Music and Design TM

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Dynamic compression due to Voice Coil Heating,

Fact or Fiction?

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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 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.

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.

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.

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).

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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:

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.

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.

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).

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.

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%.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.