MEASURE.HTM --- Part of Manual for Driver Parameter Calculator --- by Claus Futtrup.
Created 20. August 1996, last revised 15. June 2004. Ported to XHTML 1.0 on 2. October 2004. Last modified 25. October 2004.

Table of Contents:

  1. How to Measure Loudspeakers for the Driver Parameter Calculator
  2. Measuring Sd with a ruler
  3. Mass load compensation
  4. Measuring Le
  5. Other models of Le
  6. Priority on nonlinearities

How to Measure Loudspeakers for the Driver Parameter Calculator

The following chapter will explain two methods for driver data calculation:

Please note that the Driver Parameter Calculator supports the Closed Box method as well as the Added Weight method - measure as explained below, but let DPC do all the calculations for you.

Measuring in a closed box, mass load compensation is explained. It is supported by Driver Parameter Calculator.

Furthermore, measuring Le will be explained, including a special semi-inductance model, named KLe. The described methods are also supported by Driver Parameter Calculator.

You will need to measure impedance. There are several methods for measuring impedance. With the equipment below you can do the following:

I heavily believe in the latter mentioned method, the constant-voltage and constant-current setups are basically only here for completeness, backward compatibility with other authors and for academic reasons.

To approximate constant-excursion will provide you with the most persistent data, and this is so because two of the most nonlinear factors in electro-dynamic loudspeakers are Bxl and Cms. Both of these are highly dependent on the excursion (other effects are secondary). When they are dependent on excursion, then you should keep it constant at different measuring-frequencies. This is discussed further at the bottom of this file.

Minimum requirements:

        The driver
        A closed box for the driver, or some mass to add to the diaphragm
        An amplifier
        A Sine generator + frequency display
        A voltmeter (which can measure AC and DC)
        An amperemeter for constant-voltage setup (AC and DC required)
        Some resistors (values must be known quite precise)
        A pocket calculator

Please do not trust the scale on the sine generator itself, unless it includes a real counter/frequency-meter. A printed ("rigid") scale on the generator is very inaccurate and generally completely useless.

If you have a computer equipped with a sound card, then it may be capable of substituting the amplifier, the sine generator + frequency display, the AC voltmeter (you will probably still need one to measure DC) and your pocket calculator.

First you measure the driver impedance at selected frequencies, the driver is in free air (no box, no reflecting surfaces). Admittedly, this is often exagerated, for simple impedance measurements a few reflecting surfaces will not harm. Do not put the driver on the floor, with either magnet or diaphragm pointing downwards, it will not work (not reliably, anyway). Mount the driver with some wire to the seat of a chair, or to your roof or in a door opening, whereever. It is very important that the driver is held very solid so that no movement of the driver will occur, only the diaphragm should be able to move, otherwise you will measure too low values of Zres. You must perform a very stable mounting job. Holding the driver tight with your hands and between your legs should work reasonably. It will increase precision if you mount the driver on a baffle because this will include more air in the moving mass. You could for example build the test box without a rear wall. Alternatively using a vice, make sure the loudspeaker does not vibrate either by using a vice on both sides of the driver or a vice on one side and then try to tension the setup with a supporting bar.

Be aware that DPC does not qualify the radiation mass load from the air.

DPC gives you the option to compensate for change of mass load when the situation is changed from free air to a closed box, if desired. In this respect it is not necessary to mount the driver on a baffle when measuring free air, but it is easier to measure this way and omit the mass load compensation.

The setups for impedance measurements are described below:

Volt/Amp measuring

This is the good old setup, with an amperemeter between the amplifier and the driver. Then you have a voltmeter between the speaker terminals. Also try the amplifier terminals - it will give slightly different result because the amperemeter does not have zero resistance. The truth is somewhere between the two, but probably closest to the first setup (the european setup, the other method is called american setup because scientists disagree about the methods).

        R = U/I  (U in volts and I in ampere)

This method is called "constant voltage" setup, and comes close to the conditions under which a driver will operate in a system. It is the best measuring method if the driver will never be mounted in a cabinet, ie. if the driver operates in free air (for dipole and perhaps also transmission line systems).

The excursion plot: At the low measuring-frequency (fl) the excursion is at its highest. When measuring at fres the excursion drops. When measuring at fh the excursion drops further. In other words, the excursion is not constant with this method.

Volt measuring

This method is the "constant current" setup. Here you mount/solder a known resistance between the amp and the driver. Preferably the resistor is around 1 kohm (20 times the max. expected value to measure). This resistor makes the amplifier a "constant current source." You use a known resistor (must be known very accurate) to calibrate your voltmeter reading. If the resistor is 50 ohm, then you let your voltmeter read 50 millivolt. This gives you an almost constant current of only 1 mA (milliampere). Substitute the known resistor by your speaker and any reading on the voltmeter will (approximately) give you the impedance of your speaker. This is what most books recommend.

The current is very low and very far from normal (music) application. I recommend to increase the current 10 times. This means that you should let 500 millivolt show up over the 50 ohm resistor. Still not much, but at least reasonable. Now your power amplifier must be able to provide 10 volt over the total load (including the 1 kohm resistor), which most amplifiers can do, especially since it is only requiring 0.1 watt.

