Tech Design.....

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Power Matching woofers to midranges and satellite speakers.

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Power Matching woofers to midranges and satellite speakers.

When designing a full range speaker system or adding a woofer to a satellite system there are a number of factors

which should be considered. Among these are size, cost, efficiency, placement, maximum required sound

pressure, distortion, excitation of room modes, and total radiated acoustic power. Here we shall focus on this last

factor, total radiated acoustic power. One reasonable premise is that total radiated acoustic power by the speaker

system should be constant over as wide a frequency range as possibly. Indeed, this seems more than reasonable,

it seems as if dictated by common sense. However, just what does this imply about the characteristics of the

system with regard to directivity or radiation pattern? To begin we need to consider the directivity factor of a sound

source, DF(f), which is generally frequency and position dependent. For the purpose of this discussion it will be

acceptable to assume that DF is constant with frequency. Beranek [1] defines the DF as follows:

"The directivity factor is the ratio of the intensity on a designated axis of a sound radiator at a stated distance r to

the intensity that would be produced at the same position by a point source if it were radiating the same total

acoustic power as the radiator."

He continues,

"Free space is assumed for the measurements. Usually, the designated axis is taken as the axis of maximum

radiation."

The on axis, free space directivity factor for sound sources with several common free space radiation patterns are

as follows: Monopole, DF = 1.0, Dipole, DF = 3, Cardioid, DF = 3. What this means is that if these three different

sources are to radiate the same total acoustic power then if the monopole has an on axis intensity of 1.0 the dipole

and cardioid will have an on axis intensity of 3.0 or 4.77 dB greater. Conversely, if the difference sources are to

have the same on axis intensity then the dipole and cardioid will radiate 1/3 the acoustic power of the monopole.

(As an aside, for those familiar with room acoustics and reverberation this means that the critical distance from the speaker will be

greater for a dipole or cardioid than for a monopole. The critical distance is the distance at which direct and reflected sound are equal.

The level of the reflected sound, above the modal region of the room, is usually considered constant and proportional to the total

radiated power. Thus when sitting the same distance from a conventional speaker and a dipole, the dipole can potentially sound

more detailed since at the position that ratio of direct to reflected sound is greater for the dipole.) Clearly it would appear that if

DF is not frequency dependent then a speaker system would radiate constant power from which it would follow that

a monopole midrange or satellite speaker should be match with a monopole woofer, a dipole with a dipole and a

cardioid with a cardioid. Since cardioids and dipoles have the same DF it would also appear, form a power

perspective, that a dipole and a cardioid can also be matched.

Unfortunately things just aren't so simple. The above argument is keyed to one unmentioned factor, that the space

the source is radiating into is also frequency independent. Obviously, if the sources are in free space this is the

case. It would also be the case if the source were mounted in a true infinite baffle, or along the line of intersection

of to planes perpendicular to each other, or at the corner formed by three plane where are all mutually

perpendicular. These arrangements amount to what is commonly referred to as 4Pi, 2Pi, Pi, and Pi/2 space.

We are all familiar with the transition between 4Pi and 2Pi space know as the baffle step. We know that as the

transition occurs the on axis sound pressure increases by 6dB. But what about radiated power? Consider for the

moment a sealed box woofer operating at low frequency such that in free space the radiation is that of a

monopole. This corresponds to a driver radiating in the center of a sphere, as shown in Figure 1. The radiated

power is the integral of the intensity, I, over the surface of the sphere enclosing the source. If we let the radius,r,

equal 1.0 and the intensity equal 1.0, the radiated power for the monopole would be 4Pi. If the source were a

which should be considered. Among these are size, cost, efficiency, placement, maximum required sound

pressure, distortion, excitation of room modes, and total radiated acoustic power. Here we shall focus on this last

factor, total radiated acoustic power. One reasonable premise is that total radiated acoustic power by the speaker

system should be constant over as wide a frequency range as possibly. Indeed, this seems more than reasonable,

it seems as if dictated by common sense. However, just what does this imply about the characteristics of the

system with regard to directivity or radiation pattern? To begin we need to consider the directivity factor of a sound

source, DF(f), which is generally frequency and position dependent. For the purpose of this discussion it will be

acceptable to assume that DF is constant with frequency. Beranek [1] defines the DF as follows:

"The directivity factor is the ratio of the intensity on a designated axis of a sound radiator at a stated distance r to

the intensity that would be produced at the same position by a point source if it were radiating the same total

acoustic power as the radiator."

