POWER.HTM --- Part of Manual for Driver Parameter Calculator --- by Claus Futtrup.
Created 6. September 2003, last revised 15. June 2004. Ported to XHTML 1.0 on 2. October 2004. Last modified 26. October 2004.

1. Preface
2. Introduction
3. Resistance and temperature in metals
• Enhanced temperature range
4. Power compression and max temperature
5. Power compression and input power
• Automotive applications
• The knee point
6. Input signal discussion
• Expected input signal
• Todays compression level
7. Basic thermal physics
• Heat transfer
• Heat capacity
• Heat flow
• Example : Very short time perspective
• Example : Very long time perspective
8. Conclusions
9. Summary
10. References

## Preface

This document started out as a simple and short description of a couple of power handling parameters of interest. It ended up being a comprehensive introduction to power handling and to some degree the basic physics behind temperature models.

The document collects a lot of information, and references across the sections, but it should also be possible to read only the sections of interest, and get the full understanding within that particular branch of the subject.

Therefore you might want to consider which sections you are interested in, if not all. Check out the table of contents to locate if there are sections of particular interest to you.

## Introduction

This document is about voice coil temperature, power handling and heat radiation and heat storage in drivers in both a steady state (static) sense as well as in a (dynamic) time perspective.

The parameters are currently not supported by DPC, partially because driver manufacturers do not specify any of the required values, and partially because they are not that easy to measure.

Power handling is a complex issue, which this document will explore, not least because conditions can vary significantly. For example, if it is a woofer, then convection cooling is more or less forced. With tweeters the only convection cooling is through natural convection.

The applied signal is often discussed, and will have a high influence on the results of a power handling test.

It happens that users return with broken speaker systems, and sometimes it is obvious that the speakers have been (over-) loaded with a critical signal, which has ruined one or more drivers. A critical signal, which can be difficult to repeat or reproduce and therefore the circumstances might become difficult to replicate in eg. a power test.

Four kinds of mechanical failure can be the reason for failure:

A. Long term electrical power load:

• Mechanical wear/fatigue (mechanical lifetime test)

B. Short term electrical power load:

• Mechanical rupture (mechanical stress test)

Thermal overload, whether short term or long term, is due to the same mechanisms (heating up the system due to the voice coil as a heat source), and could be considered only one kind of overload. The aspect that an external heat source could overload the system is not taken into account here (like eg. automotive usage).

When the thermal overload is (very) short term, it implies that the voice coil blows - like a fuse. The overheating is not present throughout the system. When the overload is long, other components could be damaged by the heat and cause failure of the speaker.

This document only deals with the kind of mechanical failure that is related to thermal overload, primarily long term thermal overload.

Standards, like IEC 268-5, tries to tie the power handling value into a real world amplifier size recommendation, but I do not think anybody has taken it as an exact value, and you should not. In the IEC 268-5, both short term (1 second noise followed by 59 seconds of pause) and long term (1 minute noise followed by 2 minutes pause) power handling is considered. I do not know which value the short term specification has, and in this document we will consider the long term specification only. There are other power handling issues than just the thermal heat, for example IEC 268-5 also contains a lifetime test specification.

EIA-426-B is an alternative standard that alters the noise spectrum. This standard contains specification for power compression, distortion and accelerated lifetime power handling specification. The test will more often than not wear down the speaker before it dies of thermal overload. This kind of powerhandling is not considered in this document. By the way, EIA-426-B is an annoying standard which does not describe what you are doing but a lot about how to do it (step by step procedures).

Power handling is a secondary issue. What you really want for that power is output, so the parameter of interest is SPLmx. Who cares whether the power handling is higher, if it is contradicted by a lower efficiency. It is a different, more complex issue if you compromise SPLmx for eg. low frequency extension ie. lower highpass cutoff frequency, which gives the impression of more bass.

Also notice that the frequency range to be covered by the driver can significantly alter the specified power handling. For example, a 150 W tweeter is not likely to actually handle more than about 15 W steady state power consumption, but it is enough to cover the tweeter range of the spectrum in a 150 W system. This is handled in the IEC 268-5 powertest by shaping the noise signal so that the power distribution is like normal music. And do not forget that normally 99% of the power consumed by a driver is dumped as heat in the voice coil.

## Resistance and temperature in metals

All metals used for voice coils has a resistivity, a material specific resistance, defined by rho, which increases as the temperature increases. In the temperature range of 0 - 100 degC (equals 32 - 212 degF) the increase can be approximated by:

```        rho(T1) = rho(T0) * (1 + alpha * deltaT) [ohm * m]      (1)

deltaT is the temperature change = T1 - T0 [K]
T1 is a desired temperature [degC]
T0 is the reference temperature [degC]
alpha (sometimes named TCR) [1/K] is the Temperature Coefficient of
Resistance
```

Kelvin [K] and Celcius [degC] can be used arbitrarily when expressing a temperature change because they share the same scale. During this document you will find the two mixed as desired by the author.

Many metals will approximately show this linear behaviour outside the specified temperature range, but if not, a more advanced model can be applied. See the subsection on "Enhanced temperature range" at the bottom of this section.

For all conductive materials the resistance is given as:

```        R = rho * l / A [ohm]                                   (2)

rho is (still) the resistivity [ohm * m]
l is the length of the conductive material [m]
A is the crosssectional area of the conductive material [m2]
```

In other words, when the material parameter, rho, changes with temperature, so will the resistance of the voice coil. During this temperature change the geometric "l / A" part is assumed to remain constant. This is not entirely true, but see later for an explanation.

For various materials we have at 0 degC (32 degF):

```        alpha at 0 degC [1/K]           More alpha values
-----------------------         -----------------------
Aluminium (Al)  0.00430         Nickel    (Ni)  0.00670
Brass (CuZn30)  0.00210         Platin    (Pt)  0.00390
Copper    (Cu)  0.00430         Tin       (Sn)  0.00374
Gold      (Au)  0.00405         Zinc      (Zn)  0.00370
Iron      (Fe)  0.00660         Wolfram   (W)   0.00480
Silver    (Ag)  0.00410
```

Notice that alpha (the greek letter) is the same for copper and aluminium, this is not a typing error.

The alpha value changes with temperature, as can be deduced from equation 1, in the following way:

```        deltaT = 1/alpha_T1 - 1/alpha_T0 [K]                    (3)

<=>

alpha_T1 = 1/(deltaT + 1/alpha_T0) [1/K]                (4)
```

With equation 4 the values found in the table above can be converted to 20 degC (68 degF):

```        alpha at 20 degC [1/K]          More alpha values
-----------------------         -----------------
Aluminium (Al)  0.00396         Nickel    (Ni)  0.00591
Brass (CuZn30)  0.00202         Platin    (Pt)  0.00362
Copper    (Cu)  0.00396         Tin       (Sn)  0.00348
Gold      (Au)  0.00375         Zinc      (Zn)  0.00345
Iron      (Fe)  0.00583         Wolfram   (W)   0.00438
Silver    (Ag)  0.00379
```

Most sources that I have found says that alpha = 0.00393 [1/K] for aluminium and copper. There are at least two possible reasons for the deviation between my calculated value and the references, 1) I have added one decimal to the table in my source (shame on me) and 2) my reference specifies pure materials, whereas the frequently quoted value for aluminium and copper may be for voice coil alloys/impurities. The temperature coefficient of resistance, alpha, depends heavily on the amount of impurities in the metal.

