Discrete alignments cause a particular headache, which is that you cannot expect to get your system design exactly 100% fitted with an exactly matching driver Qt. When there is not an exact match, we talk about some degree of alignment error (or misalignment) in the implementation of a system design.
In previous articles, we proposed that a misalignment of Qt is ignored up to, for example 5% or maybe even 10%. If your driver Qt is about the discrete alignment requirement, your result will also be in the neighborhood. In this article, we propose a more refined approach than pure ignorance, with the purpose of extending the desired discrete target alignment to a broader range of driver Qt values.
It is well known that the bass reflex alignment is least sensitive to an error in compliance (suspension stiffness), and therefore, if your driver’s Qt is not correct, we propose that one imagines the compliance is altered in such a way that the desired target Qt is obtained — we call it a reference Qt, which is then targeted for a particular alignment. The alignment is calculated for the reference Qt, and subsequently, the bass reflex parameters alpha and h are shifted with a compliance alteration based on the real-world driver you have available and intend to use.
Historical Background
Here are several quotes from a short list of references that are relevant to this subject:
1- A. N. Thiele [1] writes (quote from page 8): “To this extent, and also in the expression for efficiency (Eq. (66)) the compliance of the loudspeaker is unimportant."
2- R. H. Small [2] analyzes sensitivity to various losses, but it is not until Part 3 of the article series [3], Figure 23, that we see how robust the bass reflex principle is to variation in driver compliance. Figure 23 only presented response curves for the Butterworth B4 alignment as an example, not a general analysis. Small writes (quote from section 13 page 5): “The relative effects are smaller for higher compliance ratios (i.e., QB3 alignments) and larger for lower compliance ratios (C4 alignments).”
3- Finally, D. B. Keele [4] provides a more general, rigorous, and extensive analysis and concludes (quote from page 9): “The low sensitivity of the Thiele alignments to shifts in driver suspension compliance is indeed quite fortunate, because it is this parameter which normally undergoes the widest variation in driver manufacturing and also is the most likely to shift with age and temperature.”
Note: Don Keele’s article was published prior to Richard Smalls’ article series. No doubt, republishing Albert Neville Thiele’s original articles in the Journal of the Audio Engineering Society (JAES) facilitated a renewed interest in his work on bass reflex alignments.
How It Works
Keele [4] systematically explored the concept of misalignment, that is, that case for which a desired alignment is not exactly realizable and so some deviation from the target must be tolerated. Keele concluded that a bass reflex design is least sensitive to misalignment in the suspension compliance (stiffness). Thus, we propose a family of alignments that begin with a reference Q, say Qt_ref, and then imagine that the compliance Cms of the driver is varied. This Cms variation will shift the resonant frequency according to ws = 1/√(Mms * Cms). Suspension compliance as an air volume equivalent is given as Vas = rho * c2 * Cms * SD2. The following equation shows the proportionality between ws (= 2 pi fs) and Qt:
where Rme = (Bl)2/Re is the electrical DC resistance transformed to the mechanical domain. The expectation for the compliance alteration method is that the frequency response will be dominantly modified in the stopband but relatively well-preserved in the passband. This expectation derives from the physical observation that driver compliance dominates in the stopband. Presumably, it is favorable to place the discrepancy in the stopband because the system has rolled off and the sound is less audible here.
Some algebra shows that the other system parameters will vary with ws according to:
The first equation describes that a shift in Qt is linearly proportional to a shift in resonance frequency. The second equation describes that a shift in alpha is an inverse quadratic function of a shift in the resonance frequency, which stems from the fact that ws = 1/√(Mms * Cms). The port tuning factor h is shifting inversely proportional to the resonance frequency. For LR4 and B4 alignments, we set href = 1.
Although you could say that the compliance alteration approach is a logical next step considering the historical background observations, the author is not aware of any prior work that outlines the method.