He continues,

"Free space is assumed for the measurements. Usually, the designated axis is taken as the axis of maximum

radiation."

The on axis, free space directivity factor for sound sources with several common free space radiation patterns are

as follows: Monopole, DF = 1.0, Dipole, DF = 3, Cardioid, DF = 3. What this means is that if these three different

sources are to radiate the same total acoustic power then if the monopole has an on axis intensity of 1.0 the dipole

and cardioid will have an on axis intensity of 3.0 or 4.77 dB greater. Conversely, if the difference sources are to

have the same on axis intensity then the dipole and cardioid will radiate 1/3 the acoustic power of the monopole.

(As an aside, for those familiar with room acoustics and reverberation this means that the critical distance from the speaker will be

greater for a dipole or cardioid than for a monopole. The critical distance is the distance at which direct and reflected sound are equal.

The level of the reflected sound, above the modal region of the room, is usually considered constant and proportional to the total

radiated power. Thus when sitting the same distance from a conventional speaker and a dipole, the dipole can potentially sound

more detailed since at the position that ratio of direct to reflected sound is greater for the dipole.) Clearly it would appear that if

DF is not frequency dependent then a speaker system would radiate constant power from which it would follow that

a monopole midrange or satellite speaker should be match with a monopole woofer, a dipole with a dipole and a

cardioid with a cardioid. Since cardioids and dipoles have the same DF it would also appear, form a power

perspective, that a dipole and a cardioid can also be matched.

Unfortunately things just aren't so simple. The above argument is keyed to one unmentioned factor, that the space

the source is radiating into is also frequency independent. Obviously, if the sources are in free space this is the

case. It would also be the case if the source were mounted in a true infinite baffle, or along the line of intersection

of to planes perpendicular to each other, or at the corner formed by three plane where are all mutually

perpendicular. These arrangements amount to what is commonly referred to as 4Pi, 2Pi, Pi, and Pi/2 space.

We are all familiar with the transition between 4Pi and 2Pi space know as the baffle step. We know that as the

transition occurs the on axis sound pressure increases by 6dB. But what about radiated power? Consider for the

moment a sealed box woofer operating at low frequency such that in free space the radiation is that of a

monopole. This corresponds to a driver radiating in the center of a sphere, as shown in Figure 1. The radiated

power is the integral of the intensity, I, over the surface of the sphere enclosing the source. If we let the radius,r,

equal 1.0 and the intensity equal 1.0, the radiated power for the monopole would be 4Pi. If the source were a

dipole or cardioid the total radiated power would be 4Pi/3,

or in general, 4PI/DF. Now consider the same source

placed at the origin of the hemisphere shown in Figure 2.

For a source other than a monopole let the axis of the

source lay in the X-Z plane dividing the sphere into a

or in general, 4PI/DF. Now consider the same source

placed at the origin of the hemisphere shown in Figure 2.

For a source other than a monopole let the axis of the

source lay in the X-Z plane dividing the sphere into a

the origin.

the origin.

hemisphere and assume that the radiation pattern is symmetric about the axis of the source. Further assume

that the same electrical power is input to the source. Since the same electrical signal is forcing the motion of

the driver the motion will remain the same. The motion of the driver will therefore displace the same volume of

air as it moves. However, this volume of air is now required to move though only 1/2 the area, the upper

surface area of the hemisphere. Conservation of mass requires that the velocity of the air passing through the

surface is equal to the volume flow of the air divided by the surface area. Since the volume of air displace by

the driver remains constant the velocity must double when the area is cut in half. The sound pressure is given

by the product of the velocity at the surface of the sphere times the specific acoustic impedance[1].

Therefore, if the velocity doubles so will the sound pressure. The intensity, I, is proportional to the sound

pressure squared. So the intensity increases by a factor of 4. Finally, the total radiated power is the integral of

the intensity over the surface area of the hemisphere or, P = 4 x 2PI/DF = 8 Pi /DF. This is twice (+3dB) the

total power radiated by the same source when radiating into 4Pi space and the doubling of the sound

pressure corresponds to a 6dB increase in sound pressure level. This is a well known result and shows that

not only does the sound pressure level increase by 6dB but also that the efficiency of a sound source doubles

when operating in a 2Pi environment [2].

