Thermal Issues in Emerging Technologies, ThETA 1, Cairo, Egypt, Jan 3-6th 2007
DJOSER: ANALYTICAL THERMAL SIMULATOR FOR MULTILAYER ELECTRONIC
STRUCTURES. THEORY AND NUMERICAL IMPLEMENTATION
Paolo Emilio Bagnoli (1), Claudio Casarosa (2), Fabio Stefani (1).
(1)
(2)
Dept. of Information Engineering, University of Pisa, Via Caruso 10, Pisa, Italy.
Department of Energetics, University of Pisa, Via Diotisalvi 2, 56100 Pisa, Italy.
ABSTRACT
This paper presents a steady-state thermal simulation
strategy called DJOSER, which is dedicated primarily but
not exclusively to packaging structures for electronic
devices. It is applicable to structures that can be likened to
a set of homogeneous layers stacked one on top of the other
and possibly separated by thermal contact resistances,
where the dissipated powers are due to two-dimensional
heat sources distributed on the interfaces between the
layers. A broad range of contour conditions and types of
dissipated powers is included to make the models as close
as possible to the typical structures of modern assembly
technologies. Flow and temperature distributions are
obtained via a system of integral equations that can be
translated directly, with the usual squaring techniques, into
a linear algebraic system.
1. INTRODUCTION
The analytical strategy for thermal simulation of electronic
packaging structures has been widely used to solve
particular problems and under given boundary conditions
both in the static and dynamic regimes [1,2,3,4,5].
The thermal simulation tool suggested in the present paper
and called DJOSER1 is based on the possibility it offers to
numerically approximate the analytic solution to a
particular conduction problem. The main model, which is a
multilayer stepped pyramidal structure, the 2D meshing at
the layer interfaces used for the description of thermal
functions and a set of boundary conditions which may be
inserted makes this simulation tool useful to accurately
simulate nearly all the assembling structures of power
devices. The approach used results in some advantages with
respect of finite-elements method (FEM), in terms of
number of variable reduction and hence of the computing
time and memory usage while is comparable in terms of
accuracy. Thanks to these advantages and the easier model
description, the present simulation system is particularly
useful for fast thermal evaluations and as a thermal solver
in cyclic electro-thermal simulations [6]. Elsewhere
presented and validated as a prototype [7,8,9], the DJOSER
program is here fully exposed in its final and complete
form, including the
FIG. 1 – Pyramidal multi-layer structure, including
thin soldering or attaching layers; To is the heat sink
temperature, eventually not uniform; Ta is the
environment one.
mathematical terms of the problem, and its solution via a
linear algebraic system.
2. THEORETICAL BASIS
The multi-layer structure shown in Fig. 1 consists of a stack
of physically
homogeneous
prismatic
elements:
rectangular-faced parallelepipeds of various thicknesses
with their edges parallel to each other. The base of each
element is smaller (or at least no larger) than the top of the
one below, and its top is larger (or at least no smaller) than
the base of the one above. The result is a sort of stepped
pyramid in which the steps are of various heights, and it is
not necessarily possible to identify symmetry planes.
The thermal analysis of the multi-layer pyramidal structure
is based on knowledge of the solution to a steady-state
heat-conduction problem formulated for a generic prismatic
element. On the lateral faces of the prism, a heat exchange
by convection with the environment at constant temperature
is established. The environment is assumed to be the zero
reference for the temperature, so that the boundary
conditions at the lateral faces are type-3 linear and
homogeneous [10]. Likewise, a convective heat exchange is
assumed for the bases of the prismatic element, but for each
base the external temperature at the interface is considered
variable from point to point. As a result, the contour
conditions are again type-3 linear, but not homogeneous. In
formal terms, assuming the coordinates shown in Fig. 2, the
problem is as follows:
∇2T = 0
for 0<x<Lx; 0<y<Ly; 0<z<Lz
∂T
− h 1T = 0 for x = 0 ; 0<y<Ly; 0<z<Lz
∂x
∂T
k
+ h 2 T = 0 for x = Lx; 0<y<Ly; 0<z<Lz
∂x
k
1
Djoser was the 3rd-dynasty Egyptian sovereign who built the
stepped pyramid at Saqqara.
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DJOSER: Analytical Thermal Simulator for Multi-layer…, P.E.Bagnoli et al.
