The 2112 Dog Box Project

For my 2112 Bass Guitar Box Project I needed to choose a suitable midwoofer to reproduce the range of frequencies from about 200 Hz to 5 kHz or above. On my short-list I had two (discontinued) 12" midwoofers, the JBL E120-8 and the JBL 2020H. Per their specifications both cover the above frequency range, they are very efficient, have high power handling, and show a rising frequency response above 1 kHz. For Hi-Fi purposes, this rising response would have to be corrected in the crossover, but for a bass guitar box this should provide some definition at no extra cost. Either one should complement the heavy cone of the large woofer very nicely.

I wanted to integrate the midwoofer into the enclosure of the woofer, both for convenience and to keep the midwoofer as close to the woofer as possible. The latter should minimize drop-outs in the off-axis response and thus make the unequal pair of drivers appear more like a single driver. To protect the midwoofer from the considerable changes in air pressure generated by the woofer, a sub-enclosure will be constructed behind the midwoofer. Colloquially, this box-in-a-box is called a dog box.

The role of my midwoofer is to complement the woofer with the part of the spectrum which it may not reproduce or for which it may lack definition (informally, the woofer sounds too slow). But the role of my midwoofer is not to try to reproduce the low B of my bass guitar (B" or B0, which is a little over 30 Hz). It doesn't have to be optimized for the low end of the spectrum. Accordingly, I should be able to use a sealed alignment. Sealed enclosures can be optimized for transient response (box Q) simply by choosing the correct (!) enclosure volume.

And herein lies the problem: Once the sub-enclosure is built and glued solidly into the main enclosure, the combination of sub-enclosure and midwoofer completely determines the resulting transient response. There are no port lengths to tune and not much of anything else to tweak. Granted, I would use the midwoofer's Thiele/Small Parameters and simulate the dog box with LspCAD. But however excellent the simulation may be, only a suitable measurement of the actual enclosure with the actual driver will give me an objective idea how close to the optimal enclosure volume—and hence transient response—I got.

I'll have to try this out with a scrap enclosure first, built purely for test purposes, or several such test boxes, if need should be. This should give me an idea how well the simulation corresponds to reality. Moreover, a test box of known internal volume will give me the ingredients for the preferred way to measure the midwoofer's actual Thiele/Small Parameters. Armed with this knowledge, I should be able to better simulate the dog box. Last but not least, with a pair of test boxes I may hear a difference between the two chosen drivers and thus be in a better position to decide which one to use in the bass guitar project.

Below are a few pictures illustrating the construction of a pair of sealed enclosures, built from 3/4 in (19 mm) MDF, and enclosing a volume of 24 L (0.85 ft3). I used MDF because it is cheaper than Baltic birch plywood but comparably well-suited for building speaker enclosures, and it comes in handy 2' by 4' (0.61 m by 1.22 m) project panels that fit into my sedan. The sawdust from MDF is fairly awkward though, avoiding to say hazardous, which becomes obvious upon routing the edges with a round-over bit, and really obvious upon sanding the enclosures. Combined with the time and cost of priming and painting the enclosures for even a rough finish make me question this decision in retrospect.

While these scrap enclosures may not have much audio jewelry appeal, they should easily serve the purpose of testing. To that end, I wanted to

determine the actual Thiele/Small Parameters,
compare a measured impedance sweep with a simulated one, and last but not least,
listen to either of the midwoofers when playing one of my 6- or 7-string bass guitars.

Measuring basic T/S parameters is not all that difficult anymore, thanks to the sheer computing power of today's personal computers. All you need is an inexpensive software package with test leads to connect your speakers to an unused USB port of your computer[1], and a suitable test enclosure[2], hence another good reason to build scrap boxes. I got the following results (rounded to 3 significant digits which should be more than enough):

Parameter [unit]	Manufacturer's Specs	Driver #1 Measurements (NOS)	Driver #2 Measurements (used)
Re [Ω]	6.3	6.64	6.38
Le [mH]	0.4	0.433	0.426
fs [Hz]	60	50.5	49.8
Qms	1.8	1.54	1.63
Qes	0.19	0.191	0.191
Qts	0.17	0.170	0.171
Mms [g]	36	41.3	39.9
Vas [L]	79.3	95.1	101

