the surfaces. The board has a square shape (L=100 mm,
10 B=100 mm, D=1.6 mm), so that we have a 2D situation (Fig.
6).
The thermal resistance R of the trace based on the mean
trace temperature T, ambient temperature Ta and the power
input shall be
R=(T-Ta)/P . (7)
Table 5: Mean Temperature of a trace (in 20 degC ambient)
as function of electrical current for traces on a thin polyimide
foil. Parameter is trace width from 0.2 mm to 10 mm.
3.3. Trace thickness other than 35 μm
The calculated diagrams above (as well as we believe it to
be the case for those in IPC-2221) are valid for a trace
thickness t=35 μm only. For a given PCB structure, the
temperature of the trace is determined by the power and the
footprint of the trace. If we double the thickness and increase Figure 6: 2D-like test geometry with constant properties.
the current by Š2, we deposit the same power (see Eqs. 3 and
4) and obtain the same temperature rise, provided the trace
Fig. 7 is showing the dependence of R as function of trace
width w remains the same. This scaling law for trace
width w. The trace is always centred with respect to the board.
thickness t other than 35 μm can be written as
At w=100 mm, the trace is of same width as the board. The
1 ŸI352 μm 1 I2 plate is either orthotropical or isotropical conducting with
Ÿ 2. (5)
35 μm w tŸw values indicated in the graph and notation of directions as in
Fig. 6. The data points in Fig. 7 are independent of power
The l.h.s. is known data from the diagram, the r.h.s. is the
input P.
desired combination of t (in μm) and I. Of course, t has to be
reasonably small, so that the trace can be considered as thin
trace. We have verified Eq. (5) by numerical simulations.
4. Interpretation of the results and scaling laws
4.1. Trace heating with constant properties
The style of the diagrams in Tables 1,3 and 5 was chosen
as to hand over them to layouters in an easy-to-use form.
From a thermal analysis point of view they should be plotted
in a different way. First, the almost parabolic shape is likely
to reflect Joule’s P=RelŸIē law, so we need to change over to
the power, to see deviations from the parabola. Second,
temperature can be included by plotting the thermal resistance
R (in K/W)
R='T/P (6)
on the vertical axis (ordinate). Third, trace width as parameter Figure 7: Numerical results for 2 pairs of constant board
should appear as independent variable on the horizontal axis properties, compared with semi-analytical equations.
(abscissa). Fourth, the conducting properties of the
For a better understanding of the results of Fig. 7, we
board/substrate should be the independent parameter.
adopt a procedure from Guenin [8]. The plate is divided into 2
To identify the scaling laws for a trace-like heated plate,
regions: the trace (Region I) and the board around it (Region
we prepare a simplified numerical computational test
II). For Region I, we assume Newton’s cooling law for the
environment which is free of temperature dependent cooling
heat flux from the footprint of the trace. For Regions IIa and
and heating effects. We apply a homogenously distributed
IIb we interpret the physical situation as cooling of the trace
fixed power in the trace (L=100 mm), allow for heat
by a heat sink fin. We should consider it as first-order
conduction only and define a cooling heat flux by Newton’s
estimate.
law with a fixed heat exchange coefficient h=10 W/mēK on
Adam, New Correlations Between Electrical Current and … 20th IEEE SEMI-THERM Symposium
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