The excursion plot: At fl and fh the excursion is very low, but at fres the excursion is suddenly many times higher. This gives excessive (unnecessary) basis for nonlinearities to show up, and the most critical parameter, fres, is measured under different conditions than the other readings. The reason why it usually is not so bad is because the current is very low, which keeps the nonlinearity changes at a very low level - only with the measurement at fres there is a risk of reaching up to the nonlinear level of excursion. This kind of measurement is far from any normal kind of application.

The advantage of this method is that you only need a voltmeter, but I think this is being too cheap.

The above description of the constant current setup is also called the voltage divider in other litterature. Any resistor could be used as a voltage divider, but most books recommend around 1 kohm, and something like 10 mA signal. A small signal model like this is only useful in an advanced model of the driver, otherwise forget about it and use a larger signal, which comes closer to real life applications. This limitation of Rs = 1 kohm is the normal choise because a limited voltage of 5 - 7 volt is required, and because of the limits on the AC voltmeter. If the voltage is increase to 20 - 30 volt, then 10 kohm is possible with only 0.1 W power consumption.

In Testing Loudspeakers, by Joseph D'Appolito, the error by this method is analyzed. If a 1k ohm resistor is used, and the speaker reaches approx 52 ohm at fres, then the reading is 50 ohm, which gives a 4% reading error. If the speaker reaches 106 ohm, then the reading is 100 ohm, a 6% reading error. The error can of course be compensated for by using the "compare resistor" technique provided that the voltmeter is accurate in both measuring ranges.

A true constant current amplifier could be applied, with the same negative sideeffects, but also with some advantages (eg. not sensitive to bad contacts because the current is simply raised), mostly simplicity. This is a sophisticated technique, not available to common people, and in my opinion completely unnecessary for measuring T/S parameters.

With a sound card and a series-resistance, eg. 10 ohm, you could actually measure drivers with an approximate "true" constant-current technique because one of the microphone inputs, which monitors the voltage over the known resistor could be held constant by a compressor function in the software.

A similar analog setup could be used, if you know how to do it.

Compare resistances

Here you mount/solder a known resistance between the amp and the driver. Preferably the resistor is around 10 ohm (around the range where you expect the driver impedance to be), 20 ohm is fine too, but principally any value will do, as long as you know it quite precise. The disadvantage being that you may have to switch operating range on your voltmeter, which gives increased inaccuracy. I recommend the 10 ohm resistor partially because it is easy for scaling in the equation below, besides this value has proven to be very suitable for several 4 ohm drivers when it comes to the requirement for linear excursion.

The resistor must be known as precise as possible, preferably less than 1%, see RESISTOR.HTM for inspiration. I have been using a resistor determined to 10.02 ohm within a precision of 0.2% which gives very good results. I can recommend this.

As you may be able to imagine, this is not a constant-voltage measurement (where the impedance is kept very low, preferably below 0.5 ohm) neither a constant-current (where the impedance is kept very high, preferably above 1 kohm). It is something in between. For this reason we get the following description of the excursion plot:

At fl the excursion is perhaps the highest, perhaps a bit lower than fres, this depends on the size of the resistor (example with 10 ohm), the Re of the driver and its other parameters. The excursion may actually be about the same at fl and fres. At fh the excursion drops a bit, but it has not dropped as much as it will from a constant-voltage setup (and much less than from a constant-current setup).

Perhaps the slight drop of excursion at fh is not so bad at all. I have tried to make constant-excursion measurements all the way up in frequency (up to above 100Hz), and it becomes very difficult at high frequencies because a lot of power is required. The power will provide an increase in Re, which will smear the measurements a bit. For this reason I like the compare-resistor measurement method.

When measuring the resistance of the driver you simply measure the voltage over it and the voltage over the resistor as well (Vknown), divide the two voltages and multiply by the resistor value (in the above example 10.0 ohm).

        R = Rknown * V/Vknown

This method can be expanded to calculate complex impedance (ie. to include phase), as described by J. L. Markwalter in Speaker Builder Magazine 2/94. In this case you must measure V and Vknown, but also a third meaurement is required, which is the voltage over both loads together, the total voltage Vt. See the following textbox:

     |                                                                    |
     |  If we would like to measure the phase of the impedance, or        |
     |  alternatively to split the impedance up into its resistive and    |
     |  reactive components (ie. complex impedance), then in both cases   |
     |  we have two numbers, which means the impedance can be             |
     |  represented by a socalled vector (an arrow) in a two-dimensional  |
     |  way. For this purpose the total voltage over both components      |
     |  must be measured (Vt). With three vector-lengths (V, Vknown and   |
     |  Vt) only one triangle can exist, and its angles may be            |
     |  calculated using socalled cosine relations:                       |
     |                                                                    |
     |  cos beta = (V^2 + Vknown^2 - Vt^2) / (2 * V * Vknown)             |
     |                                                                    |
     |  With beta being a known angle (given by arcus-cosinus to the      |
     |  above), the desired angle (the impedance-angle) can be            |
     |  calculated as:                                                    |
     |                                                                    |
     |                  alpha = 180 - beta                                |
     |                                                                    |
     |  To find the resistive and reactive parts you simply calculate:    |
     |                                                                    |
     |          R = Z * cos alpha, X = Z * sin alpha                      |
     |                                                                    |
     |  It can be quite difficult to get correct measurements and         |
     |  calculations of phase around 0 degrees.                           |
     |                                                                    |

You must measure the driver so, that the impedance curve comes very clear. This implies that you must make absolutely sure that the following data are very well determined:

Measuring Re is like measuring impedance, any of the 3 above methods can be applied. The easiest is to use compare resistance and apply a DC voltage either from a battery or an amplifier (if it can do DC). Re is desired with as many numbers as possible. 3 numbers (normally 2 decimals) is desired for accurate (better than 1%) measures of the T/S parameters.

fl is the frequency where Z is -9dB of Zfres. You can find this frequency by the following method: Zl = Zh = sqrt(Zfres*Re).

