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lhlbluesky_lhl

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filter parameter design and calculation
Dec 30^th, 2012, 5:47pm

i have some doubts about filter design.

1, for a standard two-order bandpass filter, the transfer function is vout(s)/vin(s)=H0*(W0/Q)*s/(s^2+W0/Q*s+W0^2), with Wo as center frequency, H0 as center frequency gain, Q as quality factor. if it is a three order or four order bpf, then how to calculate the parameter W0, H0, Q? for example, i have a four order bpf (two cascaded two-order bpf) with transfer function vout(s)/vin(s)=(H0₁*(W0₁/Q₁)*s/(s^2+W0₁/Q₁*s+W0₁^2))*(H0₂*(W0₂/Q₂)*s/(s^2+W0₂/Q₂*s+W0₂^2)), then what is the expression of total W0, H0, Q?

2, for twin-T notch filter(for example, c1=c2=50pF, c3=100pF, r1=r2=200k, r3=100k), Q is infinite ideally, but actually, it is not the case, in my simulation, Q=20, what is the expression of W0, H0, Q for twin-T notch filter? and what is the non-idealities?

3, if i want to get a band-rejection response with very sharp attenuation at both rising edge and falling edge, but not using multi-order structrue, is there such circuit?

thanks.

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buddypoor

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Re: filter parameter design and calculation
Reply #1 - Dec 31^st, 2012, 2:55am

lhlbluesky_lhl wrote on Dec 30^th, 2012, 5:47pm:

1, for a standard two-order bandpass filter, the transfer function is vout(s)/vin(s)=H0*(W0/Q)*s/(s^2+W0/Q*s+W0^2), with Wo as center frequency, H0 as center frequency gain, Q as quality factor. if it is a three order or four order bpf, then how to calculate the parameter W0, H0, Q?
2, for twin-T notch filter(for example, c1=c2=50pF, c3=100pF, r1=r2=200k, r3=100k), Q is infinite ideally, but actually, it is not the case, in my simulation, Q=20, what is the expression of W0, H0, Q �for twin-T notch filter? and what is the non-idealities?
3, if i want to get a band-rejection response with very sharp attenuation at both rising edge and falling edge, but not using multi-order structrue, is there such circuit?

to 1.) There is no third-order bandpass - only even orders are possible. This results from the lowpass-to-bandpass transformation.
More than that, because a fourth-order bandpass has TWO conjugate-complex pole pairs it is not possible to define a single "overall" Q value.
Normally. the center frequency of a fourth-order BP is the geometrical mean value of both pole frequencies. The gain at the center frequency depends on the used approximation (Butterworth, Chebyshev,...)

to 2.) For a notch filter only the Q value for the zero approaches infinite. The pole Q has a finite value. The values for Ho, wo and Qp can be derived directly from the transfer function (similar to the bandpass case). Of course, Ho is ideally zero.

to 3.) Such a 2nd-order design requires a very large pole Q.

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LvW (buddypoor: In memory of the great late Buddy Rich)

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raja.cedt

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Re: filter parameter design and calculation
Reply #2 - Dec 31^st, 2012, 1:47pm

hello lhlbluesky_lhl,
As buddypoor explained Quality factor defined only for a pair of complex pairs, hence for cascaded BPF, 2 Q factors will be defined. But according to razaavi Q factor can be defined for any order system, please refer APPENDIX I in the following pap.

http://www1.ee.ucla.edu/~brweb/papers/Journals/BRSept08.pdf

for your 3rd Question, try to use bessel filter which is having very sharp transition.

@buddypoor: I think you know the above pap, in theory Q can't be defined for multi order system, then how razaavi defined Q through phase and magnitude.

Thanks,
Raj.

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buddypoor

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Re: filter parameter design and calculation
Reply #3 - Jan 1^st, 2013, 2:30am

raja.cedt wrote on Dec 31^st, 2012, 1:47pm:

hello lhlbluesky_lhl,
As buddypoor explained Quality factor defined only for a pair of complex pairs, hence for cascaded BPF, 2 Q factors will be defined. But according to razaavi �Q factor can be defined for any order system, please refer APPENDIX I in the following pap.

http://www1.ee.ucla.edu/~brweb/papers/Journals/BRSept08.pdf

for your 3rd Question, try to use bessel filter which is having very sharp transition.

@buddypoor: I think you know the above pap, in theory Q can't be defined for multi order system, then how razaavi defined Q through phase and magnitude.

Thanks,
Raj.

Hi Raj,

Two comments:
* Why do you think that a Bessel response has a "sharp transition" ? What/where is the source of this (surprising) statement?
* Of course, one can define everything - however, the question is if it makes sense (or not). As to Razavis definition:
I doubt if it applies in our case because he needs a pole frequency in his paper for Eq. (13). But - what happens if we have two or more pole frequencies?
Of course, for a bandpass of any order you always can use the ratio Bandwidth-to-center frequency. But this ratio (for n>2) does not include any information on sharpness of the transition band. Thus, I think this ratio is of less importance.

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LvW (buddypoor: In memory of the great late Buddy Rich)

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raja.cedt

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Germany

Re: filter parameter design and calculation
Reply #4 - Jan 1^st, 2013, 2:58am

hello buddypoor,
sorry it's a typo..i mean to say elliptical filter.

Regarding Quality factor i also agree with you, but didn't understand physical intuition behind it. He has many Q actor definitions in one of his old pap. I guess you are aware of oscillator phase noise, he derived Q for multi pole system from there he derived phase noise from leeson equation.

In my previous post i mean to ask is there any intuition behind it...

Thanks,
Raj.

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