There are several different ways to measure the Thiele/Small parameters of a loudspeaker driver. The method described here provides a way  for the beginner and  DIY enthusiast to measure the parameters without any expensive or specialised equipment.

Definitions:

Re	Electrical resistance of voice coil
Fs	Resonant frequency of loudspeaker moving mass (in free air)
Qes	Electrical Q of loudspeaker
Qms ...	Mechanical Q of loudspeaker
Qts	Total Q of loudspeaker
Vas	Equivalent air volume of moving mass suspension

Measuring Re, Fs, Qes, Qms and Qts

To measure these parameters using the method outlined below, you'll need to have the following items:

• A power amplifier, rated at 10 Watts (RMS) or more
• Audio frequency oscillator
• Digital multimeter (with frequency measurement)
• A 5 watt resistor (any value between 4 to 10 ohms)
• Alligator clip leads - you will need 4 sets of leads (leads may be soldered instead if desired)

At resonance, the speaker impedance is pure resistance.  As the frequency increases towards resonance, the impedance characteristic is inductive.  Beyond resonance as impedance falls, the impedance characteristic is capacitive.  Within the "linear" region, the impedance is again (almost) resistive, but at the speaker's nominal impedance.  At the frequency where the inductance of the voice coil becomes significant, impedance is progressively more inductive as the frequency rises.  It is common to add a compensation network to maintain an overall resistive characteristic at these higher frequencies, so that the performance of the crossover network is not compromised.  This is not necessary with an active crossover.

The multimeter should be capable of measuring frequency, as well as AC voltage and resistance.  If it cannot, a frequency counter is highly recommended, since the frequency measurements are critical.  The amplifier must be capable of reproducing from 10 Hz to 2 kHz with no variation in output voltage.  It is imperative that it is insensitive to any load above 4 ohms.  The audio oscillator must also produce a signal with relatively low distortion, and the output voltage must not vary as the frequency is adjusted.

Measure the resistance across the speaker terminals to obtain Re.
Measure the exact resistance of the 5 Watt source resistor, Rs.

The loudspeaker driver should be suspended in free space, with no obstructions or interfering surfaces nearby.  Any boundary closer than around 600mm (about 2ft) will affect the accuracy of the measurements

Connect the circuit as shown in Figure 2, with the audio oscillator to the input terminals of the amplifier, but with the multimeter across the amplifier output terminals instead of the resistor.

Set the oscillator to approximately 100 to 200 Hz (or at least 2 octaves above resonance).

Set the output of the amplifier to between 0.5V and 1.0V (this is Vs).  You may need to experiment with different voltages, depending on the accuracy of your current readings (or calculations).  Do not be tempted to use higher voltages, as the speaker may be driven outside its linear range, which ruins the validity of the measurements.  The parameters being measured are "small signal", and it essential that a small signal is actually used.

Calculate the speaker current (Is), to make sure that you will have a reasonable current to work with.  Check that the speaker is nowhere near resonance, by changing the oscillator frequency by 50Hz or so in either direction, and measure the voltage across the speaker.  It should not change by any appreciable amount.

Is = Vs / (Re + Rs)

Adjust the frequency until the voltage across the resistor reaches a null (minimum level).  Without changing anything, carefully measure the frequency and voltage across the resistor ...
 

Frequency	Fs
Voltage across the resistor	Vm

Calculate the following ...
 

Speaker current	Im=Vm/Rs
Driver impedance at resonance	Rm=(Vs-Vm)/Im
r0	r0=Is/Im
-3dB current	Ir=(Im*Is)^0.5
-3dB Voltage	Vr=Ir*Rs

Complete the measurements for Fl and Fh, for which the voltage across the source resistor is equal to Vr, and as a sanity check (to ensure that your calculations and measurements are accurate), calculate the resonant frequency based on these last two measurements.
 

Check that ...

(Fl*Fh)=Fs

If the above checks out (within 1 or 2Hz), then Qes, Qms and Qts can be calculated as follows:

 Qms = Fs * r0/(Fh - Fl)

 Qes = (Qms/(r0-1)) * (Re/(Rs+Re))

 Qts = Qms * Qes/(Qms + Qes)

To measure Vas, use a good solid enclosure of known volume that is approximately a cube of the nominal speaker size.  For example, a 300mm driver (12") needs a box of about 28 litres (1 cu ft).  For reference, a cubic foot is 28.3168 litres, and one litre is contained by a cube of 100mm to each side.

Determine the total volume, including the speaker cut-out and that trapped by the cone with the speaker mounted on the outside of the box for easy access. Measure the resonant frequency in this situation, and use the free air space resonant frequency determined as shown above.  Determining the volume trapped by the speaker cone is slightly tricky.

Use one of the following methods ...

• Place the driver in a plastic bag, ensuring it is completely sealed.  The bag should be loose enough so that it can be pushed easily into the cone area.  Place the wrapped speaker on a flat surface, with the cone facing upwards.  The cone may now safely be filled with water, and the water carefully poured out into a measuring jug.  The resulting measurement will be a little greater than the actual volume because the cone will be depressed by the mass of the water.  The area of the speaker cutout in the cabinet must still be added.
• Take a series of measurements.  The internal cone area is measured, then divided into sections whose volume may be calculated.  For most speakers we will have two basic shapes to deal with, and although this method is not 100% accurate, it will probably give the best result in the majority of cases.

There is a flat cylinder (disc) that is formed by the outer area of the basket and the cutout in the enclosure.  The triangular ring is formed by the cone itself, as shown above.  Finally, the cylinder occupies the area over the dustcap.  The depth of the cylinder should be an approximate average of the distance from the mounting surface and the dustcap.

The volume of cylinder and disc is given  by the conventional formula ...
V_cyl =  * r²* h    (where r is the radius and h is height), and
V_disc =  * r² * h

The triangular ring's volume is given by ...
V_ring = (( * r² * h)-V_cyl)/2

The total speaker volume is simply the sum of the 3 volumes calculated above.  Box volume is calculated as one normally would, taking great care to ensure that the measurements are accurate.  The box may be braced, but must have no fibreglass or other sound deadening material inside.  Make sure that the volume occupied by any bracing is accounted for in your calculations.  Even a simple box will be sufficiently rigid at the frequencies of interest, so a completely acoustically dead cabinet is not required (although it won't hurt).

 Vas = Vb((Fb/Fs)2-1)

A "dummy" or test loudspeaker was described in another contributed article, and I have used a simulation of a similar speaker in the calculations shown.  The equivalent circuit of the speaker is shown in Figure 5.  This circuit was also used to create the impedance graph shown in Figure 1.

The following screen shot shows the values for the "speaker", and the only contrived (i.e. invented) value is for the resonance in the sealed box.  It was necessary to invent a number here, as it is not possible to simulate it.  The final figure shown is fairly typical of many such drivers, so is not too far from the truth either.

As you can see from the screen-shot, the spreadsheet will calculate everything for you, including the cone volume and Vas.  Naturally, you will get figures quite different from those shown, but the principle is exactly the same.

Page Created and Copyright  15 Dec 2000