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Thermally conditioned placement optimization of elements in hybrid microcircuits
M~croelectron. Reliab., Vol. 30, No. 1, pp. 89-103,1990. Printed in Great Britain.
0026-2714/9053.00 + .00 © 1990 Pergamon Pre~ plc
THERMALLY CONDITIONED PLACEMENT OPTIMIZATION OF ELEMENTS IN HYBRID MICROCIRCUITS BOGUS~AW WISZ Technical University of Rzesz6w, Rzesz6w, ul. W.Pola 2, Poland (Received for publication 2 May 1989)
T h e p a p e r p r e s e n t s s o m e of t h e p r o b l e m s in u s i n g t h e r m a l s i m u l a t i o n for t h e p l a c e m e n t o p t i m i z a t i o n of e l e m e n t s , which ape heat sources in hybrid microcircuits. T h e a n a l y s i s of h e a t exchange mechanisms in thick film circuits has been carried out and the mathematical m o d e l for c i r c u i t s w i t h s l n g l e - l a v e r and non-homogenous substrates has been elaborated. Analytical s o l u t i o n for d e t e r m i n a t i o n of the temperature distribution in t h e f o r m of s i n g l e F o u r i e r series has been utilized in optimization process. In the proposed procedure based upon the temperature c r i t e r i o n of o p t i m i z a t i o n and the gradient method, technological and constructional constraints have been taken into consideration. T h e e x a m p l e s of c o m p u t a t i o n results have been included and the conclusions useful in the process of computer-aided h y b r i d nhlcrocircuit d e s i g n h a v e b e e n p r e s e n t e d .
I.
I NTRODUCTI ON
The
analysis
is a s i g n i f i c a n t this
analysis
which
occur
ronment
but
physical
determining
temperature
the
aiming
mization
and at
which
process
and
adjoined
cult.
The
necessity
heat makes
sources, work
pendent
on
makes the
Consequently,
the the
and
between
dependence of
to
between
the
solving
of
place
a single of
circuit
on
the
plane
substrate electronic
at
a short
topography of
the 89
and
the
in
thermo-
heat
to
condition on
design.
plate
the
of
One
distance among
one
other
of heat.
of
microci*'--
eleme1~ts
from
of
opti-
placement
the
of that.
conditioned
that. of
unfavo0rable
iden-
facilitates
interactions
value
analy-
determining
allows
network
elements,
envi-
The
elements,
thermally is
the
changes.
engineering
[1,2]
particular
improvement
the
and
out.
effects
components
stated
the
Carrying
thermal
it
it.. T h i s
circuits the
of in
the
in
of
mainly
purposeful
construction
elements
consist
occurs
possible
the
temperature
which
causes
of
microcircuits
them.
microcircuit
optimization
the
temperature
upon
the
require
of
layer
to
designing nature
in a m i c r o c i r c u i t
regard
hybrid
the
function
interrelation~
microcircuit
problems
the
of
of
learning
the
distribution
with
condition
the
conditions
process
only
also
the
inference
not
in
work
the
inside
exchange
thermal
in
both
of
the
factor
parameters
heat
thermal
requires
sis
tify
of
being
another,
things,
power
de-
sources.
condition
can
90
B. Wmz be obtained substrate
by way of correction
surface.
of e l e m e n t s -
Solution
(heat
developing
sources)
a method
of
the heat
sources
of t h e p r o b l e m
inside
of placement,
substrate
of c a l c u l a t i n g
placement
region
temperature
on
the
optimization
consist
in:
distribution
useful
in optimization; -
determining
the criteria,
selecting
the method
and
elaborating
the
opti mi zatl o n al got i thin; - formulating them
into
technological
consideration
Th~ application designing to take most
practice
into
2.
and
HEAT
sure cated
ment
lyzed
region
method.
The
and
necessity
method.
assumptions
- stationary der atl o n ;
of
work
practise
will
the allow
thermally
the
increasing
its
parameter
and determining
out
the
selecting
model
h a s ,been
properties accepting
of
of
tame. p l a c e -~
temperature
further
at
the
the
to
advan-
the
ana-
every
point
optimization
formulated
on
the
theoretical
"permissible
conditions.
des-
feasibility
pol,~t of
and
ne-
preference
has
gradient
it
the
period
gives
en-'
compli-
and
of
repeated
when
boundary
easy
short
at a n y
temperature
of t h e r m o d y n a m i c s ,
be littIe
and
the method
the temperature
structural
should
in the process
of t h e c i r c u i t
the
but
only
the accuracy
the model
Besides,
not
is n o t
in a possibly
mathematical
the following
analysis
between
is s i g n i f i c a n t
analytical
ensure
in
OF SOLUTION
of c a r r y i n g
points
to determine
which
in particular
requirement
system
to define
of t h e l a w s
introduced
this
complexity
of t h e m i c r o c i r c u i t
analysis fying
The
programme
microcircuit
field
t i m e is a d e c i s i v e
enable
of the field,
basic
to the
the analytical
which
- METHOD
with
at n u m e r o u s
tages
the
taking
procedure.
which
in engineering
of the equation
calculations
of
at a c o m p r o m i s e
optimization.
applying
MODEL
Complying
The computation
circuits
and
of o p e r a t i o n .
sufficient
relevant
constraints
computation
the factors
of t h e t e m p e r a t u r e
arriving
of s o l u t i o n
power
conditions
EXCHANGE
as well.
cription
of h y b r i d
reliability
accuracy
cessitates
of t h e e l a b o r a t e d
work
The method
constructional
in the optimization
condsideration
favourable
stability
and
There
simplihave
been
assumptions:
conditions
of t h e
microcircuit
are
taken
into
consi-
Thermally conditioned placement optimization - dielectric heat in
-
-
rectrangular
ber
of
the
thermal
as
surface
well
is
as
are
neglected
in
the
analysis
of
oof
heat
the
substrate
sources
with
coefficient
there
is
constant
a definite
power
has
constant
and
lower
num-
value;
value
independet
heat
the
into
exchange the
flow
basic to
power
Model
conditions:
to
the is
of
function
the
here.
parts in
of
heat
leads
above
linear
taken
the
is
transfer
into
and
account
circuit parts
circuit
the
substrate
lateral
in
the
the
in
with
The
coefficient
from
assumptions
problem
layers
of
the
bei~g
surface~
circuit
constar~t:
is
omitted;
neglected. heat
transfer
influence
boundary
which
elements
thermal
elements
region
i5 ther-
is
conductivity
heat
The
CFig. l).
the
substrate
being
region
of
equation
conditions.
the
different,
microcircuit
the
upper
environment
by
homogenous
considered
of
the
heterogenous
two
both
of
from
value
dissipation
reduced
in
Cflat)
boundary
heat
heat
mal
plane
conductivity
assumed,
- the
On
layers
temperature;
complex
-
conductive
exchange;
the
of
-
and
91
sources
temperatul
T £x,y.z] p
has
can e
derided
be
been
placed
distribution
determined
by
the
expression:
T~i>Cx, y,z) P T (x,y,zD
for
O~xSA.
O~ySCI,
OSzSh
=
CID
P
T{2}Cx,y,z)
for
p
where and in
x,y,z
are
T~*~Cx,y,z] P its
the
first
the
denote
and
thickness
coordinates the
second
and
of
the
substrate
subregion,
CI(C23
O
i~
CI
considered
region],
temperature at respectively,
the
O_
widh
of
the
the
A is whole
T~S)(x,y,z) p poir*t .Cx, y , z 3
the
lenght,
CpartD
h is
substrate
yJ
C2 C1 1
A
Fig
I.
Thermal
flat. m o d e l
dimensions ficients,
of S
, S i
of
substrate, - heat 2
a heteroge*~ous k
x
circuit.
, k 2 - thermal
source~.
A,
CI,
conductivity
C~
--
coef--
92
B. WISz plate,
satisfies, t h e
conditions When
in e a c h
the heat
thermal
Laplace
with
the defined
part
of t h e
above
boundary
subregion.
source
is p l a c e d
conductivity
differential
equation
in the
coefficient
equation
k , for
the
i
substrate
function
with
T(i~Cx.y.z)
the
boundary
is c o m p u l s o r y : CaD
conditions:
6T(pt/6x = 0 for
x = O i x=A,
C3)
6Tp¢ i )~6y = 0 f o r y=O,
C4D
ai[T(pi~Cx, y,O)
- To.,] C5)
kt •6TCpt/6z =
G -
where:
a
and
~
the
p
VaT(i)Cx,y,z) = O, p with
the
are
az[T(pi'Cx, y,h)
the
heat
transfer
- Toz ] d l a
(x.y)
coefficients
. S
from
the
lower
and
z
upper
parts
of t h e s u b s t r a t e ,
density
of t h e
ambient
temperatures.
