Near vs Far Field Criteria and Thiele–Small Parameters for Tweeters?

[audio_04]

Hi everyone,





I’m currently measuring loudspeaker drivers in an anechoic chamber and I’ve run into a couple of doubts.





First, regarding the boundary between near-field and far-field: I haven’t been able to determine which criterion is the correct one. Some sources define it as a^2 / \lambda, while others use 2D^2 / \lambda. I’ve checked two different books and they seem to contradict each other. Could someone clarify which expression is appropriate, or in what context each one should be used?





Second, I’m working with both a tweeter and a woofer, and I’ve measured their impedance and frequency response. I’m using the impedance measurements to extract Thiele–Small parameters. However, after measuring the tweeter, I’m not sure whether it actually makes sense to calculate Thiele–Small parameters for it, since these are typically more relevant at low frequencies.





Within the impedance measurement, I also tried applying the added-mass method. However, for the tweeter this required an extremely small added mass in order to shift the resonance peak above its natural resonance frequency, which made the process quite impractical.





Does it make sense to extract Thiele–Small parameters for a tweeter, or should this generally be avoided? And is there a better approach in this case?

 

[Keith_W]

@NTK this one's for you.

 

[LTig]

Measuring Thiele-Small parameters usually makes no sense for tweeters. You need those parameters to design the proper housing, however almost all tweeters have their own housing embedded and there is no way to change it.

 

[LTig]

As far as I know the border between near field and far field depends on the room, as this is the position where the reflected sound equals the level of the direct sound. In an anechoic room there is no reflected sound, hence no far field.

 

[DavidR]

This would be a good starting point. I added to an old thread on the topic of 2-mic measurement, but it moved into the topic of Near-Field/Far-Field measurement and the impact on NF due to the issue of driver coalescence with distance. NTK provided some very helpful information and Python script that I modified slightly to provide data for a 10" woofer I moved to. This is the first page I would suggest to start, but the first portion relates to a small dipole midrange I tested. The 10" woofer has details later. The big issue is the NF/FF transition. You'll see useful graphs using the modified Python script that you could use yourself if you care to.

Near-Field/Far-Field 2-mic thread entry point

There is also a good reference that had been provided to an audioXpress article on driver low frequency measurement by Joe D'Appolito. I was aware of the D. B. Keele technique, had used it for years, but was unaware of the detail relating to the limit of frequency that is dependent on the driver diameter. I took it more as an absolute prior to that.

audioXpress article

This is the detail I was unaware of:

For the near-field technique to work properly, the microphone should be placed as near to the center of the diaphragm as possible. Keele shows that a microphone distance less than 0.11 times the diaphragm effective radius results in measurement errors of less than 1 dB. As an example, a 6.5" driver will typically have an effective cone diameter of 5" or an effective radius of 2.5". For this driver, the microphone should be placed within 0.275" of the driver dust cap.

 

[NTK]

The separation between near field and far field is gradual, therefore, the exact distance at which one considers near field ends and far field begins depends on one's definition.

As for the definition, the quantity used for describing the source is k×a (which is a dimensionless, or unit-less, number), where k is the acoustic wavenumber* and a is the "characteristic dimension" of the source. For a circular source like most speaker drivers, the character dimension a can be either the radius or diameter, depending on one's chosen definition.

So, if you see 2 different definitions for the far field transition point, say one as a²/λ, and another as 2d²/λ, you will need to look at how each definition is defined and what criteria are used to define far field. You can also notice that, a²/λ or 2d²/λ are simply a or d multiplied by k×a, then by some constant factors.

Usually the far field begins at a few a (a little further for larger k×a) from the source. At the cross-over region when two drivers are simultaneously active, a would be the distance spanned by both drivers.

* Note:
The acoustic wavenumber can be expressed in a few different equivalent forms: k = ω/c = 2πf/c, and since f/c = 1/λ, k = 2π/λ. [Where f is the normal frequency, ω is angular frequency = 2πf, c is the speed of sound, and λ is the wavelength.]

