Passive radiator speaker design – Box calculation example
What is a passive radiator ?
A passive radiator speaker design involves another speaker, but with no motor assembly. The magnet and voice coil are gone, but the cone and everything you can 
see from the front are all there. The passive radiator (Amazon affiliate paid link) subwoofer design is very similar to the bass reflex design, only instead of the vent or port, you have the passive bass radiator. Since they are very similar in design principle, they follow similar methodology and performance characteristics. However, certain differences do apply, some are good, some are not so desirable.
Advantages over bass-reflex :
- No more vent non-linearities (resonant pipe sounds).
- Air turbulence noise is no longer an issue, when air rapidly escapes the pipe at high volumes.
- No more high frequencies reflected out the port.
- Space efficient. Sometimes large ports are demanded which will make the enclosure that much bigger.
- Passive radiator speaker design is simpler and with fewer alignments.
Disadvantages over bass-reflex :
- Steeper roll-off.
- Less transient stability.
- Slightly higher cut-off frequency (narrower frequency bandwidth).
- Greater overall box losses (QL).
Passive radiators operate in conjunction with the active driver at low frequencies, sharing the acoustic load and reduce the excursion of the driver. Working similarly to a vent, the passive radiator will only add as much as they subtract. This means they will have the same advantages as the bass reflex port : higher power handling and lower distortion.
How does the passive bass radiator compare to the alternatives?
As we mentioned before, the response for the passive radiator speaker design is closely related to that of the bass reflex. The one thing that is unique to the passive radiator is the notch at the resonant frequency of the passive speaker. Usually, the resonance is located at the point where the response start to roll-off (f0), but for the passive bass radiator, this point is below the system’s cut-off frequency. This notch increases the slope of the driver low-frequency roll-off, and degrades the transient response.
As you can see from the graph, sealed has the smoothest roll-off, followed by bass-reflex with a steeper roll-off but with a broader frequency response. Passive radiator has the middle ground when it comes to frequency response bandwidth, but has the steepest roll-off slope. Notice the notch in the response curve, that we talked about earlier. The vented enclosure will have more low end than the PR box, but the passive radiator design will not suffer from pipe resonances and standing waves transmission problems.
Which speaker is best for a passive radiator speaker design ?
If compared to the closely related bass-reflex design, the passive radiator (PR) design is more picky when it comes to choosing you speaker. When it comes to passive radiator alignments, they are fewer options, compared to the ported design. Please read the article regarding the bass reflex alignments before continuing. When it comes to PR, you are restricted to 3rd order Quasi Butterworth (QB3), 4th order Butterworth (B4) and 4th order Chebyshev (C4). This is because any driver with a Qts higher than 0.5 will produce a C4 response with a high amount of ripple (large peak near cut-off).
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If you want to get a flat response you will need a Qts that is in between 0.2 and 0.35 values to get a nice QB3 design. Usually when you making a passive radiator speaker design, you will probably use a PR that is the same size as the active speaker. This means it has the same cone area, weight and compliance. However, if the radiator is made more compliant than the active speaker, higher Qts alignments will be more acceptable and are similar to the bass-reflex conterparts, with acceptable transient response.
Parameters used for designing the box
Before you get started you will need the following parameters, by either measuring them or by getting them from the manufacturer :
- fs – resonant frequency of the active speaker.
- Qts – Total Q.
- Vas – equivalent compliance.
- Sd – effective area of the cone.
- Vd – volume displaced by the speaker.
- δ (delta) – the compliance ratio of the passive radiator.
Using at home measurements to calculate δ is a challenging and tedious task. Hopefully the passive radiator manufacturer has provide enough information for you to calculate it without any hassle.
Please review the following parameters :
- Cab – the acoustic compliance of the enclosure.
- Cap – the acoustic compliance of the PR.
- Vb – Volume of the enclosure.
- Vas – Equivalent compliance of the driver.
δ = Cap / Cab
α = Vas / Vb
Alignments table
The following list of alignments of passive radiator speaker design takes into consideration that QL = 7 and δ = α. So to spare you any tedious calculations and complications, when you choose your passive radiator and active speaker, make sure they match each other. If they have the same Vas and Cms you are good to go and then you can use the table below. By doing this you don’t have to calculate δ.
Now let’s use a real world example of active-passive speaker combo, to make an enclosure following the alignments table. For our example, we will use two speakers from Scanspeak. Here are the links to the technical information :
Passive radiator calculator – enclosure design
As you can see, both Cms and Vas are the same for the two speakers. This means that δ = α and we can safely use the information in the alignments table. Qts = 0.31, so we use the information and that line to calculate the following parameters :
- Volume of the box : Vb = Vas / α = 95 / 2.78 = 34.17 L.
- Tuning frequency : fB = H * fs = 1.51 * 21 = 32 Hz.