On t h e o t h e r
source
hand
the
~T('}Cx,y,z) p with
the
boundary
=
with
respectively,
the surface
function
G - is
S,
and
T
the
thermal
and
T
Oi
T(Z'Cx,y,z) p
02
satisfies
power
- are
the
the e q u a t i o n :
O,
(69
conditions:
6T(p~'/6x
= 0 for
x=O
6 T p(2) /6y
= 0 for
y=Ce,
and
x=A,
(7)
C 8)
ai[T~Z)Cx,y,O)
- To1 ] (g)
On t h e
border
0 2
[C
of r e g i o n s
Cx,
y~
Cy=CID
there
occur
t h e cor,ditions:
T(i~Cx,Cl,z~ = T(2>Cx,CI,z), p P
CIO~
ks6T(S/6y = kzTp( 2 ) 76y. In t h i n of
temperature
which
affords
substrates along
the
c11> with
good
thickness
possibilities
for
thermal of
averaging
conductivity
the
plate
the d i f f e r e n c e s
are
the temperature
inconsiderable
iI~
relatioz~
Thermally conditioned placement optimization to the substrate
thickness
to the two-dimensional
and
reducing
problem
93
the searched
temperature
field
[3]:
h
TpmCX,y)
= ~ ; TpCX, y , z ) d z
= TCx,y)
Cla)
o Substituting tions
(~9
region
the
and
twice
C69
differentiated
and
using
0 S y ~ Cl,we
T(l>
+ T(*> MY
x×
expression
the adequate
CI~)
boundary
for
the equa-
conditions,
for
the
obtain:
-2T(t)
~0
= ~l
-
~2 +
F;
F=
dla Cx,y) Qi d l a C x , y )
~ E S
C139
where: 2
c~ + ~
=
1
~
2
~ T
=
k I T - ' h- '
t
+a T
Ol
~2
2
×
a n d for
+
xx
2
yy
CI
y
-<
a T 2
h
- making single
t
1
=
'
C149 +a T
Ol
¢92
2
k
2
The searched
02
h 2
solution
u s e of
of
the equations
the method
Fourier
series
making well For
ced
They)
the computational
k
2
is a l s o
to that
for
selection
termined
reasons
C3.5)
useful.
the source define
in c a s e
of
the number
of
is o b t a i n e d .
terms
of
algorithm.
of
the
t h e heat
of s o l u t i o n
Substituting
outside
it,
coefficients
as
which
is p l a -
case
coef-ana-
a n d X =X the solu1 z
method
Fourier
The number
of t e r m s
changes
source
is ii~ this
CI=C2 The
formulae
conductivity
the summed
the temperature
sol vl ng
[[3].
The method
region
and
the Fourier
conditions
of
by way
region
the t h e r m a l
homogenous
way that
determined
with
above.
calculation
been
the reprezer~tation
the substrate
presented
in such
which
+ 2 T ~xZ > C DY ) ' Cn° S C Y n m=1
distribution
the
in t h i s
for
boundary
temperature
in the part,of
have
equations
as of t h e s u i t a b l e
logou~
used
and
= TCY>o
u s e of t h e depender~ces
flcient
tion
ToCY)
the
- in t h e f o r m of
n:i co
differential
determine
of v a r i a b l e s
expressed
be
co
TCx'y)
ordinary
can
= T
=
functions
a n d C149
[4]:
T(i)Cx'y)
The
C13)
of s e p a r a t i o n
I
TC x, y)
k---'5
--< C2:
+~
k
(5
Q~
-2T(2) = Hi - [92'
T (2)
1
=
"91
--
'
t
the region
T (2)
02:
h
i
of series
automatic has
been
has
been
de-
calculated
from
se-
B. Wmz
94
ries,
as a function
a given
of t h e n u m b e r
pretermlned
value.
closed i n i n t e r v a l Temperature number basic has
:3.
sources
utilized
vement
The
PROCESS
of
heat
placement
strategy
application
of s u c h
of maximum
should
b e made.
lowest
to be the quality
the
sources
measure.
parameters
can
be assumed
that
their
of t h e w h o l e
circuit.
power.
emphasizing
There
should
Dj
of
a greater on the
solution
optimization.
solutions,
the
impro-
b y w a y of
proper
in the minlmization
and
of
steps
of
on
has
also
be found
the
has
been
to obtain
the sum
of
minimum
surface
formulated the
to ambient
the following
possibly
sources
temperature
causes
stability
the elements
The
the change
of p a r t i c u l a r Therefore function
it will
operation
introduced
to
the sources
with
the function
in-
reasons:
reliability.
of
the
of t e m p e r a t u r e
the objective
been
the significance
calculation
t h e substrat.e
s o as
and
of
direct
its
the time
reliability
However,
the
proce-
heat
of
of
the optimization
from
inner
minimization
thereby
coefficient
placement
in r e l a t i o n
durability
and
function
been
in
assuming
by the
value
the
mainly
be placed
sources
the quality
weight
of
analytical
can be obtained
temperature
work.
improve
The
has
determined
iI~ t h e m i c r o c i r c u i t .
It r e s u l t s
generated
decreasing
The obtained
in each
should
heat
of e l e c t r i c
been
pr.oblem of o p t i m i z a t i o n
of t h e i r
rise
has
coefficient
because
temperature
occurrence
constructional
condition
a quality
of p a r t i c u l a r
elements,
and
and minimum
temperature
creases
of t e r m s
than
OPTI MI ZATI O N
diversification
Thus,
Heat
to be smaller
o n t h e substrat.e s u r f a c e .
is t i m e - c o n s u m l n g
values
the
of e l e m e n t s
optimization c o n s i s t
of
field
as follows:
from
principle.
the circuit
the circuit
sources
had
the number
in the microcircuit
OF PLACEMENT
temperature
dure
resulting
in the process
Disregarding
heat
Practically,
of t h e s u p e r p o s i t i o n
been
terms
4 0 - SO.
distribution
of heat
of i t s
and
the objective higher
unlt
determined
by
the dependence: NZ
min f C x ) xEX where.x is misslbile
=
CIB)
rain ~. D j - T P j C x ) . x~_Xj=1
the vector of solutions,
the sources coordinates,
X is t h e set
NZ d e n o t e s t h e number o f s o u r c e s , T P j C x ]
of p e r is
the
Thermally conditioned placementoptimization temperature stands
for
increase the unit
The conjugate been
of t h e j - t h power
gradient
used in the search
following - there
conditions
- strict
gradient
adherence
- constraints sons
sulting
from
traints
has
resulting
assuming between and
here
Ci,v),
problem
of
with
substrate
width
and
CFig.~). cidence
of t h e l e n Q h t
function,
the
of t h e
objective
Ci)
I
rea-
[8].
of
layer.
The re-
constuctional theory
cons[B,V],
the analoqy
of
the elements
node-outset
inside
voltages
in t h e c o o r d i n a t e
of t h e d i s s e c t i o n U
placed
oi"
into a set
of
graph
Cv)
system
the rectangle rectangles
is
O
v q and
the sink
v u of t h e g r a p h
current
I
- with
the
problem
between
the geometry
and utilizing
Mapping
widh
is
the graphs
of t h e or~e-port
and
of
realization
r e p r e s e n t e d by a polar
the image
O
of
and
of
one-port
The edge
The solution
constructional
to a single
structure
terminals.
the edge
relations
on the basi~
representation
The source
and
assignment
connections
region
currents
the one-port
has
X is r e q u i r e d ;
technological
being
-
obtained.
region
concerning
resistive
of r e c t a n g l e s
R o = C U o , I o)
determination
the technological
of t h e c i r c u i t
a graphical
[5]
consideration:
o n t h e w a y to t h e m i c r o c i r c u i t
considered
mesh-loopset
j).
constraints
of o b j e c t i v e
Gj
character.
an electrical
the rectangular and
from
functional
planarity
a set
into
of a n a l y t i c a l
topography
their
been
taken
and Gmax=maxCG
linear
the optimum
to the permissible
of i t s
and
source,
with
whet e a s
n j = G j / G m a x,
value;
O n e of t h e s t a g e s
elements
method
being
a r e of l i n e a r
the design
the j-th
for
is a possibility
function
of
heat source,
95
voltage
U
the length with
the
t h e elemer~ts a n d
I
0
poles
corresponds of
with
the
the rectangle
preservation
the structure
iq
correspond
of
R
the
in-
of t h e i r
col~-
iu=l o i
is
im
RI R2
R3
f//l;///////] 4
• , ? L, ' ~ ' r
"7 I
I
V
R7
5 ~.
7_""___1 Fig. o .
Mapping:
resistive
rectangles.
one--port
,,v I u= o 0 -
pol.~r
graph
v
- dissection
into
96
B. Wmz nectlons [9],
polar
use
/ P
R
Af
and
which
the
been
making
Af in
has
graph,
area
ments
from
the
to
--/P
where
On
the of
basis
the
Ohm
and
Kirchhoff
laws
u and
Pu
diagonal
is
of
vector
u of
the
division
inside
the
the
which
circuit,
the
the
known
R
have
matrices
matrix
been
of
the
representing
particular
values
of
circuit the
e l e ~-
matrix
determined
and
P then
expressions:
values
to
of
incidence
R u inside
V
the
assigned
and
R-'
the
rectangles
reduced
rectangles
the
V
basis
dependences:
are
values
Eventually,
ly
Bf
the
substituted
V
the
the
u
placed.
elements
I
of
on
respectively,
of
are
obtained
which
the
planar
the
R
V
vectors
of the
:/P
V
.
I
618)
and
U
microcircuit
elements
structure
can of
have
substrate
be shifted the
been
has
circuit
obtained.
surface been
with
into
explicit-
a specified
topography.
4.
NUMERI CAt
CALCULATI ONS
On
the
of
zation
method
ting
and
ters
of
on
basis
the
numerical
evaluating the
temperature
function
the
expression: T = TCx
the
is
x
o
the
effect
is
o
^ , x. ~' x
of
are
heat
heat
geometrical
optimum
which
vectors
of
starting
their
ratio,
of
optimi-
investiga-
material heat
parame-
sources
and
distribution
generally
~).
