Sound Enclosure or Subwoofer Designing

Sound Enclosure or Subwoofer Designing Table of Contents: 1. Sealed System 2. Ported System 3. Passive Radiator System 4. Bandpass System 5. Transmission Line System 6. Appendices 7. References 1. Sealed System The simplest of all loudspeaker designs, the sealed enclosure system consists of a driver mounted on one side of a sealed box. The sealed enclosure system is characterized by excellent transient response, good low frequency power handling, smaller box size and lower sensitivity to misaligned parameters when compared to other alignments. However, sealed enclosure systems tend to suffer from higher cutoff points and lower sensitivity than the other low frequency systems. There are two types of sealed enclosure systems: the infinite baffle (IB) system and the air suspension (AS) system. The IB system normally uses a large enclosure where the compliance (or "springiness") of the air within the enclosure is greater than the compliance of the driver suspension. The AS system normally uses a small enclosure where the compliance of the air within the enclosure is less than the compliance of the driver's suspension by a factor of 3 or more. Box Calculations To determine the box size for a sealed enclosure system, you will need to know the following Theile/Small parameters for the driver: Vas = Equivalent air compliance (liters) Qts = total Q of the driver at Fs Fs = resonance frequency of the driver (Hz) Choose a final Qtc (total Q of system at resonant frequency) for your design. Recommended values for Qtc are from 0.6 to 1.0. Transient response degrades with higher Qtc values, but the power handling of the system increases. A Qtc of 0.7 will usually give pretty good results, but you can use a higher figure if the subwoofer has a low resonant frequency (<20 Hz) or if it's being designed for car use. Page 1 of 25

Then, Qr = Qtc/Qts Vr = Qr^2-1 Vb = Vas/Vr Fb = Qr*Fs F3 = Fb*((1/Qtc^2-2+((1/Qtc^2-2)^2+4)^0.5)/2)^0.5 If Qtc>(1/2)^.5 then dbpeak = 20*log(Qtc^2/(Qtc^2-0.25)^0.5) Else dbpeak = 0 Vb = net box volume (liters) Fb = box resonant frequency (Hz) F3 = -3dB frequency (Hz) dbpeak = maximum peak or dip in system response Frequency Response Calculations To calculate the frequency response of a sealed enclosure system, you will need to know the following: then, Fb = resonance frequency of the system Qtc = Final Q of the system at resonance Fr = (F/Fb)^2 dbmag = 10*LOG(Fr^2/((Fr-1)^2+Fr/Qtc^2)) F = frequency (Hz) dbmag = SPL 1W/1M at frequency F Power Response Calculations To calculate the power response of a sealed enclosure system, you will need to know the following: Vas = equivalent air compliance (liters) Qes = electrical Q of driver at resonance Fs = resonance frequency of driver (Hz) PEmax = maximum input power for driver (W) Fb = resonance frequency of the system (Hz) Qtc = final Q of the system at resonance Dia = effective diameter (cone + 1/3 surround) (cm) Xmax = peak linear displacement of cone (mm) Page 2 of 25

then, Sd = pi*(dia/100)^2/4 Vd = Sd*Xmax/1000 n0 = 9.64*10^(-10)*Fs^3*Vas/Qes SPL = 112 + 10*LOG(n0) K1 = (4*pi^3*Ro/c)*Fb^4*Vd^2 K2 = 112+10*LOG(K1) Amax = Qtc^2/(Qtc^2-0.25)^0.5 for Qtc >(1/2)^0.5, 1 otherwise Par = K1/Amax^2 Per = Par/n0 PeakSPL = SPL+10*LOG(PEmax) pi = 3.14159265359 c = speed of sound in air (345 m/s) Ro = density of air (1.18 kg/m^3) n0 = free-air efficiency SPL = driver output @1W/1M Par = maximum linear power output Per = electrical input required to produce Par PeakSPL = Thermally-limited SPL in passband At frequency F, Fr = (F/Fb)^2 dbmag = 10*LOG(Fr^2/((Fr-1)^2+Fr/Qtc^2)) SPLd = K2+40*log(F/Fb) Pmax = K1*((Fr-1)^2+Fr/Qtc^2)/n0 SPLt = dbmag+peakspl where SPLd = displacement-limited SPL at F (db) Pmax = power required to produce SPLd at F (W) SPLt = thermally-limited SPL at F (db) Design Notes Stuffing You can decrease the volume required for a particular sealed alignment by using a stuffing material such as fiberglass, wool, or polyfill in the cabinet. Reductions of up to 30% in volume requirements are possible. Make sure that you add the volume displaced by the driver and bracing to arrive at a final enclosure volume. If you plan to stuff the enclosure, use 0.75*Vb as the net volume for the enclosure. When you add stuffing to the enclosure, the resonance frequency should decrease. Continue adding stuffing until the resonance frequency stops decreasing. Page 3 of 25

