DATAINFO.HTM --- Part of Manual for Driver Parameter Calculator --- by
Claus Futtrup.
Created 8. March 2001, last revised 15. June 2004.
Ported to XHTML 1.0 on 2. October 2004. Last modified 25. October 2004.
This document will explain in some detail the parameters of DPC, descriptive parameters as well as data parameters, their meaning and/or interpretation, the relations to other parameters etc. Sometimes the information will be close to "chatting" and sometimes it will be boosted by scientific explanations.
Table of contents:
Bxl is named the force factor. It actually is a mixed parameter, consisting of B (magnetic field strength, magnetic induction) and l (the wirelength of the voice coil). Since B is not a given value in the entire length of the voice coil, it is actually measured and given as an average B value. This is also why the nonlinearities take effect for higher excursion.
With my x in between the two I indicate a cross-product, which is a mathematical calculation. Both B and l are given as vectors, and if they are perpendicular to each other (which you can safely assume in regular loudspeakers) then the given value is perpendicular to both (ie. pointing in the direction of the motional force). This value is multiplied by i (the electrical current) to calculate the actual force.
Strictly speaking, what one should do is to cross B and i because this is how physics work, and l is only another scale factor. B x i * l would give you the force in the true sense, named the Lorentz force. Whether you use Bxl * i or Bxi * l does not make a difference.
f4pi is the parameter in DPC, which describes the highest possible frequency, where a driver is still radiating into 4pi space.
Regarding space angles:
A sphere has the following surface area: A = 4 * pi * r^2
It is obvious to isolate the distance from the center of the sphere to the surface as a variable and name the rest Omega = A / r^2 (= 4 * pi for a sphere). Omega is the space angle. Totally we get : A = Omega * r^2 for any part of a sphere surface.
For a hemisphere (half a sphere) we have Omega = 2 * pi.
Personally I find f2pi to be a "better" figure for choosing a crossover frequency. The equation would become:
2 * c
f2pi = 2 * f4pi = ------- (because f4pi = c / 2 * pi * r)
pi * Dd
The relation between f4pi and f2pi is real easy, and this may be the reason why noone has changed concensus. Everybody just continues to look at f4pi as an indicator of a usable frequency range.
Both equations assume a flat pistonic diaphragm, of course. For further discussion on this topic one could go into a study of the directivity index.
Directivity index (DI) describes the ratio of sound intensity measured on-axis relative to the total power response sound intensity.
DI = 20 * log10(P_theta/P_0)
or
DI = on-axis frequency response (dB) - power response (dB).
If you take a look at the directivity index for a flat piston in an infinite baffle (this is a heavy mathematical simplification), DI = 3 dB at low frequencies and at f4pi the DI will have increased 10-20% which is close to nothing. Even at twice the frequency (where the piston is probably not flat anymore and the model normally does not work) the directivity index has only doubled from 3 to 6, which means that you get +3 dB on-axis frequency response relative to the total power response (in the halfplane limited by the infinite baffle).
If you on the other hand assume the piston to be at the end of a long tube (very opposite to an infinite baffle, and also a heavy mathematical simplification) you have exactly +3dB directivity at f4pi. At half the frequency you have +1 dB directivity and at twice f4pi (ie. at f2pi) you have +6 dB directivity.
In my opinion f4pi as an upper limit is very selective, and normally drivers can be used to, say, twice the frequency limit. Obviously this is very dependent on driver quality, quite individual - and yet still you have the fundamental physical properties that you cannot circumvent.
Studying directivity index and other parameters, the frequency is often specified normalized to the driver size with the ka factor, where ka is actually k * a.
f = c * ka / (2 * pi * a) = c * ka / (pi * Dd)
where
k = 2 * pi / lambda = omega / c, is the wave number
c = 343.7 m/s, is the sound speed
a = Dd/2, is the piston radius
It is necessary to look at k * a = ka because this normalizes the driver size to the frequency. For this reason it is easy to relate the ka-values to certain physical situations:
f4pi: ka = 1
f2pi: ka = 2
fmax: ka = pi
Some people specify fmax as the frequency where ka = 3, to follow the logic, while other people prefer ka = pi. I tend toward the latter because then fmax becomes physically related to the diameter of the cone.
If ka = 2 is chosen, then you get at good compromise where the directivity index is still low, but the limit frequency is increased significantly.
The 3 values of ka also relate to behaviour on the acoustic radiation load. At ka = 1 the load is mostly reactive but starting to turn toward a resistive load, at ka = 3 (or pi) the load is mostly resistive. At ka = 2 there is a turning point, which among other things alters the way the diaphragm is coupling to the air.
In other words, if you choose the limit frequency for the piston diameter instead of the piston radius then you get a realistic and "good" suggestion for the crossover frequency.
