Tech Design.....

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Distortion in Isobaric Woofer Systems.

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Distortion in Isobaric Woofer Systems.

rod then the arguments of how distortion is reduced would hold some validity. However, the compressibility of air

is not dependent on the volume of the air in the chamber or frequency although the

is, and the drivers are not connected by a light weight, rigid rod.

When the isobaric system is correctly analyzed it becomes readily apparent that the distortion canceling

mechanism of a push pull system does not translate directly to the isobaric system. Before going into any detail

some observations about the cancellation mechanism are in order. In the standard push-pull system the drivers

are independently driven and it is their acoustic output that is

components will be 180 degrees out of phase and cancel in the

individual acoustic output from each driver do not sum so the same distortion mechanism can not be at work. On

the other hand, if the drivers were connected by an infinitely stiff rod they would be forced to move with the same

motion. The forces generated by the motor and suspension would be the

motor and suspension; and with the motors wired out of phase and with a push-pull mounting, the even order

distortion in the

void of even order HD provided that no additional distortion is introduced by the air compliance of the rear box.

Note that the key here is the summation of SPL (in the standard push-pull configuration) or forces (in the isobaric

case) which leads to the cancellation. But is this summation of forces appropriate for an isobaric system? To

answer this question a very simple analysis is all that is required.

To start we must accept that for the distortion cancellation to occur in the isobaric system the motion of both

drivers must be the same. Since this is a mechanical system, if the drivers are to have identical motion then the

forces acting on each driver must also be identical. If it can be shown that these two conditions are mutually

exclusive then it will suffice to show that the cancellation mechanism doesn't stand. We begin by accounting for

the forces acting on the drivers. The figure below shows these forces. Fd1 is the force generated by the

suspension and motor of driver 1. Fd2 is the force from the suspension and motor of driver 2. K1 and K2 are

spring constants and represent the compliance of the air in the rear box and isobaric chamber. X1 and X2 are the

positions of the respective drivers relative to their rest positions. K1 and K2 are taken to be finite. If the drivers

are to undergo the same motion, Newton's Law tells us forces acting on each driver must be the same. The total

force on each driver is:

(1) Ft1 =Fd1 - K1 X1 - K2 (X1 - X2)

(2) Ft2 = Fd2 + K2 (X1 - X2)

In Equations (1&2) the term K2(X1-X2) is the coupling between the drivers by the isobaric chamber. Since for

identical motion Ft1 = Ft2,

(3) Fd1 - K1 X1 - K2 (X1 -X2) = Fd2 + K2 (X1 - X2)

with a little manipulation we find that

(4) X2 = [1 + K1/K2/2] X1 + (Fd2 - Fd1)/K2/2

We can also note that the spring constants are proportional to 1/V, where V is the volume of the box, Vbox, or

isobaric chamber, Viso. Thus,

(5) X2 ~ [1 + Viso/(2Vbox)] X1 + (Fd2 - Fd1) Viso/2

If we normalize the volumes by Vas we can write,

(6) X2 ~ [1 + α/(2β)] X1 + (Fd2 - Fd1) Vas/(2β)

where α = Vas/Vbox and β = Vas/Viso. If the system is linear (Fd2 - Fd1)=0. Equation (4) tells us that only as

K2*tends* to infinity (a perfectly ridgid rod) will X1 = X2. If K2 were indeed infinite, then we would need to rewrite

Eqs(1&2) as FT1 + FT2 = Fd1 + Fd2 - K1 X1, and X1 would equal X2 by definition. That X1 can not equal X2 for

finite K2 becomes even more obvious when we refer back to Eqs (1&2). If X1 = X2 then there would be no

coupling between the drivers, with any finite value of K2, by the isobaric chamber. Thus the forces on the drivers

would have to be different. We therefore have a contradiction; if we assume the total forces on each driver are

the same then, for any finite K2, the displacements are different; if we assume identical displacements, the

isobaric coupling is zero for any finite K2, and the forces are different. Equation (6) also gives us a hint of

where the idea that Vas for the isobaric system is 1/2 that of the single driver system since α/2 = (Vas/2)/Vbox.