This is a proof that shows how equation 3 can be derived from equation 1. This proof is not essential to this document and can be skipped. First, lets assume that T2 > T1 > T0 and start with equation 1 in various forms:

```    rho(T2) = rho(T1) * (1 + alpha_T1*(T2 - T1))                (5)

rho(T1) = rho(T0) * (1 + alpha_T0*(T1 - T0))                (6)
```

Inserting rho(T1) into the equation of rho(T2) to find rho(T2) related to rho(T0) we get:

```    rho(T2) = rho(T0) * (1 + alpha_T0*(T1 - T0)) * (1 + alpha_T1*(T2 - T1))

= rho(T0) * (1 + alpha_T0*(T1 - T0) + alpha_T1*(T2 - T1) +
alpha_T0*(T1 - T0) * alpha_T1*(T2 - T1))        (7)
```

and at the same time we may write:

```    rho(T2) = rho(T0) * (1 + alpha_T0*(T2 - T0))                (8)
```

The two equations above must equal, which means that the second part with the alphas must be equal:

```    1 + alpha_T0*(T2 - T0) = 1 + alpha_T0*(T1 - T0) + alpha_T1*(T2 - T1)
+ alpha_T0*(T1 - T0) * alpha_T1*(T2 - T1)

<=>                                                     (9)

1 + alpha_T0*(T2 - T0) - 1 - alpha_T0*(T1 - T0) = alpha_T1*(T2 - T1)
+ alpha_T0*(T1 - T0) * alpha_T1*(T2 - T1)

<=>

alpha_T0*(T2 - T1) = alpha_T1*(T2 - T1) +
alpha_T0 * alpha_T1 * (T2 - T1) * (T1 - T0)
```

Dividing both sides with alpha_T0 * alpha_T1 * (T2 - T1) we get:

```    1/alpha_T1 = 1/alpha_T0 + (T1 - T0)

<=>

deltaT = T1 - T0 = 1/alpha_T1 - 1/alpha_T0                  (3)
```

This is equation 3. End of proof.

Regarding the geometric ratio l/A assumed to stay constant, a short derivation is given. Take the voice coil wire, and unwind it from the coil. Assuming it has a round profile, you now have a strip of wire with the length l_0 and radius r_0. The ratio is:

```                     l_0
l_0/A_0 = ----------                                    (10)
r_0^2 * pi
```

The thermal length expansion coefficient, alpha_l [1/K], expresses how much the strip will expand for each dimension of the voice coil wire, when heated up.

As a sidenote, nonmetals, composites and to some extent special metal/nonmetal alloys are anisotropic. In this case an alpha_l exist for each anisotropic direction. For normal pure metals (the metal molecular structure makes these materials isotropic) the volume expansion is given as 3 * alpha_l.

For various materials we have at 20 degC (68 degF):

```        alpha_l at 20 degC [1/K]        More alpha_l values
------------------------        ------------------------
Aluminium (Al)  23.1 E-6        Nickel    (Ni)  13.4 E-6
Brass (CuZn33)  18.0 E-6        Platin    (Pt)   8.8 E-6
Copper    (Cu)  16.5 E-6        Tin       (Sn)  22.0 E-6
Gold      (Au)  14.2 E-6        Zinc      (Zn)  30.2 E-6
Iron      (Fe)  11.8 E-6        Wolfram   (W)    4.5 E-6
Silver    (Ag)  18.9 E-6
```

For the general heat expanded situation we get:

```        l_alpha = l_0 * (1 + alpha_l * deltaT)                  (11)

r_alpha = r_0 * (1 + alpha_l * deltaT)                  (12)
```

The new ratio l/A becomes:

```                              l_0 * (1 + alpha_l * deltaT)
l_alpha/A_alpha = -------------------------------------
(r_0 * (1 + alpha_l * deltaT))^2 * pi

l_0                     (13)
= -----------------------------------
r_0^2 * pi * (1 + alpha_l * deltaT)

= l_0/A_0 * 1/(1 + alpha_l * deltaT)
```

Assuming l/A is constant is wrong, l/A does not stay constant. It has changed by the factor 1/(1 + alpha_l * deltaT). This contributes to the change of the DC resistance of the voice coil.

It becomes obvious, when looking at contributions from changes in rho, that the change in l/A can be neglected because it is so small that it is smaller than the inaccuracy with which alpha is determined, given that alpha_l is a factor 1E-6 as opposed to alpha in the range 1E-3.

For example, if we select a 10 meter aluminium wire, with a round shape, diameter 0.35 mm, we get for a temperature increase of 100 degC (212 degF):

```        l = 10000 mm -> 10023.1 mm
A = 0.0962 mm2 -> 0.0967 mm2
l/A = 103938 1/mm -> 103698 1/mm
```

As can be seen the l/A ratio changes with the ratio 0.9977, ie. 0.23 %. The selected length and cross sectional area does not matter. We could in fact let: R = R(T0) * (1 + alpha * deltaT) / (1 + alpha_l * deltaT)

This little influence may also explain some of the differences sometimes seen between different references in the specification of alpha.

On the other hand, who knows what the physicists did, when they tabulized rho. Did they really maintain rho as a material property, or did they allow the density of the material to change, and therefore included the heat expansion - more or less unconsciously. If you care about this little difference, then it is important to distinguish, and be careful with your sources.

My source is DATABOG fysik kemi, page 166 and 142.

### Enhanced temperature range

One of the sources I see mentioned all the time is "Measuring the Loudspeaker's Impedance During Operation for the Derivation of the Voice Coil Temperature" by Gottfried K. Behler. He realises that the temperature of the voice coil may reach values more than 250 degC (482 degF).

Behler is concerned with the precision of his measurements, and concludes that for maximum accuracy, a second order model must be used. This is in agreement with DATABOG fysik kemi which says the upper limit for the model is at 100 degC (212 degF). So instead of only the alpha value, he uses an additional value, named beta. His equation looks like this:

```        rho(T1) = rho(T0) * (1 + alpha * deltaT + beta * deltaT^2)      (14)
```

I am not quoting Behler a 100 %, he is actually using R instead of rho, but he thinks it is rho too. He does in fact not distinguish between the two.

For aluminium (T0 = 20 degC) we have:

alpha = 0.00377 1/K beta = 1.3 E -6 1/K^2

For copper (again, T0 = 20 degC) we have:

alpha = 0.00393 1/K beta = 0.6 E -6 1/K^2

He further states that this improved second order model will give you a +/- 2.5% accuracy. Behler based his analysis on copper. More accuracy will require explicit knowledge of the material, which entails that the thermal coefficient is measured for the particular voice coil wire material in question.

No references for the 2-parameter model and material data is given.

Out of curiosity I tried to calculate which temperatures the two sets of parameters for copper and aluminium shows the same amount of power compression caused by change of Re (see next section, Power compression and max temperature). This is for 0 degC (32 degF), which is obvious since alpha and beta in this case is multiplied by 0, and for approximately 228.6 degC (443.4 degF). Between this interval, aluminium will show less power compression. The largest difference is found midways between the two temperatures, at 114.285714 degC, in favour of aluminium.

In the rest of this document, alpha is specified at 20 degC (68 degF) with the value for aluminium and copper found in most references, ie. 0.00393 [1/K], and beta is ignored, unless explicitly mentioned otherwise.

## Power compression and max temperature

All electro-dynamic loudspeaker drivers will show compression of the signal, when the temperature in the voice coil increases, because the material used in the coil has a positive temperature coefficient of resistivity, as we have seen in the previous section, usually symbolized by alpha.