Check that sqrt(fh*fl) = fres. BTW, this is pretty hard to get exact the first time. fres can be positioned a little bit to one of the sides on the impedance peak (usually to the left) due to hidden reactances, but also fl and fh can be tricky to determine exactly. The above calculation must fit pretty well, or your data are not good enough! Even if you try hard, you will find some skewness, which is the result of nonlinearities, eg. nonlinear magnetic field (Bxl), nonlinear suspension (Cms), effects from the voice coil inductance Le and/or the fact that the loudspeaker is driven with a non-constant excursion. Skewness less than 1% of fres or less than 1 Hz normally means that you are doing okay.

Comparing constant-current and constant-voltage drive (which is given by the amplifier you use), the latter mentioned gets closer to the desired constant-excursion. Using constant-current drive is sometimes easier, though. In my opinion the compare resistance method is the best method, though.

In Driver Parameter Calculator you can specify -9 dB points as well as -3 dB points. Which one is most accurate (ie. has least skewness problems) depends on your equipment. If you can only determine frequencies as a whole number (no decimals), then the distance between the -9 dB points will make this method more accurate, even if the skewness will be more problematic here. If you can read frequencies with decimals, then the -3 dB points might give you better measurements (less skewness this close to fres). Driver Parameter Calculator supports both readings, and you can choose which one you want. If you choose to specify both, then Driver Parameter Calculator will select the set of specifications, which has least skewness, based on a logarithmic frequency scale.

For a 5" - 8" Woofer the following frequencies usually displays the impedance curve quite well:

        15 Hz, 20 Hz, 25 Hz, 30 Hz, 40 Hz, 50 Hz, 65 Hz, 80 Hz, 100 Hz.

These frequencies are separated approximately one terz each. A more accurate table could be provided by first locating fres and then measure at the following frequencies:


When the volt (and perhaps ampere) are measured at these frequencies, calculate the impedances and draw a curve on some semi-logarithmic paper. Now try to determine the above frequencies (fres, fl, fh) and Zfres. Try to measure with more precise frequency steps (finally 1 Hz steps), if possible, and recalculate to make sure that you have determined the data with adequate accuracy.

Now calculate Qm = fres*sqrt(Zres/Re)/(fh-fl) and Qe = Qm/(Zres/Re-1).

The method explained above uses the -9 dB points on the impedance curve.

Qm can alternatively be calculated by the following formulas:

        find                     Res = Zres - Re

        calculate Z-3dB          Z-3dB = sqrt( Res^2/4 + (Re + Res/2)^2 )
        (also named Zl and Zh)

        find frequencies fl and fh around fres, where Z = Z-3dB = Zl = Zh

        calulate Qm as           Qm = fres/(fh - fl)

This is the method using -3 dB points on the impedance curve.

The calculation of Qe is unchanged.

Driver Parameter Calculator will handle the calculation of Qt, but for completeness:

        Qt = Qm * Qe / (Qm + Qe) = 1/(1/Qm + 1/Qe)

        Qt = ------- * sqrt(Re/Zres)
             fh - fl

Some people seems to be of the opinion that the speaker should be measured "warm," eg. after being in operation for say 12 hours with some power input. I do not recommend this because during this measuring procedure a very low amount of power is sent through the speaker, which means that the speaker will become colder - and the parameters will drift during the measurings. The reason is that the resistance of the voice coil (Re) changes and as you can see Re takes part of these calculations. You can always calculate a "warm" Re value from the "cold" Re value and rerun the above calculations, eg. you can try to increase Re with something like 10 percent.

Another concern, though, is the fact that the suspension of a brand new driver will be a bit too stiff. Various amounts of socalled burn in or break in is specified in the litterature. The fact is that Cms (and therefore fres) changes with the amount of exersize. I do not recommend longish break in, but at least you should push the diaphragm in and out as much as you can without getting violent and without breaking the driver, of course. If desired you can follow on with, say 30 minutes at a low frequency (either fres, or alternatively eg. 10-25 Hz for woofers) at moderate to high excursion / cone travel. Again, make sure that you do not bottom the driver (it will make a bad sound anyway) because it will in the long run most definitely damage it.

To illustrate this one can measure the driver out-of-the-box, then apply some power, for woofers eg. 10 Watt at eg. 20 Hz, then for 1 min. and 2 min. and so on you can re-measure fres. My colleague Henrik Adsersen has tried this with a Dynaudio 30W100 with a 17 Hz spider. He applied 10 Volt. He measured the following:

0 min.  19.83 Hz
1 min.  17.85 Hz (dropped 1.98 Hz)
2 min.  17.53 Hz (dropped 0.32 Hz)
4 min.  17.23 Hz (dropped 0.30 Hz)
8 min.  16.94 Hz (dropped 0.29 Hz)
16 min. 16.75 Hz (dropped 0.19 Hz)
32 min. 16.58 Hz (dropped 0.17 Hz)
64 min. 16.20 Hz (dropped 0.38 Hz)

Using more than 64 min. can change the resonance further, but not in a significant way.

