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An advanced study of microwave sintering

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University Mtcrotilms International
A Bell & Howell Information Company
300 North Zeeb Road Ann Arbor Ml 48106-1346 USA
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O rder N um ber 8914089
A n a d v an c ed stu d y o f m icrow ave sin te rin g
Watters, David Gray, Ph.D.
Northwestern University, 1989
C opyright © 1988 by W atters, D avid Gray. A ll rights reserved.
UMI
300 N. Zoeb Rd.
Ann Arbor, Ml 48106
NORTHWESTERN UNIVERSITY
AM ADVAMCKO STUDY OF MICROWAVE SIMTKRINO
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IH PARTIAL FULFILLMENT OF THE REQUIREMENTS
for tha dtgrtt
DOCTOR OF PHILOSOPHY
Flald of Elactrlcal Englnaarlng
By
David Gray Wattars
BVAMSTON, ILLINOIS
JUNE 1989
O Copyright by David Gray Watters 1988
All Rights Reserved
11
ABSXBASX
An Advanced Study of Mlcrwivt Sintering
David Gray Matters
An Investigation of microwave sintering of ceramics
reveals that under some conditions, anomalous results,
Including core melting and thermal "runaway" can occur.
This research explains these phenomena using theoretical
models of the electromagnetic heating process.
The most
lmpostant variable Is temperature dependent electrical
conductivity which can Increase by several orders of
magnitude during sintering.
A portion of this work Is
devoted to material characterization as a function of
temperature.
The theoretical analysis Is divided Into two cases,
electrically thick and electrically thin bodies.
A seml-
lnflnlte space Is used as the model for thick samples.
Transient analysis of the heat and wave equations, using an
explicit numerical method, reveals that low thermal
conductivity can cause subetantlal thermal gradients
resulting In core melting.
heating:
There are three regions of
Initially, low loss materials heat slowly; as
conductivity rises with temperature, the heat generation
ill
rate accelerates producing a rapid temperature rise; at high
temperatures, a "saturation” effect occurs, due to
decreasing electrical skin depth, Increasing surface
reflection and Increasing thermal radiation.
Solution of
the steady state equations Indicates a monotonlc
relationship between applied power and steady state
temperature.
The fields are assumed uniform In electrically thin
bodies, and a planar slab Is used as the model.
Numerical
analysis of transient heating reveals the possibility of
thermal Instability.
Non-dimensional analysis of the steady
state heat equation using regular perturbation theory
Indicates that there is a maximum stable steady state
temperature.
An equation Is derived relating applied power
to steady state temperature.
Above a critical applied
power level heating Is unstable.
The Inverse of this equation yields a formula for
electrical conductivity as a function of applied power and
steady state temperature.
Since both power and temperature
are measurable, material characterization Is possible.
Material characterization at room temperature is
performed using an automatic scalar network analyzer.
There
Is an optimum sample thickness, based on electrical
conductivity, dielectric constant and frequency.
Lossy
samples must be electroplated to remove gaps between sample
and sample holder.
lv
ASggfflilBgBgBXg
I want to taka this opportunity to acknowledge the
following people for their contribution to this effort.
First, I want to express ay gratitude and sincere
appreciation to ay advisers, Dr. Morris B. Brodwln, for help
In defining the alcrowave problea and Dr.
Gregory A.
Kriegsaann, for his aatheaatical Insight.
I also want to
thank the aeabers of the coaalttee, Dr. Lynn Johnson, Dr.
Carl Kannewurf and Dr. Allen Taflove for their tlae and
Interest In this work.
This work has been supported by the
Materials Research Center at Northwestern under Grant DMR85-20280.
Acknowledgement Is due Mr. Karla Mattar for his
assistance in characterization experiaents and Dr. Y.L. Tlan
for his assistance In saaple preparation and sintering
experiaents.
Finally, a special thanks to ay parents, W. Gray and
Marilyn Hatters for their encourageaent and patient support
throughout ay studies at Northwestern.
David Gray Watters, 1988
Psalas 115:1
v
TABLE OB CONTENTS
Abstract........................................
Ill
Preface.....................................................v
List of Figures................
xl
I.
Introduc t Ion........................................... 1
II.
Lltsratura Review..................................... 7
III.
Theoretical Analysis................................ 14
A.
General Theory................................... 14
1. Electromagnetic Aspects..................... IS
2. Thermal Aspects..............................23
3. Sintering Dynamics...........................24
4.
B.
Case Study................................... 28
Theoretical Heating of a Semi-Inflnlte Space
30
1. Constant Conductivity....................... 30
a.
Transient Solution..................... 35
b. Steady
State Solution.................. 37
c. Steady
State Results................... 39
d.
Transient Results...................... 42
2. Temperature DependentConductivity........... 51
a.
b.
C.
Transient Solution..................... 52
Steady
State Solution.................. 60
Theoretical Heating of a Planar Slab............. 64
vl
1. Constant Conductivity........................67
a.
Transiant Solution..................... 67
b.
Steady atata Solution..................71
2. Temperature Dependent Conductivity.......... 72
a.
Transient Solution..................... 73
b.
Steady state Solution and the Onset
of Instability......................... 77
1.
Shooting Method Solution..........78
11.
Approximate Analytical Solution...80
111.
Stability Analysis................90
3. Steady State Solution for an Insulated
Planar Slab................................. 93
4.
D.
Steady State Solution for a Thick Slab...... 99
Theoretical Heating of an Infinite Cylinder
103
1. Transient Solution..........................105
2. Steady State Solution for Constant
Conductivity............................... 105
3. Steady State Solution for Temperature
Dependent Conductivity..................... 106
4.
Steady State Solution for a Tube........... 108
5.
Steady State Solution for a Thick
Cylinder................................... 109
E.
7.
Theoretical Heating of a Sphere................. Ill
1.
Steady State Solution...................... Ill
2.
Fields In a Sphere......................... 114
Generalized Stability Formula For Thin
vii
Electrical Bodies............................... 115
G.
H.
IV.
Parameter Study................................. 118
1.
Constant Conductivity......................119
2.
Linear Conductivity........................119
3.
Quadratic Conductivity.....................120
4.
Exponential Conductivity.................. 123
5.
Other Exponential Forms................... 125
6.
Summary.................................... 129
"Hot wall" Applicator...........................131
1.
Half Space Solution........................132
2.
Planar Slab Solution...................... 134
Applicator Selection................................ 139
A.
B.
Experimental Apparatus..........................139
1.
Slab Applicator............................141
2.
Rectangular Applicator.................... 143
3.
Circular Cavity Applicators............... 149
Thermal Stability in a Rectangular Cavity
Applicator...................................... 152
V. Material Characterization............................. 175
A.
Automatic Material Characterization Equipment... 175
1.
2.
The Apparatus..............................176
a.
Theory................................ 179
b.
Results............................... 180
Measurement Range and Sources of Error
vlll
185
a.
Limitation* Based on Maxwell's
Equations.....
b.
Gap Effects........................... 198
1.
Capacitance Model................201
11.
Transverse ResonanceModel........209
111.
lv.
c.
B.
185
Techniques for Removing Gaps
214
Experimental Results.............217
Component Imperfections...............224
High Temperature Techniques.................... 232
1.
High Temperature AMCE..................... 232
2.
Other High Temperature Methods............ 233
3.
In Situ Characterization...................234
4.
A New Method for High Temperature
Characterization........................... 235
VI.
Conclusion and Recommendations for Future Nork
Notes and References............
246
249
APPENDIX A: Program for Computing Transient
Temperature Profiles In a Semi-Infinite
Space with Constant Conductivity.............257
APPENDIX B:
Program for Computing Transient Temperature
Profiles In Semi-Inflnlte Space with
Temperature Dependent Conductivity...........259
APPENDIX C:
Non-Dimensional Analysis.................... 263
lx
APPENDIX D: Damped Haraonlc Oscillator...................266
APPENDIX E: Thermal Equilibrium of Semi-Infinite solid...269
APPENDIX F: Program for Computing Transient Temperature
Profiles in a Planer Slab.................... 272
APPENDIX Q : Program for Computing Steady State
Temperature Profiles in a Planar Slab Using
a Numerical "Shooting" Method................ 274
APPENDIX H: Modes in a Dielectrically Loaded, Circular
Cavity....................................... 277
APPENDIX I: Program to Study Gap Effects Using the
Method of Transverse Resonance...............287
APPENDIX J:
Sample Resonance in Coaxial Line............ 293
K:
Electroplating Process...................... 295
Vita...................................................... 298
X
LIST OF FIGURES
1.
Temperature history of alumina sample during
sintering. Adapted from [1].......................... 16
2.
a) loss tangent and b) dielectric constant of
alumina as a function of temperature at 3.5 GHz.
Adapted from [32]..................................... 20
3.
Percent absorbed power vs. conductivity for a
planar slab of thickness d ■ 0.95 cn, f ■ 2.45 GHz,
€r ■ 10................................................ 22
4.
Thermal conductivity vs. temperature for various
ceramics. Adapted from [38]........................... 25
5.
Heat capacity vs. temperature for various
ceramics. Adapted from [38].......................... 26
6.
Ceramic particles In contact, showing a grain
boundary. Adapted from [40].......................... 27
7.
Geometry of planar half space......................... 31
8.
Percent power absorbed vs. conductivity for a
planar half space, f * 2.45 GHz, €r ■ 10...............34
9.
Transient temperature profiles In a planar half space;
o ■ .2 S/m, P - 50 W/cmz, a)500 sec, b)1500 sec,
c)steady state........................................ 44
10.
Transient temperature profiles In a planar half
space; o “ .002 S/m, P ■ 5000 W/cmz, a) 100
sec b)400 sec........................................ 45
11.
Transient temperature profiles in a planar half
space; o ■ .02 S/m, P * 500 W/cmz, a)100 sec
b )400 sec............................................ 46
12.
Transient temperature profiles in a planar half
space; o ■ .2 S/m, P ■ 100 W/cm2, a)100 sec,
b)200 sec, c )300 sec, d)400 sec...................... 47
13.
Transient temperature profiles in a planar half
space; o ■ 2 S/m, P ■ 50 W/cmz, a)100 sec
b)200 sec, c)292sec, d)1256 sec...................... 48
14.
Transient temperature profiles in a planar half
space; a)solid, o ■ .2 S/m, k ■ 20 W/m-K, P ■
✓
xi
100 W/cm2, t * 400 sec, b)dash, o * .2 S/m, k
■ 20 W/m-K, P - 200 W/cm2, t ■ 200 sec, c)dot,
o ■ .2 S/», k ■ 10 W/a-K, P » 100 W/cm2, t ■
400 sec.............................................. 50
15.
Transient temperature profiles In a planar half
space with temperature dependent conductivity,
a • .002 exp(.0026T), P ■ 500 W/cm2, a)1000
sec, b)1500 sec, c)1700 sec, d)1800 sec, e)1900
sec.................................................. 55
16.
Surface temperature vs. time for Transient
heating of a planar half space with temperature
dependent conductivity, o ■ .002exp(.0026T), P
- 1000 W/cm2......................................... 57
17. Transient conductivity profile for planar half
space after 1900 sec, o - .002exp(.0026T), P
- 1000 W/cm2......................................... 58
18.
Steady state surface temperature vs. Incident power
(W/cm2) for planar half space with temperature
dependent conductivity, o * .002exp( .0026T)......... 63
19.
Geometry of planar slab...............................65
20.
Steady state temperature profiles for a 6mm planar
slab, with a).002 S/m, b).02 S/m and c).2 S/m.
Power Is 500 W/cm2....... :.......................... 69
21.
Steady state temperature profiles for a 6mm planar
slab, with a)2 S/a, b)20 S/m and c)200 S/m.
Power is 500 W/cm2................................... 70
22.
Surface temperature vs. time for transient
heating of planar slab with temperature
dependent conductivity, o ■ .002exp(.0026T),
a) 100 W/cm2, b) 150 W/cm2 c)200 W/cm2................74
23.
Heating of sintered alumina sample, surface
temperature vs. time................................. 75
24.
Normalized steady state surface temperature vs.
normalized applied power for planar slab, with
f(v) • exp(.78v), c, ■ .002, cz ■ .00018,
using a numerical "shooting method.".................81
25.
Normalized steady state temperature profile In a
planar slab, \ - .005, f(v) « exp(.78v), ct xll
.0 0 2 ,
a n d Cj ■ . 0 0 0 1 8 ..........................................................................................8 5
26.
Noraallzad staady atate surface taaparatura vs.
noraallzad appllad power for planar slab, with
f(v) ■ axp(.78v), c, - .002, c2 ■ .00018,
using an approxlaata analytical solution............ 86
27.
Staady state surface taaparatura vs. appllad power;
un-noraallzed from Fig. 25.......................... 88
28.
Noraallzad staady state surface taaparatura vs.
noraallzad appllad power for planar slab, with
f(v) ■ exp(.78v), c, ■ .002, c2 ■ .00018,
coaparlng a)solld, approxlaata analytical
solution and b)dash, nuaerlcal "shooting"
aathod............................................... 89
29. Geoaetry of Insulated planar slab.................... 94
30.
Staady state taaparatura profile for an
Insulated slab....................................... 98
31.
Noraallzad staady state surface taaparatura vs.
noraallzad appllad power for Insulated planar
slab, with f(v) ■ exp(.78v), c, - .002, c2 ■
.00018 and a ■ 0.1, a) solid, uninsulated slab and
b)dash. Insulated slab.............................. 100
32.
Geoaetry of Infinite cylinder........................104
33.
Noraallzad steady state surface taaparatura vs.
noraallzad applied power for planar slab, with
f(v) ■ 1 + .3V2, a)a ■ .002, b)<* ■ .003,
c )ot - .004...........................................122
34.
Noraallzad staady state surface taaparatura vs.
noraallzad appllad power for planar slab, with
f(v) ■ exp(.78v), a)c, > 0 , c2 ■ .00018,
b)c, ■ .002, c2 ■ .00018, c)c, ■ .002,
c2 ■ 0 ...............................................124
35.
Noraallzad steady atate surface taaparatura vs.
noraallzad applied power for planar slab, c, ■
.002, cz - .00018, a)k -0.68, b)k - 0.78, c)k
- 0.88...............................................126
36.
Noraallzad staady state surface taaparatura vs.
noraallzad appllad power for planar slab, with
f(v) ■ 1 ♦ 600 exp(-13.3/v), c, ■ .002, c2 ■
.00018...............................................128
xlll
37. Normalized staady stats surface tsapsrature vs.
normalized applied power for planar slab, with
f(v ) - 1
378.5/V exp(-4.67/v), c, - .002,
C, - .00018......................................... 130
38. Transient hot wall heating of planar half space
with no microwave radiation, Taat ■ 1300K,
a)100 sec, b)400 sec................................ 133
39. Surface temperature vs. time for hot wall
heating of planar half space, P ■ 500 W/cm*,
o - .002 exp(.0026T), a)T, t - 300K,
b)T„t - 1300K........... ."........................... 135
40.
Surface temperature vs. time for hot wall
heating of planar slab, Ta>1 * 1300K, o ■
.002 exp(.0026T) a)P - 0, b)P - 100 W/cm*, c)P
- 500 W/cm*......................................... 136
41. Temperature profiles In hot wall heating ofan 8mm
planar slab, o ■ .002 exp(.0026T), k ■ 5 W/mK, P - 500 W/cm*, T „ t - 1300K, a)40
sec, b)140 sec, c)200‘sec, d)240 sec................ 138
42.
Block diagram of microwave sintering apparatus...... 140
43.
Slab applicator geometry.............................142
44.
Rectangular applicator geometry..................... 144
45.
T equivalent network for cylindrical rod............ 146
46.
Equivalent circuit for rectangular applicator....... 148
47.
Propagation constant for sample-filled TEllt
circular waveguide, where b/a Is the ratio of
the dielectric radius to the cavity radius.
Adapted from [69]................................... 151
48.
TE1(1 applicator geometry.............................154
49.
Equivalent circuit of cylindrical rod using
Marcuvltz model..................................... 157
50.
Variation of rod Impedance, Ra, with
conductivity........................................ 158
51.
Variation of rod Impedance, Xa, with
conductivity........................................ 159
xlv
52.
Variation of rod Impedance, R*, with
conductivity........................................ 160
53.
Variation of rod Impedance,
, with
conductivity........................................ 161
54.
Simplified circuit for low lose, thin rods..........162
55.
Cavity equivalent circuit........................... 164
56.
Normalized steady state surface temperature vs.
normalized applied power for cylindrical rod In
rectangular applicator with f(v) - exp(.78v),
c, ■ .002, cz ■ .00018, a) critical coupling
at 300K, b) free space cylinder..................... 168
57.
Normalized steady state surface temperature vs.
normalized applied power for cylindrical rod In
rectangular applicator with f(v) - exp(.78v),
c, ■ .002, cz - .00018, a) critical coupling
at 2300K, b) free space cylinder.................... 170
58.
Normalized steady state surface temperature vs.
normalized applied power for cylindrical rod In
rectangular applicator with f(v) - exp(.78v),
c, ■ .002, cz ■ .00018, a) critical coupling
at 1300K, b) free space cylinder.................... 171
59.
Percent power absorbed vs. normalized
temperature for different coupling arragements,
a)solld, 300K, b)dash, 1300K, c)dot, 2300K..........173
60.
