Sloped baffle

Consider a simple two-way speaker having a first order crossover consisting of a capacitor in series with the tweeter, and an inductor in series with the woofer. For each driver that will result in well behaved 6 db/octave rolloff characteristics, which will result in time and phase coherence if other aspects of the design are also supportive, **IF** the impedances of the woofer and tweeter are purely resistive.

However I believe Bruce has been alluding to the fact that the impedances of the drivers are not purely resistive. And it would be more accurate (if still somewhat oversimplified) to electrically model them as consisting of a resistor and an inductor in series.

So the question then becomes: Doesn't the presence of that inductive component of the driver impedance (especially in the case of the tweeter) cause a deviation from first order 6 db/octave behavior? And if so, to a degree that may audibly compromise phase and time coherence? And if so, is that or can that be compensated for in other aspects of the speaker's design?

thanx to the above question posed by Bifwynne & well enunciated by Almarg, I did some research to try to understand what the issue might be.
As I stated in my prev post, my understanding is that if the driver is used within its flat frequency response range of frequencies then that particular driver can be considered linear or purely resistive. And, some research seemed to indicated exactly that! Allow me to share:

When Bifwynne asked the above question, I discovered that it took the me towards understanding the Thiel/Small (or T/S) parameters of loudspeaker drivers. There's much to learn here but that's a subject for another day. Some info that I thought is pertinent to this subject:
There is a T/S parameter called Re (R-little-e) & a cut/paste from Wikipedia

"
Re
Measured in ohms (Ω), this is the DC resistance (DCR) of the voice coil, best measured with the cone blocked, or prevented from moving or vibrating because otherwise the pickup of ambient sounds can cause the measurement to be unreliable. Re should not be confused with the rated driver impedance, Re can be tightly controlled by the manufacturer, while rated impedance values are often approximate at best.. American EIA standard RS-299A specifies that Re (or DCR) should be at least 80% of the rated driver impedance, so an 8-ohm rated driver should have a DC resistance of at least 6.4 ohms, and a 4-ohm unit should measure 3.2 ohms minimum. This standard is voluntary, and many 8 ohm drivers have resistances of ~5.5 ohms, and proportionally lower for lower rated impedances. "

there's also a T/S parameter called Le (L-little-e)

"
Le
Measured in millihenries (mH), this is the inductance of the voice coil. The coil is a lossy inductor, in part due to losses in the pole piece, so the apparent inductance changes with frequency. Large Le values limit the high frequency output of the driver and cause response changes near cutoff. Simple modeling software often neglects Le, and so does not include its consequences. Inductance varies with excursion because the voice coil moves relative to the polepiece, which acts as a sliding inductor core, increasing inductance on the inward stroke and decreasing it on the outward stroke in typical overhung coil arrangements. This inductance modulation is an important source of nonlinearity (distortion) in loudspeakers. Including a copper cap on the pole piece, or a copper shorting ring on it, can reduce the increase in impedance seen at higher frequencies in typical drivers, and also reduce the nonlinearity due to inductance modulation. "

So, it looks like a significant source of distortion is due to voice-coil inductance modulation (variation) & not so much the fact that the voice-coil has actually a DC resistance associated with it (as Bifwynne & Almarg were thinking).
So, how to tell when viewing/reading a driver's specifications that this inductance modulation is an issue? I don't really know but I took up Roy's advice to look at driver specs on Madisound. On the Madisound I randomly selected "Seas Prestige" - Seas makes good drivers, "Prestige" seems like its upper-end line. Here's the link to one of their 8" woofer drivers:

http://www.madisoundspeakerstore.com/approx-8-woofers/seas-prestige-8-woofer-cd22rn4x-h1192

Lots of good info on this page but reading the specs might be Greek to most of us - I wanted to call your attention to the graph which shows SPL (left vertical axis) vs freq & impedance (right vertical axis) vs freq.

From a Wikipedia page on Speaker Electrical Characterisitics I learnt

".....the effective electrical impedance of the speaker to be at its maximum at Fs, shown as Zmax in the graph. For frequencies just below resonance, the impedance rises rapidly as the frequency approaches Fs and is inductive in nature.

At resonance, the impedance is purely resistive and beyond it—as the impedance drops—it behaves capacitively. The impedance reaches a minimum value (Zmin) at some frequency where the behaviour is fairly (but not perfectly) resistive over some range. A speaker's rated or nominal impedance (Znom) is derived from this Zmin value (see below)."