When he re-measured the driver the next day he got 16.68 Hz because some of the stretching gives a non-permanent change. The spider has settled again. We can conclude that this particular driver has a resonance frequency of about 16.5 Hz.

If you plot the above table in a graph, with 16 - 20 Hz on the y-scale and 0 - 64 min. on the x-scale, then you will see a nice plot of how the resonance frequency of a driver typically will change with the amount of applied burn in. The scaling will be different for different drivers, of course.

The resonance frequency drops rapidly at the beginning, then the changes becomes more rare. I believe the above measurements of fres were done with a reasonably high cone excursion, so even the first measurements is not "without" any exercise at all - in fact the resonance frequency was continuously dropping while he was measuring fres.

To determine Vas you must select one of the above mentioned methods, either closed box or added mass:

Closed Box

Put the speaker into the closed box. This closed box must be air tight! Make sure that the cabinet is air tight around the wires too, as well as around the driver. A good way to get around this is to mount the driver so that the front of the driver is playing into the box and the magnet being visible to you (ie. pointing out of the box).

An easy test is to apply a low frequency, eg. 10 - 15 Hz, and large excursion. If you can hear air whistling from the box or glue joints, then the box has too much leakage for accurate T/S parameter estimation.

Preferably you construct a test-box for all future loudspeaker projects. This box is around 10-30 liters (for woofers) and with capability of putting new front plates on top of each other for measuring of drivers of various size. Lets say the box is 30 liters, and when you need a smaller one, just fill some bricks or books or whatever into the cabinet - subtract the volume from the box-size in the calculations below! The volume we will call Vb.

It is normally preferred that the box volume Vb is made smaller than the expected Vas value, Vb < Vas or to be more accurate sqrt(Vas/Vb + 1) > 1.5, which gives Vb < 0.8*Vas, to alter fres sufficiently (in this case at least 50% up in frequency) for an accurate measurement of the T/S parameters. On the other hand, if you are going to use a larger volume in your final project, then you can try to measure the parameters in this volume.

You must know the exact box-size within five deciliter, because this will give you only 2% inaccuracy on the given data, which is acceptable. With inaccuracy on the measuring equipment we are reaching the upper limit of 5% depending of the quality of the measuring equipment, of course.

Now find the new fres (fcb) and Zres (Zfcb). Do not forget to determine fl and fh too, and try to make the three frequencies fit the above square-root for more precise calculation. Calculate the new Qm (Qmb) and Qe (Qeb). The new fres we will call fcb. Vas can be calculated as follows:

        Vas = Vb [(fcb*Qeb)/(fres*Qe)-1]

A method taking leakage losses into account, the Carrion-Isbert method, is used by the software. To do the same you must calculate the following figures:

        Res = Zres - Re;
        Ces = Qm/(2*pi*fres*Res);
        Les = Res/(2*pi*Qm*fres); --- same as : 1/((2*pi*fres)^2*Ces);
        cA  = Zfcb - Re;
        cB  = (fcb/fres)^2 - 1;
        cC  = 1/cA - 1/Res;
        cD  = 2*pi*fcb*Les/cB;
        Lb  = Les/cB*1/(1+(cD*cC)^2);
        Rl  = cC*cD*2*pi*fcb*Lb;
        Cmb = Vb/(rho*(c*Sd)^2);
        Bxl = sqrt(Lb/Cmb);
        Cms = Les/(Bxl^2);

        or more directly: Vas = Vb*Les/Lb;

Taking leakage loss into account does not imply that you can use a lossy box for the measurement, since this will under all circumstances reduce the accuracy of your measurements. The parameters of the driver will become smeared by a high leakage loss.

DPC also gives you the leakage loss as a Q value. The equation is available from several sources, but I picked the one in Jeremy Bensons A.W.A. Technical Review:

        Ql = Rl * Cmb * fcb;

As with bass reflex cabinets, Ql should be kept as high as possible. I assume leakage in closed boxes should be kept above 5, just like with bass reflex boxes, though more leakage is less devastating for the closed box system than for the bass reflex system.

Also the Carrion-Isbert method does not include absorption losses (the box-Q, it is often called Qb), which means that you get the most precise result if you make the box walls as reflective as possible and use absolutely no lining or damping inside the box. Admittedly this is sometimes exagerated. It is not a huge problem if a little bit of damping material appears in your test box, and this may make the measurements easier.

Anyway, I have never seen any focus on absorption loss. I think leakage loss will normally be a larger problem. Richard Small qualifies typical values, where Ql around 5-7 and Qb around 30. This means that the effects of leakage are dominating.

Most work on losses in boxes revolves around bass reflex systems, and all the above is taken from such articles. I cannot see that Ql and Qb values should be affected by the type of cabinet, but experience may show something else.

As you can see, more calculations are involved, but you calculate both Bxl and Cms (given one, Driver Parameter Calculator would have calculated the other for you). rho (=1.20 kg/m3) and c (=343 m/s) are involved in the calculation of Cmb. This increases inaccuracy over the Added Weight method, or you must be capable of determining them with good precision.

A method for measuring in a basreflex box has been explored by Thiele and Bullock as well, but will not be displayed here. Such a cabinet only very hardly gets appropriately air tight, but many measurements may give you a good idea of the size of the leakage.

