When it comes to speaker design, the lowest frequency driver determines the volume and therefore overall size of the enclosure. The required volume is determined by the mechanical and electrical characteristics of the driver and whether or not the enclosure is ported or sealed. A tool I like to assist me for designing enclosures is called WinISD. It is a speaker design software which allows you to input custom drivers and it will tell you things like whether or not the enclosure should be vented, the shape of the vent, and the overall volume along with ideal dimensions. It also gives frequency response, phase, SPL and other plots so you can see how changing various characteristics of the enclosure affects the performance.
EBP and Q Factor
The first thing WinISD calculates is the Efficiency Bandwidth Product or EBP of the driver. This number determines whether or not the driver(s) should be installed in a sealed or vented box. What the number actually describes is the range of frequencies for which the driver oscillates at, or the oscillation bandwidth. Several important things happen around oscillation which define the enclosure design process.
The formula to calculate EBP is:
\[EBP = \frac{f_s}{Q_{es}}\]
fs (Hz): This is the resonant frequency of a driver which is the frequency at which it most freely oscillates at. Drivers oscillate due to the relationships between their mechanical and electrical properties. It is the point where the cone velocity peaks, impedance peaks, the cone phase reverses relative to the input signal and is generally the lower cut-off frequency of the driver. As an example, below are phase and impedance plots of a Dayton Audio RS100-8 4″ midrange driver which has a resonant frequency of 92Hz:
Qes: A measure of the electrical “Q” or quality of the driver. A higher Qes indicates the driver is less electrically dampened, meaning it will oscillate at a greater amplitude over a narrower range of frequencies. The value of Qes is obtained via the formula:
\[Q_{es} = \frac{2\pi f_s M_{ms} R_e}{(Bl)^2}\]
Mms (kg): The mass of the cone, coil and other moving parts of the driver, including the force caused by the air acting against the cone.
Re (Ω): DC resistance of the voice coil.
B (T): Strength of the magnetic field present in the gap where the voice coil sits.
l (m): Length of wire that makes up the voice coil.
Looking at the formula, we can see how Qes depends highly on Bl. A larger magnet and more voice coil turns will result in a more dampened (more controlled) speaker and lower Qes.
Now back to the EBP calculation. From the EBP formula, we can see how a high Qes results in a narrow EBP since oscillation will be initially large in amplitude and reach its -3dB points quickly. A a low Qes results in a wide EBP since the peak amplitude is lower, and therefore takes longer to reach the -3dB points. The graph below shows the behavior of two oscillating systems with Q values of 1 and 4. ζ (zeta) represents the dampening, and it is clear how high Q results in lower dampening and lower EBP.
The general guideline for EBP values are:
- <50Hz – Sealed only
- 50Hz to 100Hz – Ported or sealed
- > 100Hz – Ported only
These are rule of thumb values which basically tell us that for a given resonant frequency fs, it makes more sense to put a driver with a low Q (more dampened) into a ported enclosure and a driver with a high Q (less dampened) into a sealed one.
Ported vs. Sealed
We are now faced with understanding why the above guidelines exist. To understand why, its important to understand the differences between a ported and sealed enclosure.
A sealed enclosure is of course just a box which contains some volume of air. Assuming the box is rigid, as the speaker cone moves into the box, it compresses the air inside the box. This causes a low pressure in the air outside the box, which results in audible sound since sound waves are simply waves of high and low pressure air. The air in the box also acts as a spring. As the cone pushes in, the air pushes back since its being compressed. Alternatively, as the cone moves outward, the pressure in the box drops which causes outside air to push inward on the cone. This all has the effect of dampening the response of the driver, which is helpful in terms of our EBP. The guidelines suggest that a driver with EBP less than 50Hz be in a sealed enclosure. A low EBP indicates a high Qes, which means the driver is less dampened and oscillates more intensely. A sealed enclosure will help dampen a driver of this nature and make it more controlled. One downside is it has the side effect of being less efficient and therefore requiring more power due to the fact that it’s working against the air pressure inside the box. This also results in lower SPL, or sound pressure levels at lower frequencies.
A ported enclosure has a port, which is generally a tube of some length and diameter that enters usually the front or rear of the enclosure. This in turn allows air to move into and out of the enclosure, but not freely. There is a volume of air inside the port itself, which has mass and inertia. The volume of air in the enclosure must push and pull against this mass of air and at a certain frequency the two will begin to resonate. This frequency is known as the Helmholtz resonance. The diameter and length of the port should be chosen so that the air resonates below the driver’s natural frequency of resonance. If we call the port resonant frequency fp, and the driver resonant frequency fs, then between these two points, a phase inversion occurs at the port. The sound waves from the port are in the phase with the driver output, causing them to reinforce and produce greater sound pressure levels. This has the effect of extending the low end frequency response of the driver before it drops off. One downside to the ported enclosure is that audible resonance can occur at low frequencies, causing a “smeared” bass note. This is also the reason why its best to choose drivers with a high EBP, as they have low Qes and are more dampened and controlled, making them better suited to dealing the oscillatory nature of a ported enclosure.