P is
and
Temperature
~.
the
performed,
placement
can
and
be
is
determined
by
fig) and
power,
optimum
Sr/Sp
is
k is
the
thermal
of
the
optimization
the
temperature
points, the
NZ
source
conductivity
is
sur-
and
coefficient.
the
selected
functioning parameters
an
example
presented
in
Fig. 9 h a s
planar
the
the
been
Sr/Sp.
circuit,
possible
of
configuration.
sources,
transfer
the
on
surface
illustrate of
effect
P,
the
model
have
variables
NZ,
mathematical
calculations
field
many
to substrate
To
the
of
and
number
face
the
elaborated
microcircuit
the
where
the
of
on
the
power
amplifier
been
taken
into
representations
of
the
whose
method
and
the
distribution
in
schematic
condlseration.
circuit
structure
diagram
One hes
of
the
been
pre-
Thermally conditioned placement optimization
4 ! R3
"11
o+Uz
5~",----IDR22T~T:
1
97
ii_._]1
7
.1 F i g 3.
sented
in
circuit
Fig. 4 b
can
diagram
of power- a m p l i f i e r .
i n t h e f o r m of a p o l a r
be obtained
the conductive lution
Schematic
path
making
under
of t h e e q u a t i o n
u ~ e of
the discrete
system
(17)
graph
Cthe
planarity
the possibility element
arid (I~)
of
- the diode
assuming:
of
the
conducting D).
The ~o-
I =50-i0-3m
(the
o
substrate
lenght)
and
the values
41
0
Pu = 10-4"
been graphically
slze
of t h e
the matrix
P
( i n mz) e l e m e n t s
has
]
2 1 0
consideration
of
i 2
illustrated
of c o n t a c t particular
in Fig. 4.
surfaces
elements,
for
As a r e s u l t the terminals
the numerical
(a)
of
the additional
as well
va]ues
as
of
de%ermining
the the
(b)
0
22.8
50 1 3
50
.4
3
17.5 4 26.2 3O Fig. 4.
7.8
17.2 7
5 26.6 Planar
representation
phical
solution
section
26.6"~/
34.4 of
the a m p l i f i e r
of t h e n o n l i n e a r
into rectangles,
b)
polar
circuit
equation graph.
and
system:
the graa) d i s -
B. W I S Z
98
boundaries have
of
been
the
elements
can
be
shifted
2
a.O
-< y a
<
22.4
3
3.0
_< y 3
-<
18.
4
17.3
< y4
_< 2 2 . 2
4.0
_< x 4
< 24.0
S
24.4
< y~
< ~.7.0
4,0
_< x 5
E 27.7
6
lg.
<
y6
-< 2 7 . 0
29.7
< x6
-< 3 4 . 8
7
19.5
_< 2 7 . 0
36.6
E
!
have
oi
=T
02
_< y 7
been
Cl=C2=30-10-Sm, T
5
K.
=293
O. O x 4 . 0
lO-Sm,
2.0
2.
l. O x 2 . 0
lO-Sm,
0.008
3.
O. O x 4 . 0
lO-Sm,
2.0
4.
~-.0x2.5
lO-Sm,
O. lS
W W
1.8x2.2
lO-Sm,
0.01
l O - S m ,
0 . 0 0 1
W
7.
1.8xe. 2 lO-am,
0.06
W.
which
is
been
in
the
corresponding
, . .....
heat
region
presented
_< x 2
< 28.0
31.7
_< x 3
-<
I
=k
dimensions
2
=22.75
and
x7
46.0
47.0
following W.CmK],
power~
of
data: ~=c~ +c~ = ~
in to
sources of
for
selected
permissible
Fig. 6.
The
these
positions
,. . . . . . . . . . . . . .
the
• . . . . ....
initial
solutions
temperature has
been
point
determined
distribution
in
the
illustrated
in
. . . . . . . . . . . . . . . . . . . .
1111111
• ....
3333333
1111111
3333333
1111111 1111111
2 2
)333333 3333333
44 44
66 66
77 77
55
SS
. .°
°.°
• . . . . . . . .
Fig
5.
° . . . . . . . . . . . . . . . . . . .
Initial
2
elements
W
2. O x I . S
of
24.5
the
< ~1.3
W,
~.
placement
The
< xl
for
out k
4.0
W
5.
Optimum
carried
S
h=O.V.lO-anl,
i.
.....
x, lO-Sm
-< 1 4 . 2
follows:
has
y,lO-Sm
-~ yl
was
circuit
the
3.0
W/CmZKg,
above
which
I
=42. 2 4
C F i g . 5)
:
coordin.
number
calculations
A=50-lO-Sm,
as
inside
obtained:
element
The
subregions
placement
' . . . . . . . . . . . . . . . . . .
ot
amplifier
elements,
'
.....
99
Thermallycondition~ pla~ment optimization o
.
.
o
,
J
m
,
.
,
,
,
.
o
.
.
g
,
,
.
.
o
.
.
O
,
.
,
.
.
,
,
.
.
,
,
.
.
. . . . .
,
.
.
.
,
o
,
.
.
.
.
.
.
.
o
.
.
.
.
.
.
.
.
.
.
.
.
.
2 Z
1 1 ~ 1111111
3333333 3333333
#L, ~
66
~
66 77 77
55 55 O I U O
I # O Q I t
I
0
#
I
~ e e
Fig. 6.
Fig. 7.
This
of
the
into
condiseration it
Final
ha~
is
use
As
the
of
temperature
decrease
perature
field
results
Further
. . . .
I
Q
¢
0
of
have
carried
been
and
the
fact,
heat
that
repeatedly result
have
~
•
m
0
m
4
m
i
amplifier
variability
optimization of
there
the
out
0
Q
.
*
elements.
for
tranfer
the
corrected
coefficients,
in
the
temperature
the
subroutine
when
performing
been
obtained:
the
above
Temperature surface
good
in
of
the
been
at
the
been the
vataking
function.
of
temperature
the
optimization
= 2 3 . E~6 W / C m K ) ,
X
compatibility
highest
have
selected
measured
with of
obtained
circuit
calculations
values
have
has
sources
diversification
of
obtaining
Cerror
of
called
As a r e s u l t
ter,
0 o Q
W/CmaK).
~=38.86
circuit
~ g
placement
their
made
procedure.
tally.
f
thermal' c o n d u c t i v i t y
calculation
The
O I e Q
calculations
lues
He~e,
1 6
points
the
power
has
been
use
a considerable
also
and been
the
tem-
reduced.
verified
experimen-
of
thick-film
of
real
radiation
measurements
and
thermome-
calculations
_< l O ~ ) . decrease
tained
by
placing
higher
thermal
of
the
the
temperature
sources
conductivity.
nr
field
1 and
diversificationcan
3 in
Assuming
the
part
Cl=lS-10-Sm,
of
be
substrate
C2=30-10-am,
obwith k = Z
=~aO
W/CmK),
Fig. 8,
and
Fig. 7 h a s
optimum
the been
placement
temperature shown
in
map
Fig. g.
of
heat
sources
preserving
the
has
been
temperature
presented scale
in from
B. Wmz
100
.o . . ,, , .
o .....
**l* IIII
.
-
**~ J/~:,/ l l , l -XXY
R,~RPRk .
o o,,
I'/
i!i?:rll!rllil
•
.
.
"-
+"*+'"+.,.
. % T; % "; ',: "; % ~ '- ,: "4 ; "(
•
. • . .
.
.....
.
.
,
, .....
,
"
'
'
,
,
,
, , , , , , , . . . . . . .
,
,
,
H
#
+÷+.,..+++
~ ( ~ . ' ;
, ,
. i . ~ ÷ + + + . + ~ . *
"< } '{ X ~l ~ v. ~'- '( '~ '<- ~: ~
............ ~
I
:
~
I
I
I
I
H
~
,
,
'< '< ,
,
,
,
T~
~46.8~
35~.93
3 6 5 2 3
317.26
395.3q 393.72
TMAX = 394.82
XXXX
+~++÷ .+++44+
XXXz
,
. . . .
,
,
,
,
,,
,
,
....
,
,
. . . .
.
.
.
.
.
.
, ....
'M~ fd i~ H /. lf b H t! H t/ #
l~ li ft l' ~# l~ ~l tt #l H t~ H ff H l M ~ H fl H l+ tt H # H t N#Hh~HbH i~ffq.~l###HltHl$~ ff ff if t: H i~ H H if @ ~ . . . • ......
311.09 369.~7
377.5~
403"-42 ~O~.B4
411.51 ~O9.Bg
~I~.S~
TMIN = 3 5 9 . 6 7
XXYXXXXXX XXxXX~ XXXXXX . . . .
.
.
.
.
XXXX
XXXX~ X
XXX~
W-~XXXXXX~ xYXXWX XXXXV#
++~++i44÷4~+++++++ ,++++++++++44÷+
WXxY
.
XX~X
+ ~44444÷ ++~++÷÷÷+44~
363.01
361.~9
353.31
379.47
+++4+
,
TMIN = 3~8.76
TMAX = ~ 1 9 . 5 8
+
- -
~-~7,,~.~
I'HH~HMCI~H#itlI:ItnHtt ~ , . , , , , ,i i , i #!M#HbIeH#f#'HI~IIY/II~MHH#t#¢~IIhHt'H#I.~ tl ~.f f tf fl t; H if b li P# t} tl l' #~ t, ~ #; H ll ~ l' H ~ ll li tl ll #$ t.l ~ ~ #i h lt #I tt ~tiUt:fli/M;~fftlflHt#ff~t!Nllfilllft~e~H#MI H~iit~tWliMPlP. ffit~itf,"~H~ HI fill till! I .; lt l, lt li i~ ti H ti #J l* ~ l i i/ fy H ~ l, ll U H I p ~ ' t t l u ~ . i / l ! t f ~ # # J l f : , t f l ~ J # i tl lt ti l~ ti lt li l~ ll :: tl t, it n lt ll ll #~ M f Jy t l J , u lf ll -~ h M ll fl l fY f~ H ;" f# l~ #~ ~ ~ lt : J/ ll #~ H l~ # #t #f f~ ll li #~ tl #l ~ h f+ l~ ll H if lt t# #l f l fl ~lH Ji tf t: . ## H tt H t, tl fl lt {: fl l, ~l. v M t# tt ## #4 fl l, #£ fl !ff i i H tl #'11i 11 ;: f# L it tl It tt M i,I H !1! li 11 Ii it hi it i~ It tt fl it f H H i1 ~ fl I fl #i H f ~ It H ~ H f ffl#ll#~tfllflllH:'#Jf~#lllti;;MHltf'MHllJ}Ht~ @t~tf#IflUHF'HIIHfI#~IMg~It. ###~Ht H ## ~ f; H tt ~ l; tl t tf h b l~ #~ tl M H tf t h H l' tl H t: l. n tf fl l, :" H If H L ## h H H M tt ti tl ff #f tt tI • • . ...." • • • . • . • . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . .
• ~44++ ++++++
I111 RR&~RR
XX-
, , , , ~ , , , , , , ,
3~8,
ii
XW~/YX
.....
",, ~
1 XX--
÷+++
,
, ,
~l
4444
%~.3';~
, ,.
-,,#
.,.+lxw +++X',(X-
',.~x~;:
, , , . , , , , , ,
L-IJ
,
,
,
"(?(),.'('~:'K++÷+
, , , ,
. . . . . . . . .
; i '. '. '. ; ; '. '. r
•
.
, , , . , . , . ,, ,,
..
, , , . , , , , ,
, , , ....
, ,,
~X%~7,~ .
, ......
, ,
....
,~':(;-'Y.~4
~..;(>,%;:~v.
"
o .......
$~'x:I>.~Y.'~ ;:':%,"7~,
;:W"/.X~
+++~
, , + . + + ÷ 4 + + ~
.....
}';,~;~.x'~xx~
~,;X~X~.
++++
Xl4
,,.
XX++.+x.',.';.v.~t' 7+ X'c.';:'zl
WX~W,.+÷
XXXXX
•
,..
w,+::
+÷++4444++~++÷
XXXY
~+.+.+++++++-4 ++++i+++++++*++
XXXX--
XWX
'X~X
+~+44i+~+~+,+444++++~ ++++~+÷++ ~4÷&+÷+~4+44~+~++
+4~++++ +~++44
WXXW~ XIWXXX~
"
+++*'+ +++++~
+÷+++~÷+++ ~4+44444÷+++
..... :1..414 .........
. . . . . . , . - .
Fig
7.
. . . . .
Temperature pl
ac
emen
. . . . . . . . . . . . . . . .o..
map t.
for:
a)
the
....
initial
...
. . . . . . . .
piacement,
,.,,
. . . . . . . .
b)
the
final
101
Thermally conditioned placement optimization °,
,
,
.
,
, ,
•
.
•
,
o
.
,
.
.
.
,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
°
.
.
.
.
.
.
, . ,
hl~li~4 11~II~1 111111~
.
, . .
•
,°
.
.
.
.
.
.
0.
,
333333~ 333]333 5333333 33333~3
I ~ I
~4
66
77 ?7
bb
55 55
~
~
O
m e
Fig. 8.
~
~
~
e ~
Final
I
6
~
Q
~
~
placement
~
of
g ~
~
*
i
~m~
e
elements
g
~
~
on
.
the
. . . . . . . . . . ; ; : : j j ; i : ...... . ~ 9 ~ , ~
~****÷÷÷*'
.
~
t * # * ~ * *
" ......
4
4
~
l
*
l
÷
.
*
.
.
Fig. 9.
.
.
'
.
.
.
.
.
.
o . .
- 386.29
Temperature Fig.
~.
.
.
.
.
TMIN map
•
.
.
.
.
.
.
.
.
.
.
÷
.
~
.
~
substrate.
WX~XX
**4**÷°*÷*
&
l
t
~
s
&
l
S
[xXx*****+*.
. . . . .
~
l
l
&
......
l~x~.
l
*
l
~
4
*
÷
4~I,
&
.
*
.
4
÷
4
~
.
.
*
.
4
4
4
.
.
.
.
!"" 4
~
4
*
.
.
4
.
÷
.
.
4
.
4
~
4
,
4
4
~
.
~
.
.
°
.
.
.
.
= 360.76
corresponding
with
the
final
placement
from
8.
CONCLUSIONS
The
results
of
the
the
suitability
of
conditioned
basic
elements
.
0
S ............................... ~ 4 ~
.
. . . . .
'""÷"~ r . . . . .
....
÷
:: .......
•
...... : ; : ; : ; 1 ; . . . . . . . . . . .
~XX+.*÷***+÷.÷,+**+~**xx~
XX
~
heterogenous
~-*÷.*
TMAX
th@
.
,x~..÷..,.
•
mally
.
***********************
X~XX~
~4&4÷*÷÷~*I*0
firm
.
:2:,,.'x~~
......
o
.
the
process
purpose inside
investigations
of
the
the given
of
proposed hybrid
task.
which
substrate
which
have
optimization micricircuit is
the
region.
been
method
in
designing,
placement The
conducted
con-
the
ther-
fulfilling
optimization
analysis
of
the
of oh-
102
B. Wlsz
talned i.
results
Final
leads
Location
to formulating
of
heat
the
sources
following
depends
conclusions:
on
the
position
initial
which i n d i c a t e s t h e e x i s t e n c e o f t h e o b j e c t i v e f u n c t i o n l o c a l m i n i ma, w i t h t e m p e r a t u r e v a l u e s d i f f e r i n g
most f r e q u e n t l y t o an
incon-
s i d e r a b l e degree. 2.
Thermally
significant
temperature
distribution
iS substantially
wer
plification of
of
placement
to cut upon 3.
the
the
of
down
diversified. optimization
thermally
the
of
the
optimization
variability
rotation of The
the
solution
sltiveness
of
the
the
to
Lime
the
thermal
values
k and
ness
of
the
method
functioning.
rical
system
method
ristlcs
of
temperature optimization
be
stressed,
it
is
which
the
~ is
such
however,
by
means
is
heat
Thus,
necessary
that
in
to ensure
thermally
of
sources
has
been
of
when
take
ir~to
transfer
the
[i0] after
the
This the
by
approxi-
low
seh-
coefficients
accurate
complexity
a circuit
to
transfer
the
thoroughly
repeatedly
characterized
ilustrates
heat
allows
influence
k al,d heat
to make
in
sim-
correctioh
This
allows
purposeful
distribution
the
po-
~=fCT>.
problem
not
called
and
more
in
the
a case,
negative
possible
circuit
of
and
range.
the
is
function
a practical
of
signing
any
conductivity
and
tion
when
alone.
which
conductivity
in
determinate
solution.
k=f
considered
in
COl~sist
without
thermal
placement
which
selected
elements
procedure,
changes
It s h o u l d
be
procedure
temperature
dependence of
can
temperature calculation
performing
~ in
ones
Purposeful,
obtained
of
coefficient
i.e.
significant
calculation
quality
the
sources,
in a c i r c u i t
The subroutine
account
4.
heat
determinathe
correct-
conditioned use
of
thermal
method
process
the
nume-
characte-
will of
de-
verify
placement
completed.
REFERENCES
i.
S l a g h T.D. , N a g l e in hybrid circuit
Sym@~.
H.T. , R u w e .W. , O n t h e u s e of t h e r m a l s i m u l a t i o n design. Proc. o/ ~Ae t 9 7 6 In£. M E c r o e [ e c £ r o n £ c s Oct. ll-13.1g70, Vancouver, Canada.
2.
K r a m e k J.P. , S o m e a s p e c t s of placement optimization of thermal significant c~nQnts in thlck-film hlgh-power hybrid integrated circuits. MEcroe[ec~ron~cs and ReL~ab~L(~, vol.~3,nr 3, lg83.
3.
W i s z B . , Heat sources placement o p t i m i z a t i o n i n hybrid microcirc u i t . D ( s s e r { e t ( o n , T e c h n i c a l U n i v e ~ s l t y o f Rzeszdw, Poland, Ig88.
Thermally conditioned placement optimization series
4.
Churchilll R.V. , Fourier Hc G r a ~ - H ~ L , Inc. Ig~3.
5.
Kr~glewski T. , R o g o w s k i T, R u s z c z y 6 s k i A. , S z y m a n o w ~ k i optymalizacJi w j~zyku FORTRAN. WNT, Warszawa0 i g84.
6.
S e s h u S. , R e e d M.B. , L i n e a r g r a p h s M u s s . , A d d Z s o n - W e s Z e N, Ig61.
7.
Y_/bert K. , C o m p u t e r aided planar circuit layout. Pr'oc. European Con/. C £ P c u £ L s T h e o ~ and Design, L o n d o n igv4.
IEEE
~974
8.
T u t t e W.T. , T h e d i s s e c t i o n Jo~Pna[, nr 7, I g 4 0 .
Duke
Ma£1~.
g.
Brinkmann K.D., Das Chip-Netz als ein Hilfsmittel f~r den yon MSI- und LSI-Schaltungen. N u c h ~ ' f c A £ e n £ e c A n ~ M Z. , B d . 3 0 . 1977.
of
and
and
rectangles
boundary
103 value
electrical
into
problems.
J. ,
Metody
networks.
Read£n~
squares.
Layout H. 12,
lO. N i e d z i a ~ e k T. , K a l i t a W. , Simulation of stationary temperature field in hybrid microcircuits designiz~g. Proc. o/ the ~ - Z A clnd 6--£A C o n / . o / tAe Intez'nc~t£onc~[ S o c £ e t ~ /oi- H~,br.£d H~cr.oeZectro;%£c_~ Po[cznd C h a p t e P , W r o c l a w , Ig85.