Added Weight

Do not put the speaker in a box, but add a known mass to the diaphragm, we will call it M. This mass must be known within 0.1 gram for very precise calculations, a letter weight of good quality may be sufficient. Only the new fres (fresm) must be known, but do not make life easy! Try to measure fl and fh as well. For high precision calculations the reading of fres should be very accurate, like within 0.1 Hz if measuring a woofer (50 Hz +- 1/500*fres). Then calculate Cms as follows:

        Cms = 1/(4 pi^2 M) * [((fres+fresm)*(fres-fresm))/(fres^2 * fresm^2)]

To calculate Vas you can use the formulas provided by Driver Parameter Calculator, but preferably you let the calculator calculate Vas for you:

        Vas = rho * c^2 * Sd^2 * Cms


    rho = mass density of air (typically around 1.20 kg/m3)
    c = speed of sound in air (typically around 343 m/s)

As you can see Vas is dependent on several factors, like the speed of air (dependent on the temperature, the air pressure and the humidity) as well as the mass of air (dependent of the previously mentioned factors as well). Therefore Added Weight is the prefered method in production lines, but it is not easy to provide the diaphragm with an added weight. There are some factors to be considered:

The added mass does not need to be "tightened" to the diaphragm as long as the diaphragm is not moving visibly, but the mass must lie stable on the diaphragm. It becomes increasingly difficult to mount the added mass in a stable way, when the excursion is increased.

This implies that the driver is in a vertical position, with the cone moving up and down. This can give a slightly higher fres (especially with the mass attached) than the fres you would get when the driver is playing horisontally. The reason is suspension "sagging" and DPC supports calculation of Xsag in the parameter editor to control that this mounting is feasible. It is preferable to mount the mass tightly and measure with the driver mounted horisontally, but it is not always possible.

When adding mass to the diaphragm you may run into trouble because the added weight can go into resonance with your measurements and provide you with contact-resonances that will spoil your measurements (if their contribution to the total measured value is significant). Always try to attack the added mass close to the voice coil (or dustcap) to prevent running into problems with the driver diaphragm.

The added weight must alter the parameters of the driver somewhat, or the measurings are going to be very inaccurate. If you can determine fres very accurately then a small altering of fres can be accepted (say accurate measurement down to 1/100 Hz), but if you do not have accurate reading of frequency (say only in Hz) then you need a relatively large altering of the mass.

For normal DIY speaker building equipment I recommend that the least it must alter is 25% (I mean, the new fresm <= 0.75fres), but 50% is better, if the driver suspension can carry the load, which means that the mass of the added weight should be around 3 times the mass of the diaphragm itself. 10--50 grams is usually advisable.

Some people prefer to have the added mass about the same as the moving mass of the system. This leaves too small a change in fres in my opinion for low-quality gear to accurately measure driver parameters. You would have to read the frequency with perhaps 2 decimals (ie. 1/100 Hz).

This method is usually recommended by "experienced" people because you do not have to operate with acoustical parameters. The method is useless when it comes to very soft suspended drivers, eg. the large 12" Peerless woofers. Some drivers are not made for playing up/down, the diaphragm and voice-coil will sink down a couple of millimeters and alter the parameters of the driver. Putting an additional mass on the diaphragm will make the driver more non-linear. In this case you must use the closed-box method.

I personally like the closed box method because it comes closer to the environment that the driver usuallly is going to be applied to later (therefore, use a box as close to reality as possible). Unfortunately the closed box method cannot be applied to drivers which have a closed rear, like many midrange drivers. In other words, the added weight method is great for smaller drivers (small Sd value) and/or drivers with stiff suspension (small Cms value), while the closed box method is great for larger drivers (high Mms value) and/or drivers with soft suspension (high Cms value). These "higher" and "smaller" expressions are very elastic, and somewhat dependent on the equipment you have. I recommend that you try to determine what seems likely and unlikely results yourself.

                       |     Closed box     |    Added Weight    |
        |              | Close to           | No test box        |
        | Advantage    | application        | Easy setup         |
        |              |                    |                    |
        |              | Acoustic mass will | Precision required |
        | Disadvantage | change in a box.   | Very sensitive to  |
        |              |                    | inaccuracies       |

Notice: With the implementation of closed box mass load compensation the disadvantage of the closed box method can be almost entirely neglected. The closed box method then becomes the method to be preferred no matter which intended application.

Besides, building a closed box, perhaps several closed boxes, might be more costly than having a couple of sets of magnets. I do not really think that cost can be considered a serious issue. Having a closed testbox sitting in your basement, garage or whatever may be considered less handy than a couple of small magnets.

Both methods

If both methods (Closed box and Added weight) are applied, it probably (most likely) means that the equations becomes inconsistent, since they are overdetermined, which is most certainly detected in the parameter editor.

One solution is to average the Mms value from the two methods, and then "delete" the calculated values of Qe, Cms and Vas. This could be like minimizing the errors, or getting an estimate of the inaccuracy involved in the measurement.

Another solution is to "free" one parameter, Sd, so that the measured values of Vas and Mms can match exactly. This is a method of measuring Sd, though quite indirect comparing to the normal measurement with a ruler. Hopefully the Sd-value by this method closely matches the expected value. The equation for Sd is:

        Sd = 2*pi*fres*sqrt(Mms*Vas/rho)/c

The equation can be derived from 2 equations in FORMULAS.HTM, by eliminating Cms from the relations Sd, Vas, Cms and Cms, Mms, fres. The above equation is therefore not included in FORMULAS.HTM.