Block diagram of Automatic Material
Characterization Equipment (AMCE)................... 177
61.
AMCE Sample holder, showing a) end view and b)
cross-sectional view of the modified APC-7
connector........................................... 178
62.
Reflection and transmission vs. frequency for
Stycast H1K, 0 - Is reflection data, □ - Is
transmission data................................... 182
63.
Reflection and transmission vs. frequency for
silicon, o - 0.017 S/cm, 0 - Is reflection data,
□ - Is transmission data............................ 184
64.
Sum of magnitudes and sum of squares of
magnitudes of reflection and transmission vs.
conductivity........................................ 189
XV
65.
Reflection coefficient vs. dielectric conetent
for fixed frequency and conductivity................192
66.
Transalsslon ve. conductivity for a) d ■ 4.56aa,
b) d ■ 0.3 a a ....................................... 194
67.
Reflection and tranaalsalon va. frequency for
a)o ■ .01 S/ca, b)o ■ .1 S/ca....................... 196
68.
Obaerved dielectric conetant va. true dielectric
constant for an air gap of a).l ua, b)l urn,
c)10tia, d )100 ua.................................... 200
69.
Geoaetry of the gap................................. 202
70.
Equivalent circuit for the gap using a
distributed capacitance aodel....................... 204
71.
Observed conductivity vs. true conductivity for
an air gap of a).01 ua, b).l ua, c)l ua. d)10
ua, e) 100 ua........................................ 206
72.
Observed dielectric constant vs. frequency for
an air gap of a).01 ua, b).l ua, c)l ua, d)10
Ua...................................................207
73.
Observed conductivity vs. frequency for an air
gap of R/a).l ua, b)l ua, c)10 ua, d)100 ua..........208
74.
Equivalent network for Transverse Resonance
Model (TRM)......................................... 210
75.
Observed conductivity vs. true conductivity for
a) TRM aodel, b)capacitance aodel................... 215
76.
Observed conductivity vs. frequency for a)TRM
aodel, b) capacitance aodel......................... 216
77.
The effect of air gaps on the aeasureaent of
Stycast H1K; a) dash, aeasureaent with gap, b)
solid, aeasureaent with filled gap.................. 220
78.
The effect of air gaps on the aeasureaent of
.017 S/ca silicon; a) dash, aeasureaent with gap,
b) solid, aeasureaent with filled gap...............221
79.
Transalsslon vs. frequency for high loss (ooc ■
.28 S/ca) silicon using an electroplated saaple.
Figure shows theoretical curves for .2, .25 and
.3 S/ca............................................. 223
xvi
80. Signal flow graph raprasantation of AMCE............ 225
81. Block dlagraa of In altu charactarizatIon aathod
using a tranaait/racaiva tuba....................... 236
82.
Axial taaparatura proflias in a alnterad alualna
saapla for various powar levels..................... 238
83.
Axial taaparatura profiles in a sintered
alualna-TlC (30%) saapla for various power levels...239
84.
Percent error in conductivity aeasureaent vs.
noraallzad taaparatura
for a 5% error in c,......... 242
85.
Percent error in conductivity measurement vs.
noraallzad taaparatura
for a 5% error in c,......... 243
86.
Percent error in conductivity measurement vs.
noraallzad teaperature
for a 5% error in v .......... 244
87.
Geoaetry of sample in TEMl
xv 11
cavity.................. 278
CHAPT1R I
INTRODUCTION
Microwave sintering Is a valuable aethod for processing
ceramic materials.
It produces samples with Improved
microstructure [1] at high processing rates, with greater
energy efficiency than conventional sintering techniques.
Prior work at Northwestern has demonstrated the feasibility
of sintering a variety of materials; [2] yet In some cases,
anomalous results are observed, Including core melting,
localized "hot spots" and thermal "runaway." [3]
This study has been undertaken to further our
understanding of the sintering process.
The objectives of
this work are: to develop theoretical models to predict
temperature profiles and stability In the microwave
sintering of materials of various geometries; and to
describe material characterization techniques to determine
how sample properties change during sintering.
The results
of this work apply to electromagnetic heating in general,
but we concentrate on microwave sintering.
In Chapter II, the microwave sintering literature Is
reviewed. [4-20] He survey the current progress In the
heating of ceramics with a particular emphasis on the
problem of thermal runaway.
In Chapter III, we describe the general theoretical
1
2
problea of altctroMgnttic hosting.
Wo discuss tho
toaporature dopondonco of aaterlal paraastors such as
oloctrlcal conductivity; and tho relationship between
absorbed power and conductivity.
We then consider the
solution of the heat and wave equations In soae slaple
geoaetrlea.
An explicit nuaerlcal solution Is used to solve
the transient heat equation.
The steady state solution Is
used to deteralne tne stability of the heating process.
The slaplest geoaetry is the seal-lnflnlte space.[21] We
show the laportance of theraal conductivity In deteralnlng
thereal gradients.
For aaterlals with teaperature dependent
electrical conductivity, transient analysis Indicates three
regions of heating; a) at low teaperatures the saaples heat
slowly; b) at lnteraedlate teaperatures the saaples heat
rapidly, due to increased heat generation; c) at high
teaperatures there Is a "teaperature saturation" caused by
decreased field penetration.
In the seal-infinite slab
there is a aonotonlc relationship between absorbed power and
steady state teaperature, [22] and furtheraore, the solution
is stable even when the electrical conductivity Increases
with teaperature.[23,24]
The electrically thin, planar slab geoaetry Is
considered next.
Nuaerlcal analysis of the tlae dependent
heat equation, when electrical conductivity Increases
rapidly with teaperature, Indicates the possibility of
Instability.
Analysis of the steady state equation using a
3
nuaerlcal "shooting" asthod rsvsals the prsssncs of
Instability when ths Incident power exceeds a critical
value.
A perturbation analysis of the steady state solution
yields a relationship between Incident power and steady
state teaperature.[29] Solution of the Inverse problea gives
electrical conductivity as a function of Incident power and
steady state teaperature.
Since surface teaperature and
power are aeasurable values, In situ characterization of
electrical conductivity Is possible.
In the slab geoaetry, we consider two special cases:
the perturbation solution Is applied to an Insulated saaple
to reveal stability lnforaatlon, and the solution for the
thin slab Is extended to Include slabs whose thickness Is on
the order of a wavelength.
The Infinitely long, thin, cylinder Is described,
Indicating slallar stability behavior to the planar slab.
The saae Inverse solution applies.[26] The special case of a
tubular saaple Is considered, as well as that of the thick
cylinder.
When this analysis Is applied to spherical geoaetry the
saae solution Is found.
The relationship between absorbed
power and teaperature derived for the slab holds for
cylindrical and spherical eaaples ae well.
A correction
factor Is Introduced based on the surface area to voluae
ratio of the saaple.
A general expression Is derived for
the theraal stability of electrically thin bodies.
4
A parameter atudy Is performed to determine how
electrical and thermal properties affect thermal stability.
If electrical conductivity Increases at a linear rate or
less, heating is stable.
For conductivities that vary
quadratlcally with temperature, there Is low temperature and
high temperature stability, but an Intermediate range of
temperatures that are unstable.
Exponential conductivity
variation, which models many ceramic materials, experiences
stable heating up to a maximum steady state teaperature.
This maximum value can be controlled by changing the heat
transfer conditions, or by doping the saaple to change the
teaperature dependence of electrical conductivity.
A nuaerlcal study of the "hot wall" applicator reveals
enhancement of low teaperature heating rates up to the wall
teaperature.
In Chapter IV, the experimental apparatus for
microwave sintering is discussed.
Various applicators are
reviewed, Including slab, rectangular and circular, both
TM0I0 and TEllt .
He describe the fields In the circular
applicator and show how the Impedance analysis [18] for
the rectangular applicator can be simplified to a cavity
equivalent circuit for thin low loss samples.
The thermal
stability analysis Is applied to cylindrical rods In the
rectangular cavity.
During sintering, proper control of
cavity parameters (coupling and tuning) Is essential to
maintain stable heating.
Chapter V daals with tha problem of Material
characterization.
He first deecrlbe an automatic technique
using a scalar network analyzer to determine the
permittivity and conductivity as a function of frequency and
temperature.[27,28] Uncertainties In the measurement are
discussed, Including high and low loss limits, gap effects
[29] and device Imperfections.
From Maxwell's equations we show that when conduction
current dominates, changss In displacement current
(dielectric constant) are difficult to measure; and when
displacement current dominates, changes In conduction
current (conductivity) are obscured.
Small gaps between the sample and the walls of the
sample holder produce noticeable errors.
We use a
distributed capacitance model and a transverse resonance
analysis to study the effect of the gap, and conclude that
In order to obtain accurate measurements, all gaps must be
eliminated.
We describe a method for electroplating samples
to eliminate gaps.
Multiple reflections between components produces
measurement errors.
A signal flow graph analysis Is used to
determine the most significant errors.
Some of these errors
are systematic and can be removed through analysis of the
data.
A technique is needed to measure sample properties at
high temperatures.
We review several existing methods and
6
propose an in situ ssthod based on prior theoretical
Insights.
He conclude that a sea11 spherical saaple say be
aost useful because the teaperature profile is likely to be
unifora.
The conclusion addresses areas of future work.
In the
theoretical area, this Includes expansion of the analysis to
Include skin effects In the slab, and sintering dynaalcs.
Bxperlaentally, an apparatus to perfora high teaperature
characterization should be lapleaented.
CHAPTER II
LITERATURE REVIEW
In this chapter the microwave sintering literature is
reviewed.
He first survey some of the fundamental works,
where the process la developed, and then focus on the
problem of thermal runaway.
Microwave heating In general
and sintering In particular can be performed In a variety of
applicators, both traveling wave and cavity type.
Thermal
runaway effects are observed when the electrical losses
Increase rapidly with temperature.
Surface temperature
feedback Is used to control the Input power and prevent
runaway.
Numerical analysis reveals that In some cases the
system becomes unstable.
The first known reference to microwave sintering of
ceramics Is [4].
Bertaud and Badot have examined the
sintering of refractories In a single mode cavity.
They use
an Iris coupled rectangular applicator In the TE10, mode
at 2450 MHz.
The saaple, a long rod, Is translated
vertically, parallel to the electric field vector.
is observed In the central region of the rod.
Heating
Monitoring
equipment Includes an IR pyrometer to measure surface
teaperature and a detector that is magnetically coupled to
the sidewall of the applicator to measure the resonant
behavior of the cavity.
They report a cavity Q of 3000 for
7
* cold alumina rod.
Upon firing, tha Q raducas to
approxlaataly 200 at 1000*C.
Thla lndlcataa an
lncraaaa In alactrlcal conductivity aa tha taaparatura
rlaaa.
Thay raqulra high power for Initial haatlng (400 W ) ,
but only 100 W to maintain alntarlng taaparatura.
application of 400 W cauaad tha aaapla to malt.
adjustments wars accomplishad manually.
Continual
All powar
Results show that
sintering took place In 10 to 19 minutes at an efficiency of
90*.
Mataxas and Meredith [5] discuss tha microwave haatlng
problem In detail.
Tha text provides an extensive
bibliography and dlscusaea the practical electromagnetic
aspects of the heating problem, Including applicator
selection, heating rates and numerous experimental examples.
The focus Is on the drying of materials, nevertheless the
Information Is of value for sintering efforts as well.
0
Krage [6] consider the sintering of ferrites in a
microwave oven.
The study was performed using 1.17 cm x 0.3
cm disk-shaped samplea of barium ferrite.
frequency was 2450 MHz.
The operating
The sample was placed on a
turntable within a thermally insulated container.
Sintering
wae accomplished at a heating rate of 9°C per minute up
to a stable temperature of 1230*C.
observed.
No cracking was
Total sintering time was 135 minutes, Including
burnout of the binder.
Phyelcal strength, percent
shrinkage, density and magnetic properties were comparable
to conventionally sintered ferrites.
Schubrlng [7] reports on the microwave sintering of
alumina spark plugs.
A conventional (2450 MHz) microwave
oven surrounds an Infrared cavity which Is used to contain
thermally radiated energy from the sample.
The Infrared
cavity Is fabricated from millboard and Is transparent to
lcrowaves.
Temperatures of 1600°C were easily obtained,
and feedback control was needed to prevent runaway.
The
author obtains low quality product and attributes this to a
short sintering time.
Work at Los Alamos [8] Involves the use of both
conventional microwave oven techniques at 2.45 GHz, and a 60
GHz, 200 kW gyrotron source.
Initial results presented for
alumina using the gyrotron and a TE0Z circular cavity
show final densities between 80 and 94* from green densities
between 50 and 59*.
They have also attempted to sinter
composite materials (aluminum with silicon nitride
whlskeres, etc.) In a microwave oven.
Samples are placed In
a thermally insulated enclosure within the oven.
Coupling
agents such as glycerol are needed to start the heating
process.
Temperatures near 1700°C have been obtained.
Microwave sintering research at Oak Ridge National
Laboratory [9,10] involves a 28 GHz, 200 kW gyrotron source
and a large, untuned cavity.
The higher frequency Is
10
preferred because the loss tangent is higher for many
ceramics and consequently, Initial heating Is easier.
The
untuned cavity Is large compared to the wavelength of the
source, and the measured field strength varies by only
4%.
The sample is enclosed In a thermally insulating
blanket, and a thermocouple Is used to measure sample
temperature.
Heating rates of 50°C/rain have been
obtained for alumina.
A comparison of microwave sintering
with conventional sintering shows faster processing times
and lower sintering temperatures.
Tlnga [11] reviews microwave heating and sintering up
to 1986.
He discusses the runaway problem, noting that
feedback can be used to control the heating process.
Rapid
heating Is attributed to Increasing electrical loss with
temperature, and It Is found that
the presence of small
Impurities significantly affect the loss factor.
He reviews
a number of popular applicators, Including the ring
resonantor, the meander type traveling wave applicator as
well as single mode cavities.
The use of a mode stirrer is
Important In reducing temperature extrema produced by
standing waves.
Boslslo, et al, [12] address the problem of thermal
Instabilities In the heating of Debye materials (ethanol).
They discover that measurement of the loss factor becomes
difficult above a certain power level and that localized
11
breakdown of tha liquid is tha causa.
Thay do not
discuss tha problea mathematically, but point out that
localized breakdown la enhanced by the low thermal
conductivity of the material which tends to sustain
localized teaperature increases.
An interesting approach to the runaway problem is found
in [13], where rapid heating was observed in nylon
monofilaments.
The loss factor of nylon Increases with
teaperature causing acceleration in heat generation.
This
change in loss factor produces an Impedance change in the
applicator, or load, resulting in a change in the power
delivered to the sample.
If the applicator is properly
tuned, the system becomes self-adjusting.
That is, as the
saaple heats up, the dielectric loss Increases, coupling
decreases, and the rate of heat generation
is reduced.
A
stable operating point is established at the desired
teaperature by controlling the input power and the tuning of
the load.
Roussy, et a l , [14,15] use a numerical approach to the
thermal runaway problem.
cylinder.
The geometry is an infinite
The heat equation is solved using convective heat
transfer at the boundary. They solve the steady state heat
equation analytically for the case of constant electrical
conductivity.
When conductivity Is allowed to vary
quadratlcally with teaperature, a numerical solution Is
12
needed.
The reeult Is a series of curves defining regions
of stable and unstable heating for the case of teaperature
dependent conductivity.
The meaning of stability Is not
well defined and the result depends on the convective heat
transfer constant and the polynomial coefficients of the
conductivity.
Radiation effects are not Included.
They use
their results to develop an analytical analogue for a
control system to monitor the heating of a rubber material.
Sekerka, et al, [16,17]
runaway
are interested In thermal
as It affects crystal growth.
They analyze the RF
heating of an Infinite cylinder using a shooting method.
They use an activation temperature model for the electrical
conductivity, o ■ o0exp(-TA/T) .
Their results show
regions of stable and unstable heating.
For silicon, a
sigmoidal curve Is derived, Indicating a region of lower
temperature stability and an upper branch of stable heating,
separated by a region of unstable heating.
The upper
branch for silicon has been observed experimentally.
More
complex curves are used to describe the melting process and
the solld-llquld Interface.
To summarize, these papers Indicate that 1) microwave
sintering is feasible, offering decided advantages over
conventional sintering;
2) alumina is almost universally
used as the model material; 3) thermal runaway is attributed
to electrical conductivity that increases rapidly with
13
temperature; and 4) runaway aay ba controllad by faadback
tachnlquaa If tha ayataa la atable.
Our work builds on prior studlas at tha Mlcrowava
Tharaal Processing Laboratory at Northwastarn. [2] Aranata,
at al, [18,19] hava davalopad an lapadanca analysis for rods
In ractangular wavagulda, and have verified tha
slnterablllty of a variety of aaterlals Including a and 0alualna, N10 and ZnO.
Recant work has Included studlas In
single aoda applicators Including, tha slab applicator, the
ractangular applicator, tha TEItl [3] circular cavity
applicator and tha T M ^ circular cavity applicator.
The circular cavities feature alrrored walls to reflect
radiated heat back to tha sample, resulting In high
sintering temperatures and reduced thermal gradients.
Tlan,
et al, [1,20] have succeeded in producing very fine grain
structure In alumina and have sintered alumlna-TlC.