Specified and measured T/S parameters of two JBL E120-8 samples

Parameter [unit]	Manufacturer's Specs	Driver #1 Measurements (used)	Driver #2 Measurements (used)
Re [Ω]	4.8	4.74	4.85
Le [mH]	0.02	0.161	0.165
fs [Hz]	66	66.0	68.6
Qms	5	4.91	4.68
Qes	0.26	0.335	0.330
Qts	0.25	0.314	0.309
Mms [g]	44	43.4	43.8
Vas [L]	51.5	53.1	48.5

Specified and measured T/S parameters of two JBL 2020H samples

How close are these measurements in practice? For Re (DC resistance) JBL specifies ±10% for their model E120-8, a tolerance I have seen for many of their products. Re can be determined very easily with a simple multi-meter, as well, since what we are measuring is essentially the resistance of a long wire. An Re close to 0 would indicate a short circuit, while a value close to ∞ would indicate an open voice coil—both conditions of a broken speaker. Less dramatic deviations may indicate a worn voice coil (leakage between adjacent turns of the coil) or an aftermarket replacement.

Le (voice coil inductance) is trickier. Strictly speaking, it is not a constant for all frequencies, and while it is often specified at 1 kHz, the particular speaker may not have any appreciable inductance at that frequency. In particular, for their model 2020H, JBL indicate that the driver is purely resistive at 1 kHz (this is related to the exceptionally low distortion levels of this driver and is one of the reasons the driver made it onto my short-list). Both the specified and the measured impedance sweep corroborate this fact. The value of 0.16 mH likely represents the attempt of my test software to extract a constant out of the measured impedance[3].

fs (free air resonance) is the frequency at which the combination of cone plus suspension resonates, measured while suspended in free air (unobstructed by enclosures, walls, the floor, etc.). It shows up as a peak in the impedance sweep which should be easy for the test software to identify[4]. For a typical factory tolerance Wikipedia suggests ±15%. A measured fs below spec would indicate that the suspension (spider plus surround) has loosened up with time, while fs above spec would indicate tightening of the suspension. For both sealed and vented enclosures, fs is proportional to f3, the frequency at which the response of the respective box will be 3 dB down[5]. Hence for a [sub-]woofer you'll want to make sure that fs is not significantly above spec!

For Vas Wikipedia suggests ±20 to 30% as acceptable tolerance. Informally, Vas denotes the volume of air that has the same springiness as the suspension of the cone (spider plus surround). More formally, for a targeted Qtc (box Q) of a sealed enclosure and a given Qts, Vas is proportional to the enclosure volume VB[6]. Accordingly, a higher Vas would require a larger box volume VB to keep everything else the same. Conversely, keeping the box volume VB the same, a higher Vas would yield a higher Qtc and in turn alter transient response.

For the remaining T/S parameters I do not yet know what would constitute acceptable tolerances. Also, the above list of T/S parameters is not complete. I left out some parameters because they require to test the driver with a large signal (e.g. to determine Xmax, the maximum linear excursion of the cone), which my software tool cannot do. The remaining parameters are derived from other parameters by mathematical formulae, which doesn't contribute new information about the health of a driver under test.

To come to a conclusion how healthy a driver is, I'll put the T/S parameters to practical use by calculating the volume VB, enclosure resonance fB, and cut-off point f3 of a sealed enclosure and compare them with the respective results calculated from the specs[7]. In the process, I'll target Qtc = 0.5 since this should give me the best transient response. I got the following results:

Sealed enclosure parameter for Qtc = 0.5	Manufacturer's Specs	Driver #1 Measurements (NOS)	Driver #2 Measurements (used)
VB [L]	10.4	12.4	13.4
fB [Hz]	176	149	146
f3 [Hz]	274	231	226