2. Ported System A ported enclosure system consists of a driver mounted on one side of a box that has an open tunnel or port which allows the passage of air in and out of the box. At low frequencies, the vent contributes substantially to the output of the system. The ported enclosure system is characterized by lower distortion and higher power handling in the system's operating range, and lower cutoff frequency than a sealed enclosure system using the same driver. Distortion rapidly increases below the cutoff frequency however as the driver become unloaded and the transient response of a ported enclosure system is usually inferior to that of a sealed enclosure system using the same driver. However, the lower cutoff frequency and better power handling within the system's passband often makes ported systems the alignment of choice for many speaker builders. Ported enclosure systems are much more sensitive to misaligned parameters than sealed enclosure systems, which makes their construction more difficult for the beginners. Almost any driver can be used in a ported enclosure system; however, only drivers which have a Qts value between 0.2 to 0.5 will generally give satisfactory results. If the driver has a Qts above 0.4, try using it in a sealed enclosure or single reflex bandpass system instead. Box Calculations To determine the optimum box size and tuning for a ported enclosure system, you will need to know the following Thiele / Small parameters for the driver: then, Vas = Equivalent air compliance (liters) Qts = total Q of the driver at its resonant frequency Fs = resonance frequency of the driver (Hz) Dv = internal diameter of port (cm) Vb = 20*Qts^3.3*Vas Fb = (Vas/Vb)^0.31*Fs F3 = (Vas/Vb)^0.44*Fs dbpeak = 20*LOG(Qts*(Vas/Vb)^0.3/0.4) Vb = net box volume (liters) Fb = box resonant frequency (Hz) F3 = 3dB frequency (Hz) dbpeak = maximum peak or dip in system response Page 4 of 25

Frequency Response Calculations To calculate the frequency response of a ported enclosure system, you will need to know the following: then, Vb = net box volume (liters) Fs = driver resonance frequency (Hz) Qts = driver Q at system resonance Fb = box tuning frequency (Hz) Ql = box losses (Ql=7 can be assumed for most cases) Fn2 = (F/Fs)^2 Fn4 = Fn2^2 A = (Fb/Fs)^2 B = A/Qts+Fb/(Fs*Ql) C = 1+A+(Vas/Vb)+Fb/(Fs*Qts*Ql) D = 1/Qts+Fb/(Fs*Ql) dbmag = 10*LOG(Fn4^2/((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2)) Power Response Calculations To calculate the power response of a ported enclosure system, you will need to know the following: then, Vas = equivalent compliance of driver (liters) Qes = electrical Q of driver at resonance Fs = resonance frequency of driver (Hz) PEmax = maximum input power for driver Fb = resonance frequency of the system F3 = frequency at which response is down by -3dB Dia = effective driver diameter driver (cm) Xmax = peak linear displacement of driver cone (mm) Sd = pi*(dia/100)^2/4 Vd = Sd*Xmax/1000 n0 = 9.64*10^(-10)*Fs^3*Vas/Qes SPL = 112 + 10*LOG(n0) K1 = (4*pi^3*Ro/c)*Fs^4*Vd^2 K2 = 112+10*LOG(K1) Par = 3*F3^4*Vd^2 Per = Par/n0 PeakSPL = SPL+10*LOG(PEmax) Page 5 of 25

