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Copyright © 2013 by Wayne Stegall
Updated April 20, 2015. See
Document History at end for details.
(1) 
Thermal Resistance (ºC/W) =  Temperature (ºC) Heat Flow (W) 
(2) 
P_{OUT} = 
V_{OUTPEAK}^{2}
2R_{L} 
(3) 
V_{OUTPEAK} ≤ V_{CC} – V_{DO} 
(4) 
P_{IN} = V_{CC} × I_{OUTAVE} 
(5) 
P_{IN} = V_{CC}  2
π 
I_{OUTPEAK} 
(6) 
P_{IN} =  2
π 
×  V_{CC} × V_{OUTPEAK}
R_{L} 
(7) 
Eff = 
P_{OUT}
P_{IN} 
= 

(8) 
Eff =  π
4 
×  V_{OUTPEAK}
V_{CC} 
(9) 
P_{DISS} = P_{IN} – P_{OUT} =  2V_{CC} × V_{OUTPEAK}
πR_{L} 
–  V_{OUTPEAK}^{2}
2R_{L} 
(10) 
P_{DISS} =  4V_{CC} × V_{OUTPEAK}
– V_{OUTPEAK}^{2}
2πR_{L} 
V_{CC}  
28V 

P_{OUT}  
68W 
T_{JMAX} 
150ºC 
R_{θJC}  1ºC/W 
(11) 
V_{OUTPEAK} =  2P_{OUT}R_{L}  = 
2 × 68W × 4Ω  = 23.3238V 
(12) 
P_{IN} =  2
π 
×  V_{CC} × V_{OUTPEAK}
R_{L} 
= 
2
π 
×  28V × 23.3238V
4Ω 
= 103.939W 
(13) 
P_{DISS} = P_{IN} – P_{OUT} = 103.939W  68W = 35.9387W 
Figure
1:
Thermal
circuit representing one semiconductor device mounted
to a heat sink 
(14) 
T_{J} – T_{A} = P_{DISS}R_{θJC} + P_{DISS}R_{θCS} + P_{DISS}R_{θSA} 
(15) 
R_{θSA} =  T_{J} – T_{A} – P_{DISS}(R_{θJC} + R_{θCS})
P_{DISS} 
(16)  R_{θSA} =  150ºC – 25ºC – 35.9387W(1ºC/W +
0.2ºC/W)
35.9387W 
(17)  R_{θSA} =  2.27814ºC/W 
(18)  R_{θSA} =  150ºC – 50ºC – 35.9387W(1ºC/W +
0.2ºC/W)
35.9387W 
(19)  R_{θSA} =  1.58252ºC/W 
Figure
2:
SPICE
model of thermal circuit of figure 1 
* Thermal circuit
for LM3886 gainclone * SPICE version: Spice Opus 2.31 i1 0 tj dc 35.9387 rthjc tj tc 1 rthcs tc ts 0.2 rthsa ts ta 1.58252 v1 ta 0 dc 25 .end .control op print all .endc 
Figure 3: SPICE output for model of figure 2. 
ta = 2.500000e+001
tc = 8.906145e+001 tj = 1.250002e+002 ts = 8.187371e+001 v1#branch = 3.593870e+001 
(2) 
P_{OUT} = 
V_{OUTPEAK}^{2}
2R_{L} 
(20) 
P_{IN} = 2V_{CC} × I_{BIAS} 
(21) 
If I_{BIAS} = 
V_{OUTPEAK}
2R_{LMIN} 
(22) 
P_{IN} =  V_{CC} × V_{OUTPEAK}
R_{LMIN} 
(23) 
Eff = 
P_{OUT}
P_{IN} 
= 

(24) 
Eff =  V_{OUTPEAK} × R_{LMIN}
2V_{CC} × R_{L} 
(25) 
P_{DISS} = P_{IN} – P_{OUT} 
(26) 
P_{DISSMAX} = P_{IN} 
(27) 
P_{DISSMIN} = P_{IN} – P_{OUTMAX} 
P_{D} 

