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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 closedbox. 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.
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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 
27th June 2006, 10:22 PM  #5  
diyAudio Member
Join Date: Feb 2004
Location: Stockholm

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

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 highpass 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/[(1r^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/[13.91^2)+j(3.91/0.793)]=1.0115 jc 
28th June 2006, 06:55 PM  #7  
diyAudio Member
Join Date: Feb 2004
Location: Stockholm

Quote:
G(jw) = [  ((w/wo)^2)]/[ 1  ((w/wo)^2) + j[(w/wo)/Qt] " The minus signs comes from squaring j. 

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? 
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/(1r^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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