pi = 3.14159265359 c = speed of sound in air (345 m/s) Ro = density of air (1.18 kg/m^3) n0 = free-air efficiency of driver SPL = SPL of driver @1W/1M Par = maximum linear power output Per = electrical input required to produce Par PeakSPL = Thermally-limited SPL in passband Maximum output at a given frequency F can be calculated as follows: Fn2 = (F/Fs)^2 Fn4 = Fn2^2 A = (Fb/Fs)^2 B = A/Qts+Fb/(Ql*Fs) C = 1+A+(Vas/Vb)+Fb/(Fs*Qts*Ql) D = 1/Qts+Fb/(Fs*Ql) E = (97/49)*A dbmag = 10*LOG(Fn4^2/((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2)) Pmax = (K1/n0)*((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2) /(Fn4-E*Fn2+A^2) SPLmax = K2+10*LOG(Fn4^2/(Fn4-E*Fn2+A^2)) SPLtherm = PeakSPL+dBmag SPLmax = displacement-limited SPL at frequency F (db/1m) SPLtherm = thermally-limited SPL at frequency F (db/1m) Port Calculations Port Length The port length required to tune a volume of air to a specific frequency can be calculated by using the following equation: Lv = (23562.5*Dv^2*Np/(Fb^2*Vb))-(k*Dv) Dv = port diameter (cm) Fb = tuning frequency (Hz) Vb = net volume (liters) Lv = length of each port (cm) Np = number of ports k = end correction (normally 0.732) Page 6 of 25

The value for k, the end correction, can be fine-tuned by using the following values to derive the appropriate end correction figure for each end of the port, then adding them together Flanged End: 0.425 Free End: 0.307 e.g. if both ends were flanged, k = 0.425 + 0.425 = 0.850 if one flanged, one free, k = 0.425 + 0.307 = 0.732 if both ends were free, k = 0.307 + 0.307 = 0.614 Normally, k=0.732 is assumed In practice, it's best to use ports that are slightly longer than predicted by the above equations, and then adjust their length until the correct tuning is achieved. It is much easier to shorten a port than to lengthen it! Minimum Port Diameter To calculate the minimum diameter of the port required to prevent port noises, you will also need to know the following: Xmax = maximum linear displacement (mm) Dia = Effective diameter of driver (cm) Np = number of ports Calculate the minimum port diameter from the following equations: Sd = pi*(dia/100)^2/4 Vd = Sd*Xmax/1000 Dmin = 100*(20.3*(Vd^2/Fb)^0.25)/Np^.5 Dmin = minimum port diameter (cm) Note: You CAN use ports that have a smaller diameter than that given by the equation above, especially if the ports are flared at both ends. However, at higher volumes, you may notice some port noise caused by the air rushing through the ports. Page 7 of 25

Slot Ports If you wish to use a slot port, first determine the diameter of a round port that has the same crosssectional area as the slot. The following equation can be used to do this: Dv' = 2*((W*H)/pi)^0.5 Dv' = diameter of equivalent round port W = width of slot H = height of slot Once you've calculated Dv', you can use it in the equation for Lv above to determine the required length of the slot port to tune the enclosure to the required frequency (Fb). Design Notes Choosing an alignment The alignment (combination box size and tuning) given on the design equations is one possible alignment for your driver. This alignment will produce the flattest response within the passband for a ported system using the driver. However, this may not be the best alignment for you. Examine carefully the cutoff frequency and box size to determine whether or not this alignment lives up to your expectations. Driver with low Qts If the driver has a very low Qts value, try using a larger box size and/or lower tuning. Use the frequency and power response equations (or the supplied spreadsheet) to examine the results. It may be possible to build a design that has a lower cutoff point, the tradeoff being a slight ripple in the passband. A ripple of less than 1 db in the passband will probably be unnoticeable. High power applications If you're designing your system for high-power applications, use a slightly lower tuning frequency. This should produce better results at higher volumes. You may also want to use an infrasonic ("rumble") filter to prevent the driver's cone from flapping around at very low frequencies. Car Audio applications If you're designing a system for car-audio use, remember that the interior of the car is going to boost the bass by about 12dB/octave below 60~80 Hz. A ported box that sounds flat in open air may sound boomy and flabby in a vehicle. Try using a lower tuning frequency than that predicted by the equations - this will lower the cutoff point and reduce any boominess that might occur. Check that driver! Make sure that the driver is suited for the enclosure that you have in mind! For example, just because software or equations predict that the driver may perform great in ported enclosure doesn't mean that it WILL perform well. Read the literature included with the driver (or talk to the manufacturer) to determine what's the best enclosure for the driver. When the design equations predict very big enclosures with very low tuning for your driver, that's usually a sign that the driver really wants to live in a sealed system. Page 8 of 25

