This type of subwoofer is quite easy to calculate. If the Thiele and Small parameters are known (they can also be measured) the cabinet volume is easy to calculate.
In general the cabinet should be tuned to a fitting Q factor of Qtc=0.71. Should the bass be particularly dry and tight, a somewhat smaller Qtc value could be chosen.
This leads to a bigger cabinet and a considerably more gentle level decrease towards lower frequencies, whereby sloping starts already at higher frequencies.
The electrical Q factor increases: Qes = Qes.previous * (Rv + Re) / Re, where Re is the DC resistance of the bass driver.
As a result the drive system may be weakened and bass reproduction in the lower bass region could be increased, requiring a bigger cabinet.
But, what is the effect of a capacitor in series to the driver?
At a glance you would say that this is a 6 dB high pass, allowing only high frequencies through. As an approximate explanation this is correct.
For a more detailed explanation we need to use the complex number plane. For, resistance is not just a one-dimensional value that may simply be added up, they, rather, consist of several components that should be entered in the complex number plane in different directions. To the left we have the simple case of having a driver that acts as a resistor, with a capacitor in series.
The red arrow to the right indicates the driver's resistance (since we are talking about voltage, the driver's resistance - like any other resistance - is to be multiplied with the current I).The blue arrow corresponds to the capacitor, of which the resistance is dependant on the frequency. This is expressed mathematically as jω whereby ω = 2πf, i.e. 6.28 x the frequency f.
You will notice that the smaller f (or ω) or the smaller the capacitance C, the smaller the denominator resp. the longer the blue arrow.
If the blue arrow is longer, then the ratio of the lengths ULS (voltage applied to the driver) divided by Uges (total voltage on input terminals) is smaller.
Therefore: when the frequency drops or using a smaller capacitor value, less and less voltage will reach the driver.
However, we had made the assumption that the driver only has an effective resistance.
But, in reality that's not the case.
The driver has: - An effective resistance (the voice coil's resistance), marked red - An inductive reactance of the voice coil jωL, upwards in the picture - And an induced voltage part (by the movement of the voice coil within the magnetic field, represented by a black arrow)
The sum of all individual voltages result in ULS, i.e. the voltage applied to the loudspeaker terminal.
In addition, the capacitor is going to be wired in series (blue arrow downwards).
You will notice in the picture that the voltage applied to the driver ULS is bigger (i.e. the arrow is longer) than the voltage Uges applied to the loudspeaker terminals (turquoise arrow).
In other words: the capacitor causes an upward transformation, i.e. the capacitor increases the voltage applied to the driver.
Now the question comes up, when this happens.
Answer: whenever the capacitor helps to move the arrow of the total voltage closer to the axis to the right (effective / loss voltage).
What are the implications in practise?
If you have a measuring instrument that is capable of measuring complex impedance (possibly your sound card?), then it's easy to calculate how a capacitor changes the level.
Below, we have done this for the Alcone AC12 SW4 (first five columns).
In column 6 (inductive part C) the inductive impedance of a 800 μF capacitor was added (since the impedance is capacitive the values are negative).
Column 7 shows the sum of all inductive parts. If this value is smaller than the inductive part in column 5, then the level rises, indicated by the two last columns.
What causes the rising levels?
Area 1 (subsonic, yellow): At low frequencies the capacitor's impedance is so high that the driver's level is lowered.
Area 2 (below the resonance frequency of the bass driver, light-green): From 22 Hz on, where the inductive part of the impedance is higher than the sum of the inductive and capacitive part, the level continues to rise up to 49 Hz.
Area 3 (above the resonance frequency of the bass driver, light-blue): At 49 Hz the driver reaches its resonance; at this point the phase changes as is typical. The capacitor now starts reducing the level; a slight change initially since the driver's impedance is high. When the frequency continues to rise the imaginary impedance of the capacitor decreases, but much faster than the driver's impedance, which is why the capacitor causes the level to decrease up to 70 Hz.
Area 4 (higher frequency range, reddish): When the frequency continues to rise the capacitor's impedance becomes so small that this component doesn't play an important part anymore. Again, some small level increase may occur.
The calculated values correspond closely to the measuring results. If the capacitor's value is increased the bass boost starts at a lower frequency but turns out to be somewhat gentler: - at 1000 uF: boost from 20.5 Hz / increase of 5.5 dB - at 1200 uF: boost from 19 Hz / increase of 4.7 dB - at 1600 uF: boost from 17.5 Hz / increase of 4 dB
The capacitor's value, however, should not be too small, otherwise the driver's pulse response suffers.
This is important since otherwise the diaphragm needed to much bigger (several square metres are needed for 20 to 50 Hz).
The disadvantage: The bass reflex tube is a resonating object and therefore, shifts the phase and reproduces bass with less accuracy.
In case the driver's magnet is too strong it can be weakened by employing a resistor.
Or a low bass tuning has to be applied. In that case the driver has absolute control over its own diaphragm; but then, the amplifier module needs to restrict the bass at higher frequencies. The Alcone subwoofer Sub 10-60 work according to this system.
The method to boost low bass with a capacitor provides less control, since every energy storing component (like a capacitor) reduces pulse response. The advantage of the bass reflex port (using the rear radiation of the diaphragm) doesn't come into effect here. We, therefore, don't regard the solution to employ a capacitor for less than optimal (only disadvantages, no advantages!).
We prefer the little known tuning method by employing a resistor in series (see above or more detailed here) to the method of using a capacitor. The serial resistor solution is totally unconventional and is rejected by many designers due to reduced damping of the amplifier. However, if this unconventional method is employed correctly, there won't be any disadvantages, only advantages - refer to interview.
Further interesting solutions are the space consuming horn loaded bass, it needs at least a room corner. Or the transmission line speaker, trying to combine the advantages of the closed cabinet with advantages of the bass reflex cabinet. But, here other disadvantages become evident, like e.g. the TML gap.
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