Copyright © 2013 by Wayne Stegall
Updated August 23, 2016. See Document History at end for
details.
Thermal Design
Part
2:
Thermal design for onebend amplifier is used to illustrate
theory advanced to include thermal capacitance.
Introduction
Because datasheets ordinarily only specify thermal resistance for
semiconductors, it would seem that thermal design would stop
there. However, there is another factor involved. Matter
stores heat like a capacitor stores charge. Presumptions that
average power dissipation would predict junction temperature also
presumes thermal mass in the entire system high enough to smooth out
all variations of temperature. Certainly, with real thermal
masses the junction temperature would be expected to vary to some
degree with the amplified signal. Therefore, it is sometimes
desirable to include thermal capacitance in a SPICE simulation.
Thermal Capacitance
Because thermal resistance is defined in units of ºC/W, compatible
units must be used for thermal capacitance for RC calculations of
thermal circuits to be valid. First consider the thermal
equivalent of the
capacitor equation.
(1)

P_{DISS} = C_{θ}

dT
dt

If it is only desired to establish comparable units at this point,
solve for C then substitute known units.
(2)

C_{θ} =

P_{DISS}dt
dT 
gives units of

W⋅s
ºC 
or

J
ºC 
Also because Kelvin and Celsius have the same sized units, K can be
substituted for ºC here.
Specific Heat
Thermal capacitance is not often specified for objects of
interest. However, it can be calculated from a known property of
materials: specific heat. Specific heat is thermal
capacitance per unit weight and in our system is in units of
If specific heat is given in other units a conversion must be
made. For example, I found the specific heat of aluminum given as
(3)

0.214

cal
ºC⋅g 
, which is the same as 0.214 
kcal
ºC⋅kg 
Then conversion from kilocalories to joules requires multiplication by
a constant varying from 4.18 to 4.2 depending on the exact branch of
science involved.
(4)

Specific heat_{Al} = 4.2 
J
kcal

× 0.214 
kcal
ºC⋅kg 
= 0.8988

J
ºC⋅kg 
Once specific heat is known, thermal capacitance can be calculated by
multiplying by mass.
(5)

C_{θ} = specific
heat × mass

Example SPICE Thermal Simulation  Onebend Amplifier
Creation of model
Because I expected junction temperature of power MOSFETs would vary
with the amplified signal, I decided that the onebend amplifier needed
a more
thorough SPICE simulation of the thermal design. Fortunately, I
found that Vishay had published thermal models of their IRFP240 MOSFETs
in a separate datasheet.
Figure
1 below shows the model I chose to use of the two presented.
Figure
1:
Thermal
model
for
IRFP240
and
presumed
for
IRFP9240
as
well.


RF1 

60.8721mΩ 
CF1 

1.5231mF 
RF2 

125.9014mΩ 
CF2 

6.5766mF 
RF3 

295.8460mΩ 
CF3 

11.5740mF 
RF4 

347.3805mΩ 
CF4 

103.6350mF 
Next I looked at heat sinks. One promoted for 0.1ºC/W with
natural
convection turned out instead to be for forced cooling instead.
Therefore it would be underweight at 0.9kg for one using natural
convection. As a result, I estimated a hypothetical weight for a
natural convection heat sink to be a minimum of 6.4kg. Calculate
thermal capacitance for this mass.
(6)

C_{θSA} = specific
heat × mass = 
0.9

J
ºC⋅kg 
× 6.4kg = 5.76

J
ºC 
Figure 2 below combines this
heat sink data with that given for the MOSFETs.
Figure
2:
Full
thermal
model


Note:
Z_{θJCx}'s represent entire MOSFET thermal
model from figure 1 above 
For the actual SPICE model, I modeled the MOSFETs paralleled into
groups of four by multiplying the capacitances and dividing the
resistances each by four to produce a combined model for each parallel
NMOS and PMOS bank.
Figure
3:
SPICE
thermal
model 
* thermal circuit of 8 MOSFET
follower stage
* set ambient temp
v1th ta 0 dc 25
* calculate instantaneous power dissipation
b1th 0 tjp i=abs((i(vpos2:x1))*(v(vdd2)v(vout)))
b2th 0 tjn i=abs((i(vneg2:x1))*(v(vout)v(vss2)))
* lumped model of positive NMOS bank
xthpjc tjp tcp ta thirfp240_4
rthpcs tcp ts 60m
* lumped model of negative PMOS bank
xthnjc tjn tcn ta thirfp240_4
rthncs tcn ts 60m
* heat sink model mass=6.4kg
rthsa ts ta 0.1
cthsa ts ta 5.76
* individual transistor model included for reference
.subckt thirfp240 tj tc ref
rf1 tj t1 60.8721m
cf1 tj ref 1.5231m
rf2 t1 t2 125.9014m
cf2 t1 ref 6.5766m
rf3 t2 t3 295.8460m
cf3 t2 ref 11.5740m
rf4 t3 tc 347.3805m
cf4 t3 ref 103.6350m
.ends
* lumped transistor model actually used
.subckt thirfp240_4 tj tc ref
rf1 tj t1 15.218m
cf1 tj ref 6.0924m
rf2 t1 t2 31.4753m
cf2 t1 ref 26.3064m
rf3 t2 t3 73.9615m
cf3 t2 ref 46.296m
rf4 t3 tc 86.8451m
cf4 t3 ref 414.54m
.ends

The thermal model was included in the overall amplifier model and the
following analyses were run.
SPICE
amplifier
model
SPICE
thermal
model
SPICE
power
supply
model
Simulation results
Figure 4: Operating point
analysis. 
ta = 25
tcn = 57.51615
tcp = 57.51632
tjn = 83.46631
tjp = 83.46706
ts = 50.01249

Figure 5: Transient analysis at zero output shows smooth rise to
temperature.. 

Figure 6: Transient
analysis driving one watt into 8Ω.


Figure 7: Transient
analysis driving 30W into 8Ω.


Figure 8: Transient
analysis driving 60W into 4Ω. 

Comments on simulation
The results turned out well. Peak junction temperatures only
reached 91ºC, only 7.5 degrees higher than the 83.5ºC operating point
without an input signal. The 50ºC heat sink temperature that
resulted was fortunate because that is about the highest temperature
that you can hold with your hand without feeling that the metal will
burn
you. Perhaps it should have been a design objective as well in
addition to junction temperatures well below the 150ºC maximum.
Document History
August 17, 2013 Created.
November 27, 2015 Corrected some wording.
August 23, 2016 Restored missing subfile to SPICE model.