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


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/W].

          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


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/W    = 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 + 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.