FORMULAS.HTM --- Part of Manual for Driver Parameter Calculator --- by
Claus Futtrup.

Created 17. July 1996, last revised 3. November 2003.
Ported to XHTML 1.0 on 2. October 2004. Last modified 25. October 2004.

This document intends to describe the "data connections" mentioned in MANUAL.HTM by formulas and describe each parameter in Driver Parameter Calculator, among these the parameters given in the following list of data connections (from MANUAL.HTM):

* Qm, fres, Cms, Rms * Qe, fres, Mms, Re, Bxl * Rme, Bxl, Re * Rme, fres, Mms, Qe * Qt, Qe, Qm * Df, Qt * Dd, Sd * Mcost, Rme, Xmax, Hc or Hg * Mpow, Bxl, Re * Mpow, Rme * f4pi, Sd * f2pi, Dd * f2pi, f4pi * fpist, Dh * fmax, Dd * Cms, Sd, Vas * Mms, Cms, fres * Mair, Dd * Mvac, Mair, Mms * Zres, Re, Bxl, fres, Qm, Cms * EBP, fres, Qe * Bxl, Mms, Gamma (the acceleration factor) * no, fres, Vas, Qe * no, Mms, Bxl, Re, Sd * Pn, SPL (sound pressure level) * no, SPL * SPLmx, Pe, SPL * USPL, Sd, Bxl, Re, Mms * SPL, USPL, Re * Hc, Hg, Xmax * Gloss, Xmax, fres * Vd, Xmax, Sd * Dvol, Dd, Depth, MagDpt, Magnet, VCd * Dvol, Outer (simplified method) * Res, Zres, Re * Res, Rms, Bxl * Ces, Mms, Bxl * Ces, Qm, fres, Res * Les, Cms, Bxl * Les, fres, Ces * Les, Res, Qm, fres * Res, Re, Qm, Qe * Le, fLe, KLe (descriptive data, no datachecking) * Znom, Re (simple connection without datachecking) * NomDia, Outer (simple connection without datachecking) * NomDia, Dd (further simplified method)

In alphabetical order we have the following:

B : Magnetic Induction/Magnetic Flux Density in [T], an average of the flux density available across the voice coil. Basket : Basket diameter (the hole to cut in the baffle) in [mm] BoltD : Bolt diameter (the diameter where the bolts for fastening the driver is located) in [mm] Brand : The Brand name of the driver Bxl : Magnetic Induction crossed with wire length in the airgap in [N/A] (crossing = cross product, a mathematical vector-operation) Ces : The electrical equivalent of Mms, the moving mass, in [uF] Cms : Compliance of driver (inverse of spring stiffness) in [mm/N] Date : A description of the date you entered the information, or if you're quoting a manufacturers datasheet, you may specify the date it was made (see datasheet for futher information). Dd : Diameter of Diaphragm in [cm]. I recommend that Dd is measured with at least one decimal. Df : The total damping factor, corresponding to Qt. Df = 1 means critically damped, higher numbers equals more damping. Depth : Depth of the driver in [mm] Dh : Diaphragm height in [mm], the height/depth of the diaphragm Dvol : Driver volume in [dm3] (liters). Approx. the box volume occupied by the driver, when mounted with magnet pointing into the box. EBP : Efficiency-Bandwith-Product in [Hz] f4pi : Max frequency in [Hz] where a circular diaphragm radiates as a point source piston into 4 pi space (4 pi space is all the way around - a sphere) in [Hz]. Above this figure the driver starts to show circumferential breakup modes (first mode). f2pi : Max frequency in [Hz] where a circular diaphragm radiates with minor directivity so that there is no energy loss into 2 pi space (a half sphere). The f2pi value is normally a good indicator of a crossover frequency in multi-way speaker systems of reasonably high quality. At this frequency a driver has passed several modes of vibration. fcb : Resonance frequency in a closed test-box in [Hz] fh3 : Frequency higher than fres in [Hz], where Z has dropped 3 dB from Zres-Re fl3 : Frequency lower than fres in [Hz], where Z has droped 3 dB from Zres-Re fh9 : Frequency higher than fres in [Hz], where Z has dropped 9 dB from Zres-Re fl9 : Frequency lower than fres in [Hz], where Z has dropped 9 dB from Zres-Re fLe : The frequency at which Le and KLe is to be determined in [Hz] fmax : Max frequency in [Hz] before breakups occur diagonally across the diaphragm---this is where the trouble starts and you can get double breakup modes. Most drivers show a somewhat nonlinear response above this frequency. fmin : The frequency between fres and ZLe where the impedance is at its lowest in [Hz] fpist : Max frequency in [Hz] where the diaphragm can be considered to be a flat circular piston (but not necessarily a point source). fpist is calculated if you specify Dh, the diaphragm height and it is defined as the frequency where 10% of its wavelength equals the height of the diaphragm. From a conical shaped diaphragm phase trouble will start here. As the listener gets farther away from the driver, the phase problems will be less apparent, because things will blend. fres : Free air resonance frequency of driver in [Hz] Gamma : the acceleration factor (acceleration per ampere) in [m/(s2*A)] Hc : Height of coil in [mm] Hg : Height of airgap in [mm] KLe : Voice coil semi-inductance in [H*sqrt(Hz)], after Vanderkooy Le : Voice coil inductance in [mH] Les : The electrical equivalent of Cms in [mH], the suspension compliance Mms : Mechanical Mass of the vibrating part of the driver in [g] including air load. Mair : Mass of air attached to the diaphragm in [g], when the driver is playing in free air, ie. not in a box or on a baffle. Mvac : Mechanical Mass of the vibrating part of the driver in [g] excluding air load, as if moving in vacuum. Most places this variable is named Mmd, but here I have made an exception regarding the naming convention for parameters in DPC. Magnet : Magnet diameter in [mm] MagDpt : Magnet depth/height/thinkness (cylinder height) in [mm] Mcost : Motor cost-factor in [N*s/m] (or [kg/s]). Mcost expresses how powerful the motor system is (based on Rme, Xmax and either Hc or Hg depending on whether the voice coil is overhung or underhung), and the Xmax value includes an indicator of how much efficiency is "lost" in the design. This factor is therefore a description of how expensive the motor system is. This is an indicator on the price of the driver, but please forget about the unit. Other factors comes in, like diaphragm material, manufacturing tolerances etc. This version of Mcost (instead of using Rme) is based on an extension suggested by T. L. Clarke, where the cost of getting a high Bxl at low impedance must be even higher when the driver is significantly overhung or underhung. Mpow : Motor power-factor in Newton per square-root Watt [N/ûW]. Similar to Mcost. I have seen Rme as a measure of motor power, but this is simply the square-root of Rme, and it provides a simple measure in Newton, which I prefer, and which seems to relate the actual (subjectively perceived) power in a linear way. The square-root Watt unit can be difficult to understand, but should be interpreted as square-root of Volt * Ampere. In this respect it becomes clear that Mpow is independent of the drivers impedance level, and therefore does not prioritize high or