Influence Coefficient Method for Calculating Discrete Heat Source
Temperature on Finite Convectively Cooled Substrates
Y.S. Muzychka†
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
St. John’s, NF, Canada, A1B 3X5
Phone: (709) 737-8944
Fax: (709) 737-4042
Email: yuri@engr.mun.ca
Ts
Tf
Xc , Yc
ABSTRACT
A simple method is developed for predicting discrete
heat source temperatures on a finite convectively cooled
substrate. The method is based on the principle of superposition using a single source solution for the mean
or maximum contact temperature of an eccentric uniform heat source on a rectangular substrate. By means
of influence coefficients, the effect of neighboring source
strength and location may be assessed. It is shown that
the influence coefficients represent localized thermal resistances, which must be weighted according to source
strength. For a system of N heat sources, there exists N
effects of source strength and position on any one heat
source. This includes a self effect (source of interest) and
N-1 influence effects (neighboring sources). The method
is developed for isotropic, orthotropic, and compound
systems. Convection in the source plane is addressed
for isotropic and orthotropic systems. Expressions are
developed for both mean and centroidal temperature.
βmn
δn
γ
θ
θ
θ̂
θs
κ
λm
φ
ζ
k, k1 , k2
m, n
N
Q
R
t, t1 , t2
=
=
=
=
=
=
† Assistant
linear dimensions, m
Fourier coefficients
Modified Fourier coefficients
influence coefficient, K/W
contact conductance or
film coefficient, W/m2 · K
thermal conductivities, W/m·K
indices for summations
number of heat sources
heat flow rate, W
thermal resistance, K/W
total and layer thicknesses, m
Professor
0-7803-8357-5/04/$20.00 ©2004 IEEE
p
eigenvalues, ≡ λ2m + δn2
eigenvalues, nπ/b
p
transform variable ≡ kz /kxy
temperature excess, ≡ T − Tf , K
mean temperature excess, ≡ T − Tf , K
centroidal temperature excess, ≡ T̂ − Tf , K
surface temperature excess, ≡ Ts − Tf , K
relative conductivity, k2 /k1
eigenvalues, mπ/a
spreading function
dummy variable, m−1
=
=
=
=
=
=
=
=
=
=
=
Subscripts
eff
i, j
xy
z
=
=
=
=
effective value
denotes the ith and j th sources
xy-plane
z-plane
Superscripts
NOMENCLATURE
=
=
=
=
=
surface temperature, K
sink temperature, K
heat source centroid, m
Greek Symbols
Keywords:
Conduction, Electronics Cooling, Heat
Spreaders, Compound Systems, Heat Sinks, Spreading
Resistance, Orthotropic Properties
a, b, c, d
A0 , Am , An , Amn
B0 , Bm , Bn , Bmn
fˆ, f
h, h1 , h2
=
=
=
(·)
ˆ
(·)
=
=
mean value
centroid value
INTRODUCTION
Calculation of the mean or centroidal value of discrete
heat sources on a rectangular subststrate is of interest
in electronic packages, circuit boards, and heat sink systems. In the simplest level of analysis, the total heat
dissipated from all sources may be lumped together and
evenly distributed over the substrate giving rise to the
lowest system temperature. Further refinements in analysis may be made by treating this lumped heat source
as a single discrete heat source with an area equivalent
to the total area of all heat sources. In this case, the
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discrete lumped formulation will usually give rise to the
highest system temperature, depending upon the distribution and size of individual heat sources. In most cases,
these two approaches will yield useful information for preliminary sizing of cooling systems. A more refined discrete heat source analysis is often desired to minimize
hot spots and evenly distribute heat flows. This paper
presents a simplified method of analysis for systems with
multiple discrete heat sources, which enables the effects
of neighboring source location and strength to be determined.
Presently, a number of methods are widely used for examining systems with multiple packages or heat sources,
see Fig. 1. These include the TAMS method of Ellison
[1], the GENPaK model of Culham et al. [2], and full numerical solutions using Finite Element Methods (FEM),
among others. The analytic methods are based on a
Fourier series solutions to Laplace’s equation in isotropic
or multilayered systems. They differ primarily in how
the local heat source is introduced into the analysis. In
the TAMS method, the source is specified through the
governing partial differential equation, whereas in the
GENPaK method, the heat source is specified through
the die plane boundary condition. The present method
is based on the latter approach with a significant simplification of the remaining boundary conditions and a
limitation on the number of layers. This simplified system was recently addressed by Muzychka et al. [3], who
obtained a managable Fourier series based solution for a
single eccentrically located heat source on an isotropic or
compound substrate, which is convectively cooled with
a uniform film coefficient or contact conductance. Using the principles of superposition, this solution may be
used for multiple discrete sources. The present analysis considers both heat loss through the sink plane and
die plane. The method of Ellison [1] and Culham et al.
