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More BAT Phase-Flat Crossovers
Yes, more crossovers. Why? More crossover choices are always welcome to the loudspeaker designer. Having more crossover topologies to choose from is like a photographer having more camera lenses in his kit. In addition, new phase-flat crossovers are special, special in that they preserve the music's waveform; non-phase-flat crossovers only preserve waveforms of continuous sinewaves.

Now, the secret to my staggering topological fecundity is that each new idea, which can be either a new circuit topology or a new design technique, begets more new ideas, which in turn give rise to other new ideas. An example was my use of a difference amplifier to find the needed target frequency and phase plots to create what the midrange driver's frequency plot and phase plot would need to be, with 3rd-order woofer and tweeter filters, to establish both a flat frequency response and flat phase response when all three drivers were summed. The missing output delivered a target for me to hit with passive components; actually, two targets: a phase plot and a frequency plot.

I decided to use this technique of finding the needed output and then designing the passive network that delivers that output to a 2nd-order high-pass filter two-way passive loudspeaker crossover. SPICE simulations make this an easy, well relatively easy, task. The tweeter sees a 2nd-order high-pass filter with a Q of 0.5 (i.e. the same as Yamanaka and Linkwitz-Riley).

The target output from the OpAmp-based differential amplifier reveals what the woofer needs to contribute to the combined output to deliver both a flat phase and flat frequency response.

All I had to do was contrive a passive network (i.e. some combination of capacitors, inductors, and resistors) to deliver the woofer's needed signal. Since the needed output contained a hump, I knew that that the network must entail an insertion loss of a few dB, which could be cancelled by using a higher SPL woofer relative to the tweeter and by inserting a series resistor.

Here is the target frequency response for a 1kHz crossover and my attempt at hitting it.

According to my mathematical analysis, the resistor's value needed to provide the optimum insertion loss is equal to 8 ohms divided by 3, or 2.666… Thus, I used a 2.7-ohm resistor in my SPICE simulations, as it is the closest readily available resistor value. I hit the required frequency response, but this is not enough as I had to also hit the phase plot. Fortunately, if I hit one target, I also hit the other target.

Now for the big reveal, the passive network that delivered the needed frequency and phase plots is the following:

Low-frequencies down to DC are attenuated due to the resistor and the woofer defining a two-resistor voltage divider. At high-frequencies, the capacitor shunts the resistor boosting the highs until the inductor imposes a 1st-order low-pass filter. This is how we get our needed frequency hump. Of course, the capacitor, inductor, and resistor values are critical. Fortunately, I have gift wrapped it for you:

By the way, do not restrict your mind to the notion that only woofers and tweeters are used in a two-way loudspeaker system, as we could use a woofer with a fullrange driver or a horn whose low-frequency response extended down deep into the midrange frequencies. In addition, bear in mind that resistor R1 will decrease the woofer's Qes, as the resistor is in series with the woofer at low-frequencies. This is only a problem if we failed to acknowledge it and fail to incorporate it in our calculations. In fact, it could prove a feature in a dipole loudspeaker, as most woofers are too damped for effective dipole loading, resulting in a pinched bass response.

Here is a 1kHz design example:

Of course, when actually building this crossover, we would use 10µF, 120µF, 2.5mH, 0.85mH part values. Actually, I take the last value, 0.85mH, back, as we can subtract the woofer's own internal series inductance (Le) from the 0.85mH value. The 2.7-ohm resistor must dissipate 25% of the heat that the woofer does at low-frequencies. In other words, think power resistor. I would use at least a 20W resistor with a 100W power amplifier, just to play it safe.

(If your mind is as devious as mine, you realize that since the formula states that the resistor value must equal 1/3 that of the woofer, you could use three of the same woofer type in parallel, which could be located on the rear of a tall loudspeaker cabinet. Why? The three backfiring woofer would deliver two stunning attributes: canceling the single front firing woofers motions—see Newton's 3rd Law of motion—and undoing diffraction loss at low frequencies.

The three woofers in parallel each see one third the current that the front-firing woofer sees, thus each produces one third the output of the front woofer. In other words, unity, as 1/3 x 3 = 1. The rear woofers present three times the mass of the front woofer, but only move one third as far as the front woofer. Once the signal frequency rises up to where the 117.9µF capacitor kicks in, the rear woofers' output drops off with 1st-order low-pass filter slope of -6dB per octave and -20dB per decade. In this example, the -3dB frequency is around 340Hz, which is just about perfect for undoing low-frequency diffraction-loss (aka baffle step response). If only I were a patent attorney.

This brings up an interesting aspect of this BAT phase-flat crossover, namely at what frequencies are the reactive parts (the capacitors and inductors) tuned?