This means that relative to the resistance at room temperature, usually 20 degC (68 degF), the resistance at an increased temperature is given by:

```        Re(T1) =  Re(T0) * (1 + alpha * deltaT) [ohm]           (15)

deltaT = temperature change = T1 - T0 [K]
T1 = temperature of interest [degC]
T0 = reference temperature, usually 20 degC
Re(T0) = The nominal resistance at the reference temperature [ohm]
Re(T1) = The new resistance at the temperature of interest [ohm]
alpha = Temperature coefficient of resistivity [1/K] of the voice coil
magnet wire
```

Often equation 15 is used the other way around so that the temperature change, deltaT = T1 - T0, is measured by measuring the Re(T1) value. See Behler, Measuring the Loudspeaker's Impedance During Operation for the Derivation of the Voice Coil Temperature. This way it is possible to determine several power handling parameters.

```                 Re(T1) - Re(T0)
deltaT = ---------------                                (16)
alpha * Re(T0)
```

Equation 16 can be formulated in several alternative ways.

The power compression occurs when Re is increased, because Re is in the equation for reference efficiency, no, and Voltage sensitivity, USPL. From FORMULAS.HTM we have:

```                                  2
rho_air * (Sd*Bxl)
no = -------------------------- [-]                     (17)
2
Mms  * 2 * pi * c_air * Re

rho_air : air density, ca. 1.2 kg/m3, see AIR.HTM
Sd  : diaphragm area [m2]
Bxl : force factor [Tm]
Mms : moving mass [kg]
c_air : sound speed, ca. 343 m/s, see AIR.HTM
Re  : DC resistance [ohm]
```

Notice that equation 17 can be multiplied by 100 to get a percentage value for the reference efficiency, no, but this is omitted here. All parameters follow the SI-units for compatibility.

```        SPL = SPL_ref + 10 * log10( no ) [dB/W/m]               (18)
```

SPL_ref : The reference sound pressure level, typically ca. 112.2 dB, but it varies with the air density and speed of sound, which is dependent on climatic conditions (temperature, pressure and molecular composition of the air - eg. humidity).

```                          / USPL_Volts * rho * Sd * Bxl \
USPL = 20 * log10( ----------------------------- )      (19)
\   2 * pi * pref * Re * Mms  / [dB/2.83V/m]

USPL_Volts : The reference voltage, typically 2.83 Volt
pref : The reference pressure, 20 uPa
```

Obviously, when Re is doubled due to excessive heat, the reference efficiency is cut in half, which equals a drop of power sensitivity of 3 dB, and the voltage sensitivity, USPL, drops 6 dB.

When DPC calculates SPLmx, it assumes a max 3 dB drop in power sensitivity, ie. a doubling of Re. In reality this figure is chosen more or less arbitrarily. It can be seen that the esteemed manufacturer, Peerless, shows that they allow a 3 dB drop on their data sheets.

Personally my experience also is, that a driver which stays within a 3 dB drop will survive without a permanent change of the driver. This must be the target since a permanent change of the driver implies some kind of damage.

It is a good question which voice coil temperature is actually associated to this. Douglas J. Button, Maximum SPL in Direct Radiator Loudspeakers, quotes a max temperature of 270 degC (518 degF) describing it as a knee point. In the LEAP software, power compression calculations are based on the assumption that the voice coil can handle up to 250 degC (480 degF).

What Button means by "knee point" is a point where less power input will give you less output, but more input will not provide you with more output. This happens when an increase in input voltage (USPL_Volts) will give you a similar increase of the resistance (Re), reducing the current flow, so that the sound level (USPL) does not increase. See equation 19.

When a user experiences such a knee point, the user will turn the volume back down - or at least not attempt to turn it further up - because he or she does not get the desired extra power output. See section Power compression and input power - The knee point, where this power is derived.

If we want the above equation 15 for Re(T1) = 2 * Re(T0) we get that alpha * deltaT = 1. Since alpha = 0.00393 we can calculate deltaT for doubling Re, it is 254 degC (489 degF), since T0 = 20 degC then T1 = 274 degC. In other words this temperature gives us the power compression 10 * log(Re(T1)/Re(T0)) = 10 * log(2) = 3.0103... dB.

As a side note, a speaker with a voice coil made of a material with a smaller alpha value would obviously provide lower compression at the increased temperature, T1, and a material with higher alpha values would show more power compression.

It becomes obvious that even for more rugged drivers, like PA and other pro drivers, the max temperature dictates ca. 3 dB compression. I have seen specifications as high as 4 - 6 dB power compression, in an article by Douglas J. Button, Maximum SPL for Direct Radiator Loudspeakers, but I am sure that he was talking about voltage sensitivity - though this does not really make sense, since power input and power sensitivity are directly related.

For tweeters that use ferrofluid which is hydrocarbon based, the ferrofluid will start to boil at 170 degC (338 degF), and then this value should be chosen as an upper limit, and please do not forget to add a safety margin. For ester based ferrofluids it is slightly higher, like 200 degC (392degF). Soft dome tweeters are not very durable when it comes to high temperatures anyway. Alu dome tweeters on the other hand are supplied with a nice heat dissipating dome surface.

Notice that with a lower limiting temperature, the user does not experience the knee point to the same extent, and may not become warned that the system is pushed to its limits.

I have seen a single reference quoting 400 degC (752 degF) as max temperature, but it must have been a mistake and probably should have been degrees Fahrenheit. There is the possibility for the voice coil itself to survice 400 degC, because the melting point for aluminium is at 660 degC (1220 degF) and for copper it is 1085 degC (1985 degF), but I would not dare to use these values for anything. It leaves no safety margin anywhere. The insulation coating will be damaged, and the metal will be damaged too.

Take care of your drivers solder joints etc. which may have a melting point as low as ca. 183 - 188 degC (361 - 370 degF) for normal Pb/Sn solder tin, typical mix ratio of 63/37 to 60/40. Working range 200 - 350 degC (392 - 662 degF). Normally these temperature limits are not crucial, since the joints are kept away from the voice coil, and in a place where the heat is less extreme, but with the voice coil reaching 400 degC (752 degF), everything will be very hot.

Voice coil wire is usually specified at 155 or 180 degC (311 - 356 degF), higher grades exist, but these figures from the manufacturers are meant to last in eg. transformers with lifetime of 10000 hours or so. If you take some fresh wire and load it up to around 300 degC (572 degF), then the color of the wire will change to brown because of structural changes in the metal.

The conclusion is that a short burst of a high temperature will wear down the voice coil just as much as many years of usage with a lower thermal limit. This short term high temperature is called the quench temperature. It frames a short term versus long term power handling. Keep the temperature down if you want your speakers to last.

When Re changes, other parameters relevant for box tuning are affected, like Qe - and therefore Qt. This is of course very undesirable, but not of my concern in this document. This is easy to confirm with DPC. Pick any driver and try to double Re to see the effects. Box simulation software with this particular feature will show you the effect.

Obviously the temperature in the voice coil is not the same as the temperature in the magnet gap, or in the magnet, because for each new material/space where the heat flows, there will be a temperature difference. In case the signal is stopped and everything cools off, the voice coil might have a slightly higher or lower temperature than the surrounding magnet structure.

Most neodymium magnet systems risks overheating, ie. the temperature will reach the curie temperature (Tc) of the magnet material and then it will demagnetize. Normal grade goes to 120 degC (248 degF) and higher grade goes to 150 degC (302 degF). With permanent, irreversible, damage. For a neodymium magnet specified to 120 degC I would not dare to go much above 90 degC (194 degF), just to keep a safety margin.

