Considerations on group delay of different cabs loaded with same driver

[agedhorse]

I think I've determined that for basic sealed and ported cabs (did not study any others), the frequency response, phase response, and group delay curves are all joined at the hip, below about 250 Hz. Two speakers with the same frequency response will have the same group delay, and vice versa. And you can't have a sealed cab with the same group delay as a ported cab.

My proof, for anybody who enjoys math: Invalid Link Removed

Agreed, sealed boxes will tend to have similar shaped group delays and ported boxes will have similar shaped group delays. Group delays will tend to be quite different between a family of sealed boxes and a family of ported boxes.

Where the differences show up is in the raw frequency response of the sealed boxes (since there are limited other parameters to alter in practice". In ported boxes, the group delay will differ based on the tuning of the system... both the frequency response as well as the damping of the response. In theory, the response of a ported box follows the filter equations of a high pass filter, and the filter can be over damped, critically damped, or under damped. The phase response changes significantly with damping (or filter Q). The easiest way to visualize the phase response is to generate a bode plot of the equiv. filters, then overlay the two plots corresponding to the different tunings and compare phase plots.

[rant on] As electrical engineers, we work with this same (basic) concept in great detail when designing power amplifiers, as the phase response is an important part of defining the phase margin under various global feedback conditions. Without understanding the concept of phase margin, which combined with loop gain, is the basis the Nyquist Stability Criteria analysis to insure that an amplifier (or control system) remains stable throughout it's operating range. This is the sort of stuff that real engineers deal with on a daily basis of course, and understanding the analogy between filter math and speaker math really helps when designing a line of speakers that play well together. Way too many "speaker designers" (armchair and otherwise) do not have the background to understand and apply these principles to their work, so it's just easier for them to say that combining different speaker sizes can't work well rather than doing the math (and it is a lot of work) to insure that they do work together by design. [/rant off]

 

[ThisBass]

I don't feel your comment is a rant adressed to the average technician or DIY folks with average skills in science.
To fully understand the backround needs to be trained in engineering science. There is lots of math (complex most of the time) and transformations into frequency domain (Laplace transformation) neccessary to rate relationships AND then figure solutions for a given task.
Electrical engineers mostly act in the frequency domain rather then on time domain processes. IMO that is the main reason why so many folks allthough trained with some average (time domain processed) skills can't understand the full backround of things.

Once a system is described in the frequency domain then it is possible to evaluate the system on properties like stability and transfer. In the frequency domain it's also possible to decide which kind of controller fits the needs of a dynamic system to hold the process variable (nearly) constant and evaluate the stability of control loop.

In the frequency domain filters (active or passive) can be easily evaluated by their filter coefficients.
Most common in audio are Butterworth and Linkwitz-Riley alignments.
A 2nd order Butterworth alignment is given when filter coefficients:
a=1.41
b=1

Note that these are pure constants and don't carry any unit like L or C neither U or I, neither feet or inch.
It's possible to realize a 2nd order Butterworth by passive network with L and C and R, but it's also possible to realize the Filter with R and C as an active filter design.

The filter coefficient "a" may be replaced with the reciproke of damping 1/Q
a=1/Q

Q directly corresponds to Qtc which is a well known parameter on sealed cab designs.
So if a sealed cab design shall show Butterworth alignment the number for Qtc has to be 0.71

Does a pure number of 0.71 tells anything about driver size? Probably not!
So let's see to find a driver that fits the needs for Qtc to equal 0.71.

The essential driver parameters to determine the filter alignment for a given cab volume are:
Vas, f0, Qts,
To know only these three driver parametrs reaches the needs to estimate a 2nd order high pass alignment for a given cab volume. Once a driver is given then the cab volume has to be adjusted to get a desired alignement, and conversaly.
Neither Vas nor f0 neither Qts carry any information about driver size. So if a 12" driver and a 10" driver would provide identical parameters then the resulting 2nd order filter would be exactly the same for just the same cab size.

Wasn't it on TB forum where so many folks enhance that driver size has nothing to do with response at the low end?
So why stressing that driver size may cause issues whereas some little math tells there can't be an issue because the math don't bother about driver size. The math does not bother about driver size like TB forum as well does not bother about it.


As Andy already mentioned the math for cabs may grow into very complex stuff. The transfer function for a ported cab looks like as follows:

Sorry, could not get a more "selfexplaining" formula. It was the only G(s) transfer function that I could get for ported. But it can be easily seen that there is an obvious analogy with normalized G(s) of 4th order high pass, written in normalized form that looks like:

No need to discuss this stuff here but, there is absolutely no dependence on any driver size or inch (nor any C nor any L or anything).

