# Understanding Sallen-Key LPF

#### RLT

Joined Aug 8, 2014
19
I am trying to understand S-K LPFs. Using Okawa’s online calculations there is a provision for evaluating oscillating frequencies. First question, if the poles are located at 45 degrees in quadrants 3 and 4, how can the system oscillate? As an example: Non-inverting input, Av (K) = 1, R1 = R2 = 102.3K ohms, Cf = 44 nf and Cg = 22 nf; Fc = 50 cps, Fo ~ 35 cps, both damping and Q are 0.707.

My second question, what is the formula for calculating oscillating frequency in an S-K LPF?

THANK YOU FOR YOUR TIME AND EFFORT!!!

#### OBW0549

Joined Mar 2, 2015
3,566

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#### RLT

Joined Aug 8, 2014
19
Thank you for your response. However, I do not see how the TI paper addresses the two questions I asked. It does, in fact, states oscillation would occur if the poles were in quadrants 1 and 4, which is absolutely understandable. Did you try the example I stated? If so, you would see what I said regarding Okawa's results, in particular, oscillation frequencies and pole locations. I am certainly not disagreeing with Okawa...I just do not understand the results.

Thanks again.

#### Papabravo

Joined Feb 24, 2006
14,689
I don't know the answer for sure, but if I had to guess it would involve the use of a non ideal opamp which turns for example a 3 rd order filter into a 5th order filter by introducing a low frequency pole on the negative real axis and a high frequency pole on the negative real axis at the unity gain bandwidth. If those two poles are taken into account, and a root locus is constructed there may be combinations of gain and phase that will produce instability. They will be where the root locus crosses the jω-axis. I have never encountered this behavior in practice so I can't be sure about this. I also have not seen an explicit discussion in standard textbooks, but it is possible that I just missed it.

#### Irving

Joined Jan 30, 2016
1,070
One example (from experience) is where the op amp is driving a capacitive load, e.g. an unmatched transmission line (in my case about 6m of RG59 coax). The circuit, a 4-pole 2-stage S-K LPF with a <25Hz cut-off and >40+dB down at 50Hz worked perfectly on the bench, but the 600pF approx line had it squawking at about 600kHz until I swapped the op amp for one with a much lower Gain.Bandwidth . I couldn't tell you where the poles were, not having a bode plot analyser to hand, and I've no idea how to calculate the oscillation freq (and intuitively I can't see it oscillating at thar low a frequency anyway).

#### LvW

Joined Jun 13, 2013
1,127
I am trying to understand S-K LPFs. Using Okawa’s online calculations there is a provision for evaluating oscillating frequencies. First question, if the poles are located at 45 degrees in quadrants 3 and 4, how can the system oscillate? As an example: Non-inverting input, Av (K) = 1, R1 = R2 = 102.3K ohms, Cf = 44 nf and Cg = 22 nf; Fc = 50 cps, Fo ~ 35 cps, both damping and Q are 0.707.

My second question, what is the formula for calculating oscillating frequency in an S-K LPF?

THANK YOU FOR YOUR TIME AND EFFORT!!!
I must admit that I do not understand the last question.
A low pass is a filter with two complex poles and cannot oscillate - unless you shift the poles to the imag. axis (pole quality factor Qp infinite).
Question: What is your aim? Two design a filter or an oscillator?

When you provide a link to the Okawa page, we could see what they mean with "oscillating frequency". Perhaps the frequency of "damped" oscillations as a step response of the filter?

#### Papabravo

Joined Feb 24, 2006
14,689
One example (from experience) is where the op amp is driving a capacitive load, e.g. an unmatched transmission line (in my case about 6m of RG59 coax). The circuit, a 4-pole 2-stage S-K LPF with a <25Hz cut-off and >40+dB down at 50Hz worked perfectly on the bench, but the 600pF approx line had it squawking at about 600kHz until I swapped the op amp for one with a much lower Gain.Bandwidth . I couldn't tell you where the poles were, not having a bode plot analyser to hand, and I've no idea how to calculate the oscillation freq (and intuitively I can't see it oscillating at thar low a frequency anyway).
You can infer approximate pole locations for an opamp. The low frequency pole is where Aol starts to roll off. The high frequency pole is where the open loop gain is reduced to unity, and is approximated by the GBW parameter.

