Music and Design TM
__________________________

Dynamic compression due to Voice Coil Heating,
Fact or Fiction?
It is well known the power dissipated in the voice coil of a dynamic driver generates heat which results in an increase
in the voice coil temperature and resistance. Over a long time the voice coil reaches a quasi-steady temperature for
which the rate of heat generation in the voice coil is balanced by heat transferred to the motor structure and
surroundings. Since the power dissipated in the voice coil is not constant but will vary as the level of sound varies,
the voice coil temperature will meander about this quasi-steady state value reflecting the changes
in the average power dissipated  with time. In
addition, it has been conjectured that there can
potentially be large spikes in temperature due to
short term, high SPL dynamic  transients. If these
transients do indeed give rise to significant increases
in voice coil temperature these transients could be
dynamically compressed as a result of the increases
in voice coil resistance which accompanies these
temperature increases.

The temperature dependence of the voice coil
resistance can be expressed as:

Re(T) = Reo x (1 + a (T(t)-To))

where Reo is the voice coil resistance at ambient
temperature, To. T(t) is the actual, time dependent
voice coil temperature and "a" is the temperature
coefficient of resistance.  
Since the efficiency of a dynamic driver can be expressed as

n = constant x (BL x Sd / Mms)^2 / Re

it is apparent that an increase in Re reduces efficiency. Since the sensitivity is related to the efficiency as

Sp = 112.2 + 10 x Log (n)

it is further apparent that the sensitivity will vary with Re such that

Sp = 112.2 + 10 x Log (n) + 10 Log (Reo/Re(T))

This last "correction" terms shows that if Re doubled in value due to heating during a dynamic event, the sensitivity
would be reduced by 3dB and the event would thus be dynamically compressed. For this to happen, the voice coil
temperature must rise exceeding fast.
Additionally, if the voice coil temperature does not
return to the meandering level before the next
dynamic event occurs, the increase in temperature
and reduction in efficiency may compound itself, as
shown to the left, leading to more severe dynamic
compression of the latter events.

It has been suggested that drivers with higher
efficiency will not suffer from such dynamic
compression to as great an extent as lower
efficiency drivers. This seems logical since the rate
of heat generation in the voice coil is given as

Q' = Real (I x E)

where I is the current flowing though the voice coil
and E is the voltage across it. Since power is
dissipated only due to voice coil resistance,

Q' = E^2 / Re.
If a high efficiency driver is nominally 10dB more sensitive than a typical "HiFi" driver then the reach to same SPL
10dB less power must be dissipated in the voice coil. That is to say, the heat generated in the voice coil during a
dynamic event would be 1/10 that dissipated in the HiFi driver. This would seemingly result in a significant
reduction in the increase of the voice coil temperature, reducing the thermal compression, and therefore provide a
more dynamic transient.

What we see here is both the rate of heating and cooling of the voice coil are important. However, the rate of
heating is the dominating factor in dynamic thermal  compression. The reason is simple; since the dynamic events
are of short duration, the rate of heating must be sufficiently fast to generate a significant increase in temperature
over the duration of the event. If it is not, Re will not change significantly during the event, and there will be no
compression arising from thermal effects. Since after the transient is over the power is reduced to that associated
with the meandering level, no additional excess, "dynamic heat" is generated. Thus, before worrying about how the
voice coil cools and relaxes back to the meandering level after the dynamic event ends, first we can look at how
fast the voice coil heats. This greatly simplifies the problem. Cooling of the voice coil is a complex problem
involving transfer of heat by both convection and radiation from the voice coil to the VC former, the motor
structure, the pole piece and any other structures that may act as a heat sink.  On the other hand, all the heating
effects arise form the power dissipated in the voice coil.

So, let us proceed. We begin by assuming the all the heat generated in the voice coil gives rise to an increase in
temperature. We shall ignore any heat transferred to the surrounding. The result of this assumption is that we will
establish an upper limit on the rate of heating. If cooling were considered, heat would be removed form the voice
coil and its temperature could not rise as quickly.

The equation governing the rate of temperature increase is given as

(ρ cp V) dΔT/dt = Q’

Here ρ is the density of the voice coil material, cp the specific heat and V the volume of material contained in the
voice coil. ΔT is the difference between the temperature of the voice coil at the beginning of a dynamic event, To,
and the instantaneous tempeture during the event, T(t).