Example Linkwitz-Riley (LR4CA)
We can summarize the steps to generate a compliance alteration alignment based on LR4, which we refer to as the LR4CA alignment, for an arbitrary driver Qt and loss Ql:
Note: Any correlation with the resonance frequency ws is replaced by the equivalent correlation with Qt because for alignments, it is Qt that we are deemed to know in advance.
Summary: We define the driver’s actual Qt as well as the reference (known from when we calculate an LR4 alignment). All the calculations are made with the reference, then shifted due to compliance alteration at the end. Here illustrated with a piece of Scilab code:
Ql = 10
Qt = 0.39 // Replace this value with your own driver’s Qt-value
Qt_ref = 1./(sqrt(8)-1./Ql)
alpha_ref = 2 - 1./(Ql * Qt_ref)
h_ref = 1
alpha = alpha_ref * (Qt_ref/Qt).^2
h = h_ref * Qt_ref ./ Qt
Qt = 0.39 // Replace this value with your own driver’s Qt-value
Qt_ref = 1./(sqrt(8)-1./Ql)
alpha_ref = 2 - 1./(Ql * Qt_ref)
h_ref = 1
alpha = alpha_ref * (Qt_ref/Qt).^2
h = h_ref * Qt_ref ./ Qt
With this algorithm, you are free to set the Qt-value of the driver at hand, as well as the Ql-value, and use it as input to the calculations.
This method is particularly interesting if your driver Qt is a bit too high, because with the compliance alteration technique, the box calculation is then treated as if the suspension is a bit too stiff. Fortunately, the driver suspension will experience aging (or burn-in) over time and will soften. When this happens, the provided equations dictate that your system will, over time, move toward the desired LR4 target response, and if softened enough to reach Qt_ref, it drops into place and becomes a correct Linkwitz-Riley response without any error (except for the error created by the leakage loss Ql).
Prior to the error-free situation, see Figure 1, the deviation from the desired target response is found in the compliance-dominated range, i.e., below the driver’s resonant frequency, i.e., in the stop-band, which is convenient because this is where the deviation will likely cause the least audible effect. As we move our point of interest from the -3dB point (the roll-off region) and into the passband, the mass-controlled range, this quasi-alignment will provide approximately the correct response in both magnitude and phase (and group delay).
Since this alignment is a discrete alignment, it is faced with the same issues as the LR4 alignment, that we cannot expect to match Qt exactly, we propose to use the same solution; to utilize compliance alteration and define a B4CA alignment, which is not a discrete alignment but allows for some flexibility regarding driver Qt.
Ql = 10
Qt = 0.36 // Replace this value with your own driver’s Qt-value
Qt_ref = 1./(1./cos(3*%pi/8)-1./Ql)
a3 = sqrt(8)*cos(%pi/8)
alpha_ref = sqrt(2) - (1./Ql.^2) .* (a3.*Ql-1)
h_ref = 1
alpha = alpha_ref .* (Qt_ref./Qt).^2
h = h_ref * Qt_ref ./ Qt
Qt = 0.36 // Replace this value with your own driver’s Qt-value
Qt_ref = 1./(1./cos(3*%pi/8)-1./Ql)
a3 = sqrt(8)*cos(%pi/8)
alpha_ref = sqrt(2) - (1./Ql.^2) .* (a3.*Ql-1)
h_ref = 1
alpha = alpha_ref .* (Qt_ref./Qt).^2
h = h_ref * Qt_ref ./ Qt
Specify Ql as you wish and specify the Qt value of the driver you want to apply. If you specify Qt in the provided Scilab code as a vector (array) with values in the vicinity of the reference Qtref, for example ±5%, which yields [0.38, 0.39, 0.40, 0.41, 0.42], then you can define a small Qt range in which you can obtain something close to the B4 frequency response characteristic. The point is, if you specify Qt as an array of input values, then the code will produce alpha and h containing arrays of results accordingly.
In this example, we have specified Qt 10% too low, and as the suspension softens, this will only become worse. But remember, the alignment will work well in the mass-controlled region, so this is still a feasible approach, although it seems counterintuitive to wish for a higher Qt than what you have available. Remember to investigate what deviation this will cause compared with your target response. It is worthwhile to consider opting for one of the other available alignments with a Qt_ref closer to the available driver’s Qt; at least explore the possibility. The nearest option in this case would be a QB3 alignment because it still targets a maximally flat frequency response.
