turn an inductor into a capacitor if you wanted.

All LTI filter systems are formed from R, L and C impedances. All we are doing with active filters is substituting the Ls for actively converted Cs, and practical considerations make it easy to do this with a C impedance attached, so we do.

• I do not think that is correct to say that in active filters we "would simulate an inductive impedance using only capacitors." Active simulation of inductors is ONE out of several other methods to design 2nd order filter blocks. For example, in S+K stages (under discussion) we create a conjugate-complex pole pair using active RC-feedback stages - but we do NOT replace any inductor in a passive RLC-structure.. As another example, think of the state-variable filters containing integrator stages (KHN, Fleischer-Tow, Tow-Thomas). – LvW Feb 7 '18 at 15:40
• I agree with the above, a gyrator is a circuit that seeks to mimic an inductor and a sallen key filter doesn't fall into that category. The whole premise of your question appears to built on this fallacy. – Andy aka Feb 7 '18 at 18:06

One pretty fundamental reason, not mentioned so far, is the underlying mathematics. You can decompose any polynomial into a product of quadratic equations (and a linear equation for odd-order polynomials).

I have only dim recollections of doing this by hand in high school mathematics, but remember it being tedious but tractable.

It follows from this that you can take any desired filter transfer function expressed as a polynomial, and translate it into a set of quadratic equations - if necessary, by hand.

These quadratic equations are, of course, second order filter sections on paper; translating them into circuitry is a relatively easy step.

So, historically, this has been the only way to deal with the mathematics of filter design. And electronic practice has developed around the underlying mathematics. Now we have numerical methods for synthesis, but standard building blocks like Sallen&Key sections dedicated to this older approach are usually just easier.

In terms of poles, a filter such as a 2nd order sallen key has either: -

1. Complex conjugate poles (producing a resonance in the bode plot)
2. A single real (non complex) pole ($\zeta$ = 1)
3. A pair of real poles ($\zeta$ > 1)

A more complex filter (higher order) will have multiple versions of the above. For instance a 6th order butterworth filter will have poles positioned like so: -

The red X's mark one conjugate pair; the green marks another conjugate pair etc.. For a butterworth filter all poles sit on a circle of magnitude $\omega_n$, the natural resonant frequency (in radians per second). For different types of even-order filters poles will still occur as conjugate pairs but won't necessarily be confined to the circle (synonmous with Butterworth filters).

Why would we use 2nd order filters to build a higher order filter

It makes sense to use a circuit that naturally generates a pair of conjugate poles and then "stack" them to produce multiple conjugate pairs to realize a higher order filter.

Is the oscillation problem in higher order filters?

As poles approach the vertical axis there are greater damped oscillations when the filter input is an impulse or step. This is what would be expected of course. Providing the poles are correctly positioned then sustained oscillation should not occur.

One thing I have found when designing high order filters is to have the lower Q conjugate pair circuit as the input and the higher Q conjugate pairs following that stage. The advantage this gives is that there isn't excessive ringing that might cause op-amp rail clipping because any transient edge has been sufficiently filtered by lower Q preceding stages.

• To explain the phrase "it makes sense": It is the easiest synthesis method because we can thus design and tune the variuos 2nd order stages separately. However, this is only one aspect - another important criterion is the sensitivity of the filter to parameter tolerances. And In this respect the cascade method is not optimum- – LvW Feb 7 '18 at 12:12

You don't necessarily have to limit an active filter to a 2nd order and cascade Op Amps, but it makes it easier to control source impedances or equations, and it does reduce phase margin. Here with a 6th order LPF, I got a Q of 100 in peaking but I could eliminate that with some compensation gain on the switch. < Simulation

Why would we use 2nd order filters to build a higher order filter, like a Sallen & Key and others?

Because you can. Also you can't make more than a 2nd order active RC filter.

Is the oscillation problem in higher order filters?

No.

Not unless you start incorrrectly applying negative feedback between many multiple stages.

The advantage of many 2nd order and a 1st order filters cascaded is that the breakpoint and Q of each stage can be staggered to fit a polynomial like Chebychev 0.5dB ripple passband with a brickwall bandstop.

Here a 5th order Chebychev filter can attenuate 62 dB @2 octaves up as well as a linear phase ( constant group delay ) 8th order LPF.

Notice the max Q of each filter type.

An 8th order filter only takes one Quad Op Amp and all these filters take less than a minute to design with free software and practice, knowing how to define filter specs..

The beauty of this TI owned software is you can choose any tolerance of R and C from perfect , 1%,2%, 5% to 20%and the change is instant and interactive. It does LPF,BPF,HPF and APF and provides a Bill of Material (BOM) which you may export to Excel.