ΔT = T(t) - To

Q' is the rate of heat generation,

Q' = E^2 / Re(t)

where, from above,  

Re(t) = Reo x (1 + a (T(t) - To))

Thus we have

(ρ cp V) dΔT/dt = E^2 / (Reo x (1 + a ΔT ))

Rearranging:

d (ΔT + (a/2) ΔT^2 ) /dt = E^2 / ( ρ cp V Reo)

Letting

C = 1/ ( ρ cp V Reo)

we have

d (ΔT + (a/2) ΔT^2 ) /dt = C x E^2

Integrating,

ΔT + (a/2) ΔT^2  = C ∫ E^2 dt

Now, if we assume the dynamic event is of the from of an impulse of duration td, such that E is constant over
duration we can find the value of  ΔT at any time between t=0, the start of the impulse, and td by finding the root of
the equation,

(a/2) ΔT^2 +  ΔT - C E^2 t = 0    for  0 < t< td

The assumption that E is constant is, again, a conservative estimate which will yield greater increases VC in
temperature than a true transient spike with the same magnitude as indicated in the sketch to the left. The
equation above is a simple quadric equation and the desired solution is given as:

ΔT(t)  = [1- sqrt( 1 + 2 a C E^2 t )] / (2a)

This equation will give a good prediction of the maximum rate of increase in temperature the voice coil can
experience. All we need do is insert the correct physical properties.

For a copper wire voice coil

cp = 0.385 J / K

ρ = 8.9g/cm^3

a = 0.0039 / K

Calculations for a Midrange Driver:

For a typical HIFi driver, such as th SEAS W22 the voice coil diameter is 39 mm, the height is 16 mm with Re
specified as 6 ohms. Form this information the volume of copper in the voice coil can be determined  to be 0. 297
cm^3 assuming a single layer voice coil. The assumption of a single layer is conservative in that it minimized the
volume of the voice coil, thus maximizing the rate of temperature increase. The driver has a quote sensitivity of
90.5 dB and Reo = 6 ohms at room temperature of To = 20 degree C.  Setting Reo to the cold, DC value also
maximizes the rate of heating thus is again very conservative. Starting at a higher temperature would reduce
increase the initial value of Re and reduce the value of Q'.

To see how the voice coil heats up under the influence of a dynamic pulse we can make the pulse arbitrarily long.
We need only consider the behavior out to the time of interest. If we choose to stop the pulse at any time the
temperature would begin to cool thereafter. Thus the first plot presented below shows how the VC heats up if 2.83
volt is applied across the terminals (1W / 8 ohms).
To the right we see that with 2.83V
applied across the driver it takes 100
seconds for the voice coil
temperature to rise to 5.4 times it's
initial value, To, room temperature
(20 degrees C). The voice coil
resistance increases to 9 ohms and
the sensitivity drops by 1.52 dB after
100 sec. Recall, this is with out
considering any cooling due to heat
transfer to the surroundings.
Next, consider what happens when the magnitude of the pulse is increase to 28.3 volts or 100 W / 8 ohms which
would yield an SPL spike 20 dB above the 90.5 dB reference sensitivity at 110.5dB
Note that the time scale in the figure to
the left, for a 28.3V pulse, only extends
to 1.2 sec. The figure shows that when
the amplitude is increases by a factor
of 10 the rate of increase is 100 time
as fast. This makes sense since the
heat generation rate also increases by
a factor of 100. Still, a true dynamic
pulse would not likely last for a second
or longer. In all likelihood such a
transient would be measured in msec,
or fractions there of. After 20 msec the
compression is less than 0.05 db, the
VC temperature has risen 2.6 degrees.
and Re has increased form 6 to 6.06
ohms.
Raising the input power level to the
equivalent of 1000 W /8 ohms, as
shown below, increase the rate of
temperature by another factor to 10.
Even so, after 20 msec there is only a
0.4 dB reduction in sensitivity. The
temperature has risen 25 degrees to
45 C and the resistance to 6.58 ohms.
Calculations for a HiFi Tweeter:

For this example a SEAS tweeter with 2.6cm diameter voice coil, 0.15 cm VC length and Re = 4.8 ohms was
considered. The pulse is set at 100 W / 8 ohms. Reo was taken at room temperature, 4.8 ohms with a reference
sensitivity of 90.5 dB.
Increasing the pulse to 1000 w / 8 ohms.
For the sake of comparison the 1000 W pulse case was performed assuming that the VC temperature at the beginning of
the pulse 250 C. This yields Reo = 11.38 ohms and the sensitivity at the start of the pulse would be 87.71 dB.
When starting from a temperature of
250 degrees after 20 msec the
reduction in sensitivity is 0.22 dB, the
VC temperature has risen 13 degrees
from 250 C to  263 C and Re has
increase  11.98 ohms.