Looking at Figure 2, we have modified Qt a lot, ca. ±20%, which is equivalent to an alignment error in Cms of about ±50%. These alternatives are calculated with compliance alteration, which means alpha and h are changed accordingly, but the actual box volume and port tuning are the same; only compliance is changed. The B4 alignment is remarkably insensitive to even quite large alignment errors (misalignment), provided this is due to compliance. This figure reproduces Small’s Figure 23 [3], although the curve for higher Qt does not exactly match Small’s observation from his simulator. Maybe he simulated a driver with an even higher Qt value, for example, Qt = 0.58.
As you can see from these examples, we calculate the original alignment with the theoretically perfect Qt-value and output reference alpha and h values. Then in the last two lines, we apply a shift of h and alpha, which accounts for the error in Qt with a corresponding shift in Cms (or Vas).
Please note that as we apply Compliance Alteration, the poles are no longer located on exactly the same position of the unit circle. In general, this observation is valid for all alignments with compliance alteration. It moves the poles to different locations, which means the polynomial coefficients change and the target response is not exactly met.
The “Compliance Alteration” technique is our answer to dealing with any discrete alignment but is also feasible in general to compensate for expected changes in suspension stiffness over time for any other alignment, such as QB3, BB4, and others. The background for compliance alteration is found spread out in the cited references, but here it is applied in a new context to our advantage.
Compliance alteration takes advantage of the insensitivity to compliance variations. Although compliance alteration is not the only way to modify the box tuning towards a desired target response function, it has the advantage that it works with the same parameters, notably the driver’s Qt value, which is still the only input parameter of the driver, plus the new Qt_ref parameter, which stems from your target response. To find out how close (or not) your target is matched with your specific driver, you must run a complete simulation to study the frequency and phase response (group delay as a derivative of phase is often easier to study).
It is interesting to consider the case where the driver Qt at hand is slightly higher than the target Qt because the suspension is initially too stiff. As the driver experiences aging (or burn-in) and the suspension softens, the frequency response will move toward the target alignment.
You may consider compliance alteration as a “fix” to a problem, one that you add onto whatever algorithm you applied beforehand, so it doesn’t target an explicit direction for the fix. Wherever you land with this correction, it becomes an implicit tuning. On the other hand, studies show that this practical approach is surprisingly good at keeping your box tuning frequency response linear (if it was linear to begin with) in the passband. There are other ways to work with misalignment, but compliance alteration moves the alignment error into the stopband, where it is presumably less audible. VC
References
[1] A.N. Thiele, “Loudspeakers in Vented Boxes: Part 1 - 2,” Proceedings of IREE (Australia), Volume 22 (1961). Republished in the Journal of the Audio Engineering Society (JAES), Volume 19, May and June 1971. www.aes.org/e-lib/browse.cfm?elib=2173 and https://www.aes.org/e-lib/browse.cfm?elib=2163
[2] R. H. Small, “Vented-Box Loudspeaker Systems - Part I: Small-Signal Analysis,” Journal of the Audio Engineering Society (JAES), Volume 21, Issue 5, pp. 363-372; June 1973.
www.aes.org/e-lib/browse.cfm?elib=1967
[3] R. H. Small, “Vented-Box Loudspeaker Systems - Part III: Synthesis,” Journal of the Audio Engineering Society (JAES), Volume 21 Issue 7 pp. 549-554; September 1973.
www.aes.org/e-lib/browse.cfm?elib=1951
[4] D. B. Keele, Jr., “Sensitivity of Thiele’s Vented Loudspeaker Enclosure Alignments to Parameter Variations,” Journal of the Audio Engineering Society (JAES), Volume 21, Issue 4, pp. 246-255, May 1973.
www.aes.org/e-lib/browse.cfm?elib=1983
This article was originally published in Voice Coil, October 2024