In the investigation to follow, we will perform a
theoretical analysis of microwave heating for both
electrically thick and electrically thin bodies to describe
temperature profiles during heating and to better
understand the runaway phenomena.
choice for a model material.
Alumina is a logical
CHAPTER III
THEORETICAL ANALYSIS
In this chapter the theoretical aapecta of nlcrowave
sintering are explored.
We describe the general
sintering process which Includes a discussion of Maxwell's
equations, the heat equation and sintering dynamics.
These
equations are then applied to some simple geometries to
learn more about electromagnetic heating.
We first consider
the seml-lnfInlte space, considering the cases of constant
electrical conductivity and temperature dependent
conductivity.
A discussion of the planar slab geometry
reveals the possibility of temperature Instability, and a
relationship Is developed between steady state temperature
and applied power.
Cylindrical and spherical geometries,
and the "hot wall" applicator problem are also considered.
A.
Qeneral Theory
The basic Ideas Involved In microwave sintering can be
pictured In the following way.
Consider a sample of
arbitrary shape Irradiated by microwaves.
Loss mechanisms
within the sample (such as electrical conductivity) cause
the microwave energy to be converted to heat producing a
temperature rise In the sample.
14
This generated heat Is
redistributed via thermal conduction, and lost to the
surroundings via convection and radiation.
As the
temperature rises, the saaple undergoes physical changes
including denslflcation, thermal expansion, etc.
In a
controlled sintering environment, the temperature will rise
rapidly to a stable "sintering temperature" where it remains
until sintered, then it is allowed to cool.
A typical
sintering run is pictured in Pig. 1 [1].
It is Important to note the temperature and temporal
dependence of the material properties during sintering.
As
temperature rises dielectric constant, electrical
conductivity, thermal conductivity, etc. change.
With time,
the density and microstructure of the material change
altering the electrical and thermal properties as well.
1.
Electromagnetic Aspects
The electromagnetic problem reduces to the questions:
what are the fields everywhere within the applicator and
more specifically within the sample, and what is the total
power absorbed by the sample?
Por an arbitrary sample,
lnhomogeneous and isotropic, Maxwell's Equations are (for a
source free region): [30]
7 * H ■ J ♦
(la)
16
(* C )
3300
PHCHCAT1NC
STACC
3000
I
STAfiLC SINTTKlNC
STAGE
transition
ST ACE
TEMPERATURE
I
1300
1000
t
t
/
r
;
'I
300
I
I
^ i i i i M
TIME
H g u r * 1.
I. l r
a A - l l l l* l^lq 11- kU,*^ 2
1 1 1 •
j ilx J
(MINUTE)
Tsnpsrstur# history o * aluaina aaapl* during
•Intaring. Adapted froa [1].
17
7 x E ■
(lb)
7 •D - 0
(lc)
7 •B - 0
(Id)
where D - £E , B ■ uH
and £ Is permittivity,
,J ■ oE,
u Is permeability and o
la
conductivity.
Since electrical properties change with time as the
sample Is heated, these equations cannot In general be
simplified.
However,
temperature change to
If we assume the time scale for
be much slower than themicrowave
period, the electromagnetic problem can be reduced to the
sinusoidal steady state over a finite time Interval.
(This
Is reasonable because electrical effects are measured In
nanoseconds, while thermal effects are measured In
milliseconds.)
In this case, Maxwell's equations become
7 x H - (o +ju£)E
(2a)
7 x E ■- jwuH
(2b)
7 • (€E) - 0
(2c)
7 • (UH) - 0
(2d)
where
cj
Is the angular frequency.
18
Heating of the sample is Induced by material losses,
namely conductive, dielectric and magnetic.
In terms of
observables, dielectric loss (due to dipole action) and
conductive loss (due to collisions) are Indistinguishable.
We write the loss terms as follows:
(3a)
(3b)
where £0, the permittivity of free space, Is 8.854 x
10~‘zF/m and n0, the permeability of free space, Is
4ir x 10‘7H/m.
In general, these losses vary with temperature.
To solve for the fields In the sample, we must solve
the non-homogeneous wave equation. [31]
7ZH ♦ <i>2t V H + 7^H •
(7 x H) ] x (7t‘) - 0
(4a)
For homogeneous samples, this reduces to the familiar wave
19
equation.
7*H ♦ w V u ’H ■ 0
(5a)
72E + w 2€*u'E ■ 0
(5b)
Note that materials which are homogeneous at room
temperature
can become lnhomogeneous astemperature
due to the temperature dependence
rises
of materialparameters and
the presence of thermal gradients.
In this Investigation, ceramic materials are of
particular Interest.
They are typically non-magnetlc, and
their dielectric properties change with temperature.
In
Fig. 2, Westphal [32] shows the variation of permittivity
and conductivity as a function of temperature for alumina.
The dielectric constant Increases slowly with temperature,
while the conductivity (proportional to the loss tangent)
Increases dramatically with temperature.
Ho [33] attributes
the Increase in dielectric constant to the Increase In
volume, and thus polarlzablllty, due to thermal expansion.
In high grade ceramics, loss Is attributed to low quantities
of Impurities.
The power absorbed by the sample can be computed from
the Poynting Theorem, and Is given as a time average of the
absorbed power. [34]
20
0.010
0.000
oooo
Temperoture (C)
<*4o odd 666 " i5
Temperature (C)
71gure 2
a) loss tangent and b) dielectric constant of
ainalna ae a function of teaperature at 3.6 QHz.
Adapted froa [32].
21
P.>. “ * / o | E | * d v ♦
^ V
(6)
{ Ur" IH 12dv
*V
In an actual system, incident power will be partially
absorbed and partially reflected.
If we consider a slab
sample backed by a short-circuit, the reflection coefficient
Is [35]:
(1 - jc7)e™ + (1 + J67)e-y«
(7)
(1 ♦ J O e y‘ ♦ (1 - £7)e"v<’
where d Is the thickness of the slab and Y Is the
propagation constant in the sample.
The absorbed power, In terms of the Incident power and
the reflection coefficient, p, is
(8 )
P.*. - (1 - lPl‘)P,Be
Figure 3 shows percent absorbed power as a function of
conductivity.
For low conductivities, there is low
absorption since there Is no means of generating heat; the
energy is transmitted through the sample.
For high
conductivities, there Is low absorption since there is high
22
1.00
Power
A bsorbed
0.80
0.60
0.40
0.20
0.00
10
'*
10"*
10
Conductivity
Figure 3.
1
1
(S /cm )
Percent absorbed power vs. conductivity for s
planar slab of thickness d ■ 0.95cs, f - 2.45
GHz, €r ■ 10.
23
reflection from the front eurface and weak field
penetration.
The conductivity of maximum absorption depends
on the thickness of the sample.
Note that transverse
variations in the electric field (such as occur in
rectangular waveguide) have been neglected.
2.
Thermal Aspects
The heating of a sample is governed by the heat
equation.
The general equation is [36]
7 • (k 7T)
(9)
where k Is the thermal conductivity,
c, is the heat
capacity and n is the density.
When thermal conductivity is constant this becomes
(1 0 )
Heat is generated from the absorbed microwave power and at a
point in the sample it is
(ID
Heat energy is lost to the surroundings through
convection and radiation [37].
The boundary condition is
24
where T0 Is the ambient temperature, h Is the convection
constant, s Is the Stefan-Boltzmann constant and £> is
the emlsslvlty.
The first term on the right hand side Is due to
convective heat transfer and the second Is attributed to
radiation. Initial conditions require a uniform temperature
and a homogeneous sample.
Thermal parameters vary with temperature [38].
The
thermal conductivity for many ceramics Is shown to decrease
with Increasing temperature, Fig. 4.
Figure 5 shows
specific heat as a function of temperature.
At high
temperatures it approaches the 3Nkb limit, or 5.96 cal/ga'tom°C. [39]
3.
Sintering Dynamics
Figure 6 shows spherical particles In contact.[40] In
the region of contact, mass transport occurs causing the
formation of a neck at the boundary between grains.
Mass
transport occurs via several mechanisms: lattice diffusion,
grain boundary diffusion, surface diffusion and plastic
flow.
Those processes which produce denslflcatlon are
25
004
M
E
U
u
E
W
003
Smgl«cry$t»l TiO.
II to r axit
I
$
T
*
V
Potycry»t»llmt Al2()
o 17m grain tin
o 9m grain tin
£ 002
1
1
001
Polvcryttallina TiO.
200
Figure 4.
400
800
1000
1200
Thermal conductivity v s . taaparatura for varloua
caraalca. Adapted froa [38].
26
(o. UJ0|» 1/1*}) li.}*<J«} )**H
sc
MrO
Al . 0
0
Temperature (*C)
Figure 5.
Heat capacity va. taaparatura for various
caraalca. Adapted froa [38],
27
Cram boundary
Figure 6.
Ceramic partlclee In contact, allowing a grain
boundary. Adapted from [40].
28
lattice diffusion, grain boundary diffusion and plastic
flow. Coarsening Is produced by surface diffusion.
Conventional sintering favors surface diffusion at low
temperatures because of Its low activation energy. Rapid
sintering techniques have a shorter dwell time at the lower
temperatures, and tt\us tend to inhibit excessive grain
growth. [41]
In a typical sintering operation, a green sample will
have an Initial density of 30-70% theoretical density.
Particle size Is on the order of a micron or less.
Sintered
samples show 95% theoretical density or better. [1]
He note that as density Increases, shrinkage occurs,
causing a change In surface area, which affects heat
transfer.
A change in density affects both electrical and
thermal parameters, and should result In a higher electrical
conductivity, dielectric constant and thermal conductivity.
4.
Case Study
In the theoretical Investigation to follow, results are
applied to particular examples.
material.
We use alumina as a model
The physical constants listed below will be used
throughout the next section, unless otherwise noted.
[42]
29
f - 2.45 GHz
€r - 10
o0 * .002 S/m
(C « 20 W/m*K
cp - 1000 J/kg-K
P ■ 3970 kg/m?
h - 10 W/m2*K
a ■ 5.67x10"’ W/m2-K’
(„ « 0.6
where f Is operating frequency, €r Is dielectric
constant, o0 Is electrical conductivity,
k
Is
thermal
conductivity, cp Is heat capacity, p Is density, h Is
convection constant, s Is Stefan-Boltzmann constant and
Is emlsslvlty.
The melting point of alumina Is approximately
2050°C, but this condition will be relaxed In the
theoretical analysis to follow.
Temperature dependent electrical conductivity for
sintered alumina Is determined from published data. [32,43]
30
B. Thtorttlctl H a t i n g of a S— 1-Inflnlte Space
We consider the simplest geometry, a semi-infinite
space.
The goal is to determine temperature profiles as a
function of incident power and material parameters.
The
analysis is divided into two parts; 1) constant electrical
conductivity and 2) temperature dependent electrical
conductivity.
Both steady state and transient solutions are
developed.
The geometry of the system is shown in Fig. 7.
A
uniform plane wave is Incident from the left upon a planar
surface.
Part of the signal is reflected, and the remainder
penetrates the space.
The transmitted wave heats up the
material and a temperature distribution with time and
position is the result.
Lt
Constant Conductivity
The temperature profiles are determined by solving the
heat equation with appropriate boundary conditions.
The
time dependent heat equation in one dimension is:
X > 0
Heat is lost to the surroundings by convection and
(13a)
UNIFORM
PLANS
WAVS
RADIATION
rigur* 7.
I
CERAMIC SAMPLE
Gto M t r y of planar half spaca.
32
radiation at the aurface.
icfj - h(T - T..,) ♦ a€K(T4 - T ^ )
.
x - 0
(13b)
The flret tera on the right Is due to convection and the
second to radiation.
In the U n i t of Infinite depth,
C m T ■ T„ < oo .
(13c)
The space Is Initially at a reference tenperature of 300K.
Heat Is generated electronagnetlcally, and Is
proportional to the absorbed power,
P(x) - Jo(x) | E(x) | *.
(14)
The total absorbed power Is the Incident power less the
reflected power.
P0 - (1 - |p|*)P1Be
(15)
If constant conductivity Is assumed, the electric field
within the saaple Is: [44]
E(x) - E0e y"
(16a)
33
where v -
(16 b)
end
(16c)
| B(x) | * - E(x)-E*(x) - E^e'2"
where a * $ 4 r [ i ♦
]
sln( ^ tan'1^ )
<16d>
The generated heat le [40]
Q, - no0P0e-*" -
no0(l - |PI*)Plnee‘*“"
(17)
The reflection coefficient from a planar homogeneous surface
Is: [46]
l - rf~t
P - ---- 1
i ♦ £7
•
(18)
He consider the variation of transmission as a function
of conductivity In Pig. 8 where the percent absorbed power
Is plotted as a function of conductivity for a fixed
dielectric constant of 10.
For low conductivities, the
absorbed power Is determined by the dielectric constant.
For high conductivities, above 2 S/m, the material becomes
more reflective, approaching the limit of total reflection
by a perfect conductor.
34
POWER
1.00
0.80
ABSORBED
0.60
0.40
0.20
0.00
1
-3
10
■*
10-
1
10
10
CONDUCTIVITY ( S / M )
Flour* 8.
Parcant powar abaorbad v*. conductivity for a
planar half spaca, f ■ 2.45 OH*, €r - 10.
35
a. TriMltnt Solution
The previous aquation la aolvad nuaarlcally to
dataralna taaparatura aa a function of tlaa and position,
ualng an axpllclt aathod [47].
Wa represent the derivatives
by finite differences:
all
-irJL
- 2T
3x*
-t T
(dx)*
- T
dt
(19a,b)
whers ths subscripts represent Increments In position and
the superscripts, Increments In time.
The temperature at
time t ■ j+1 can be expressed In terms of temperatures at
time t ■ J:
T
f i
s')
j
Q„dt
- r T
+ T
♦ (1 - 2r)T +
•
V ‘“ I IM /
1
P*
where
(20)
r ■ —
— =c^P (dx)
The boundary conditions at the eurface are linearized
and recomputed at each time Increment.
where hr ■ h ♦ e€xT*
The grid le extended one point beyond the boundary.
(22a)
(22b)
(22c)
The Infinite region Is approximated by a finite grid.
It la necesaary to determine boundary conditions In the
Interior to minimize error propagation from the rear surface
where the grid la truncated.
The temperature gradient Is
extrapolated one point beyond the grid.
(23)
The grid spacing Is constant.
A multipoint method for
predicting the end value could further Improve the accuracy.
37
[48]
The •xplicit method Is not unconditionally stable.
an Interior node, we auet choose r < 0.8.
For
At the surface,
the stability criteria Is: [49]
h Ax
—ljj— ) < 0 . 5
r( 1
(24)
Since hr can becone large due to radiation we choose r <
0.25.
Before examining the transient temperature profiles, we
first consider the steady state solution as a function of
electrical conductivity, Incident power and thermal
conductivity.
b, Steady State Solution
As time Increases to Infinity, the heat equation
becomes
(25)
The same boundary conditions apply.
The solution of (25) yields temperature as a function
of position.
38
Q
T -
- •"*••] ♦ D
(26)
D Is found froa the solution of a quartlc aquation.
D '
- [ssfe:+
+
♦ T-.*] - 0
Tha taaparatura at tha surfaca, x ■ 0, la D.
i” )
Tha
taaparatura daap within tha aatarlal la
T- - D ♦
(28)
and la navar lass than tha surfaca taaparatura.
Wa consldar two Halting cases, low and high electrical
conductivity.
In the first case, o -* 0, and In this Halt,
. _Q2_
24T
%
(29a,b)
2€IP 'm
(4^7 ♦ 1 )*S4k
-
4€ *P
WKCJC ♦ 1J
(29c,d)
and ws sea that steady state surfaca taaparatura Is
Independent of electrical and thsraal conductivity, and Is a
39
strong function of Incident power, heat transfer at the
boundary, and dielectric constant.
In the Interior, the
asymptotic value of temperature Increases Inversely with
electrical and thermal conductivity.
In the second case, o -» oo, and we have
(30a,b)
(30c.d)
Note that the surface temperature Is Independent of
electrical conductivity and dielectric constant, and that
the Internal temperature limit approaches the surface
temperature as the electrical conductivity becomes large.
The thermal gradient Is strongly related to the electrical
skin depth and the thermal conductivity.
c. steady Stmts Results
He evaluate these equations using a model material,
which approximates the material parameters of aluminum
oxlds. The physical constants are listed In Section A.4 of
this chaptsr.
He use different values of electrical
conductivity to understand the role of skin depth in the
constant conductivity model.
The external temperature Is
»
40
equal to the initial taaparatura of tha aatarial and is
300K.
Tabla I shows how tha surfaca and daep tanperaturaa
vary as a function of physical paraaatars.
Tha first two
coluans raprasant tha alactrlcal and tharaal conductivities.
Tha next* coluans list tha assuaed incident and absorbed
power densities.
Tha fifth and sixth coluans are tha
surfaca and deep taaparatures.
Tha final coluan is a
■assure of tha rata of rise of interior taaparatura as
raprasantad by tha depth at which the taaparatura rises froa
tha surfaca taaparatura to 90% of tha total rise.
Taaparatures above tha aalting point of aluaina (2000°C)
are theoretical only.
At a vary low electrical conductivity, Line A, there is
a large rise in taaparatura as one proceeds a considerable
distance into tha aatarial.
Increasing conductivity, B,
decreases the interior taaparatura and shortens the length
of tha taaparatura gradient.
