MATHEMATICAL DESCRIPTION OF THERMAL SYSTEMES (distributed linear RC systems)

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MATHEMATICAL DESCRIPTION OF THERMAL
SYSTEMES(distributed linear RC systems)
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Introduction

• Linearity is assumed
• later we shall check if this assumption was
  correct
• Thermal systems are
• infinite
• distributed systems
• The theoretical model is distributed linear RC
  system
• Theory of linear systems and some circuit theory
  will be used

For rigorous treatment of the topic see
V.Székely "On the representation of
infinite-length distributed RC one-ports", IEEE
Trans. on Circuits and Systems, V.38, No.7, July
1991, pp. 711-719
Except subsequent 12 slides no more difficult
maths will be used
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Introduction

• Theory of linear systems

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Introduction

• Theory of linear systems

• The h(t) unit-step function is more easy to
  realize than the d(t) Dirac-delta
• h(t)
  a(t),
• a(t) is the unit-step response function

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Step-response

• The a(t) unit-step response function is another
  characteristic function of a linear system.
• The advantage of a(t) the unit-step response
  function over W(t) weight function is that a(t)
  can be measured (or simulated) since it is the
  response to h(t) which is easy to realize.

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Thermal transient testing

• The measured a(t) response function is
  characteristic to the package. The features of
  the chippackageenvironment structure can be
  extracted from it.

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Step-response functions

• The form of the step-response function
• for a single RC stage

characteristic values R magnitude and t
time-constant

• for a chain of n RC stages

characteristic values set of Ri magnitudes and
ti time-constants
If we know the Ri and ti values, we know the
system.
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Step-response functions

• for a distributed RC system

characteristic R(t) time-constant spectrum
If we know the R(t) function, we know the
distributed RC system.
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Time-constant spectrum

• Discrete RC stages discrete set of
  Ri and ti values
• Distributed RC system continuous R(t)
  function
• If we know the R(t) function, we know the
  system.
• R(t) is called the time-constant spectrum.

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Practical problem

• The range of possible time-constant values in
  thermal systems spans over 5..6 decades of time
• 100ms ..10ms range semiconductor chip / die
  attach
• 10ms ..50ms range package structures beneath the
  chip
• 50ms ..1 s range further structures of the
  package
• 1s ..10s range package body
• 10s ..10000s range cooling assemblies
• Wide time-constant range ? data acquisition
  problem during measurement/simulation what is
  the optimal sampling rate?

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Practical problem (cont.)
Measured unit-step response of an MCM shown in
linear time-scale
Nothing can be seen below the 10s range

• Solution equidistant sampling on logarithmic
  time scale

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Using logarithmic time scale
a(z)
Measured unit-step response of an MCM shown in
linear time-scale
z ln(t)
Details in all time-constant ranges are seen

• Instead of t time we use z ln(t) logarithmic
  time

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Step-response in log. time

• Switch to logarithmic time scale a(t) ? a(z)
  where
• z ln(t)
• a(z) is called
• heating curve or
• thermal impedance curve
• Using the z ln(t) transformation it can be
  proven that

Sometimes P?a(z) is called heating curve in the
literature.
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Step-response in log. time

• Note, that da(z)/dz is in a form of a convolution
  integral

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Extracting the time-constant spectrum in practice
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Extracting the time-constant spectrum in practice
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Using time-constant spectra

• The time-constant spectrum gives hint for the
  time-domain behavior of the system for experts
• Time-constant spectra can be further processed
  and turned into other characteristic functions
• These functions are called structure functions

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Break!
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INTRODUCTION TO STRUCTURE FUNCTIONS
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Example Thermal transient measurements
heating or cooling curves
Evaluation
Network model of a thermal impedance
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How do we obtain them?
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Structure functions 1

• Discretization of R(z) ? RC network model in
  Foster canonic form
• (instead of ? spectrum lines, 100..200 RC stages)

• A discrete RC network model is extracted ? name
  of the method NID - network identification by
  deconvolution

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Structure functions 2

• The Foster model network is just a theoretical
  one, does not correspond to the physical
  structure of the thermal system
• thermal capacitance exists towards the ambient
  (thermal ground) only
• The model network has to be converted into the
  Cauer canonic form

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Structure functions 3

• The identified RC model network in the Cauer
  canonic form now corresponds to the physical
  structure, but

• it is very hard to interpret its meaning

• Its graphical representation helps

• This is called cumulative structure function

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Structure functions 4
The cumulative structure function is the map of
the heat-conduction path
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Structure functions 6

