The Subwoofer DIY Page v1.1
Transmission Line Systems : Design Notes
last updated: 15 February 2018

Introduction:
For these design notes, I will step you through the design process. My tool of choice for designing transmission lines is Hornresp.  There may be better tools out there, but Hornresp is simple to use (once you learn how to use it!) and the results are close enough to be useful for most purposes.

Choosing the driver:
For this example, I'm going to use the Dayton Audio PA310.  This is a cheap but decent quality 12" pro audio driver with a nice low Fs and Qts, making it suitable for a number of different designs. 

The specifications of the PA310 are as follows:
Sd = 530.9 cm^2
Re = 5.2ohms
Fs = 39 Hz
Vas = 99.27 l
Qes = 0.31
Qms = 7.51
Qts = 0.30
Le = 1.4 mH
Xmax = 5 mm

"Classical" transmission line theory suggests that an appropriate transmission line configuration for this driver would be one that is long enough to resonate at 39 Hz, and has a constant cross-sectional area of 530.9 cm^2.  But what would the response of such a design be like?

Enter Hornresp, which can be used to simulate such a build. 

The following image shows the parameters for a Hornresp sim of this "classical" type of transmission line.  Note: to enter Cms, Mmd, BL and Rms for the driver, simply double-click on the "Sd" text box - a dialog box will pop up asking you for the t/s parameters for the driver, and Hornresp will use these to calculate the other parameters it uses.  Note also that I have selected an "OD" (offset driver) simulation for this build - the reason for this will be explained later!

Hornresp sim - strailght TL

A brief outline of what's shown in the sim above:
Ang - for comparison purposes, this shouild always be set at "2.0 x Pi", which basically represents a speaker sitting on the floor
Eg - the voltage that's being applied to the sim, 2.83V
Rg - the resistance of the cable (which I've set to zero for the purpose of this sim)
S1 - the cross-sectional area at the start of the transmission line (in cm^2)
S2 - the cross-sectional area where the driver is located (in cm^2)
Par (1) - the distance between S1 and S2 (in cm)
S3 - the cross-sectional area at the end of the transmission line (in cm^2)
Par (2) - the distance between S2 and S3 (in cm)

So far, so good.  At this point, we can select "Tools...Loudspeaker Wizard", and check what this sim is going to look like

Hornresp sim - classical TL schematic

Ok, that looks like what a "classical" TL would look like, with the driver (the red circle) at one end and the vent at the other. Now, let's see what the predicted response of this sim looks like...

Horrnresp sim - classical TL - response

Ugh, that looks pretty bad.  The resonance frequency is where we expect it to be, but there's this massive notch in the resonse between 100 Hz and 200 Hz.  Why is that?  Well, because of resonances along the transmision line.  That's because the transmission line does not resonate at one frequency - it resonates at many frequencies. We can see what the result of this looks like by looking at "Output 1", not the "Combined" output...

Hornresp sim - classical TL - vent response

So, we're now looking at the response from the vent alone.  See that big peak between 100 Hz and 200 Hz? that's one of the resonance frequences of the straight transmission line, and if look closely you'll see that it's three times the lowest resonance frequency, 39 Hz, which works out to 117 Hz.  This resonance is out of phase with the driver's output, which results in the big notch that you see in the "Combined" response.  This is why building a transmission line subwoofer with a constant cross-section with the driver at one end and the vent at the other is so bad - it introduces this huge notch in the response that basically limits the effective passband of the subwoofer.

So, how do we deal with this notch? Well, one effective way is to simply locate the driver partway down the line rather than right at the end.  Let's see what happens when we do that...

Hornresp sim - classical TL - offset driver

So here's what the response looks like with the driver moved to 73.1 cm down the line (now do you see why I chose the "Offset Driver" type of Hornresp sim?). The notch between 100 Hz and 200 Hz has now been eliminated, and the sim shows a response that extends up to 200 Hz before the next notch appears.  Note that there is a peak at 200 Hz, but in reality this peak will not be so high as Hornresp does not include the impact of box losses in its sims.

At this point we could stop here, but I want to refine this build a bit.  First of all, a resonance frequency of 39 Hz is a bit too low for this driver (it only has 5mm Xmax, and will run out of excursion fairly quickly in its passband in such a large box).  There is also a small hump in the response at the lower end of the passband, which suggests that the response of the subwoofer will be slightly "under-damped" at that point. Some people don't mind this.  I do, so I tend to adjust my sims to eliminate it.  So, let's adjust the sim to bring the lowest resonance point up to 48 Hz, and see if we can remove that slightly under-damped response as well. To do the former, we need to shorten the path, and to do the latter we need to decrease the cross-sectional area of the transmission line.  

Hornresp sim - classical TL - 48 Hz

Ok, this looks a bit better.  The under-damped response at lower frequencies is now gone, and the resonance frequency has been moved up to 48 Hz.  The takeaway for this is (1) you don't HAVE to set the resonance frequency of the transmission line to be the same at the resonance frequency of the driver, and (2), you just might get better results using a smaller cross-sectional area than "classical" transmission line theory suggests (in this case 411 cm^2 instead of 530.9 cm^2), and end up with a smaller box with better performance to boot. 

The challenge at this point is going to be trying to fold up this transmision line into a box that puts the driver in the correct location down the path. Sometimes this can turn out to be quite difficult to do, as the driver might end up being having to be placed around a corner to be at the correct distance from the start of the path, and this might be impossible to implement. In this case, the driver will have to be located 59.3 cm from the start of a path that's a total of 168.6 cm long, so if the line is folded to have the vent come out on the same side of the box as the driver, we might run into this difficulty.  So, what can we do?

Hornresp sim - classical TL - addtional segment

To tackle this problem of driver placement, I've modified the Hornresp simulation a bit.  I've split it into three sections, so instead of S1, S2 and S3, I now haveS1, S2, S3 and S4, where S4 now represents the cross-sectional area at the vent.

I'm now going to change the transmission line from a straight transmission line to a tapered one by adjusting S1 and S3 and load it with a straight vent represented by section S3-S4.  This converts it into a mass-loaded transmission line. Note that there is an "Auto" right next to S2 - this means that Hornresp is automatically calculating the value of S2 for me (to keep the taper from S1 to S3 constant). The default is "Manual" (self-explanatory), but by double-clicking on S2, I've changed it to "Auto".

Hornresp sim - MLTL (1)

Note now that the driver is now at 49.6 cm from the start of the transmission line, which may be short enough to ensure that it doesn't have to be located on a corner (if the line is folded). So, what does the response of this modified version of the transmission line look like?

Hornresp sim - MLTL (2)

That actually looks a bit better.  Not only does the response at the low end remain the same, but the response at high frequencies is a bit smoother - the notch in the response at 300 Hz has been moved up to around 350 Hz.  In fact, you will find that if you increase taper (by increasing S1) and adjust the rest of the parameters to ensure that the resonance frequency remains the same, the aberrations at the upper end of the passband will move further up in frequency.  The downside here is that you will likely have to reduce the cross-section of the vent in the process, and this will reduce the output of the subwoofer at high SPL levels.  To prevent this "vent compression" effect, I recommend not reducing the vent cross-sectional area any less than 2/3rd of the driver's Sd, and keeping it above that if possible.

Note that so far I have simulated an unstuffed transmission line.  Hornresp does include a way to simulate the effect of stuffing part or all of the box.  I'll leave it up to you to figure out how to use this feature.