0026-2714/9053.00 + .00 © 1990 Pergamon Pre~ plc
THERMALLY CONDITIONED PLACEMENT OPTIMIZATION OF ELEMENTS IN HYBRID MICROCIRCUITS BOGUS~AW WISZ Technical University of Rzesz6w, Rzesz6w, ul. W.Pola 2, Poland (Received for publication 2 May 1989)
T h e p a p e r p r e s e n t s s o m e of t h e p r o b l e m s in u s i n g t h e r m a l s i m u l a t i o n for t h e p l a c e m e n t o p t i m i z a t i o n of e l e m e n t s , which ape heat sources in hybrid microcircuits. T h e a n a l y s i s of h e a t exchange mechanisms in thick film circuits has been carried out and the mathematical m o d e l for c i r c u i t s w i t h s l n g l e - l a v e r and non-homogenous substrates has been elaborated. Analytical s o l u t i o n for d e t e r m i n a t i o n of the temperature distribution in t h e f o r m of s i n g l e F o u r i e r series has been utilized in optimization process. In the proposed procedure based upon the temperature c r i t e r i o n of o p t i m i z a t i o n and the gradient method, technological and constructional constraints have been taken into consideration. T h e e x a m p l e s of c o m p u t a t i o n results have been included and the conclusions useful in the process of computer-aided h y b r i d nhlcrocircuit d e s i g n h a v e b e e n p r e s e n t e d .
I.
I NTRODUCTI ON
The
analysis
is a s i g n i f i c a n t this
analysis
which
occur
ronment
but
physical
determining
temperature
the
aiming
mization
and at
which
process
and
adjoined
cult.
The
necessity
heat makes
sources, work
pendent
on
makes the
Consequently,
the the
and
between
dependence of
to
between
the
solving
of
place
a single of
circuit
on
the
plane
substrate electronic
at
a short
topography of
the 89
and
the
in
thermo-
heat
to
condition on
design.
plate
the
of
One
distance among
one
other
of heat.
of
microci*'--
eleme1~ts
from
of
opti-
placement
the
of that.
conditioned
that. of
unfavo0rable
iden-
facilitates
interactions
value
analy-
determining
allows
network
elements,
envi-
The
elements,
thermally is
the
changes.
engineering
[1,2]
particular
improvement
the
and
out.
effects
components
stated
the
Carrying
thermal
it
it.. T h i s
circuits the
of in
the
in
of
mainly
purposeful
construction
elements
consist
occurs
possible
the
temperature
which
causes
of
microcircuits
them.
microcircuit
optimization
the
temperature
upon
the
require
of
layer
to
designing nature
in a m i c r o c i r c u i t
regard
hybrid
the
function
interrelation~
microcircuit
problems
the
of
of
learning
the
distribution
with
condition
the
conditions
process
only
also
the
inference
not
in
work
the
inside
exchange
thermal
in
both
of
the
factor
parameters
heat
thermal
requires
sis
tify
of
being
another,
things,
power
de-
sources.
condition
can
90
B. Wmz be obtained substrate
by way of correction
surface.
of e l e m e n t s -
Solution
(heat
developing
sources)
a method
of
the heat
sources
of t h e p r o b l e m
inside
of placement,
substrate
of c a l c u l a t i n g
placement
region
temperature
on
the
optimization
consist
in:
distribution
useful
in optimization; -
determining
the criteria,
selecting
the method
and
elaborating
the
opti mi zatl o n al got i thin; - formulating them
into
technological
consideration
Th~ application designing to take most
practice
into
2.
and
HEAT
sure cated
ment
lyzed
region
method.
The
and
necessity
method.
assumptions
- stationary der atl o n ;
of
work
practise
will
the allow
thermally
the
increasing
its
parameter
and determining
out
the
selecting
model
h a s ,been
properties accepting
of
of
tame. p l a c e -~
temperature
further
at
the
the
to
advan-
the
ana-
every
point
optimization
formulated
on
the
theoretical
"permissible
conditions.
des-
feasibility
pol,~t of
and
ne-
preference
has
gradient
it
the
period
gives
en-'
compli-
and
of
repeated
when
boundary
easy
short
at a n y
temperature
of t h e r m o d y n a m i c s ,
be littIe
and
the method
the temperature
structural
should
in the process
of t h e c i r c u i t
the
but
only
the accuracy
the model
Besides,
not
is n o t
in a possibly
mathematical
the following
analysis
between
is s i g n i f i c a n t
analytical
ensure
in
OF SOLUTION
of c a r r y i n g
points
to determine
which
in particular
requirement
system
to define
of t h e l a w s
introduced
this
complexity
of t h e m i c r o c i r c u i t
analysis fying
The
programme
microcircuit
field
t i m e is a d e c i s i v e
enable
of the field,
basic
to the
the analytical
which
- METHOD
with
at n u m e r o u s
tages
the
taking
procedure.
which
in engineering
of the equation
calculations
of
at a c o m p r o m i s e
optimization.
applying
MODEL
Complying
The computation
circuits
and
of o p e r a t i o n .
sufficient
relevant
constraints
computation
the factors
of t h e t e m p e r a t u r e
arriving
of s o l u t i o n
power
conditions
EXCHANGE
as well.
cription
of h y b r i d
reliability
accuracy
cessitates
of t h e e l a b o r a t e d
work
The method
constructional
in the optimization
condsideration
favourable
stability
and
There
simplihave
been
assumptions:
conditions
of t h e
microcircuit
are
taken
into
consi-
Thermally conditioned placement optimization - dielectric heat in
-
-
rectrangular
ber
of
the
thermal
as
surface
well
is
as
are
neglected
in
the
analysis
of
oof
heat
the
substrate
sources
with
coefficient
there
is
constant
a definite
power
has
constant
and
lower
num-
value;
value
independet
heat
the
into
exchange the
flow
basic to
power
Model
conditions:
to
the is
of
function
the
here.
parts in
of
heat
leads
above
linear
taken
the
is
transfer
into
and
account
circuit parts
circuit
the
substrate
lateral
in
the
the
in
with
The
coefficient
from
assumptions
problem
layers
of
the
bei~g
surface~
circuit
constar~t:
is
omitted;
neglected. heat
transfer
influence
boundary
which
elements
thermal
elements
region
i5 ther-
is
conductivity
heat
The
CFig. l).
the
substrate
being
region
of
equation
conditions.
the
different,
microcircuit
the
upper
environment
by
homogenous
considered
of
the
heterogenous
two
both
of
from
value
dissipation
reduced
in
Cflat)
boundary
heat
heat
mal
plane
conductivity
assumed,
- the
On
layers
temperature;
complex
-
conductive
exchange;
the
of
-
and
91
sources
temperatul
T £x,y.z] p
has
can e
derided
be
been
placed
distribution
determined
by
the
expression:
T~i>Cx, y,z) P T (x,y,zD
for
O~xSA.
O~ySCI,
OSzSh
=
CID
P
T{2}Cx,y,z)
for
p
where and in
x,y,z
are
T~*~Cx,y,z] P its
the
first
the
denote
and
thickness
coordinates the
second
and
of
the
substrate
subregion,
CI(C23
O
i~
CI
considered
region],
temperature at respectively,
the
O_
widh
of
the
the
A is whole
T~S)(x,y,z) p poir*t .Cx, y , z 3
the
lenght,
CpartD
h is
substrate
yJ
C2 C1 1
A
Fig
I.
Thermal
flat. m o d e l
dimensions ficients,
of S
, S i
of
substrate, - heat 2
a heteroge*~ous k
x
circuit.
, k 2 - thermal
source~.
A,
CI,
conductivity
C~
--
coef--
92
B. WISz plate,
satisfies, t h e
conditions When
in e a c h
the heat
thermal
Laplace
with
the defined
part
of t h e
above
boundary
subregion.
source
is p l a c e d
conductivity
differential
equation
in the
coefficient
equation
k , for
the
i
substrate
function
with
T(i~Cx.y.z)
the
boundary
is c o m p u l s o r y : CaD
conditions:
6T(pt/6x = 0 for
x = O i x=A,
C3)
6Tp¢ i )~6y = 0 f o r y=O,
C4D
ai[T(pi~Cx, y,O)
- To.,] C5)
kt •6TCpt/6z =
G -
where:
a
and
~
the
p
VaT(i)Cx,y,z) = O, p with
the
are
az[T(pi'Cx, y,h)
the
heat
transfer
- Toz ] d l a
(x.y)
coefficients
. S
from
the
lower
and
z
upper
parts
of t h e s u b s t r a t e ,
density
of t h e
ambient
temperatures.