Where should I put the ports? Ports should be placed at least one diameter away from any adjacent walls. If this is not possible to do this, the tuning frequency for a given port length will be lower than that predicted by the equations, and this may adversely affect the results. Lining the enclosure One layer of lining on every wall for each ported alignment will generally give better results. Ensure that no lining obstructs the ports. Fiberglass will work here, but make sure that none's located near the port entrance, as air turbulence can rip thinks of it off the walls and eject it through the port. Port size Use the largest port possible for the ported designs. This will reduce power compression effects and port noise caused by turbulence. Dual-Chamber ported systems Dual-Chamber ported systems are systems that have the following physical characteristics: The system consists of two separate volumes, connected together by a port. Each volume is vented to the outside by a port. All ports are typically of the same size and length. One volume, typically the larger one, contains the driver. The primary advantage of the dual-chamber system over a simple ported system using the same driver is a further reduction in driver excursion, caused by the addition of a second port resonance within the passband of the system. This reduction in excursion results in increased power handling within the system's passband, making this type of system particularly suitable for use for drivers with low Xmax. However, because of the additional chamber and venting requirements, this system is considerably more difficult to design and build when compared to a simple vented system. Design Notes: To design a dual-chamber system, first start by selecting a standard ported alignment for the driver in question. Then, select two ports of equivalent length and diameter to tune the system to the required frequency. Split the required volume for the alignment into two, placing one port in the smaller chamber (1/3 total volume), and the other port and driver in the larger chamber (2/3 total volume). Finally, join the two chambers together with a port that's the same size and length as the ports used to vent the chambers to the outside. Page 9 of 25

3. Passive Radiator System Passive radiator systems are very similar in operation to ported systems. However, instead of a port, the passive radiator system uses a passive radiator (also known as a "drone cone") to extend the system's low frequency response. The response of a passive radiator system is similar to that of a ported system using the same driver. However, the cutoff (-3dB) frequency is slightly higher, and the cutoff slope is deeper, mostly due to the presence of a "notch" in the frequency response corresponding to the passive radiator's resonance frequency. However, this notch is normally located far outside of the passband of the system, and therefore usually of little audible significance. The larger the passive radiator, the lower the passive radiator's resonance frequency (for the same target Fb), and the further the notch is out of the passband. To design a passive radiator alignment, start with a simple ported alignment using that driver that provides the desired box size and frequency response. Then, use the diameter of your chosen passive radiator as the "port diameter", and use this to calculate the required port length. Work out the volume occupied by this port and then use this to calculate the mass of air occupied by this port. The result is the required mass of the passive radiator. If it is too small, use a larger passive radiator and repeat the calculations. Example: Driver: Vas: 2 cu.ft. Qts: 0.30 Fs: 30 Hz Diameter: 8 in. Ported Alignment (QB3): Vb = 0.70 cu.ft. Fb = 39.4 Hz Now, we need to select an appropriately-sized passive radiator. ALWAYS use a passive radiator that is larger in diameter than the active driver, as the displacement of the passive radiator usually has to be 1.5 to 2 times that of the driver. If it's not possible to use one large passive radiator, then you can use two or more smaller ones, and tune them by working out the effective diameter from the combined surface area of the radiators. Page 10 of 25