∂T
− h 3 T = 0 for 0<x<Lx; y = 0; 0<z<Lz
∂y
∂T
k
+ h 4 T = 0 for 0<x<Lx; y = Ly; 0<z<Lz
∂y
(1)
k
∂T
− h 0 T = -f(x, y) for 0<x<Lx; 0<y<Ly; z = 0
∂z
∂T
k
+ h 5 T = g(x, y) for 0<x<Lx; 0<y<Ly; z = Lz
∂z
k
where for now the functions f(x,y) and g(x,y) are
considered assigned to the two faces.
The problem can be solved if we first identify two subproblems, each with homogeneous contour conditions,
except for one of the two bases, i.e. z=0 or z=Lz.
If we apply the variables-separation method to the two subproblems, it is easy to see that they share both eigenvalues
and eigenfunctions. In particular, once we introduce the
nondimensional Biot's numbers:
Bi(j)=hjLj/k
where L0 =L5 =Lz , L1 = L2 = Lx , L3 =L4 =Ly
(2)
the eigenvalues for Bi(j)+Bi(j+1) ≠ 0 are solutions for the
following transcendent equation:
ctg(ξ ) =
Bi(j) ⋅ Bi(j + 1)
ξ
−
Bi(j) + Bi(j + 1) [Bi(j) + Bi(j + 1)] ξ
j = 1 ξ n = β n Lx
j = 3 ξ m = µ mLy
(3)
while for Bi(j)+Bi(j+1) = 0 they are:
FIG. 2 – Diagram of the individual layer: (a) and (b)
are the section and plan, showing the boundary
conditions; (c) is the section with the known variables
ˆ , P , P*) and the unknown variables (T, q).
( q̂ , T
ξ = 0, π , 2π , 3π ,.......(n − 1)π .
The corresponding eigenfunctions are:
X(β n x) = β n L x cos(β n x) + Bi(1) sen (β n x)
(4)
Y(µ m y) = µ m L y cos(µ m y) + Bi(3) sen (µ m y)
T(x,y,z) =
and the respective norms are [5]:
N(β n ) =
N(µ m ) =
1 
2
2
 (β n L x ) + Bi(1)
2 
[
=



Bi(2)
⋅ 1 +
+ Bi(1)
2
2 
(β
L
)
+
Bi(2)


n x

]
[
]



1 
Bi(4)
2
2
+ Bi(3) 
 (µ m L y ) + Bi(3) ⋅ 1 +
2
2 
2
(µ m L y ) + Bi(4) 



[
]
[
]
(5)
Starting from the eigenfunctions and particularly from their
well-known orthogonal properties, we construct the
solution [4] that contains a double series obtained from the
Fourier transform of the function f(x,y) or g(x,y),
depending on the base at which we have positioned the
nonhomogeneous boundary condition to which the subproblem refers. The sum of the two solutions is the solution
to the starting problem. Assuming
υ n, m = β 2n + µ 2m
[
]
S(n, m) = (υ n, m L z ) + Bi(0) ⋅ Bi(5) ⋅ sinh(υ n, m L z ) +
2
the solution is:
+ [Bi(0) + Bi(5)](υ n, m L z ) ⋅ cosh(υ n, m L z )
(6)
Lz ∞ ∞ X(βn x) ⋅ Y(µm y)  υn, mLz cosh υn, m (Lz -z) + Bi(5)sinh υn, m (Lz -z)
⋅ ∑∑
⋅

k n=1 m=1 N(βn ) ⋅ N(µm ) 
S(n,m)
1
Lx Ly
⋅
1
Lx Ly
Lx Ly
∫ ∫ f(x′,y′)X(β x′)Y(µ
n
m
y′) dx′dy′ +
υn, mLz cosh(υn, mz) + Bi(0)sinh(υn, mz)
0 0
Lx Ly
∫ ∫ g(x′,y′) ⋅ X(β x′) ⋅ Y(µ
n
0 0
m
S(n,m)

y′) dx′dy′

(7)
In order to extent the result for one element to the thermal
analysis of the pyramidal structure, we must first identify
the functions f(x,y) and g(x,y), starting from the physical
schematization of the structure. While convective heat
exchange with the external environment occurs on the
lateral faces of a generic prismatic element, the factors
present on the bases will generally be as follows:
on base z = 0: a plane heat generation per unit of surface
and time; the heat flux that crosses contact interface à from
the element above the one we are examining; and
convective heat exchange, which is present only on the part
(A–Ã) of the surface that is in direct contact with the
external environment;
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DJOSER: Analytical Thermal Simulator for Multi-layer…, P.E.Bagnoli et al.
on base z = Lz: heat exchange by conduction to the
underlying element, possibly with the presence of a contact
resistance between the two elements; and, if such a contact
resistance is present, another plane source of heat
generation per unit of surface and time.