[2] also allow for convective cooling in the die plane in
addition to the sink plane.
Heat Sources



Heat Sink
Fig. 1 - Multi-Source System [2].
kn, tn
kn-1, tn-1
..
.
k2, t2
k1, t1
In this work, the solution of Muzychka et al. [3] is
re-cast in terms of influence coefficients, which allow the
effects of neighbouring heat sources to be easily assessed.
A temperature for each heat source may be computed
in terms of these influence coefficients which shows that
the total temperature excess of any given heat source is
comprised of a self effect in addition to the sum of all
induced effects due to neighboring sources. Using the
influence coefficients it is shown that a unique thermal
resistance for each heat source cannot be defined in the
presence of other heat sources. The present approach
does allow for more efficient computation of heat source
temperatures.
LITERATURE REVIEW
A review of the literature reveals that several approaches for computing the thermal spreading resistance
and/or heat source temperature have been developed for
a rectangular substrate with single or multiple discrete
sources.
In the case of multiple heat sources several approaches
are found in the open literature. Hein and Lenzi [4] obtained a solution for an IC package using Fourier transforms. In their development, the heat source is specified
by means of a Poisson equation using a piecewise function to model discrete heat sources. Both the die plane
and sink plane are convectively cooled using uniform
heat transfer coefficients. Later, Kokkas [5] obtained a
Fourier/Laplace transform solution for a multi-layer substrate containing discrete heat sources. The substrate
base was assumed to be attached to a heat sink of fixed
temperture. Discrete heat sources were dealt with using
the die plane boundary condition. Ellison [1] developed
a method refered to as TAMS. This method is similar to
that of Hein and Lenzi [4], but considers multiple layers. More recently, Culham et al. [2] developed a three
dimensional Fourier series model for an electronic packaging system. There model is very general and allows for
the specification of a mixed boundary condition in the die
plane. Heat sources are specified through the boundary
condition in the die plane. Due to the complex nature
of the die plane boundary condition, numerical analysis
is required to complete the solution. In all of the above
methods significant effort is required to code the analysis.
In the case of a single discrete heat source, several
approaches are readily found in the open literature.
Kadambi and Abuaf [6] obtained steady and transient
solutions for a central heat source on an isotropic rectangular substrate which was convectively cooled in the
sink plane. Later, Krane [7] obtained a steady solution
for a similar system in which the sink plane is at a constant temperature. More recently, Yovanovich et al. [8]
and Muzychka et al. [3] obtained solutions for a compound convectively cooled rectangular substrate, containing a central and eccentric heat source, respectively.
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Finally, Muzychka et al. [9] extended these solutions to
orthotropic systems, while Muzychka et al. [10] obtained
a solution for a central heat source on an isotropic convectively cooled rectangular substrate with edge cooling.
where ζ is replaced by λm , δn , or βmn , accordingly. The
spreading parameter accounts for the effects of conductivity, thickness, and convection cooling.
MATHEMATICAL MODELLING
The system of interest in the present work is idealized
as a rectangular substrate which may be either isotropic,
orthotropic, or compound in nature. For the time being we will only consider an isotropic system, see Fig.
2. Later, the effects of adding a conductive layer to promote the spreading of heat, and system orthotropy will
be examined. In the present system all of the edges are
assumed be adiabatic, a reasonable assumption in many
electronics applications where edge area is significantly
less than the area of the source and sink planes. Finally,
there is no heat loss through the source plane, such that
all heat is dissipated through the sink plane by means of
a uniform film coefficient, i.e., thermal wake effects are
neglected. The addition of convection in the source plane
is dealt with in a separate section.
Single Source Solution
The single source solution of Muzychka et al. [3] for a
single eccentric uniform heat source on an isotropic substrate has the following form:
Fig. 2 - Single Eccentric Heat Source [3].