On t h e o t h e r
source
hand
the
~T('}Cx,y,z) p with
the
boundary
=
with
respectively,
the surface
function
G - is
S,
and
T
the
thermal
and
T
Oi
T(Z'Cx,y,z) p
02
satisfies
power
- are
the
the e q u a t i o n :
O,
(69
conditions:
6T(p~'/6x
= 0 for
x=O
6 T p(2) /6y
= 0 for
y=Ce,
and
x=A,
(7)
C 8)
ai[T~Z)Cx,y,O)
- To1 ] (g)
On t h e
border
0 2
[C
of r e g i o n s
Cx,
y~
Cy=CID
there
occur
t h e cor,ditions:
T(i~Cx,Cl,z~ = T(2>Cx,CI,z), p P
CIO~
ks6T(S/6y = kzTp( 2 ) 76y. In t h i n of
temperature
which
affords
substrates along
the
c11> with
good
thickness
possibilities
for
thermal of
averaging
conductivity
the
plate
the d i f f e r e n c e s
are
the temperature
inconsiderable
iI~
relatioz~
Thermally conditioned placement optimization to the substrate
thickness
to the two-dimensional
and
reducing
problem
93
the searched
temperature
field
[3]:
h
TpmCX,y)
= ~ ; TpCX, y , z ) d z
= TCx,y)
Cla)
o Substituting tions
(~9
region
the
and
twice
C69
differentiated
and
using
0 S y ~ Cl,we
T(l>
+ T(*> MY
x×
expression
the adequate
CI~)
boundary
for
the equa-
conditions,
for
the
obtain:
-2T(t)
~0
= ~l
-
~2 +
F;
F=
dla Cx,y) Qi d l a C x , y )
~ E S
C139
where: 2
c~ + ~
=
1
~
2
~ T
=
k I T - ' h- '
t
+a T
Ol
~2
2
×
a n d for
+
xx
2
yy
CI
y
-<
a T 2
h
- making single
t
1
=
'
C149 +a T
Ol
¢92
2
k
2
The searched
02
h 2
solution
u s e of
of
the equations
the method
Fourier
series
making well For
ced
They)
the computational
k
2
is a l s o
to that
for
selection
termined
reasons
C3.5)
useful.
the source define
in c a s e
of
the number
of
is o b t a i n e d .
terms
of
algorithm.
of
the
t h e heat
of s o l u t i o n
Substituting
outside
it,
coefficients
as
which
is p l a -
case
coef-ana-
a n d X =X the solu1 z
method
Fourier
The number
of t e r m s
changes
source
is ii~ this
CI=C2 The
formulae
conductivity
the summed
the temperature
sol vl ng
[[3].
The method
region
and
the Fourier
conditions
of
by way
region
the t h e r m a l
homogenous
way that
determined
with
above.
calculation
been
the reprezer~tation
the substrate
presented
in such
which
+ 2 T ~xZ > C DY ) ' Cn° S C Y n m=1
distribution
the
in t h i s
for
boundary
temperature
in the part,of
have
equations
as of t h e s u i t a b l e
logou~
used
and
= T
u s e of t h e depender~ces
flcient
tion
ToCY)
the
- in t h e f o r m of
n:i co
differential
determine
of v a r i a b l e s
expressed
be
co
T
ordinary
can
= T
=
functions
a n d C149
[4]:
T(i)Cx'y)
The
C13)
of s e p a r a t i o n
I
TC x, y)
k---'5
--< C2:
+~
k
(5
Q~
-2T(2) = Hi - [92'
T (2)
1
=
"91
--
'
t
the region
T (2)
02:
h
i
of series
automatic has
been
has
been
de-
calculated
from
se-
B. Wmz
94
ries,
as a function
a given
of t h e n u m b e r
pretermlned
value.
closed i n i n t e r v a l Temperature number basic has
:3.
sources
utilized
vement
The
PROCESS
of
heat
placement
strategy
application
of s u c h
of maximum
should
b e made.
lowest
to be the quality
the
sources
measure.
parameters
can
be assumed
that
their
of t h e w h o l e
circuit.
power.
emphasizing
There
should
Dj
of
a greater on the
solution
optimization.
solutions,
the
impro-
b y w a y of
proper
in the minlmization
and
of
steps
of
on
has
also
be found
the
has
been
to obtain
the sum
of
minimum
surface
formulated the
to ambient
the following
possibly
sources
temperature
causes
stability
the elements
The
the change
of p a r t i c u l a r Therefore function
it will
operation
introduced
to
the sources
with
the function
in-
reasons:
reliability.
of
the
of t e m p e r a t u r e
the objective
been
the significance
calculation
t h e substrat.e
s o as
and
of
direct
its
the time
reliability
However,
the
proce-
heat
of
of
the optimization
from
inner
minimization
thereby
coefficient
placement
in r e l a t i o n
durability
and
function
been
in
assuming
by the
value
the
mainly
be placed
sources
the quality
weight
of
analytical
can be obtained
temperature
work.
improve
The
has
determined
iI~ t h e m i c r o c i r c u i t .
It r e s u l t s
generated
decreasing
The obtained
in each
should
heat
of e l e c t r i c
been
pr.oblem of o p t i m i z a t i o n
of t h e i r
rise
has
coefficient
because
temperature
occurrence
constructional
condition
a quality
of p a r t i c u l a r
elements,
and
and minimum
temperature
creases
of t e r m s
than
OPTI MI ZATI O N
diversification
Thus,
Heat
to be smaller
o n t h e substrat.e s u r f a c e .
is t i m e - c o n s u m l n g
values
the
of e l e m e n t s
optimization c o n s i s t
of
field
as follows:
from
principle.
the circuit
the circuit
sources
had
the number
in the microcircuit
OF PLACEMENT
temperature
dure
resulting
in the process
Disregarding
heat
Practically,
of t h e s u p e r p o s i t i o n
been
terms
4 0 - SO.
distribution
of heat
of i t s
and
the objective higher
unlt
determined
by
the dependence: NZ
min f C x ) xEX where.x is misslbile
=
CIB)
rain ~. D j - T P j C x ) . x~_Xj=1
the vector of solutions,
the sources coordinates,
X is t h e set
NZ d e n o t e s t h e number o f s o u r c e s , T P j C x ]
of p e r is
the
Thermally conditioned placementoptimization temperature stands
for
increase the unit
The conjugate been
of t h e j - t h power
gradient
used in the search
following - there
conditions
- strict
gradient
adherence
- constraints sons
sulting
from
traints
has
resulting
assuming between and
here
Ci,v),
problem
of
with
substrate
width
and
CFig.~). cidence
of t h e l e n Q h t
function,
the
of t h e
objective
Ci)
I
rea-
[8].
of
layer.
The re-
constuctional theory
cons[B,V],
the analoqy
of
the elements
node-outset
inside
voltages
in t h e c o o r d i n a t e
of t h e d i s s e c t i o n U
placed
oi"
into a set
of
graph
Cv)
system
the rectangle rectangles
is
O
v q and
the sink
v u of t h e g r a p h
current
I
- with
the
problem
between
the geometry
and utilizing
Mapping
widh
is
the graphs
of t h e or~e-port
and
of
realization
r e p r e s e n t e d by a polar
the image
O
of
and
of
one-port
The edge
The solution
constructional
to a single
structure
terminals.
the edge
relations
on the basi~
representation
The source
and
assignment
connections
region
currents
the one-port
has
X is r e q u i r e d ;
technological
being
-
obtained.
region
concerning
resistive
of r e c t a n g l e s
R o = C U o , I o)
determination
the technological
of t h e c i r c u i t
a graphical
[5]
consideration:
o n t h e w a y to t h e m i c r o c i r c u i t
considered
mesh-loopset
j).
constraints
of o b j e c t i v e
Gj
character.
an electrical
the rectangular and
from
functional
planarity
a set
into
of a n a l y t i c a l
topography
their
been
taken
and Gmax=maxCG
linear
the optimum
to the permissible
of i t s
and
source,
with
whet e a s
n j = G j / G m a x,
value;
O n e of t h e s t a g e s
elements
method
being
a r e of l i n e a r
the design
the j-th
for
is a possibility
function
of
heat source,
95
voltage
U
the length with
the
t h e elemer~ts a n d
I
0
poles
corresponds of
with
the
the rectangle
preservation
the structure
iq
correspond
of
R
the
in-
of t h e i r
col~-
iu=l o i
is
im
RI R2
R3
f//l;///////] 4
• , ? L, ' ~ ' r
"7 I
I
V
R7
5 ~.
7_""___1 Fig. o .
Mapping:
resistive
rectangles.
one--port
,,v I u= o 0 -
pol.~r
graph
v
- dissection
into
96
B. Wmz nectlons [9],
polar
use
/ P
R
Af
and
which
the
been
making
Af in
has
graph,
area
ments
from
the
to
--/P
where
On
the of
basis
the
Ohm
and
Kirchhoff
laws
u and
Pu
diagonal
is
of
vector
u of
the
division
inside
the
the
which
circuit,
the
the
known
R
have
matrices
matrix
been
of
the
representing
particular
values
of
circuit the
e l e ~-
matrix
determined
and
P then
expressions:
values
to
of
incidence
R u inside
V
the
assigned
and
R-'
the
rectangles
reduced
rectangles
the
V
basis
dependences:
are
values
Eventually,
ly
Bf
the
substituted
V
the
the
u
placed.
elements
I
of
on
respectively,
of
are
obtained
which
the
planar
the
R
V
vectors
of the
:/P
V
.
I
618)
and
U
microcircuit
elements
structure
can of
have
substrate
be shifted the
been
has
circuit
obtained.
surface been
with
into
explicit-
a specified
topography.
4.
NUMERI CAt
CALCULATI ONS
On
the
of
zation
method
ting
and
ters
of
on
basis
the
numerical
evaluating the
temperature
function
the
expression: T = TCx
the
is
x
o
the
effect
is
o
^ , x. ~' x
of
are
heat
heat
geometrical
optimum
which
vectors
of
starting
their
ratio,
of
optimi-
investiga-
material heat
parame-
sources
and
distribution
generally
~).