Notice that this calculation of Sd is legitimate, also when the Carrion-Isbert method for leakage loss is applied, but it requires recalculation of the driver parameters Bxl and Cms as well as the box leakage loss Ql. A new leakage loss value is not calculated in DPC because it is not stored for future reference because it is not a driver parameter, just use the previously given Ql value to indicate the amount of leakage, or rerun the Closed Box method entirely, with the new calculated value of Sd.

Measuring Sd with a ruler

The normal method for calculator Sd is to apply a standard ruler. You place the ruler on the top of the diaphragm/surround and measure the diameter Dd to the middle of the surround. Then Sd = (Dd/2)^2 * pi.

In physics we have been taught that a helix shaped coil spring, where one end is attached and the other moving freely up and down (eg. with a moving mass at that end), one third of the coil mass should be added to the moving mass. A similar rule may apply for cone surrounds.

This means that, to get a more accurate measure of Sd, one should measure the inside (by the diaphragm) diameter ID and outside (by the basket) diameter OD of the surround roll. From this you can calculate the area of the surround alone:

        Sd_surround = Outer_area - Inner_area
        Outer_area = (OD / 2)^2 * pi
        Inner_area = (ID / 2)^2 * pi

The moving part of the surround then is

        Sd_moving_surround = (Outer_area - Inner_area) / 3

The total moving area, diaphragm including surround is

        Sd = Inner_area + Sd_moving_surround

Unfortunately, when I have tried this more advanced method, I have not found much deviation from the simpler method initially mentioned.

Perhaps with the more extreme subwoofers, where the surrounds can become a large part of the radiating surface, there is a larger (noticeable) difference in calculations. In this case you could try the method mentioned in the previous section for calculating Sd based on measurements using "both methods" (closed box and added mass).

Mass load compensation

As mentioned above, the closed measuring box will change the acoustic air load on the diaphragm. This change of mass can be compensated for, if desired, so that the parameters derived fits a driver in free air perfectly.

I have implemented 2 methods, which I have simply named method 1 and method 2.

Method 1

This method takes advantage of the fact that a change in fres should also change Qe. This means that the method requires additional impedance data for the in-box situation, for calculation of Qe in the box, named Qeb. DPC gives you the option to specify -3 dB points as well as -9 dB points exactly as with the calcuation of Qm and Qe.

The change of Mms is specified with the following ratio:

        Rm = fres * Qeb / (fcb * Qe)

The new moving mass is given as Mms*Rm, but a calculation of the mass itself is not necessary, since the additional mass load can be removed from the in-box resonance frequency directly:

        fcb = fcb_measured/sqrt(Rm)

With this new fcb value, the Carrion-Isbert calculations are rerun.

Method 2

This method is a more "direct" method, taking advantage of approximate theories on the radiation load provided when the driver is baffled. The area of the baffle must be specified, as Sbaffle. Already given the area of the diaphragm, Sd, the following ratio describes the radiation load:

        Km = 10^-(0.462 * Sd / Sbaffle + 0.057)

The Km "ratio" describes how much radiation load should be included on the rear side of the diaphragm (the side toward the inside of the box).

        Mmr_rear = 0.667 * Km * Sd^1.5

The radiation load on the front is:

        Mmr_front = 0.408 * Sd^1.5

The total radiation load in the box becomes:

        Mmr = Mmr_front + Mmr_rear

As opposed to the free air radiation load:

        Mair = 8/3 * rho * (Sd/pi)^1.5

Based on these calculations the mass load change, Mch, can be calculated as:

        Mch = Mmr - Mair

Notice that Mair < Mmr < 2*Mair, DPC checks this for you in a similar equation: 0 < Mch < Mair. Please also notice that the model does not cover all kinds of special cases. It only deals with a traditional closed box. For example a special case like in-wall box or on-wall box or large open baffle mounting gives a different radiation load, which will be something like Mmr = 2*Mair.

The Mmr_front value is only reasonably accurate for medium sized boxes, eg. speakers less than 230 liter, which should cover most cases.

Based on the calculation of Mms, a well known equation, which is including the extra air load, we have:

        Mms = 1/((2*pi*fres)^2*Cms)

The new, adjusted Mms value, where the extra air load is subtracted, becomes:

        Mms_new = Mms - Mch

And the new, compensated in-box resonance becomes:

        fcb = 1/(2*pi*sqrt(Cms*Mms_new))

With this new fcb value, the Carrion-Isbert calculations are rerun.

Besides the two methods I have implemented a third option, where both methods are executed, and an average of the two results is calculated:

        fcb = (fcb_method1 + fcb_method2)/2

If you have had success with both methods, which implies that they are in agreement, then the accuracy of the calculations can be improved by utilizing the average of both methods.

At the end you will be presented with a nice information screen, where all fcb values are displayed, for your information. If something looks suspicious, ie. the two methods are not in agreement, then you will have to decide which one of the two methods is wrong.

I such situations, all you have to do is to re-enter menu item 2, ie. calculation of Bxl, Cms and Vas in a closed box. Reset the fcb value to the measured value, and exit the menu. You will again have the option to choose either method, both methods, or exit without compensation. Choose as you please.