The final Fourier coefficients Am , An , and Amn were
obtained by taking Fourier series expansions of the
boundary condition in the source plane, z = 0. This
yielded the following expressions for the Fourier coefficients:
θ(x, y, z) = A0 + C0 z+
∞
X
Am cos(λm x) [cosh(λm z) − φm (λm ) sinh(λm z)] +
Am
=
h ³
´
³
´i
2Q sin (2X2c +c) λm − sin (2X2c −c) λm
=
h ³
´
³
´i
2Q sin (2Yc2+d) δn − sin (2Yc2−d) δn
a b c k1 λ2m φ(λm )
m=1
∞
X
An cos(δn y) [cosh(δn z) − φn (δn ) sinh(δn z)] +
An
n=1
∞ X
∞
X
Amn cos(λm x) cos(δn y)∗
[cosh(βmn z) − φmn (βmn ) sinh(βmn z)]
(1)
p
where λm = mπ/a, δn = nπ/b, and βmn = λ2m + δn2
are the eigenvalues. The origin of the coordinate system
is taken to be the lower left corner of the substrate.
The general solution contains four components, a uniform flow solution and three spreading (or constriction)
solutions which vanish when the heat flux is uniformily
distributed over the entire source plane, z = 0. The genral solution is a linear superposition of each component.
Application of the boundary conditions in the through
plane direction yields solutions for one half of the unknown constants and gives rise to the following expression for the spreading parameter φ:
ζ sinh(ζt1 ) + h/k1 cosh(ζt1 )
φ(ζ) =
ζ cosh(ζt1 ) + h/k1 sinh(ζt1 )
16Q cos(λm Xc ) sin( 21 λm c) cos(δn Yc ) sin( 21 δn d)
a b c d k1 βmn λm δn φ(βmn )
(3)
where Xc and Yc are the coordinates of the centroid of an
arbitrarily placed heat source with respect to the lower
left corner of the substrate as shown in Fig. 2.
Finally, values for the coefficients in the uniform flow
solution are given by
Amn
m=1 n=1
(2)
a b d k1 δn2 φ(δn )
=
A0 =
Q
ab
µ
t1
1
+
k1 h
¶
(4)
and
C0 = −
Q
k1 ab
(5)
Centroidal Source Temperature
The maximum or centroidal heat source temperature
may be determined from Eq. (1) when x = Xc , y = Yc ,
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and z = 0. This gives:
θ̂ = A0 +
∞
X
Am cos(λm Xc ) +
m=1
∞ X
∞
X
may be computed using Eq. (1) evaluated at the surface
∞
X
An cos(δn Yc )+
θi (x, y, 0) = Ai0 +
n=1
Amn cos(λm Xc ) cos(δn Yc )
∞ X
∞
X
(6)
Mean Source Temperature
The mean heat source temperature is obtained by integrating the local source temperature over the source
area, i.e.,
ZZ
1
θ=
θ(x, y, 0)dA
(7)
A
This leads to the following result for the mean temperature excess of a single eccentric heat source:
∞
X
m=1
∞
X
n=1
An
Am
Ain cos(δn y)+
n=1
Aimn cos(λm x) cos(δn y)
(10)
The Fourier coefficients are now evaluated at each of the
ith heat source characteristics, i.e. ci , di , Xc,i , Yc,i and
Qi .
Centroidal Source Temperature
The maximum or centroidal temperature is now the
sum of all heat source contributions at the point of interest. Thus, using Eq. (10) evaluated at the centroid of
the j th heat source, we may write
θ̂j =
N
X
θ̂i (Xc,j , Yc,j , 0) =
N
X
θ̂ij
(11)
i=1
i=1
where
4 cos(δn Yc ) sin( 21 δn d) cos(λm Xc ) sin( 21 λm c)
λ m c δn d
m=1 n=1
(8)
The results given by Eqs. (6) and (8) may now be
used to analyze systems containing multiple heat sources.
These expressions may also be used as a fundamental
surface element for analyzing irregularly shaped heat
sources, by discretizing the region into several rectangular strip sources.
θ̂ij = Ai0 +
Amn
MULTIPLE DISCRETE HEAT SOURCES
If more than one heat source is present (see Fig. 3), the
solution for the temperature distribution on the surface
of the circuit board, heat sink, or chip substrate may be
obtained using superposition. Both the centroidal and
mean heat source temperatures will be obtained for each
heat source.