From the analytic standpoint, the greatest difficulty arises
on the upper base z = 0 of the prismatic element, where – as
we have seen – the contour conditions are mixed: one of the
second type on (x,y)∈Ã and the other of the third type on
(x,y)∈(A–Ã) [5]. If we use the term P(x,y) to denote the
plane heat generation due, for example, to the Joule effect
related to the passage of current in semiconductors
junctions or in metal interconnections; q̂(x,y) for the heat
flow from the overlying element, nil for (x,y) ∈ (A–Ã); and
T(x,y,0) for the temperature on the top of the prismatic
element we are examining, then the function f(x,y) on base
x=0 will be:
~
f(x, y) = P(x, y) + qˆ (x, y) + h 0 T(x, y,0) for (x, y) ∈ A , z=0
~
(8)
f(x, y) = P(x, y) for (x, y) ∈ (A − A) ; z =0
T(x, y,0) = Bi(0) ∫∫ T( x ′, y′,0) G11 (x ′, y′ | x, y)dx ′dy′ +
~
A
L
+ z
k
[
∫∫ [P(x ′, y′) + q̂(x ′, y′)] G
A
11
(11a)
(x ′, y′ | x, y)dx ′dy′
]
+ ∫∫ R *P* (x ′, y′) + T̂(x ′, y′) G12 (x ′, y′ | x, y)dx ′dy′
A
-------------------------------------
q(x, y, L z ) = h 0 ∫∫ T( x ′, y ′,0) G 21 (x ′, y ′ | x, y)d x ′dy ′ +
~
A
+ ∫∫ [P( x ′, y′) + q̂(x ′, y ′)] G 21 (x ′, y′ | x, y)d x ′dy ′ A
k
−
Lz
∫∫ [R
]
* *
P (x ′, y′) + T̂ (x ′, y ′) G 22 (x ′, y ′ | x, y)d x ′dy ′
A
(11b)
where the functions Gij(x’,y’x,y) are the Green ones
structured as follows for this case:
1
Lx Ly
G11 (x ′, y′ | x, y) =
D11 (n, m) X(β n x ′) ⋅ Y(µ m y′)
⋅
⋅
N(β n ) ⋅ N(µ m )
n =1 m =1 S(n, m)
∞
∞
∑∑
⋅ X(β n x) ⋅ Y(µ m y)
--------------------------
Accordingly, when h0 ≠ 0, the boundary condition for face
z=0 implies a convective exchange identically nil on the
(x,y)∈Ã part and, with the environment temperature at 0,
not nil on the remaining part, (x,y) ∈ (A–Ã). This position
obviously does not solve the difficulty created by the
existence of mixed contour conditions. In fact, even
assuming that the functions P(x,y) and q̂(x,y) are assigned,
the temperature T(x,y,0) remains altogether unknown,
which implies that an integral equation is present in the
double Fourier series. The situation on the lower base of the
element, z=Lz is simpler. If we use the term P*(x,y) to
denote the further plane heat generation due to the aforesaid
phenomena, T̂(x, y) for the temperature of the interface
with the underlying layer, and R* for the contact thermal
resistance between the element being examined and the one
below it, the function g(x,y) will be:
Tˆ (x, y) for (x, y) ∈ A , z =Lz
g(x, y) = P (x, y) +
R*
*
1
Lx Ly
∞
∞
∑∑ υ
n,m
Lz ⋅
n =1 m =1
Bi(5) X(β n x ′) ⋅ Y(µ m y′)
⋅
S(n, m) N(β n ) ⋅ N(µ m )
⋅ X(β n x) ⋅ Y(µ m y)
--------------------------
(12)
G 21 (x ′, y ′ | x, y) = G 12 (x ′, y ′ | x, y)
-------------------------
G 22 (x ′, y ′ | x, y) =
⋅
1
LxLy
∞
∞
∑∑υ
n, m
Lz ⋅
n =1 m =1
D 22 (n, m)
⋅
S(n, m)
X( β n x ′) ⋅ Y( µ m y ′)
⋅ X( β n x) ⋅ Y( µ m y)
N( β n ) ⋅ N( µ m )
since we have assumed that
D11 (n, m) = υ n, m L z cosh (υ n, m L z ) + Bi(5)sinh (υ n, m L z )
D 22 (n, m) = υ n, m L z sinh (υ n, m L z ) + Bi(0)cosh(υ n, m L z ) .