Now we ask, what happens if the signal to the driver is adjusted so that the on axis intensity and sound

pressure are the same as when the driver is radiating into 4Pi space? This requires that the sound pressure

be reduced to 1.0 which in turn reduces the intensity back to 1.0 in which case the total radiated power into

2Pi space will be 2Pi/DF. This is -3dB relative to a source with the same DF radiating at the same on axis level

in to 4Pi space. The physics is quite simple. The intensity is constant but the area over which it radiates is

reduced by 1/2 when radiating into 2Pi space so the total power is also reduced by 1/2.

**The important conclusion is that when a source with a given DF is radiating into 2Pi space it will **

radiate 3dB less total acoustic power than when radiating in to 4Pi space provided the on axis

intensity or sound pressure level are identical.

This is an interesting observation but you may be asking what this has to do with integrating a woofer system

with a midrange driver or a satellite speaker system when considering uniformity of the radiated power

response? The missing element is a function describing how the radiation space varies with frequency and

placement of the sources. If we consider a single source positioned at some distance above a ground plane

(GP) or floor, and we consider only the effect of the floor,**the effective space into which it radiates will**

undergo a transition from 4Pi to 2Pi as the wave length of the radiated sound becomes longrelative

to the distance between the source and the GP. It follows from the conclusion presented above that,**if the **

source has constant DF and is designed to radiate flat on axis frequency response it must

necessarily radiate 3dB less acoustic power at low frequency than it does at high frequency. This is

shown pictorially in Figure 3, to the left. Thus the supposition that a speaker system with constant directivity

and flat frequency response will radiate constant power at all frequencies is not correct unless the space into

which the speaker radiates is constant. This will not be the case unless the speaker has limited low frequency

extension such that the lowest frequency radiated has a wave length less than the distance to a boundary

(floor, wall ceiling, etc). This may be the case for satellite speakers mounted on stands, or for in wall speakers

where the distance to an adjacent wall, the ceiling or floor is greater than the wave length of the lowest

frequency the system produces.

However, if we consider a typically floor standing 3-way speaker system it would not be unusual to find the

woofer located close to the floor and the midrange at a height of 2 to 3 feet above the floor. If we further

assume that other than the floor the speaker is positioned far from other surfaces then we can see that

somewhere across the frequency range of the system a transition form 4PI to 2Pi radiation will occur.

Furthermore, since the woofer and midrange are at different distances from the floor this GP related transition

can occur at different frequencies for different drivers. If we assume that the wave length is long compared to

the distance between the source and the GP when the wave length is 3 times that distance, then a midrange

driver positioned 3 feet from the floor would be operating in a 4Pi environment above approximately 125 Hz

(wave length 9 feet). This is a reasonable lower limit for the crossover point of a midrange driver. Thus we

may consider the midrange to operate in 4Pi space and have uniform radiated total power as long as its DF

remains constant (and we are below the baffle step frequency for a monopole system). By the same

arguments, the woofer system, being close to the floor, say 1 foot above the GP, will be operating in 2Pi

space at frequencies as high as 375 Hz. Thus the woofer can be considered to operate in 2Pi space over its

entire frequency range. Now, returning to the conclusion reached above we can see that if the midrange and

the woofer have the same DF and the system is designed to have flat on axis response over its entire

frequency range,**then the total acoustic power radiated by the woofer will be 3dB below the power **

radiated by the midrange. This result is independent of the system type,be it a full range monopole,

dipole or cardioid.

Since it has been shown that**a speaker system with constant directivity is not the solution to **

constant total radiated power we can ask ourselves what can be done to realize the constant power

solution, at least if we restrict the argument to the presence of a ground plane. One potential solution would

be using sources with different DF. For example, with a 3-way system of the format above if the midrange

were a dipole or cardioid the radiated power would be 4Pi/3. Since the woofer is operating in 2Pi space its

radiated power would be 2Pi/DFw where the subscript indicates the DF of the woofer. If DFw = 1.5 then the

woofer would radiate the same total acoustic power as the midrange when both had the same on axis intensity.

Before discussing how such a woofer could be constructed it is instructive to**considerer what the result **

would be if a speaker with a dipole midrange were mated with a monopole woofer system.