Surface Temperature Distribution
For N discrete heat sources the maximum temperatures occur in the source plane. The surface temperature
distribution is obtained from
Ts (x, y, 0) − Tf = θs =
∞
X
m=1 n=1
2 cos(λm Xc ) sin( 21 λm c)
+
λm c
2 cos(δn Yc ) sin( 21 δn d)
+
δn d
∞ X
∞
X
Aim cos(λm x) +
m=1
m=1 n=1
θ = Ao +
∞
X
N
X
θi (x, y, 0)
∞
X
Aim cos(λm Xc,j ) +
m=1
∞ X
∞
X
∞
X
Ain cos(δn Yc,j )+
n=1
Aimn cos(λm Xc,j ) cos(δn Yc,j )
m=1 n=1
(12)
The present notation θij , denotes the effect of the ith
heat source in the region of the j th heat source.
Mean Source Temperature
The mean heat source temperature of an arbitrary
rectangular patch of dimensions cj and dj , i.e. the j th
heat source, located at Xc,j and Yc,j , may be computed
by integrating Eq. (7) over the region Aj = cj dj , i.e.,
1
θj =
Aj
1
θdAj =
A
j
Aj
ZZ
ZZ
N
X
θi (x, y, 0)dAj (13)
Aj i=1
which may be written as
θj =
(9)
ZZ
N
N
X
X
1
θi (x, y, 0)dAj =
θij
Aj
Aj
i=1
i=1
(14)
i=1
where θi is the temperature excess for each heat source
by itself. The temperature excess of each heat source
Using Eqs. (10) results in the following expression for
the mean temperature excess contribution of the ith heat
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source in the region of the j th heat source:
θij = Aio +
2
∞
X
n=1
Ain
which may be written as:
cos(λm Xc,j ) sin( 21 λm cj )
2
Aim
+
λm cj
m=1
∞
X
∞ X
∞
X
cos(δn Yc,j ) sin( 21 δn dj )
+4
Aimn ∗
δ n dj
m=1 n=1
θ̂j = Q1 fˆ1j + Q2 fˆ2j + · · · + QN fˆN j
(17)
N
X
(18)
or
θ̂j =
cos(δn Yc,j ) sin( 21 δn dj ) cos(λm Xc,j ) sin( 21 λm cj )
Qi fˆij
i=1
λm cj δn dj
(15)
Equation (14) represents the sum of the effects of all
sources over an arbitrary region cj dj . Equation (15)
is evaluated over the region of interest cj dj located at
Xc,j , Yc,j . The coefficients Ai0 , Aim , Ain and Aimn are then
evaluated at each of the ith source parameters.
where
fˆij = B0 +
∞
X
i
Bm
cos(λm Xc,j ) +
m=1
∞ X
∞
X
∞
X
Bni cos(δn Yc,j )+
n=1
i
Bmn
cos(λm Xc,j ) cos(δn Yc,j )
m=1 n=1
(19)
where
B0
i
Bm
Bni
INFLUENCE COEFFICIENT METHOD
The present results may now be used to define an influence coefficient. Influence coefficients were first proposed
by Negus and Yovanovich [11] for semi-infinite domains
and later applied by Negus et al. [12] and Negus and
Yovanovich [13,14] for multiple sources on a half space.
The concept of an influence coefficient for a finite substrate was partially addressed by Hein and Lenzi [4]. Influence coefficients offer an insightful assessment of the
effect of neighbouring heat sources on thermal resistance,
and hence the mean or centroidal temperature excess of
each discrete heat source.
We begin by examining Eq. (12) and Eq. (15), for the
centroidal and mean temperature excess θ. Beginning
first with Eq. (11) we may write
θ̂j = θ̂1j + θ̂2j · · · θ̂N j
µ
t1
1
+
k1 h
¶
=
h ³
´
³
´i
2 sin (2X2c +c) λm − sin (2X2c −c) λm
=
h ³
´
³
´i
2 sin (2Yc2+d) δn − sin (2Yc2−d) δn
a b c k1 λ2m φ(λm )
a b d k1 δn2 φ(δn )
16 cos(λm Xc ) sin( 21 λm c) cos(δn Yc ) sin( 21 δn d)
a b c d k1 βmn λm δn φ(βmn )
(20)
are modified Fourier coefficients, since Qi has now been
factored out. Once again it is noted that the coefficients
are evaluated at each of the ith heat source characteristics, i.e. ci , di , Xc,i , an Yc,i . Thus, the influence coefficients are only functions of the substrate properties and
dimensions and of heat source geometry and location.