(13)
(9)
It should be noted that the contact thermal resistance entails
a linear heat exchange like the convective one, which is
thus:
R* = 1 / h5
G 12 (x ′, y′ | x, y) =
(10)
Based on the physical schematization of the structure with
which the functions f(x,y) and g(x,y) were identified, it is
clear that while the plane heat generation terms P(x,y) and
P*(x,y) are assigned functions, the terms q̂(x,y) and T̂(x, y)
– related respectively to the examined element's upper and
lower interface – are unknown; to the contrary, they depend
on the thermal fields of the overlying and underlying
elements.
The next step in the thermal analysis of the pyramidal
structure is to replace the functions f(x,y) and g(x,y) in the
solution T(x,y,z) and its derivative for z with functions (8)
and (9). We then operate so as to express the temperature in
z=0 and the heat flow in z=Lz in the following form:
As mentioned above, (11a) is a second type Fredholm
linear integral equation [11], where the temperature
T(x,y,0) acts as an unknown function and the Green G11
function as a nucleus. At any rate, it is easy to see that the
equation would remain an integral one even if Bi(0)=0.
Indeed, both (11a) and (11b) are by nature integral, because
the functions q̂(x,y) and T̂(x, y) in the integrals on their
right-hand sides are unknown.
We assign a numerical index to each of the stack's generic
prismatic elements, from 1 for the top element to n for the
bottom one. The index appears to the lower right of the
various magnitudes or, in parentheses, to the upper right.
With these notations, the conservation of the heat flow and
the continuity of the temperature on the interfaces of a
generic "i-th" element translate into the following
equations:
qˆ (x, y) := qˆ i -1 (x, y) = Pi*-1 (x, y) + q i-1 (x, y, L(iz-1) ) for
z=0
Tˆ (x, y) := Tˆ i+1 (x, y) = Ti +1 (x, y,0)
(14)
for
z= L(i)z
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DJOSER: Analytical Thermal Simulator for Multi-layer…, P.E.Bagnoli et al.
In this way, for a generic prismatic element we have a pair
of linear integral equations in the following four functions:
q̂ i -1 ( x, y) ; q̂ i ( x, y) ; T̂i ( x, y) ; T̂i + 1 ( x, y) . In the resulting
system of linear integral equations, the number of equations
is equal to the number of unknown functions. If we use the
letter n to denote the number of layers contained in the
structure, a more accurate count, however, gives 2(n-1) as
the number of equations and of unknown functions. In fact,
since the parallelepiped at the top of the stack does not
underlie any other, its upper face lacks the à part for which
we would otherwise have to calculate the temperature
integrals T(x,y.0). As a result, this temperature disappears
from the right-hand side of the pair of equations for the first
element, and both the function and the equation
corresponding to it, being nonessential, are excluded from
the system. The same conclusion is reached for the heat
flow through the lower face of the bottom element, for it is
easy to see that this flow does not appear on the right-hand
side of the pair of equations for any other element. Thanks
to these observations, both the number of functions and the
number of equations decrease by two. Lastly, while the
flow q̂(x,y) is obviously nil for the top layer in the stack,
the function T̂(x, y) at the lower interface of the bottom
layer is the temperature of the heat sink on which the stack
rests; that is, an assigned To(x,y) function.
Based on the foregoing, the system of linear integral
equations is:
q̂1 (x, y) = P1* (x, y) +
∫∫ P (x′, y′) G
1
(1)
21
(x ′, y′ | x, y)dx ′dy′ +
(15a)
A(1)
[
]
k1
* *
(1)
− (1)
∫∫ R1 P1 (x′, y′) + T̂2 (x′, y′) G 22 (x′, y′ | x, y)dx′dy′
L z A(1)
------------------------------------------------------------------------------------------(j)
T̂i (x, y) = Bi i (0) ∫∫ T̂i (x ′, y′) G11
(x ′, y′ | x, y)d x ′i dy′ +
FIG. 3 – Diagram showing the indices of the layers,
coordinates and grids used in the numerical
treatment.
coordinates in equations that refer to different elements are
usually not identical: while the axes in the systems they
describe are all parallel, their origins do not necessarily
coincide.
In principle, the system's solution provides the heat flux
and temperature functions at the interfaces. Once these
factors are known, we can determine the f(x,y) and g(x,y)
functions for each prismatic element. The solution to the
conductive problem can thus be used with convective
boundary conditions factored in, and the thermal field
throughout the structure can be calculated layer by layer.
As it would be unthinkable to try to solve the integralequation system analytically, what we need is an
approximate solution. If we calculate the integrals
numerically, we can convert the integral system into a
linear algebraic system, which we can then solve with the
aid of the usual electronic computing tools.