150W @ T_{C} = 25ºC  
T_{J}  
–55ºC – 150ºC 
R_{θJC}  0.83ºC/W  R_{θCS}  0.24ºC/W, flat surface with
thermal grease 
(28) 
P_{IN} = 2V_{CC} × I_{BIAS} 
(29)  P_{IN} = 2 × 35V × 4A 
(30)  P_{IN} = 280W 
(31) 
P_{DISS} = P_{IN} = 280W 
Figure 4: Thermal circuit representing four semiconductor devices mounted to a heat sink 
(32) 
P_{DISS/DEVICE} = P_{IN}/#devices 
(33)  P_{DISS/DEVICE} = 280W/4 devices 
(34)  P_{DISS/DEVICE} = 70W 
(35) 
T_{J} – T_{A} = P_{DISS/DEVICE}R_{θJC1} + P_{DISS/DEVICE}R_{θCS1} + (#devices × P_{DISS/DEVICE})R_{θSA} 
(36) 
R_{θSA} =  T_{J} – T_{A} – P_{DISS/DEVICE}(R_{θJC1} + R_{θCS1})
#devices × P_{DISS/DEVICE} 
(37)  R_{θSA} =  150ºC – 50ºC – 70W × (0.83ºC/W +
0.24ºC/W)
4 × 70W 
(38)  R_{θSA} =  0.0896429ºC/W 
Figure 5: Thermal circuit representing eight semiconductor devices mounted to a heat sink 
(39) 
P_{DISS/DEVICE} = P_{IN}/#devices 
(40)  P_{DISS/DEVICE} = 280W/8 devices 
(41)  P_{DISS/DEVICE} = 35W 
(42) 
R_{θSA} =  T_{J} – T_{A} – P_{DISS/DEVICE}(R_{θJC1} + R_{θCS})
#devices × P_{DISS/DEVICE} 
(43)  R_{θSA} =  150ºC – 50ºC – 35W × (0.83ºC/W +
0.24ºC/W)
8 × 35W 
(44)  R_{θSA} =  0.223393ºC/W 
Figure
6:
SPICE
model of thermal circuit of figure 5 
* Thermal circuit
for eight transistor class a output stage. * SPICE version: Spice Opus 2.31 i1 0 tj1 dc 35 i2 0 tj2 dc 35 i3 0 tj3 dc 35 i4 0 tj4 dc 35 i5 0 tj5 dc 35 i6 0 tj6 dc 35 i7 0 tj7 dc 35 i8 0 tj8 dc 35 rthjc1 tj1 tc1 0.83 rthcs1 tc1 ts 0.24 rthjc2 tj2 tc2 0.83 rthcs2 tc2 ts 0.24 rthjc3 tj3 tc3 0.83 rthcs3 tc3 ts 0.24 rthjc4 tj4 tc4 0.83 rthcs4 tc4 ts 0.24 rthjc5 tj5 tc5 0.83 rthcs5 tc5 ts 0.24 rthjc6 tj6 tc6 0.83 rthcs6 tc6 ts 0.24 rthjc7 tj7 tc7 0.83 rthcs7 tc7 ts 0.24 rthjc8 tj8 tc8 0.83 rthcs8 tc8 ts 0.24 rthsa ts ta 0.223393 v1 ta 0 dc 25 .end .control op print all .endc 
Figure 7: SPICE output for model of figure 6. 
ta = 2.500000e+01 tc1 = 9.595004e+01 tc2 = 9.595004e+01 tc3 = 9.595004e+01 tc4 = 9.595004e+01 tc5 = 9.595004e+01 tc6 = 9.595004e+01 tc7 = 9.595004e+01 tc8 = 9.595004e+01 tj1 = 1.250000e+02 tj2 = 1.250000e+02 tj3 = 1.250000e+02 tj4 = 1.250000e+02 tj5 = 1.250000e+02 tj6 = 1.250000e+02 tj7 = 1.250000e+02 tj8 = 1.250000e+02 ts = 8.755004e+01 v1#branch = 2.800000e+02 

Document History
July 12, 2013 Created.
July 12, 2013 Corrected R_{θSA} in SPICE model of
figure 6, regenerated corresponding SPICE output, and made minor
improvements to wording.
April 20, 2015 Corrected some misspellings.