Similarly, we may obtain an expression for the mean
temperature excess of the j th heat source using Eq. (14):
i
Bmn
Fig. 3 - Multiple Heat Sources [3].
=
1
ab
=
θj = θ1j + θ2j · · · θN j
(21)
which may be written as:
θj = Q1 f 1j + Q2 f 2j + · · · + QN f N j
(22)
N
X
(23)
or
(16)
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2004 Inter Society Conference
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where
f ij = Bo +
∞
X
m=1
∞
X
n=1
Bni
i
Bm
same as the potential at Xc,j , Yc,j due to a unit heat input
at Xc,i , Yc,i . This property also holds upon integration
over a finite region. The reciprocity of the influence coefficients was also observed by Negus and Yovanovich [11]
for semi-infinite regions.
2 cos(λm Xc,j ) sin( 21 λm cj )
+
λm cj
∞ X
∞
2 cos(δn Yc,j ) sin( 21 δn dj ) X
i
+
Bmn
∗
δ n dj
m=1 n=1
Thermal Resistance
Finally, if we consider defining a thermal resistance
Rj = θj /Qj , for each heat source, it can be shown that
4 cos(δn Yc,j ) sin( 21 δn dj ) cos(λm Xc,j ) sin( 21 λm cj )
λm cj δn dj
(24)
When i = j, the contribution is a self effect, i.e., the
effect of the source acting alone. When i 6= j the contribution to the temperature excess is an influence effect. The self effect fii , is merely the single source thermal resistance. The influence effects fij , are affected by
two factors: source strength and the location and size of
neighboring sources, i.e. a geometry effect. The influence
coefficients fij are clearly functions only of the location
of the neighboring sources.
Finally, we may write the temperature excess in the
following matrix form:

 


θ1 
f11 f12 · · · f1N
Q1




 θ2 
  f21 f22 · · · f2N   Q2 



 





θ3
=  f31 f32 · · · f3N   Q3  (25)





.
.
.
.
.
.

.. 
..
..
..   .. 


 ..







θN
fN 1 fN 2 · · · fN N
QN
or
{θ} = [Fij ][Q]
(26)
where Fij is the matrix of influence coefficients.
The influence coefficient method offers a number of
advantages. First, it becomes obvious what the effect a
neighboring heat source has on the thermal resistance of a
particular heat source. Examination of Eqs. (18) or (23)
reveals that an influence effect arises by virtue of proximity and strength. In otherwords, a remote and/or weak
heat source has little influence on another heat source.
Second, it can be shown that the influence coefficients
also possess reciprocity for the case when i 6= j,
fji = fij
(27)
This property significantly reduces computation for systems where more than five sources are present. In general, for a system of N sources, a symmetric N × N matrix results for the influence coefficients. As a result of
this symmetry only (N 2 +N )/2 coefficients need be computed. An upper triangular matrix is all that is needed
to compute the temperature excesses. Thus the influence method offers a substantial savings in computation
over the use of Eq. (9). The reciprocity is a result of
the property of Greens functions [15], i.e. the potential
at Xc,i , Yc,i due to a unit heat input at Xc,j , Yc,j is the
R̂j =
N
X
Qi ˆ
fij
Qj
i=1
(28)
Rj =
N
X
Qi
f
Qj ij
i=1
(29)
or
The above equations clearly demonstrate that the concept of thermal resistance is not strictly applicable in
multiple source systems, since the total resistance of any
given source depends on both proximity of the neighboring heat source, i.e. fij , and the relative strength ratio,
i.e. Qi /Qj . Changing location or strength of any source
leads to a new value of thermal resistance.
CONVECTION IN THE SOURCE PLANE
Convection in the source plane may now be dealt with
using results of Hein and Lenzi [4]. Comparison of the
solution of Muzychka et al. [3] with that of Hein and
Lenzi [4] shows that coefficient Bo becomes:
µ
¶
1 t1
1
+
ab k1 h2
B0 =
(30)
h1 h1 t1
1+
+
h2
k1
where h1 denotes the film coefficient in the source plane
and h2 denotes the film coefficient in the sink plane. Further, the spreading function φ becomes:
µ
¶
µ
¶
h1
k1 ζ
h1
+
sinh(ζt1 ) + 1 +
cosh(ζt1 )
k1 ζ
h2
h2
φ(ζ) =
k1 ζ
cosh(ζt1 ) + sinh(ζt1 )
h2
(31)
Both Eqs. (30) and (31) reduce to Eqs. (2) and (20),
when h1 = 0, i.e. adiabatic source plane.