The crossover frequency is 1kHz, but none are tuned at 1kHz—however, all are tuned relative to the crossover frequency. Well, the important question to ask is how well does this new crossover work? Very well indeed.

Since all loudspeaker frequency response plotlines look good with 10dB vertical decrements, we must examine the close up.

Not only is this not bad, it's crazy good. The frequency response deviates by only 30 millibells! The phase response is close to ±0.1 degree from 10Hz to 100kHz. Bear in mind, if I had used a 2.666… ohm resistor, the plot lines would be flatter still. Next, we see a four-cycle 1kHz tone burst (the hardest test for any loudspeaker to pass).

Okay, is this novel crossover perfect? No crossover is perfect. As I see it, the biggest potential issue is that the impedance plotline is not flat.

Well, at least the impedance doesn't dip too much. If an OTL power amplifier drives this loudspeaker impedance, we win, as OTL tube-based amplifiers are current-limited, not voltage-limited. Most of the energy in recorded music is found below 600Hz, so the OTL power amplifier can swing higher voltage into the higher impedance just where it matters most: the bass.

In contrast, solid-state power amplifiers are voltage-limited, not current-limited, which put the solid-state amplifier at a disadvantage with the higher impedance at low frequencies, as it can deliver fewer watts into the higher impedance. One possible workaround would be to use a 4-ohm woofer with an 8-ohm tweeter, which would result in lower impedance at low frequencies.

Not a bad load for a solid-state power amplifier, as the impedance does not dip below 4 ohms. We could use two 8-ohm woofers in parallel in place of a single 4-ohm driver, which would help in gaining the needed 2.5Db higher SPL per watt and would allow for an easy D'Appolito (MTM) arrangement, with the tweeter in the center of the two woofers. (The 2.7-ohm crossover resistor would be replaced with a 1-ohm resistor in series with a 0.33-ohm resistor.) Imagine a fullrange driver between two woofers with a crossover frequency of 400Hz. I would love to hear it.

Keep in mind that in SPICE simulations resistors were used in place of loudspeaker drivers. Real speaker drivers, with the possible exception of planar drivers, present true horror-show impedance plotlines. In fact, I recommend using a "series-notch filter" network across the woofer leads, which will undo the impedance at the woofer's resonant frequency. Sadly, such a network often requires huge capacitor and inductor values.

What will complicate the situation is that the crossover's series resistor will increase the woofers Qes and Re, which are variables in the network's equation—or maybe not, as the network appears after the series resistor.

What if we want to give the woofer the 2nd-order low-pass filter instead, thereby forcing the tweeter (or fullrange to perform the phase correction? I assumed that it would be a simple inversion of the formulas. It wasn't. I started with my test-bed setup with a differential amplifier to find the needed tweeter output.

Once again, output from the OpAmp-based differential amplifier reveals what the tweeter needs to put out in addition to the woofer's output to deliver both a flat phase and flat frequency response. Here is the needed target plot in green and my attempt to meet it in blue.

Next, the required phase plot.

Note how the phase swings positive at low frequencies, in contrast with the previous phase target that swung negative with high-frequencies. Here is the schematic and the needed math.

Here are the frequencies at which the reactive parts are tuned.

Note how the same frequencies appear as before, but differently located; in fact, inverted. How well does this new crossover work?

Once again, we do a zoom in at the crossover frequency of 1kHz.

This is crazy good, which explains how the crossover passes a tone burst without distortion.

Note that four cycles go in ad four come up. Also note that both the woofer and tweeter continue moving after the termination of the input signal, but their summed output cancels. This is transient-perfect reproduction on display. The graph remaining is the impedance plotline.

Active-Passive Crossovers
After my recent torrent of phase-flat/transient-perfect crossovers, I wondered if I could not create an active-passive loudspeaker crossover that delivered a steep (2nd-order) high-pass filter to the tweeter (or fullrange driver) and yet still produced a flat phase response. I showed something like this back in Post 585, where the crossover was a 1st-order two-way with the two woofers being driven by both the external power amplifier and an internal power amplifier.

Here is the explanation from that post on what is going on here:

The way this 1st-order 500Hz two-way crossover works is that, at high frequencies, the inverting amplifier (within the speaker enclosure) ceases to be an inverting amplifier, becoming instead a unity-gain non-inverting amplifier. With high frequencies coming from the external power amplifier, the woofer sees the exact same signal at both its terminals, both in amplitude and in phase. No difference, no sound.

At low frequencies, in contrast, the internal power amplifier inverts the external amplifier's output signal. Since a loudspeaker driver is an intrinsically differential device, the woofer only responds to low frequencies, as that is the only time when it sees as a differential signal across its terminals.