#### RLT

Joined Aug 8, 2014
19
I don't know the answer for sure, but if I had to guess it would involve the use of a non ideal opamp which turns for example a 3 rd order filter into a 5th order filter by introducing a low frequency pole on the negative real axis and a high frequency pole on the negative real axis at the unity gain bandwidth. If those two poles are taken into account, and a root locus is constructed there may be combinations of gain and phase that will produce instability. They will be where the root locus crosses the jω-axis. I have never encountered this behavior in practice so I can't be sure about this. I also have not seen an explicit discussion in standard textbooks, but it is possible that I just missed it.
Thank you PapaBravo for your response. I am still in development, nothing has been constructed. We certainly agree if it crosses the imaginary axis instability would occur. I assume Okawa used an ideal opamp since no real device was stated. When I entered the values in the online calculator I was VERY surprised to see the oscillation results. LaPlace does not offer anything regarding oscillation frequency that I can see.

#### LvW

Joined Jun 13, 2013
1,127
..........When I entered the values in the online calculator I was VERY surprised to see the oscillation results. LaPlace does not offer anything regarding oscillation frequency that I can see.
So - you have entered some parts values to design a lowpass (2nd-order ?) ?
And which results were surprising? What do you mean with "...to see the oscillation results..." ?

#### RLT

Joined Aug 8, 2014
19
I must admit that I do not understand the last question.
A low pass is a filter with two complex poles and cannot oscillate - unless you shift the poles to the imag. axis (pole quality factor Qp infinite).
Question: What is your aim? Two design a filter or an oscillator?

When you provide a link to the Okawa page, we could see what they mean with "oscillating frequency". Perhaps the frequency of "damped" oscillations as a step response of the filter?
Thank you LvW for your response. This for an Fc = 50 cps LPF. I revisited Okawa's calculator and removed the transient analysis options and recalculated...the oscillation frequency remained ~ 35 cps. I certainly agree with you, I cannot see how this can oscillate either. Maybe I am being too critical...maybe there is an issue the calculator...that is what I am trying to determine.

Thanks again.

#### RLT

Joined Aug 8, 2014
19
So - you have entered some parts values to design a lowpass (2nd-order ?) ?
And which results were surprising? What do you mean with "...to see the oscillation results..." ?
If you enter the values into Okawa's calculator as stated previously (Non-inverting input, Av (K) = 1, R1 = R2 = 102.3K ohms, Cf = 44 nf and Cg = 22 nf), on the left side are the calculated results, one of which is oscillation frequency, ~35 cps. This does not make sense to me; I just do not understand why that is there.

#### LvW

Joined Jun 13, 2013
1,127
RLT - as I suspected before - the "oscillation frequency" is the frequency which would be theoretically observed without any damping.
This frequency is identical to the imaginary part of the pole(s).
However, from the theoretical system point of view it is formally possible to calculate this "frequency".
But is has no practical meaning for the lowpass function.

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#### Irving

Joined Jan 30, 2016
1,070

#### Papabravo

Joined Feb 24, 2006
14,689
If you enter the values into Okawa's calculator as stated previously (Non-inverting input, Av (K) = 1, R1 = R2 = 102.3K ohms, Cf = 44 nf and Cg = 22 nf), on the left side are the calculated results, one of which is oscillation frequency, ~35 cps. This does not make sense to me; I just do not understand why that is there.
Have you tried asking the company?

#### Papabravo

Joined Feb 24, 2006
14,689
I agree...

I ran the same design on a different site: https://www.beis.de/Elektronik/Filter/ActiveLPFilter.html
and got the same values.