Surfaca taaparatura and power
absorption are unchanged since reflection is doalnatad by
tha dielectric constant.
A further increase in conductivity, C, further
decreases tha interior taaparatura and shortens tha
gradient.
Decreasing tharaal conductivity, D, only
increases interior taaparatura.
Increasing alactrlcal
conductivity, at tha original tharaal conductivity, E,
41
o (S/a) K(W/aK) P lM <*/«») *P.H
T0(C)
T.(C)
.9T_(a)
A.
0.002
20
200
73.0
1899
98900
9.70
B.
0.02
20
200
73.0
1899
11600
0.97
C.
0.2
20
200
72.6
1898
2869
0.097
D.
0.2
10
200
72.6
1898
3840
0.097
E.
2.0
20
200
56.8
1857
1962
0.0115
P.
20.0
20
200
21.4
1805
1827
0.0027
0.
20.0
20
100
21.4
1505
1516
0.0027
H.
20.0
20
900
21.4
2283
2339
0.0027
Tabla 1.
Taaparatura profllaa in a aaal-lnflnlta aolld.
42
decreases absorbed power (due to Increased reflection),
lowers the surface temperature slightly, reduces temperature
lnhoaogenelty and shrinks the gradient.
A further Increase,
F, shortens the gradient and further laproves the
hoaogenelty.
Changee In applied power, Lines G and H affect
the temperature and not the gradient.
d. Tranelent Results
Transient teaperature profiles are determined by
numerical calculation.
The computer program le given In
Appendix A. The same material parameters are used.
Again,
the external temperature Is 300K.
Figure 9 shows teaperature within a half-space as a
function of depth.
The electrical conductivity Is 0.2 S/m
and the Incident power Is 50 W/cm*.
1mm and the time step Is 0.04 eec.
The grid spacing Is
Figure 9(a) shows the
temperature profile after the 500 sec.
We notice that heat
Is transferred to the surface where It Is lost to the
surroundings by convection and radiation.
Heat Is
transferred to the Interior via conduction, and after 1500
sec, Fig. 9(b), the teaperature has penetrated more deeply
Into the material.
The maximum temperature Is In the
interior of the material.
Given sufficient time, Ignoring
the melting point, the temperature profile will approach the
43
steady state value, Fig. 9(c).
In the figures to follow we
will truncate the grid near the surface and observe only In
the near Interior heating effects.
The following sequence of figures, Figs. 10-13, show
the transient teaperature profiles at constant
conductivities varying from .002 S/m to 2 S/m.
extended 200 mm Into the material.
The grid Is
This has been shown to
be sufficient for an observation area of 100 mm by comparing
these results with a grid extending 300 mm Into the
material.
These results become less accurate as time
progresses.
A typical transient profile Is shown In Fig. 10 where
the conductivity Is 0.002 S/m and the Incident power Is 5000
W/cm:.
Reflections reduce the power by 21% to 3650 W/cm:.
This power level Is not practical In a laboratory, but
serves to Illustrate the heating phenomena.
The transient
profile after 100 s shows pratlcally uniform heating.
After
400 s, there Is a temperature gradient near the surface due
to heat transfer to the surroundings.
Heat Is also
conducted to the Interior as evidenced by the shallow
gradient on the right hand side of the figure.
All
transient profiles have these characteristics.
The next figure, Fig. 11, Is for a conductivity of 0.02
S/m and an Incident power of 5000 W/cnr .
At 100 s and
400 s the profiles are almost Identical to Fig. 10.
This
44
TEMPERATURE
(C)
5000
40001
3000
2000
melting
point
- ■
10001
0
100
200
300
400
500
DISTANCE (MM)
Figure 9.
Transient teaperature profiles In a planar half
space; o ■ .2 S/a, P - 50 tf/ca*, a)500 sec,
b)1500 sec, c) steady state.
45
(C)
3000 -]
m _ii
TEMPERATURE
2000
1•**
1000H
0
20
40
60
80
100
DISTANCE (MM)
Figure 10.
Transient taaparatura profiles In a planar half
space; a ■ .002 S/a, P - 5000 W/ca*. a) 100
sac b)400 sac.
(C)
46
TEMPERATURE
2000
. ... ■ •M in i
p * >nt
1000
20
40
60
80
100
DISTANCE (MM)
Plgura 11.
Transient taaparatura profllaa In a planar half
apaca; o ■ .02 S/a, P ■ 500 W/ca*, a)100 aac
b)400 aac.
»1
y'
tr *
K.
.
X%
fV ^
***• ’**0°
*'»»c
»w #
v •* '
*• * - 4'I40°
mC
< :
«.;
-'i0°
«*e* * -*c •
48
(C)
3000n
TEMPERATURE
2000
1000
-
20
30
40
50
DISTANCE (MM)
Plgure 13.
Transient taaparatura profiles In a planar half
space; o ■ 2 S/a, P • 80 W/ca*, a)100 aac
b)200 aac. c)292sec, d)1256 sac.
49
demonstrates that hsat is being generated at the same rate
In both cases.
It would take 4000 s for the previous
example to reach these temperatures If the incident power
was 500 W/cm2.
Since condutlvlty has been Increased 10-
fold , power must be reduced proportionally to maintain the
same heating rate.
Figures 12 and 13 are for conductivities of .2 and 2
S/m respectively.
figures.
Incident power is 100 W/cm2 in both
These plots show the reduced field penetration by
the decreased heating of the interior.
Reflection from the
surface has Increased.
Figure 14 shows the effect of changes in power,
observation time, thermal conductivity and skin effect upon
temperature non-uniformity, for an electrical conductivity
of 0.2 S/m.
The solid curve at 100 W/cm2 is observed after
400 seconds and exhibits a small gradient near the surface.
At a higher power, 200 W/cm2, the dashed curve, we achieve
similar temperatures in a shorter time, 200 s.
near the surface has been Increased.
The gradient
If we decrease thermal
conductivity for the 100 W/cm2 - 400 s curve, we have the
dotted curve.
Note the poorer uniformity arising from
reduced heat flow into the interior.
Note also the relative values of surface versus
interior temperature.
For the solid and dotted curves,
which differ only in thermal conductivity, the surface
50
TEMPERATURE
(C)
3000 n
2000
1000
20
40
80
100
DISTANCE (MM)
Figure 14. Transient teaperature profiles In a planar half
space; a)solId, o - .2 S/a, k • 20 W/a-K, P ■
100 W/ca*, t ■ 400 sec, b)dash, o » .2 S/a, k
- 20 W/a-K, P - 200 W/ca*, t - 200 sec, c)dot,
o - .2 S/a, k - 10 W/a-K, P - 100 W/ca*, t 400 sec.
51
temperatures are practically the same whereas the internal
peak temperatures are significantly different.
Upon
comparing the solid and dashed curves, with only the power
Increased, we see the expected Increase In surface
teaperature as a result of the Increase In absorbed power.
The dotted curve coincides with the dashed curve In the deep
interior.
**
It should be noted that steady state temperature
distributions are well off the graph.
In normal sintering
practice high power Is applied to heat the sample to the
sintering temperature, then power Is reduced to maintain a
constant teaperature.
2. Temperature Dependent Conductivity
The time dependent heat equation In conjunction with
the time harmonic wave equation are used In the temperature
dependent conductivity problem.
In one dimension they are
, x > 0
“ h <T - T.«<> ♦ *x(T« - O
, x - 0
(31a)
(31b)
(31c)
fjj ♦ -r„E - 2 %
. x - o
Old)
52
where 0 ■
an<* vo
*he free epace
propagation conetant. Since the nonuniformity le In the
direction of propagation, and TE waves are assumed,
dielectric gradient terms arising from (4) can be neglected.
The electrical conductivity Is a function of
temperature
a - o0f(T)
(32)
The function f Is defined to be unity at the background
temperature, i.e. f(T.„t) ■ 1.
conductivity case f(T) Is 1.
For the constant
To complete the mathematical
statement of the model problem, conditions are required on T
and E as x approaches Infinity.
|E| - 0
(33a)
T - T„ < oa
(33b)
where T_ Is a bounded constant.
Transient Solution
The transient solution to the heat equation remains the
same as before.
However, it Is necessary to modify the
solution to the wave equation.
We note that
53
e„
♦ r*E - 0 .
(^
♦ in fa s
(34)
■ i r >B - 0
In the constant conductivity case there Is no backward
tranvsling wave In the saterlal.
In the teaperature
dependent conductivity case, we aust Include forward and
backward traveling waves as part of the solution.
We do
this In discretized fora, allowing each region of the grid
to be coaprlsed of both coaponents, representing aultlple
reflections In that grid region.
We assuae that at soae
point deep within the saaple, there Is no reflected wave.
By proceeding backwards along the grid, the electric field
can be deteralned at each point in the saaple. [50]
where
(35b)
To reduce computation time, we can assume that at low
temperatures the conductivity is nearly constant.
Reflections between layers can be Ignored and the simple
attenuation equation of the constant conductivity case can
be applied, using the new conductivity of each grjld element.
At higher temperatures, where conductivity gradients are
significant, the full model must be used.
A computer
program using this algorithm Is given In Appendix B.
We apply the transient analysis to our model material,
using an exponential conductivity law, where o ■
.002exp(.0026T), which is a good fit to the conductivity vs.
temperature data for alumina (Honeywell A-203, 95*). [32]
Figure 15 shows the the temperature profile vs. depth
for a power input of 500 W/cmz.
After 1000 seconds, the
material Is at a uniform temperature of 400 C to this depth.
After another 500 seconds, the teaperature has doubled, and
continues to accelerate due to the exponential conductivity
variation.
The surface teaperature tends toward a limiting
value, while heat Is conducted Into the Interior.
Again,
30CC
r*
— 1 2000
1800 sec
1700 sec
60
Figure 15.
80
Transient taaparatura prof1las in a planar half
spaca with taaparatura dependent conductivity,
o • .002 exp(.0026T), P - 500 H/ca*, a)1000
sac, b)1500 sac, c)1700 sac, d)1800 sac, a)1900
sac.
56
the M l ting point is ignored.
This is bsttsr illustrated in Fig. 16, where the
surface teaperature is plotted vs. tlae.
regions of heating.
He see three
Region I is initial heating, the
teaperature rises slowly as the conductivity is only weakly
dependent on teaperature.
Region II is rapid heating, where
the heat generation rate accelerates exponetlally; and
Region III, teaperature saturation, where the skin effect
has H a l t e d the field penetration into this high
conductivity aaterlal and surface reflection has Increased,
reducing the total absorbed power.
Figure 17 shows the conductivity as a function of depth
in this Halting region.
At low teaperatures, the theraal
gradients are saall, and consequently, there is little
variation in conductivity.
In the teaperature saturation
region, however, a conductivity gradient is evident.
This
reinforces the need for the aore exact solution to the wave
equation.
In the constant conductivity section, we discussed
truncation errors —
that is, a finite grid must be used to
approxlaate the infinite solid.
result froa the grid spacing.
length be
s m
Inaccuracies can also
It is laportant that the grid
II enough to resolve detail in both the theraal
and electrical regiaes.
The theraal length is defined as a
ratio between conduction and surface transfer constants:
67
(C )
4000
TEMPERATURE
3000
2000
000
500
iME
Flgur* 16.
1000
1500
(S E C )
Surface taaparatura vs. tlao for Trsnslsnt
hosting of s plsnsr half spscs with taaparatura
dspsndsnt conductivity, o ■ .002sxp(.0026T), P
■ 1000 W/ca*.
58
10 1
uo
>-
o
3
o
z:
o
o
1
0
DEPTH
Flour* 17.
40
20
(MM)
Tranaiant conductivity £ » * “ • tor planar half
gpicc aftar 1900 **c« with o ■
.0 0 2 axp(.0026T), P ■ 1000 W/ca*.
99
At zero degrees, the theraal length le 2 aetere.
Ae
teaperature rleee, the theraal length decreaees.
At 600*C,
lt - 1 a; at 1500*C, 1, le .1 a; at 3800“C,« lt Is .Ola; and
at 8200°C, lx la .001 a.
Since the grid apaclng la .001a,
the explicit aethod le unatable above thla teaperature.
The electrical length la referenced to the akin depth,
which la
If we uae the aaae exponential law for conductivity, we
notice that at 0°C, 1. la .2a; at 23904C, o Is .1 S/a and 1,
le .Ola; at 4161°C, o le 100 and la Is .001, which la on
the order of the grid apaclng.
So we eee that In the
teaperature dependent conductivity caee, the electrical
length la auch leae than the theraal length at high
teaperaturee.
To resolve greater detail In the field
distribution, a saaller grid spacing Is needed.
Unfortunately, to aalntaln stability, the tlae scale aust be
reduced In proportion to the square of the length, greatly
Increasing processing tlae.
A variable grid aethod aay be
valuable In this case, and is outlined in [51].
60
fe, Steady 3t>ti Solution
In this section we summarize the results of technical
report [23] and a publication [24] which
show that a steady
state solution does exist for the sesl-lnflnlte space, and
furthermore, the relationship between incident power and
steady state temperature Is monotonlc, and unconditionally
stable.
The solution of the mathematical problem posed by these
two coupled differential equations is Impossible to find
analytically, since the temperature and heat flux are
related non-llnearly.
However, we can obtain an approximate
solution by exploiting the smallness of thres dlmenalonless
paramsters.
=. * KT • c« ‘
* zzfc
(37»-c)
Ths constant c, Is the square of the ratio of the skin depth
to wavelength In the slab at t-0.
Ths constants c, and cz
are the ratios of thermal lengths to wavelengths in the slab
at t-0, and are difficult to physically lntsrpret in the
seml-lnflnlte solid.
Thess paramsters are derived from a
non-dimensional analysis of the heat and wave equations In
Appendix C.
The non-dimensional forms of the these
equations are:
61
B„ ♦ (1 - Jc,f(u)))E ■ 0 , x > 0
(38a)
-JT *. ♦ J« - 2j , x - o
(38b)
u„ - - Xf(u) I E(x) I
(38c)
, X > 0
U, - C, U ♦ CjUu-fl)4 - 1), X - 0
where X -
(38d)
, u - ^ ^ Tfl
In [23] a formal asymptotic analysis was performed In
the limit as c, -* 0
and bounded.
with c,/c, and c,/c, fixed
In this case the conductivity was assumed
exponential, I.e.
f(u) - exp(u).
An approximate solution
was obtained In the fora of a power series In ct,
T ~ T0 + c,T, ♦ (c,)*!, + ...
(39a)
E 'v E0 ♦ c,E, ♦ (ci)tE2 ♦ ...
(39b)
where explicit formulae wereobtained forT0,T,,
E0 and E, .
In particular, the surface teaperature T0(0)was found
function of the incident microwave power, P0.
as a
However, for
sufficiently high power, the series became invalid because
the correction terms c,T, and c,E, became larger than T0 and
Eq , respectively.
This nonunlforalty was removed by seeking
a different asymptotic expansion of T and E which Involved a
generalized power series, using the method of multiple
scales [52].
(A similar effect Is demonstrated in Appendix
62
D.)
By using ths method of matched asymptotic expansions
[53] a composite result was obtained which yielded a uniform
approximation for T and E for any power level, P0.
The
functional relationship between the surface teaperature
T, and the Incident power P0 Is stated below.
P
- M T . - Tq) ♦ »€k (t / 1 -
where 1 - |p|* ■
V)
| p|*
____________
( 1 ♦ J2(r(x+l) ♦(rx)<j2(x-l)
+ c.
Por low Incident powers, where c, << l, this reduces to
h(T - T0) ♦ st (T * - T04)
P„ - - L-J----- ^ — .-5^* 1----- ^
4^7
<i ♦ JC)2
(40b)
Por high powers, where c, >> 1, we have
P0 - ^ c ~ 3 s€kT.\
(40c)
Plgure 18 shows surface temperature vs. Incident power
where the material parameters are the same as ths constant
conductivity case, except the electrical conductivity Is
63
3000
j
(C)
J
1
-l
H
Temper at ur e
-4
2000
i
1
i
i
/
J /
J
Surface
1ooc -I
1/
-II
HI
f I I II I I I I I f I I I I II I II |'l I I I I I I T | I I I I I I I I I I I M I'IT T I II
0
200
ncident
Figure 16.
*00
600
800
1000
Power ( W / c m 2 )
Steady state surface teaperature vs. Incident
power (W/ca*) for planar half space with
teaperature dependent conductivity, o ■
.002exp(.0026T).
64
.002exp(.0026T) S/m.
Notice that lt raqulras ralatlvaly
•■all power to achlava modest temperatures, but that a
•light Increase In teaperature at the high end requires a
large Increase In power.
We note that the solution Is
monotonic, a one-to-one correspondence exists between
Incident power and steady state surface temperature.
The
stability of this solution Is proved In [23] Indicating that
this system is unconditionally stable, every temperature can
be achieved by simply adjusting the Incident power level.
Theoretical Heating of a Planar Slab
The next geometry we consider Is the planar slab, which
Is shown In 71g. 19.
sample from the left.
A uniform plane wave Impinges on the
A portion of the signal is reflected
from the sample and a portion Is transmitted through the
•ample.
The remainder of the signal Is absorbed by the
sample as heat.
The analysis, however, Is based on the
fields In the sample and Is not related to the Incident
signal.
In the discussion to follow we consider both
transient and steady state solutions to the heat equation,
and derive a model to predict the onset of temperature
Instability.
