Hello,
I would like to show the design philosophy and practical implementation of a phase-accurate WMTMW Baekgaard loudspeaker here.
The Baekgaard loudspeaker dates back from the seventies with Erik Baekgaad working for Bang&Olufsen. The AES publication is here, dating back from March 1975 : http://www.aes.org/tmpFiles/elib/20100925/2480.pdf
The Beovox M70 is one B&O commercial implementation in 1975 : Beovox M70 Passive Loudspeakers
The Beovox S45 is one B&O commercial implementation in 1976 : Beovox S45 Passive Loudspeakers
The Beovox S45-2 is one B&O commercial implementation in 1978 : Beovox S45-2 Passive Loudspeakers
The S45 should be a WMTMM loudspeaker but unfortunately it went WMT for cost reasons, maybe.
The M70 is a S45 with a specialized deep-bass driver, hence bigW-WMT.
I own 2 pairs of S45 and one pair of M70 in original state.
The S45 sounds terrible (bad) by nowadays standard, agressive high range and non-existing deep bass. They were quoted as "bright" when launched.
The MT70 sounds better with a smooth medium and high range. The M70 has SEAS dome tweeters instead of Philips dome tweeters in the S45. The M70 deep bass is decent.
The Beovox S45 should not be confused with the Beovox S22, S25, S30, S35 and S40. Those two are conventional 2-way designs.
S30 in 1975 : Beovox S30 Passive Loudspeakers
S22 in 1977 : Beovox S22 Passive Loudspeakers
S25 in 1977 : Beovox S25 Passive Loudspeakers
S35 in 1977 : Beovox S35 Passive Loudspeakers
S40 in 1980 : Beovox S40 Passive Loudspeakers
The Baekgaard is an elegant crossover approach based on reassuring & straightforward maths. Let us consider three 2nd order system. Let us say they share the same denominator in their transfer function. Their denominator = s**2 + (Q**-1)*s + 1.
The s variable tracks the relative frequency in complex notation. When s = 1, you are at the central frequency of the filter.
Q = 0.707 for a Butterworth shape factor (restricted bandpass bandwith).
Q = 0.500 for a Bessel shape factor (regular bandpass bandwith)
Q = 0.333 for a very soft shape (extended bandpass bandwith)
Those are fine values for Q in a Baekgaard approach.
Now, look how everything simplifies down if :
- the Baekgaard highpass has the Numerator = s**2
- the Baekgaard bandpass has the Numerator = (Q**-1)*s
- the Baekgaard low pass has the Numerator = 1
What's happening when you sum the three ouputs ?
You get the numerator equal to s**2 + (Q**-1)*s + 1.
Knowing the denominator is s**2 + (Q**-1)*s + 1.
This means a perfect unity transfer function.
This is the beauty of the Baekgaard.
Amplitude gets thus perfectly reconstructed.
Phase gets thus perfectly reconstructed.
The group delay of the reconstructed signal is thus constant across the entire frequency range.
But wait a minute, what is the difference between this and a conventional 2-way 2nd order crossover ?
- in the Baekgaard, you get the medium driver as "filler" unit or "missing link" between the woofer and the tweeter
- the medium unit is an absolute necessity because at the crossover frequency, the woofer and the tweeter have the same amplitude (-3dB or -6dB depending on Butterworth or Bessel) but relative 180 degree phase shift. They cancel themselves. At the exact crossover frequency (s=1), the acoustic output is thus only coming from the medium unit. If the medium unit was not there, there would be a huge amplitude dip at the crossover frequency.
- in a conventional 2-way 2nd order crossover, one usually puts the tweeter in reverse phase. This way there is no more dip. In a conventional 2-way 2nd order crossover, if you use Q=0.707 (Butterworth) and if you put the tweeter in reverse phase, you get a +3dB boost instead. If you double the filtering like cascading two Q=0.707 (Butterworth), you get the well known 4th-order Linkwitz-Riley arrangement and perfect amplitude flatness. But, of course, an ugly phase distorsion with 360 degree phase error between low frequency and high frequency.
In a nutshell, the Baekgaard is the native implementation of a 2nd-order frequency splitter. Being 2nd-order, it is natively 3-way, as the math above do explain.
No need to say, I very much like this crossover approach.