The assumption that the driver moves as a flat piston is not true. The real situation is (normally) that breakup modes will appear, and the (visco-) elastic diaphragm means that the effective piston diameter is smaller than the nominal size of the driver. The breakup has other sideeffects though, e.g. the sound starts further back in the driver, and there is a kind of horn effect from the shape of the diaphragm (as a whole). The horn effect has bad sideeffects on the sound quality. In other words, either the diaphragm is hard and the assumptions are true, or the diaphragm is soft(er) and progressively other sideeffects show up. This is why f2pi is still a good indicator of a crossover frequency, even though the initial assumptions are not valid.
Summary on the diaphragm-describing parameters:
f4pi, f2pi and fmax are all frequencies, which are basically giving you the same information, namely the area of the diaphragm, but each of the frequencies represents a new "event" - where something new happens in the mechanical system. It will always be so that f4pi is a smaller figure than f2pi, which again is a smaller figure than fmax.
The assumption that f4pi is the frequency where the system behaves omnidirectional also assumes that the driver is mounted in a box, and not in a wall. In this situation and in free space, then the -3 dB point will be at 180 degrees (ie. right behind the speaker).
To use any of the 3 frequencies as a clear indicator of good or bad crossover frequency can definately be debated. If directivity is not a concern, then remember that the breakup modes are strictly dependent on whether the air is capable of bending the diaphragm. How high you can go in frequency will depend partially on the choice of material. Nonetheless it can be seen by using modal analysis that the first breakup mode is often located around f4pi, say at a frequency within 10-20% of f4pi.
Talking breakup modes it must be realized that there are several kinds of breakup. One kind is due to bending of the material while another kind is due to streching/compression in the material. At higher vibration modes / frequencies the picture becomes more complex (mixed), but the first 3 modes are clean.
Measuring the frequency response with an accelerometer attached around the voice coil (or at the center) versus measuring with a microphone (nearfield) will show identical curves until the diaphragm is no longer a rigid piston, then a difference between the two curves will occur.
fpist is a different parameter though, which depends entirely on the depth of the diaphragm, which depends on engineering choices, but it is normally directly related to the cone bending stiffness - either as a function of material or geometry.
fpist can be a very small figure if the diaphragm is deep (indicating that the driver is for low frequencies only) or a very high frequency for a flat diaphragm. Theoretically, if it is a flat panel, then fmax becomes infinity.
This section became too large for this document and has been moved into a separate file, see XMAX.HTM for further information.
This frequency is the frequency used in DPC when calculating the value for Le and KLe. The value is given as:
Le = 1/(2*pi*fLe)*sqrt(Z10k^2 - Re^2)*1000 in milliHenry
KLe = Le * sqrt(2*pi*fLe)
The lowest frequency for measuring of Le should be:
fLe>= fmin^2 / fres
Where fmin and fres are characteristic points (local extrema) on the impedance curve.
This point of fLe means that the distance to fmin is just as large as the distance from fres to fmin. This means that the mutual influence is approximately equal.
fmin is the point where Ces (basically without influence from Les) and Le "resonate" to a local minima, ie. they have equal influence.
Subtracting inductance contribution from the measured impedance would give (almost) just as big an error on determining T/S parameters as if the Le data had not been subtracted. Only ALMOST as bad.
As a special detail, if your T/S data are known, and if you want to calculate Le at the fmin frequency, a local impedance minimum above fres, then you can use the following equation:
Le = 1 / ((2*pi*fmin)^2 * Ces)
where Ces = Mms / Bxl^2
Of course the Le value found is, as usual, only valid at the measuring frequency. A discussion on more accurate models of the voice coil inductance is found in MEASURE.HTM, section "Other models of Le."
The above simplifications do not hold water when the impedance notch at fres is small, because then Les and/or Res will still have a noticeable amount of influence at fmin on the level of Zmin.
All this fiddling back and forth basically is a result of one big headache - what to calculate first. The best option is to measure a lot of data and then do a mathematical curvefit based on the mathematical model (T/S) you wish to apply. For simple models this can be done in a straight-forward fashion, eg. in a spreadsheet.
It is my opinion, though, for simple calculations, that it would be better to keep fLe >= 2 * fmin^2 / fres, then the benefits of measuring Le and subtracting from impedance before calculating T/S parameters will be increased.
The parameters ZLe and Zres are both irrelevant, since they (Le versus Ces) will decline about equally at the respective ends of the frequency spectrum.
On the other hand one could say that the impedance curve is also characterized by fl and fh. Either of these could be chosen instead of fres. Choosing fl would be more discriminating (improve on the independence), whereas choosing fh would be more relaxing.
Based on the implementation of KLe, which is more accurate at fmin, one could write the following equation (possibly isolating any unknown variable):
2*pi*fmin = (2 / (KLe*Mms/Bxl^2) )^(2/3)
and given fmin an approximate value of Zmin can be calculated, but complex number calculations is required. Generally we have:
Z(f) = ZKLe + ZCes + Re
where
ZKLe = KLe * sqrt(i*2*pi*fmin)
ZCes = 1/(i*2*pi*fmin*Ces)
The value of fmin is approximate because though the derivation by Vanderkooy is much more accurate than the pure inductive value Le, this is not an exact science - more accurate models of the voice coil inductance are available, one of the best available is the model by J. R. Wright.