In this case the midrange would still radiate a power of 4Pi/3, but the woofer would radiate a power of 2Pi

when the on axis response is matched to the midrange level. This results in the woofer radiating 1.5 time the

power or +1.76dB more power than the midrange. While not a perfect match, it is certainly better than the

-3dB result obtained by using a dipole or cardioid woofer system, or by using a monopole midrange with a

monopole woofer.

**So how can this power mismatch be corrected? **That depends on the type of speaker system. If we look

at the case of a monopole midrange there is a fairly easy solution. Such a system will typically already have a

4Pi to 2Pi transition at higher frequency dictated by the baffle step. Thus, all that need be done is to make the

baffle sufficiently large that the baffle step is below the low frequency cut off of the midrange. In this case the

midrange will radiate constant power into 2Pi space as long as its DF remains close to 1, i.e. below the

frequency at which the midrange driver becomes directional. Both the monopole woofer and the midrange will

radiate the same acoustic power when the on axis response is flat. While a potential solution, it is in conflict

with the current trends of narrow baffles. We should also note that**a monopole midrange radiating into **

2Pi space radiates only 1.76 dB more total power than a dipole midrange radiating into 4Pi space,

the typical dipole midrange format. Perhaps the trend towards narrow baffles is actually a step backwards for

conventional speakers?

**If we consider a speaker with a dipole or cardioid midrange response there are a number of **

potential solutions. Considering a dipole system first, if we restrict ourselves to a dipole woofer with its axis

aligned with the midrange then the only way to match the midrange power is to double the on axis intensity of

the woofer. Unfortunately this corresponds to a 3dB increase in the on axis sound pressure level. However,

recall form the definition of DF that it is*usually* defined on the axis of maximum radiation. For a dipole DF will

vary as we move off axis from the maximum of 3 on axis to 0 at 90 degrees off axis. At 45 degrees DF for a

dipole is 1.5. Thus if the woofer system is rotated such that the woofer axis is at 45 degrees relative to the axis

of the midrange, which we shall take as the axis of the speaker system, the response of the woofer on the

system axis would match that of the midrange when it is radiating the same total acoustic power. With a stereo

pair of woofers rotated inward towards the listening position such an arrangement could potentially yield a

more uniform low frequency sound field to the left and right of the listening position since as the listener

moved to the left the drop off of the left woofer, due to the dipole polar response would be compensated for

by the increase on the output from the right side woofer. Such an arrangement is shown in Figure 4. The

dipole woofers are rotated 45 degrees relative to the speaker (and midrange) axis. The green lines show the

increased bass level radiated form the left woofer could potentially compensate for the drop

that the same electrical power is input to the source. Since the same electrical signal is forcing the motion of

the driver the motion will remain the same. The motion of the driver will therefore displace the same volume of

air as it moves. However, this volume of air is now required to move though only 1/2 the area, the upper

surface area of the hemisphere. Conservation of mass requires that the velocity of the air passing through the

surface is equal to the volume flow of the air divided by the surface area. Since the volume of air displace by

the driver remains constant the velocity must double when the area is cut in half. The sound pressure is given

by the product of the velocity at the surface of the sphere times the specific acoustic impedance[1].

Therefore, if the velocity doubles so will the sound pressure. The intensity, I, is proportional to the sound

pressure squared. So the intensity increases by a factor of 4. Finally, the total radiated power is the integral of

the intensity over the surface area of the hemisphere or, P = 4 x 2PI/DF = 8 Pi /DF. This is twice (+3dB) the

total power radiated by the same source when radiating into 4Pi space and the doubling of the sound

pressure corresponds to a 6dB increase in sound pressure level. This is a well known result and shows that

not only does the sound pressure level increase by 6dB but also that the efficiency of a sound source doubles

when operating in a 2Pi environment [2].