• Cumulative (integral) structure function

Calculate dC/dR ?
air
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What do structure functions tell us and how?
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A hypothetic example for the explanation of the
concept of structure functions 1
An ideal homogeneous rod
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A hypothetic example for the explanation of the
concept of structure functions 2
An ideal homogeneous rod
Ideal heat-sink at Tamb
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A hypothetic example for the explanation of the
concept of structure functions 3
An ideal homogeneous rod
This is the network model of the thermal
impedance of the rod
Ideal heat-sink at Tamb
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A hypothetic example for the explanation of the
concept of structure functions 4
Let us assume DL, A and material parameters such,
that all element values in the model are 1!
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A hypothetic example for the explanation of the
concept of structure functions 5
Let us assume DL, A and material parameters such,
that all element values in the model are 1!
It is also very easy to create the differential
structure function for this case. Again, we
obtain a straight line y1
Rth_tot
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A hypothetic example for the explanation of the
concept of structure functions 6
What happens, if e.g. in a certain section of the
structure model all capacitance values are equal
to 2?
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A hypothetic example for the explanation of the
concept of structure functions 7
What would such a change in the structure
functions indicate?
It means either a change in the material
properties
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A hypothetic example for the explanation of the
concept of structure functions 8
What would such a change in the structure
functions indicate?
or a change in the geometry or both
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A hypothetic example for the explanation of the
concept of structure functions 9
What values can we read from the structure
functions?
Cumulative structure function
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A hypothetic example for the explanation of the
concept of structure functions 10
What values can we read from the structure
functions?
V3/cv1
V2/cv2
K2 A22cv2?2
V1/cv1
Cumulative structure function
K1 A21cv1?1
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Structure functions 5 Differential structure
function

• The differential structure function is defined as
  the derivative of the cumulative thermal
  capacitance with respect to the cumulative
  thermal resistance
• K is proportional to the square of the cross
  sectional area of the heat flow path.

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Some conclusions regarding structure functions

• Structure functions are direct models of
  one-dimensional heat-flow
• longitudinal flow (like in case of a rod)
• Also, structure functions are direct models of
  essentially 1D heat-flow, such as
• radial spreading in a disc (1D flow in polar
  coordinate system)
• spherical spreading
• conical spreading
• etc.
• Structure functions are "reverse engineering
  tools" geometry/material parameters can be
  identified with them

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Some conclusions regarding structure functions
In many cases a complex heat-flow path can be
partitioned into essentially 1D heat-flow path
sections connected in series
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IC package assuming pure 1D heat-flow
Chip
Base
...and create its model in form of the cumulative
structure function
Cold-plate
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IC package assuming pure 1D heat-flow
The heat-flow path can be well characterized e.g.
by partial thermal resistance values
Differential structure function
The RthDA value is derived entirely from the
junction temperature transient. No thermocouples
are needed.
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Example of using structure functions DA testing
(cumulative structure functions)
Reference device with good DA
Identify its structure function
Identify its structure function
This change is more visible in the differential
structure function.
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Example of using structure functions DA testing
(differential structure functions)
Unknown device with suspected DA voids
Reference device with good DA
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Some conclusions regarding structure functions

• In case of complex, 3D streaming the derived
  model has to be considered as an equivalent
  physical structure providing the same thermal
  impedance as the original structure.

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Specific features of structure functions for a
given way of essentially 1D heat-flow

• For ideal cases structure functions can be
  given even by analytical formulae
• for a rod
• for radial spreading in a disc of w thickness and
  l thermal conductivity

Section corresponding to radial heat spreading in
a disk
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Accuracy, resolution

• Structure functions obtained in practice always
  differ from the theoretical ones, due to several
  reasons
• Numerical procedures
• Numerical derivation
• Numerical deconvolution
• Discretization of the time-constant spectrum
• Limits of the Foster-Cauer conversion
• 100-150 stages
• Real physical heat-flow paths are never sharp
• Physical effects that we can try to cope with
• There is always some noise in the measurements
• Not 100 complete transient / small transfer
  effect
• In reality there are always parasitic paths
  (heat-loss) allowing parallel heat-flow

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Accuracy, resolution

• Comparison of the effect of the numerical
  procedures

• Resolution of structure functions in practice is
  about 1 of the total Rthja of the heat-flow path

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Use of structure functions

• Plateaus correspond to a certain mass of material
• Cth values can be read
• material ? volume
• dimensions ? volumetric thermal capacitance

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Use of structure functions partial thermal
resistances, interface resistance

• Origin junction, singularity ambient
• Rthja and partial resistance values
• interface resistance values (difference between
  two peaks)