But as the first G(s) function represents the real ported cab as a 4th order high pass there are all essential cab properties determinable.
Amplitude response, Phase response, Group delay and damping and poles can be estimated for a given driver by a given cab volume and given port tuning.


It's not my intention nothing at all to teach here about filter technique or the discipline of systems theory. But to show something of the stuff engineers have sometimes to keep busy with, I think that can't be wrong instead of hitting phrases that is a lack of any engineering science.

That all said, different cab designs may cause issues with phase and group delay although loaded with same driver. Issues with phase and group delay are more dependent on different cabs rather then different driver size.
Doing the math (alignment) for cabs means a pure number of driver size does not matter.
Essential driver parameters, these are parameters to do the math for alignment on cabs don't carry any information about driver size.

 

[agedhorse]

There are many different techniques that are used to model such speaker systems. Each has strengths and weaknesses but correct, none of them look at driver size but various math models of performance.

The more accurate you wish to get, the more complicated the formulas become. Ideally you simplify the model enough that it becomes possible to solve but accurate enough to be useful

 

[ThisBass]

As you already mentioned driver parameters may change on batch production. And sometimes published TS-parameters are changed (or corrected) by a manufacturer. So it's legal to do simplifications to the math of the cab model. Never the less the 4th order high pass characteristic still remains.


Typical example how it may happen on a forum anywhere.

I would have a smaller vented 210 cab loaded with Beta 10's. As the low end lacks a little bit on beef I'm caused to consider about an additional cab because I'm considering a practical stack for bigger locations. And so I ask the forum a question what kind of driver or cab may improve the lows around 50Hz.

I myself consider a 115 could be a good choice to hit the goal because I know the 10"s of my cab are weak at the lows.

"...driver size means nothing about response at the lows, there are 10" drivers available which bite most of any 15"s out there ..."

Ok, there is a nice 10" at ******* available with the suffix LF that should fit all the needs I'm looking for. The driver is a 10" so it fits the demand for matched driver size and also promisis pretty low response.

The cab simulation shows a balanced response down to around 50Hz, so no problem at all to present 1st Overtone of a B-string.

The response of the existing cab loaded with Betas

There is some hump at midbass, -3dB around 70Hz, no doubt about it the cab needs help from something like a sub or similar thing.

Both response graphs at normalized amplitude based on 0dB

Looks not so bad. It seems like there is a very noticable improvement around 50Hz-70Hz.

Next graph shows the so called "customized" amplitude response. Note the response now depends on input power

Midbass and Lowmids will be dominated by the Betas, Treble will be dominated by the Betas.
At the lows the improvement of lows starts at least at 70Hz, the LF cab provides enlarged level around 50Hz-70Hz of appro +3dB.

Although the graphs above look fine but, there is something outstanding like phase and group delay that is not considered by now:
Both cabs are ported cabs tuned to 50Hz each, group delay looks as follows,

Group Delay difference at 50Hz of 7msec corresponds to a phase shift of 125 degree
Group Delay difference at 60Hz of 4msec corresponds to a phase shift of 86 degree

I would not claim a muscian is unhappy with this combination per se. Having the cone surface doubled may cause psychoacustic mechanism that covers lacks on design
But the goal to enlarge low response obviosly would fail in this case due to group delay issues.
Would a player really recognize the issue? Probably not if the player reads carefully enough at forum discussions. The omnipresent dogma that same speaker size does not cause problems, but mixing speaker size does, may lead to analogies and issues (because dogma is the issue) like seen in the past where some myths refused to die over several decades.

What I would do to get rid of this cab issue with group delay?
I would probably throw away the Beta's cab and keep the Beta's for future use. Then looking for different 10"s that fit the goals in a more better way.

To be fair, design for the 10"s Beta cab was done to be somewhat less than perfect. By the other hand there are so many cabs out there beeing far awy from perfectly aligned. Keep in mind that in the 70's and 80's many cabs where build by widely ignoring filter alignments. Lot's of these cabs are still in use, and may be somebody may ask a question the next day on a forum for help to improve low response.

 

[agedhorse]

This example illustrates nicely how similar LOOKING speakers might not be a great solution for this specific example.

One place where mixed driver sizes and/or driver types would be where one driver was selected based on robust low frequency performance and the other driver selected based on a particular mid voicing sacrificing low frequency performance. A case where a single drive can not accomplish both tasks adequately. In this case, mixed driver sizes, along with appropriate performance parameters (which would include tuning possibilities with appropriate group delay profile) might be a really good choice for a particular overall system response.