Both methods are in compliance with my sources:

L. L. (Leo Leroy) Beranek, Acoustics, McGraw-Hill, 1954, later published by American Institute of Physics, Inc. 335 East 45 Street, New York, New York 10017, for the Acoustical Society of America, 1986 edition (2. edition), second printing 1987, ISBN 0-88318-494-X.
For the Sbaffle method see chapter 5 and chapter 8 (page 216ff).

Joseph D'Appolito, Testing Loudspeakers Audio Amateur Press, Publishers, Peterborough, New Hampshire, 1. edition, 1998, ISBN 1-882580-17-6 See page section 3.1.2, page 38, for a description of the Sbaffle method (method 2). See page 40 for a description of the Qb method (method 1).

I have thoroughly tested the code, and made appropriate limitations to prevent division by zero and other runtime errors. If any anomalies are observed, please contact the author.

Measuring Le

When measuring Le, the result is dependent on what frequency is used when it is measured, because the inductor is lossy. For making a meaningful value, it is best to measure it at a frequency, where you would want to conjugate the impedance to linear.

This could be around the upper limit frequency of the driver (eg. for a 6,5" driver around 2.5-3.5 kHz). Alternatively where you want a low impedance in the filter, which would be at the tweeter resonance (eg. 1kHz for a 1" tweeter).

Due to this weird situation with several desired measuring points I recommend that the value entered into the Driver Parameter Calculator be calculated at a relatively high frequency (which is standard around the world), namely 10 kHz and that the speaker builder himself recalculates this value, if he/she want's an optimum conjugate circuit at another frequency. 10 kHz should be high enough to suit most drivers, also it is high enough to get rid of nonlinear effects from the drivers suspension.

This measuring method uses the same setup to measure impedance, as described earlier. Then find the impedance at 10 kHz (called Z10k) and calculate Le = 1/(2*pi*10000)*sqrt(Z10k^2 - Re^2)*1000 in milliHenry.

This method is somewhat inaccurate, but considering the easy setup and the above discussion on preferred measuring frequency, I think it will suffice to keep a database at a high quality. Le is an informative parameter only, not used for any of the interrelations. For this reason more accurate data, or data calculated at another frequency are welcome as well (something is better than nothing).

DPC supports measurements / calculations with any frequency of your choice, this frequency is named fLe (the frequency at which Le is measured), for your information it is stored in DPC.

When selecting a frequency, it is recommended never to choose a frequency lower than : fLe = fmin^2 / fres, preferably at least twice this value should be chosen. fres is the resonance frequency, whereas fmin is the frequency above fres where the impedance is at minimum (this minimum immpedance level is named Zmin). The above equation represents the idea that the distance from fres to fmin should be the minimum distance from fmin and to the measuring frequency fLe, because fmin is (approximately) the point where Ces and Le "resonate" to a local minima, ie. they have equal influence. See DATAINFO.HTM for a further discussion on fLe.

Other models of Le

Other, more accurate methods:

One is a quite accurate empirical model by Wright, but this method cannot "associate" the found data to a single Le value (it would be frequency dependent) and is therefore not useful with the Driver Parameter Calculator (considering its purpose, on the other hand it is a very useful method for filter optimization software). The model by Wright is shortly described in DATAINFO.HTM, section fLe.

Another consist of an electrical equivalent circuit of three components + the DC-resistance, by Victor Staggs. This method could be useful for design of conjugate circuits, but requires (to the best of my knowledge) curve-fitting with a numerical optimization routine.

This method can be generalized with 5, 7 or more components until a fit within the desired frequency range is obtained.

A model by Vanderkooy suggests that for hi-fi units one could assume 3 dB increase from the coil. Such an assumption could simplify the numerical calculations (from second method above). Driver Parameter Calculator supports this method.

Normally an inductor is given by L = z/(2 * pi * f) which gives z = L * 2 * pi * f. Vanderkooy gives a different equation : z = K * sqrt(2 * pi * f) (+Re, of course) for a semi-inductor (half inductive component) in a loudspeaker magnetic gap. K is the value of the component (to replace / substitute the traditional Le value). Notice that the semi-inductor K has a different unit than an inductor, it is H / sqrt(s), or H*sqrt(Hz) instead of simply Henry (or milliHenry).

This information is based on old studies and well founded in physics, and it was actually known facts before he wrote the article, it just happended that (almost) everybody had forgotten about this old research. The model is less accurate than the other models (which are both empirical / experimental), but has the a big advantage that only one parameter is given (whereas the other models require 3 or 4 parameters).

If a given value for Le for a given frequency (named fLe) is available, then the new parameter, which I will name KLe may be calculated as follows:

        KLe = Le * sqrt(2 * pi * fLe)

Example, fLe = 10 kHz, Le = 0.8 mH:

        KLe = 0.8 / 1000 * sqrt(2 * pi * 10000) = 0.2 H*sqrt(Hz)

You can try to compare the Le model and the KLe model by calculating their respective impedances at 10 kHz, where they should be equal. They both give 50 ohm. The KLe model will provide you with better impedance data at lower and higher frequencies.

DPC supports the model by Vanderkooy, both measurement section and calculation section. You can read more about this in DATAINFO.HTM in the fLe section.