- –3 db point : f3 = (f3/fs) * fs = 1.59 * 21 = 34 Hz.
- PR displacement : Vpr = (Vpr/Vd) * Vd = 2.35 * 0.00044 = 0.001034 m3
- Vd = Sd * Xmax = 0.0352 m2 * 0.0125 m = 0.00044 m3
Of course you don’t have to follow these alignments and make the enclosure to your desired response and tuning frequency. To alter the tuning frequency, you do it similar to the bass reflex enclosure. Instead of modifying the length of the port you modify the mass of the passive radiator. Adding clay or swaping the dust cap for a different material. Here is a Seas passive radiator, which shows how adding mass to the cone changes resonant frequency.
Conclusion on the passive radiator subwoofer design
The passive radiator is intended to be used as an alternative to the ported design. Pro’s and con’s have already been discussed and you should read them carefully to make a good decision if the passive radiator speaker design is for you. Also take note of the price. The passive speaker will be cheaper than the active one because they are similar in most aspects, but the passive one is missing the motor design. While this is true, there is no question that a passive speaker is more expensive than a piece of plastic pipe (reference to bass reflex).
References
- Loudspeaker Design Cookbook 7th Edition by Vance Dickason (Audio Amateur Pubns, 2005). (Amazon affiliate paid link)
- Audio Engineering Explained by Douglas Self (Taylor & Francis, 2012). (Amazon affiliate paid link)
- Image source : link.
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12 comments
Awesome explanation!
Why is there a notch for passive radiator response compared to bass reflex, doesnt the response from the passive radiator add up similarly to the active speaker and vent in case of bass reflex?
The notch shouldn’t be of much concern as it happens way below the system cut-off (at least if it’s well designed). The only thing that is important is that PR roll-off is steeper than bass reflex. This means bass reflex will have an extended frequency response. While the 2 have similar concept design, the passive radiator has a suspension, which gives it a certain stiffness. Bass reflex uses the air inside the port which moves “freely”.
If you are interested it’s because of the way the active and passive driver couple basically as you pass across the resonant frequency of the pr the phase alignment of the drivers changes from in phase below resonance to anti phase above. This creates either constructive or destructive interference depending on the phase. The notch occurs due to this destructive interference
I have an idea. What about a hybrid combination port/PR design. For example, use a port plus a PR in an enclosure to get the best of both worlds. The port could be separate from the PR, or part of it. The port could even be in the center of the PR. I have no recollection of any mention of this concept in any reading ever. I think it is a concept worth exploring.
Bass reflex port design has only 80Hz to 300Hz of bass frequency.. Sealed soundbox speaker has a 15Hz to 200Hz of bass.. Passive radiator soundbox has 20Hz to 200Hz of bass.. So the Sealed box and Passive radiator box are the two Best design.. Got it.. Tight design soundbox.. Bass reflex sound ugly and sounds like open-baffle bass, not a tight bass.. Bass reflex is not good, only 80Hz, and bass reflex doesnt have the 40Hz sound.. Dull bass sound.. Not rich bass.. The Aiwa subwoofer design in 1998 is very excelent..
Yes. You right. I built my speakers based on Silicon Chip (if I’m not mistaken July edition) 2014. Using a 15″ woofer. It’s two-way. It’s a leaky closed speaker bcs thr’s a bit gap to let the air pressure to come out. It can reach 15 Hz according to the magazine test report. I can easily hear the bass from contra bass in a classical music jst at volume level 1 from my 30 W amp.
What you’re saying doesn’t make any sense. A Bass reflex box can be tuned to 20Hz, just as a passive radiator box can be tuned at 80Hz. It all depends on the area and length of the port (or size and weight of the passive radiator).
I think you are confused about the formulas and the table which gives a specific result. Those are alignments. Bass reflex has alignments also. If you want a Butterworth response (Maximum flat response) , or a Linkwitz Riley (best transient response) you simply follow the numbers.
OR … you do it like you said, you tune to whatever frequency by altering the port and box dimensions.
What if you make a closed box with a normal driven bass speaker and exactly opposite facing in the back of this closed box a passive radiator like a KEF BD139. Now from the outside of the radiator you start to make a horn, with an average length around 2m folded off course, like an floor standing box. Would this be a nice idea? The horn will be driven with air waves generated by the passive radiator, these air waves travels through the horn resulting rising SPL and depth in frequency….. Make horn length 1/4 lambda of your lowest frequency and voila!! I go experiment with some KEF drivers that I have laying around!!
I don’t see why you couldn’t do that. Horns are not enclosures, they are acoustic couplers. You can add horns to anything : speakers, ports, passive radiators etc.
Good explanation of PR design, etc. But what the fuck is with some of these comments? If you like sealed, fine…but sealed isn’t the end all design…just ask the bass horn guys. These audio articles shouldn’t allow comments…..too many douchebags with ignorant opinions.