A system based on ferrite magnet(s) will experience a 15 % loss in output when the magnet becomes 100 degC (212 degF), but the loss is reversible, ie. the magnet power is restored when the temperature drops back down. When hot, the magnet system is more sensitive to external magnetic fields, though, and can become demagnetized.

One danish manufacturer of active speakers, where temperature sensing is to be implemented has found, that it is always the voice coil temperature that sets the limit. Their maximum service temperature is set to 210 degC (410 degF), ie. a safety margin is applied. In this case the spider and magnetsystem is around 100 degC (212 degF). The spider can get as hot as 90 - 95 degC (194 - 203 degF) in spite of its cloth material, ferrite magnet systems without any temperature control can get up to 110 degC (230 degF). Generally speaking the spider seems to end with a temperature ca. 20 degC (68 degF) lower than the magnet system, when the speaker is pushed to its limits.

Their limit for a neodymium magnet specified at 120 degC (248 degF) is set at 90 degC (194 degF). Anyway, the glue joints holding the magnet system together is loaded to its max service temperature at this point. There are special issues with active speakers, though, because active speakers is another ballgame than passive speakers. Compression, limitation etc. can alter conclusions very quickly.

When considering automotive applications, where the ambient temperature may reach up to 60 - 70 degC (140 - 158 degF), such a temperature may easily appear in the magnet system, and perhaps an even higher safety margin should be applied. See the EN 60721-3-5 standard for more information.

To have a specification for the max allowed temperature for a voice coil, Tmax, would be quite a nice (steady state) parameter to have for drivers. It could be used for simple comparison of the durability of the voice coil and surrounding materials. Combined with material information (or alpha specified directly) you could calculate Re(Tmax) = Rmax, and then the max power compression as well as the max SPL level.

I would not expect a DIY'er to measure Tmax, since the test must permanently damage the driver to determine the actual limit.

## Power compression and input power

In the LEAP software and probably also other applications, the above definitions are converted into a different value:

```        Theta = deltaTmax/Pe [K/W]                              (20)

deltaTmax = Tmax - T0 [K]
Pe = Max power handling [W]
```

The way I read the LEAP manual, Tmax = deltaTmax = 250 degC (482 degF), which means that LEAP assumes T0 = 0 degC (32 degF). Any reference temperature can be chosen, so long that the same reference temperature is chosen for all calculations, and the accompanying constants are given for that reference temperature.

The electrical power, P (or Pe for the max allowed power), is easily calculated as P = R * i^2 or P = U^2 / R. Remember that R must be a resistive value, not a reactive impedance value, to represent a power consumption. Reactive loads only stores energy, and there is no conversion of power from electrical to thermal power. This means that R must be the drivers Re value.

The way LEAP utilizes equation 20, the power compression is tied to the power handling (Pe) instead of to Tmax. In LEAP it is assumed that all drivers will handle the same temperature in the voice coil wire, but that different drivers are not equally good at dissipating the heat, due to variations in Pe, and therefore will have different load handling, ie. different Pe specifications.

I would personally prefer to have Tmax specified, but since most users of LEAP probably would not have access to such data, the complication was omitted by assuming deltaTmax = 250 degC (482 degF) for all drivers.

Theta is the power-temperature coefficient. For a given input power, the temperature can be calculated:

```        T1 = T0 + Theta * P [degC]                              (21)

T1 = temperature for the given power load [degC]
T0 = reference temperature [degC]
P = electrical power [W] dumped in the voice coil
```

The heat power, Ph, is defined as:

```        Ph = P * (1 - no)                                       (22)

P = Electrical input power [W]
no = the reference efficiency [-]
```

This equation ignores that a small part of the loss in a speaker is due to friction losses, which is not converted to heat in the voice coil, but ignoring this means that it is assumed dumped in the voice coil.

I assume that LEAP is using the above equation 21 put into equation 15 from the power compression and max temperature section and gets:

```        Re(P) =  Re(T0) * (1 + alpha * Theta * P) [ohm]         (23)
```

For the LEAP case, Re(T0) must be given at 0 degC (32 degF), but if you calculate Theta with a different reference temperature, like 20 degC (68 degF), then Re(T0) must also be at that reference temperature (in this case 20 degC).

Notice that when you specify a given input power in the above equation 20, the change in Re is "instantaneous." The reason is that the above value is assuming a steady state conditions, where heat energy has accumulated over time. So things needs to stablize for this equation to become true.

In other words, if Theta is calculated from measurements, it should also be determined for steady state conditions. One method could be to apply the full power, Pe, to the driver - and then after some hours, measure Re(Pe). Assume it has reached its max temperature and calculate Tmax.

This section is clearly based on information from the LEAP manual (for DOS). Later in this document, section Basic Thermal Physics - very long time perspective, a physical explanation for Theta will be derived.

### Automotive applications

In automotive applications the ambient temperature may well be up to 60 - 70 degC (140 - 158 degF) with passengers in the car, for a short period of time, until the air condition unit has cooled down the car cabin. During this period of time, obviously the power handling will be lower, and the max allowed temperature increase equally lowered. Theta stays constant so that we get:

1. calculate Theta at normal room temperature, and redefine:

```        Theta = deltaTmax / Pe = (Tmax - T0) / Pe,0             (24)
```

2. based on Theta = (Tmax - Ta) / Pe,a (alternative ambient temperature) calculate Pe,a (the power handling at the alternative temperature):

```                                            Tmax - Ta
Pe,a = (Tmax - Ta) / Theta = Pe,0 * ---------           (25)
Tmax - T0

Ta = the alternative ambient temperature [degC]
Pe,a = the power handling [W] at Ta
```

For example, if we have Pe = 100 W, Tmax = 250 degC and T0 = 20 degC we get:

```        Theta = (250 - 20)/100 = 2.3
```

Assuming Ta = 60 degC (140 degF), we get Tmax - Ta = 190 degC (374 degF) and we get:

```        Pe,a = 100 * (250 - 60)/(250 - 20) = 82.6 W
```

We can conclude that the drivers powerhandling, Pe, at this temperature is reduced from 100 Watt to 82.6 Watt.

### The knee point

As mentioned earlier, there is a knee point where increased input voltage does not increase the output power. It would be interesting to locate this knee point.

Based on equation 19, which we will repeat here for convenience, we can see that only one parameter depends on temperature, it is Re:

```                          / USPL_Volts * rho * Sd * Bxl \
USPL = 20 * log10( ----------------------------- )      (19)
\   2 * pi * pref * Re * Mms  / [dB/2.83V/m]

USPL_Volts : The reference voltage, now a knee point voltage
pref : The reference pressure, 20 uPa
```

What we want to find is the situation where USPL remains unchanged. This happens when the change in USPL_Volts is balanced by an equally large relative change in Re.

```        dUSPL_Volts/USPL_Volts = dRe/Re                         (26)
```

When this equation is fulfilled, an increase in input voltage will not provide an increase in output power.

First we start to derive dRe/Re by using equation 23, which we will repeat here for convenience:

```        Re(P) =  Re(T0) * (1 + alpha * Theta * P) [ohm]         (23)
```

The expression dRe (or deltaRe, if you must) is calculated as the difference between two power inputs:

```        Re(P2) = Re(T0) * (1 + alpha * Theta * P2) [ohm]        (27)
```

and

```        Re(P1) = Re(T0) * (1 + alpha * Theta * P1) [ohm]        (28)
```

We define that the input power P2 = P1 + dP (a small increment).

```        Re(P2) - Re(P1) = Re(T0) * (1 + alpha * Theta * P2)     (29)

- Re(T0) * (1 + alpha * Theta * P1) [ohm]

= Re(T0) + Re(T0) * alpha * Theta * P2

- Re(T0) - Re(T0) * alpha * Theta * P1

= Re(T0) * alpha * Theta * (P2 - P1)

= Re(T0) * alpha * Theta * (P1 + dP - P1)

= Re(T0) * alpha * Theta * dP

= dRe(P)
```

This means that we can write:

```        dRe     Re(T0) * alpha * Theta * dP
--- = --------------------------------                  (30)
Re    Re(T0) * (1 + alpha * Theta * P)

alpha * Theta * dP
= ---------------------
1 + alpha * Theta * P
```

This equation is valid for any input power.