COMPOUND AND ORTHOTROPIC SYSTEMS
The results developed earlier may be easily adapted
to compound and orthotropic systems with little effort.
In a recent paper, Muzychka et al. [9], applied the necessary transformations to show the relationship between
isotropic and orthotropic systems. Further, using the
results of Yovanovich et al. [8], one may modify the
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isotropic model to effectively model a resistive or conductive layer placed on a rectangular substrate. Each
modification is discussed below.
This modification can only be applied to the case when
there is no convection in the source plane.
Orthotropic Systems
If the rectangular flux channel is orthotropic such that
the in plane and through plane conductivities are different, i.e. kxy 6= kz , then the following transformations
may be made to apply the present method to such systems (Muzychka et al. [9]):
APPLICATION OF RESULTS
k → keff =
p
kxy kz
(32)
where, kxy and kz represent the in-plane and throughplane thermal conductivity, and
t → teff =
t
γ
(33)
p
where γ = kz /kxy is the conductivity ratio of the orthotropic system. The orthotropic transformation is also
valid for a substrate which is convectively cooled in the
source plane.
Compound Systems
The effect of an additional layer was also examined by
Muzychka et al. [3]. It was shown that the effect of an
additional layer (see Fig. 4) may be handled by means
of the modified spreading parameter given by:
¡ 4ζt
¢
¡
¢
αe 1 − e2ζt1 + % e2ζ(2t1 +t2 ) − αe2ζ(t1 +t2 )
¡
¢
φ(ζ) =
(αe4ζt1 + e2ζt1 ) + % e2ζ(2t1 +t2 ) + αe2ζ(t1 +t2 )
(34)
where
%=
ζ + h/k2
ζ − h/k2
and α =
1−κ
1+κ
with κ = k2 /k1 , and ζ is replaced by λm , δn , or βmn ,
accordingly. Further, the coefficient B0 is now given by:
µ
¶
1 t1
t2
1
B0 =
+
+
(35)
ab k1
k2
h
The results may now be applied to a simple system.
Three cases will be examined: isotropic, compound, and
orthotropic. The thermal property and component thicknesses are given in Table 1. In all three cases, the
heat source layout summarized in Table 2 is used along
with the following substrate properties: a = 200 [mm],
b = 100 [mm], and h = 100 [W/m2 K].
In the first case, an isotropic substrate which is cooled
in teh sink plane is considered. Next, the effect of a heat
spreader is examined through the addition of a conductive layer. Finally, the effect of orthotropic properties is
examined. This gives rise to t = teff = 31.623 [mm] and
k = keff = 31.623 [W/mK] using the properties in Table
1.
Maple V Release 8 [16] was used to perform the necessary calculations. The simple code is given in the Appendix for the isotropic case. To ensure convergence, 100
terms were used in each of the single summations and
50 terms in the double summation. The results of each
of the six runs are summarized in Tables 3-5, which report the centroidal and mean temperature excess for each
case. In general, convergence is much slower for the centroidal temperature excess due to an alternating series.
In the case of the mean temperature excess, convergence
is much more rapid since all terms are positive.
The results illustrate the effect that a conductive layer
or layers have on the temperature. The addition of a
thin conductive layer in Case B, reduces the overall temperature level in addition to flattening the temperature
distribution. Similar results are also obtained for the orthotropic case where the in-plane conductivity is higher
than the through plane. In both cases thermal spreading
is promoted due to the presence of a higher conductivity material. The results illustrate the ease with which
discrete heat source temperatures may be determined.
Typical computation times ranged between 100 and 400
seconds depending on whether an isotropic or compound
system was considered and whether the centroid or mean
value of temperature was computed.
Table 1 - Case Studies
t1
t2
k1
k2
[mm]
kxy
kz
[W/mK]
Case A
10
-
10
-
-
-
Case B
2
10
100
10
-
-
Case C
10
-
-
-
100
10
Fig. 4 - Compound System [3].
