FIGURE 4. Using Equations 4-6, we
calculate the thermal resistance in a
1 x 1 cm PCB with 2. 5 oz Cu model in
the stack-up. The horizontal Rtherm =
20 °C/W and vertical Rtherm = 31 °C/W.
The vertical Rtherm includes the
thermal resistance of the PCB surface
to the air ( 28 °C/W). Empirical data
was used to obtain this value and
matched to simulated values.
References
[1] LTSpice/SwCad III. www.linear.com/
designtools/software.
[ 2] Azar, K. and Graebner, J.E.
“Experimental Determination of Thermal
Printed Wiring Boards.” IEEE Semi-Therm
Symposium. 1996.
power from all resistors (Equation 3) is
fed into the thermal square model. Then,
we simulate the model and are able
to probe the voltage points which
correspond to temperate rise.
Results/Conclusion
EQUATION 5:
κ =⎛ ⋅⎛ Tk ⎜⎜ 3. 23⎜⎜1− Tk ⎞−1 CU ⎞ ⊥ ⎝ ⎝ Tk ⎟⎟+0.0026⋅ CU Tk ⎟⎟ PCB⎠ PCB⎠
The thermal resistivity is calculated by:
EQUATION 6:
R l therm = SA ⋅ κ
Similar to the resistive square model,
a thermal resistance square model
is applied using LTSpice. The thermal
resistances are plugged into the model as
shown in Figure 4, and a current source
is used to inject the analog to the power
dissipated in the electrical square, where
mA equals mW, ohms correspond to
thermal resistance [°C/W], and voltage
corresponds to temperature. We use a
dependent current source where the
With a 50A current source, the
simulation models a temperature rise
of 10-11°C near the current source. As
the heat spreads through the board,
the heat dissipates, dropping to 3-4°C
at the right edge. This is a very
comforting number and we’ll take it!
Depending on your design, the results
can help determine parameters like
stack-up, board area, and copper size.
Our simulations by no means
replace costly analysis tools as the model
makes basic approximations about the
thermal resistivity and neglects other
parameters like air flow (e.g., fans) and
vias. While the model doesn’t use fancy
dynamic models, you can create and
model arbitrary size boards and different
loads. With the added advantage, you
can do this quick analysis for free. NV
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SPICE (Simulation Program with
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64
November 2007
