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 Transfer Function
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 22nd June 2006, 04:47 PM #1 diyAudio Member   Join Date: Jun 2006 Transfer Function Hi All, I'm new here. Looks like an interesting and well run place. I need help with equation (19) from Richard Small's article in the J.A.E.S, December 1972. It is the response function for a driver in a closed-box. G(s) = s^2 Tc^2 / s^2 Tc^2 + s Tc / Qtc + 1 The first term on the right s^2 Tc^2 is the numerator. Values are: fc (resonance) = 58.31 Hz f = 228 Hz Qtc = 0.793 Also could someone explain the modus operandi of the equation, and is one term negative? Thanks jclouse
 23rd June 2006, 12:10 AM #2 diyAudio Member     Join Date: Jun 2002 Location: USA, MN Tc = 1/Fc Search on 'transfer function' , 'frequency response' , 'magnitude' , 'phase' , etc.... on google and you should be set. or just go to www.diysubwoofers.org and see how to solve them, the math is all boiled down to its simplest form there. __________________ Our species needs, and deserves, a citizenry with minds wide awake and a basic understanding of how the world works. --Carl Sagan Armaments, universal debt, and planned obsolescence--those are the three pillars of Western prosperity. —Aldous Huxley
 23rd June 2006, 07:46 AM #3 diyAudio Member   Join Date: May 2002 Location: Switzerland This is a 2nd order Highpass transfer function and should be written as G(s) = s^2 Tc^2 / (s^2 Tc^2 + s Tc / Qtc + 1) in order to be correct. It is not that difficult to imagine what happens: If s ( i.e. j* 2 * Pi * f) is approaching infinity then the function approaches 1. At the pole frequency the term s^2*Tc^2 becomes -1 and therefore the the function behaves as - 1/ (s Tc / Qtc). Hence we have humps for high Q values and premature rolloff at low Q values. The closer the frequency moves to zero the more the function approaches the behaviour s^2 Tc^2 / 1 hence the 2nd order rolloff. Regards Charles
 27th June 2006, 07:14 PM #4 diyAudio Member   Join Date: Jun 2006 To Ron E and Phase Accurate Thanks. For sinusoidal signals G(s) becomes G(jw), so that G(jw) = [(w/wo)^2]/[((w/wo)^2 -1) + j[(w/wo)/Qt] jc
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Quote:
 Originally posted by jclouse To Ron E and Phase Accurate Thanks. For sinusoidal signals G(s) becomes G(jw), so that G(jw) = [(w/wo)^2]/[((w/wo)^2 -1) + j[(w/wo)/Qt] jc
I'd say

G(jw) = [ -(w/wo)^2]/[ 1 - (w/wo)^2 + j[(w/wo)/Qt]
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 28th June 2006, 06:04 PM #6 diyAudio Member   Join Date: Jun 2006 Svante wrote: " G(jw) = [ -(w/wo)^2]/[ 1 - (w/wo)^2 + j[(w/wo)/Qt] " This gives a negative value in the passband, and being a high-pass function, should not be below unity. Note although 1 - w/wo is negative, when squared it becomes positive. IMO the following is OK (converting to w/wo=r for simpler presentation). G(jw)=r^2/[(1-r^2)+j(r/Qt)] With the values cited (f=228 Hz, fo=58.31 Hz, Qt=0.793) this gives magnitude of G(jw) = 3.91^2/[1-3.91^2)+j(3.91/0.793)]=1.0115 jc
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Join Date: Feb 2004
Location: Stockholm
Quote:
 Originally posted by jclouse Svante wrote: " G(jw) = [ -(w/wo)^2]/[ 1 - (w/wo)^2 + j[(w/wo)/Qt] " This gives a negative value in the passband, and being a high-pass function, should not be below unity. Note although 1 - w/wo is negative, when squared it becomes positive. IMO the following is OK (converting to w/wo=r for simpler presentation). G(jw)=r^2/[(1-r^2)+j(r/Qt)] With the values cited (f=228 Hz, fo=58.31 Hz, Qt=0.793) this gives magnitude of G(jw) = 3.91^2/[1-3.91^2)+j(3.91/0.793)]=1.0115 jc
Ok, to clarify what I meant:

G(jw) = [ - ((w/wo)^2)]/[ 1 - ((w/wo)^2) + j[(w/wo)/Qt] "

The minus signs comes from squaring j.
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 28th June 2006, 07:41 PM #8 diyAudio Member   Join Date: Jun 2006 Svanre wrote: "Ok, to clarify what I meant: G(jw) = [ - ((w/wo)^2)]/[ 1 - ((w/wo)^2) + j[(w/wo)/Qt] " The minus signs comes from squaring j. " Squaring the ratio of two frequencies should not make the result negative. jc
 28th June 2006, 09:12 PM #9 diyAudio Member     Join Date: Feb 2004 Location: Stockholm Ok, let's go back to the original equation: G(s) = s^2 Tc^2 / (s^2 Tc^2 + s Tc / Qtc + 1) which should be the same as G(jw) = (jw/w0)^2 / [ (jw/w0)^2 + (jw/w0) / Qt + 1] which becomes G(jw) = - ((w/w0)^2) / [ - ((w/w0)^2) + (jw/w0) / Qt + 1] which can be rearranged to G(jw) = [ - ((w/wo)^2)]/[ 1 - ((w/wo)^2) + j[(w/wo)/Qt] Right? __________________ Simulate loudspeakers: Basta! Simulate the baffle step: The Edge
 28th June 2006, 10:01 PM #10 diyAudio Member   Join Date: Jun 2006 " Ok, let's go back to the original equation: G(s) = s^2 Tc^2 / (s^2 Tc^2 + s Tc / Qtc + 1) which should be the same as G(jw) = (jw/w0)^2 / [ (jw/w0)^2 + (jw/w0) / Qt + 1] which becomes G(jw) = - ((w/w0)^2) / [ - ((w/w0)^2) + (jw/w0) / Qt + 1] which can be rearranged to G(jw) = [ - ((w/wo)^2)]/[ 1 - ((w/wo)^2) + j[(w/wo)/Qt] Right? " Unsure. Polarity was my concern at the beginning. You stated: "The minus signs comes from squaring" and my reply was not addressed per se, i.e. "Squaring the ratio of two frequencies should not make the result negative." Also the equation: G(jw)=r^2/(1-r^2)+j(r/Qt) from physics books does not show r in the numerator as negative. This contradicts the numerator being negative. Note the last term may be written as j(2dr), where damping d=1/2Qt, jc

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