The next step in understanding the behavior of the isobaric system is to develop a suitable model. Without

presenting the details, I have developed a model from the ordinary differential equations which describe the

driver's motions as influenced by the sealed rear box and the isobaric chamber. The details can be downloaded

here. This model was solved numerically. The model includes a rigorous accounting for nonlinearity due to the

compression/expansion of air in the rear box and in the isobaric chamber. It also includes a simple nonlinear

compliance model as shown below. The suspension compliance can be symmetrical

are to undergo the same motion, Newton's Law tells us forces acting on each driver must be the same. The total

force on each driver is:

(1) Ft1 =Fd1 - K1 X1 - K2 (X1 - X2)

(2) Ft2 = Fd2 + K2 (X1 - X2)

In Equations (1&2) the term K2(X1-X2) is the coupling between the drivers by the isobaric chamber. Since for

identical motion Ft1 = Ft2,

(3) Fd1 - K1 X1 - K2 (X1 -X2) = Fd2 + K2 (X1 - X2)

with a little manipulation we find that

(4) X2 = [1 + K1/K2/2] X1 + (Fd2 - Fd1)/K2/2

We can also note that the spring constants are proportional to 1/V, where V is the volume of the box, Vbox, or

isobaric chamber, Viso. Thus,

(5) X2 ~ [1 + Viso/(2Vbox)] X1 + (Fd2 - Fd1) Viso/2

If we normalize the volumes by Vas we can write,

(6) X2 ~ [1 + α/(2β)] X1 + (Fd2 - Fd1) Vas/(2β)

where α = Vas/Vbox and β = Vas/Viso. If the system is linear (Fd2 - Fd1)=0. Equation (4) tells us that only as

K2

Eqs(1&2) as FT1 + FT2 = Fd1 + Fd2 - K1 X1, and X1 would equal X2 by definition. That X1 can not equal X2 for

finite K2 becomes even more obvious when we refer back to Eqs (1&2). If X1 = X2 then there would be no

coupling between the drivers, with any finite value of K2, by the isobaric chamber. Thus the forces on the drivers

would have to be different. We therefore have a contradiction; if we assume the total forces on each driver are

the same then, for any finite K2, the displacements are different; if we assume identical displacements, the

isobaric coupling is zero for any finite K2, and the forces are different. Equation (6) also gives us a hint of

where the idea that Vas for the isobaric system is 1/2 that of the single driver system since α/2 = (Vas/2)/Vbox.

The next step in understanding the behavior of the isobaric system is to develop a suitable model. Without

presenting the details, I have developed a model from the ordinary differential equations which describe the

driver's motions as influenced by the sealed rear box and the isobaric chamber. The details can be downloaded

here. This model was solved numerically. The model includes a rigorous accounting for nonlinearity due to the

compression/expansion of air in the rear box and in the isobaric chamber. It also includes a simple nonlinear

compliance model as shown below. The suspension compliance can be symmetrical

about x/xmax = 0.0, or asymmetry can be introduced through an offset. A symmetric nonlinearity introduces only

odd order harmonics. The asymmetry introduced by the offset generates even order distortion components and

reversed polarity/push-pull can be mimicked by introducing a positive offset in one driver and an equal

magnitude, negative offset in the other. Consideration of the nonlinear air compliance is always asymmetric.

Both the nonlinearity of the air compliance and that of the suspension model can be switched on and off to

investigate the effects of each separately or in combination. The code can also be run as two isolated drivers to

model a true push-pull woofer (first figure).

To demonstrate the effects nonlinear distortion a number of simulations were performed. These were all based

on a driver with the T/S parameters similar to those of a 10" Peerless XLS woofer. The woofer was assumed to

be in a box suitable for a Qtc = 0.707. All calculations were performed at fc, approximately 40 Hz. A suitable

input signal level was used to push the drivers well into the nonlinear excursion region to emphasize the

distortion levels. Results are presented as plots of distortion.

