### THE DESIGN PROTOCOL: PART 2b,VB-LOW FREQUENCY TUNING

This part is dedicated to Vented Box (VB) design. We have already chosen our loudspeaker drivers and therefore have their datasheets along with their dimensions and TS (Thiele Small) parameters. We also have a rough estimate of the required (internal) volume of our VB enclosure. There are several pieces of software that simulate the low frequency behaviour of woofers in Vented Box (Bass Reflex) designs, giving estimates of system SPL responses in the range 10Hz-1000Hz.

Some of them also attempt to simulate the effect of the diffraction SPL step (or loss) that rises from the finite (not very large as in the case of the infinite baffle condition) dimensions of the enclosure.

-Can we just start using such a simulation program and let all the theory of the past decades aside ?

The answer is no. Because using such a software involves decision-making concerning too many possible changes in enclosure dimensions and driver placement coordinates. Deep knowledge of the underlying theory helps us understand which parameter must be altered to get the desired change.

The following table gives a general picture of what is needed for this part of the design protocol and what is expected to be derived:

 PART 2: - INPUT CONSTRAINTS: - MAJOR DELIVERABLES: - driver dimensions ? -> net enclosure volume - woofer TS parameters ? -> internal enclosure dimensions - enclosure thickness ? -> external enclosure dimensions - power handling capacity ? -> driver placement - approx. enclosure volume requirement ? -> vent dimensions and placement - MINOR DELIVERABLES: -> max RMS power due to displacement limitation

The heart of our approach is the fact that the shape of the SPL response at very low frequencies is that of a Fourth Order High Pass System as explained in a previous tutorial. We want this shape to be either flat or with a moderate ripple. Fourth Order responses are not described by a quality factor as in the case of Second Order Systems. So we can not adopt a target quality factor for Vented Box speakers.

 VENTED BOX, STEP 1: ENTER VARIOUS PARAMETERS -> NET SPEAKER VOLUME & min VENT DIAMETER -> SPL RESPONSE ESTIMATES and VENT DIMENSIONS woofer datasheet fsb: woofer suspension resonance frequency (Hz) Vb: net enclosure volume (m3)  &  min Dv: minimum vent diameter for low turbulance noise (m) fb: estimated speaker resonance frequency (Hz) Vas: suspension compliance equivalent volume  (m3) Qts: total quality factor of woofer driver f3: estimated speaker cutoff (-3dB) frequency (Hz) Sd: equivalent cone diaphragm area (m2) SPL ripple: estimated ripple value (dB) with respect to flat alignment. Xp: max. positive cone displacement (m) max RMS power: estimated maximum RMS power input to speaker due to woofer cone displacement limitation (Watt) Vent Diameter: Dv (m) Vent Length: Lv (m)

 As explained in the diagram on the right, parameters in red are input while those in blue are derived by the design formulas presented. These formulas were produced by Garry Margolis in 1981 in JAES (Journal of the Audio Engineering Society) and to our opinion constitute the most practical design procedure we can get from mathematical analysis of woofer suspension mounted on a speaker enclosure. There is a net air volume Vb estimate for the fourth order SPL response to be flat. If this value exceeds the respective value associated with our speaker category we better stick to the latter. A small SPL response ripple won't harm us. a is an intermediate variable that helps us derive box resonance fb and cutoff f3 frequencies. Then we derive the expected SPL response ripple and the maximum RMS power this design can withstand in order for the vibration amplitude not to exceed the driver's limit Xp. If this RMS limit in Watts is more than what the woofer driver can 'thermally' undertake we must not take it into account. Then we come to the noise (due to air turbulance) issue associated with bass reflex ports. Minimum vent diameter is a very helpful estimate if combined to the list of available vent pipe diameters. L' is a parameter related to vent length and should be combined to Lv calculation. The last formula for the vent's length concludes this design phase. If Lv is very low, less than the vent pipe's radius, we will have a port with irregular operation so we must reconsider. We must ask for a longer vent, ie a lower resonance frequency which leads to a larger cabinet volume. We must therefore choose a larger Vb value and run our formulas again beginning with the new value of 'a'. Minimum Dv calculation can be omitted for it is not expected to enforce a new selection of vent diameter.

If we choose to work differently and set Vb to an arbitrary value we can still use these formulas with the same order beginning with the evaluation of 'a'.

Now we have to deal with cabinet's (inner) dimensions and driver/vent placement :

 We create a drawing of our baffle dimensions with the minimum values we can have. We call it the minimal baffle configuration. As shown on the picture on the right, we place our tweeters, woofers and ports as close as possible to enclosure walls and to each other. We define a proximity clearance for all baffle objects : q=5mm. Aesthetics and Wave Physics ask for a larger clearance of port with respect to side and bottom walls. We set this clearance equal to half of the inner width. It is better visualized as a red hemisphere around port's longitudinal axis of symmetry. Minimal baffle (inner) width is evaluated by the largest driver chassis diameter (in this case the woofer's diameter) plus 2 x q: Minimal Baffle Width = max{driver diameters)+2q In a similar sense we evaluate a minimal baffle (inner) height as : Minimal Baffle Height = sum{driver diameters)+3q+ Dv/2+Min.Baffle Width/2 As a result we can have an estimate of the (inner) Area of our minimal baffle configuration: Minimal Baffle Area= Min.Width x Min.Height This in turn can lead to the (inner) enclosure Depth which will produce the required net volume specification : Required Enclosure Depth= Vb / Minimal Baffle Area. The idea is simple: This value is actually the largest depth we can have in this speaker's cabinet design because we have assumed the smallest baffle configuration.We want to make sure that this maximal Enclosure Depth is feasible.