Put that into Spice and look at the damped step response and its around 30-odd mS settling time which is ~35Hz

View attachment 209376View attachment 209376
The confirmation that this is indeed the explanation would be to reduce the damping factor further to the point where in "rings like a bell" and see if the frequency matches. The reason I've never encountered this phenomena is that I usually am not concerned with the step response since the step has much higher frequency components than the corner frequency.

#### LvW

Joined Jun 13, 2013
1,127
If you enter the values into Okawa's calculator as stated previously (Non-inverting input, Av (K) = 1, R1 = R2 = 102.3K ohms, Cf = 44 nf and Cg = 22 nf), on the left side are the calculated results, one of which is oscillation frequency, ~35 cps. This does not make sense to me; I just do not understand why that is there.
From the theoretical point of view, the poles are located at p1,2=-(sigma) (+-) SQRT (wn).
Hence, wn is the imginary part of the pole.
The frequency wn is the natural frequency which can be observed as a damped oscillation within the step response.

OKAWA says: Imag. part of the pole is 22......cps (identical to the given "oscillation frequency").
Where do you see a frequency of about 35 cps?

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#### RLT

Joined Aug 8, 2014
19
So - you have entered some parts values to design a lowpass (2nd-order ?) ?
And which results were surprising? What do you mean with "...to see the oscillation results..." ?
If you enter the values into Okawa's calculator as stated previously (Non-inverting input, Av (K) = 1, R1 = R2 = 102.3K ohms, Cf = 44 nf and Cg = 22 nf), on the left side are the calculated results, one of which is oscillation frequency, ~35 cps. This does not make sense to me; I just do not understand why that is there.
From the theoretical point of view, the poles are located at p1,2=-(sigma) (+-) SQRT (wn).
Hence, wn is the imginary part of the pole.
The frequency wn is the natural frequency which can be observed as a damped oscillation within the step response.

OKAWA says: Imag. part of the pole is 22......cps (identical to the given "oscillation frequency").
Where do you see a frequency of about 35 cps?
The only thing that makes sense to me is the damping issue. I did not think of it. If the system is damped, which of course it is, why include damping as a valid parameter? Rhetorical question. To answer the question, where did I see 35 cps, attached is a screen shot of Okawa's data.

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#### Papabravo

Joined Feb 24, 2006
14,689
35 Hz is just the imaginary part of the pole location defined by the corner frequency: $$50/\sqrt{2}\;\approx\;35$$

#### RLT

Joined Aug 8, 2014
19
35 Hz is just the imaginary part of the pole location defined by the corner frequency: $$50/\sqrt{2}\;\approx\;35$$
AGREED!!! I think this is the undamped imaginary value. Since the system is damped I do not see any value in obtaining the oscillating frequency.

I would like to thank all for your help. I certainly learned something and hope this helps others!

PapaBravo, Marion County as well!

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#### LvW

Joined Jun 13, 2013
1,127
AGREED!!! I think this is the undamped imaginary value. Since the system is damped I do not see any value in obtaining the oscillating frequency.

I would like to thank all for your help. I certainly learned something and hope this helps others!

PapaBravo, Marion County as well!
RLT - just for clarification: I have used two resistors 162.3k instead of 102.3 k.
That is the reason for the discreapancy between our results. My fault.
As I have suspected, OKAWAs "oscillation ffrequency" is the imag. part of the pole pair (wn) - and it can be observes as a damped oacillation for larger Q values only (several periods...).

One final remark (real opamps): If you are going to realize this filter you will see that the damping properties of the circuit are not as good as shown in OKAWAs simulation. The reason is that each opamp has a finite output resistance - and for larger frequencies (far above the pole) there will be a rising portion of the input signal which arrives DIRECTLY via the feedback capacitor at the opamp output and will produce an unwanted output signal across the output resistance (which, unfortunately, will even rise with frequency). Hence, the maximum possible attenuation in the stop band is perhaps only 30...40 dB.
This is one of the drawbacks of this filter configuration.

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