Now we ask, what happens if the signal to the driver is adjusted so that the on axis intensity and sound

pressure are the same as when the driver is radiating into 4Pi space? This requires that the sound pressure

be reduced to 1.0 which in turn reduces the intensity back to 1.0 in which case the total radiated power into

2Pi space will be 2Pi/DF. This is -3dB relative to a source with the same DF radiating at the same on axis level

in to 4Pi space. The physics is quite simple. The intensity is constant but the area over which it radiates is

reduced by 1/2 when radiating into 2Pi space so the total power is also reduced by 1/2.

radiate 3dB less total acoustic power than when radiating in to 4Pi space provided the on axis

intensity or sound pressure level are identical.

with a midrange driver or a satellite speaker system when considering uniformity of the radiated power

response? The missing element is a function describing how the radiation space varies with frequency and

placement of the sources. If we consider a single source positioned at some distance above a ground plane

(GP) or floor, and we consider only the effect of the floor,

undergo a transition from 4Pi to 2Pi as the wave length of the radiated sound becomes long

to the distance between the source and the GP. It follows from the conclusion presented above that,

source has constant DF and is designed to radiate flat on axis frequency response it must

necessarily radiate 3dB less acoustic power at low frequency than it does at high frequency.

shown pictorially in Figure 3, to the left. Thus the supposition that a speaker system with constant directivity

and flat frequency response will radiate constant power at all frequencies is not correct unless the space into

which the speaker radiates is constant. This will not be the case unless the speaker has limited low frequency

extension such that the lowest frequency radiated has a wave length less than the distance to a boundary

(floor, wall ceiling, etc). This may be the case for satellite speakers mounted on stands, or for in wall speakers

where the distance to an adjacent wall, the ceiling or floor is greater than the wave length of the lowest

frequency the system produces.

However, if we consider a typically floor standing 3-way speaker system it would not be unusual to find the

woofer located close to the floor and the midrange at a height of 2 to 3 feet above the floor. If we further

assume that other than the floor the speaker is positioned far from other surfaces then we can see that

somewhere across the frequency range of the system a transition form 4PI to 2Pi radiation will occur.

Furthermore, since the woofer and midrange are at different distances from the floor this GP related transition

can occur at different frequencies for different drivers. If we assume that the wave length is long compared to

the distance between the source and the GP when the wave length is 3 times that distance, then a midrange

driver positioned 3 feet from the floor would be operating in a 4Pi environment above approximately 125 Hz

(wave length 9 feet). This is a reasonable lower limit for the crossover point of a midrange driver. Thus we

may consider the midrange to operate in 4Pi space and have uniform radiated total power as long as its DF

remains constant (and we are below the baffle step frequency for a monopole system). By the same

arguments, the woofer system, being close to the floor, say 1 foot above the GP, will be operating in 2Pi

space at frequencies as high as 375 Hz. Thus the woofer can be considered to operate in 2Pi space over its

entire frequency range. Now, returning to the conclusion reached above we can see that if the midrange and

the woofer have the same DF and the system is designed to have flat on axis response over its entire

frequency range,

radiated by the midrange. This result is independent of the system type,

dipole or cardioid.

Since it has been shown that

constant total radiated power

solution, at least if we restrict the argument to the presence of a ground plane. One potential solution would

be using sources with different DF. For example, with a 3-way system of the format above if the midrange

were a dipole or cardioid the radiated power would be 4Pi/3. Since the woofer is operating in 2Pi space its

radiated power would be 2Pi/DFw where the subscript indicates the DF of the woofer. If DFw = 1.5 then the

woofer would radiate the same total acoustic power as the midrange when both had the same on axis intensity.

Before discussing how such a woofer could be constructed it is instructive to

would be if a speaker with a dipole midrange were mated with a monopole woofer system.

when the on axis response is matched to the midrange level. This results in the woofer radiating 1.5 time the

power or +1.76dB more power than the midrange. While not a perfect match, it is certainly better than the

-3dB result obtained by using a dipole or cardioid woofer system, or by using a monopole midrange with a

monopole woofer.

at the case of a monopole midrange there is a fairly easy solution. Such a system will typically already have a

4Pi to 2Pi transition at higher frequency dictated by the baffle step. Thus, all that need be done is to make the

baffle sufficiently large that the baffle step is below the low frequency cut off of the midrange. In this case the

midrange will radiate constant power into 2Pi space as long as its DF remains close to 1, i.e. below the

frequency at which the midrange driver becomes directional. Both the monopole woofer and the midrange will

radiate the same acoustic power when the on axis response is flat. While a potential solution, it is in conflict

with the current trends of narrow baffles. We should also note that

2Pi space radiates only 1.76 dB more total power than a dipole midrange radiating into 4Pi space,

conventional speakers?

potential solutions.

