Everything you always wanted to know about cardioids* (*but were afraid to ask)

[HammerSandwich]

I can't promise this thread will have all that, actually. But I'll start with a few answers to questions from the Buchardt S400 thread, and hope that the ASR members who have theoretical & practical experience also post. There are at least a few of you...

The cardioid effect occurs over a limited bandwidth, with the controllable frequency range related to the spacing between the drivers. Easy example: because the Kii's designers wanted ~3 octaves of cardioid response, they placed the side-mounted drivers closer to the front-mounted mid than to the woofers on rear. IOW, your hands are tied if using 2 drivers in a single cabinet.

Interesting, thanks. Can you provide further info (math) or references on this?

AFAIK, Olson's Gradient Loudspeakers was the first real study. Geddes discusses cardioids, with math obviously, in his Audio Transducers book, which should be in PDF on the Gedlee site. Might be worth looking at the archives for Linkwitz & MusicandDesign (aka Nao), too. You can find a bunch of info - in various levels of complexity - from the pro-sound folks. Finally, @kimmosto has a ton of practical results, plus excellent simulation software, at kimmosaunisto.net.

 

[HammerSandwich]

We can understand the concepts a bit without getting too deep. Like much in acoustics, it's fundamentally about distance versus wavelength, connected by the speed of sound.

Because the 2 sources are in fixed locations, their phase relationship is different at every wavelength/frequency AND also across the polar pattern. This develops exactly the same sort of complex interference pattern as monopoles or dipoles; it's only the exact details that change.

I think I've set this up correctly in VituixCAD.

• A pair of imaginary drivers with perfectly flat frequency response & omnidirectional dispersion, i.e., ideal point sources.
• 200mm spacing between them. This is good to experiment with, but 200mm covers a useful span of midrange.
• Because the 2nd driver is 200mm farther from the imaginary microphone, its amplitude is increased 0.9dB to match the main driver's. (In the real world, you need to EQ to match FR as well.)
• We'll keep all of those things constant while altering the 2nd driver's delay & polarity.

Monopole - no delay, positive polarity. This shows essentially perfect omnidirectional behavior up to 500Hz. Higher, we see massive comb filtering on the main axis & serious lobing elsewhere.

Dipole - no delay, inverted polarity. Massive cancellation below ~250Hz, an octave below the monopole's transition. A nice dipole pattern covers the 2 octaves from 300Hz to 1.2kHz. The same combing & lobing above, but nulls & peaks switch. No surprise that this matches the monopole's results only reversed.

Cardioid - 580us delay (200mm spacing, or 1 wavelength at 1.7kHz), inverted polarity. The gradient behavior again drops by an octave or so, giving a good pattern from roughly 150Hz to 700Hz. Combing & lobing yet again, though a very narrow null directly behind the speaker remains clean.

The driver spacing causes LF reinforcement when both have the same polarity, while cancelling in designs with one inverted. Just as in a dipole, the cardioid's secondary driver nulls the main output at all angles, once the wavelength becomes long enough to overwhelm the geometry. (Note that both Kii & Dutch^2 use monopole subs below 100Hz or so.)

Regardless of the configuration, 2 drivers with this spacing will exhibit major lobing at higher frequencies. The waves simply cannot combine properly when the sources are a significant fraction of a wavelength apart.

So, covering 2 octaves isn't too hard, 2.5 appears reasonable. More than 3? You're gonna need 2 cardioids or another way to control directivity. (Note that this bandwidth fits a 4-way, which is pretty much where all of the serious DIY - read "doesn't need to be marketable" - builders end up. If you're beginning to suspect that physics & our hearing have conspired just to annoy loudspeaker designers, we're on the same page.)

 

[HammerSandwich]

After seeing the polar maps, it's hard to believe that any of those designs could work, right? But remember that both drivers were run full range. In the bass, simply roll off the nulling driver with your crossover. To avoid the lobing, hand those frequencies off to a waveguided tweeter. (What? You went to the effort of making a cardioid mid and didn't plan to use a directional tweeter?!??)

So, cardioid + DSP. I added one peaking filter to flatten the axial response, then spent a few minutes playing with XO frequencies. Everything else is the same as the previous post's cardioid.

BAM! About 2 octaves of clean cardioid, well controlled to +/- 90 degrees or so. Omnidirectional bass, starting around 100Hz.

Now, where can I buy 2 Platonic point sources?

 

[HammerSandwich]

See Low Frequency Directivity Control at Aalborg University.

 

[q3cpma]

Since you seem to know your stuff, do you know how ME Geithain manages to be completely cardioid while others only manage down to 100Hz in the best case? I guess there's some patent stuff going on.

 

[HammerSandwich]

FWIW, MEG's products all claim "cardioid characteristic in the bass range." Checking just a couple different models, I see ~30-250Hz, so not full-range. Getting enough bass without making the speakers enormous must be the challenge!

Anyhow, I am very ignorant about passive cardioids, but they do seem to manage wider bandwidth.