The 47nF capacitor and 6.8k resistor in series set the 500Hz crossover frequency. The formula is easy enough:

       C = 159155/F/R

Where C is in µF and F is the crossover frequency.

As far as the external power amplifier is concerned, it is connected to an 8-ohm woofer that is in series with a loss-less inductor. Yes, the internal power amplifier mimics an inductor. Think about it: the higher the frequency above the crossover frequency, the less current the external amplifier delivers into the woofers, just as an inductor would bring about.

This setup yields both a flat phase and flat impedance output, along with the required flat frequency response. Well, as flat as the drivers allow. One way of thinking about this arrangement is that the internal power amplifier matches the power delivered by the external power amplifier into the woofer, making this a semi-powered loudspeaker design. What will be the resulting two-woofer SPL? Since we doubled the woofer radiating surface, we gain 6dB in SPL, so two 88dB woofers will produce 94dB of SPL at one meter with 2.828Vrms (4Vpk) signal from the external power amplifier.

I had two goals in this design: I wanted to emulate the electrostatic Quad 57 dipole loudspeaker by using a 500Hz crossover frequency and two bass drivers and one high-frequency driver in the middle; and I wanted to give the bass output a 6dB boost relative to the fullrange driver, while still retaining the loudspeaker's nominal 8-ohm impedance.

An added bonus was no crossover inductor, which are expensive and introduce DCR. Actually, I do not believe that power amplifier's damping factor is all that important; period. Many, however, do believe it is critical, so losing the series inductor is real bonus. Even the $107 Jantzen Audio 10mH 14 AWG C-coil toroidal crossover inductor presents 0.102 ohms of DC resistance, while the $92 air-core 1.2mH 14 AWG copper-foil tape inductor imposes 0.293 ohms of series resistance, which will reduce a solid-state power amplifier's damping ratio from something like 400 to 27.

To see how this setup works, we must realize that the two woofers are intrinsically differential devices. If we apply the same voltage to both driver terminals, we get zero cone movement—even if that voltage is 100V. On the other hand, if we apply 99V to one terminal and 100V to the other, the driver sees a 1V difference and the cone move accordingly. Let's look into this differential functioning in detail.

Each woofer sees the differential voltage as the other, so their net SPL is 6dB higher, which brings their combine output up to that of the fullrange driver. If the external power amplifier puts out a DC voltage, the following voltage relationships occur:

The 40µF crossover capacitor blocks the DC voltage to the fullrange driver. The internal power amplifier inverts the +1V into -1V. Next, we examine a high-frequency signal leaving the external power amplifier. The 0.1µF and 40µF capacitors pass the high-frequency AC signal, and the internal power amplifier no longer inverts the external amplifier's output signal.

Since the two woofers see the same in-phase signal voltage on both their terminals, they ignore it. In order for this active-passive setup to work, the internal power amplifier must be able to deal as much power as the external power amplifier. The assumption here has been that a relatively low-power tube-based external power amplifier would be used. Since modern class-D power amplifiers are both powerful and cheap, probably less expensive than a high-end woofer inductor, we can force this arrangement to work with 200W solid-state external power amplifiers.

In Post 585, I also show a version that gives the tweeter a 2nd-order high-pass filter.

I wondered what a version would look like that used a single 8-ohm woofer. No internal power amplifier phase inversion would be needed, but the internal amplifier would still have to deliver the same amount of power as the external power amplifier. The woofer will see a slightly bigger voltage swing near the crossover frequency due to the phase difference between the two amplifiers' output.

The internal power amplifier has been replaced by a unity-gain power buffer. Only two crossover parts are needed, the 20 µF capacitor and 5mH inductor. As a proxy for low-frequencies, we start with 3Vdc leaving the external power amplifier.

The unity-gain power buffer acts as a virtual ground and must absorb all the current flow through the woofer, i.e. 3V/8 or 0.375A. Next, we examine the high-frequency AC example.

The unity-gain power buffer passes on the AC signal that the tweeter sees in phase. Since the woofer sees the same in-phase AC signal voltage on both its terminals, it ignores it. How well does this setup work?

Next Time: 3-Way BAT
After coming up with the phase-flat, two-way, two-driver crossover, I wondered if I could devise a phase-flat, three-way, three-driver crossover that gave both the woofer and tweeter 2nd-order cut-off slopes. I could. Here is the topology:

Next, we see a design example:

The crossover frequencies are 500Hz and 3kHz. In this example, the midrange driver must be 2.7dB more efficient in SPL than the woofer or tweeter. How close did I come to hitting my two targets, flat frequency response and flat phase? Dang Close.