Por the steady state analysis, we assume that
the sample Is electrically thin, that Is, the fields are
uniform.
We will show In section D of this chapter that the
65
UNXrOftH
PLANE
MAVE
Flgur# 19.
Qaoastry of planar slab.
CEMNXC
SAMPLE
66
solution for s thin cyllndsr only dlffsrs from ths solution
for s slab by a constant.
Therefore, tha slab analysis,
which Is slaplsr aathaaatlcally, Is prafarrad.
Tha taaparatura distribution Is found by solving tha
hast aquation subjact to approprlata boundary conditions.
In tha slab gaoaetry, tha haat aquation Is
- Q , -d/2 < x < d/2
(41a)
The boundary conditions are:
-K§| + h(T - T0) ♦ e€K(T4 - T0) - 0 , x - -d/2
+ h(T - T0) *
“
T o>
" 0 '
x
" d/2
(41b)
<41c>
Haat Is lost to tha surroundings through convection and
radiation.
We assuae that tha aablant taaparatura Is 300K
and tha theraal conductivity Is constant.
Tha ganaratad
haat Is
Q “ *0P 1B€o ( x ) ( l
-
IP I* -
|T|*)Ig ^
*
“ inc
(42)
67
Lx
Corntant Conductivity
For tha constant conductivity casa, tha aatarlal la
homogeneous.
Tha raflactlon and transmission coafflclents
can ba axprasaad axactly.
If the sample la Illuminated by
TE waves, and Is terminated by a matched load (l.*e. free
space), the reflection and transmission coefficients are:
[54]
p
m
T
-
(1 - 6r*)slnh Vd
f
(€r* + l)slnh yd ♦ 2
cosh yd
----------- --------------- =
(€r ♦ l)slnh yd ♦
------- :
(43a)
(43b)
------------------------
cosh yd
If the sample Is terminated by an Ideal short circuit, the
net transmission Is zero and the reflection coefficient Is
given by (7).
TraMient Solution
We use the sane technique described In the semllnflnlte space presentation to numerically determine the
transient temperature profiles In the planar slab.
modification Is that there are two boundaries.
The only
The right
hand boundary Is given by (22b) and the left hand boundary
Is
68
hrT J - (hT
f
0
*"*
♦ sc^T ' ) - -^fl
*
*1
n ♦I
- T ‘ 1
n-l J
(44)
For computational purposes, ths grid extends fros 0 to n,
where ndx “ d, the thickness of the slab.
Appendix 7
lists e progres for an explicit numerical solution for
transient heating of a planar slab.
sample la In free space.
We assume that the
We now use the transient analysis
to study the steady state temperature profiles In a slab
with physical properties outlined in Section A.4 of this
chaptsr.
Figure 20 shows ths stsady state teaperature
distribution for conductivities of .002, .02 and .2 S/m with
constant power Input of 900 W/ca2.
While more heat Is
gensratsd for ths lossler samples, ths dlffsrence between
surface and Interior temperatures Is small.
thermal conductivity.
This Is dus to
As the thickness of ths slab Is
reduced, for fixed power, thermal gradients will decrease.
71gure 21 shows the steady state temperature
distribution for conductivities of 2, 20 and 200 S/a and
power of 500 W/ca2.
symmetry Is reduced.
Am the conductivity Increases, ths
7or Infinite conductivity, we have
surface heating, with a linear decrease In temperature
towards ths other boundary.
Note that as ths conductivity
69
2500
i
.2 S /r
temperature
(C)
i
200C "
1500
.02
S /m
1000
.0 0 2
500
^'rn—r~r- n
0.00
S /m
-I--T—
r~r~i i ' i > i ' > r
2.00
position
Fiaura 20.
t t v rT T-n ~ T- n ~i ■;
*00
(mm)
6.00
traiiMrif
t
Stwady atata taaparatura P*0*1***
c\ 2
planar •lab. with a).002 S/a. b).02 S/a and c).2
S/a.
70
(C)
3500-
300C
temperature
20
S /m
2500
2000
200 S /
1500 i i ? » i
0.00
t—i—n
—|—i—i—r i i—i i i i—r i i i i "i—r—i—i—r - |
2.00
position
Figure 21.
4.00
6.00
(mm)
Steady atata taaparatura profIlea for a 6aa
planar alab, with a)2 S/a, b)20 S/a and c)200
S/a.
71
rises, absorbed power diminishes because of Increased
reflection.
Me conclude fros these figures that the heating of a
planar slab with constant conductivity Is quite uniform.
The maximum teaperature is In the Interior, and the
difference between the central maximum and the surface
teaperature Is determined by the theraal constants for
conduction, convection and radiation.
In the heating of
constant conductivity materials, measurement of surface
temperature Is sufficient to determine the average
temperature of the material.
Stmmdv Stmts Solution
Por thin samples, we can assume that the field
distribution within the sample Is uniform.
The steady state
heat equation Is
£
■ -
I
If the electric field Is constant, the solution must be
symmetric about the center of the sample, T(-d/2) ■ T(d/2)
and the boundary condition at zero is sufficient,
(41b).
Solving (49), the steady state temperature as a
function of position Is
72
(46)
where B la a solution to ths quartlc
(47)
The temperature at the surface Is T(d/2) ■ T(-d/2) - B; the
temperature at the center Is
(48)
In the absence of radiation, the ratio of the central
teaperature to the surface teaperature Is
(49)
which Is small for thin samples.
The assumption of symmetry Is Invalid for samples with
high conductivity, since the field distribution Is not
uniform.
A 16th order equation must be solved, because each
boundary must be treated Independently.
Lt
*— perature Dependent Conductivity
In most ceramics, the electrical conductivity Increases
73
rapidly with taaparatura.
We apply thla crltarla to tha
slab geoaetry consldarlng both tha transiant and staady
atata solution.
L
Tr«nyi«n<; Solution
Tha nuaarlcal analysis for tha taaparatura dependant
conductivity case Is Identical to that for tha constant
conductivity planar slab, except that tha field distribution
aust be recalculated as taaparatura rises.
Wa use tha
prograa listed In Appendix F to obtain transient taaparatura
profiles in a planar slab.
Figure 22 shows surface taaparatura as a function of
tlae for an elactrlcal conductivity of .002 exp(.0026T) S/a,
and various power levels.
Whan tha Incident power Is low,
tha transiant solution converges to a steady state, (a)
and (b).
Whan tha Incident power Is high, (c), runaway
occurs.
This Is conslstsnt with experlasntal observations.
Figure 23 is a plot of surface taaparatura vs. tlae for a
sintered alualna saapla of .5 ca dlaaeter.
Tha aeasureaents
ware taken with a pyroaeter using ths rectangular
applicator.
Tha power was turned on to tha aaxlaua level
(approxlaataly 700 W) and tha saapla was allowed to haat up.
In Fig. 23, wa sea slow Initial heating, followed by a
region of rapid rise.
Tha power aust be dlalnlshed to
74
2000
(C)
200 W
surface
temperature
1500
150 W
1000100 W
500
1000
time
Figurs 22.
2000
3000
(sec)
Surfscs traptratur* vs. tlas for trsnsisnt
hsstlng of plsnsr slab with tsapsrstur*
dspsndsnt conductivity, a ■ .002sxp(.0026T), s)
100 W/ca*, b) 150 W/ca* c)200 W/ca*.
9*1
M
CJ
rt SC
• •
■ 9
92
2 <o
rt
(S O
9
< M.
• 9
*1 *%
rt
•
r t **
6
s
M*
s
c
*1
Ml
»
n
SORTACE TEMPERATURE (C)
prevent destruction of the sample.
steady state heating Is achieved.
When power Is reduced,
The periodic variation In
the data la due to a slight misalignment of the rotating
sample relative to the pyrometer.
Steady state can be
maintained up to about 1500-1800°C, above which, heating
appears to be uncontrollable.
In the steady state analysis
to follow, we consider the question of Instability In
greater detail.
77
b. >ttidv « i t i Solution and the Onset of Inata bllltv
In this section, we demonstrate that Instability can
sxlst In tha heating of a planar slab.[25] Ha first
raprasant tha haat aquation In non-dlmenslonal fora, then wa
find a solution to tha steady state haat aquation using a
numerical "shooting" method.
An approximate analytical
solution using regular perturbation theory
Is seen to match
tha numerical result whan Important constants
aresmall.
Tha result Is that whan conductivity Increases rapidly
enough with temperature, there Is a critical power level
above which steady state cannot be attained.
In a later
section, He study the control of the stable/unstable regions
by varying the temperature dependence of conductivity and
thermal constants.
He assume that the sample Is thin enough
so that the electric field Is uniform.
The heat equation can be expressed In non-dlmenslonal
fora as shown In Appendix C.
The length Is scaled to half
the thlclcneas of the slab.
u„ - ut ■ -\f(u) , -1 < x < 1
(50a)
-u.
(50b)
c,u ♦ c, ((u+1)4 - 1) ■ 0 , x ■ -1
uB ♦ c,u ♦ ct((u+l)4 - 1) * 0
where
, x ■ 1
(50c)
78
and x - 2X/d.
Tha paraaeter c, la tha Blot nuaber, [49] which and
coaparaa tha affacta of convactlon and conduction. Whan c,
<< 1, tha raalstanca to haat conduction within tha aaapla la
auch laaa than tha raalatance to convection across the
boundary, and tha taaparatura within tha saapla Is nearly
unlfora.
Tha constant ct can be viewed as tha radiative
equivalent of tha Blot nuabar, and tha saaa rule applies
whan lt Is saall.
For thin saaples and low power levels
(and hence low teaperatures), both c, and c, are saall;
typical values are .002 and .00018, respectively.
1>
Shooting Hethod Solution
Tha solution to tha steady state aquationIs found
ut ■ 0.
whan
Because of Its nonlinear nature, a nuaerlcal
solution aust be found.
Tha goal Is to express tha non-
dlaenelonal taaparatura,
u, as a function oftha Incident
power, X., with ct and c. as paraaatars.
Me use a nuaerlcal "shooting" aethod to deteralne this
solution. [55] Tha approach Is to convert tha steady state
boundary value problea Into an Initial value problea and
solve using tha Runge-Kutta aethod.
79
* ” +\f(«)
-0
4»(-1) ■ a
(51a)
ifc'(-l) ■ b • c,a ♦c2((a+l)4 - 1)
g(X) - U»'(1) ♦ c.wd) ♦ c2((*(1)+1)4
whan g(X)
(51b,c)
- 1)
(51d)
■0, than tit ■ u
Ha can aolva for X using Newton's aathod [47]
V. - V whara
<»2>
la found from solving
K" +
U»J-1) - 0
- -*(*)
(53a)
vD/(-l) - 0
gf - U»x*(1) + c,^(l)
If X << 1 then
a -
(53b,c)
♦ 4ct[U»(l) ♦ 11 30/k <1)
(53d)
, which constitutes a good first
guess for tha numerical solution.
Tha aquations are solved
numerically using tha Runge-Kutta method. [47] Tha two
second order differential equations are separated to form a
system of four first order differential equations.Appendix
G contains a computer program that Implements this solution.
Ut" ♦ Xf(tlt) - 0
(54a)
80
(54b,C)
(54d,e)
(55a)
^
-
S
- *(#)
0
(55b,c)
(55d,e )
* 0
Figure 24 shows the solution fdr f(v) ■ exp(.78v), where
normalized power vs. normalized temperature are the axes.
It Is evident that for low power levels, there Is a
corresponding steady state temperature.
There Is a critical
power level, above which, steady state cannot be maintained.
Power levels above this critical value will produce sample
melting.
Points on the upper branch are not physically
realizable.
11, Approximate Analytical Solution
We note that In many applications the constants c. and
cz are small, e.g. ct - .002, cz - .00018.
We shall
exploit the smallness of these values to derive an
approximate analytical solution using regular perturbation
theory [52].
The approximate solution compares favorably
with results In the prior section,
when the thermal
81
nor mal i zed
sst at e
temp
10.00 -3
8.00
6.00
4.00
2.00
0.00
■
0.000
0.001
0.002
0.003
normalized
Figure 24.
0 .0 0 4
0.005
0.006
power
Normalized steady stats surface teaperature vs.
normalized applied power for planar slab, with
f(v) ■ exp(.78v), c, ■ .002, ct • .00018,
using a numerical "shooting msthod."
82
a parameter, we redefine the other constants.
a - ct/cl , 0 ■ X/c,
(56)
where a and 0 are at most order one quantities.
We
find that the steady state temeperature distribution,
v (x ;cj) , satisfies the nonlinear boundary value problem.
v„. " " c,0f(v)
(57a)
-V„(-1) + ctv(-l) + «c(((v(-l)-H)4 - 1) ■ 0
(57b)
v, (1) + c,v (1) + ac, ((v (1) +1)4 - 1) - 0
(57c)
We shall now determine an asymptotic approximation of v
in the limit as c, -* 0 with a and 0 held fixed.
First we
assume that v has the representation
(58)
v - 2 vn(x )Ci"
Inserting this into (50), expanding the nonlinear terms
using Taylor series, and equating to zero the coefficients
of the powers of c, yields an infinite set of equations.
These sequentially determine the vn(x).
We shall just list
the first few which are sufficient to deduce the leading
order behavior of v.
They are:
83
a) leading ordar approxlaation, v ■ v0
v0‘ ■ 0 , -l < x < l
(59a)
- v 0 ■ 0 , x - -1
(59b)
v0 - 0 , x - 1
(59c)
b) first ordar approxlaation, v ■ v0 ♦ c,v,
v,* - -flf(v0) , -1 < x < 1
(60a)
♦ v0 ♦ <*((v0+l)4 -
(60b)
-V ,-
v,
1) - 0 ,x - -1
♦ v0 ♦ or( (v0+l )4 - 1) - 0 ,
Naxt we solve (59) for v0
x- 1
(60c)
andfind that It Is a
constant, which Is unknown for the moment.
Then we solve
(60a) and find that
v, » A, ♦ B,x - J x*f(v0)
(61)
Inserting this result Into the boundary conditions
(60b) and (60c), and eliminating the constant Bt we obtain
v0 ♦ <x((v0+l)4 - 1) - |f(v0)
(62)
This gives v0 lapllcltlly as a function of X, the scaled
power parameter, and hence the first term of the expansion
(58) Is determined.
84
In suaaary, we have found that
v - v0 4- 0(C,)
C j V 4- C j ( ( V4-1 ) * -
x "
(63a)
1)
(63b)
r w i ---------
Thus, to leading order, the teaperature within tfre thin slab
la unifora and Its value la given lapllcltly as a function
of power by (63b).
This can be re-expressed In teras of physical
paraaeters by reversing the normllatlon procedure.
For an
8ma alualna slab, this Is accoapllshed by a) aultlplylng the
normalized teaperature, v, by 300 to obtain temperature In
degrees Celsius; and b)multlplylng the normalized power, X,
by 50,000 to obtain power In Watts per square centlmenter.
The revised equation Is:
p ■ 5 3 5 ^ ? r f h <1 - T«» +
- V ) ]
(64)
Figure 25 shows the steady state temperature
distribution within the slab and Is derived from (61).
The
applied power level, \, Is .005, f(v) - exp(.78v), ct *
.002 and ct ■ .00018.
We note that Interior teaperature
variation Is saall, and that aaxlaua teaperature occurs In
the Interior.
Figure 26 displays the relationship between power and
86
nor ml ai z ed
temperature
3.60 -i
3 .5 5
-
3 .5 0
3 .4 5
-
3 .4 0
-
1.00
-
0 .5 0
0.00
normalized
Pigure 25.
0 .5 0
1.0 0
distance
Normalized steady state teaperature profile In a
planar slab, X ■ .005, f(v) ■ exp(.70v), ct *
.002, and c, ■ .00010.
86
nor mal i zed
temperature
10.00
unstable
8.00
m elting
point
6.00 H
4.00 d
2.00 H
stable
o.oo
o.ooo
0.002
normalized
Figure 26.
0.004
power
0.006
cri tical
Normalized steady etate surface teaperature vs.
normalized applied power for planar slab, with
f(v) ■ exp(.?8v), c, - .002, c, - .00018,
using an approximate analytical solution.
87
teaperature for a slab with tha saae alactrlcal propartlas
aa Fig. 25.
We nota that v0 aa a function of power la
aultlvalued.
Wa ahall show In Sactlon 111 that tha lower
branch la atabla whereas tha uppar branch la unatabla.
In
tha praaant aodal wa obaarva that ralalng tha powar abova
Xa raaulta In uncontrollad haatlng.
Plgura 27 la ldantlcal to Flgura 26, axcapt tha unnoraallzed axpraaalon, (64) was usad to coaputa tha curva.
It shows that for this aodal, tha aaxlaua stabla teaperature
for alualna Is approxlaataly 1250°C.
This valua Is
lower than experlaental observations, Indicating that batter
conductivity data as a function of teaperature la needed.
Plgura 28 coapares tha results of tha "shooting" aathod
solution (a) with tha approxlaate analytical aolutlon (b).
It Is seen that for low powar levels and low taaparaturas
tha results are nearly ldantlcal.
A higher ordar
approxlaatlon Is needed to aatch tha solution for high
taaparaturas.
We digress hare for a aoaent to sake an laportant
observation. Tha reeults given In (63b) allow us to deduce
f(v) froa powar and teaperature aeasureaents.
valuable In characterizing slab aaterlals.