Dutch 8C covers a full 3 octaves, for example. The 8C's ports are closer to the mid than in my example, so that moves the high end up. There are also several variables with the passive filtering for the rear wave, and I believe this explains most of the bass control. Easy example: if your damping progressively low-passes the rear wave as it travels away from the cone, you can make higher frequencies leak out closer to the front with only bass escaping farther back. IOW, the low-pass filter can act as a variable-distance control for the nulling output. It's a neat trick, because you effectively create >1 cardioid, 100% passively!

This is way out of my realm. Kimmo's pages linked up top have a lot of examples, and @Martijn Mensink documented much of the 8C's development. I promise he's already forgotten more about passive nulling than I ever knew...

Regardless, you won't see many octaves of consistent cardioid/dipole performance with just a couple of drivers. Once baffle width & driver beaming effects appear, everything changes, which explains the LX521's wild baffle. You won't see any of that with the ideal point sources I simmed above. I managed a decent cardioid mid in an hour, but that's a far cry from building a real one with many more interactions in play.

 

[HammerSandwich]

BTW, building an active cardioid is a totally doable experiment. If you're interested in the topic, try it!

1. Start with 2 matched speakers, preferably small. A stereo bookshelf pair is ideal.
2. Place them fairly close to each other to start, but plan to experiment with this.
3. Measure the center-to-center distance between the woofers.
4. Now you need a way to duplicate a channel & time-delay one of them. EQ APO is an easy answer for Windows.
5. Send a normal signal to the main driver.
6. Invert the second driver's signal AND delay it by #3's distance divided by the speed of sound. (344m or 1130' per sec.)
7. Add EQ and gain, if you're trying for a better null.

It won't be pretty, but this method will demonstrate the effect.

 

[HammerSandwich]

...do you know how ME Geithain...

Sound and Recording tested the MEG RL 944K1, and the results provide some details. Halfway down the page, you can scroll through the measurement graphs.

1. FR is pretty flat, both individual drivers & summed. The HF bumps aren't related to the cardioid.
2. Whoa! Not only is the woofer running hotter than the other drivers in its passband, it sees 12dB of boost. "Getting enough bass" was the challenge, all right.
3. Raw driver response confirms this, with rolloff starting about 100Hz.
4. THD versus SPL surprised me. This isn't a loud loudspeaker, but it manages more than I'd expected from the earlier data. Perhaps a limiter to reduce the bass boost when needed?
5. Skip to the horizontal polars, and we find two more details. First, the 944's cardioid works significantly higher than the 250Hz spec. Second, it reduces off-axis SPL around 3-9dB from 100-400Hz. This is useful - far better than a typical 8" box - but less than Dutch or Kii manage. IIRC, those speakers produce 12-15dB reductions.
6. The final chart of polar maps confirms this. The cardioid works deeper into the bass, but less effectively.

How does MEG do it? The same way everybody else does - with a whole bunch of engineering tradeoffs.

 

[q3cpma]

Well, the 944K (even more the K1) IS a compromise and with sealed enclosures, you're gonna get that kind of driver boost. Note the clean cumulative spectral decay, by the way. The RL901K is where it's at (https://www.me-geithain.de/en/rl-901k2.html); since we now see that the official specs are quite accurate, this makes it quite better: -10 dB at 32 Hz and -12 dB over 80 Hz.
Also, I REALLY like that manly utilitarian look.

 

[q3cpma]

By the way, at what frequency point is the box big enough to contain the waves so that it becomes naturally cardioid? Must the wavelength be compared to the box depth or its largest dimension?

 

[Soniclife]

Second, it reduces off-axis SPL around 3-9dB from 100-400Hz. This is useful - far better than a typical 8" box - but less than Dutch or Kii manage. IIRC, those speakers produce 12-15dB reductions.

Do you have a source for the 12 - 15 dB?

 

[HammerSandwich]

By the way, at what frequency point is the box big enough to contain the waves so that it becomes naturally cardioid? Must the wavelength be compared to the box depth or its largest dimension?

Sounds like you're talking about baffle step. That transition occurs over several octaves, centered (-3dB) where the baffle's dimension is about 1/3 of a wavelength.

Do you have a source for the 12 - 15 dB?

Audio.de's 2015 Kii review shows FR directly behind the speaker. Audioxpress included a polar map in their 2017 review.

For the 8C, I've trusted the polar map in the official spec sheet, as well as the DIYAudio thread linked above. All of the other specs seem reasonably confirmed by Soundstage/NRC & Stereophile, so that feels safe to me. If someone knows of an independent polar map, please post it.

Polar results for other cardioids would be good, too! We know that MEG shows some. Many pro-audio companies should have CLF data or similar. Who else? Amphion? Gradient?

 

[dc655321]

do you know how ME Geithain manages to be completely cardioid while others only manage down to 100Hz in the best case?

Design and implementation efficacy. Have a look in the pro domain - different goals and designs to suit.