__Testing the Code: Isolated drivers and standard push-pull configurations (Figure 1)__

First a series of test were performed to verify the operation of the code. To begin, the code was run in separate

driver mode. In separate driver mode*Front Driver* data* *corresponds to either driver in isolation and its distortion

odd order harmonics. The asymmetry introduced by the offset generates even order distortion components and

reversed polarity/push-pull can be mimicked by introducing a positive offset in one driver and an equal

magnitude, negative offset in the other. Consideration of the nonlinear air compliance is always asymmetric.

Both the nonlinearity of the air compliance and that of the suspension model can be switched on and off to

investigate the effects of each separately or in combination. The code can also be run as two isolated drivers to

model a true push-pull woofer (first figure).

To demonstrate the effects nonlinear distortion a number of simulations were performed. These were all based

on a driver with the T/S parameters similar to those of a 10" Peerless XLS woofer. The woofer was assumed to

be in a box suitable for a Qtc = 0.707. All calculations were performed at fc, approximately 40 Hz. A suitable

input signal level was used to push the drivers well into the nonlinear excursion region to emphasize the

distortion levels. Results are presented as plots of distortion.

driver mode. In separate driver mode

characteristics. *Rear Driver *data corresponds

to the summed acoustic output of the two

drivers.

The figure to the immediate left is the result

for the case where no nonlinearities are

included. No distortion is present in either the

individual driver or summed responses.

The result presented at the right is for the

case where the suspension nonlinearity is

considered, but is taken as symmetric about

zero displacement. Note that only odd order

distortion is present. Also note that the

summed response is identical to the individual

driver's response. There is no reduction in

odd order distortion.

The third result, presented to the left, shows

the rise of even order distortion when a 1mm

offset is introduce in the suspension

compliance nonlinearity. This case represents

the distortion behavior when the drivers are

NOT in a push-pull configuration.

The result at the right is for identical

conditions as those at the left except the

drivers are now in push-pull configuration.

The even order distortion components are

eliminated. (The residual 2nd and 4th order

components are a result of loss of significant

figures due to importing and post processing

the generated simulation data in Excel).

This last result for the isolated driver case

(left) shows what happens to the classical

push-pull configuration when the nonlinearity

of the box air compliance is included. The

even order HD is reduced, but not eliminated

because the nonlinear air compliance

generated distortion is not dependent on how

the driver is mounted.

These results show that the simulation code is

operating correctly.

to the summed acoustic output of the two

drivers.

The figure to the immediate left is the result

for the case where no nonlinearities are

included. No distortion is present in either the

individual driver or summed responses.

The result presented at the right is for the

case where the suspension nonlinearity is

considered, but is taken as symmetric about

zero displacement. Note that only odd order

distortion is present. Also note that the

summed response is identical to the individual

driver's response. There is no reduction in

odd order distortion.

The third result, presented to the left, shows

the rise of even order distortion when a 1mm

offset is introduce in the suspension

compliance nonlinearity. This case represents

the distortion behavior when the drivers are

NOT in a push-pull configuration.

The result at the right is for identical

conditions as those at the left except the

drivers are now in push-pull configuration.

The even order distortion components are

eliminated. (The residual 2nd and 4th order

components are a result of loss of significant

figures due to importing and post processing

the generated simulation data in Excel).

This last result for the isolated driver case

(left) shows what happens to the classical

push-pull configuration when the nonlinearity

of the box air compliance is included. The

even order HD is reduced, but not eliminated

because the nonlinear air compliance

generated distortion is not dependent on how

the driver is mounted.

These results show that the simulation code is

operating correctly.

Qtc and Fc. The rear box volume is 29L, 1/2 that of required for the individual driver case and the volume of the

isobaric chamber was specified as 4L which is reasonable for a face to face, push-pull configuration allowing for

surround clearance and cone motion. For the first simulation, presented directly below, the nonlinear

suspension and air compliance were switched off. Here I have also presented the front and rear driver

excursion and the pressure variation in the isobaric chamber. They are typical. The front driver is that which

actually radiates the sound. As with the isolated driver case, there is no distortion generated when the model is

linear. However, it is observed that the front driver excursion exceeds that of the rear driver. This is consistent

with the contradiction we observed in the simplified analysis, above. We also see that the pressure in the

isobaric chamber is anything but constant. In fact, it is approximately 1/2 the amplitude of the pressure variation

in the rear box. Again, this makes sense since if the driver motions are close to the same, which they are, then

since the motor and suspension forces are identical for both drivers the pressure forces must be about the

same too. The front driver is subject to a pressure force of 1/2. The rear driver has a pressure force of 1 acting

on the back face and -1/2 on the front face for a net of 1/2. Note that these are approximate and while the

excursions and pressure forces are close to the same, they are different, and as will become apparent these

small differences are important when considering nonlinear distortion and cancelation.