A fourth model on Le exist, made by W. Marshall Leach. This model expands on Vanderkooys model, so that KLe = Le power(2 * pi * fLe,n). It is not only a squareroot (like when n = 0.5), but n can be of any value between 0 and 1. If no=0 the inductance is totally lossy (purely resistive) and if n=1 then the inductance is without loss (purely inductive).

This means that the model by Vanderkooy is expanded from one parameter into two parameters, KLe and n. You could name "n" the loss factor.

Leach goes one step further and proposes a calculation method for the 3-parameter model by Victor Staggs, and an easy method for approximating the three electrical equivalent circuit components based on only the two input parameters. This is quite clever, since it only takes two input parameters (and the frequency range you want to approximate, specified as the lower frequency f1 and upper frequency f2) to specify the 3-parameter model by Staggs.

The reason that Leach can tie the 3-parameter model down into 2 variables is because he applies a restriction, namely that the slope of the curve is constant. When applying the 3-parameter model directly (raw, so to say), the slope of the curve can vary over the measuring range of the voice coil inductance. This should make the 3-parameter model more adaptable to input of any kind, but apparently Leach has found that the slope is constant with most real-world speakers.

To achieve KLe and n in the first place, you must support the model with quite a few measurement points (impedance magnitude and phase), and make a numerical approximation to those points, in other words optimization / curve fitting is required.

The model is attempted to be expanded, so that the lossy voice coil is simulated by 5 instead of 3 components, but the example shows that this larger model is usually not required (at least not when working with normal drivers in the normal frequency range).

With the restricted 3-parameter model by Leach, he has found that 0.5 < n < 0.7 are typical values, unlike the conclusion by Vanderkooy.

The model is not suitable for software like DPC, since it requires the user to input quite a bit of impedance data, not just single-point measurements.

If I was to make a software that reads a file with impedance measurements, then my software would attempt to optimise the entire model to the measurements.


J. R. Wright, An Empirical Model for Loudspeaker Motor Impedance, JAES vol 38, no. 10, October 1990, pages 749-754 + comments JAES vol. 40 no. 1/2 January/February 1992, pages 42-43.

Victor Staggs, Exploring Loudspeaker Impedance, Speaker Builder Magazine, vol 15, 5/94 pages 28-39 + 63-64.

W. Marshall Leach, Loudspeaker Voice-Coil Inductance Losses: Circuit Models, Parameter Estimation, and Effect on Frequency Response. JAES vol 50, no. 6, June 2002, pages 442-450.

J. Vanderkooy, A Model of Loudspeaker Driver Impedance Incorporating Eddy Currents in the Pole Structure, JAES vol 37, no 3, March 1989, pages 119-128.

Priority on nonlinearities

Drivers are inherently nonlinear. Thiele knew it, Small knew it. Today we have quite some insight in the way these nonlinearities affect driver T/S parameters.

The two most significant nonlinearities are only dependent on cone displacement and in this respect static. They are 1) Force factor (Bxl) and 2) Suspension compliance (Cms) in order of priorities - Bxl is more important than Cms which can also be seen from how often Bxl occurs in the various equations in DPC - 10 times, whereas Cms only occurs 5 times.

Since they are displacement-related it is the most natural choice to keep displacement constant for the frequencies where you measure T/S parameters, ie. when reading impedance data for fres, fl and fh. Only this way will the data be taken under equal conditions.

The secondary nonlinearities are less understood, and they are friction related, or visco-elastic related (creep). They depend on the speed of stretching and contraction, or the air flow speed (aerodynamic drag). One effect is material stiction (like a static friction - it is the force required to cause one body in contact with another to begin to move). One other effect is thixotropy (a gel which becomes fluid when stirred/shaked/disturbed, and freezes back into a gel when the disturbance is stopped). Many weird effects arises from the use of textiles, plastics etc. (eg. the resin used in spiders). These nonlinearities have not been well understood/weighted in the field of acoustics, but as a mechanical engineer I can understand and relate their secondary significance.

The secondary effects will only be of a reasonable (not neglectable) importance if one is trying to simulate a driver, and wants the response to be very correct in the time domain. For T/S parameters I would not put significance to these effects. They are quite difficult to measure, and to separate them from one another definately requires destruction of the driver to be tested, or simply to have the components measured individually.

Since cone velocity is not constant for fres, fl and fh when displacement is kept constant, these secondary nonlinearities will become more significant at higher frequencies (when measuring fh as opposed to fl and fres). For this reason it could be advisable to reduce the cone displacement/excursion when measuring fh, if it does not compromise the static nonlinearities of Bxl and Cms too much. This is exactly what the recommended compare resistance method does.

Another reason to reduce cone excursion is, that the power consumption in the voice coil increases with increasing frequency, when cone displacement is kept constant. This will increase the temperature in the voice coil and corrupt T/S measurements. It could be compensated for by monitoring the voice coil temperature at each of the measuring frequencies, but it complicates the measurement process. Reducing excursion at higher frequencies will reduce this problem.

One of these models of Le should be considered before concerning with dynamic nonlinearities like creep in the suspension. Curvefitting data to Le and subtracting the calculated impedance from measurements from fl to fh will improve on the quality of the calculated T/S parameters.

Besides, do not forget that inclusion of the dynamic nonlinearites will make the loudspeaker model significantly more complex, and alter the understanding / interpretation of the parameters.

You can read more about nonlinearities in XMAX.HTM.