| Pros | Cons | |
|---|---|---|
| Ported | -Very high efficiency -High SPL -Low cutoff frequency | -Large enclosure volume (nearly twice sealed) -Very large cone excursion below cutoff frequency -Possible port noise -Worse transient response, possible 'smearing' of bass notes |
| Sealed | -Less enclosure volume -Better transient response | -Very low efficiency -Low SPL -Higher cutoff frequency |
SPL and -3dB Point
SPL, or sound pressure level, measured in decibels, describes the intensity of the sound the driver can produce at some power. Most all drivers are measured at a power of 1W, with the measuring instrument 1 meter away from the driver. For example, an Alpine SWR-10D2 10″ subwoofer has an SPL of 82dB at 1W and 1m.
The -3dB point, also known as the half power point, is the frequency at which the acounstic output power of the driver is half of what it normally is. In terms of SPL, its the point where the SPL is 3dB down from what it normally is. The results in about a 50% audible reduction in volume. Decibels are a logarithmic representation of sound intensity, which is why only -3dB equals 1/2 power loss. The actual formula to find the change in SPL based on a change in power level is:
\[\Delta SPL = 10 \log{\frac{P}{P_{ref}}}\]
Pref (W): The initial power input to the system.
P (W): The changed power level.
Below shows a plot from WinISD which shows a driver with an EBP of 57 in a ported and sealed configuration. The yellow plot is the sealed enclosure which has a volume of 19.5L and a -3dB point of 43Hz. The green plot is the ported enclosure which is tuned to 24Hz, has a volume of 38L and a -3dB point of 22Hz.
We can calculate just how inefficient a sealed enclosure is at lower frequencies. At the -3dB frequency of the ported enclosure, 22Hz, the sealed enclosure is 12dB down. Let’s say the amp is outputting 500W into the ported enclosure. If we arrange the previous formula, it can be written to solve for P:
\[P = P_{ref} 10^{\frac{\Delta SPL}{10}}\]
We want to reach -3dB, so our \(\Delta SPL = 9dB\) and \(P_{ref} = 500W\). Therefore the required input power is \(P = 3972W\). Nearly 8 times the power requires to reach the same dB level as the ported enclosure!
Additional Q Factors
Qms: This is the mechanical Q or quality of the driver. Higher Qms indicates less mechanical losses and therefore a more compliant surround and spider and higher diaphragm mass. Once again, a high Q indicates a less dampened system. The formula for Qms is:
\[Q_{ms} = \frac{2\pi f_s M_{ms}}{R_{ms}}\]
Qts: The combined electrical and mechanical Q of the driver. This value typically ends up between 0.2 and 0.5:
\[Q_{ts} = \frac{Q_{es} Q_{ms}}{Q_{es} + Q_{ms}}\]
Qtc: The total Q of the driver and enclosure together
As we can see, Qtc depends on Vas, but Vas is an unknown. Since Qtc is a Q value and Q values are well characterized, we can select an ideal Q factor to maximize performance of our enclosure. Below is a unit step-response plot which shows how a system behaves and recovers when a pulse of magnitude 1 is applied and held. This is called a unit step. It shows several values of ζ (zeta), the dampening factor, which is directly related to Q by:
\[Q = \frac{1}{2 \zeta}\]
When ζ is < 1, the system is said to be underdamped meaning it resonates before reaching a steady state. When ζ is = 1, the system is critically damped, meaning there is no overshoot and it reaches steady state as fast as possible. For ζ > 1, the system is over damped, and takes longer than it should to reach steady state. For the most part, a ζ of \(\frac{1}{\sqrt{2}}\) is ideal as this is the lowest value of ζ for which the system does not resonate, has minimal overshoot and reaches steady state quickly. This corresponds to a Q of the same value, \(\frac{1}{\sqrt{2}}\) = 0.707.
Sealed Enclosure Formulas
Enclosure volume:
\[V_b = \frac{V_{as} N_d}{\Big(\frac{Q_{tc}}{Q_{ts}}\Big)^2 – 1}\]
Nd: The number of drivers you are using.
Vas (L): This is the inverse of stiffness of the driver’s suspension, also known as compliance, while it is mounted in free air. The unit is in liters. What this value represents is a volume of air that when acted upon by a piston of the same area as the driver’s cone, the stiffness of said air is the same as the stiffness of the driver’s suspension.
When calculating enclosure volume for sealed or ported enclosures, the volume scales linearly per driver. So 2 identical drivers results in twice the volume. Alternatively, when using multiple drivers, one could partition the inside of the box as to have an isolated chamber for each driver. Another important key here is that when calculating internal volume, in order to be accurate you must take into account the volume of the portion of the driver’s that protrude into the enclosure itself. Subtract anything inside the enclosure that isn’t air or stuffing, including internal bracing, parts of the driver’s, crossover networks, etc.