Notice that the angular frequency (in time) is ω = 2πf = 2π/τ, where τ is the period. Compare ω to k = 2π/λ where λ is the wavelength, it can be seen that k is the spatial equivalent of ω. Therefore, k is the angular frequency in space.

The reason why dimensionless quantities such as k×a are used is the subject of dimensional analysis and similitude. Similitude let us use test (or analysis) results from scaled models of the same geometry and apply to the current problem, such as wind tunnel testing of a scaled model of a plane and apply the results to the full sized plane. In this case, by matching the k×a values, we can determine the far field transition point of a driver by using the known results of another driver of similar geometry (circular) but with a different diameter.

 

[fragzone]

The separation between near field and far field is gradual, therefore, the exact distance at which one considers near field ends and far field begins depends on one's definition.

As for the definition, the quantity used for describing the source is k×a (which is a dimensionless, or unit-less, number), where k is the acoustic wavenumber* and a is the "characteristic dimension" of the source. For a circular source like most speaker drivers, the character dimension a can be either the radius or diameter, depending on one's chosen definition.

So, if you see 2 different definitions for the far field transition point, say one as a²/λ, and another as 2d²/λ, you will need to look at how each definition is defined and what criteria are used to define far field. You can also notice that, a²/λ or 2d²/λ are simply a or d multiplied by k×a, then by some constant factors.

Usually the far field begins at a few a (a little further for larger k×a) from the source. At the cross-over region when two drivers are simultaneously active, a would be the distance spanned by both drivers.

* Note:
The acoustic wavenumber can be expressed in a few different equivalent forms: k = ω/c = 2πf/c, and since f/c = 1/λ, k = 2π/λ. [Where f is the normal frequency, ω is angular frequency = 2πf, c is the speed of sound, and λ is the wavelength.]

Notice that the angular frequency (in time) is ω = 2πf = 2π/τ, where τ is the period. Compare ω to k = 2π/λ where λ is the wavelength, it can be seen that k is the spatial equivalent of ω. Therefore, k is the angular frequency in space.

The reason why dimensionless quantities such as k×a are used is the subject of dimensional analysis and similitude. Similitude let us use test (or analysis) results from scaled models of the same geometry and apply to the current problem, such as wind tunnel testing of a scaled model of a plane and apply the results to the full sized plane. In this case, by matching the k×a values, we can determine the far field transition point of a driver by using the known results of another driver of similar geometry (circular) but with a different diameter.

Sorry random question, do you recommend any textbooks that deal with the fundamentals of speakers/audio science? I'm reading Floyd tool's book which is great but it's seemingly more to do with the empirical side of listening preference. Thanks.

 

[Keith_W]

Woah @NTK your post went into far more detail than I have ever seen. My definition of the transition distance between nearfield and farfield (the Rayleigh distance rR) is very simple:

Where a is the diameter of the driver in question and S is the surface area (for non-circular drivers).

Going back to Don Keele's paper as mentioned by @DavidR, where the mic needs to be spaced "0.11x the diameter of the driver", let us take a 25cm subwoofer driver. The recommended mic spacing would be (0.11 * 0.25) = 0.0275m / 27.5mm. Now we work out the distance according to the Rayleigh distance for 20Hz. r = pi * (0.125)^2 / 17.15 = 0.00286m / 2.86mm. That's well within the 0.11 mic spacing distance.

Now let's take a 25mm tweeter and calculate for 20kHz. The recommended minimum mic spacing would be (0.11 * 0.025) = 0.00275m / 2.75mm. Using the given Rayleigh formula for 20kHz, r = pi * (0.0025)^2 / 0.01715 = 0.001145m / 1.14mm. Also within the 0.11 mic spacing distance.

So IMO Keele's recommendation of 0.11x speaker diameter would seem to hold.