This Is
Proa pyroaatar
aeasureaents of tha surface teaperature, for a given
Incident powar, wa can obtain a functional relationship
between v0 and X, Independent of (63b), v0 ■ 0(X) .
88
3000
temperature
(c)
unstable
2000 J
1000
melting
point
-
stable
100
power ( W / c m 2)
Figure 27.
300
200
c r it ic a l
Steady etate eurface teaperature v s . applied
power; un-noraallzed from Fig. 25.
89
nor mal i zed
sstate
temp
10.00
8.00
6.0C -3
4.0C
2.00
0.000
0.001
0.002
0.003
normalized
Figure 28.
0.004
0.005
0.006
oower
Normalized ataady atata aurfaca taaparatura va.
normalized applied powar for planar elab, with
f(v) ■ exp(.78v), c, ■ .002, cg - .00018,
comparing a)aolld, approximate analytical
aolutlon and b)daah, numerical "ehooting"
method.
90
Inserting this into (12b) we find
c.Q(X) ♦ c, ((Q{\)+l)* - l
— ----F(X) - f (v0(X )) - -1 -- ---- Y
which gives conductivity as a function of X.
(65)
Pros these
parametric descriptions, 7(X), v0(X), we can deduce f[v0(X)]
as a function of v0
ili-t
Stability Analysis
In the previous section we have computed an
approximation to the steady state teaperature distribution
and now we will study its stability to small perturbations.
If thess perturbations decay to zero as t -» oo, the
solution is stabls.
If thess perturbations grow without
bound as t -* ®, then the solution is unstable and will
never be realized in the laboratory.
Accordingly, we add a small time dspsndent perturbation
to the steady state
u(x,t) - v(x;c,) ♦ 6e"*“9(x)
(66)
where 6 is a measure of the size of the perturbation and
& << ctz.
Inserting this into (50), using Taylor ssrles and
91
(32), we find after omitting nonlinear tarns In e.
♦«. ♦ [u ♦ X f (v)]d - 0 , -1 < x < 1
(67a)
“♦« * (ci + 4c1(v+l)3)e - 0
(67b)
♦ (CI ♦ 4Ct(V+l )*)• ■ 0
, x - -1
(67c)
, X • 1
This Is a classical Stura-Llouvllle eigenvalue problem.[56]
We shall now shorn that the smallest eigenvalue, ut, Is
positive (negative) when
> 0
< oj.
That Is,
when the response curve la monotonlc, the steady state Is
stable because the perturbation will decay to zero.
Similarly, If the response curve has a branch where
^
< 0, then this steady state Is unsta'ble.
To prove this assertion we fora the "variational"
equations corresponding to (50) by taking the derivatives of
these equations with respect to X.
Denoting
by
,
we have
♦ Xf'(v)K» - -f(v) , -1 < x < 1
(68a)
-Hi. ♦ (c, + 4cI(v+l)3)U) ■ 0 , x - -1
(68b)
♦ (c, ♦ 4ct(v+1 )3)Ui ■ 0 . x - 1
(68c)
Note that (67) and (68) have the same boundary conditions.
Next, we multiply (67a) by «, and (68a) by e, take
92
their difference, end Integrate over the length of the eleb.
*[♦„ ♦ <u ♦ Xf'(v) )e] - e[u»„ ♦ xf (v)«i» ♦ f(v)] -
/
* /
-1
UU»6dx ■
-1
f
o
ef(v)dx
(69a)
(69b)
-1
*
where
j
(*♦..
- 6*„)dx ■
|' -
0
(69c)
and we have
U,
f
6,il>dx ■
* -i
f
f (v)e,dx
(70)
■* -■
Since f(v) la always positive, and the lowest eigenfunction
of a regular Stura-Llouvllle problem, e,, Is always
positive, the right hand side of (69) le also positive.
From this we deduce that ut Is positive (negative) when
& > ° ( &<<>) •
Proa this result we see, for example, that the lower
branch ehown In Fig. 26 is stable whereae the upper ie
unstable.
When ut ■ 0, we have
■ 0 and this defines the
"critical power level," \e.
A etudy of the effect of variation In thermal and
electrical parametera is performed In Section Q.
93
3. 8 f dv 8tit» Solution for an Insulated Planar Slab
In son* cases, sicrowsvs heating Is perforaed on
Insulated samples.
In this section, the approxlaate
analytic aethod Is applied to the Insulated planar slab to
detsralna Its hsatlng behavior.
The result Is similar to
the uninsulated case except that the critical power Is
lower.
The sample configuration Is shown In Figure 29.
A thin
sample, Region II, Is bounded on both sides by an thin layer
of Insulation, Region I.
Region II Is subject to microwave
heating, while Region I Is a window material, Inert to
microwave heating.
Region II extends. In normalized units
from -1 to 1.
Region I has thickness a, on the same order
as Region II.
For simplicity, symmstrlcal hsatlng will be
assumed.
We write the heat equation In each region In nondlmenslonal form as developed In Appendix C:
Region I
There Is no source of heat In Region I, so the heat
equation Is:
I: INSULATION
II: CERAMIC SAMPLE
Plgura 29.
OtOHtry
of lnsulatad planar alab.
95
V,„
- 0 , -a-1 <
v , - ^v,
x
<
-1 ,
1 < X <
♦ ct((v,-fl)4
-1)
,x
"vi.« " civi ♦ cf((v,-fl)4 - 1) ,
where c, - ^
1-fa
(7la)
- -a-1
(71b)
x
(71c)
-
a-f1
, c, -
Tha conatanta c, and cs ara acalad with raapact to the
thlckneee of region II.
conaldar
Because of syaaetry, wa only need
-a-1 < x < -1.
Tha aolutlon to (71a) la alaply
v, - Ax ♦ B
(72)
Ualng (72), wa aaa that at tha boundaries
v,(-l) - B-A ; v, ,(-1) - A
(73a,b)
v, (-a-1) - B-aA-A ; v, „(-a-l) - A
(73c,d)
Substituting (73c,d) Into (71b) wa have
A - c,(B - A - aA) ♦ c, ((B - A - a A + l ) 4 - l )
(74)
96
Region II
Heat Is generated In Region II and transferred across
the boundary to Region I.
The boundary conditions require
the continuity of teaperature and energy flow.
The
appropriate equations are:
vi:.«« " “ ^ fvri)
(75a)
V ,,(-l)
(75b)
-
V ,(-l)
(75c)
To solve the heat equation In Region II, we perform the
regular perturbation analysis discussed earlier.
He use
only the first two terms of the expansion:
(76)
We can rewrite the heat equation by substituting (76) Into
(74a),
v, - -8f(v0)
(77)
and the solution Is:
v» " ' ff<v0)(l - x») ♦ B,
(78a)
vt,. ■ " 8*(v0)x
(78b)
97
where K, la a constant of Integration.
Froa (75b) and (74) we gat
A - c,(v0 - aA) ♦ c4((v0 - aA ♦ 1)4 - 1)
„
(79)
A „ °.v. - c,((v,+.)’ - 1 )
1 ♦ c,a ♦ 4ac,(v0+l)
o
Froa (75c) and (73a), wo obtain an expression for A:
K..\f(v)
A - -STE-C
—
k7
(81)
Equating (80) and (81) yields
Ate,
k 7JT(v T
_
ic, c ,v ♦ c2( (v+1)4 - 1)
K,,f(v) j + c a + 4ac,(v-fl)3
(82)
When a - 0, this reduces to (63b), the solution for the
uninsulated planar slab.
When the thickness of the
Insulating layer Is on the order of the saaple thickness,
the aaxlaua stable teaperature does not chango, however,
less power Is needed to achieve the sane teaperature.
Figure 30 shows a noraallzed steady state teaperature
profile In the Insulated slab.
Ths distribution Is linear
In the insulated region, where no source Is present and
quadratic In the Interior, where electroaagnetlc heating
98
3.005
<D 2.995
CL
~D
• - 2.985
2.975
- 0 .9
- 0 .4
normalized
Plgura 30.
0.1
0.6
1.1
thickness
Staady atata taaparatura profila for on
lnaulatad slab.
99
occurs.
Vlgurs 31 compares ths uninsulated and Insulated
samples, assuming that the thermal conductivity of region I
Is half that of region II.
Ths dashed curve represents ths
Insulated case, showing that Instability still occurs above
a critical power level.
While the critical temperature
remains largely unchanged, the power necessary to achieve
this operating point Is reduced.
As the thermal
conductivity ratio between the two regions Is diminished,
the critical power level Is also decreased.
This
demonstrates ths value of an insulated sample In reducing
ths energy consumption of the system.
Lx
Stsady Stats Solution for a Thick Slab
The approximate analytical solution Is based on the
assumption that the electric field Is uniform In ths sample.
This Is valid when sample thickness Is much less than a
wavelength and much less than the skin depth.
In this
section we develop an Improved solution when sample
thickness Is on ths order of a wavelength, but still much
less than ths skin depth.
Ths field Inside a homogeneous sample, backed by a
short-circuit one-quarter wavelength away is derived from
(35) :
nor mal i zed
temperature
100
8.00
6.0C
2.00
-4
0.00
0.000
0.001
0.002
0.003
normalized
Figure 31.
0.00*
0.0 0 5
0.006
oower
Normalized steady state surface teaperature ve.
normalized applied potter for Insulated planar
elab, ttlth f(v) - exp(.78v)f c, ■ .002, ct .00018 and a ■ 0.1; ajaolld, uninsulated elab,
b)dash, Insulated elab.
101
.
(£7 - 1 ).-'“-
,x’ ‘
[ST'. »>••*
* (J<7 ♦ iK ''*"’
♦ (J*7 - 0.-‘'4
and tha reflection coefficient le given by (7).
'
’
The heat
equation becoaea
v.. - -\f(v)(1 - |p |*) | E(x) | * - -Xf (v)g(x)
(84)
Uelng the perturbation analysla, we write
v l>>B - -6f(v0)g(x)
(85)
Integrating (85) once, we obtain
v, , - -aftvj
f
g(T)dT
(8 6 )
• _■
Subetltutlng (86) Into the boundary conditions (60b,c) we
find that
e.v ♦ c ((v+1 )* - l)
*-------------,
f(v) J
g(x)dx
» .
-Ii
\ « —
(B7)
102
where the expression in the denominator represents an
average value for the fields.
It should be emphasized that the analysis of the planar
slab Is valid only when the field distribution In the sample
le Independent of temperature.
When skin effects, produced
by teaperature dependent conductivity, reduce field
penetration, the heat and wave equations cannot be de­
coupled.
simple.
The question of stability (formally), Is no longer
The prediction Is that the planar slab will re-
stablllze at high temperatures.
Whsn skin depth Is
sufficiently small, the slab will resemble the seal-lnflnlte
space, which Is always stable.
103
IL
Theoretical Heating of an Infinite Cylinder
The next geoaetry of Interest Is the thin, Infinite
cylinder pictured In Tig. 32.
A unifora redial wave
laplnges on the saaple, and a fraction la abeorbed.
We will
consider the constant conductivity case, a aethod for
transient analysis, and steady state heating for the
teaperature dependent conductivity reglae.
We consider the
special cases of a tubular saaple and a thick cylinder.
In
general, the results for the cylinder are slallar to the
results for the slab.
The heat equation In cylindrical coordinates Is:
K (Trr ♦
t t,>
- C,PT* ■ - £ o0 | B(r) | * , 0 < r < a
(88a)
The boundary conditions are:
ICTr ♦ h(T - T0) ♦ s£k(T4 - T04) - 0 , r - a
(88b)
and we require the teaperature to be bounded at the origin;
T(0) Is finite.
UNIFDRA
■>
PLANE
WAVE
Plgura 32.
CERAAIC
SAAPLE
A
/
Qaoaatry of lnflnlta cyllndar.
105
1. Twntltnt Solution
The explicit sethod used to obteln teeperature profiles
for the half-space and the planar slab aust be modified In
the case of the Infinite cylinder.
If we fora a difference
equation based on (88), we have a singularity at the origin.
At the origin, we average adjacent points, yielding [57]:
A prograa to aodel the transient solution for the cylinder
has not been iapleaented.
2. Steady State Solution for Constant Conductivity
To determine the steady state solution to the constant
conductivity case, we restate the heat equation:
(90a)
which has for a solution:
-5)
♦„
(90b)
106
where D la a root of tha quartlc,
D' + & D - [ $ ;
♦ £
to
♦ v ]
- o
(
91
)
Thla reault la alallar to (47) and (48), tha aolutlon for
tha planar alab.
3. Steady State Solution for Temperature Dependent
Conductivity
We conaldar tha atablllty of the cylinder when the
conductivity lncreaaea with temperature.
Tha regular
perturbation analyale la applied to the ateady atate heat
equation aa daacrlbed In Section II.C.
The non-dlaenalonal heat equation, from Appendix C, la:
r3rlr SrJ m 'Xf<v > . 0 < r < 1
(92a)
vr ♦ c,v ♦ ct((v+l)* - 1) * 0 , r ■ 1
(92b)
where c, - ^
, c. - — -g8 - , X -
Baaed on the aaauaptlon that ct and c, are eaall, we
fora the aolutlon ualng an aayaptotlc expanalon (76).
Equation (92a) becoaea
107
(93)
and the solution la
(94)
The constant A must be zero, since the temperature Is finite
at the origin. Applying the solution at r * 1 gives
v(l) - v0
(95a,b)
Substituting for the boundary conditions,
(92b), we have
This Is virtually Identical to (63b), the solution for the
planar slab.
The difference Is In a scaling constant.
Heating a cylinder to a desired temperature requires twice
the power per unit volume as a slab of equal thickness.
Inverse solution,
The
(63b) applies to the cylinder as well,
Indicating that cylindrical samples could be used to
characterize temperature dependent conductivity using steady
state temperature and power measurements.
loe
L,
S t M d Y S t t f Solution for a Tubs
He extend the discussion of the thin cylinder to
Include the case of e tubular geometry.
He flret assume
that there Is no convection in the tube and as a consequence
there Is no heat flux across ths Inner boundary.
Then we
consider the effect of forced convection In the center of
the tube.
He use the scaled heat equation (92a,b), with an
additional boundary condition at r ■ b < 1:
vr - 0 , r - b
(97)
Hhen c, and ct are small, the solution to the heat
equation, (94), still applies. In this case, however, A Is
not zero. In fact,
(98)
Inserting (94) and (98) Into (92b), we have
x -
2 (c,v ♦ Cj ((v+1) - 1)
f(v)(l - b‘)
(99)
Hhen b - 0, this reduces to (96). If b is nearly one, X
becomes large, and the analysis Is not valid.
If we allow air flow through the center of the tube,
109
(97) b«COM«
• vr - c',v , r * b
(100)
.
h.a
where c , *
The convection constant Mill bs greater in the interior
when there is forced convection.
(92b),
get
2(c.v ♦ fbc .v ♦ c, ((v+1)4 - 1)
2— --- :-------------------------------------------f(v)(l - b*)
- - 1 - i ----- 1---- i
-
\
ms
Using (94) and (100) in
This increases the heat lost to the surroundings, increasing
the power required to heat the sample to a desired
temperature.
h
Interior thermal gradients are also Increased.
Steady etate Solution for a Thick Cylinder
As in the planar slab, we have assumed that the
electric field within the cylinder Is uniform.
When the
cylinder is on the order of a wavelength in diameter, this
assumption is no longer valid.
For low conductivities, the
field distribution looks like Ja, the Bessel function of the
first kind, order zero [58].
The approximate analytical
solution to the steady state solution applies, using an
110
X » a (c1v »
* (V)
(v+1) - 1)
f0'g(r)rdr
C,(
(1 0 2 )
As conductivity rlsss with temperature, ws have the sase
difficulty as we encountered in the slab; naaely that the
sanple becomes lnhoaogeneoua, and the heat equation must be
coupled with the wave equation for an accurate solution.
When the fields within the sample do not vary with
temperature, we can consider three types of conductivities:
a) low conductivity, field distribution dominated by J0;
b) Intermediate conductivity, Bessel functions of complex
arguments are needed; and c) high conductivity, where J0
can be represented by ber and bel functions [58].
a) E -
(103a)
b) E - J0([fl - Jo]r)
(103b)
where v ■ a ♦ Jfl
(103c)
c)
7or large enough conductivities, the fields decay
exponentially as thsy do In planar geometry.
We can conclude from this analysis that the heating
behavior of the cylinder closely resembles the heating
behavior of the slab, and therefore analysis using a planar
Ill
•lab geometry Is sufficient to pradlct heating effects in a
cyllndar.
L
Theoretical Heating of a Sohara
The last class of geometries we consider is the
spherical sample.
He discuss the steady state solution to
the heat equation based on the perturbation study, we
consider the Inverse problem of characterization, and we
describe the fields In a sphere as a function of
conductivity.
Li
Steady State Solution
The steady state heat equation, assuming uniform
fields, and using the non-dimensional analysis in Appendix
C , Is :
(104)
The boundary conditions are the same as the Infinite
cylinder, (92b), where the scaling Is with respect to the
radius of the sphere.
Again, the temperature at the center
Is bounded; u(0) Is finite.
Applying the series expansion
(76) to ths solution, and assuming c, and cz are small, we
112
obtain:
flf l v I r 1
v,
----'-f1—
4- A/r ♦ B
where A must ba zaro.