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2004 Inter Society Conference
on Thermal Phenomena
Table 2 - Source Layout
Q
c
d
Xc
Yc
[W ]
[mm]
[mm]
[mm]
[mm]
Source 1
10
20
20
40
30
Source 2
15
30
40
95
30
Source 3
25
30
70
155
45
Source 4
10
50
10
55
75
orthotropic systems. Modification of the basic equations
for the case where heat is dissipated in the source plane
was also discussed.
ACKNOWLEDGMENTS
The author acknowledges the support of the Natural Sciences and Engineering Research Council of
Canada (NSERC). The author also thanks Prof. M.M.
Yovanovich for comments given during manuscript preparation.
Table 3 - Isotropic Substrate Results
[◦ C]
θ̂
θ
Source 1
52.59
47.37
Source 2
51.98
47.45
Source 3
52.62
46.85
Source 4
43.57
39.88
REFERENCES
Table 4 - Compound Substrate Results
[◦ C]
θ̂
θ
Source 1
37.55
36.16
Source 2
40.14
38.58
Source 3
40.40
38.30
Source 4
34.66
33.56
Table 5 - Orthotropic Substrate Results
[◦ C]
θ̂
θ
Source 1
38.06
36.67
Source 2
37.59
36.52
Source 3
37.79
36.59
Source 4
35.65
34.79
SUMMARY AND CONCLUSIONS
A simple method for predicting mean and centroidal
heat source temperature was developed by means of an
influence coefficient. It was shown that this coefficient
is only a function of source location and size. It was
also shown that discrete heat source thermal resistance
is weighted according to the relative source strength ratios. Further, it was also shown that the influence coefficients lend themselves to more efficient computation due
to the reciprocity property. Several examples were computed to demonstrate the ease of application. Finally,
the method was developed for isotropic, compound, and
[1] Ellison, G., Thermal Computations for Electronic
Equipment, Krieger Publishing, Malabar, FL, 1984.
[2] Culham, J.R., Yovanovich, M.M., and Lemczyk, T.F.,
“Thermal Characterization of Electronic Packages Using
a Three-Dimensional Fourier Series Solution,” Journal of
Electronic Packaging, 2000, Vol. 122, pp. 233-239.
[3] Muzychka, Y.S., Yovanovich, M.M., and Culham,
J.R., “Thermal Spreading Resistance of Eccentric Heat
Sources on Rectangular Flux Channels,” Journal of Electronic Packaging, Vol. 125, pp.178-185, 2003.
[4] Hein, V.L. and Lenzi, V.D., “Thermal Analysis of
Substrates and Integrated Circuits,” pp. 166-177, Electronics Components Conference, 1969.
[5] Kokkas, A., “Thermal Analysis of Multiple-Layer
Structures,” IEEE Transactions on Electron Devices,
Vol. Ed-21, No. 14, pp. 674-680, 1974.
[6] Kadambi, V. and Abuaf, N., “Analysis of Thermal
Response for Power Chip Packages,” IEEE Trans. Elec.
Dev., Vol. ED-32, No. 6, 1985.
[7] Krane, M.J.H., “Constriction Resistance in Rectangular Bodies,” Journal of Electronic Packaging, Vol. 113,
1991, pp. 392-396.
[8] Yovanovich, M.M., Muzychka, Y.S., and Culham,
J.R., “Spreading Resistance of Isoflux Rectangles and
Strips on Compound Flux Channels,” Journal of Thermophysics and Heat Transfer, Vol. 13, 1999, pp. 495-500.
[9] Muzychka, Y.S., Yovanovich, M.M., and Culham,
J.R., “Thermal Spreading Resistances in Compound and
Orthotropic Systems,” Journal of Thermophysics and
Heat Transfer, In Press, 2003.
[10] Muzychka, Y.S., Culham, J.R., and Yovanovich,
M.M., “Thermal Spreading Resistances of Rectangular
Flux Channels: Part II Edge Cooling,” 36th AIAA Thermophysics Conference, Orlando, FL, 2003.
[11] Negus, K.J. and Yovanovich, M.M., “Thermal Resistance of Arbitrarily Shaped Contacts,” Numerical Methods in Thermal Problems, Proceedings of the 3rd International Conference, Seattle, WA, 1983, pp. 1072-1082.