System Resonant Frequency (Hz):
\[f_c= \frac{Q_{tc} f_s}{Q_{ts}}\]
Even a sealed enclosure will resonate at some frequency, but it doesn’t have much of an effect compared to ported. The resonance will be well controlled because of the dampened nature of the sealed enclosure. From the formula, we know that Qts is a fixed value, but Qtc can be modified as we please. Lowering it will cause the resonant frequency to drop, but lowering below 0.5 will cause the enclosure to be over dampened leading to poor transient response.
-3dB Point (Hz):
\[f_3 = \sqrt{\frac{\Big(\frac{1}{Q_{tc}^2} – 2\Big) + \sqrt{\Big(\frac{1}{Q_{tc}^2} – 2\Big)^2 + 4}} {2}} f_c\]
Ported Enclosure Formulas
Enclosure volume (L):
\[V_b = 20 {Q_{ts}}^{3.3} V_{as} N_d\]
The volume of a ported enclosure is almost twice that of a sealed one. This can be an important factor if space is a concern, like say the interior of a vehicle. Something to note, like with a sealed enclosure, subtract anything from the volume which isn’t air or stuffing material. This most importantly includes the volume of the ports. The ports are considered to be external, so the air inside of them do not contribute to the internal enclosure volume.
Tuning frequency of the enclosure (Hz):
\[f_b = \Big(\frac{V_{as} N_d}{V_b}\Big)^{0.31} f_s\]
This is the frequency at which the port will be tuned to resonate at and is necessary for finding port dimensions.
-3dB Point (Hz):
\[f_3= \Big(\frac{V_{as} N_d}{V_b}\Big)^{0.44} f_s\]
Port Calculations:
After spending some time running numbers and collecting data, the existing formulas for port diameters and air velocities are unsatisfactory so I will not be including them here. This is where it is necessary to use a program such as WinISD pro to find port dimensions. The key to figuring out port dimensions is the air velocity through the port. The speed sound in air is 343m/s. Something traveling at 343m/s is said to be traveling at Mach 1. The general rule of thumb is to keep the the port velocity below 0.2 mach , or 68.6m/s to prevent audible port noise, although some swear to never go over 0.05 mach. Now this is not always possible since while a larger port diameter reduces air velocity, it also increases their length, making some configurations impossible to fit into the enclosure. It’s always best to use flanged port ends as they allow for faster air velocity without the port noise.
Below is plot provided by WinISD Pro which shows the port velocity for a JBL P1020 subwoofer running at 350W tuned to 24Hz. The port dimensions are 3″ diameter x 18.5″ length, and the air peaks at 55m/s which is 0.16 mach, so it falls within our desirted guidelines.
Dampening and Fill
It is best to stuff a sealed enclosure with some kind of filler, such as polyester or wool, as this has the effect of increasing the apparent volume by up to 15% and lowering the Q through dampening.
Ported enclosures benefit from some kind of material lining the interior panels to prevent internal reflections escaping out the ports and causing distortion. Materials include fiberglass, felt, wool, dense foam, polyfill and anything else that is absorptive. An inch or two will suffice on about 50% of large exposed surfaces inside the enclosure. Too much and it can over dampen it. It is also not advisable to stuff a ported enclosure, as this has the effect of lowering the Q and dampening resonance, which is the opposite of what we want from a ported speaker.
The way any dampening material works is via friction; as a pressure wave encounters some kind of dampening material, the air molecules are pushed into it and lose kinetic energy due to friction. For ported enclosures, this means the wave will be highly attenuated if the dampening material is mounted to a reflective surface. For sealed enclosures, this results in the air being harder to compress and lowers the cabinets Q factor.
Variables
\(Bl = \small\text{Product of magnetic field strength in voice coil gap and length of wire in voice coil (T m)}\)
\(M_{ms} = \small\text{Mass of the driver’s diaphram and coil including acoustic load (kg)}\)
\(N_{d} = \small\text{Number of drivers in the enclosure}\)
\(f_b = \small\text{Tuning frequency of the enclosure (Hz)}\)
\(f_s = \small\text{Resonant frequency of the driver (Hz)}\)
\(V_{as} = \small\text{The compliance of the driver suspension (L)}\)
\(V_b = \small\text{Enclosure volume (L)}\)
\(Q_{es} = \small\text{Electrical Q of driver}\)
\(Q_{ms} = \small\text{Mechanical Q of driver}\)
\(Q_{tc} = \small\text{The total Q of the driver and enclosure together}\)
\(Q_{ts} = \small\text{The combined electrical and mechanical Q of the driver}\)
\(R_e = \small\text{DC resistance of voice coil } (\Omega)\)
\(R_{ms} = \small\text{Mechanical resistance of a driver’s suspension (N/sm)}\)
\(X_{max} = \small\text{Maximum excursion of the speaker cone in one direction (m)}\)