Of course, the phase plot is in degrees. To be flat within ±0.01 degrees is stunning. Moreover, a speaker that is frequency flat within ±3dB is considered very flat. If it were flat within ±0.06dB, we would have to come up with a new metaphor, as rulers are not that flat.

The problem with this crossover design is that I haven't been able to come up with the neat formulas as I had done so with the previous two-way design. (Math is a young man's game, alas. I am way past my math prime.) Instead, I came up with an array of crossover frequency ratios. For example, this design example holds a ratio of 1 to 6, which 200Hz and 1.2kHz also conforms to, as does 600Hz and 3.6kHz. The 1.1-ohm resistor remains constant, but all the other parts must be altered by a 500Hz/200Hz ratio in the first case and by a 500Hz/600Hz ratio in the second case.

In other words, if we wanted the 200Hz and 1.2kHz crossover frequencies, we would multiply all the capacitor and inductor values by 500/200 or 5/2 or 2.5, leaving the 1.1-ohm resistor in place; the 600Hz and 3.6kHz crossover, by 500/600 or 5/6 or 0.8333. This procedure might qualify as "gift wrapped," just without the fancy bow.

In my next post, I will display a table of part values and ratios.

Music Recommendation: Kodo's Mondo Head
Long I have kept short lists of audio tracks with which to test stereo systems, with some tracks falling off and other tracks being added over time. One track that continuously appeared was "Daraijin," by the Japanese drum group, Kodo. Yet, I haven't play the track for many, many years—that is not all the way through; instead, I start the track and get bored about 30 seconds in and switch to another demo-grade track. A huge mistake. I recently allowed the test playlist to play through. About 45 seconds into "Daraijin," I just about fell out of my chair. Mercy! What a demo track. It puts your subwoofers into overdrive and sends your cat into hiding.

Mickey Hart, of Grateful Dead and world-music and "Dafos" fame, produced this 2002 album, which goes a long way to explaining the captured sonic magic. Beyond deep stomach-punching bass, the stereo imaging is spectacular. I understand that the SACD of the album must be heard to be believed. Nonetheless, the CD is amazing. (Considering that I bought this CD at a garage sale for a dollar, the sonic splendor per penny is insane.) I just found today that Qobuz sells a 24-bit, 96kHz download version of the album, which I just bought for about $18—that in its self constitutes a big recommendation from me.

//JRB

PS
While making a PDF of this post, I noticed the AI assistant on the right side of Adobe Acrobat stating that it could make a summary of the PDF. What the hell, I clicked go for it. Here are the results:

This document, titled "Post 624," dated August 8, 2025, discusses advancements in loudspeaker crossover design, focusing on phase-flat crossovers. ​ The author explores the importance of preserving the music's waveform and introduces new crossover topologies that achieve both flat frequency and phase responses. ​ Key highlights include:

• Phase-Flat Crossovers : The document explains the design of 2nd-order high-pass filters for two-way passive loudspeaker crossovers, using SPICE simulations to achieve precise frequency and phase targets. ​ The author emphasizes the importance of resistor values, such as the use of a 2.7-ohm resistor, to achieve optimal insertion loss and frequency response. ​
• Design Techniques : The author details the use of passive networks (capacitors, inductors, and resistors) to shape the woofer's output, ensuring a flat frequency and phase response. ​ The document also discusses the impact of resistor values on woofer characteristics like Qes and Re. ​
• Active-Passive Crossovers : The author explores hybrid designs combining active and passive elements to achieve steep high-pass filters and flat phase responses. ​ Examples include setups with internal power amplifiers mimicking inductors and designs that eliminate crossover inductors to improve damping. ​
• 3-Way Crossovers : The author introduces a three-way, three-driver crossover design with 2nd-order cut-off slopes for both the woofer and tweeter. ​ While achieving flat frequency and phase responses, the design lacks simplified formulas and relies on crossover frequency ratios for part value adjustments. ​
• Music Recommendation : The author recommends the track "Daraijin" by Kodo from the album Mondo Head as an excellent demo for testing stereo systems, praising its deep bass and stereo imaging. The album, produced by Mickey Hart, is available in high-resolution formats. ​

The document is a deep dive into advanced loudspeaker crossover design, offering insights, examples, and practical tips for audio enthusiasts and engineers. ​

I am impressed. How long before AI can rewrite my post in Japanese or French or German? I am sure it can do now, but I wouldn't trust it. Next year? Maybe.

Did you enjoy my post? Do you want to see me make it to post 1,000? If so, think about supporting me at Patreon.

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