4th order bandpass calculator – With box design example
How to build a 4th order bandpass subwoofer box in 7 steps?
The 4th order bandpass box design is not recommended for the first time builder. You need some experience before you get into these complicated designs. Actually, if you are a good listener and/or are savvy with enclosure design software, you shouldn’t have much of a hassle when making this kind of project. The problem is, even if you conceive a good 4th order bandpass design, but have slight building errors, the result will be sub-par. Now that you are all encouraged, let’s learn more about these bandpass enclosures.
What is a 4th order bandpass box ?
The term bandpass is also used in passive crossovers. Usually used for midrange drivers, to filter low frequencies and high frequencies at the same time, only to let the frequencies between the crossover points pass.How is this passive filter example relevant to the bandpass speaker enclosure ? It is, because when you look at the frequency response of a bandpass subwoofer, it looks exactly like the frequency response of a speaker with a bandpass filter on it. Imagine the frequency response of sealed or bass reflex enclosure. For the very low frequencies, when the resonant frequency is reached, the response starts to roll-off. Now, for the higher frequencies, the response will naturally start to roll-off, at different points, depending on the speaker. Because of the size of the woofer, it will be impossible for it to play high frequencies at some point, so it naturally rolls-off.
However, the 4th order bandpass design acts differently. It is composed by two chambers : a rear sealed chamber and a front vented chamber. You can think of it like a normal closed box system, but the vented chamber in front of the speaker acts like an acoustic filter, a low pass filter to be more precise, so higher frequencies are filtered depending on how it is designed. Because of this filtering, the frequency response bandwidth is quite narrow. You can make it broader, but you will sacrifice efficiency. Or you can make it even more narrower and increase efficiency by a lot. This seems appealing for those who want to play a narrow frequency range at very high volumes (like car audio SPL competitions). Sealed and bass reflex can make this kind of trade-off also, but bandpass has a better “yield”.
How to make a bandpass box ?
Further on, I will explain how to design your box without any design software. Your pen and paper will be the actual 4th order bandpass calculator. This will get a bit technical, but I’m sure you will figure it out. Before you start off your project, you should first select a woofer. Anything that will work well in a sealed box, will work well in 4th order bandpass design as well. If you want to crunch numbers when choosing your speaker, you can divide fs / Qes , and the smaller the number, the better (lower than 60). This means a good bandpass speaker will have a low fs, or high Qes, or both.
First of all, let’s define some terms, so we can speak the same language :
- fB – Tuning frequency of the port in the vented chamber.
- R – The radius of the port in centimeters.
- Lv – Length of the port in centimeters.
- fL – f3 of the low frequency roll-off.
- fH – f3 of the high frequency roll-off.
- Qbp – Q of the sealed rear chamber.
- Vf – Volume of the front chamber (the acoustic filter chamber) in liters.
- Vr – Volume of the rear chamber (the sealed chamber) in liters.
- S – Passband ripple (How many ±db does the frequency response deviate from linear frequency response).
- Vt – Total volume (Vf+Vr) in liters.
As you can see from the graph above, fL and fH are positioned -3 db after the response starts to roll-off (for low and high frequencies). The passband ripple measures the amount of variation from linear response. High amount of ripple will result in peaks / dips in the frequency response. Judging from our graph the ripple looks like around ±0.5 db.
4th order bandpass calculator in 7 steps
We are going to explain, step by step, what are you supposed to do in designing a great sounding box. To make things more clear, we are going to use a real life woofer, and design a bandpass subwoofer box for it. The woofer I chose is the JL 10TW3-D4 . There is no particular reason for my pick, other than it has good specs for a sealed enclosure, which means it will be good for bandpass as well.
I’m going to list the Thiele / Small parameters of the JL woofer for convenience :
- Fs = 32.3 Hz.
- Qes = 0.656 .
- Qms = 11.35.
- Vas = 19.82 L.
- Qts = 0.62.
- Xmax = 15.2 mm.
Step 1 : Know what do you want
You have to understand that there is no perfect bandpass enclosure for each kind of woofer. You have to make compromises between linear frequency response, how wide the frequency response is, and efficiency. General rule of thumb is that the louder it is, the narrower the frequency response is. Depending on what you want to achieve, you will have to find a balance between these 3 aspects, so that you will end up with a 4th order bandpass design that suits your needs.
Step 2 : Determine how linear the frequency response should be
Remember, at the beginning of the article, we talked about some parameters that are relevant to this project. One of them is S, which is the passband ripple. This ripple, describes how many ±db, the frequency response will digress from linear response. In a perfect world the ripple would be ±0 db, which is achievable. However, that is at the expense of the other 2 characteristics we need to take care of, in the later steps.
There are 3 values of S, which give an acceptable amount of ripple, and have the following characteristics :
- Best transients for S = 0.7, and 0 db ripple.
- S = 0.6 , somewhat degraded transients , ±0.35 db ripple.
- S = 0.5 , worse transients than S = 0.6 , ± 1.25 db ripple.