3. ALGEBRAIC FORMULATION
A(i -1)
+
+
L(i)z
ki
(i)
∫∫ [Pi (x ′, y′) + q̂ i -1 (x′, y′)]⋅ G11 (x′, y′ | x, y)dx′dy′ +
(15b)
A(i)
∫∫ [R
]
* *
i
(i)
Pi (x ′, y′) + T̂i +1 (x ′, y′) G12
(x ′, y′ | x, y)d x ′dy′
A(i)
---------------------------------------------q̂ i (x, y) = Pi* (x, y) + h (i)
0
∫∫ T̂ (x′, y′) G
i
(i)
21
(x′, y′ | x, y)dx′dy′ +
A(i-1)
(15c)
(i)
∫∫ [Pi (x′, y′) + q̂i-1 (x′, y′)] G21 (x′, y′ | x, y)dx′dy′ +
A(i)
−
∫∫ [R P (x′, y′) + T̂
ki
L(i)z
* *
i
i
i +1
]
′ ′
′ ′
(x′, y′) G (i)
22 (x , y | x, y)dx dy
A(i)
------------------------------------------------------------------------------------------Tˆ n (x,y) = Bi n (0)
∫∫
+ ∫∫
A(n)
∫∫ Φ(x′, y′) ⋅ G
(n)
Tˆ n (x ′,y′) G11
(x ′,y′|x,y)dx ′dy′ +
A(n-1)
L(n)
+ z
kn
A single Fredholm linear integral equation, being
formulated on a finite interval, can be reduced to a system
of algebraic equations. The equation must be verified at a
finite number of points, and the defined integral must be
approximately squared with the values assumed by the
unknown function at the same points (5). This method can
be directly extended to systems of linear integral equations;
in particular, it can be applied to system (15), to which it
provides an approximate solution.
Before proceeding, we note that all the integrals appearing
in the system's equations are of the same type and that for a
generic function Ф, they can be expressed as follows:
∞
ij
n =1 m=1
A
(15d)
∫∫ [ Pn (x′,y′) + qˆ n-1 (x′,y′)] ⋅ G (x′,y′|x,y)dx′dy′ +
(n)
11
A(n)
(n)
 R *n Pn* (x ′,y′) + To(x ′,y′)  G12
(x ′,y′|x,y)dx ′dy′


It must be kept in mind that each of the above equations
refers to one base of one element, and that the x,y
⋅
1
Lx Ly
∞
(x′, y′ | x, y)dx′dy′ = ∑∑Ci j (n,m) X(βn x)Y(µ m y) ⋅
Lx Ly
∫ ∫ Φ(x′, y′) X(β x′ )Y(µ
n
m
y′)dx ′dy′
(16)
0 0
Once we have chosen the technique of approximate
squaring in two dimensions and identified in domain A an
appropriate grid of N points on which to calculate the
function Ф, we have:
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DJOSER: Analytical Thermal Simulator for Multi-layer…, P.E.Bagnoli et al.
1
Lx Ly
LxLy
∫ ∫ Φ(x′, y′) X(β x′ )Y(µ
n
N
m
y′)dx ′dy′ ≅ ∑ Bn,m (x k , y k ) ⋅ Φ(xk , y k )
k =1
0 0
(17)
where the coefficients Bn,m depend on the chosen type of
squaring as well as on the values assumed at the grid points
by the primitives of the eigenfunctions. For example, if we
use the parallelepiped-squaring technique on a rectangle
whose sides are labeled 2a and 2b, we approximate the
function Ф with the value it assumes at the centre point
obtaining:
B n, m (x k , y k ) = 4 ⋅ sen( β n a)sin( µ m b)[cos( β n x k ) +
+
Bi(1)
Bi(3)
sin( β nx k)] ⋅ [cos( µ m y k ) +
sin( µ m y k )]
βnLx
µmLy
(18)
Fig. 4 – Diagram of the algebraic system (4 layers). I
is the identity matrix; T.N. is the column vector of
known terms. The blank submatrices are null
matrices.
In any case, assuming
∞
∞
φij (x k , y k | x, y) = ∑∑Cij (n, m) ⋅ Bn,m (x k , y k ) ⋅ X(β n x)Y(µ m y)
n =1 m=1
(19)
we can always write
∫∫ Φ(x′, y′) ⋅ G
A
N
ij
(x ′, y′ | x, y)dx ′dy′ ≅ ∑ φ ij (x k , y k | x, y) Φ(x k , y k )
k =1
(20)
To introduce this approximation into the system of integral
equations (15), we must clarify an important point: Each
element in the stack has a system of coordinates of its own
to which the points on faces z=0 and z=Lz(i) refer, but the
squaring grids identified on the two geometrically identical
faces usually differ in mesh size and number of points (see
Fig. 3). At the contact interfaces, though, the grids on two
contiguous elements must be identical, matching perfectly
with each other.