aligned with the midrange then the only way to match the midrange power is to double the on axis intensity of

the woofer. Unfortunately this corresponds to a 3dB increase in the on axis sound pressure level. However,

recall form the definition of DF that it is

vary as we move off axis from the maximum of 3 on axis to 0 at 90 degrees off axis. At 45 degrees DF for a

dipole is 1.5. Thus if the woofer system is rotated such that the woofer axis is at 45 degrees relative to the axis

of the midrange, which we shall take as the axis of the speaker system, the response of the woofer on the

system axis would match that of the midrange when it is radiating the same total acoustic power. With a stereo

pair of woofers rotated inward towards the listening position such an arrangement could potentially yield a

more uniform low frequency sound field to the left and right of the listening position since as the listener

moved to the left the drop off of the left woofer, due to the dipole polar response would be compensated for

by the increase on the output from the right side woofer. Such an arrangement is shown in Figure 4. The

dipole woofers are rotated 45 degrees relative to the speaker (and midrange) axis. The green lines show the

increased bass level radiated form the left woofer could potentially compensate for the drop

with frequency for a constant

directivity source with flat on axis

frequency response as it undergoes a

transition between a 4PI and 2Pi

environment due to the proximity to a

ground plane.

off of the right woofer. However, with

the speakers canted in towards the

listening position by 30 degrees, as

is typical, the dipole woofer is

rotated 75 degrees relative to the

front to back axis of the listening

room. And of course, there would be

complications with the crossover

from midrange to woofer in addition

to very different stimulation of room

modes. All in all a novel solution that

requires considerable more

investigation. (Note, such an

arrangement could also be

experimented with for midrange

reproduction.)

In the case of a cardioid woofer

system, a similar arrangement could

be made, however the rotation of the

woofers would have to be greater

since the DF = 1.5 point a cardioid is

close to 70 degrees off axis [3].

the speakers canted in towards the

listening position by 30 degrees, as

is typical, the dipole woofer is

rotated 75 degrees relative to the

front to back axis of the listening

room. And of course, there would be

complications with the crossover

from midrange to woofer in addition

to very different stimulation of room

modes. All in all a novel solution that

requires considerable more

investigation. (Note, such an

arrangement could also be

experimented with for midrange

reproduction.)

In the case of a cardioid woofer

system, a similar arrangement could

be made, however the rotation of the

woofers would have to be greater

since the DF = 1.5 point a cardioid is

close to 70 degrees off axis [3].

yield flat on axis response and flat power response.

In view of the somewhat bizarre nature of these solutions to the power

response problem it is reasonable to look for a more appealing solution

where the axis of the woofer system is aligned with the axis of the

speaker. For this to be the case we must develop a woofer system with

an on axis, free space DF = 1.5. This can be accomplished by combining a

dipole woofer with a monopole. If we start with a dipole woofer and being to

sum its response to that of a monopole we find a gradual deformation of

the dipole radiation pattern into a cardioid at the point where the isolated

dipole and monopole on axis sound pressures are equal. Since the

cardioid still has a DF = 3, as for the dipole, it is unlikely that any of the

intermediate polar response patterns would result in our target DF = 1.5.

However, if we continue to increase the monopole strength above that of

the dipole the null to the rear of the cardioid begins to fill in. As the strength

of the monopole continues to increase relative to the dipole, the polar

pattern becomes that of a monopole with DF = 1. This variation in polar

response is shown in Figure 5. Clearly, somewhere between the cardioid

and the monopole there exists a situation where the relative strength of the

monopole and dipole are such that DF = 1.5 can be obtained on axis. Such

a polar pattern would be the desired response to achieve constant radiated

power between a dipole midrange operating in fee space, 4Pi, and a

woofer system operating in 2Pi space, as when the woofer is positioned

close to a ground plane. Further details of the design of a woofer system

with va**r**ious DF can be found here.

response problem it is reasonable to look for a more appealing solution

where the axis of the woofer system is aligned with the axis of the

speaker. For this to be the case we must develop a woofer system with

an on axis, free space DF = 1.5. This can be accomplished by combining a

dipole woofer with a monopole. If we start with a dipole woofer and being to

sum its response to that of a monopole we find a gradual deformation of

the dipole radiation pattern into a cardioid at the point where the isolated

dipole and monopole on axis sound pressures are equal. Since the

cardioid still has a DF = 3, as for the dipole, it is unlikely that any of the

intermediate polar response patterns would result in our target DF = 1.5.

