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Re: Tom Danley at AES Chicago

82.182.186.41

Posted by svante on February 24, 2007 at 02:07:04

In Reply to: Re: Tom Danley at AES Chicago posted by tomservo on February 23, 2007 at 06:49:51:

Hi there I am Svante, and I am new to this forum. I am a teacher in electro acoustics at university level, and the writer of the Basta! loudspeaker simulation software, just to give you an idea of my background. I just had to jump in on this because this post was pointed out to me and that I find the statements in Tom's post fundamentally wrong (no offence intended).

The idea of a radiation resistance that is proportional to frequency ie Ra~f^2 comes from the radiation impedance of a baffled piston. Here folows an explanation, skip to **** if you already know it.

The analytical expression for this impedance is rather complicated and involves Bessel fuctions. Fortunately, for low frequencies, the impedance is in essence identical to that of the pulsating sphere, and this impedance in turn has a rather simple expression:

Za=rho0*c/S*jkr/(1+jkr)

k=w/c, c=344 m/s, rho0=1.2kg/m3, S= surface of the sphere, r= radius of the sphere, j=sqrt(-1).

The interesting part is jkr/(1+jkr).

Now if we want to see this impedance as a series connection of a resistive part and a reactive (mass) part we can multiply both the numerator and denominator by (1-jkr), and the result then becomes jkr(1-jkr)/(1+(kr)^2)=(jkr+(kr)^2)/(1+(kr)^2).

The real part of this expression is (kr)^2/(1+(kr)^2) which for low frequencies can be approximated by (kr)^2, ie the resistive part is proportional t f^2 (since k = w/c).

****

Now, this line of thinking would produce a resistor, the radiation resistance, which would absorb an amount of power that corresponds to the power radiated to the air. The model is not, however, suited to find the phase relation between the applied velocity of the cone and the pressure in the air, since there also is a reactive part of the radiation impedance present that affects the pressure.

The solution to find this phase relation is to see the radiation impedance as a parallel circuit instead of a series circuit. Doing so it will have the admittance:

Ya = 1/Za = S/rho0*c*(1+jkr)/jkr.

Here the part (1+jkr)/jkr is the interesting one, and the real part of it is 1, ie it is frequency independent. In other words, the radiation impedance can also be seen as a parallel circuit with an acoustic mass, and an acoustic resistance, and in this case both are well-behaved ordinary impedances.

If this is done, the analog for the loudspeaker in an infinite baffle can be set up, transformed to the acoustical side and reduced for the mass controlled range like in the attached image.

From the reduced analog it is clear that the end result is a pressure (~voltage) divider, with normal well-behaved impedances, and it can be seen that the pressure (voltage) across the radiation impedance is in phase with the driving pressure.

This is also supported by simple measurements of complex waveforms with the main energy in the driver's mass-controlled range.

Now, I was not present at the AES meeting where you presented this 90 degree off measurements, but I cannot see how you managed to get them. In my experience, measurements with a square wave or similar on a single driver typically results in a a square wave from the microphone, any small discrepancies can usually be explained with the frequency response of the driver.

My bottom line here is that there is NO phase shif introduced by the combination of the radiation resistance and the mass reactance. The acoustic signal and the electrical signal are identical and in phase(neglecting other errors in the driver).

This post is made possible by the generous support of people like you and our sponsors:

Schiit Audio

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