# Measuring the cutoff frequency with Q >1

#### Junus2012

Dear friends,

below is the image of the 4th order sallen key LPF, I have a deviation from the Butterworth approximation as it is noticeable.

While in ideal butterworth the cutoff frequency is calculated at the - dB frequency, but what about this case ?

Thank you

#### barry

Not sure what you mean " cutoff frequency is calculated at the - dB frequency ".

Regardless, you've got 0 dB gain in the passband, the -3dB cutoff frequency is at -3dB, although I have read in some literature that the cutoff frequency is 3dB down from the highest passband gain. I'm not sure I agree with that.

Junus2012

### Junus2012

points: 2

"Normally" one thinks of Q as being applied to BP or bandstop or Notch filters which
generally have distinct BW and center freq.

Q being ratio of Fcenter / BW @ -3db

So practically speaking your plot really does not possess a - 3db reference. Same
for equaripple Chebyshev, what ripple number constitutes a focus for measuing Q....

There are LPF filters that have large peaking and one could think of a Q of the peak.
But of what use that would be is strictly up to user interpretation.

Regards, Dana.

Junus2012

### Junus2012

points: 2

#### BigBoss

-3dB is valid for $A_{v}\leq0$ only.Your circuit has a feedback..

Junus2012

### Junus2012

points: 2

#### sutapanaki

You usually measure the cut-off frequency at the -3dB frequency. This is by assumption. If system needs, you can measure, for example at the -1dB but largely accepted is the -3dB frequency.

As for the Q - it is measured as the peak of the frequency response with respect to DC gain. This is correct for high Q, because the highest point of the peak is at the resonant frequency of the bi-quad. This is not quite the case for you because the peak is not large. So, unless you do some math, you can only measure it approximately. Since you talk about Sallen-Key filter, I assume you have two second order sections and you can look at the transfer function of each and try to map to what you expect form Butterworth and see where the discrepancy comes from.

Junus2012

### Junus2012

points: 2

#### FvM

##### Super Moderator
Staff member
I don't agree with some statements.

1. cut-off frequency of a low-pass is often measured as - 3dB gain point relative to low frequency pass-band gain. No matter if the filters are implemented with feedback (= active filters) or e.g. passive LC filters and what's the pass band gain. -3db criterion is usually applied to filters with monotonous transfer characteristic, e.g. Bessel or Butterworth. For filters with ripple in the transfer characteristic like Chebyshev, the cut-off frequency is either specified as -3dB point, or more common as the point where the gain falls below the ripple band.

2. Q is a parameter of the second order building block ("biquad") respectively a complex pole pair. A 4th order filter can be characterized by two Q values.

The best way to characterize your degenerated butterworth filter is to determine an approximate 4th order transfer characeteristic, either by determing the transfer characteristic analytically or by fitting a filter to the empirical measurements. If the deviation is caused by amplifier GBW, you can expect that the exact transfer characteristic is of 6th or higher order.

Junus2012

### Junus2012

points: 2

#### sutapanaki

Butterworth filters are maximally flat. Some of the individual bi-quads in the filter realization do have higher Q, while the others don't and when put together they compensate each other such that the result is a maximally flat response. If you get peaking in the frequency response, that might suggest that the characteristics of the biquads don't align properly or they have higher individual Qs than they are supposed to have. For example, if you get excess phase shift in the integrators because the opamp's non-dominant pole is too close to the unity gain cross-over frequency of the loop, that will cause Q enhancement for the biquads.

Junus2012

### Junus2012

points: 2

#### Dominik Przyborowski

3dB is an convention defining passband as FWHM of signal spectrum (as -3dB means half of power). Depending on spectrum it can be defined relatively to max magnitude (narrowband PB filters) or to magnitude at central frequency (for wide passband), low freq for wide LPF or high freq for HPF.

For some applications, cut-off frequency is even not defined, as filter can simply has not passband at all (like CR-RC^n) and is described by it impulse response in time domain only.

Junus2012

### Junus2012

points: 2

#### Junus2012

Dear friends,

Thank you very much for your contribution to my post
for the moment I will postpone the comments on the Q measurement or the reasons caused it, and I will discuss about the cutoff frequency measurement.

Yes for sure for Butterworth LPF, the cutoff frequency is generally measured at -3dB (mistakenly written -db in my first post), however here the assumption is for a flat response with maximum output magnitude at DC (in my case 0 dB).

In the case of the transfer I showed, I have a small peak (or even in case if it is larger),

Now I need your help to decide one of your presented solutions as :

1. fc is at -3dB from the DC passband gain

2. fc is at is 3dB down from the highest passband gain (-3dB from the peak magnitude)

3. fc is at the point where the gain falls below the ripple band (like the case of Chebyshev ).