(105)
Subatltutlng (105) Into (92b)
tha
following ralatlonahlp batwaan powar and taaparatura la
obtalnad:
3 (c,v ♦ Cj ((v+1)4 - 1)
>- -
(106)
Thla dlffara froa (96), tha aolutlon for tha cyllndar, by a
conatant, Implying that tha eaaa stability phanoaana la
oboarvad In tha cyllndar.
Ha can compare tha powar
taaparatura relatlonehlps for planar, cylindrical and
spherical geoaetrlea.
C.V
+
C. ( ( V4-1 )* -
SLAB: X - -»-------CYLINDER: X -
They are:
2( C . V
1
(107a)
4- C. ( ( V 4 - 1 ) 4 -
1)
:----- -
3 ( C . V 4- C. ( ( V4- 1 ) 4 -
(107b)
1)
SPHERE: X - — L-*---- --------------
(107c)
For the cyllndar and sphere, c, and cz are unchanged.
Note that tha same functional ralatlonahlp Is seen In
all three gaoaatrlas.
This Indicates that tha basic
stability relationship Illustrated by Plgure 28 Is still
true.
Tha constant by which (107a,b,c) differ is simply the
113
ratio of normalized surface area to volume of the sample.
Heat transfer to the surroundings Is Influenced by this
ratio, and hence more power per unit volume Is required to
heat the sphere to the same temperature as a cylinder of
equal thickness, and more power per unit volume Is required
to heat the cylinder to the same temperature as a slab.
In terms of un-normal1zed parameters, the surface area to
volume ratio will depend on the size of the bodies.
Important Information about the conductivity of the
sample can be obtained from the Inverse solution of these
equations, as given by (64).
Experimentally, this
requirement of nearly uniform field distribution and uniform
temperature limits the application of (64).
A planar slab,
Inserted in a rectangular waveguide will have a transverse
variation In the electric field proportional to sln^.
Visual observations Indicate that a cylindrical rod in a
rectangular waveguide can have a substantial axial
temperature gradient.
The sphere, due to Its compact
nature, Is likely to have a uniform temperature profile and
thus is the preferred geometry for characterization
experiments.
To determine the temperature dependence of conductivity
using a sphere, it Is necessary to know the power absorbed
by the sphere and the surface temperature.
The power
absorbed can be calculated by subtracting the reflected
114
power fro* the Incident power, •■•using no wall loss**.
Is Independent of applicator gaosatry.
It
Bacauaa tha
taaparatura distribution la naarly uniform, whan c, and cz
arc asall, [49] tha aurfaca taaparatura la a good aaasura of
tha avaraga taaparatura In tha aphara, and can ba obtalnad
by a pyroaatar.
Additionally, knowledge of tha heat
transfer coefficients la Important In determining tha
constants c, and ct.
This characterization method la
dlacuaaad In graatar detail In Chapter V.
2, Fields in a Sphere
As In tha slab and tha cyllndar, as the thickness
Increases, tha assumption of uniform flelda la no longer
valid.
Tha fields In a thick slab are given In (83), and
for a cylinder In (103).
The fields In a sphere are
described using spherical Bessel functions.
The electric
field distribution, for the fundamental TM mode ls:[S9]
(104a)
(104b)
Note that this mode yields fields which are a maximum at the
origin, which Is consistent with the field distribution In
ths slab and cylindrical gsomstrles.
For higher
115
conductivities, in ths hoaogeneous sphsrs, spherical bessel
functions with coaplex arguaents aust be used.
Hhen
conductivity Is very high, functions analogous to the ber
and bel functions (103c) are derived and strong attenuation
Is observed.
Hhen the fields In the saaple are described by
(104a,b), the heat equation aust be aodlfled to include
angular variation:
The solution to this equation Is not sought here.
Li
generalized Stability Toraula for Thin Electrical Bodies
A general relationship between Incident power and
steady state teaperature based on the geoaetry of the systea
has been discovered.
He notice that the power necessary to
achieve a desired teaperature varies with geoaetry.
The
aaxlaua stable teaperature does not vary with geoaetry.
He know that at steady state, for soae arbitrary solid,
the heat generated In the aaterlal aust equal the heat
leaving the aaterlal.
This can be expressed as [60]:
116
$o0t (T) 18 |*dv - <f> K(7T-n)do
(106)
(Applying the divergence theorea to the left hand side
yields the faalllar heat equation.)
rron the boundary conditions we describe the heat
leaving the aaterlal by heat transfer conditions.
(107a)
(107b)
We constrain this arbitrary solid In order to nondlaenslonallze the equations.
First, It Is assuaed that
the solid Is convex, that Is, all tangent lines or planes
touch the surface at a single point.
It Is sufficient for
the heat transfer conditions at the boundary to assuae that
the noraal vector at every point on the surface, S, does not
re-lntersect the surface, so that the saaple transfers heat
to a uni fora background.
Furtheraore, we assuae a polar
coordinate systea for two-dlaenslonal solids and a spherical
coordinate systea for three-dlaenslonal solids so that the
length nay be scaled to soae aean radius.
Non-dlaenslonal
power can now be expressed In teras of non-dlaenslonal
steady state teaperature.
117
\
X
f
(108)
f(v)|l|*dv
V
Por electrically eaall bodies, the fields are uniform
In the solid and we can make a leading order approximation,
assuming that c, and cz are small.
(109)
V
The normalized surface area to volume ratio provides a
geometric correction factor.
It Is 1, 2 and 3 for the slab,
cylinder and sphere respectively.
Consequently, the power
required to achieve a desired steady state temperature is
dependent on geometry.
We determine the maximum stable steady state
teaperature by evaluating the derivative of the power with
respect to temperature and setting It to zero.
Por the
leading order approximation, where the field distribution
does not change with temperature, this becomes
0 - [ct ♦ 4cf (v+1 )3]f (v) - [ctv ♦ ct((v+l)* - 1) jfv (v )
which Is Independent of geoaetry.
We conclude that the
(111)
118
maximum taaparatura la not a function of geoaetry,
eventhough tha critical powar la gaoaatry aansltlva.
A
stability analysis Is naadad to foraally prova this
assartIon.
0.
Parameter Study
In this section we will axaalna how tha critical power,
\e, and tha corresponding taaparatura, v, change as
functions of tha electrical conductivity and thermal
constants.
Tha critical powar Is found by taking tha derivative of
(63b) and setting It to zero.
This yields
(ct ♦ 4clv*)f(v) - (ctv ♦ c,v4)gj - 0.
(112)
Tha root of this aquation Is tha maximum stable taaparatura,
v.If tha electrical conductivity Is aonotonlcally
decrsaslng with taaparatura,
(112) indicates that heating is
always stable since there Is no positive solution.
However,
In aost caraalcs, [40] tha conductivity Increases with
temperaturs, which allows ths possibility of Instabilities.
We shall now axaalna some specific cases within ths present
theoretical framework.
119
l-i
Constant Conductivity
Sine* the electrical conductivity has been scaled with
respect to 9s$*0,
we take without
loss of generality,
f(v) - 1 ' SJ "
(113>
Substituting (113) Into (112), we obtain
v3 ♦ ^
- 0.
(114)
Since the teaperature Is always positive, we
conclude that the heating of a planar slab with constant
conductivity follows a aonotonic law. The solution Is
stable.
2. Linear Conductivity
The slaplest aodel Is a linearly Increasing
conductivity, and
f (v ) - l + kv .
- l
(115)
Substituting (115) Into (112), we have
3c,kv* ♦ 4c2v* ♦ c, ■ 0
(116)
120
Again, thara la no poaltlva aolutlon, tharafora tha haatlng
of a planar alab with llnaarly Increasing conductivity
la
unconditionally atabla.
In tha absence of radiation, c, - 0, the solution Is
still stable, since c, Is positive.
3_!
gmdrfltjc
Conductivity
It Is convenient to aodel soaa aaterlals by using a
quadratic conductivity law: [14]
f(v ) - 1 + 2k,v * k,v* ,
- k, ♦ 2kjV
(117)
Substituting (117) Into (112), we find that v satisfies
(c, ♦ 4Cj (v+i )3)(1 ♦ k,v + kjV2)
- (c,v ♦ Cj( (v+1)4 - 1) )(k, + 2k,v) - 0
(118)
Since a positive solutlon(s) can exist, Instability Is
possible.
Suppose the power level Is low enough to neglect
radiation effects, that Is, c2 - 0. Equation (118) yields
(119)
121
This yields ths aaxlaua stable teaperature, vB, with
corresponding critical power
X- ’ *, * l2 Q Z '
,12°»
Stable heating Is possible only when \ < X,, and cannot be
achieved above this power.
If, on the other hand, the radiative effects doalnate
the convective ones, then c, ~ 0, Instead, and (118) becoaes
In this H a l t
2k,v* ♦ (3kt ♦ 4k, )v4 ♦ 4 (2k, + ljv3
2(6 ♦ 3k, - 2k, )v* ♦ 12v ♦ 4 + 2k,
(121)
For high teaperatures, (v > 1), there Is no soltulon to
(121), and stability Is therefore assured.
These two cases suggest that radiative effects will
restablllze the systea at high enough teaperatures.
general,
roots.
In
(118) can be shown to have 0, 1 or 2 positive
He shall only consider the case when k, ■ 0, which
Is Illustrated in Fig. 33.
Here teaperature, v, Is plotted
as a function of \ for different values of ot (where we
recall that a Is c,/c,).
If there Is no Intersection,
/
or,, the solution Is aonotonlc and stable.
If there Is
one lnteresection, <*,, there Is an Inflection point, and
again the solution Is stable.
For saall enough a, there
122
normalized
tem perature
10.00
8.00
6.00
4 .0 0
2.00
0.00
0.000
0.001
0.002
0 .0 0 3
normalized
Plgure 3 3 .
0 .0 0 4
0 .0 0 5
power
Normalized steady state surface teaperature vs.
normalized applied power for planar slab, with
f ( v ) - 1 ♦ .3 v * , a)a - . 0 0 2 , b)a - . 0 0 3 ,
c)a ™ . 0 0 4 .
123
are two stable branch**, oi, saparatad by an unatabla ona.
In thla casa tha radlativ* affacta hava rastablllzad tha
systaa at high taaparatura*.
4. Ixponantlal Conductivity
An axaalnatlon of conductivity data In [32,43,61]
shows that an axponantlal raprasantatIon Is more appropriate
for ceramic materials than a quadratic law.
f(v) - exp(kv) , ££ - k exp(kv)
(123)
Substituting (123) Into (112) wa find that the critical
taaparatura la a root of
c,(kv - 1) ♦ c,[ (v+l)*(k(v+l) - 4)] ■ 0
(124)
Thera la a positive solution to this aquation,
Indicating that there Is a range of steady state
temperatures that cannot be realized.
Furthermore, when cz
Is 0, this range Includes all teaperatures greater than
1/k.
If ct la 0 Instead, stability Is maintained until vB Is
somewhat less than 4/k.
In the presence of both convection
and radiation, the maximum stable teaperature Is between
these two values. Figure 34 shows the power vs. teaperature
curves for several values of c, and ct.
The
124
nor mal i zed
temperature
10.00
8.00
6.00
4.00
2.00
0.00
{Til111111111 t i rr>n fr11 m 11»i; n n 11111 11 ■1111111 |m 111 11111
0.000
0.001
0.002
0.003
normalized
Flgura 34.
0.004
0.005
0.006
power
Normallzad ataady atata aurfaca taaparatura va.
normallzad appllad powar for planar alab, with
f(v) • axp(.?8v), a)c, • 0, ct ■ .00018,
b)c, - .002, c, ■ .00018, c)c, ■ .002,
c, • 0.
129
corresponding critical powar levels, \e, ars also shown.
By adjusting thermal parameters, the maximum stable
teaperature can be controlled up to a maximum value
determined by the c, ■ 0 curve.
Turther control of the maximum teaperature la possible
by adjusting the teaperature dependent conductivity through
doping.
In Fig. 35', the power vs. teaperature curves for
sevsral valuss of k ars shown.
The lower the value of k,
the higher the maximum teaperature.
This Is consistent with our experiments with
A1203-T1C. [1] Ths conductivity of alumina Is low at
room teaperature, but It Increases rapidly with teaperature.
Thermal Instability has been observed at high teaperatures.
On the other hand, TIC has a aodsrats loss at room
temperature which Increase slowly with teaperature. Thermal
runaway has not besn observed In TIC.
Mixtures of
AljOj-TIC with 10% TIC or lsss show thermal runaway.
Mixtures of greater than 30% TIC sinter stably.
9. Other Exponential Forms
While the exponential law has besn sufflclsnt for our
preliminary work, there are more complex relations which
bstter fit the experimental data. We examine two such
sxprssslons below.
126
nor mal i zed
tem perature
10.00 -I
8.00
6.00
0.68
.78
4.00
0.88
2.00
0.002
0.004
0.006
normalized
rigur* 35.
0.008
0.010
power
Normallzsd st«ady stats surfacs taaparatura vs.
normallzsd appllsd powar for planar slab, c, ■
.002, C| - .00016, a)k - 0.68, b)k - 0.78, c)k
- 0 .88 .
127
f(v) - 1 4 k0« x p ( ^ 1) ,
exp(^)
(125)
Substituting (125) into (112), yields
(c, ♦ *c2(v*l)’)[ 1 ♦ k o s x p ( ^ ) ]
- (c,v ♦ c2((v+l)4 -
•xp(‘^ r O ]
■ 0
(126)
Again the possibility for instability exists, but in
this case, for large values of v, stability is regained.
Figure 36 is the power teaperature curve where k,, is 600 and
k, is 13.3.
The constants wsre derived froa a curve fit to
the data in [32].
The curve shows a stable low teaperature
region, an lnteraedlate range of instability, and high
teaperature stability.
Another useful fora for expressing the teaperature
dependence of conductivity is:
f(v) - 1 ♦ ^ - e x p ( ^ )
,
^
•xp(^) [ v
- l]
(127)
128
normali zed
tem perature
10.00
8.00
6.00
4.00
2.00
0.00 -fr0.000
0.005
0.010
normalized
Flgura 36.
0.015
0.020
power
Normallzad ataady atata aurfaca taaparatura va.
normallzad appllad powar for planar alab, with
f (v) - 1 ♦ 600 axp(-13.3/v), c, - .002, ct ■
.00018.
129
Substituting (127) into (112), w* obtain
(c, ♦ 4c, (v-fl )*) [ i ♦ V« x p( - rr O ]
- (c,v ♦ c,((v+l)4 - 1 ) [ ^ « x p ( ^ )
“ l]]
- 0
(128)
As in ths prior csss, high tsmpsrsture stability
sxlsts, yet an intermediate region of instability can be
found.
Figure 37 shows the power teaperature curve modeled
after ft-alualna. [62] k, is 378.S and k, is 4.67.
a
«»—
This section shows the importance of teaperature
dependent electrical conductivity on the stability of the
systea.
The functional fora for the conductivity will vary
with the material.
Ideally, a curve fit to actual data
points could bs used to establish the stability of a
particular systea.
Suitable dopants might be added to alter
the conductivity teaperature curve to Improve system
stability.
Control of the maximum stable temperature is
also possible by adjusting heat transfer constants.
Ths results of this section are consistent with
[16,17], where a numerical study of Instabilities in the RF
heating of ceramics is performed.
A coll surrounds a thin
130
nor mal i zed
tem perature
6 .0 0 -i
4.00
.00
-
0.00
0.000
0.002
0.004
0.006
normalized
Vlgur# 37.
0.008
0.010
power
Norullzsd steady stats surface tnptratur* vs.
noraallzsd sppllsd power for planar slab, with
f(v) - 1 ♦ 378.8/v sxp(-4.67/v), ct ■ .002,
ct * .00018.
131
cylindrical saapla. Inducing RF currant In tha saapla,
generating haat.
An activation taaparatura aodal la uaad
for tha alactrlcal conductivity, o0axp(-Ta/T) , and tha
raault la a algaoldal curva ovar power and taaparatura with
atabla and unatabla branchaa.
Experimental atudlaa on
alllcon achlava atabla oparatlon on tha uppar branch, aa
wall aa low temperature atablllty.
"Hot Wall" Applicator
From our analyala of tha half>space, wa notad that
thara ara thraa dlatlnct raglona of haatlng.
Thaaa ragIona
ara a raault of tha taaparatura depandanca of tha alactrlcal
conductivity.
At low tamparaturaa, haatlng la alow, dua to
tha low conductivity of tha aatarlal.
At lntaraadlata
taaparaturaa, tha accalaratad haatlng rata producaa a rapid
taaparatura rlaa; at high taaparaturaa, aurfaca raflaction,
dacraaaad field penetration and lncraaaad radiation
contribute to a taaparatura aaturatlon or Halt.
In thla aactlon wa ara concerned with tha low
taaparatura region, where taaparatura rlsea slowly.
If tha
background taaparatura ware fixed at a taaparatura above
that of tha saapla, wa could rely on convective haatlng as
wall as alcrowava haatlng, to "wara up" tha saapla.
Tha use of "hot-wall" applicators Is not unknown to tha
132
microwave haatlng literature [7,10,63].
The general
approach la to place a eaaple In an Insulating container In
a microwave oven.
sample.
A susceptor may be In contact with the
The microwaves heat up the susceptor, and the
susceptor In turn heats the sample.
