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]

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]

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

 

[MJK]

If r = jw then r^2 = - (w^2).

 

[Ron E]

To get frequency response magnitude and phase, you need to group real and imaginary terms in the numerator and denominator, then consider the imaginary component as Y and the real component as X - the magnitude is then the vector sum. For both the numerator and denominator, You square both the real and imaginary groups, and take the square root of the whole.

To get phase you consider them as vector components again and figure out the angle formed by the numerator and denominator, and subtract the angle of the denominator from the angle of the numerator.

Just do a search on transfer function magnitude and phase and there should be a tutorial out there somewhere.

 

[jclouse]

To Svante:

Looking back, you wrote:
"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] "

Ambiguity re Qt + 1 , therefore:
G(jw)=[(w/wo)^2]/[((w/wo)^2 -1) + jw/woQt]

Changing w/wo to r for ease of presentation,
G(jw) = r^2/[(r^2-1)+j(r/Qt)]
G(jw) = 3.910^2/(3.910^2-1)+j(3.910/0.793)=1.0115

What do you think?

 

[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? "


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

Ok, so there are two sign issues here. In your first post the denominator was (essentially):

r²-1+jr/Q

I corrected that to (also essentially):

1-r²+jr/Q

The other issue is regarding the sign of the numerator.

I proposed -r² and you proposed r²

Definitely there are systems of both kinds. In the loudspeaker case you only need to change the polarity of the connectors and you have the other one.

The nice thing about having the negative sign is that the transfer function becomes 1 for high frequencies, but of course there are also inverting systems, and they don't have the negative sign in the numerator.

 

[MJK]

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

OK, I have a few minutes to take a crack at this question. First lets rewrite the equation with a few more parenthesis

G(s) = (s^2 Tc^2) / (s^2 Tc^2 + s Tc / Qtc + 1)

The definitions of s and Tc are shown below

s = jw = j(2 pi f)

Tc = 1 / wc = 1 / (2 pi fc)

Substituting

G(jw) = [- (w/wc)^2] / [(1 - (w/wc)^2) + j(w/wc) / Qtc]

The negative signs come from squaring the imaginary number j, j^2 = -1. This result matches Svante's.

Now

w/wc = f/fc = 228/58.31 = 3.91

Qtc = 0.793

So substituting into G(jw) and doing the calculations (watch out for the imaginary number in the denominator) you get a complex result

G(jw) = [-(3.91)^2] / [(1 - (3.91)^2) + j (3.91) / 0.793]

G(jw) = 0.956 + j0.330

which can be expressed as a magnitude and a phase

magnitude = 1.011

phase = 19.039 degrees

That is my solution, hopefully I didn't make any mistakes.

 

[jclouse]

Martin said:

"OK, I have a few minutes to take a crack at this question. First lets rewrite the equation with a few more parenthesis

G(s) = (s^2 Tc^2) / (s^2 Tc^2 + s Tc / Qtc + 1)

The definitions of s and Tc are shown below

s = jw = j(2 pi f)

Tc = 1 / wc = 1 / (2 pi fc)

Substituting

G(jw) = [- (w/wc)^2] / [(1 - (w/wc)^2) + j(w/wc) / Qtc]

The negative signs come from squaring the imaginary number j, j^2 = -1. This result matches Svante's.

Now

w/wc = f/fc = 228/58.31 = 3.91

Qtc = 0.793

So substituting into G(jw) and doing the calculations (watch out for the imaginary number in the denominator) you get a complex result

G(jw) = [-(3.91)^2] / [(1 - (3.91)^2) + j (3.91) / 0.793] (a)

G(jw) = 0.956 + j0.330

which can be expressed as a magnitude and a phase

magnitude = 1.011

phase = 19.039 degrees "

---------------

Thank you.

For reference:
fo = resonance = 58.31 Hz
f = 228 Hz
Qt = 0.793

You note above:

G(jw) = [-(3.91)^2] / [(1 - (3.91)^2) + j (3.91) / 0.793]

Then is this written correct for magnitude?

|G(jw)|= -3.91^2 / root [(1-3.91^2)^2 + (3.91/0.793)^2] = 1.011

with phase angle of

arc tan [(2*d*3.91)/(1-3.91^2)]

where
d = damping ratio = 1/(2*Qt) = 1/(2*0.793) = 0.6305

My original calculation was based on the magnification
factor or amplitude ratio from books on vibration where

G(jw) = r^2 / [ (1-r^2)+j(r/Q) ]


angle = arc tan [ (2 d r) / (1-r^2) ]

where r = w/wo

jc

 

[jclouse]

Ok, so there are two sign issues here. In your first post the denominator was (essentially):

r²-1+jr/Q

I corrected that to (also essentially):

1-r²+jr/Q

The other issue is regarding the sign of the numerator.

I proposed -r² and you proposed r²

Definitely there are systems of both kinds. In the loudspeaker case you only need to change the polarity of the connectors and you have the other one.

The nice thing about having the negative sign is that the transfer function becomes 1 for high frequencies, but of course there are also inverting systems, and they don't have the negative sign in the numerator.

OK. Thank you.

 

[MJK]

If I use your equation for magnitude

|G(jw)|= -3.91^2 / root [(1-3.91^2)^2 + (3.91/0.793)^2] = 1.011

I get -1.011.

If I use your equation for the phase, I get -19 degrees. In my opinion you are making sign errors which probably arise from the imaginary terms and algebra mistakes. I believe my results are correct.

While the numerical values are correct, the sign errors are going to catch up when you try and use the results to design and integrate an enclosure with other drivers. Sign errors will get even worse with a ported box transfer function.

 

[Svante]

If I use your equation for magnitude



I get -1.011.

Hehe, the equation in itself is a contradiction, it is a ratio between an always negative number and an always positive number, and that is supposed to be the absolute value of something. Of course the minus sign should be removed from the numerator, if we want the absolute value.

However, this is not a big deal, the error is so obvious that it does not matter...

 

[kelticwizard]

I couldn't possibly comment on the mathematics of this, it is far beyond my range.

But Svante-whatever happend to your old avatar? It was so much more fun than the latest one. And with summer here, the idea of feet sticking out of water....


My apologies for being utterly irrelevant.