Let us now proceed step by step for a year 2010 implementation :
a) We shall implement a Baekgaard crossover at 1700 Hz with Q=0,5. This is a phase accurate crossover (perfect reconstructed phase), however not very selective (2nd order only - quality drivers are thus mandatory), however delivering substantial inter-driver phase shifts in the overlap region (a very compact WMTMW arrangement is thus mandatory).
b) The Baekgaard crossover is easy to implement both in active form and in passive form. You may thus mount your passive crossover in an external box for ensuring compatibility with an active crossover approach.
c) As we need a bass-reflex W section having a -3dB natural bandwith of 35 Hz to 3400 Hz, our W drivers shall not exceed 16 cm in diameter.
d) In a domestic listening environment, two 16 cm diameter W drivers can deliver enough acoustic pressure, provided they have a decent excursion like 4 mm.
e) As we need a medium M section having a -3dB natural bandwith of 425 Hz to 6800 Hz, our M drivers can be 8 cm or 10 cm in diameter with low excursion and low power handling.
f) As we need a tweeter section having a -3dB natural bandwith of 850 Hz to 15 kHz, a high quality 1 inch dome tweeter with built-in rear chamber would be the best choice, with decent power handling.
g) Later on, for increasing the maximum sound pressure level at 35Hz, we can build two or four W---W columns, same height as the main WMTMW colums, assisting them in the deep bass range. The crossover between the W---W column and WMTMW column shall be active, avoiding expensive big coils and unreliable big caps. The subwoofer filter may thus be 1st order at 120 Hz. The 1st order is chosen in order to keep the relative phase shift below 90 degree. One may opt for a Lipshitz-Vanderkooy phase-compensated 2nd order Bessel crossover for the subwoofer, providing a zero relative phase shift, but causing a 180 degree reconstructed phase shift error in the 60 Hz to 240 Hz range. One may opt for a digital Lipshitz-Vanderkooy delay-compensated 4th order Bessel crossover for the subwoofer, providing a virtually zero relative phase shift, and causing no reconstructed phase shift error in the 60 Hz to 240 Hz range. With this active crossover approach for the subwoofer option, we can use a few inexpensive power amps in the 35 Hz to 120 Hz range.
Let me know if you feel interest for such WMTMW Baekgaard arrangement. If yes, let us select the drivers, running some simulations.
Regards,
Steph
I would like to show the design philosophy and practical implementation of a phase-accurate WMTMW Baekgaard loudspeaker here.
The Baekgaard loudspeaker dates back from the seventies with Erik Baekgaad working for Bang&Olufsen. The AES publication is here, dating back from March 1975 : http://www.aes.org/tmpFiles/elib/20100925/2480.pdf
The Beovox M70 is one B&O commercial implementation in 1975 : Beovox M70 Passive Loudspeakers
The Beovox S45 is one B&O commercial implementation in 1976 : Beovox S45 Passive Loudspeakers
The Beovox S45-2 is one B&O commercial implementation in 1978 : Beovox S45-2 Passive Loudspeakers
The S45 should be a WMTMM loudspeaker but unfortunately it went WMT for cost reasons, maybe.
The M70 is a S45 with a specialized deep-bass driver, hence bigW-WMT.
I own 2 pairs of S45 and one pair of M70 in original state.
The S45 sounds terrible (bad) by nowadays standard, agressive high range and non-existing deep bass. They were quoted as "bright" when launched.
The MT70 sounds better with a smooth medium and high range. The M70 has SEAS dome tweeters instead of Philips dome tweeters in the S45. The M70 deep bass is decent.
The Beovox S45 should not be confused with the Beovox S22, S25, S30, S35 and S40. Those two are conventional 2-way designs.
S30 in 1975 : Beovox S30 Passive Loudspeakers
S22 in 1977 : Beovox S22 Passive Loudspeakers
S25 in 1977 : Beovox S25 Passive Loudspeakers
S35 in 1977 : Beovox S35 Passive Loudspeakers
S40 in 1980 : Beovox S40 Passive Loudspeakers
The Baekgaard is an elegant crossover approach based on reassuring & straightforward maths. Let us consider three 2nd order system. Let us say they share the same denominator in their transfer function. Their denominator = s**2 + (Q**-1)*s + 1.
The s variable tracks the relative frequency in complex notation. When s = 1, you are at the central frequency of the filter.
Q = 0.707 for a Butterworth shape factor (restricted bandpass bandwith).