Sealed enclosure parameters for two JBL E120-8 samples

Sealed enclosure parameter for Qtc = 0.5	Manufacturer's Specs	Driver #1 Measurements (used)	Driver #2 Measurements (used)
VB [L]	17.2	34.6	30.0
fB [Hz]	132	105	111
f3 [Hz]	205	163	172

Sealed enclosure parameters for two JBL 2020H samples

Good thing I made a pair of scrap enclosures first before committing to a final dog box size. What these numbers tell me is that for both 2020H samples I would not have achieved a box Q of 0.5, even with an oversized enclosure: Remember that the enclosures have a volume of 24 L. Subtract about 2 L for the volume displaced by the front-mounted driver, and per the specs I should have an enclosure that is 4.8 L larger than the 17.2 L necessary to achieve Qtc = 0.5. I was planning on adding bricks or similar to reduce the enclosure volume, at least for measuring purposes. But instead, per the T/S parameters measured for driver #1, my enclosure size should be more than twice the size I calculated from the specs.

To confirm these calculations in practice I used my software tool to take a free air measurement of driver #2 installed in the enclosure. In this mode of operation, the tool will measure fB in lieu of fs, and likewise Qtc in lieu of Qts. Following is a screen shot of this measurement, along with the LspCAD simulation of the measured T/S parameters:

Impedance sweep of a JBL 2020H (driver #2) in a 24 L sealed enclosure

Simulation of the impedance of the above JBL 2020H in a 24 L sealed enclosure, using the previously measured T/S parameters

That's remarkable! The measured enclosure resonance fB (118 Hz) is in very close agreement with the simulated resonance (119 Hz) and within a few percent of fB (123 Hz) I calculated for a 22 L (net) enclosure[8]. Likewise, the measured Qtc (0.560) is very close to the calculated Qtc (0.554). If nothing else, this would suggest that simulating reality in LspCAD using actual T/S parameters is not a purely academical exercise[9].

At this point I hesitate to decide if these drivers are healthy or not. Let me try another practical approach first. For this test, I assume I have already built the enclosures to spec, and I'll use the measured T/S parameters to predict Qtc, fB, and f3[10]. Here are the numbers:

Sealed enclosure parameter for VB = 10.4 L	Manufacturer's Specs	Driver #1 Measurements (NOS)	Driver #2 Measurements (used)
Qtc	0.5	0.542	0.560
fB [Hz]	176	161	163
f3 [Hz]	274	250	253

Sealed enclosure parameters for two JBL E120-8 samples

Sealed enclosure parameter for VB = 17.2 L	Manufacturer's Specs	Driver #1 Measurements (used)	Driver #2 Measurements (used)
Qtc	0.5	0.635	0.604
fB [Hz]	132	133	134
f3 [Hz]	205	207	208

Sealed enclosure parameters for two JBL 2020H samples

Wow! For both samples of the 2020H, fB and f3 are within about 1.5% or better. Even the deviations of the E120-8 samples are less than 10%. To put these 10% into perspective, in musical terms 10% is less than a full note apart. From this angle, these drivers look remarkably healthy. That is, if I can sleep well at night knowing that I would have missed the targeted Qtc.

Now then, which driver will I use as midwoofer in my 2112 Bass Guitar Box Project? I don't know yet. I should probably try to get both test boxes to the same Qtc. That way at least I would be comparing apples with apples. The test box with the E120-8 currently measures at Qtc = 0.392. Applying its T/S parameters for a Qtc of 0.56 yields a VB of 10.4 L. That would be quite a few bricks to get it up to 22 L though. It looks like I will need to make more sawdust...