As a rule, the coordinates of a point we think of as being on
the bottom surface of the overlying element will naturally
differ from the coordinates of the same point when we
think of it as being on the top surface of the element
underneath. Accordingly, since each of the equations in
system (15) applies to all the points on the face to which
the equation refers, each equation needs to be verified only
for the grid points used for the squaring. This process gives
rise to an equal number of algebraic equations. Their
ensemble constitutes the linear algebraic system into which
the integral system has been converted.
The notations x(i) and y(i) refer to the coordinates of a
generic element in the stack, to highlight the fact that they
are usually not identical: xt(i) , yt(i) are the points on the top
surface z=0 of the i-th element, and xb(i), yb(i) are the points
on the bottom surface z=Lz(i) of the i-th element. Moreover,
to make it clear that the summations refer to integration
grids that generally differ, the symbols k and p are used for
the N(i) and M(i) points on the grids drawn, respectively,
on surfaces Z=0 and z=Lz(i) of the i-th element, and the
symbol j is used for the contact interface A between two
contiguous elements.
The points on the interface are M(i-1), because, in general,
the grid on surface z=Lz(i-1) of the overlying element
matches the grid on surface z=0 of the i-th element only in
part; of all the N(i) points on the grid on surface z=0 of the
i-th element, only the M(i-1) points match exactly. The
notations are shown in Fig. 3.
Using the notations described above and equations (19) and
(20) for converting the dual integrals, the equation (15)
system is transformed into an algebraic system made up of
2[M(1)+M(2)+....+M(n-1)] equations for the same number
of unknowns. Its normal form, written concisely and with
the unknown variables isolated on the left-hand side, is as
follows:
M(1)
∑ω
N(1)
M(1)
k =1
p =1
*(1)
(1)
(1)
* *(1)
⋅ Tp(2) + q (1)
+ ∑ φ (1)
b = Pb
kb ⋅ Pk − ∑ ω pb ⋅ R 1 ⋅ Pp
(1)
pb
p =1
with b =1,2,3...M(1)
-------------------------------------------------------------------------------------------------------------------------------------
Tt(i) −
M(i -1)
∑h
(i)
0
M(i)
M(i -1)
p =1
j=1
(i +1)
ψ (i)j t ⋅ Tj(i) + ∑ φ (i)
−
p t ⋅ Tp
j=1
N(i)
M(i)
k =1
p =1
∑ψ
(i)
jt
⋅ q (ij-1) =
(i)
(i)
* *(i)
= ∑ ψ (i)
k t ⋅ Pk + ∑ φ p t ⋅ R i Pp
with t =1,2,3...M(i-1)
≤ N(i)
------------------------------------------------------------------M(i -1)
−
∑h
(i)
0
M(i)
M(i -1)
p =1
j=1
(i +1)
φ (i)j b ⋅ Tj(i) + ∑ ω (i)
−
p b ⋅ Tp
j=1
N(i)
M(i)
k =1
p =1
∑φ
(i)
jb
⋅ q (ij-1) + q (i)
b =
* *(i)
(i)
(i)
= Pb*(i) + ∑ φ (i)
k b ⋅ Pk − ∑ ω p b ⋅ R i Pp
with b =1,2,3...M(i)
-------------------------------------------------------------------------------------------------------------------------------------
Tt(n) −
M(n -1)
M(n -1)
j=1
j=1
∑ h (0n ) ψ (n)j t ⋅ Tj(n) −
N(n)
M(n)
k =1
p =1
∑ψ
(n)
j t
[
⋅ q (nj-1) =
(n)
(n)
* *(n)
= ∑ ψ (n)
+ Fp
k t ⋅ Pk + ∑ φ p t ⋅ R n Pp
with t =1,2,3...M(n-1)
≤ N(n)
]
(21)
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DJOSER: Analytical Thermal Simulator for Multi-layer…, P.E.Bagnoli et al.