Q = 0.500 for a Bessel shape factor (regular bandpass bandwith)
Q = 0.333 for a very soft shape (extended bandpass bandwith)
Those are fine values for Q in a Baekgaard approach.
Now, look how everything simplifies down if :
- the Baekgaard highpass has the Numerator = s**2
- the Baekgaard bandpass has the Numerator = (Q**-1)*s
- the Baekgaard low pass has the Numerator = 1
What's happening when you sum the three ouputs ?
You get the numerator equal to s**2 + (Q**-1)*s + 1.
Knowing the denominator is s**2 + (Q**-1)*s + 1.
This means a perfect unity transfer function.
This is the beauty of the Baekgaard.
Amplitude gets thus perfectly reconstructed.
Phase gets thus perfectly reconstructed.
The group delay of the reconstructed signal is thus constant across the entire frequency range.
But wait a minute, what is the difference between this and a conventional 2-way 2nd order crossover ?
- in the Baekgaard, you get the medium driver as "filler" unit or "missing link" between the woofer and the tweeter
- the medium unit is an absolute necessity because at the crossover frequency, the woofer and the tweeter have the same amplitude (-3dB or -6dB depending on Butterworth or Bessel) but relative 180 degree phase shift. They cancel themselves. At the exact crossover frequency (s=1), the acoustic output is thus only coming from the medium unit. If the medium unit was not there, there would be a huge amplitude dip at the crossover frequency.
- in a conventional 2-way 2nd order crossover, one usually puts the tweeter in reverse phase. This way there is no more dip. In a conventional 2-way 2nd order crossover, if you use Q=0.707 (Butterworth) and if you put the tweeter in reverse phase, you get a +3dB boost instead. If you double the filtering like cascading two Q=0.707 (Butterworth), you get the well known 4th-order Linkwitz-Riley arrangement and perfect amplitude flatness. But, of course, an ugly phase distorsion with 360 degree phase error between low frequency and high frequency.
In a nutshell, the Baekgaard is the native implementation of a 2nd-order frequency splitter. Being 2nd-order, it is natively 3-way, as the math above do explain.
No need to say, I very much like this crossover approach.
Let us now proceed step by step for a year 2010 implementation :
a) We shall implement a Baekgaard crossover at 1700 Hz with Q=0,5. This is a phase accurate crossover (perfect reconstructed phase), however not very selective (2nd order only - quality drivers are thus mandatory), however delivering substantial inter-driver phase shifts in the overlap region (a very compact WMTMW arrangement is thus mandatory).
b) The Baekgaard crossover is easy to implement both in active form and in passive form. You may thus mount your passive crossover in an external box for ensuring compatibility with an active crossover approach.
c) As we need a bass-reflex W section having a -3dB natural bandwith of 35 Hz to 3400 Hz, our W drivers shall not exceed 16 cm in diameter.
d) In a domestic listening environment, two 16 cm diameter W drivers can deliver enough acoustic pressure, provided they have a decent excursion like 4 mm.
e) As we need a medium M section having a -3dB natural bandwith of 425 Hz to 6800 Hz, our M drivers can be 8 cm or 10 cm in diameter with low excursion and low power handling.
f) As we need a tweeter section having a -3dB natural bandwith of 850 Hz to 15 kHz, a high quality 1 inch dome tweeter with built-in rear chamber would be the best choice, with decent power handling.
g) Later on, for increasing the maximum sound pressure level at 35Hz, we can build two or four W---W columns, same height as the main WMTMW colums, assisting them in the deep bass range. The crossover between the W---W column and WMTMW column shall be active, avoiding expensive big coils and unreliable big caps. The subwoofer filter may thus be 1st order at 120 Hz. The 1st order is chosen in order to keep the relative phase shift below 90 degree. One may opt for a Lipshitz-Vanderkooy phase-compensated 2nd order Bessel crossover for the subwoofer, providing a zero relative phase shift, but causing a 180 degree reconstructed phase shift error in the 60 Hz to 240 Hz range. One may opt for a digital Lipshitz-Vanderkooy delay-compensated 4th order Bessel crossover for the subwoofer, providing a virtually zero relative phase shift, and causing no reconstructed phase shift error in the 60 Hz to 240 Hz range. With this active crossover approach for the subwoofer option, we can use a few inexpensive power amps in the 35 Hz to 120 Hz range.
Let me know if you feel interest for such WMTMW Baekgaard arrangement. If yes, let us select the drivers, running some simulations.
Regards,
Steph
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