This Seas driver seems to have a 6.1 Ohms impedance at, say, 150Hz. Using the info from the Wikipedia site, the driver must be mostly resistive at 150Hz to give its minimum impedance at that frequency. Look at this driver's frequency response from 90Hz - 400Hz: practically ruler flat & look at the impedance variation over this same range - goes from 6.1 Ohms to 10 Ohms on both sides of 6.1Ohms, which is a small change in driver impedance compared to the change over the entire 20Hz-20KHz. The driver appears to be mostly resistive in this frequency range.
I *think* the answer to Bifwynne's question is that if you use this driver in the 90Hz-400Hz range, you will get a mostly resistive driver whose impedance varies very little (between 6-10 Ohms), it's frequency response will be flat/linear & the phase distortion will be minimal meaning that the voice-coil inductance modulation/variation (which is a significant source of distortion) will be negligible.

Roy, please correct me if I'm wrong. Thanks.

Roy, what about the 90 degree phase shift of a single pole crossover?

Hi to all,

Bombaywalla, you ask-
"So the question then becomes: Doesn't the presence of that inductive component of the driver impedance (especially in the case of the tweeter) cause a deviation from first order 6 db/octave behavior? And if so, to a degree that may audibly compromise phase and time coherence? And if so, is that or can that be compensated for in other aspects of the speaker's design?"

My answer is YES, but we and others use simple Zobel networks on woofer, mid and tweeter. These offset the changes in impedance at high frequencies.

Normally, the impedance of a woofer, mid or tweeter (= 'driver') becomes 9, 12, 30 Ohms as we go higher and higher up the scale (= "inductance"), instead of staying constant at say, a steady six Ohms, which is what any type of crossover circuit wants to see-- a flat impedance 'curve', so that all of its capacitors and inductors do what they are supposed to do. Any change in impedance literally turns some of those circuit parts off, with the result 'not measuring right' to the microphone.

A Zobel circuit offsets a voice-coil's rise in impedance with increasing frequency, and is quite simple to construct: a capacitor is connected to a resistor (= 'in series'). Those two are then placed in parallel with the driver's +/- wires, the capacitor connected usually to the "+" and the resistor to the "-". I hope that is clear!

With those two Zobel parts placed 'across' a woofer, mid or tweeter, the result is literally a 'Y-adaptor' to the signal coming from that driver's crossover, because two paths now exist for the signal. One goes through the driver back to the crossover as usual, and the other through that Zobel cap, then its resistor, and thence back to the crossover.

That woofer, mid or tweeter's impedance is still going up and up the higher up the scale we go, but our Zobel's capacitor has an impedance that is going down and down by the same amount/at the same rate. This is its 'correction', with the resistor limiting/shaping the amount of correction provided.

When those two paths are made to change by 'equal and opposite amounts', that driver's crossover circuit then sees 'no change in the impedance at any frequency', so goes the theory.

Where most every designer goes wrong is by making the assumption that the electrical impedance curve of the voice coil is what one is measuring and correcting. Not so, unfortunately. There are many other impedance curves that overlay, thus hide, the real electrical curve one is looking to flatten. These other impedances include:

- the mechanical impedance of the driver's suspension and any ferrofluid used.
- the acoustic impedance from how each driver is coupled to the air in front of it, and behind.
- any cone/dome flexing (= mechanical impedance changes).
- the mechanical impedance changes caused by the size of the boxes used for woofer and mid, and a tweeter's rear-chamber.
- what happens in the various types of fibers placed behind drivers to absorb their rear waves.

Then (!) most of those change with loudness, especially with 'average' drivers. Some of those also change when a voice coil is moving inwards versus out, again especially on 'average' drivers. Visit the Klippel company's website to see some of their measurements for these problems, now done automatically by their unique computer and programming- such a smart designer! I had to perform them manually, darn it. On our website, I describe much of what can be done to minimize or avoid these last issues.

Finally, a few years ago, I found a way to use the values of cap and resistor in that simple Zobel circuit to perfect our final acoustic phase down to near Zero at all frequencies. While I still must update our website about this, it's never been accomplished by anyone, as seen in the values they still use in their Zobels. If we sought a patent, I'd have to reveal how to come up with 'the right numbers'.