At the left the result for nonlinear but

symmetric suspension compliance is

presented. Only odd order components are

present but note the difference between this

isobaric case and the separate, isolated

drivers. Here the distortion is different for the

front and rear drivers and the magnitude of

the higher order components is actually

higher for the front driver.

When asymmetry in introduced in the

suspension compliance by a 1 mm offset the

result is as shown to the right. This is for a

front to back, not a push-pull configuration so

no cancellation would be expected. The even

order distortion components are now present

and again, the higher order components are

greater for the front driver, which radiates the

sound.

Directly to the left is the result when the

drivers are mounted in a push-pull

configuration. In the isolated driver case the

even order distortion canceled completely

(except for small numerical error). Here, while

we do see a reduction in the 2nd order HD

the 4th order HD actually increased over the

non push-pull case.

To the right is the result when nonlinear air

compliance for both the rear box and the

isobaric chamber is considered as well. A

small but significant increase in distortion is

observed. This last figure should be

compared to the last figure for the*standard*,

isolated driver, push-pull configuration above.

The dark blue, large data points to the left

should be compared to the smaller pink data

symmetric suspension compliance is

presented. Only odd order components are

present but note the difference between this

isobaric case and the separate, isolated

drivers. Here the distortion is different for the

front and rear drivers and the magnitude of

the higher order components is actually

higher for the front driver.

When asymmetry in introduced in the

suspension compliance by a 1 mm offset the

result is as shown to the right. This is for a

front to back, not a push-pull configuration so

no cancellation would be expected. The even

order distortion components are now present

and again, the higher order components are

greater for the front driver, which radiates the

sound.

Directly to the left is the result when the

drivers are mounted in a push-pull

configuration. In the isolated driver case the

even order distortion canceled completely

(except for small numerical error). Here, while

we do see a reduction in the 2nd order HD

the 4th order HD actually increased over the

non push-pull case.

To the right is the result when nonlinear air

compliance for both the rear box and the

isobaric chamber is considered as well. A

small but significant increase in distortion is

observed. This last figure should be

compared to the last figure for the

isolated driver, push-pull configuration above.

The dark blue, large data points to the left

should be compared to the smaller pink data

points for the standard push-pull case. It will be observed that compared to the standard push-pull configuration the isobaric push-pull

configuration offers very little in terms of reduced even order distortion. Finally, below to the right is the result for the same isobaric

woofer system with asymmetric suspension compliance and nonlinear air compliance when not in a push-pull configuration.

configuration offers very little in terms of reduced even order distortion. Finally, below to the right is the result for the same isobaric

woofer system with asymmetric suspension compliance and nonlinear air compliance when not in a push-pull configuration.

Compared to the result directly above very little change in distortion is observed further

demonstrating the lack of significant cancellation of even order distortion in isobaric

woofer system using a push-pull driver format.

One last simulation is presented below to show the effect of reducing the isobaric

chamber volume to 1L from the previous 4L. The lower left plot shows the complete

nonlinear air and compliance result in standard format. At the lower right is the result

when the drivers are placed in push-pull configuration. Improvement in even order

distortion is apparent, however reduction of the isobaric volume to 1L or less is unrealistic.

demonstrating the lack of significant cancellation of even order distortion in isobaric

woofer system using a push-pull driver format.

One last simulation is presented below to show the effect of reducing the isobaric

chamber volume to 1L from the previous 4L. The lower left plot shows the complete

nonlinear air and compliance result in standard format. At the lower right is the result

when the drivers are placed in push-pull configuration. Improvement in even order

distortion is apparent, however reduction of the isobaric volume to 1L or less is unrealistic.