low impedance drivers. Mpow is purely a motor system power-factor. Model nr: Complements Brand, tells you which driver from Brand we are looking at no : Efficiency (n should be the greek letter "eta") in percent [%] NomDia : Nominal Diameter of driver in [in] Outer : Outer diameter (the space to make room for on the baffle for the driver) in [mm] Pe : Thermal limited max. continuous electric power handling in [W]. If a driver is driven continuously above Pe, then it will eventually fail. Pn : Efficiency in [W], like no or SPL, but given as required power to reach 96 dB SPL, without considering power compression Producer: The full name of the producer of the driver Brand, which is the manufacturer, ie. company behind Provider: A description of you (who provided the information) Qe : Electrical Q (=damping), lower value means higher damping. It describes a drivers ability to resonate at fres based on electrical means. Qeb : New Qe in testbox Qm : Mechanical Q (=damping, lower value means higher damping). It describes a drivers ability to resonate at fres based on mechanical means. Qt : Total damping (parallel coupling of Qm and Qe) Re : DC-resistance of voice coil in [ohm] Res : The electrical equivalent of Rms in [ohm] Rme : Electromagnetic Damping Factor in [N*s/m] (the unit for viscosity), gives you the mechanical control/damping of the diaphragm arising from the electro-magnetic motor system. Rme is related to Qe in a way similar to how Rms is related to Qm. Rme is often used as a measure for power of the magnetic motor system, see Mpow and Mcost. Rms : Mechanical damping in [kg/s] (the unit for friction), (or [N*s/m] as given in MKSA.CU) gives you the mechanical damping of the diaphragm arising from mechanical friction, including the resistive part of the radiation load. Rms can be compared to Rme, and Rms is similarly related to Qm. Larger Qm gives smaller Rms. For woofers this is normally desired because the suspension then operates closer to a perfect spring. SPL : Efficiency in deciBell (SPL = sound pressure level) in [dB] per Watt into 8 ohm load, directly related to no, but definately not an "accurate" figure in applications. In other words, if a speaker driver is specified by the manufacturer to some other value, do not use that value for DPC unless you need it to calculate some Thiele-Small parameters and approximate values are better than no values at all. DPC assumes distance 1 meter, radiation into halfspace (2*pi), and voltage drive into 8 ohm. Values for other situations may be calculated from the following formula: SPL = SPLref + 20 * log10(2.828/Volts) + 20 * log10(meter/1). SPL is the socalled power-sensitivity, not really related to application, normally voltage amplifiers are used, but can become relevant if you want to compare two similar drivers with different nominal impedance levels. The SPL value is valid in the midband, perhaps reasonably accurately determined if assumed to be in the frequency range from fmin to f4pi. SPLmx : Maximum thermal limited SPL in [dB] (at maximum Pe, assuming power compression = 3 dB) playing into 2pi space USPL : efficiency in deciBell (SPL = sound pressure level) in [dB/2.83V] dB per 2.83 Volt (similar to 1W into an 8 ohm load). The actual voltage can be changed in the configuration file. This SPL-measurement is similar to SPL (see above), but gives different values. This shows you the difficulties about matching drivers. With 8 ohm drivers 2.83 Volt gives you 1 Watt and the two figures (SPL and USPL) will be similar, but at lower impedance levels the USPL level will increase. USPL is the socalled voltage sensitivity and is closer to application with voltage amplifiers. To a limited extent you could match drivers for a loudspeaker system with this factor. Sd : Surface-area of Diaphragm in [cm2]. Thick : The thickness of the basket plate in [mm] Vas : Equivalent Volume of air to Cms in [l] by specified pressure etc. Vb : Volume of testbox (a closed box for testing) in [l] (liters) VCd : Voice coil diameter in [mm] Vd : Volume Displacement in [cm3], how much air the driver can move in its linear range Weight : The weight of the driver in [kg] Xmax : Max peak linear excursion in [mm], usually calculated as abs(Hc-Hg)/2, and sometimes multiplied by a factor (1.15 or 0.87, depending on how much distortion is accepted) in [mm]. Driver Parameter Calculator does not multiply with any of these factors. Some manufacturers erroneously gives you peak-to-peak Xmax, which will be twice the Xmax that Driver Parameter Calculator will need. Driver Parameter Calculator uses center-to-peak values. Some manufacturers erroneously gives you Xmax as the damage limit, see Xlim. Gloss : The relative (to Xmax) sagging in [%] that will occur when the driver is mounted vertically. Sagging will produce additional distortion in the speaker, unless the speaker is made for this mounting explicitly. This is why the parameter is called Gravity loss (Gloss). Xlim : Damage limit excursion in [mm], also a center-to-peak value. Zfcb : Impedance at resonance frequency in closed test-box in [ohm] ZLe : The impedance for calculating Le (related to fLe) in [ohm] Zmin : The impedance at fmin in [ohm] Znom : Nominal impedance of the driver in [ohm] Zres : Impedance at free air resonance frequency in [ohm]