The calculation of fmin and Zmin from the above equation is not supported by DPC because I have found them to be way too inaccurate. An fmin value 100% off from the measured value is not unusual.
When Vanderkooy documented his analysis in JAES 1989, he showed the range of which the KLe model would work. With a normal loudspeaker magnet system I believe he found it to work down to 170 Hz. Below this frequency the system does not behave like a semi-inductor. The transisition frequency is of course greatly dependent on the actual magnet and voice coil system design.
W. Marshall Leach expands on Vanderkooys model, by letting the semi-inductor with its square-root behave with a power different from 0.5. This means that the semi-inductor no longer is required to increase with 3 dB/octave, but can be stronger or weaker, which gives room for better curve fitting - and one more parameter to regulate.
The J. R. Wright model for voice coil inductance (excl. Re) goes as follows:
Z(f) = Rem + i * Lem * 2 * pi * f
= Kr * (2 * pi * f)^Xr + i * Ki * (2 * pi * f)^Xi
As can be seen, even though the purely resistive part is excluded, a resistive part, Rem, shows up within the inductor at higher frequencies (starting a 0 ohm at DC). Besides the reactive part, Lem, is specified. In fact both Rem and Lem are frequency-dependent variables.
The Wright model is a quite hefty curvefitting, with 4 parameters - thought they are well under control with 2 dedicated for the real part and 2 for the imaginary part, both in an exponential fashion.
To measure and determine the four constants, Kr, Xr, Ki and Xi, one would have to measure complex impedance at 2 (high) frequencies, I suggest that they are eg. 1 octave from each other. Then:
log10 (Re(Z1)/Re(Z2)) log10 (Im(Z1)/Im(Z2))
Xr = --------------------- and Xi = ---------------------
log10 (f2 / f1) log10 (f2 / f1)
With this in mind it is possible to simply calculate K-values as follows:
Re(Z1) Re(Z1)
Kr = ------------ and Ki = ------------
(2*pi*f1)^Xr (2*pi*f1)^Xi
Theoretically one should be able to measure the complex impedance with the compare resistance method described in MEASURE.HTM, but Wright warns against this approach because small errors in the reading of the modulus can create large deviations in the imaginary part and therefore wrong modeling of the motor system.
The AES paper by Wright, though probably representing the best model available today, documents that his model may show errors below fmin (at Zmin), where the motor system becomes more pure inductive than rather semi-inductive. The model by Wright is very good for crossover designs, but less perfect for extraction of loudspeaker fundamental parameters incl. T/S parameters.
As defined in FORMULAS.HTM, Mms is the mechancial mass of the part of the driver, which vibrates, including air load on the diaphragm. Over time this mass has had different names, sometimes Mas (perhaps to imply that it is the mass of the system including acoustical loads).
The additional air load in a box is normally neglected. When measuring Vas it can be accounted for in an approximate manner, as described by L. L. Beranek, Acoustics. DPC supports this compensation in the measuring section (main menu item 1).
The question is - should you compensate or not? It depends. First of all I dislike that measuring driver parameters is mixed with intended application. Driver parameters should be measured without considering application - and be comparable across different electro-dynamic driver brands (and their individual application). On the other hand, if you have a specific application in mind, why not use this knowledge when measuring T/S parameters. Normally the reason for measuring T/S parameters is the following application in some kind of box/cabinet.
If your software for box calculations does not include an advanced model, then it may be better not to subtract additional air load during measurement of T/S parameters, because then the way you measured T/S parameters is closer to application, and your target alignment will be better approximated. For better approximation to the box mounting, even the free-air measurements could be made in the test box, but with the rear wall removed.
Mms can be split into a number of masses, which could be defined as follows:
Mvac is often named Mmd, but then it includes the weight of the voice coil (which is not what the name indicates. This is why I invented the term Mvac.
Mvc can be a valuable parameter, when combined with knowledge about the voice coil material. This gives an indication on short term power handling because the heat capacity in the voice coil can be calculated.
Furthermore the weight of the coil and the weight of the diaphragm is connected through the voice coil former, which can act as a spring between the two masses. This can create a resonance problem if the driver is poorly designed, which then shows up on the drivers frequency response.
Besides these mechanical parameters one can calculate the changes in Mair due to radiation load changes when the driver is loaded in a box. In this case the front and the rear of the driver diaphragm is specified separately:
DPC does not support the above mentioned masses, but radiation load corrections can be made during measurement of a driver with the closed box method. Since release 0.5.0 the Mair and Mvac parameters have been supported by DPC.
The air load, Mair, changes with the weather condition because Mair is dependent on the air density, rho. The way this is handled in DPC is by always giving Mms at the specified weather condition. If you want to calculate the new Mms value at a different weather condition, then all you have to do is set various parameters constant (like Cms) and let Mms (and fres) be adjustable. Then you exit and go into the Setup menu (menu 5 from the main menu) and further into the Air properties. Here you can change rho, either directly or by specifying weather conditions. When you exit and re-enters the parameter editor (menu 3 from the main menu), then the new data will be calculated, and this includes the new moving mass.