With the input voltage things must be handled slightly different. To tie the equation into something dependent on input power, we must expand the equation:

```        dUSPL_Volts       1        dUSPL_Volts
----------- = ---------- * ----------- * dP             (31)
USPL_Volts    USPL_Volts        dP
```

Then we use the fact that

```        P = U^2/Re <=> U = sqrt(P * Re)                         (32)
```

This is inserted, both into the expression for U = USPL_Volts, and into the part, where USPL_Volts is differentiated by P on the right side of equation 31:

```            1        dUSPL_Volts             1         d(sqrt(P * Re))
---------- * ----------- * dP = ------------ * --------------- * dP
USPL_Volts        dP            sqrt(P * Re)         dP

(33)
```

We differentiate the middle section of the left side of equation 33 separately, to make it easier to see what is going on:

```        d(sqrt(P * Re))   1
--------------- = - * sqrt(Re / P)                      (34)
dP          2
```

Inserting this into equation 33 and shorting the left side, like equation 31, we get:

```        dUSPL_Volts        1         1
----------- = ------------ * - * sqrt(Re / P) * dP      (35)
USPL_Volts    sqrt(P * Re)   2

sqrt(Re / P)
= ---------------- * dP
2 * sqrt(P * Re)

1
= ----- * dP
2 * P
```

This equation is valid for any input power.

For one special situation, the knee point, the equations 30 and 35 are the same:

```          1            alpha * Theta * dP
----- * dP = ---------------------                      (36)
2 * P        1 + alpha * Theta * P
```

We could of course proceed to integrate both sides, but it will not change the outcome. It is much easier to divide the equations on both sides by dP and then isolate P. After removing dP we multiply with the dividend under both fractions:

```        1 + alpha * Theta * P = 2 * P * alpha * Theta           (37)

<=>

1 = 2 * P * alpha * Theta - alpha * Theta * P

<=>

1 = P * alpha * Theta

<=>
1
P = -------------
alpha * Theta
```

Since the initial equation 37 is only valid for P at the knee point, this electrical input power defines Pknee:

```                      1
Pknee = -------------                                   (38)
alpha * Theta
```

There are several interesting observations. First, notice that there is only one mathematical solution.

The solution describes the knee point with one material parameter, alpha, and one parameter dependent on the power handling, which is dependent on the specific driver (engineering implementation dependent).

If the input power for Theta is defined as the IEC 268-5 power handling limit, Pe, then the knee point, Pknee, is given for the same conditions.

If Pknee is smaller than Pe, then the user is warned by the knee point when he turns up the volume and observes that he cannot hear any increase in output.

We could, based on Pknee, calculate a theoretical Tmax:

```        Tmax = T0 + deltaTmax = T0 + Theta * Pknee              (39)
```

If Pknee is larger than Pe, then you will damage your speakers permanently if you attempt at such high power levels and temperatures.

If we insert Pknee into the initial equation for Re(P) we get:

```        Re(Pknee) = Re(T0) * (1 + alpha * Theta * Pknee)        (40)

= Re(T0) * (1 + alpha * Theta / (alpha * Theta))

= Re(T0) * (1 + 1) = 2 * Re(T0)
```

It can be seen that doubling Re gives you the knee point, always. We already know that this gives a 3 dB power compression, and that this is what is usually considered the limit for what a speaker will handle.

Is this really a coincidence, or is there something wrong with the derivation? No, there is nothing wrong with the equations, but in reality the knee point is not as sharp as in the model. The model does not behave correctly because of the simplifications, but the results are usable indicators of the system performance anyway.

## Input signal discussion

As mentioned in the introduction, power handling is a complex issue, some times also creating a hefty discussion among experts. The above two power compression sections are pretty easy to handle. The way I see it the time perspective is the reason for all controversies, and all discussions are founded in the time perspective on the expected input signal. The problems appear when people do not realize this.

### Expected input signal

The old school uses IEC 268-5, and it describes a testing method where the input signal from IEC 268-1 is utilized in a specific manner, to create a given power handling specification. It is based on DIN 45573 Section 3.

The IEC 268-1 signal consists of a pink noise signal, but filtered in the bass and tweeter range to simulate music signal spectrum. This noise signal probably was quite representive in the 70'ties. Unfortunately since then, a lot has happened on the consumer market scene, for example the CD media showed up and since then MP3, DVD-Audio and SACD.

Pink noise is a signal which has more energy at the lower end of the frequency spectrum than at the upper end, but in such a way that if you look at a logarithmic frequency scale it will be evenly distributed.

Peter John Chapman from Bang & Olufsen (B & O) has written an AES paper with the title "Programme Material Analysis" which concludes that modern music has a higher content of high-frequency sounds, but that the low end, the bass range, is approximately the same as from the old IEC 268-1 test signal. Personally I see more bass at 50 - 60 Hz. This paper also shows the differences in power spectrum for different classes of music.

Chapman also concludes in his paper, that the IEC 268-1 Simulated Programme Material no longer is representative when it comes to the peak / RMS ratio, it should be bigger.

Unfortunately, who cares about "average" music, when the driver you design should perhaps handle much more (or much less) than this, depending on what you are targeting at.

Please notice that the sounds were catgorized into the following 10 groups: Symphonic, Chamber, Opera, Pop, Heavy, Hip-Hop, Jazz, Blues, Folk and Speech. This means that Rock must be fit into the other categories. It also means that Home Theatre is not considered, because there is no Movie-Track category - a category that is most likely not going to fit into any of the above mentioned 10 categories.

Regarding thermal power handling, the low end is not so interesting because, to a certain point, the loudspeaker system you make will be excursion limited, and not thermally limited. Good point, except that more excursion also provides more forced convection cooling, which again means that the thermal power handling is increased. It can be concluded that the test signal affects the power handling results.

An early reference for thermal behaviour including forced convection is the JAES article written by Douglas Button (JBL), Heat Dissipation and Power Compression in Loudspeakers. This reference is based on work by Clifford A. Henricksen (Electro-Voice), see "References" in the section at the bottom for the article Heat Transfer Mechanisms in Loudspeakers: Analysis, Measurement and Design.

To focus entirely on the thermal behaviour, one option is to exclude forced convection cooling from your further work, at least to the larger extent. This is a legal step for tweeters and dedicated midranges, which applies crossovers, where you instead must decide for a crossover frequency and slope. For application in passive crossover circuits, the crossover for power handling measurements should also be a passive circuit to simulate a real world situation. For LFE and dedicated subwoofers, this is a bad restriction, the power handling will be too conservative, because all output from such speakers are in the relatively high-excursion range.

In the world of loudspeakers it seems to become increasingly difficult to move into a field not already explored by Wolfgang Klippel, utilizing his nonlinear models. In one of his AES papers, Klippel shows how the thermal behaviour can include forced convection cooling. It requires knowledge of how the nonlinear excursion of the speaker is behaving, which basically means that two models are interacting. Wolfgang Klippel also showed his studies at an ALMA symposium, perhaps with a slightly different perspective, and definitely put in a way that makes it is easier to understand.