This result confirms that starting at
room temperature conditions results in
the maximum increase in T and
maximum compression.
Form the above examples it would appear that dynamic compression arising from thermal effects is not likely to be
significant for a typical HiFi midrange driver. Rather, the results suggest that repeated short term transients, as seen
in music, lend to a gradual increase in VC temperature. This may then results in thermal compression over an
extended period of listening.  In the above example a 1000 w pulse, translating to a 120 dB peak level, only began to
show any significant effect. In all probability, in a home audio system such power levels would not be available and
the source of dynamic compression would more likely arise form other factors, including insufficient amplifier power.
The tweeter results show that after 20
msec the sensitivity has dropped 0.7
dB, the VC temperature has by risen
44degrees to 64 C, and Re has
increased to 5.64ohms. At first glance
this might seem like a significant result
and the thermal compression is
significant. However, we must consider
that a tweeter will typically be
connected to a high pass filter and
would never see the complete
spectrum form a 20 msec pulse. Thus
we must examine what the electrical
signal reaching the tweeter would look
like if a 20 msec or longer pulse were
applied to the input of the crossover
filter.
The figure to the left shows in red the
pulse seen by a tweeter when
subjected to a 20 msec pulse filtered
though a 1 K Hz, 2nd order
Linkwitz/Riley, high pass filter. The
actually signal reaching the tweeter
lasts no longer than about 1.25 msec.
Thus we must examine the result above
for t less than 1.25 msec.
Tweeter short time response.
This figure show the same result as
above but for time less the 2.5 msec.
Clearly, when the pulse is filtered
through the crossover resulting in only
a short duration pulse reaching the
tweeter, no significant thermal
compression is observed.
The results presented above were somewhat
surprising in that they indicate that dynamic
compression due to thermal effects seems
unlikely. Rather they suggest that any
compression is the result  of longer term heating
of the voice coil due to sustained high power
levels, and dynamic compression is likely a result
of the reduction in driver sensitivity due to the
long term heating coupled with limited amplifier
power. One last figure is presented at the right.
This figure shows the degree of compression as
a function of voice coil temperature assuming
copper wire. Note that at the melting point of
copper a maximum of 7 dB compression is
possible. If we assume that under long term, high
power operation the voice coil temperature
reached 300 C, then less than a 2dB
compression from the drivers reference
efficiency would be observed. To achieve
another 2 dB compression during dynamic
transients would require the temperature
reaching over 600 C ( 1110 F), which is getting a pretty good red glow on. The melting point of aluminum, used as VC
formers in many tweeters, is 637 C. Additionally, checking into the characteristics of
Ferrofluids used in audio indicates
that their application should be limited to temperature transients  in the range of 200 C and longer exposure above 110 C
should be minimized. Thus it would seem unreasonable to expect a ferrofluid tweeter to operate at temperatures above
these  limits.

Including Cooling

When cooling is included, the equation for the time rate of change in voice coil temperature can be expressed as

(ρ cp V) dΔT/dt = E^2 / (Reo x (1 + a ΔT )) - k A (Tc - Ts) / wg - σ A ε (Tc^4 – Ts^4)

The first additional term represents heat lost by conduction: k in the thermal conductivity of the material in the gap and A
the effective surface area for conduction, Tc is the voice coil temperature, Ts the temperature of the heat sink (motor
structure, etc) and wg is the gap width. The second additional term represents heat lost by radiation. Here, σ Stefan-
Boltzmann constant, A is the surface area of the voice coil, and ε is the emissivity of the voice coil. This form of the
radiation loss is appropriate for a convex object (the voice coil) surrounded by a large concave surface and represents
an upper limit of heat rejection by radiation. Examining the radiation term under the assumption that Tc is on the order of
200 C and Ts = 20 C shows that radiation effects are very small and can be dropped. Thus the equation for the voice coil
temperature can be expressed as

(ρ cp V) dΔT/dt = E^2 / (Reo x (1 + a ΔT )) - k A (Tc - Ts) / wg

It is recognized that the heat conduction term is actually much more complex that the simple terms presented here.
However, it is reasonable to assume that the heat rejection (cooling) is dominated by conduction across the gap, both to
the motor structure and pole piece. Additionally, since the thermal conductivity of the motor structure and pole piece is
typically much greater than that of the material in the gap (air or Ferrofluid), and since the thermal mass motor structure
and pole piece is much greater than that of the voice coil, it is also reasonable to assume that the temperature of the
sink, Ts, is constant and at room temperature. In reality Ts would increase over time, but at time scales much, much
longer than those associated with the heating and cooling cycles of the voice coil due to transients, or even under longer
periods. For example, the motor will heat up over hours compared to the voice coil which will response in seconds.