Footnotes

I use Dayton Audio's Woofer Tester 3 available from Parts Express for just under $100. This tool was developed by John L Murphy, a physicist of True Audio fame. Like any tool, you'll have to learn how to use it: Follow the manual about warm-up time, calibration, and don't forget to turn the volume on your Windows speaker dialog all the way up to the max. If you don't, and if you make any noise during the measurement, the speaker talking back to your computer will show up on the impedance sweep! After all, a speaker is a pretty good microphone. When measuring the free-air parameters, don't leave the speaker face down on the floor. If you do, the speaker will behave as if it were in a small sealed if slightly leaky enclosure formed by the air trapped between the speaker's cone and the floor. Also, like any measurement in physics, you'll have to learn how to interpret the results. To that end, it helps to understand how the various T/S parameters are related to each other, which parameters are actually measured, and which ones are derived by computation (cf. right below for determining Vas), and hence how measuring errors propagate to the derived parameters.
Per the software, there are 3 methods to determine Vas. Method 1 requires a test box of known internal volume, and a good estimate of the volume displaced by the driver when mounted. The estimate gets a little easier when front-mounting the driver in reverse, facing into the enclosure. This adds a small volume, formed by the baffle cut-out and the volume bordered by the cone. The latter can be broken down into a short cylinder formed by the speaker's flange, plus the volume of a truncated cone (assuming the speaker's cone is actually conical and not somehow curved), and minus (!) the volume of the center dome. Method 2 requires to temporarily attach a known mass, preferably symmetrically around the center dome. Without an appropriate scale, it may be tempting to tape stacks of quarters to the cone, since each quarter weighs exactly 5.67 g (1/80 lbs). This will not be very accurate, however, since the quarters will decouple from the cone depending on the applied frequency. The cone plus added mass no longer behaves like a single object. Method 3 requires to know the SPL measured at 1 m for 1 W input. I excluded this method from the beginning because I don't know what frequency to use in the process and because I think my Radio Shack SPL meter would not be accurate enough.
To get an idea how this may be extracted, consider the total impedance Z of a resistor (Re) in series with an inductor (Le):
Z = Re + j·ω·Le.
Its magnitude is
|Z| = (Re2 + ω2·Le2)½.
Assume you have measured Re and the magnitude of Z at 1 kHz. Now solve for Le. If Re and |Z| barely differ from each other, a small measurement error of either of these values may drown the value for Le.
Woofer Tester 3 measures the impedance by taking 30,000 samples between 1 Hz and 20 kHz. Assuming the samples are spaced at equal intervals on the logarithmic scale, each interval will differ from the adjacent one by about 0.033%. Accordingly, if a sample were taken at e.g. 60 Hz, adjacent samples would be taken at 60 Hz ± 0.033% or approximately 59.98 Hz and 60.02 Hz, respectively, and recursively. Accurate enough to find that frequency with almost 4 significant digits—at least to the degree that the sampled impedances are accurate.
If you are not afraid of mathematical formulae, e.g. for a sealed enclosure
f3 = fB·[(a + (a2 + 4)½)/2]½
with the resonant frequency of a sealed enclosure
fB = fs·(Qtc/Qts),

a = 1/Qtc2 − 2,
and Qtc the targeted box Q.
Vas is related to VB equivalently by
Vas = VB·[(Qtc/Qts)2 − 1] = VB·[(fB/fs)2 − 1].
I suspect this formula is used for determining Vas in Method 1 described in Footnote 2 above, given that the software first determines Qts in free air, and subsequently Qtc with the driver installed in (or on) the enclosure and with VB typed in.
I used the above
Vas = VB·[(Qtc/Qts)2 − 1]
and solved for
VB = Vas·Qts2/(Qtc2 − Qts2).
In this representation it is easy to see that Qts should be less than Qtc, in fact, quite a bit less than Qtc or else the enclosure will get very large: As Qts → Qtc, VB → ∞. fB and f3 are calculated as introduced in Footnote 5 above. With Qtc = 0.5, a = 2, and f3 = 1.5538·fB.
I used
Vas = VB·[(fB/fs)2 − 1]
and solved for
fB = fs·(Vas/VB + 1)½.
Subsequently I used
fB = fs·(Qtc/Qts)
and solved for
Qtc = Qts·(fB/fs).
LspCAD has several loudspeaker driver properties it will try to simulate, such as B·l non-linearity, voice coil heating, and Cms non-linearity. For the presented illustrations I turned all these simulations off, hoping that this scenario would more closely correspond to the simulation being taken at small signal levels. Turns out it did!
The required math is playing musical chairs with the previous formulae.