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FYI, it is a big mistake to flatten the low-frequency change in impedance caused by a woofer or mid's box-size and a tweeter's rear-chamber size, by using a different type of Zobel circuit. Those who do this type of correction do not understand what these impedance rises at resonance actually represent electrically, mechanically and acoustically. The result is poor sound, and usually a very difficult speaker to drive in the bass.

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Addressing another question of yours right above: Thiel/Small parameters are a guide only to box tuning, nothing else. However, those equations turned out to give, at best, only an approximation of the correct box size, because the impedance values plugged into it are not 'right' because we have left out all of those other impedances I just described.

Box-modeling software relies on those simple equations, so they cannot give you the exact box size for a woofer or a mid. One must build several test boxes to determine the actual 'best size'.

FYI, Seas' best drivers are their Excel line. Even so, have a look at the high-frequency cone-breakup resonance in their best metal-cone woofer. Designers believe a high-order crossover and a notch filter 'fix' that problem. Not true, as that cone resonance is also triggered by lower-frequency sounds and 'noises'.

Example- your car's dashboard buzzes from the low-frequency 'thump' of a pothole. This concept and its math are taught in high-school level physics, which is what most speaker designers never study. It is why those metal-cone drivers still sound 'metallic'. Stereo magazines and reviewing websites never mention these facts, but then again, it makes sense how they do not want to upset any advertisers!

For tweeters having a strong ultrasonic resonance from their metal dome breakup, the same thing still happens,with the ultrasonic HF resonance ringing out. However, what is heard instead is its effect on the audible-treble tones. That is a 'modulation distortion', and sounds like perhaps a 'zing' to the treble, or again, a metallic sound. These are all factual statements supported by physics theory and math, and by measurements. They are not 'Roy's opinions'.

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Omsed,
The phase shift of that one inductor used on a woofer with its Zobel produces -45 degrees of phase shift (= time delay) at the crossover point. The single capacitor used for the tweeter's first-order crossover gives the opposite shift of +45 degrees (= time 'advance'). The difference between these two is 90 degrees.

When a website or text makes this mistake, that writer had never looked at the simple math involved, which any competent electrical engineer should have learned in their first Filter Theory class. Only hearsay is being passed on to you, including the non-existent 'downwards tilt to a first-order speaker's radiation pattern'. A totally bogus claim. There, the math was completely mis-interpreted.

The important aspects of this 90-degree DIFFERENTIAL produced by a simple first-order crossover, proper Zobels and really good drivers are
a) it remains a CONSTANT 90-degree difference between those two drivers as we go up or down the scale, and
b) that constant difference of 90 degrees allows the sonic outputs of those two drivers to always add up to the one original wave, having no added time delay, which is totally non-intuitive.

A 'perfect summation' happens because those two drivers are operating 'in quadrature' (90 degrees being one-fourth of 360) at every frequency. The math involved shows their outputs, one lagging, one leading, really do combine to make only the one original wave having neither lag nor lead. Weird.

No higher-order crossovers can maintain this CONSTANT phase differential, so they produce a time delay, a phase shift, that changes with frequency, perhaps 'linearly' but always changing.

This varying time-delay is what DEQX-type components are trying to correct, and what regular digital crossover circuits never attempt to correct (offering only fixed time delays, such as one millisecond). To correct the varying time delay, a heck of a computer is required, hence the high cost of DEQX type of gear.

Measurement issues and limitations still confuse DEQX type of gear, for two reasons- we cannot (yet) program that computer to how we actually hear on music, and that a measurement microphone cannot resolve the (countless) reflections off the front of a cabinet. If I had spent money on a DEQX, I would first place an "F-11" pure wool felt all around the tweeter, and then run the calibration routine.

Best,
Roy

I forgot to mention a couple of things:

When a driver is being run full-range with no crossover or Zobel, its changing impedance curve has no effect on its tone balance when using solid-state amps, but only on tube amps via interaction with their much higher output impedances. For a tube amp running a 'full range' driver, a voice-coil Zobel circuit on that driver would return its tone balance to 'factory spec'.

When a speaker has a flat impedance curve, that does not indicate if this speaker is time-coherent. From the outside, all we can see is how the many different impedance curves I described above combine into one curve.

Best,
Roy

Wouldn't it be easier just to tilt the speaker slightly backwards? I just want to listen to good music