Never the less, it is difficult to model the details of the heat rejection terms without detailed knowledge of the mechanical
structure of the motor and without resorting to highly complex modeling approaches. However, it is not necessary to know
such details to understand the behavior. What is needed is some knowledge of the maximum allowable voice coil
temperature under sustained conditions. Fortunately, many driver manufactures provide specifications for the thermal
power limits of their drivers. For example, the maximum long term power of the SEAS 27TDFC tweeter is given as 90
watts. 90 watts across 4.8 ohms is about 20.8 volts.  Other insight can be gained for the ferrofluid specifications which
recommend long term temperature not exceeding 100 C, transients of 200 C, boiling point somewhere above 260 C.
Since the ferrofluid is in direct contact with the VC these would seem to translate to the range of acceptable VC temps.
Knowing this, the equation for the voice coil temperature under long term steady state conditions can be expressed as:

E^2 / (Reo x (1 + a ΔT )) = k A (Tc - Ts) / wg

If it is further assumed that Ts = To = room temperature


E^2 / (Reo x (1 + a ΔT )) = k A ΔT / wg

Since ΔT = (Tcmax - To), Reo and E are known it is possible to find

k A / wg = E^2 / (Reo x (1 + a ΔT )) / ΔT

The quantity, (k A / wg), is the effective convective heat transfer coefficient for the driver. While this does not provide
details of the heat transfer process, it does provide a reasonable estimate of the rate of heat transfer to the surrounding
from the voice coil and allows solution to the transient heating/colling problem. Additionally, various solutions can be
made assuming different values for the maximum allowable voice coil temperature to study what effect this would have on
the behavior of transients and dynamic thermal compression.

Dynamic results for a tweeter:
Here the max VC temp is
assumed to be allowed to
reach 220 C, a 200 C
increase form room temp, 20
C, when subject to the
equivalent of 100W/8 ohms. It
shows the rise takes about 1/2
sec and cooling takes about
0.75 sec.
Next, the max VC temp was
allowed to reach 320 C. Here
the rise of 320 degrees takes
about 1 sec and cooling takes
about 1.5 sec.
Lastly, the VC temp is
allowed to reach 400 C. The
rise takes about 1.25 sec
and cooling takes 1.9 sec (or
slightly less).
From the Ferrofluid specs it would be expected that max VC temps in excess of 200 degrees C are not anticipated under
normal operation. But note that in all cases the ratio of fall to rise is about 1.5. That is, it takes roughly 1.5 times as long
to cool as it does to heat up. Also note that since when the VC starts to heat up the initial rate is the same in all cases
(you can not cool something that isn't hot). So what is different in these cases that when the max temp is limited to lower
values the rate of heating decreases faster during the heating cycle for lower allowed max temp.