Note that the effective diameter of the radiator is approximately equivalent to the diameter of the passive radiator's face plus 1/3 of the surround. If unsure, use the quoted Sd for that radiator, then use the following equation to determine the effective radius: R = (Sd/PI)^0.5 In this case, we choose to use a passive radiator that has an effective radius of 5 inches (roughly corresponding to a "12-inch" passive radiator). "Port" Radius = 5 in. Required Port Length = 186.1 in. "Port" Volume = (PI*R^2)*h = (3.14 *5^2)*186.1 = 14609 cu.in. = 8.45 cu.ft. = 0.2393 m^3 Mass = "Port" Volume * Density of Air = 0.2393 * 1.21 = 0.289553 kg = 290 g The passive radiator therefore has to have a weight of 290g. To achieve this, start with a passive radiator with lower mass, then add weight to make up the difference. To measure the resonance frequency of the passive radiator, install it in a free-air baffle (e.g. the box it's going in, without the driver in place), then hold a driver, driven by a sine wave generator, as close as possible to the passive radiator, then vary the frequency. At the passive radiator's resonance frequency, you should see the greatest peak to peak excursion of the passive radiator. Like their ported cousins, passive radiator systems are much more sensitive to misaligned parameters than sealed enclosure systems, which make their construction more difficult for the beginners. Almost any driver can be used in a passive enclosure system; however, only drivers which have a Qts value between 0.2 to 0.5 will generally give satisfactory results. If the driver has a Qts above 0.4, try using it in a sealed enclosure or single reflex bandpass system instead. Page 11 of 25

4. Bandpass Systems a. 4th Order Bandpass Systems The 4th order or sealed rear chamber bandpass system is basically a sealed enclosure system with the addition of an acoustic filter in front of the driver. The resulting system usually provides a lower cutoff frequency, the tradeoff being a larger enclosure. The enclosure can be reduced in size by using two drivers in an isobaric configuration. 4th order bandpass systems usually demonstrate better power handling characteristics than the other main systems considered here. Its transient response is second only to the sealed enclosure systems, making it a good choice for subwoofer applications. As all of the output of the 4th order bandpass system is via the port, the largest port diameter possible for the enclosure should be used in order to minimize port noises. The ports should be flared whenever possible, for the same reasons. The 4th order bandpass system rarely exhibits a perfect bandpass response - there is usually some out-of-band noise present in its output. A simple notch filter can be used to reduce this noise if it is audible. Alternatively, a low-pass filter can be used in series with the driver, but the in-band response of the system may also be affected if this approach is taken. Box Calculations To use the following calculations, you will need to know the following: Vas = equivalent air compliance for the driver (liters) Fs = driver resonance frequency Qts = driver Q at Fs The following equations will allow you to design a 4th order bandpass system with a desired low frequency limit or a desired gain. You will need to choose a value for "S" that suits your requirements. 4th order bandpass systems where S is less than 0.7 will have a degraded transient response, but wider bandwidth and smaller box requirements. if S = 0.7, then b = 0.7206, passband ripple = 0.00 db if S = 0.6, then b = 0.9560, passband ripple = 0.35 db if S = 0.5, then b = 1.2712, passband ripple = 1.25 db 4th order bandpass system with desired low frequency limit Choose a value for Fl, the lower 3dB cutoff frequency, then, Fl' = (Fl*Qts)/Fs Fh = (Fl'+b)*Fs/Qts Page 12 of 25