[12] Negus, K.J., Yovanovich, M.M., and DeVaal, J.W.,
“Development of Thermal Constriction Resistance for
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2004 Inter Society Conference
on Thermal Phenomena
Anisotropic Rough Surfaces by the Method of Infinite Images,” National Heat Trnasfer Conference, Denver, CO,
1985.
[13] Negus, K.J. and Yovanovich, M.M., “Transient Temperature Rise at Surface Due to Arbitrary Contacts on
Half Spaces,” Transactions of the CSME, 1987, Vol. 13,
pp. 1-9.
[14] Negus, K.J. and Yovanovich, M.M., “Thermal Computations in a Semiconductor Die Using Surface Elements and Infinite Images,” International Symposium on
Cooling Technology in Electronic Equipment, Honolulu,
HI, 1987, pp. 563-574.
[15] Morse, P.M., and Feshbach, H., Methods of Theoretical Physics, Part I, McGraw-Hill, New York, 1953.
[16] MapleT M Release 8, Waterloo Maple Inc., Waterloo,
ON, 2002.
> f[i]:=value(B[0]+2*add(Bm[i]*cos(lambda*X[j])
*sin(1/2*lambda*c[j])/(lambda*c[j]),m=1..100)
+2*add(Bn[i]*cos(delta*Y[j])*sin(1/2*delta*d[j])
/(delta*d[j]),n=1..100)+4*add(add(Bmn[i]*cos(
lambda*X[j])*sin(1/2*lambda*c[j])*cos(delta*Y[j]
)*sin(1/2*delta*d[j])/(lambda*c[j]*delta*d[j]),
m=1..100), n=1..100)):
Input System Parameters
> baseparameters:={
a=0.2,b=0.1,k=10,h=100,t=0.01};
> sourceparameters:={
c[1]=0.02,d[1]=0.02,X[1]=0.04,Y[1]=0.03,Q[1]=10,
c[2]=0.03,d[2]=0.04,X[2]=0.095,Y[2]=0.03,Q[2]=15,
c[3]=0.03,d[3]=0.07,X[3]=0.155,Y[3]=0.045,Q[3]=25,
c[4]=0.05,d[4]=0.01,X[4]=0.055,Y[4]=0.075,Q[4]=10};
APPENDIX
Calculate Influence Coefficients
Simple Maple Release 8 code for Case A results.
Define Influence Coefficient
> restart;
> lambda:=m*Pi/a;
> delta:=n*Pi/b;
> beta:=sqrt(lambda^2+delta^2);
> phi:=zeta->(zeta*sinh(zeta*t)+h/k*
cosh(zeta*t))/(zeta*cosh(zeta*t)+
h/k*sinh(zeta*t));
> B[0]:=1/(a*b)*(t/k+1/h);
> Bm[i]:=2*(sin((2*X[i]+c[i])*lambda/2)sin((2*X[i]-c[i])*lambda/2))
/(a*b*c[i]*k*lambda^2*phi(lambda));
> Bn[i]:=2*(sin((2*Y[i]+d[i])*delta/2)sin((2*Y[i]-d[i])*delta/2))
/(a*b*d[i]*k*delta^2*phi(delta));
> Bmn[i]:=16*(cos(lambda*X[i])*sin(1/2*
lambda*c[i])*cos(delta*Y[i])*sin(1/2*delta*d[i])
)/(a*b*c[i]*d[i]*k*lambda*delta*beta*phi(beta));
> f1s:=[seq(evalf(subs(j=1,i=n,baseparameters,
sourceparameters,f[i])),n=1..4)];
> f2s:=[seq(evalf(subs(j=2,i=n,baseparameters,
sourceparameters,f[i])),n=1..4)];
> f3s:=[seq(evalf(subs(j=3,i=n,baseparameters,
sourceparameters,f[i])),n=1..4)];
> f4s:=[seq(evalf(subs(j=4,i=n,baseparameters,
sourceparameters,f[i])),n=1..4)];
Calculate Source Temperature Excesses
> Source1Theta:=subs(j=1,sourceparameters,
add(Q[i]*f1s[i],i=1..4));
> Source2Theta:=subs(j=2,sourceparameters,
add(Q[i]*f2s[i],i=1..4));
> Source3Theta:=subs(j=3,sourceparameters,
add(Q[i]*f3s[i],i=1..4));
> Source4Theta:=subs(j=4,sourceparameters,
add(Q[i]*f4s[i],i=1..4));
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2004 Inter Society Conference
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