The calculation presented here are not

intended to be exact representations of a real

drivers or woofer systems. Rather they are

intended as qualitative representations of the

behavior of these types of woofer systems.

The principles upon which the models are

founded are reasonable and the results are

consistent with reasonable expectation of the

trends of real systems.

intended to be exact representations of a real

drivers or woofer systems. Rather they are

intended as qualitative representations of the

behavior of these types of woofer systems.

The principles upon which the models are

founded are reasonable and the results are

consistent with reasonable expectation of the

trends of real systems.

1. V. Dickason, *Loudspeaker Design Cookbook*, 5th edition, Audio Amateur Press, Peterborough, NH.

2. M. Colloms,*High Performance Loudspeakers*, 5th edition, john Wiley and Son, New York.

2. M. Colloms,

One commonly used method to reduce harmonic distortion in woofer systems is through

the use of two woofers mounted in a push-pull format as shown in the figure to the right.

When one driver is connected with inverted polarity in push-pull format the result is that

even order distortion components will be 180 degrees out of phase and should cancel in

the summed*acoustic* response. This is fundamentally true and can be realized for dipole

or infinite baffle woofers, and other woofer systems provided the air mass load on both

sides of the woofers is identical. However, in boxed woofers there is an additional

distortion generating element which is unaffected by the driver mounting. This is the

nonlinearity of the compliance due to the air in the box. If the box is large, or the

excursion of the woofers is small, this source of distortion may also be small, thus

cancellation of the majority of even order distortion is possible.

This distortion canceling approach has been routinely adopted to isobaric or compound

woofer systems as well. The lower figure to the right shows an isobaric woofer system

with the drivers mounted face to face. It has been stated [1], [2] that this configuration

benefits from the same even order distortion canceling mechanisms. However, this is not

the case. The premise behind the isobaric configuration is that such an arrangement

allows the box volume to be 1/2 that of a single woofer configuration, and that the air in

the isobaric chamber remains at constant pressure. If such were the case, then the outer

woofer would have no air spring effect on it and it would operate as if mounted in free

air. We know this is doesn't happen. In actuality the outer woofer moves, approximately,

as if it were mounted by itself in a box of twice the volume. In [2] the following

explanation is offered:*"The small air chamber between the drivers is essentially *

incompressible at frequencies below 150 Hz. Hence the diaphragms may be regarded

as closely coupled, as if by a lightweight rod. Now the analysis is simple. Conventionally

connected, with the motor coils connected in parallel, the composite dual driver has the

following characteristics if compared with the single device: twice the moving mass;

the use of two woofers mounted in a push-pull format as shown in the figure to the right.

When one driver is connected with inverted polarity in push-pull format the result is that

even order distortion components will be 180 degrees out of phase and should cancel in

the summed

or infinite baffle woofers, and other woofer systems provided the air mass load on both

sides of the woofers is identical. However, in boxed woofers there is an additional

distortion generating element which is unaffected by the driver mounting. This is the

nonlinearity of the compliance due to the air in the box. If the box is large, or the

excursion of the woofers is small, this source of distortion may also be small, thus

cancellation of the majority of even order distortion is possible.

This distortion canceling approach has been routinely adopted to isobaric or compound

woofer systems as well. The lower figure to the right shows an isobaric woofer system

with the drivers mounted face to face. It has been stated [1], [2] that this configuration

benefits from the same even order distortion canceling mechanisms. However, this is not

the case. The premise behind the isobaric configuration is that such an arrangement

allows the box volume to be 1/2 that of a single woofer configuration, and that the air in

the isobaric chamber remains at constant pressure. If such were the case, then the outer

woofer would have no air spring effect on it and it would operate as if mounted in free

air. We know this is doesn't happen. In actuality the outer woofer moves, approximately,

as if it were mounted by itself in a box of twice the volume. In [2] the following

explanation is offered:

incompressible at frequencies below 150 Hz. Hence the diaphragms may be regarded

as closely coupled, as if by a lightweight rod. Now the analysis is simple. Conventionally

connected, with the motor coils connected in parallel, the composite dual driver has the

following characteristics if compared with the single device: twice the moving mass;