A linear approximation of the excursion is probably doomed to fail for most drivers, at medium to high excursion levels. This is why I recommend not to include excursion in the model. Leaving it out will provide you with a more conservative power handling rating. As mentioned earlier, this is only a real restriction on subwoofers, which are going to be too limited by this rating. On the other hand, subwoofers are usually not thermally limited at all. Subwoofers and LFE channels are restricted by mechanical excursion.

Perhaps I should mention, that IEC 268-5 describes how the IEC 268-1 noise signal is manipulated for power handling measurements. First it ensures that the crest factor is low, ie the Peak voltage / RMS voltage is 2:1. This is done by clipping the pink noise signal until the condition is fulfilled. This means that the input signal is not producing high excursion in the drivers, but at maximum a moderate excursion in woofers. The advantage is that you do not need an enormous amplifier capable of delivering extreme peak power values. Furthermore, to provide a reasonable ratio between power input and real music, the driver plays for 1 minute, then rests for 2 minutes. This is repeated 10 times at each power testing level. It gives a peak / RMS ratio of 10 * log (2 * 3) = 7.78 dB power spectrum.

In todays average music and sounds, the compression of the music signal is so that lots of music has a peak / RMS ratio of 10 dB (power spectrum), see P. J. Chapman, Programme Material Analysis. With commercials on TV, and cartoon sound hitting the bottom, the peak / RMS ratio may be as low as 6 dB (power spectrum).

With the dynamics in music reaching higher than the IEC standard predicts, it also means that the recommended amplifier power from this test method will be smaller than what your drivers can handle with average modern music, long term.

This of course shoots a hole in the IEC 268-5 standard, and nowadays it can be considered a conservative rating of the recommended amplifier size, but that is the whole nature of this standard, ie. being qualitative in a relative way. I personally find it to be nice that the rating is conservative, and find it to be quite useful.

The other way around when it comes to highly compressed sounds, the average level will be higher and may slowly overheat your drivers voice coil. You can be rest assured, though, that unless you have a mismatch between your loudspeakers sensitivity and the size of your amplifier, then you would never want to hear commercials and cartoons at the full power level of your system.

Please notice, that we are only considering IEC 268-5 long term power handling. The IEC standard also provides a method for specifying the short term power handling.

JBL adds a classification to the IEC standard, where the speaker is placed in one of the 3 classes:

1. For carefully monitored applications, where peak transient capability must be maintained, a system can be powered with an amplifier up to twice the IEC rating. Such high-dynamic sound reproduction will give an average power level below the limit of the speaker.
2. For routine application where high continuous, but non-distorted, output is expected, the IEC rating specifies the max size of the amplifier.
3. For musical instrument application, where distorted (overdriven) output could be a requirement for the "right sound" an amplifier not larger than half the size of the IEC rating is recommended.

I think this is a very reasonable classification, which also shows the limitation (or initial purpose) of the IEC standard.

One special consideration, a point made by my colleague Krestian Pedersen from B & W, is that socalled eddy currents in the magnet system is a problem at high frequencies. Here the inductance of the coil approaches a semi-inductance (as described by Vanderkooy and others). Energy is absorbed due to the eddy currents, and the energy becomes heat in the magnet system. A lot of heat, that is. He claims that just one Watt of input power can heat up the drivers magnet system considerably.

At the other end of the spectrum, at DC, we also find that even very durable speakers will handle almost no power. Here the voice coil is heated without anything but pure radiation to keep it cool. A very limited input power can heat up the drivers voice coil considerably.

My point is, that the input signal is of very high importance for the power handling specification. Of course very high frequencies (>20 kHz) or very low frequencies (DC) are important, but in my point of view this frequency range is normally irrelevant.

Both high-frequency and DC output are usually symptoms on amplifier malfunction (clipping, power supply instability, etc). They are irrelevant when it comes to specification of drivers, but highly relevant to take into account for a good implementation of a system. For example automotive systems are notoriously famous for high power junk amplifiers, installed in the aftermarket.

### Todays compression level

The CD media has given us a higher dynamic range, and much lower noise level, than what was previously the norm when the IEC 268-1 and 268-5 standards were made, but the current level of compression is still surprisingly high.

The reason for the relatively high level of compression in modern music is, that this means the artist is played louder over the radio, you will have an easier time hearing it at the shopping mal, in your car or wherever you may be. This means that the artist is more noticed, probably becomes more popular and sells more records.

And of course it has been made possible because of better compression algorithms, better implementation of compressors etc. so that the compression is not as noticeable as would have been the consequence before the evolution of the digital studios. The improved understanding of our own hearing perception (ie. psychoacoustics) fuels this development. The new features are used by everybody, even esteemed bands like Pink Floyd etc.

## Basic thermal physics

It is possible to make a thermal model of the driver, which describes the behaviour in a true time-perspective. An electrical equivalent circuit model of the driver can be created, which seems to be the preferred approach.

Based on this model, any time function of power input can be given and the model will predict the temperature at different positions of the driver model - assuming a given heat path. Take note about the previous section and its discusion on exactly which signal is representative. One must realize that the models are hard facts, but garbage input gives you garbage conclusions.

In the models, pairs of constants are represented by capacitors showing the heat storage and resistors showing the heat flow. We will start with the basic physics and then derive the components for the models for some simple conclusions.

### Heat transfer

A heat flow is always a result of one of the following three physical types of processes:

1. Conduction
2. Convection

Conduction is when material properties allows the heat energy to flow through the material itself and/or several materials in which case one must consider the boundaries between the materials (like when you cook on the stove). The conductivity of materials is given by lambda, which based on a one-dimensional (1D) direction specifies the energy transferred per temperature difference (the gradient).

Convection is when a carrier material allows the heat to be transmitted off a surface of the material (like air flowing across and heated up by a radiator or cooled down by a window). Forced convection is when the air flow is managed by eg. a fan (like a car engine). The convective heat transfer is given by h, which based on a two-dimensional (2D) area speficies the the energy transferred per temperature difference.

Radiation does not require a media to flow, it is electromagnetic radiation (like sun shine or the light from a light bulp). Radiation becomes a significant factor when the temperature is high, but for the rest of this document we will neglect it. The radiation exchange is given by the Stefan-Boltzman constant, sigma.

To neglect radiation is obviously a limitation, since the energy balance is the sum of all three contributions.

### Heat capacity

The heat capacity describes the amount of energy stored in a given object, per unit temperature increase. Or, the other way around, it describes the materials capacity to store heat energy, for a given temperature change.

Each material has its own capability of storing heat, called specific heat capacity, and when multiplied by the mass of the object, a heat capacity of the object is calculated.

```        dQ
---- = C = m * c_p                                       (41)
dT

Q : Heat energy [J]
dQ/dT : Heat energy increase per temperature increase [J / K]
T : Temperature [degC]
C : Heat capacity [J / K]
m : mass of the object [kg]
c_p : Specific heat capacity [J / (kg * K)] at constant pressure (a
material specific constant)
```

Notice how the heat capacity does not specify a volumetric value, like what can be contained, but rather the capability to handle heat energy, like a capacitance.

The specific heat capacity, c_p, can be picked from the table below:

```        c_p at 20 degC
[J / (kg * K)]                  More c_p values
-------------------             --------------------
Aluminum (Al)   896             Nickel    (Ni)   444
Gold     (Au)   129             Platin    (Pt)   133
Iron     (Fe)   452             Tin       (Sn)   226
Copper   (Cu)   387             Zinc      (Zn)   389
Brass (CuZn33)  390             Wolfram   (W)    135
Silver   (Ag)   235
```

My source is DATABOG fysik kemi, page 142.