Thank you once again
Regards

#### FvM

##### Super Moderator
Staff member
Clearly the first criterion, because you are still trying to implement a Butterworth filter.

Junus2012

### Junus2012

points: 2

#### barry

You are really arguing semantics. Saying the cutoff frequency is at 3dB or 5 dB or 100 dB is pretty irrelevant. What are your requirements? What attention do you need at what specific frequency? What pass band ripple/peaking can you tolerate?

Junus2012

### Junus2012

points: 2

#### sutapanaki

Look at it this way. You want to characterize a response. You have a DC gain, you have a peak and you have a slope with which it rolls off. So, measure peak with respect to DC gain, measure slope with its proxy the -3dB frequency with respect to DC gain. As was said before, instead of -3d, you can measure at -5dB for example. But -3dB is accepted.

Junus2012

### Junus2012

points: 2

#### Junus2012

Dear friends,

Thank your very much for your help, I think I can move in discussion toward the Q,

I am suspecting the reason of the peak and veviation from ideal Q = 0.707 for two reasons

1. Limited gain bandwidth of the amplifier, which is a natural concern for people designing active RC filter and thier main reason to move toward gm_C filter when fc is high as the case I designed for

2. It is due to the design of the feedback elements that control Q

Now my concern is how to measure the Q resulted from my graph, I have read this part from TexasInstrument, the method is based on measuring the distance (in liner) between the peak and the DC gain, however, they say that this method is not accurate for the case if Q < 3

#### FvM

##### Super Moderator
Staff member
Did you notice that the peaking transfer curves in the figure are that of the partial (2nd order) filters. As already explained in post #6, Q is only defined for 2nd order filter. In so far, the Q measurement isn't applicable for the problem in post #1.
--- Updated ---

To analyze the transfer function which magnitude is plotted in post #1, you'll determine the location of its poles, respectively f0 and Q of its partial 2nd order filters.

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Junus2012

### Junus2012

points: 2

#### Junus2012

Did you notice that the peaking transfer curves in the figure are that of the partial (2nd order) filters. As already explained in post #6, Q is only defined for 2nd order filter. In so far, the Q measurement isn't applicable for the problem in post #1.
Dear Fvm, I got your point, you are absolutely right, I just understand this point, Q is a parameter of single biquad (2nd order), hence to characterise the higher-order by the meaning of Q is only by characterizing the individual Q of the individual biguads.

#### sutapanaki

Also, as I mentioned in #5 and #7 it is not only necessary to measure individual Qs. You will have to see what discrepancy there is with the ideal Butterworth because although it is maximally flat as a whole, the individual biquads have peaks i.e. Q and droops that have to aligned with each other to get the final result.

#### Junus2012

Also, as I mentioned in #5 and #7 it is not only necessary to measure individual Qs. You will have to see what discrepancy there is with the ideal Butterworth because although it is maximally flat as a whole, the individual biquads have peaks i.e. Q and droops that have to aligned with each other to get the final result.
Dear Suta, I am sorry I overlooked the posts 5, thank you for reminding me, completely agreed with the TEXAS explanation about measuring the Q.

As you explained, the complete filter Q is a composite of individual Qs of the biquads stages, and yes indeed the second biquad in this design has more than should be (Q1 should be 0.545 and Q2 = 1.3)

I want to ask you my friends,
If I would live with a peak, what is the maximum peak value (in dB) I should accept? actually when I use the TEXAS web tool to build the filter the maximum peak that can be accepted is 3 dB, why?

Thank you once again

#### sutapanaki

I think this is quite specific to the application. There is usually a mask that sets the boundaries for the pass band, the stop band and the transition between the two. For example, Chebishev I type filers can have, say, 0.25dB of pass band ripple, or 0.5db or more. Which one you choose depends on your application. Similarly for Butterworth. Also, maybe a variation in the peak also varies the phase response and hence the overshoot of the step response. A model of your filter here can go ways because you can explore different scenarios faster.

Junus2012

### Junus2012

points: 2

#### Junus2012

I think this is quite specific to the application. There is usually a mask that sets the boundaries for the pass band, the stop band and the transition between the two. For example, Chebishev I type filers can have, say, 0.25dB of pass band ripple, or 0.5db or more. Which one you choose depends on your application. Similarly for Butterworth. Also, maybe a variation in the peak also varies the phase response and hence the overshoot of the step response. A model of your filter here can go ways because you can explore different scenarios faster.
Dear Suta,

I understand that designing the filter for specific approximation for example Butterworth, doesn't mean that if I am having a slight deviation it means 'I have a serious problem, I understand that if this deviation still accepted by the application then it is ok.

In my post #1 the transfer function is something between Butterworth and Chebyshev