However, if we continue to increase the monopole strength above that of

the dipole the null to the rear of the cardioid begins to fill in. As the strength

of the monopole continues to increase relative to the dipole, the polar

pattern becomes that of a monopole with DF = 1. This variation in polar

response is shown in Figure 5. Clearly, somewhere between the cardioid

and the monopole there exists a situation where the relative strength of the

monopole and dipole are such that DF = 1.5 can be obtained on axis. Such

a polar pattern would be the desired response to achieve constant radiated

power between a dipole midrange operating in fee space, 4Pi, and a

woofer system operating in 2Pi space, as when the woofer is positioned

close to a ground plane. Further details of the design of a woofer system

with va

to cardioid to monopole.

In conclusion it has been shown that constant directionality, or uniformity in the polar response pattern does

not result in uniform power unless the sound source in question is restricted to radiating into a uniform

space as well. While this may be the case for a limited bandwidth system mounted on stands or in walls, the

restriction of uniform space will typically be violated for larger, full range speaker systems regardless of type.

The discussion has shown that for a typical speaker system with woofer close to the floor and midrange 2 to

3 feet above the floor, the woofer system should have a free space directivity factor, measured on axis, that

is 1/2 that of the midrange if constant radiated power with frequency is desired, assuming the midrange is

operating in 4Pi space. This result is independent of the midrange type, monopole, dipole or cardioid. For a

monopole speaker it was shown that this problem can be rectified by using a wide baffle, placing the baffle step

below the crossover point, so that the midrange driver is operating in 2Pi space over its entire frequency range.

It was also shown that for systems using conventional formats mating a dipole midrange with a monopole woofer

will yield more uniform power response than when using a dipole woofer. Finally, it was shown that for constant

power a woofer system to be mated to a dipole midrange should have a polar response lying somewhere

between that of a cardioid and that of a monopole to achieve the required directionality factor of 1.5. The

discussion has been limited to that of constant radiated power. Other factors which will impact system

performance, such as room modes, room pressurization below the room fundamental and crossover between

different sources have been neglected. These topics are addressed in other discussions (see the Tech menu).

____________________

References:

1. Beranek, L. L., Acoustics, McGraw Hill Book Company, NY, 1954.

2. D'Appolito, J., Testing Loudspeakers, Audio Amateur Press, Peterborough, NH, 1998.

3. Olson, H.F., Gradient Loudspeakers, JAES, March, 1973. Also, Loudspeakers an Anthology, Vol. 1, 1980.

not result in uniform power unless the sound source in question is restricted to radiating into a uniform

space as well. While this may be the case for a limited bandwidth system mounted on stands or in walls, the

restriction of uniform space will typically be violated for larger, full range speaker systems regardless of type.

The discussion has shown that for a typical speaker system with woofer close to the floor and midrange 2 to

3 feet above the floor, the woofer system should have a free space directivity factor, measured on axis, that

is 1/2 that of the midrange if constant radiated power with frequency is desired, assuming the midrange is

operating in 4Pi space. This result is independent of the midrange type, monopole, dipole or cardioid. For a

monopole speaker it was shown that this problem can be rectified by using a wide baffle, placing the baffle step

below the crossover point, so that the midrange driver is operating in 2Pi space over its entire frequency range.

It was also shown that for systems using conventional formats mating a dipole midrange with a monopole woofer

will yield more uniform power response than when using a dipole woofer. Finally, it was shown that for constant

power a woofer system to be mated to a dipole midrange should have a polar response lying somewhere

between that of a cardioid and that of a monopole to achieve the required directionality factor of 1.5. The

discussion has been limited to that of constant radiated power. Other factors which will impact system

performance, such as room modes, room pressurization below the room fundamental and crossover between

different sources have been neglected. These topics are addressed in other discussions (see the Tech menu).

____________________

References:

1. Beranek, L. L., Acoustics, McGraw Hill Book Company, NY, 1954.

2. D'Appolito, J., Testing Loudspeakers, Audio Amateur Press, Peterborough, NH, 1998.

3. Olson, H.F., Gradient Loudspeakers, JAES, March, 1973. Also, Loudspeakers an Anthology, Vol. 1, 1980.