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Some examples of using structure functions
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Measurement of the package/heat-sink interface
resistance
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Measurement of the package/heat-sink interface
resistance
The transient responses
STRUCTURE FUNCTIONS WILL HELP
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Measurement of the package/heat-sink interface
resistance
See details in A. Poppe, V. Székely Dynamic
Temperature Measurements Tools Providing a Look
into Package and Mount Structures, Electronics
Cooling, Vol.8, No.2, May 2002.
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Example The differential structure function of a
processor chip with cooling mount

• The local peaks represent usually reaching new
  surfaces
• (materials) in the heat flow path,
• their distance on the horizontal axis gives the
  partial thermal
• resistances between these surfaces

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Example FEM model validation with structure
functions
Courtesy of D. Schweitzer (Infineon AG), J.
Parry (Flomerics Ltd.)
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Structure functions summary

• Structure functions are defined for driving point
  thermal impedances only. Deriving structure
  functions from a transfer impedance results in
  nonsense.
• Structure functions thermal resistance
  capacitance maps of the heat conduction path.
• Connection to the RC model representation as well
  as mathematically derived from the
  heat-conduction equation.
• Exploit special features for certain types of
  heat-conduction (lateral, radial).

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SUMMARY of descriptive functions

• Descriptive functions of distributed RC systems
  (i.e. thermal systems) are
• the a(t) or a(z) step-response functions
• the R(t) time-constant spectrum
• the structure functions
• CS(RS) cumulative
• K(RS) differential
• Any of these functions fully characterizes the
  dynamic behavior of the thermal system
• The step-response function can be easily measured
  or simulated
• The structure functions are easily interpreted
  since they are maps of the heat flow path

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SUMMARY of descriptive functions

• Descriptive functions can be used in evaluation
  of both measurement and simulation results
• Step-response can be both measured and simulated
• Small differences in the transient may remain
  hidden, that is why other descriptive functions
  need to be used
• Time-constant spectra are already good means of
  comparison
• Extracted from step-response by the NID method
• Can be directly calculated from the thermal
  impedance given in the frequency-domain (see e.g.
  Székely et al, SEMI-THERM 2000)
• Structure functions are good means to compare
  simulation models and reality
• Structure functions are also means of
  non-destructive structure analysis and material
  property identification or Rth measurement.

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SUMMARY of descriptive functions

• The advanced descriptive functions (time-constant
  spectra, complex loci, structure functions) are
  obtained by numerical methods using sophisticated
  maths.
• That is why the recorded transients
• must be noise-free and accurate,
• must reflect reality (artifacts and measurement
  errors should be avoided),
• must have high data density.

• since the numerical procedures like
• derivation and
• deconvolution
• enhance noise and errors.

Besides compliance to the JEDEC JESD51-1
standard, measurement tools and methods should
provide such accurate thermal transient curves.
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PART 3APPLICATION EXAMPLESFailure analysis/DA
testingStudy of stacked diesPower LED
characterizationRthjc measurementsCompact
modeling
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TESTING OF DIE ATTACH QUALITY basics
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Die attach quality testing
The die attach is a key element in the
junction-to-ambient heat-conduction path
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Detecting voids in the die attach of single die
packages

• Experimental package samples with die attach
  voids prepared to verify the accuracy of the
  detection method based on thermal transient
  testing
• (acoustic microscopic images, ST
  Microelectronics)

See M. Rencz, V. Székely, A. Morelli, C. Villa
Determining partial thermal resistances with
transient measurements and using the method to
detect die attach discontinuities, 18th Annual
IEEE SEMI-THERM Symposium, March 1-14 2002, San
Jose, CA,USA, pp. 15-20
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Main time-constants of the experimental samples
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Measured Zth curves of the average samples
Already distinguishable
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Differential structure functions of the
experimental samples
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The principle of failure detection

• Take a good sample as a reference
• Measure its thermal transient
• Identify its structure function
• Take sample to be qualified
• Measure its thermal transient
• Identify its structure function
• Compare it with the reference structure function
• Locate differences
• A difference means a possible failure
• If needed, quantify the failure (e.g. increased
  partial thermal resistance)

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The principle again
Unknown device with suspected DA voids
Reference device with good DA
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TESTING OF DIE ATTACH and SOLDER QUALITY case
studies
A power BJT mount Stacked die packages
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Measurement of a power BJT mount failure analysis
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Measurement of a power BJT mount failure analysis
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Measurement of a power BJT mount failure analysis
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Measurement of a power BJT mount failure analysis