For example let us take the case of a small VB design where this depth is found to be 105mm whereas the required vent's length is 145mm. Obviously our vent will not fit. The Required Enclosure Depth is not feasible : our minimal baffle configuration is too large : our drivers/ports are too large for the net volume Vb we need.

-How do we check for the feasibility of the Required Enclosure Depth ?

 We create a drawing of the side view of our enclosure with the minimum dimensions we can have. We call it the minimal depth configuration. As shown on the picture on the right, we place our tweeters, woofers and ports as in the minimal baffle setup and define a proximity clearance p between the enclosure's back wall and the port's end (rarely we expect this clearance to be applied to a woofer driver). Though p can be arbitrarily selected we choose its value to be 1-2cm for drivers. For vents it better be set equal to Dv. We also observe that for each driver a part of its overall chassis depth is actually fitted in the baffle's wall thickness t. So within the enclosure only a length equal to (driver depth - t) really exists. The same holds for vent pipes flush-mounted on most speaker baffles. We form our minimum accepted value of (inner) enclosure depth as follows : Minimal Enclosure Depth = max{driver depth,Lv)-t+p This is the absolutely minimum depth value we can get with these drivers and/or ports. In that sense this value should not exceed the Required Enclosure Depth we evaluated in the previous step. If this happens no design can be achieved unless : a) we increase the required net volume Vb, b) change our selection of drivers, c) shorten our vent length. Again we must not forget that crossover PCB's and binding post terminals also need need space to be accounted for. That is why very small speakers are very difficult to design.

-How do we apply all these considerations to a VB speaker having two or more woofer drivers (and a midrange in a three way system) ?

Well it is not difficult to form the geometrical equations governing the minimal baffle and depth configurations. We present two figures illustrating the idea:

The case of a very small VB speaker can also be addressed with a different geometrical perspective:

Apparently such a design needs intensive elaboration.

-What about determining the final enclosure dimensions?

We have stopped our design process at the points where the notions of minimal baffle and minimal depth configuration were introduced and checked for. It is a good idea to use the values of Minimal Baffle Width,
Minimal Baffle Height and Required Enclosure Depth as a start.

Changing any of these three dimensions should respect the following 'set of the eight rules' :

### The Eight Rules of Vented Box Baffle Design

1.Width, height and depth are inversely proportional variables for the net volume Vb to be kept constant. Therefore an increase of any of these three variables should be compensated by a decrease of anyone of the other two.
2.Diffraction step loses its strength when baffle dimensions increase. Small baffle widths lead to loss of woofer very low frequency SPL response and to increased tweeter SPL irregularities.
3.Enclosure (inner) depth should never equal baffle (inner) width. Standing waves within the enclosure become very strong for the absorbing material to deal with effectively. It is also a matter of speaker cabinet elegance. We usually make speaker depth more than 20% larger than its width (Depth/Width more than 1.20 and less than 1.80).
4.The distances of a tweeter's center from the three nearby baffle edges should not be equal. This would create a strong diffraction and thus a very irregular SPL response. For this reason tweeters are usually placed in an asymmetrical way on the front baffle.
5.Tweeter's center defines speaker reference axis which is generally expected to point to the typical listening position. Therefore tweeter's height should match listener ears'  height from the floor. It is a good idea to set that height to 105-115 cm from the floor. Needless to say that in special design situations this height can be varied accordingly.
6.Woofers and vents should not be placed at very low heights above the floor. Their interaction with it creates severe artifacts at very low frequencies. In addition the overall speaker impression is not nice (to put it in a gentle way).
7.Keeping woofer, midrange and tweeter centers as close as possible eliminates spatial image smearing at high quality recordings of human voices and/or solo wind instruments.
8.In baffle design whenever technical reasoning compromises speaker elegance, failure is the outcome.

If we consider the formation of standing waves within a speaker's cabinet and the respective resonance SPL peaks being radiated out of the port's mouth we must pay attention to the baffle heights where woofer and vent centers are located.

### The Rules of Standing Waves in Vented Box Speakers

1. Woofers must lie as close as possible to baffle's middle height.
2. Vents must also lie as close as possible to baffle's middle height.
3. Rules 1 and 2 can be combined if a woofer and a vent are placed as close as possible and in a symmetrical way around the baffle's middle point.
4. If two woofers are included in a VB design a W-V-W configuration should be adopted : (Woofer-Vent-Woofer).

At this point the low frequency design procedure comes to its end. Inner and overall enclosure dimensions have been determined and placement of all front baffle objects (drivers) has been concluded.

An accurate technical drawing of our speaker cabinet should be issued and printed as an image or a .pdf document. All dimensions, distances, details, material specifications and special assembly instructions (if any) should be included even if this turns out to be a multiple page document.

It will save valuable time and money resources to help your 'wood expert' not provide his/her own solutions to questions raised during the cabinet construction process. Apparently your drawing should not allow for multiple interpretations at any point or detail.

Even if it doesn't sound well I strongly advice that your drawings have a version number and a date of issue. A drawing that evolves is a natural thing in speaker design.

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