Assisted Alignments of Order 5, 6 and 7
All alignments having n > 5 were designed using the Matlab/Octave software for assisted alignments. In all the following examples having odd n, fp is the pole frequency in Hz of the first-order factor of the transfer function of the external filter.
Example 6: Fifth-Order Sub-Chebyshev Assisted Alignment
A plot of the dB SPL at 1 meter and displacement vs. frequency of a fifth-order assisted alignment is shown below in Figure 9.
Performance Summary for Example 6
The fifth-order sub-Chebyshev assisted alignment has the following data:
VB = 9.7ft3
fB = 19.4Hz
System f3 = 21.1Hz
Reference SPL = 108.6dB
dB ripple = 0dB
k value = 1.3390
Filter fp = 15.79Hz
The graph of displacement vs. frequency shows that the high-pass filter doesn't have enough low-frequency attenuation to equalize the amplitude of the displacement peaks. Since the preamp output amplitude is adjusted to achieve a 14mm peak displacement at the highest peak, this configuration has somewhat lower reference SPL than the unassisted alignments with added high-pass filter.
Example 7: Sixth-Order Chebyshev Class 1 Assisted Alignment
A plot of the dB SPL at 1 meter and displacement vs. frequency of a sixth-order Chebyshev class 1 alignment is shown below in Figure 10.
Performance Summary for Example 7
The sixth-order Chebyshev class 1 assisted alignment has the following data:
VB = 11.2ft3
fB = 15.0Hz
System f3 = 10.2Hz
Reference SPL = 91.6dB
dB ripple = 0.06dB
k value = 0.4399
Filter undamped resonant frequency = 10.46Hz
Filter Q = 4.2718
This alignment suffers from a very poor reference SPL, 9.6dB worse than even the unassisted alignment without any high-pass filter. Figure 11 shows the cause of the problem.
The lower pane of Figure 11 shows the loudspeaker displacement in mm vs. frequency for a constant voltage amplitude of 1Volt peak at the loudspeaker terminals. The upper pane is the frequency response of the external high-pass filter. The filter has a boost of 12.7dB at 10.6Hz. This would not be so bad by itself, but the boost occurs below the box tuning frequency where the displacement has already risen considerably above its value at the secondary peak at 22.4Hz. The combined effect is one of a very large displacement peak near 10.4Hz, making this a very risky design, especially if used for low-frequency effects in a home theater.
This alignment also points out the risks of blindly using the synthesis software without also analyzing the system behavior using simulation, especially in regard to the displacement vs. frequency. The 10.2Hz -3dB frequency looks too good to be true, and in this case it is, due to the displacement problem.
Example 8: Sixth-Order Sub-Chebyshev Class 2 Alignment
A plot of the dB SPL at 1 meter and displacement vs. frequency of a sixth-order sub-Chebyshev class 2 alignment is shown below in Figure 12.
Performance Summary for Example 8
The sixth-order sub-Chebyshev class 2 assisted alignment has the following data:
VB = 12.5ft3
fB = 19.8Hz
System f3 = 20.6Hz
Reference SPL = 113.4dB
dB ripple = 0dB
k value = 1.1904
Filter undamped resonant frequency = 18.76Hz
Filter Q = 0.6530
This is a usable alignment whose behavior is similar to the unassisted alignment with second-order Butterworth filter added, although its box volume is a bit larger than the unassisted one.
Example 9: Sixth-Order Sub-Chebyshev Class 3 Alignment
A plot of the dB SPL at 1 meter and displacement vs. frequency of a sixth-order sub-Chebyshev class 3 alignment is shown below in Figure 13.
Performance Summary for Example 9
The sixth-order sub-Chebyshev class 3 alignment has the following data:
VB = 8.9ft3
fB = 20.1Hz
System f3 = 23.1Hz
Reference SPL = 112.1dB
dB ripple = 0dB
k value = 1.6872
Filter undamped resonant frequency = 13.99Hz
Filter Q = 0.5063
This is also a usable alignment whose box volume is somewhat smaller than the unassisted alignment, in return for a somewhat higher system -3dB frequency.
Example 10: Seventh-Order Chebyshev Class 1 Alignment
A plot of the dB SPL at 1 meter and displacement vs. frequency of a seventh-order Chebyshev class 1 alignment is shown below in Figure 14.
Performance Summary for Example 10
The seventh-order Chebyshev class 1 alignment has the following data:
VB = 9.8ft3
fB = 17.0Hz
Filter f3 = 11.5Hz
System f3 = 14.1Hz
Reference SPL = 104.7dB
dB ripple < 0.01dB
k value = 0.6635
Filter fp = 21.31Hz
Filter undamped resonant frequency = 14.34Hz
Filter Q = 3.3391
Like the sixth-order class 1 alignment, this alignment has a compromised reference SPL due to excessive boost of the external filter at frequencies below the box tuning frequency. It's not compromised as badly as the sixth-order class 1 alignment though, and it does appear marginally usable. The 14.1Hz system f3 may appeal to those not overly concerned with high SPL.
Example 11: Seventh-Order Sub-Chebyshev Class 2 Alignment
A plot of the dB SPL at 1 meter and displacement vs. frequency of a seventh-order sub-Chebyshev class 2 alignment is shown below in Figure 15.
Performance Summary for Example 11
The seventh-order sub-Chebyshev class 2 alignment has the following data:
VB = 10.2ft3
fB = 21.6Hz
Filter f3 = 26.6Hz
System f3 = 23.7Hz
Reference SPL = 115.5dB
dB ripple = 0dB
k value = 1.4825
Filter fp = 15.99Hz
Filter undamped resonant frequency = 19.58Hz
Filter Q = 0.6549
This is a usable alignment with very good reference SPL, but the system -3dB frequency could be improved.
Example 12: Seventh-Order Sub-Chebyshev Class 3 Alignment
A plot of the dB SPL at 1 meter and displacement vs. frequency of a seventh-order sub-Chebyshev class 3 alignment is shown below in Figure 16.
Performance Summary for Example 12
The seventh-order sub-Chebyshev class 3 alignment has the following data:
VB = 8.4ft3
fB = 20.5Hz
Filter f3 = 24.0Hz
System f3 = 24.2Hz
Reference SPL = 114.2dB
dB ripple = 0dB
k value = 2.0185
Filter fp = 11.99Hz
Filter undamped resonant frequency = 12.95Hz
Filter Q = 0.5140
Performance is similar to Example 11, but with a slightly higher system -3dB frequency and smaller box volume.