Notice how it requires more energy to heat up aluminium than copper, or any of the other metals.

By the way, c_p and c_v are not that different from each other, for these solid metals, very unlike air constituents where gamma = 1.4.

When equation 41 is multiplied on both sides by the potential temperature increase, and the differential notation is converted into delta-values, the potential heat energy stored in the object is isolated:

```        deltaQ = m * c_p * deltaT = C * deltaT                  (42)

deltaQ : Heat energy difference [J] = Q1 - Q0 = change of stored heat
energy for the given temperature change
m : mass of object [kg]
c_p : specific heat capacity of object [J / (kg * K)]
C : heat capacity [J / K]
deltaT : Temperature difference to the surrounding environment [K]
= T1 - T0 = the temperature change for the given change in heat
energy
```

### Heat flow

The heat flow goes from a high temperature area toward a lower temperature area. In thermodynamics it is described with a physical law, saying that the total entropy must increase. If a heat pump is installed, then the heat can flow in the opposite direction, but the total entropy is still increasing (because you must add/waste energy to reverse the natural direction, like eg. in refridgerators which use electrical energy to reverse the direction).

Each material has its own capability of transporting heat, called the specific heat conductivity, which Fourier described in his one-dimensional heat conductivity differential equation:

```        dQ                  dT
---- = - lambda * A ----                                 (43)
dt                  dx

dQ/dt : The Heat energy flowing per unit time [J/s = Watt]
lambda : Heat conductivity of the material in the heat path [W/(m*K)]
A : Cross section area of heat path [m2]
dT/dx : The Temperature increase per unit length of the heat path [K/m]
Q : Heat energy [J]
t : time [s]
T : Temperature [K]
x : distance [m]
```

Notice that the heat flows in the opposite direction of the temperature increase, and that is why the minus sign is present in equation 43. The dT/dx will also be negative, because the positive direction of x is the same positive direction of Q.

We can conclude that the heat conducted per unit time is the product of 3 sizes. The area A perpendicular to the heat flow, a material parameter lambda called the specific heat conductivity, and the temperature gradient dT/dx, which is the temperature increase per unit length of the heat path.

The heat conductivity, lambda, can be picked from the table below:

```        lambda, 20 degC [W/(m*K)]       More lambda values
-------------------------       -------------------------
Aluminium (Al)        238       Nickel    (Ni)         91
Brass (CuZn33)         96       Platin    (Pt)         72
Copper    (Cu)        401       Tin       (Sn)         67
Gold      (Au)        318       Zinc      (Zn)        116
Iron      (Fe)         80       Wolfram   (W)         174
Silver    (Ag)        427       Air (not moving)    0.024
```

My source is DATABOG fysik kemi, page 142.

For the above metals, heat conductivity is quite high. When the temperature is increased, then lambda increases too. The difficulties in measuring lambda means, that there are large variations in measuring results and reference tables (it is worse for low lambda values, though, ie. non-metals). It is noteworthy that the metals which are the best electrical conductors are also the best heat conductors, see explanation later.

In fact, for a certain temperature range they are proportional, and the proportionality is called the Lorenz number, L. The behaviour of the ratio L * T = lambda/rho is sometimes called the Wiedemann-Franz Law (not being strictly true). At some point the electrical conductivity drops with temperature, whereas the heat conductivity increases.

For materials which har not homogenous, different lambda values can exist for the different directions. Metals, like the pure metals above, will behave isotropic.

For a steady state situation, equation 43 could be transformed into:

```        Ph = - lambda * A/l * deltaT                            (44)

Ph : Heat power (=dQ/dt) input
```

In electrodynamic loudspeakers almost all the electrical power, P, is converted into the heat power, Ph. The difference is given by the reference efficiency, no, which describes the part converted into acoustical power. We get:

```        Ph = P * (1 - no)                                       (22)
```

### Example : Very short time perspective

The very short time perspective solely considers the first heat storage, which is the voice coil itself. It means that virtually no transfer of heat has happened yet. We are talking about a time period of less than a second, perhaps only 1/10 of a second.

The stored heat energy is calculated as follows, from equation 42:

```        deltaQ = Ph * t = m * c_p * deltaT                      (45)

deltaQ = change of stored heat energy [J] for a given temperature change
Ph = Heat power [W]
t = time [s]
m = mass of heat storage [kg]
c_p = specific heat capacity [J / (kg * K)]
deltaT = the temperature change [K] for the given change in heat energy
```

Equation 45 can be rearranged and rephrased as follows:

```        Phmax = m * c_p * deltaTmax / t                         (46)

m = mass of voice coil
c_p = specific heat capacity of the voice coil material
deltaTmax = Tmax - T0 is the max temperature change
t = time duration for power input
Phmax is the max allowed heat power before destroying the driver, see
equation 22 for the relation between heat power and electrical
input power
```

As can be seen, the input power really reaches infinity as the time is going towards zero. This means that any driver (almost) can be specified for a short term load of 1000 Watt, or higher if desired, even tweeters - just keep the time duration below a few milliseconds. This is of course only true in real life if glue joints and other parts can handle the short high-energy load, we are only talking thermal loads right now.

For example, if the mass of the voice coil wire is 3 gram (0.003 kg), the material is aluminium (c_p = 896 [J / (kg * K)]) and the deltaTmax = 200 degC (392 degF), then for half a second, the driver will handle:

```        Phmax = 0.003 * 896 * 200 / 0.5 = 1075 W                (47)
```

If the heat power is converted into input power, the efficiency of the driver must be known. Say that no = 1 %, then the max applied power to prevent damages would be:

```        Pe = Phmax / (1 - no) = 1075 * (1 - 0.01) = 1086 Watt   (48)
```

As can be seen, even a tweeter may achieve very high power specifications, if the duration which this power is applied is sufficiently small.

Equation 46 may be converted to isolate the time, t, as a function of power, so as to see how long a driver can withstand a given power input. The value will be very conservative, because the heat flow is neglected.

Physically there is a limit, of course. If an extremly high amount of energy is shot into the system, a transient pulse of energy, then it will take some time for the heat to spread. This is called diffusivity. Transient heat transfer is normally dealt with in mathematical Finite-Element models. There are 3 diffusional type of heat transfer processes:

```        1) Energy, a [m^2/s]
2) Mass, D [m^2/s] (molecule exchange includes transfer of heat)
3) Momentum, v [m^2/s] (or kinematic viscosity diffusion
coefficient)
```

Energy transfer is the one relevant for solid materials, and relates directly to heat conduction:

```             lambda
a = --------- [m^2 / s]
rho * c_p

a : the thermal diffusivity (some references use alpha for the notation)
lambda : heat conductivity (at the temperature before the transient)
rho : density of the material (at the temperature before the transient)
c_p : specific heat capacity of the material
```

The product of rho * c_p is called the volumetric specific heat capacity, and is, if not a constant per se, then at least grouped for almost all materials in a very limited band, around 3 * 10^6. This means that materials with high conductivity also have a high diffusivity.

The almost constant rho * c_v (equals rho * c_p) can be understood by noting that a solid containing N atoms has 3*N vibration modes. Given Bolzmann's constant and the fact that almost all solids contain a certain amount of atoms per cubic meter, gives us the 10^6 factor.

Diffusivity expresses how fast the heat spreads through a material. Metals is the material group with the highest thermal diffusivity, so normally this is not considered a problem with voice coils.

