1. Fine trace limit 2. Broad trace limit
The heat flux from the trace is spread (in x-direction) by the For broad traces on low conductivity boards, Region I is
board and the board will do the bulk of the total heat transfer. dominating (note RI is independent of k)
The classical heat sink fin formula [9] for a homogenously 1 1
Rtotv h w , (14)
heated thin fin with (cf. Fig. 6) fin height B/2, fin length L
1 1 1 2v 1 2 1 2
'T v h 1w w t I h w t I , (15)
and fin thickness D applied to Regions IIa and IIb is giving as
thermal resistance Iv h1/2w 1t/2'1T 12/ . (16)
1 1 1 For high conductivity boards the width of the board B should
R (8a)
II 2 kmDL B replace w in Eq. (14) with the corresponding consequences of
tanh(m )
2 proportionalities in I and 'T.
with To verify Eq. (10), disregarding form factors, we perform
some more parameter variations, as shown in Fig. 8. Within a
h
m 2 . (8b) factor of Š2 the fine trace limit and the broad trace limit is
kD
fulfilled.
The first factor 1/2 in Eq. (8a) is because the resistances of
Regions IIa and IIb are parallel.
1. Broad trace limit
For broad traces, which cover most of the board, Region I is
dominating. The thermal resistance is given by the area of the
footprint and the heat exchange coefficient according to
Newton’s cooling law
1
R 1
I . (9)
2 hwL
The factor 1/2 is because the resistances of the trace on top
face and the thermal image of the footprint at the bottom face
are parallel.
2. The total thermal resistance finally is (approximately)
1 1 1
. (10)
Rtot R R
I II
Equations (8a,b), (9) and (10) are plotted as lines into Fig.
7 for the parameters given in the legend. The agreement is
well regarding the crude mathematical modelling. The
deviations between the points and the lines are of the order of
Figure 8: Testing the scaling laws by parameter variations.
Š2, reflecting uncertainties in defining fin cross-section and
fin surface area for Eqs. (8a) and (8b): Assuming, that only
4.3. Thermal resistance of the traces on board models
the top face of a bad heat spreading board will dominantly
The board models of Sect. 4.1 are not as homogeneous as
heat and cool, then RII would be about Š2 times larger.
they are in Sect. 4.2 but have some thermal isolation between
4.2. Scaling laws trace and first heat spreading copper layer. Moreover, Joule
Strict analytic solutions, although with necessary heating and external heat transfer is not independent of
simplifications, of the 2D heat conduction equation in form of temperature or power. In order to test the findings of Section
Fourier series are given by Ling [10]. However, the 4.1 we have re-plotted in Fig. 9 some graphs from Table 1,
dependence of 'T(I) on h, k, D and w is hidden behind but with ordinate Rtrace='T/P as function of trace width. The
complicated equations. Our simplified approach is a much power P is taken from Eqs. (3) and (4) with T as the mean
easier route. trace temperature.
1. Fine trace limit In Fig. 9 there are two observations to be made. First,
there is a bigger scatter in the data and, second, boards 2,4,6,
For fine traces where Region II is dominating (note RII is
which have the same amount of copper, are shifted in
independent of w), the thermal resistance of the trace is
ordinate. The scatter is certainly partially due to the non-
1/2 /21 1/2
Rtotv h k D . (11)
constant heat transfer, e.g., at the left end of the abscissa at
If we introduce Eq. (1) into 'T=Rtot*P together with Eq. (3) w=10 mm, the lower values of R correspond to the high
and (4), the desired scaling laws are simply currents (high temperatures). In Appendix A we derive some
' T v 1/2 /21 1/2 1 1 I2 (12) approximate effective heat transfer coefficient heff as function
h k D w t
of power density q related through the expression heff=9 q0.14.
or
According to Eq. (11), the thermal resistance should scale
/4 1/4 1/4 1/21/2 12/
I v h1 k D w t ' T . (13)
with h-1/2, namely as q-0.07, namely as P-0.07.
Adam, New Correlations Between Electrical Current and … 20th IEEE SEMI-THERM Symposium
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