P is
and
Temperature
~.
the
performed,
placement
can
and
be
is
determined
by
fig) and
power,
optimum
Sr/Sp
is
k is
the
thermal
of
the
optimization
the
temperature
points, the
NZ
source
conductivity
is
sur-
and
coefficient.
the
selected
functioning parameters
an
example
presented
in
Fig. 9 h a s
planar
the
the
been
Sr/Sp.
circuit,
possible
of
configuration.
sources,
transfer
the
on
surface
illustrate of
effect
P,
the
model
have
variables
NZ,
mathematical
calculations
field
many
to substrate
To
the
of
and
number
face
the
elaborated
microcircuit
the
where
the
of
on
the
power
amplifier
been
taken
into
representations
of
the
whose
method
and
the
distribution
in
schematic
condlseration.
circuit
structure
diagram
One hes
of
the
been
pre-
Thermally conditioned placement optimization
4 ! R3
"11
o+Uz
5~",----IDR22T~T:
1
97
ii_._]1
7
.1 F i g 3.
sented
in
circuit
Fig. 4 b
can
diagram
of power- a m p l i f i e r .
i n t h e f o r m of a p o l a r
be obtained
the conductive lution
Schematic
path
making
under
of t h e e q u a t i o n
u ~ e of
the discrete
system
(17)
graph
Cthe
planarity
the possibility element
arid (I~)
of
- the diode
assuming:
of
the
conducting D).
The ~o-
I =50-i0-3m
(the
o
substrate
lenght)
and
the values
41
0
Pu = 10-4"
been graphically
slze
of t h e
the matrix
P
( i n mz) e l e m e n t s
has
]
2 1 0
consideration
of
i 2
illustrated
of c o n t a c t particular
in Fig. 4.
surfaces
elements,
for
As a r e s u l t the terminals
the numerical
(a)
of
the additional
as well
va]ues
as
of
de%ermining
the the
(b)
0
22.8
50 1 3
50
.4
3
17.5 4 26.2 3O Fig. 4.
7.8
17.2 7
5 26.6 Planar
representation
phical
solution
section
26.6"~/
34.4 of
the a m p l i f i e r
of t h e n o n l i n e a r
into rectangles,
b)
polar
circuit
equation graph.
and
system:
the graa) d i s -
B. W I S Z
98
boundaries have
of
been
the
elements
can
be
shifted
2
a.O
-< y a
<
22.4
3
3.0
_< y 3
-<
18.
4
17.3
< y4
_< 2 2 . 2
4.0
_< x 4
< 24.0
S
24.4
< y~
< ~.7.0
4,0
_< x 5
E 27.7
6
lg.
<
y6
-< 2 7 . 0
29.7
< x6
-< 3 4 . 8
7
19.5
_< 2 7 . 0
36.6
E
!
have
oi
=T
02
_< y 7
been
Cl=C2=30-10-Sm, T
5
K.
=293
O. O x 4 . 0
lO-Sm,
2.0
2.
l. O x 2 . 0
lO-Sm,
0.008
3.
O. O x 4 . 0
lO-Sm,
2.0
4.
~-.0x2.5
lO-Sm,
O. lS
W W
1.8x2.2
lO-Sm,
0.01
l O - S m ,
0 . 0 0 1
W
7.
1.8xe. 2 lO-am,
0.06
W.
which
is
been
in
the
corresponding
, . .....
heat
region
presented
_< x 2
< 28.0
31.7
_< x 3
-<
I
=k
dimensions
2
=22.75
and
x7
46.0
47.0
following W.CmK],
power~
of
data: ~=c~ +c~ = ~
in to
sources of
for
selected
permissible
Fig. 6.
The
these
positions
,. . . . . . . . . . . . . .
the
• . . . . ....
initial
solutions
temperature has
been
point
determined
distribution
in
the
illustrated
in
. . . . . . . . . . . . . . . . . . . .
1111111
• ....
3333333
1111111
3333333
1111111 1111111
2 2
)333333 3333333
44 44
66 66
77 77
55
SS
. .°
°.°
• . . . . . . . .
Fig
5.
° . . . . . . . . . . . . . . . . . . .
Initial
2
elements
W
2. O x I . S
of
24.5
the
< ~1.3
W,
~.
placement
The
< xl
for
out k
4.0
W
5.
Optimum
carried
S
h=O.V.lO-anl,
i.
.....
x, lO-Sm
-< 1 4 . 2
follows:
has
y,lO-Sm
-~ yl
was
circuit
the
3.0
W/CmZKg,
above
which
I
=42. 2 4
C F i g . 5)
:
coordin.
number
calculations
A=50-lO-Sm,
as
inside
obtained:
element
The
subregions
placement
' . . . . . . . . . . . . . . . . . .
ot
amplifier
elements,
'
.....
99
Thermallycondition~ pla~ment optimization o
.
.
o
,
J
m
,
.
,
,
,
.
o
.
.
g
,
,
.
.
o
.
.
O
,
.
,
.
.
,
,
.
.
,
,
.
.
. . . . .
,
.
.
.
,
o
,
.
.
.
.
.
.
.
o
.
.
.
.
.
.
.
.
.
.
.
.
.
2 Z
1 1 ~ 1111111
3333333 3333333
#L, ~
66
~
66 77 77
55 55 O I U O
I # O Q I t
I
0
#
I
~ e e
Fig. 6.
Fig. 7.
This
of
the
into
condiseration it
Final
ha~
is
use
As
the
of
temperature
decrease
perature
field
results
Further
. . . .
I
Q
¢
0
of
have
carried
been
and
the
fact,
heat
that
repeatedly result
have
~
•
m
0
m
4
m
i
amplifier
variability
optimization of
there
the
out
0
Q
.
*
elements.
for
tranfer
the
corrected
coefficients,
in
the
temperature
the
subroutine
when
performing
been
obtained:
the
above
Temperature surface
good
in
of
the
been
at
the
been the
vataking
function.
of
temperature
the
optimization
= 2 3 . E~6 W / C m K ) ,
X
compatibility
highest
have
selected
measured
with of
obtained
circuit
calculations
values
have
has
sources
diversification
of
obtaining
Cerror
of
called
As a r e s u l t
ter,
0 o Q
W/CmaK).
~=38.86
circuit
~ g
placement
their
made
procedure.
tally.
f
thermal' c o n d u c t i v i t y
calculation
The
O I e Q
calculations
lues
He~e,
1 6
points
the
power
has
been
use
a considerable
also
and been
the
tem-
reduced.
verified
experimen-
of
thick-film
of
real
radiation
measurements
and
thermome-
calculations
_< l O ~ ) . decrease
tained
by
placing
higher
thermal
of
the
the
temperature
sources
conductivity.
nr
field
1 and
diversificationcan
3 in
Assuming
the
part
Cl=lS-10-Sm,
of
be
substrate
C2=30-10-am,
obwith k = Z
=~aO
W/CmK),
Fig. 8,
and
Fig. 7 h a s
optimum
the been
placement
temperature shown
in
map
Fig. g.
of
heat
sources
preserving
the
has
been
temperature
presented scale
in from
B. Wmz
100
.o . . ,, , .
o .....
**l* IIII
.
-
**~ J/~:,/ l l , l -XXY
R,~RPRk .
o o,,
I'/
i!i?:rll!rllil
•
.
.
"-
+"*+'"+.,.
. % T; % "; ',: "; % ~ '- ,: "4 ; "(
•
. • . .
.
.....
.
.
,
, .....
,
"
'
'
,
,
,
, , , , , , , . . . . . . .
,
,
,
H
#
+÷+.,..+++
~ ( ~ . ' ;
, ,
. i . ~ ÷ + + + . + ~ . *
"< } '{ X ~l ~ v. ~'- '( '~ '<- ~: ~
............ ~
I
:
~
I
I
I
I
H
~
,
,
'< '< ,
,
,
,
T~
~46.8~
35~.93
3 6 5 2 3
317.26
395.3q 393.72
TMAX = 394.82
XXXX
+~++÷ .+++44+
XXXz
,
. . . .
,
,
,
,
,,
,
,
....
,
,
. . . .
.
.
.
.
.
.
, ....
'M~ fd i~ H /. lf b H t! H t/ #
l~ li ft l' ~# l~ ~l tt #l H t~ H ff H l M ~ H fl H l+ tt H # H t N#Hh~HbH i~ffq.~l###HltHl$~ ff ff if t: H i~ H H if @ ~ . . . • ......
311.09 369.~7
377.5~
403"-42 ~O~.B4
411.51 ~O9.Bg
~I~.S~
TMIN = 3 5 9 . 6 7
XXYXXXXXX XXxXX~ XXXXXX . . . .
.
.
.
.
XXXX
XXXX~ X
XXX~
W-~XXXXXX~ xYXXWX XXXXV#
++~++i44÷4~+++++++ ,++++++++++44÷+
WXxY
.
XX~X
+ ~44444÷ ++~++÷÷÷+44~
363.01
361.~9
353.31
379.47
+++4+
,
TMIN = 3~8.76
TMAX = ~ 1 9 . 5 8
+
- -
~-~7,,~.~
I'HH~HMCI~H#itlI:ItnHtt ~ , . , , , , ,i i , i #!M#HbIeH#f#'HI~IIY/II~MHH#t#¢~IIhHt'H#I.~ tl ~.f f tf fl t; H if b li P# t} tl l' #~ t, ~ #; H ll ~ l' H ~ ll li tl ll #$ t.l ~ ~ #i h lt #I tt ~tiUt:fli/M;~fftlflHt#ff~t!Nllfilllft~e~H#MI H~iit~tWliMPlP. ffit~itf,"~H~ HI fill till! I .; lt l, lt li i~ ti H ti #J l* ~ l i i/ fy H ~ l, ll U H I p ~ ' t t l u ~ . i / l ! t f ~ # # J l f : , t f l ~ J # i tl lt ti l~ ti lt li l~ ll :: tl t, it n lt ll ll #~ M f Jy t l J , u lf ll -~ h M ll fl l fY f~ H ;" f# l~ #~ ~ ~ lt : J/ ll #~ H l~ # #t #f f~ ll li #~ tl #l ~ h f+ l~ ll H if lt t# #l f l fl ~lH Ji tf t: . ## H tt H t, tl fl lt {: fl l, ~l. v M t# tt ## #4 fl l, #£ fl !ff i i H tl #'11i 11 ;: f# L it tl It tt M i,I H !1! li 11 Ii it hi it i~ It tt fl it f H H i1 ~ fl I fl #i H f ~ It H ~ H f ffl#ll#~tfllflllH:'#Jf~#lllti;;MHltf'MHllJ}Ht~ @t~tf#IflUHF'HIIHfI#~IMg~It. ###~Ht H ## ~ f; H tt ~ l; tl t tf h b l~ #~ tl M H tf t h H l' tl H t: l. n tf fl l, :" H If H L ## h H H M tt ti tl ff #f tt tI • • . ...." • • • . • . • . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . .
• ~44++ ++++++
I111 RR&~RR
XX-
, , , , ~ , , , , , , ,
3~8,
ii
XW~/YX
.....
",, ~
1 XX--
÷+++
,
, ,
~l
4444
%~.3';~
, ,.
-,,#
.,.+lxw +++X',(X-
',.~x~;:
, , , . , , , , , ,
L-IJ
,
,
,
"(?(),.'('~:'K++÷+
, , , ,
. . . . . . . . .
; i '. '. '. ; ; '. '. r
•
.
, , , . , . , . ,, ,,
..
, , , . , , , , ,
, , , ....
, ,,
~X%~7,~ .
, ......
, ,
....
,~':(;-'Y.~4
~..;(>,%;:~v.
"
o .......
$~'x:I>.~Y.'~ ;:':%,"7~,
;:W"/.X~
+++~
, , + . + + ÷ 4 + + ~
.....
}';,~;~.x'~xx~
~,;X~X~.
++++
Xl4
,,.
XX++.+x.',.';.v.~t' 7+ X'c.';:'zl
WX~W,.+÷
XXXXX
•
,..
w,+::
+÷++4444++~++÷
XXXY
~+.+.+++++++-4 ++++i+++++++*++
XXXX--
XWX
'X~X
+~+44i+~+~+,+444++++~ ++++~+÷++ ~4÷&+÷+~4+44~+~++
+4~++++ +~++44
WXXW~ XIWXXX~
"
+++*'+ +++++~
+÷+++~÷+++ ~4+44444÷+++
..... :1..414 .........
. . . . . . , . - .
Fig
7.
. . . . .
Temperature pl
ac
emen
. . . . . . . . . . . . . . . .o..
map t.
for:
a)
the
....
initial
...
. . . . . . . .
piacement,
,.,,
. . . . . . . .
b)
the
final
101
Thermally conditioned placement optimization °,
,
,
.
,
, ,
•
.
•
,
o
.
,
.
.
.
,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
°
.
.
.
.
.
.
, . ,
hl~li~4 11~II~1 111111~
.
, . .
•
,°
.
.
.
.
.
.
0.
,
333333~ 333]333 5333333 33333~3
I ~ I
~4
66
77 ?7
bb
55 55
~
~
O
m e
Fig. 8.
~
~
~
e ~
Final
I
6
~
Q
~
~
placement
~
of
g ~
~
*
i
~m~
e
elements
g
~
~
on
.
the
. . . . . . . . . . ; ; : : j j ; i : ...... . ~ 9 ~ , ~
~****÷÷÷*'
.
~
t * # * ~ * *
" ......
4
4
~
l
*
l
÷
.
*
.
.
Fig. 9.
.
.
'
.
.
.
.
.
.
o . .
- 386.29
Temperature Fig.
~.
.
.
.
.
TMIN map
•
.
.
.
.
.
.
.
.
.
.
÷
.
~
.
~
substrate.
WX~XX
**4**÷°*÷*
&
l
t
~
s
&
l
S
[xXx*****+*.
. . . . .
~
l
l
&
......
l~x~.
l
*
l
~
4
*
÷
4~I,
&
.
*
.
4
÷
4
~
.
.
*
.
4
4
4
.
.
.
.
!"" 4
~
4
*
.
.
4
.
÷
.
.
4
.
4
~
4
,
4
4
~
.
~
.
.
°
.
.
.
.
= 360.76
corresponding
with
the
final
placement
from
8.
CONCLUSIONS
The
results
of
the
the
suitability
of
conditioned
basic
elements
.
0
S ............................... ~ 4 ~
.
. . . . .
'""÷"~ r . . . . .
....
÷
:: .......
•
...... : ; : ; : ; 1 ; . . . . . . . . . . .
~XX+.*÷***+÷.÷,+**+~**xx~
XX
~
heterogenous
~-*÷.*
TMAX
th@
.
,x~..÷..,.
•
mally
.
***********************
X~XX~
~4&4÷*÷÷~*I*0
firm
.
:2:,,.'x~~
......
o
.
the
process
purpose inside
investigations
of
the
the given
of
proposed hybrid
task.
which
substrate
which
have
optimization micricircuit is
the
region.
been
method
in
designing,
placement The
conducted
con-
the
ther-
fulfilling
optimization
analysis
of
the
of oh-
102
B. Wlsz
talned i.
results
Final
leads
Location
to formulating
of
heat
the
sources
following
depends
conclusions:
on
the
position
initial
which i n d i c a t e s t h e e x i s t e n c e o f t h e o b j e c t i v e f u n c t i o n l o c a l m i n i ma, w i t h t e m p e r a t u r e v a l u e s d i f f e r i n g
most f r e q u e n t l y t o an
incon-
s i d e r a b l e degree. 2.
Thermally
significant
temperature
distribution
iS substantially
wer
plification of
of
placement
to cut upon 3.
the
the
of
down
diversified. optimization
thermally
the
of
the
optimization
variability
rotation of The
the
solution
sltiveness
of
the
the
to
Lime
the
thermal
values
k and
ness
of
the
method
functioning.
rical
system
method
ristlcs
of
temperature optimization
be
stressed,
it
is
which
the
~ is
such
however,
by
means
is
heat
Thus,
necessary
that
in
to ensure
thermally
of
sources
has
been
of
when
take
ir~to
transfer
the
[i0] after
the
This the
by
approxi-
low
seh-
coefficients
accurate
complexity
a circuit
to
transfer
the
thoroughly
repeatedly
characterized
ilustrates
heat
allows
influence
k al,d heat
to make
in
sim-
correctioh
This
allows
purposeful
distribution
the
po-
~=fCT>.
problem
not
called
and
more
in
the
a case,
negative
possible
circuit
of
and
range.
the
is
function
a practical
of
signing
any
conductivity
and
tion
when
alone.
which
conductivity
in
determinate
solution.
k=f
considered
in
COl~sist
without
thermal
placement
which
selected
elements
procedure,
changes
It s h o u l d
be
procedure
temperature
dependence of
can
temperature calculation
performing
~ in
ones
Purposeful,
obtained
of
coefficient
i.e.
significant
calculation
quality
the
sources,
in a c i r c u i t
The subroutine
account
4.
heat
determinathe
correct-
conditioned use
of
thermal
method
process
the
nume-
characte-
will of
de-
verify
placement
completed.
REFERENCES
i.
S l a g h T.D. , N a g l e in hybrid circuit
Sym@~.
H.T. , R u w e .W. , O n t h e u s e of t h e r m a l s i m u l a t i o n design. Proc. o/ ~Ae t 9 7 6 In£. M E c r o e [ e c £ r o n £ c s Oct. ll-13.1g70, Vancouver, Canada.
2.
K r a m e k J.P. , S o m e a s p e c t s of placement optimization of thermal significant c~nQnts in thlck-film hlgh-power hybrid integrated circuits. MEcroe[ec~ron~cs and ReL~ab~L(~, vol.~3,nr 3, lg83.
3.
W i s z B . , Heat sources placement o p t i m i z a t i o n i n hybrid microcirc u i t . D ( s s e r { e t ( o n , T e c h n i c a l U n i v e ~ s l t y o f Rzeszdw, Poland, Ig88.
Thermally conditioned placement optimization series
4.
Churchilll R.V. , Fourier Hc G r a ~ - H ~ L , Inc. Ig~3.
5.
Kr~glewski T. , R o g o w s k i T, R u s z c z y 6 s k i A. , S z y m a n o w ~ k i optymalizacJi w j~zyku FORTRAN. WNT, Warszawa0 i g84.
6.
S e s h u S. , R e e d M.B. , L i n e a r g r a p h s M u s s . , A d d Z s o n - W e s Z e N, Ig61.
7.
Y_/bert K. , C o m p u t e r aided planar circuit layout. Pr'oc. European Con/. C £ P c u £ L s T h e o ~ and Design, L o n d o n igv4.
IEEE
~974
8.
T u t t e W.T. , T h e d i s s e c t i o n Jo~Pna[, nr 7, I g 4 0 .
Duke
Ma£1~.
g.
Brinkmann K.D., Das Chip-Netz als ein Hilfsmittel f~r den yon MSI- und LSI-Schaltungen. N u c h ~ ' f c A £ e n £ e c A n ~ M Z. , B d . 3 0 . 1977.
of
and
and
rectangles
boundary
103 value
electrical
into
problems.
J. ,
Metody
networks.
Read£n~
squares.
Layout H. 12,
lO. N i e d z i a ~ e k T. , K a l i t a W. , Simulation of stationary temperature field in hybrid microcircuits designiz~g. Proc. o/ the ~ - Z A clnd 6--£A C o n / . o / tAe Intez'nc~t£onc~[ S o c £ e t ~ /oi- H~,br.£d H~cr.oeZectro;%£c_~ Po[cznd C h a p t e P , W r o c l a w , Ig85.