where
(i)
(i)
(i)
(i)
; (i) ˆ i (x (i)
; (i)
Tk(i) ≡Tˆ i (x (i)
k , y k ) Pk ≡ P i (x k , y k )
k , y k ) q k ≡q
(i)
(i)
; *(i) ≡ Pi*(x(i)
;
Pp*(i) ≡ Pi* (x(i)
p , y p ) Pp
p , yp ) Fp ≡ F(xp , y p )
(22)
and
ψ (i)
kt ≡
L(i)z (i) (i) (i) (i) (i)
φ11 (x k , y k | x t , y t )
ki
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
φ (i)
p t ≡ φ 12 (x p , y p | x t , y t ) = φ 21 (x p , y p | x t , y t )
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
(i)
φ (i)
kb ≡ φ 21 (x k ,y k |x b ,y b ) = φ 12 (x k ,y k |x b ,y b )
k i (i) (i) (i) (i) (i)
(23)
ω (i)
φ 22 (x p , y p | x b , y b )
pb ≡
L(i)z
The quantities defined by (23) are the coefficients of the
individual rectangular submatrices in the principal matrix
of the algebraic system, organized as shown in Fig. 4. The
system's solution enables us to know the grids of flow and
temperature values on all of the structure's inner interfaces.
With these data in hand, we can then calculate the flow and
temperature at every point of the assembly, thereby
obtaining a complete three-dimensional map.
4. THERMAL SIMULATION TESTS
In order to demonstrate how the simulator works and how
the temperature calculation procedure converges, some
simulation tests were performed on purposely designed
virtual samples, representing a typical assembling structure
of an electronic power device. Fig. 5 shows the top and
lateral views of the two samples together with the power
generation map (top of figure) localized on the top surface
of the second layer. Table I also reports the geometrical and
material data for the layers of the two samples.
TABLE I: Geometrical and material data of the layers
composing the samples A and B used for the simulation
tests.
LAYER
PLASTIC
SILICON
METAL
Lx
mm
8
8
12
Ly
mm
4
4
8
Lz
µm
800
600
2000
k
W/m°C
0.3
135
100
R*
mm2°C/W
2
10
The layer sequence is the following starting from top: a
plastic-epoxy coverage layer, a silicon die and wider
generic metal heat spreader with thermal conductivity k =
100 W/m°C. In order to figure the presence of soldering or
attaching layers, two contact thermal resistances per unit
area R* were placed below the silicon layer and the metal
heat spreader in contact with the underlying heat sink. The
two layers have the same power generation density located
on the top silicon surface, organized in three islands, each
uniformly dissipating 8 W. since the center island has a
larger area, its power density is lower than the others. In
order to slightly increase the heat transfer by convection,
the bottom heat sink temperature was set to To = 20° C
FIG. 5 – Top and lateral views of the virtual samples
A and B. The power generation on the top silicon
surface is shown on the top of the figure.
while that of the surrounding environment was set to Ta =
0 °C.
The B sample differs from A for the position of the upper
layer only: sample A is fully axisimmetrical while in B the
plastic and silicon layers are aligned with the south-west
corner of the metal spreader. This difference was set in
order to show how the temperature distributions are
influenced not only by the layer thickness and thermal
conductivity but also by their relative positions.
Fig. 6 shows the color-scaled temperature maps of the three
layers of sample A calculated using the DJOSER simulator
under the adiabatic condition. In each map the
corresponding maximum and minimum temperature are
reported. As expected, the maximum temperature of the
system is on the top silicon surface. Fig. 7 also shows the
comparison between the temperature plots of samples A
(red) and B (blue) along the medium horizontal (a) and
vertical (b) symmetry axes of the silicon. The continuous
lines are for silicon while the dashed ones are for the top
plastic layer. These plots clearly demonstrate the effects of
the geometrical configuration of the layer below the power
dissipation plane on the upper temperature distributions.
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DJOSER: Analytical Thermal Simulator for Multi-layer…, P.E.Bagnoli et al.
FIG. 7 – Temperature plots along the medium
horizontal (a) and vertical (b) symmetry axes of
samples A and B, for the top silicon surface and for
the plastic one in the adiabatic condition.
FIG. 6 – Color-scaled maps of temperatures (dark
red are the higher values, dark blue the lowers)
calculated for all the layers of sample A in the
adiabatic condition.
The plots of sample B are not only higher but they have also
different shapes, notwithstanding the power generation
density is the same. This behavior is fully due to the
different position on the metal spreader, inducing a different
heat flux displacement at the silicon-metal interface. In fact,
a fully asymmetrical position on the metal is a net
geometrical disadvantage for the heat flux spreading, hence
causing a temperature increase in the layers above.