Note that these results have considered only that 100w/8 ohms was applied to the tweeter. The next series of figures
look at the effects of frequency and how heating affects distortion. The temperature is allowed to come the quasi-steady
value determined as the temperature it would reach if the power were delivered at DC. Then the effect of frequency on
the oscillation of T about the quasi-steady value  and the harmonic distortion in the current flowing through the VC are
examined. The frequencies are chosen to show the ability of the temperature to follow the input power and are not
intended as realistic frequencies to drive the tweeter.
10 Hz: The temperature is able to follow the power to some degree but the thermal response lags the power. As a result, the VC resistance and
driver sensitivity also follow temperature. While there is no change in amplitude form cycle to cycle the plot to the right shows that there is harmonic
distortion generated due to the temperature fluctuations. Only odd order harmonics are present indicating that the temperature variation is
symmetric about the mean level. 3rd odder HD runs about 1%.
100 Hz: With the frequency increased to 100 Hz the temperature variation shows that it is just about constant. The frequency is sufficiently high
that the VC temperature can not follow the power variation and the temperature remains very nearly at the quasi-steady value. Sensitivity and VC R
thus remain constant as well. Thus, while the VC heating results in compression from the cold Re value, there is no significant dynamic effect on
SPL. However, even though the dynamic compression is not significant, there are distortion components generated, as shown to the right. In this
case the 3rd order HD is about 0.1%, a factor of 10 reduction with the factor of 10 increase in frequency.
1 k Hz: At 1 k Hz there are no variation in any parameter visible on the scale of the figure. However, the distortion plots show that there is still
some small variation with 3rd order HD at a level of 0.01%, another 10 fold reduction with at 10 fold increase in frequency. Obviously, increasing
the frequency to 10 K Hz would result in another order of magnitude reduction in HD.
Amplitude Modulated: The next figure is a little more complex. In this case the input signal is a 1 k Hz amplitude modulated at 125 Hz. The
figure shows that there is some small dynamic effect on temperature but nothing that would likely be audible in terms of SPL. However, the
distortion plot tells a different story. In this figure the red diamonds with black out line represent the harmonic content of the output current.
Notice the two dots at 875 and 1125 Hz have black centers. The black dots represent the harmonic content of the input signal (sum and
difference. 1000 +/- 125). Thus the remaining dots w/o black centers represent distortion arising from thermal effects. Note also in this figure
that the temperature level is overall lower that that for a pure sine wave. This is as expected as the mean power is also lower although the peak
power remains the same.
The simulations presented above lead to the conclusion that dynamic compression in a typical hifi tweeter, in the sense that a sharp
transient will be reproduced at a reduced SPL due to VC heating by the transient signal, is not significant. Rather the VC temperature
will follow a meandering values associated more closely with the time variation of the mean power level over a much longer time
scale. However, distortion associated with the transients may have components which are highly dependent on the actual form of the
signal. Of course, there are those would will dismiss the above arguments and simulations since they do not represent music. In that
regard the next series of figures show the results of simulation using sampled music signal as the driving function. In all cases the
simulations were started at a VC  temperature representative of what would be expected if the driver were reproducing this music for
an extended period of time. In all cases the signal was sampled after passing through a Linkwitz/Riley 1k Hz, 4th order high pass filter,
thus representing the signal that would reach the tweeter. Each result is preceded by the name of the source material. All sample
were made at a rate of 61440/sec. The tweeter is assumed to be a typical Hi Fi tweeter. Levels have been set arbitrarily and do not
necessarily represent true playback levels.
Soft Rock (Steve Winwood, Higher Love)
For this case the temperature (red
line) follows the average power level. A
spike equivalent to 440W shows no
significant effect on Re or sensitivity,
but would likely be compressed in a
home environment due to limited
amplifier power.
Jazz (Earl Klugh, Across the Sand)
Again, there is no sign of any dynamic
thermal compression. The tweeter VC
is not capable in thermally following the
variations in input power.
Chopin, Polonaise
The results remain similar.  
Symphonic, (Stravinsky, The Rite of Spring)
The results remain similar.  


All the results show that the VC temp
does not follow the transient, but
rather response slowly to changed in
the mean power level. There is no
short term dynamic compression due
to thermal effects.
The figure to the left shows the thermal time
constant for the simulated tweeter as a function of
the maximum allowable VC temperature. As
previously stated, the properties of audio ferrofluids
would seem to dictate that the max VC temperature
would be in the 200 C range with a time constant
about 0.146  sec. This would correspond to around
7 Hz. Note that to cool by 5% takes a time of 0.05
time constants which at 200 degrees would be
7.3msec or 135 Hz. A 5% change in temperature
for 200 degrees C would be only 10 degrees and a
change in VC resistance of 3.9% or Re. Higher
allowable VC temperature requires longer time
constants.
The above  analysis and sample simulation show that the typical Hi Fi driver should not suffer from dynamic compression arising from
thermal effects. While the model includes an estimated rate of cooling term, cooling is not to dominate aspect. Faster cooling would result in
less heating. If the cooling rate approached infinity, then the VC would never get hot. The rate of heating is determined by the power
dissipated in the VC, the mass of the VC and the heat capacity of the VC, all of which can be well defined. The maximum rate of heating
always occurs at the onset of a transient. Thus if the rate of heating at the onset of a transient is insufficient to generate a significant
increase in temperature even if maintained over the duration of the transient, then it would not be possible for such a transient to suffer
thermally generated dynamic compression. The last figure below shows the maximum rate of temperature increase for an incremental
change in power for the simulated tweeter. For example, if the system were suddenly subjected to an increase in applied power of 100 watts,
then the maximum rate of temperature increase would be about 1.5 C / msec. This maximum rate would only exist at the instant the power
level changed and after that the rate of increase would decrease with the same time constant as the cooling. In other words, if the cooling
time constant is 0.15 sec, then the heating rate would drop decrease by 36% after 0.15 sec of sustained power application. While cooling
plays a roll in reducing the rate on increase after the application of elevated power, the primary roll of cooling is to limit the maximum
temperature the voice coil may obtain under sustained operation with it its  rate power limits.