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Woofer measurements

1 - Impedance
Much can be learned from a measurement of the impedance of the completed subwoofer. If a long speaker cable is used, then include it, so that the measurement represents the load seen by the amplifier. Only four numbers are needed to determine F0 and Q0. These are Rdc, Rmax, F1 and F2. The two frequencies are measured at the impedance value of the geometric mean for Rdc and Rmax. It only takes a signal generator and an ac voltmeter to determine all necessary quantities.( f0Q0.gif )

Once F0 and Q0 are known we can predict the acoustic frequency response curve, because we then know the location of the two poles of the 2nd order acoustic highpass filter, that is formed by a driver in a closed box. ( 12db-hpf.gif )

The calculation of the pole frequencies and the component values of the simplified impedance model is carried out with closed-box1.xls. Block 1 of the [Test] sheet is used for this. 
Block 2 gives the model parameters. A value for Le could be estimated from the measured impedance at a high enough frequency, where the effect of jwLe becomes significant. The model for calculating F0 and Q0 though assumes Le = 0. The impedance minimum above Fmax in the above graph is higher than Rdc , because jwLe is added. The resonance curve becomes skewed and the values measured for F1, F2 and Rmax are not exactly those that are assumed in the model. Consequently, all the calculated values have some associated uncertainty and should be taken as approximate. 

2 - Frequency response
The subwoofer frequency response is measured with the microphone very close (0.5") to the driver dust cap.

The straight line asymptotes of 12 dB/oct and 6 dB/oct slope for the calculated poles at 19 Hz and 67 Hz give only a marginal fit to the measured response. A better approximation appears to be with poles at 30 Hz and 46 Hz, as in the following graph.

These values will be taken as starting frequencies for equalization. Note that the calculated values on the [Spec's] sheet, 18 Hz and 72 Hz in block 2a, correspond closely to those derived from the impedance measurement (19 Hz and 67 Hz) above. It shows that the models used are consistent with each other, but they do not fully represent reality. Still, they are useful tools to get close to it. In many cases I have found close agreement between impedance and frequency response measurement results, but usually Q0 is greater than 0.4.

3 - Non-linear distortion
Non-linear distortion limits the maximally useable output of any subwoofer. It adds new spectral components, harmonics and intermodulation products, to the original signal and thereby falsifies the sound. Though some people have developed preferences for certain types of low frequency distortion it is my objective to keep it to a minimum.
The non-linear behavior of a subwoofer increases rapidly as cone excursions become large, because force factor Bl, compliance Cms, and voice coil inductance Le, change with displacement from the resting position. Distortion also changes with frequency, even when the peak-to-peak displacement is held constant. This can be seen by comparing 30 Hz and 20 Hz harmonic distortion measurements of THOR for constant p-p excursion of 0.5" (13 mm).
(All sound pressure levels are relative in these tests. They were measured outdoors with the speaker on the ground. The microphone was at 4" from the plane defined by the driver's rubber surround and on-axis with the dust cap.)

Distortion is higher at 20 Hz than at 30 Hz even though the displacement is the same. This is a curious result. The fundamental drops 40log(30/20) = 7 dB, as expected. The 2nd and 3rd harmonics, though, remain at nearly the same sound level, as the voice coil swings through the same range for each of the excursion dependent non-linear parameters. Thus the distortion percentage increases. 

Repeating the measurement with twice the drive voltage for a targeted 1" p-p excursion clearly shows the limits of output capability. 

While 1" p-p is marginally useable at 30 Hz it is clearly leading to gross distortion at 20 Hz. At most 0.5" p-p can be counted on at 20 Hz. The single input tone generates a rich spectrum of harmonics, which extends into the midrange regardless of any crossover. While there is plenty of sound output it is a poor representation of a 20 Hz tone.

It is interesting to note that a 830500 driver in an open baffle has somewhat lower distortion at the same excursions and frequencies as above. The enclosed air in the THOR box stiffens and linearizes the driver suspension, but it also causes more roll-off in low frequency response. Thus the distortion generated harmonics are of higher amplitude for the closed box than the open baffle. In addition, linearizing the compliance does not necessarily lead to better overall driver performance, because it might change the degree of cancellation that existed between different distortion mechanisms. The acoustic output at the same excursion, though, is higher for the closed box. 

A 10-cycle shaped tone burst at 20 Hz with cosine envelope gives a picture of the time waveform distortion as the excursions build up to 1" p-p and then decrease towards zero.

The rapid changes in the waveform indicate the many high frequency spectral components. A reduction in amplitude to 0.5" p-p gives a better replica of the input, but distortion is still recognizable by the slightly triangular shape at large amplitudes.

The large signal behavior of a driver and its effects upon the radiated sound quality are quite complicated. In the case of THOR one should not count on more than 0.5" excursion at 20 Hz or 0.75" p-p at 30 Hz. For accurate very low frequency reproduction it will almost always be necessary to place at least two THOR units in a room to obtain sufficiently low distortion levels.

The importance of low distortion at very low frequencies can be deduced from the equal loudness contours. The threshold of hearing is around 70 dB SPL at 20 Hz. This is at the level of normal conversation. With increasing frequency the threshold drops rapidly. The loudness contours have an initial slope of 80 dB/dec, or 24 dB/oct, at low perceived volume levels (phon).

This means that if the 40 Hz 2nd harmonic of a 20 Hz tone is at a 24 dB lower level, then it will sound equally as loud as the fundamental. This corresponds to 6% 2nd harmonic distortion. The 3rd harmonic distortion would have to be below 1%, or over 38 dB down, in order that it is less loud than the 20 Hz fundamental. It all leads to very low distortion requirements. The fundamental frequency sound pressure level needs to be above 70 dB to even become audible and it should not be masked by higher frequency distortion products. 
For a detailed investigation of requirements see: Louis D. Fielder & Eric M. Benjamin, "Subwoofer performance for accurate reproduction of music", JAES, Vol. 36, Number 6, pp. 443 (1988).

The equal loudness contours of the graph above are bunched more tightly together at low frequencies. Re-drawing such graph in terms of perceived loudness (phon) versus sound pressure level in dB SPL for different frequencies, it shows us the sensitivity we have to changes in low frequency level adjustment. 

A change of 10 phon corresponds to a perceived doubling of loudness. At 1 KHz the SPL must change (by definition) 10 dB to obtain this doubling, but at 20 Hz it takes only about 5 dB for the same effect. This is the reason why changes of only a few dB in subwoofer level setting have a large perceived effect. Adding a second subwoofer for 6 dB more SPL, or for lower distortion at the initial SPL, can give the subjective benefits of up to 12 phon.

| Introduction | Estimates | Design | Measurement | Equalization | THOR-ORION xo |
| Supplies | SPL limits |