These are "simple to understand" descriptive parameters:

Driver Type (woofer, midrange, tweeter, woof/mid, coaxial, fullrange etc.) Voice coil material (aluminium, cupper, silver etc.) VC former material (aluminium, kapton, paper, glass fiber etc.) Number of Coils (typically 1, eg. alternatively 2 = Dual Voice Coil) If 2 or more coils is specified, please indicate whether the data is given for the coils in serial or in parallel. If no indication is given, then it must be assumed that the coils are connected in parallel. Number of VC layers (typically 2 or 4, 1-layer coils also exist) Basket material (aluminium, cast magnesium, sheet steel, etc.)

(VC stands for Voice Coil)

What the units stand for:

cm = centimeter dB = deciBell g = gram Hz = Hertz (1 Hz = 1 cps = 1 s^-1 = 1/s, where s=seconds) l = liter mm = millimeter mm/N = millimeter per Newton m/(s2*A)= meter per square-second per Ampere (Acceleration per Ampere) mH = milliHenry ohm = ohm cm2 = square-centimeter cm3 = cubic-centimeter T = Tesla N/A = Newton per Ampere N/ûW = Newton per square-root Watt uF = microFarad V = Volt W = Watt

The equations:

* Qm, fres, Cms, Rms Qm := 1/(2*pi*fres*Cms*Rms); * Qe, fres, Mms, Re, Bxl Qe := 2*pi*fres*Mms*Re/power(Bxl,2); * Rme, Bxl, Re Rme := power(Bxl,2)/Re; * Rme, fres, Mms, Qe Rme := 2*pi*fres*Mms/Qe; * Qt, Qe, Qm Qt := 1/(1/Qm+1/Qe); * Df, Qt Df := 1/(2*Qt); * Dd, Sd Sd := pi/4*power(Dd,2); * Mcost, Rme, Xmax, Hc or Hg Mcost := Rme*(H+Xmax)/H; where H is the smallest of either Hc or Hg * Mpow, Bxl, Re Mpow := Bxl/root(2,Re); * Mpow, Rme Mpow := root(2,Rme); * f4pi, Sd f4pi := c*root(2,1/(4*pi*Sd)); * f2pi, Dd f2pi := 2*c/(pi*Dd); * f2pi, f4pi f2pi := 2*f4pi; * fpist, Dh fpist := c/(10*Dh); * fmax, Dd fmax := c/Dd; * Cms, Sd, Vas Cms := Vas/(power(Sd,2)*rho*power(c,2)); * Mms, Cms, fres Mms := 1/(power(2*pi*fres,2)*Cms); * Mair, Dd Mair := rho/3*power(Dd,3); or (when measuring), Mair := 8/3*rho*power(Sd/PI,1.5); * Mvac, Mair, Mms Mvac := Mms - Mair; * Zres, Re, Bxl, fres, Qm, Cms Zres := Re + power(Bxl,2)*2*pi*fres*Qm*Cms; * EBP, fres, Qe EBP := fres/Qe; * Bxl, Mms, Gamma (the acceleration factor) Gamma := Bxl/Mms; * no, fres, Vas, Qe no := power(2*pi,2)/power(c,3)*power(fres,3)/Qe*Vas; * no, Mms, Bxl, Re, Sd no := rho*power(Sd*Bxl,2)/(power(Mms,2)*2*pi*c*Re); * Pn, SPL SPL := 96 - 10*log10(Pn); * no, SPL SPL := SPL_ref + 10*log10(no); where SPL_ref = approx. 112.2 dB in 2*pi space is given as: SPL_ref := 10*log10(rho*c/(2*pi*power(pref,2))) where pref = 20 uPa reference pressure for 0 dB SPL. * SPLmx, Pe, SPL SPLmx := SPL + 10*log10(Pe) - 3; * USPL, Sd, Bxl, Re, Mms USPL := 20*log10(USPL_Volts*rho*Sd*Bxl/(2*pi*pref*Re*Mms)); where USPL_Volts = 2.83 Volt (equals 1 watt into 8 ohm load), but can be changed to eg. 1 Volt by modifying DPC.INI. * SPL, USPL, Re SPL := USPL + 10*log10(Re/power(USPL_Volts,2)); * Hc, Hg, Xmax Xmax := abs(Hc-Hg)/2; Here abs means absolute value, ie. the value must be positive * Gloss, Xmax, fres Gloss := (g/(2*pi*fres)^2)/Xmax; where g = 9.81 m/s2 but can be changed to eg. 9.82 g by modifying DPC.INI. * Vd, Xmax, Sd Vd := Xmax*Sd; * Dvol, Dd, Depth, MagDpt, Magnet, VCd Dvol := pi/4*((Depth-MagDpt)/3*(power(Dd,2)+power(VCd,2)+Dd*VCd)+MagDpt*power(Magnet,2)); * Dvol, Outer (simplified method) Dvol := 0.4*power(Outer,4); * Res, Zres, Re Res := Zres - Re; * Res, Rms, Bxl Res := power(Bxl,2)/Rms; * Ces, Mms, Bxl Ces := Mms/power(Bxl,2); * Ces, Qm, fres, Res Ces := Qm/(2*PI*fres*Res); * Les, Cms, Bxl Les := Cms*power(Bxl,2); * Les, fres, Ces Les := 1/(power(2*PI*fres,2)*Ces); * Les, Res, Qm, fres Les := Res/(2*PI*fres*Qm); * Res, Re, Qm, Qe Qe := Qm*Re/Res; * Le, fLe, KLe KLe := Le*root(2,2*pi*fLe);

All these formulas are checked in DPC's calculation routine in all possible combinations of known and unknown variables.

The following formulas are based on descriptive data, and cannot be assumed tightly connected and/or 100% correct:

* Znom, Re Znom := 2*ROUND(0.75*Re); (approximate) or Znom := ROUND(1.5*Re) for low values of Re * NomDia, Outer NomDia := Outer * NomDia, Dd NomDia := Dd*1.27

These calculations are exceptions. They are not checked in any way, and the user may "overrule" anything DPC comes up with without problems and/or warnings. Some of the equations only work one-way, for approximations, like calculation of Znom.