Fig. 8 shows the results of the simulation tests of the
structures in presence of a heat exchange by convections at
the external surfaces in contact with the external
environment whose temperature was set at 0 °C. Fig. 8a is
referred to the medium x symmetry axis of the top silicon
surface in sample A at which the same heat convection
coefficient H was applied to all the surfaces. The
progressively decreasing curves correspond to H values
ranging fro 0 to 5000 W / m2 °C. However only the curves
of the last five H values were drawn; in fact the cooling
effect on silicon of the convection exchange is quite weak
in comparison to the thermal conduction path tough the heat
sink, so that lesser H values practically produces curves not
distinguishable from the adiabatic one. Instead Fig. 8b is
referred to the top plastic layer of sample B (medium x
symmetry axis) at which the H coefficient was applied to
the southern and western lateral surfaces only of all the
three layers. This is to demonstrate the capability of the
implemented simulation system to impose different
boundary conditions at the model surface. The temperature
FIG. 8 – a) Temperature plots for sample A along the
x-axis and for several values of the convection
coefficient H applied to all the exposed surfaces. b)
Temperature plots of top plastic surface along the xaxis of sample B; the convection was here applied to
the southern and western lateral surfaces only.
plots are fully consistent with this particular boundary
condition, since the cooling effect is evident on the western
side only. Furthermore, in this last case in which the plastic
surface is strongly isolated from the main thermal
conduction path, the used H range is one order of
magnitude lower than in the previous experiment, allowing
observing the cooling effect also with low convective
exchange.
The aim of the last simulation experiment was to verify the
convergence of the calculation method and to evaluate the
temperature error as a function of the two main free
parameters of the calculation procedure. The first one is
Nmn, that is the maximum number of the eigenvalues in the
x and y directions used for the calculation of the harmonic
series in equations (16), (19) and hence (23). This
parameter must be set for every layer in the structure. In the
calculation we always used an ensemble of eigenvalues
forming a square matrix with Nmn as dimension.
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DJOSER: Analytical Thermal Simulator for Multi-layer…, P.E.Bagnoli et al.
based on the numerical implementation of a full analytical
strategy, instead of the usual 3-D discretization of the heat
conduction equation. For the first time its all mathematical
and numerical details were completely exposed.
The demonstrative simulation tests showed a rather good
convergence characteristic which, in any case, is a function
of some internal parameters. As far as the calculation time
is concerned, we must emphasize that the calculation
procedure, implemented in not compiled MATLAB 7.0
system, is still far to be optimized from this point of view
and much work must be done to improve it. However, for a
1.6 GHz laptop computer, the best compromise for the
above parameters, which can be Nmn = 70 and a cell
density of 9 mm-2, needed a calculation time of about 200
seconds whose 65% was devoted to the coefficient matrix
calculation: this matrix can be memorized and used for
different tests with other power charges since it does not
contain the power data. These performances are at least
encouraging to make the DJOSER simulation strategy faster
and more convenient of the finite-element-based programs,
at least for the thermal simulation of electronic devices.
FIG. 9 – Maximum and average absolute temperature
error bar-diagrams calculated for sample A as a
function of the maximum number for side of used
eigenvalues (a) and of the cell densities used on the
silicon and metal surfaces.
The second source of error is the grid cell densities used at
the layer interfaces to describe the temperature and flux
functions. These grids were used to numerically calculate
the 2-D integrals with the rectangular method. The increase
of both these two parameter generally leads to a
quadratically increasing of the whole procedure computing
time. In order to evaluate the temperature errors due to
these parameters we obtained the silicon temperature maps
of sample A using a very high number of eigenvalues
(Nmn=180) and a very high cell densities for all the layers
(100 mm-2), assuming the temperature data of this test as a
reference. Then we performed several simulation tests with
lower values calculating the maximum and average error
percentages with respect to the reference data over the
whole silicon surface. The results of these tests are reported
in the Fig. 9 in terms of bar diagrams. Fig. 9a shows the
results as a function of the number of eigenvalues Nmn,
using cell densities of 9 mm-2. As can be seen the errors are
decreasing functions with increasing Nmn and are well
below 1% for numbers greater than 50. However the bardiagram does not seem to be monotonic, showing two peaks
perhaps due to a sort of error resonance among the various
layers. Instead Fig. 9b shows the results for the tests as a
function of the cell densities using a number of eigenvalues
Nmn = 70. Also in this case, the error percentages are
decreasing functions of the cell density, although the errors
are much more consistent, showing the importance of this
factor in limiting the temperature accuracy of the
calculation procedure.
6. CONCLUSIONS
In this paper a steady-state thermal simulation method for
particularly shaped multilayer structures was presented. It is
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