For any cardioid one needs:
1) a secondary source 180 deg out of phase with the primary source
2) amplitude matching between sources
3) a source of delay (distance, electronics, etc)

Dutch 8C covers a full 3 octaves, for example. The 8C's ports are closer to the mid than in my example, so that moves the high end up. There are also several variables with the passive filtering for the rear wave, and I believe this explains most of the bass control. Easy example: if your damping progressively low-passes the rear wave as it travels away from the cone, you can make higher frequencies leak out closer to the front with only bass escaping farther back. IOW, the low-pass filter can act as a variable-distance control for the nulling output. It's a neat trick, because you effectively create >1 cardioid, 100% passively!

That "neat trick" is exactly what I was referring to in the Buchardt thread with my "semi-clever DSP" comment.
One needs a filter on the secondary source with a group delay that varies such that destructive interference is achieved across a bandwidth of interest. It is also the same principle used in the link you provided above, albeit actively, via DSP. From that thesis' abstract:

An analytical study of traditional setups for controlling the low frequency directivity is made, completed by Finite-Difference Time-Domain simulations. Two solutions to improve the directivity control are proposed. One based on an all pass filter delaying the signal by different amounts at different frequencies and another base on the difference of gain of the signal applied on the different subwoofers.

Looks like the author experienced some unexpected results. An interesting read none the less.
Thanks @HammerSandwich for that link and the other information you've provided. Much appreciated.

To come back to the D&D 8c approach, here again is that measurement I previously posted. The image is from a patent application for the predecessor to the 8c, a pro-sound design with waveguide, compression driver, and 12" woofer (pictures of that are in the patent and on diy audio somewhere. can probably dig out if anyone is interested).

"204" is the on-axis measurement in front of the DUT, "202" is the bandwidth of interest, and "206, 208, 210" are measurements at 180 deg as a function of depth of inner panel: i.e. the depth behind the (cardioid) driver, internal to the enclosure. It is interesting to note that there is an optimum depth, "210", that lies between the closest back panel position (206) and the farthest position (208).

Other than back-wave damping material and thickness, this tuning of resonant condition via back-panel depth is one of the few control knobs afforded via a passive cardioid design. The uncertainties in those variables is, to me at least, what gives an active cardioid design slightly more appeal. There is no denying the elegance of the passive approach though...

 

[HammerSandwich]

That "neat trick" is exactly what I was referring to in the Buchardt thread with my "semi-clever DSP" comment.

Apologies for underestimating the question. Honestly, I need to reread all of this stuff.

There is no denying the elegance of the passive approach though...

Right on! You have to respect Martijn's comments about avoiding cabinet issues by filtering/leaking them, especially when done without adding more parts. Successful "less is more" always impresses.

 

[q3cpma]

Here's an interesting technical interview with ME Geithain's boss about the subject: https://www.me-geithain.de/en/assets/media/products/studio/RL901K/en/PPThomsen-RL901K-02-05-eng.pdf

 

[dc655321]

Apologies for underestimating the question. Honestly, I need to reread all of this stuff.

No apology required - it was a pretty opaque comment on my part.
F'ing hindsight...

 

[sarumbear]

Here's an interesting technical interview with ME Geithain's boss about the subject: https://www.me-geithain.de/en/assets/media/products/studio/RL901K/en/PPThomsen-RL901K-02-05-eng.pdf

That is very interesting and explained (at least to me) how Geithain does it; they use an extremely complex acoustic filter. No design I know of - and I have an MA on acoustics - can achieve this. There is even a second enclosure inside the main one! The designer is still not giving away much information but it’s clear that they developed an acoustic filter that was not yet seen anywhere else.

Thank you for the link.

If you thought it was very simple to achieve cardioid design at the bottom octaves do please read the article.

 

[Kvalsvoll]

Using acoustic ports and tuned internal chambers is not unique, though it seems very few designs utilize what is possible to achieve - not only for radiation pattern, but also for better internal resonance control and more controlled acoustic port output.

It is possible to design such enclosures to match a chosen radiation pattern, and so far this is one alternative to very large horns, when pattern control beyond high-mid frequencies is desired. It is possible to design a very small speaker with controlled pattern down into the bass range.

 

[sarumbear]

Using acoustic ports and tuned internal chambers is not unique, though it seems very few designs utilize what is possible to achieve - not only for radiation pattern, but also for better internal resonance control and more controlled acoustic port output.

It is possible to design such enclosures to match a chosen radiation pattern, and so far this is one alternative to very large horns, when pattern control beyond high-mid frequencies is desired. It is possible to design a very small speaker with controlled pattern down into the bass range.

You seem to know a lot on the subject. Care to show us how as I can’t see such a design among your products.

 

[Kvalsvoll]

You seem to know a lot on the subject. Care to show us how as I can’t see such a design among your products.

The F205 has radiation control with acoustic ports and several internal chambers, with ports inside the cabinet. There is a rendering (picture below "Technology" section) which shows some of the internal construction, without damping materials.

The purpose of those internal chambers and ports and damping material is to shape the frequency response and delay of the sound emitted from the acoustic ports. This all works together in a rather complex way, that is only possible to see properly using simulation of the acoustic system.

The cabinet with the ports and internals is one part of the design - the choice of radiation pattern is also a part of the design process. A desired pattern is defined, and then the cabinet is optimized to create this pattern, by placement and size of acoustic ports, and tuning of the internal chambers.