Qbp = (Fl'*(Fl'+b))^0.5 Fb = Qbp*Fs/Qts Vf = (2*S*Qts)^2*Vas Vr = Vas/((Qbp/Qts)^2-1) Pa = -40*LOG(1/(Qbp*2*S)) Fh = upper -3dB cutoff frequency (Hz) Qbp = Qtc of sealed chamber Fb = resonance frequency of vented chamber(hz) Vf = net volume of vented chamber (liters) Vr = net volume of sealed chamber (liters) Pa = gain (db) 4th order bandpass system with desired gain Choose a value for Pa, the gain in efficiency, then, Qbp = ((10^(-Pa/40))*2*S)^-1 Fl = ((-b+(b^2+4*qbp^2)^0.5)/2)*(fs/qts) Fh = Fl+(b*Fs/Qts) Fb = Qbp*Fs/Qts Vf = (2*S*Qts)^2*Vas Vr = Vas/((Qbp/Qts)^2-1) Fl = lower -3dB cutoff frequency (Hz) Fh = upper -3dB cutoff frequency (Hz) Qbp = Qtc of sealed chamber Fb = tuning frequency of vented chamber (Hz) Vf = net volume of vented chamber (liters) Vr = net volume of sealed chamber (liters) Pa = gain (db) Page 13 of 25

Frequency Response Calculations To calculate the frequency response of a 4th order bandpass system, you will need to know the following: Vf = net front volume (liters) Ff = front volume tuning frequency Vr = net rear volume (liters) Fs = driver resonance frequency (Hz) Qts = driver Q at system resonance Ql = box losses (Ql=infinite (10000) can be assumed for most cases) then at frequency F, A = (1/Ff)^2*F^4 B = ((1/Ql+(Fs/Ff)/Qts)/Ff)*F^3 C = (((1+Vas/Vr+Vas/Vf)*Fs/Ff+(1/Qts)/Ql)*Fs/Ff+1)*F^2 D = ((1/Qts+(Fs/Ff)/Ql*(Vas/Vr+1))*Fs)*F E = (Vas/Vr+1)*Fs^2 G = A-C+E H = -B+D dbmag = 20*log(F^2/(G^2+H^2)^.5) Power Response Calculations To calculate the power response of a 4th order bandpass system, you will need to know the following: then, Vas = equivalent compliance of driver (liters) Qes = electrical Q of driver at resonance Fs = resonance frequency of driver (Hz) PEmax = maximum input power for driver Fb = resonance frequency of the system F3 = frequency at which response is down by -3dB Dia = effective driver diameter driver (cm) Xmax = peak linear displacement of driver cone (mm) Vr = net volume of sealed section (liters) Vf = net volume of vented section (liters) Qts' = Qts*((Vas+Vr)/Vr)^0.5 Fs' = Fs*Qts'/Qts Qes' = Qes*Qts'/Qts Vas' = Vas*Vr/(Vas+Vr) Vb' = Vf Sd = pi*(dia/100)^2/4 Vd = Sd*Xmax/1000 Page 14 of 25

n0 = 9.64*10^(-10)*Fs'^3*Vas'/Qes' SPL = 112 + 10*LOG(n0) Par = 3*F3^4*Vd^2 Per = Par/n0 PeakSPL = SPL+10*LOG(PEmax) K1 = (4*pi^3*Ro/c)*Fs'^4*Vd^2 pi = 3.14159265359 c = speed of sound in air (345 m/s) Ro = density of air (1.18 kg/m^3) n0 = free-air efficiency of driver SPL = SPL of driver @1W/1M Par = maximum linear power output Per = electrical input required to produce Par PeakSPL = Thermally-limited SPL in passband Maximum output at a given frequency F can be calculated as follows: Fn2 = (F/Fs')^2 Fn4 = Fn2^2 A = (Fb/Fs')^2 B = A/Qts'+Fb/(Ql*Fs') C = 1+A+(Vas'/Vb')+Fb/(Fs'*Qts'*Ql) D = 1/Qts'+Fb/(Fs'*Ql) E = (97/49)*A dbmag = 10*LOG((A*Fn2)^2/((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2))) Pmax = (K1/n0)*((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2)/(Fn4- E*Fn2+A^2) SPLmax = SPL+dBmag+10*LOG(Pmax) SPLtherm = PeakSPL+dBmag SPLmax = displacement-limited SPL at F (db/1m) SPLtherm = thermally-limited SPL at F (db/1m) Note: The port calculation is similar to the ported system. Page 15 of 25