Transfer Function

[jclouse]

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

[Ron E]

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.

[phase_accurate]

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

[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

[Svante]

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]

[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

[Svante]

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.

[jclouse]

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

[Svante]

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?

[jclouse]

" 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