Picking a certain value for S, will narrow your possibilities for the other factors we need to figure out. Another guideline which is useful when choosing S, is that if you go for a bigger value (0.7), the frequency response will be narrower. On the other hand, a lower value S (0.5), would translate into a wider frequency response. So, choose the ripple you like, but keep this in mind when doing so.
Here is a table with the values of S, and the values of the frequency response and sensitivity, corresponding to that certain S value. Get familiar with this table, as we will need it for the next step as well.
- For our woofer we will choose S = 0.6.
Step 3 : Determine the desired frequency response bandwidth and sensitivity
Now you have to determine the values of fL and fH. This means you will determine the – 3 db points, when the response starts to roll-off, for both low frequency and high frequency roll-offs. This will effectively set your frequency response bandwidth between the two values. You figured out the value of S, at the 2nd step, now you will chose the values of fL and fH, corresponding to that S. After that, the value of the sensitivity is chosen for you. If you choose a certain value for the sensitivity, then the values of fL and fH are chosen for you. This is the balance I talked about, that you need to make, to find the sweet spot.
To find the values of fL and fH you have to do the following :
- fL = Fs / Qts * (fL factor)
- fH = Fs / Qts * (fH factor)
Now let’s do some number crunching for our woofer :
- I’m choosing the sensitivity to +5 db because I want a loud bandpass enclosure.
- This means that the start and end of the frequency response are chosen for me.
- fL = 32.3 / 0.62 * 0.7317 = 38 Hz.
- fH = 32.3 / 0.62 * 1.6877 = 88 Hz.
So, we have figured out that the frequency response will be from 38 Hz to 88 Hz, with a +5 db boost and ±0.35 db ripple.
Step 4 : Calculate the volume of the front enclosure
Calculate the front chamber volume using the following formula:
Vf = (2S * Qts)2 * Vas
- Vf = (2 * 0.6 * 0.62)2 * 19.82 = 11 L
Step 5 : Calculate the volume of the rear enclosure
Calculate the rear chamber volume using the following formula:
Vr = Vas / ((Qbp / Qts)2 – 1)
- Vr = 19.82 / ((1.1113 / 0.62)2 – 1) = 19.82 / (3.21 – 1) = 9 L
Step 6 : Calculate the tuning frequency of the front chamber
Calculate the tuning frequency using the following formula:
fb = Qbp * ( Fs / Qts)
- fb = 1.1113 * ( 32.3 / 0.62) = 1.1113 * 52.1 = 58 Hz
Step 7 : Calculate the dimensions of the port
The radius of the port (R), should be as large as possible, to minimize port non-linearity. Understand that making the port bigger, will mean that the length of the port will be longer. This means there are certain limitations to how large you can go.
- For our project I’m choosing a port radius of 5 cm (10 cm in diameter).
- R = 5 cm.
Calculate the port length using the following formula:
Lv = ((94250 * R2) / (fb2 * Vf)) – (1.595 * R)
- Lv = ((94250 * 52) / (582 * 11)) – (1.595 * 5) = (2356250 / 37004) – 7.98 = 63.68 – 7.98 = 55.7 cm
Results for our 4th order bandpass calculator
Now we have finished making a 4th order bandpass design box for our 10″ JL woofer, and the dimensions are as follows :
- Front chamber = 11 L.
- Rear chamber = 9 L.
- Port diameter = 10 cm.
- Port length = 55.7 cm.
Please bear in mind, that these are the net volumes. This means that you will have to add the volume displaced by the speaker, to the volume of the rear chamber. Also the volume displaced by the port needs to be added to the volume of the front chamber. Add any other elements to the total volume (like bracing).
The frequency response will look like the following graph :
Variations of 4th order bandpass enclosure
In the example above, we used just one speaker, and made a 4th order bandpass design. If we use 2 speakers, we can place them in different positions so that we end up with different variations of bandpass enclosures. Here are some variations of the 4th order bandpass enclosure:
- Single driver bandpass subwoofer box.
- Dual driver push / pull bandpass enclosure.
- Push / pull compound bandpass box design.
- Triple chamber bandpass subwoofer box.
- Push / pull triple chamber bandpass box design.
For the dual driver push / pull variation, calculate the volume of the front and rear chamber, for each individual driver, and then add them up. For the triple chamber bandpass subwoofer box, the rear chambers are separated, and therefore calculate them normally. The 2 drivers are sharing the center chamber. So calculate the volume of the front chamber for each driver individually, and add them up, to get the volume of the center chamber. When there is a push / pull configuration, remember to connect one of the drivers out of phase electrically (reverse polarity).
Conclusion on bandpass box design
The 4th order bandpass box design is definitely an interesting solution. If you do not need a wide frequency response, and want a boost in output, you should seriously consider it. However, the enclosure can get quite big, and you don’t have direct access to the speaker. As a result, if you need to replace the speaker, you have to tore open the enclosure. The design and build difficulty can be a let down for the inexperienced builder, but if done properly, the 4th order bandpass design can be quite impressive.
- Loudspeaker Design Cookbook 7th Edition by Vance Dickason (Audio Amateur Pubns, 2005).
- Introduction to Loudspeaker Design: Second Edition by John L. Murphy (True Audio, 2014).
- Image source : link.