The diffusivity can be measured directly by observing the temperature decay when a heat source is switched off. The diffusivity is usually differentiated by the load time, t, in the differential equations for heat transfer.

The range of both lambda and a reflects the mechanisms of heat transfer in each class of solids. Electrons conduct the heat in pure metals, such as copper, silver and aluminium. Here the conductivity is described by lambda = 1/3 * C_e * c * l, where C_e is the electron specific heat per volume unit, c is the electron velocity (2 * 10^5 m/s) and l the electron mean path. l is typically around 10^-7 for pure metals, whereas for alloys (eg. steel) it reduces to 10^-10, which reduces the heat conductivity, lambda, and the diffusivity, a, considerably.

As a sidenote, heat is not conducted by electrons in non-conductive materials. Instead the heat is transmitted by lattice vibrations of short wavelengths (phonons). This determines the phonons mean free path, l. The same equation applies (C_e is substituted by rho * c_p).

A temperature shock resistance, deltaTshock [K], sometimes called the quench temperature, can be determined, where the material is damaged and does not maintain its mechanical strenght. A number of cycles with shock will also be able to destroy the material strength. Again, I do not think this is a problem with loudspeakers. Considering that the magnet wire almost instantly changes color just above 300 degC (572 degF), this could be considered the maximum shock temperature.

The conclusion is, there is a limit to how fast you can load a speaker voice coil with energy. Diffusivity is normally not an issue. Remember that a voice coil per definition will hold back the energy because it acts as a low-pass filter. At very high transients it can be observed that the input side of the voice coil is the hot side, whereas down at the other end of the voice coil is the cold side.

### Example : Very long time perspective

In this case a steady state situation has stabilized. This means that the temperatures at various positions are constant, all heat capacities are charged to the given steady state level. Since the heat can flow into the surrounding environment, where the ambient temperature is kept constant, a stable heat flow is present, which also requires a stable input power to feed this constant heat flow.

Looking globally on the heat flow, let Ph = dQ/dt be the constant energy flow, ie. the heat power load, then for steady state heat transfer, and constant input heat power, the following relation can be made between the temperature difference between two objects and the heat flow:

```        deltaTdrop = Ph * R_T [K]                               (49)

R_T : Heat flow resistance [K/W] of the total heat chain
deltaTdrop : The temperature drop [K], as a positive (absolute) value
```

The energy flow, Ph, flows in the direction of a dropping temperature, in an attempt to even out the energy state in the system.

Does equation 49 sound familiar? Yes indeed, just check out equation 20 to realize that R_T * (1 - no) = Theta. But also notice the connection to equation 25 and 42 as mentioned earlier. Rephrasing equation 44, by isolating deltaT and changing to a temperature drop, we get:

```        deltaTdrop = l / (lambda * A) * Ph                      (50)

deltaTdrop = Temperature drop [K] across the heat path, as a positive
(absolute) value
```

This means that if (and only if) the heat path in a driver consisted of a single material and a single length of heat path, and not the real world complex heat path, then:

```        R_T = Theta / (1 - no) = l / (lambda * A)               (51)
```

You could say that R_T (and Theta) expresses a simple parameter for the total heat path with its varying geometric shape and material parameters. This parameter describes in a global sense how well the total heat path has been engineered, when compared to other similar designs.

## Conclusions

Well, who said that power handling is a simple issue?

The model requiring alpha and Theta is a simple model. They could be supported by DPC, but they are difficult to support unless the parameters are specified by manufacturers. Guessing values of Tmax is not a desired feature of DPC. The parameters are not very important for DPC in itself, but relevant for box simulation software because it helps the user in the decision process, and here guessing Tmax is a more suitable solution to lack of input data.

You should not put a great deal of value in power handling or the accompanying alpha and Theta parameters, partially because it is of much higher value to look at the output side instead of the input side, but they do give you the option to explore the behaviour of your loudspeaker system, when heavily loaded, since not only the output level is affected but also the electrical Q, Qe, and therefore misalignment will appear, again parameters more relevant to box simulation software.

Different box alignments might show more or less sensitivity to misalignment, and this is a nice way to study such temporary misalignment based on long term power input. It would be interesting if driver manufacturers would provide data on Tmax directly or indirectly, preferably the question should be:

```        "When you specify the IEC 268-5 power handling, what
is the Re-value at the given max power input?"
```

You could name this value Re(Pe) or Re(Tmax). Besides this, you need the voice coil material to achieve the material constants to establish the simple models.

Talking amplifier size, you can use the Re(Tmax) = Re(Pemax) value to calculate the max amplifier voltage output (Umax) based on the standard equation P = U^2/R:

```        Umax = sqrt( Re(Tmax) * Pemax )                         (52)
```

With this value in your hands you can see whether the recommended amplifier will be able to blow your driver (without crossover).

Even though a crossover is applied, the amplifier will usually be able to deliver a high-frequency power output above the power handling limit of the midrange and considerably above the power handling limit of the tweeter. In these cases the thermal models become relevant, but you must be able to define an appropriate input signal.

It is an interesting point that a certain amount of power compression is desired, so that the user can determine the limiting power, bordering overload, and the risk of destroying the speaker is reduced.

It is also interesting to see, that a second order behaviour, like the model suggested by Behler, is prefered over a linear behaviour, because it leaves more available working range with low power compression, to quickly at a certain temperature turn into higher compression. Here aluminium is preferred over copper, with its more pronounced second order model, and a more clearly defined knee point.

Finally, if the model data are supposed to be right, then it is important to know whether alpha is based on the resistanse R, which implies that l/A is included, or the resistivity rho (then l/A could be added separately).

Personally I feel that the risk of confusion is minimized, if the model is for Re directly, but admittedly material data are somehow scientifically nicer.

## Summary

To summarise on this document I would like to clarify which parameters are needed and which are convenient, to apply a simple model for power compression in DPC. The necessary minimum amount of parameters are:

• alpha (initialized to 0.00393 1/K, valid for aluminium and copper, but to be user specified)
• Tmax and T0 (initialized to 270 degC and 20 degC respectively)
• Theta (calculated when Pe is specified)

Notice that the Tmax temperature definitely is not a conservative figure, if it is left unchanged, but this value is the value that will provide ca. 3 dB power compression. This is consistent with the specified power handling levels following IEC 268-5 and practical experience with what drivers can handle during this test.

The alpha value and reference temperature T0 should probably be specified in DPC.INI, and then if T0 equals the "temperature" initialized no convertion of alpha values will be necessary during initialization of a driver datasheet. Conversion is otherwise necessary since the specified Re-value is assumed given at room temperature.

alpha and T0 should be changeable in the parameter editor, with conversions applied to various parameters. Tmax is driver specific and should be changeable too. The parameters belong with the power compression and/or SPLmx parameters in the parameter editor.

For convenience it would be nice to have the following parameters explicitly in DPC:

• beta, if a 2-parameter resistivity model is applied [1/K^2]
• Re(T0), the resistance [ohm] at the reference temperature (= Re if T0 is the same temperature as in the initial setup)
• Rmax = Re(Tmax) is the max resistance [ohm] at the max temperature
• PComp = 10 * log (Rmax/Re(T0)) is the power compression [dB] as a positive value (must then be subtracted in the equations)
• alpha_l, the heat length expansion coefficient [1/K] if alpha and alpha_l are not to be treated as a combined variable

With these parameters the user will be able to play around with the various dependent variables while DPC calculates the free and unknown variables, and study manufacturers more or less conservative specifications.