We examine the "hot-wall" problem
from a theoretical
perspective, comparing the case of a background temperature
of 1000*C to a background temperature of 0*C.
cases, the sample Is Initially at 0°C.
In both
The numerics are
the same as before except that when the sample Is colder
than ambient, heat flows Into the sample.
The programs In
Appendix A,B and 7 are used with the appropriate background
temperature.
Lt
B f U - f P tf f f
Figure 38 shows transient conventional heating of the
half-space, where the temperature as a function of position
Is plotted at selected time Increments.
Incident microwave energy.
There Is no
Notice that the maximum
temperature Is at the surface, and that the heating rate Is
quite slow.
An analytic expression for the transient
profile has been described In [64]
133
—
temp
(C)
100
5 0 --
0
10
20
depth
Figure 38.
30
40
50
(mm)
Transient hot wall haatlng of planar half space
with no alcrowave radiation, T.., ■ 1300K,
a)100 sec, b)400 sec.
134
- .xp(
^
) .rfc( ^
^
)
(129)
where a -
Figure 39 ehowe the coablned effect of alcrowave and
conventional heating, where an exponential conductivity law
hae been asauaed, f(T) - exp(.0026T).
The surface
teaperature la plotted as a function of tlae.
the low teaperature heating rate Is Increased.
Observe that
Interaedlate
and high teaperature heating Is still doalnated by alcrowave
effects.
cases.
The saae aaxlaua teaperature Is achieved In both
A decrease In saaple voluae should laprove the
conventional heating rate.
Ls
Pl«iwr Slab solution
Figure 40 shows the heating of a planar slab, surface
teaperature vs. tlae, for different power levels.
background teaperature Is fixed at 1000*0.
The
Steady state is
achieved at 1000°C, when there Is no Incident alcrowave.
When alcrowave power is applied, we know that Instability
136
s urface
temp
(C)
3000--
2000
1000
- -
-
-
0
400
200
time
Flgurs 39.
600
(sec)
Surfacs traptratur* vs. tias for hot wall
hosting of planar half spaca, P ■ 500 W/ca*,
o ■ .002 sxp(.0026T), a)T
■ 300K,
b)T..t - 1300K.
(C)
136
Surface
Temp
1000
500
200
Time
Plgurs 40.
400
600
(sec)
Surfacs tsapsratur* vs. tla* for hot wall
hsstlng of planar slab, T#>t ■ 1300K, o ■
.002 sxp(.0026T) a)P - 0, b)P - 100 W/ca*. c)P
- 500 W/ca*.
137
can occur If this lsvsl Is too high.
Flgurs 41 shows ths cross-sectional variation In
temperature for ths saae slab, at various tlass.
As long as
ths teaperature of the saaple Is less than the background,
the aaxlaua teaperature Is at the surface, since there Is a
net flow Qf heat Into the saaple.
Microwave heating Is
voluaetrlc and yields a central teaperature aaxlaua.
The hot wall applicator Bay prove particularly useful
in sintering aaterlals whose low teaperature conductivity Is
quite saall.
As ths teaperature rises, through "hot wall”
heating, the conductivity rises.
Above a certain value,
alcrowave heating will becoae possible at reasonable power
levels.
138
(C)
1500
Temp
1000-
Surface
2000
500-
Ml t
Distance(mm)
Plguro 41.
• l(k kMllkiry
T
Tooporoturo profiles In hot wall hosting of on
8oo plonor olob, o ■ .002 oxp(.0026T), K m
5 W/n-K, P - 500 W/co*, T„, - 1300K,
0)40 ooc, b)140 ooc, c)200 ooc, d)240 ooc.
CHAPTER IV
APPLICATOR SELECT IOil
Microwave sintering la studied using a variety of
applicator techniques at the Microwave Thereal Processing
Laboratory, at Northwestern.[65] In this chapter we discuss
the experlaental apparatus.
This Includes the slab
applicator, the rectangular applicator, the TEtll
circular cavity applicator and the TM,,,,, circular cavity
applicator.
Por the rectangular applicator we convert the
existing lapedance analysis Into a cavity equivalent circuit
and exaalne theraal stability as a function of cavity
paraaeters.
Li
Experalaental Apparatus
The equipaent used In our alcrowave sintering
experlaents Is shown In the slapllfled block diagram, Pig.
42.
A aagnetron operating at 2.49 GHz, with a aaxlaua
output power of 2000 H Illuminates a saaple within an
applicator.
Between the source and the applicator Is a
circulator and a slotted line.
The purpose of the
circulator la to protect the alcrowave source froa large
reflections.
The power reflected froa the applicator Is
directed to a calorimeter which measures the magnitude of
139
140
f*Mkuk
----------TEMPERATURE
*-y
PLOTTER
STRIP-CMART
RECOVER
CONTROLLER
A
-i
CALORIMETER
POUR-PROBE
STSTBI
PYROMETER
MICROWAVE
T
GENERATOR
♦
_L
-H
CIRCULATOR
Flgurs 42.
3LOTT® WAVEGUIDE
APPLICATOR
Block dlagraa of aicrowava sintering apparatus.
141
the reflected power.
The four-probe eyetea conelete of four
dlodee, epeced at eighth-wavelength Intervale, that extend
Into the waveguide.
The combined output of the theee
detectore yields the real and Imaginary parts of the
reflection coefficient, which can be plotted on a Smith
Chart.
They reveal changes In applicator Impedance while
sintering.
A variety of applicators Is available.
Typically a rectangular or circular resonant cavity Is used
In order to couple energy to the saaple most efficiently.
Saaple surface teaperature Is monitored by a pyrometer.
The
output of the pyrometer Is fed back to the magnetron In
order to regulate the sintering temperature.
1. Slab Applicator
The simplest applicator Is the slab applicator In
TE,0 rectangular waveguide, and Is pictured In Pig. 43.
This design places a slab sample of thickness "d" a
quarter-wavelength In front of a short circuit where the
electric field Is a maximum.
The Input Impedance of the
applicator la: follows: [19]
- vufcd]
where v ■ >(S)
<” 0>
- w*u( ♦ Jwiio
142
V
V
d
r
*
slob
rigur* 43.
Slab applicator gaoaatry.
\
i
short
143
and "a" is tha width of ths wavsguids.
Knowledge of the input lapedance is useful in
deteraining the total power absorbed by the saaple.
It does
not, however, give the distribution of this power within the
saaple.
For the slab geoaetry this can be laportant, since
the electric field distribution for the TEt0 aode in
rectangular waveguide is proportional to slnf^P), it is
zero at the waveguide walls and aaxlaua at the center.
Because of this uneven field distribution, nonunlfora
heating is observed.
The principle advantage of this applicator is its
simplicity, both in analysis and design (tuning not
required).
Its disadvantages are uneven field distribution
and large saaple size if the operating frequency is 2450
MHz.
h
Kactmiular Applicator
The next applicator we will discuss is the rectangular
applicator, shown in Pig. 44.
Here, the slab has been
replaced by a rod so that the entire saaple is in the region
of aaxlaua electric field. [4] The short circuit is now
adjustable, and an adjustable iris (inductive) has been
added to control coupling into the structure.[66] In
144
K T O C CUTOFF T\JBT
tC N -O M T A C T I» e M F T
Flgura 44.
Ractangular applicator gaoaatry.
145
ganaral, tha applicator can ba rapraaantad by an aqulvalant
Input lapadanca, whara tha rod la aodalad aa a T aqulvalant
natwork.
For low loas aaaplaa wa hava a TEIOn cavity.
Wa praaant tha lapadanca daacrlptlon of tha applicator aa
daacrlbad by Aranata [19] and latar show that for low loaa
aaaplaa a parallal RLC circuit la an accaptabla
alapllf1catIon.
A variational mathod, daacrlbad by Schwlngar [67] waa
uaad by Marcuvltz [68] to aodal wavagulda obataclaa,
Including thin dlalactrlc roda.
Aranata axtandad tha
Marcuvltz approxlaatlon ualng hlghar ordar taraa, hla
raaulta ara valid for thick roda and roda at raaonance.
Tha
rod la modalad aa a T aqulvalant natwork, Pig. 45, whara tha
lapadancea ara In ganaral coaplax. Tha aquatlona that ralata
tha circuit paramatera to tha cantarad dlalactrlc poat ara
atatad balow.
A flrat ordar (* x 1) approximation haa been
uaad. [19]
f
3 , ODD
►
(131a)
146
Z
11
- Z
12
Z
- Z
11
12
cz::.3—i—i---- v
12
rigura 45.
T aqulvalant natwork for cylindrical rod.
147
g 11
It ' k
9 , ODD
I - ![(¥)■ ♦ ( & ) ’ * W ] ,
fiJJflJYJa) - *JjB)YJa) ~[
TRT L «3,(®)J0(a) - ai0(fl)Jt («} J
«*k*r
(131b)
where
T - .5772 , a - kR , fl* - 6rV
, k - w/c
and R Is the radius of tha rod.
When this rod Is placsd In the applicator, we aust
Include the Iris and the short In ths equivalent circuit, as
shown In Fig. 46.
When the rod Is placed a sultlple of a
quarter wavelength froa the iris, and ths short is ideal,
the Input adalttance of the applicator Is
<Z„ - Z^MZ,, ♦ Z„) ♦ J Z , , t a n f ^ O
(132)
Z„ *
* -
where Y, Is the adalttance of the Iris and lt Is ths
distance between the rod and the short.
The adalttance of
148
Z
11
- Z
Z
12
11
- Z
12
IN
XftXS
o
Plgure 46.
*•>
<----
Equivalent circuit for rectangular applicator.
149
•n inductlve iris, with no loos la derived by Marcuvltz.
[68]
For aaall aparturaa, w, this Is
Ths rectangular applicator has sssn widest uss In our
alcrowavs slntsrlng sfforts.
1*
Circular Cavltv Applicators
Ms consider two types of cylindrical cavity
applicators.
Ths TEtll and the TM0I0 cavities.
The
purpose of the cylindrical geoaetry Is to Increase the
aaxlaua teaperature of ths saaple by using a theraally
reflecting wall. [3] Ths aaxlaua saaple teaperature Is
prlaarlly H alted by heat radiated froa the saaple which
Increases as teaperature to the fourth power.
By silver-
plating ths walls of a cylindrical cavity, this radiated
energy can be reflected back onto ths saaple reducing heat
loss and thus Increasing saaple teaperature.
The TEllt cavity Is preferred because of ease in
tuning— ths end caps are adjusted to bring ths systsa Into
resonance.
However, there are soae disadvantages.
One
disadvantage of the TE,,, cavity Is field non-unlforalty.
The electric field varies axially ( s ln ( ^) , where
ISO
d la tha cavity height) reaching a aaxlaua at tha cantar and
la noraal to tha rod.
Soaa unuaual axparlaantal raaulta hava baan obtalnad
with tha TE,,, cavity.
Whan haatlng SIC, tha hot zona
algrataa to tha anda of tha cavity, rather than raaldlng In
tha cantar Ilka S13W4 or A1Z0,.
Thla aay Indicate
that tha conductivity of SIC la high enough that haatlng
occura at aagnatlc field maxima, reeultlng In IZR loaaaa,
rather than at electric field maxima (oE2 loaaaa).
Thla
ahlft In hot zona doaa not appear In tha other appllcatora
alnca aamplaa ara always placed In an electric field
maximum.
Tha TE(II cavity la relatively lnaeneltlve to
the preaanca of email aaaplaa.
A perturbation analyela [69]
of dlelectrlcally-loaded cylindrical waveguide shows little
change In tha propagation constant for thin samples, Fig.
47.
In tha TM^ q cavity, tha electric field la z-dlrected,
with radial variation.
The electric field aaxlaua la at the
cantar of tha cavity, analogous to tha TEI0 rectangular
waveguide.
Tha resonant frequency la a function of radius
and tha cavity la tuned by partially Inserting sapphire
tubas (low loss) to change tha cavity volume.
Tha )TMo|o
cavity la currently under development.
The field distribution for both these cavities,
151
t-o
I
1-0
4
Plgura 47.
Propagation constant for sanpla-fIliad T*llt
circular wavagulda, whara b/a la tha ratio of
tha dlalactrlc radius to tha cavity radius.
Adaptad froa [69]
152
neglecting the coupling aperture, can ba dataralnad by
solving a straightforward boundary value problea —
Appendix
H. (Neglecting tha coupling aparture la Justified
by tha close agraaaant between theoretical and exparlaental
values of Q and resonant frequency.)
We discuss the
THolo cavity first, because Its solution Is slapler.
In the eapty T ^ (0 cavity,
we have an axial electric
field and an azlauthal Magnetic field.
They are given as
follows:
Et(r) - E0 J0(kr)
(134a)
H*(r) - JE0n J, (kr)
(134b)
where
When the rod Is Inserted Into the cavity the field
coaponents reaaln the saae, but their values must change to
satisfy the additional boundary conditions at the rod/alr
Interface.
The fields In the two regions are given as
follows:
(135a)
E.| - E0 J0(k,r)
E„ - E# (J„(k,r) H., - J B , , ^ J.llt.r)
Y0(k,r)
(135b)
(135c)
1S3
(13Bd)
The propagation conatants k, and k2 ara In ganaral complex.
Tha TEI(1 cavity, rig. 48, can ba traatad In a
almllar aannar.
For thin roda a parturbatlonal approach may
ba uaad to datarmlna tha propagation conatant of tha
partlally-fIliad cavity.
A mora ganaral approach la the
hybrid mode analyals. [31] It racognlzaa that In tha empty
cavity wa have radial and azimuthal E-flelda and axial Hflelda; but whan tha aampla la praaant, tha boundary
condltlona require an axial E-fleld aa wall.
Although In
moat caaaa thla component la weak, tha mode excited In tha
aample-fIliad cavity la neither tranaverae electric nor
tranaveraa magnetic.
Appendix H.
A complete derivation la given In
In tha fundamental mode, tha electric field
lnalde tha aampla la:
J,<*„,) J c o m
■»» "
« l ny
1J, (k..p) ]alne eln^f
(136a)
(136b)
(136c)
154
iT U t
•X
• M tT M
M IT
COM
»
Vlgura 48.
TBU1 applicator gaoaatry.
*11
155
where A, and B, art given by (H-14) and (H-18) raspactlvaly.
Hlghar sintering taapaaturea hava baan obtalnad
axparlaantally In tha TBII( cavity than In tha rectangular
applicator. [1] As expected, samples whose electrical
conductivity Increases rapidly with frequency are still
difficult to control.
ft;
[2 0 ]
Thermal Stability In a Rectangular Cavltv Applicator
In this section we use the expression relating Incident
power to steady state teaperature In the Infinite cylinder
(63) to study the thermal stability of a cylindrical rod In
the rectangular applicator.
It Is found that stability can
be controlled to some degree by proper Iris selection.
He first derive a cavity equivalent circuit and
calculate the coupling coefficient at resonance to determine
the absorbed power as a function of sample conductivity.
This relationship Is then assimilated Into (63) to fora a
modified expression for Incident power and steady state
temeperature.
It Is assumed that the cavity Is lossless,
that is, all available power Is either reflected by the
cavity or absorbed by the sample.
Por thin rods, (128a and b) reduce to the equations
developed by Mareuvltz, [6 8 ).
The equivalent circuit Is re­
expressed in terms of normalized reactances, which may be
196
lbssy, and la shown In rig. 49.
Tha approprlata
ralatlonahlpa ara:
r
tjir 6l3J«)3l(6) -
«J0U)J,(c«)
(137a)
z. - iz. • 5?:
[ ( n* ”
2 2
)
" /I
■a]
n-*.ODD
irR
(137b)
a'JAb)
JTta- y ^ r r J T T E r ^ W T T T T ^ y
• 2
Figures 50-53 show the real and laaglnary parts of the
complex lapsdancs as a function of conductivity, using
(131), where Z. ■ R. ♦ JX. and Z* ■ R* + JX^.
These figures
show that for low conductivities, X, dominates, so we can
neglect Z» and view the circuit as a shunt reactance with a
series resistance, Fig. 54.
For very high conductivities,
the approximate solution for thin metallic posts given by
Mareuvltz [6 8 ] Is obtained.
The empty rectangular applicator is modeled as a
resonant cavity using a parallel RLC circuit, while the Iris
Is modeled as a transformer.
This circuit Is shown in Fig.
157
JXb
JXb
-JX
Figure 49.
m
Equivalent circuit of cylindrical rod using
Mareuvltz nodal.
158
’° 1
1
10
3 t
10
» M iiim
'*
i iiittbi~'i rrn wi
10 ’4 10
■*
10
i 1 1 hi t * m u m ~ i r i m n
10
conductivity
Figura 50.
1
10
i m i n n ~ i t mttoi
10*
(S /m )
Variation of rod lapadanca, R., with
conductivity.
10 3
159
5.0C 3
-1
o
4.0C
i
3.0C
1
2.00
X
-
3
3
j
.oc ■3
3
T
H
3
0.00 J
3
H
3
3
3
1.00 — I'TTIIIW i
’0
-»
r»
-«
iiiiirnj 'i 11iirai TTTTTWr
-a
»
1
10 “* 1
10 “*m 10
111 inn
conductivity
Figura 51.
iiiim
i mini i i mint
10 .
10 2
10
(S /m )
Variation of rod lapodanco, X , with
conductivity.
160
10
_Q
-sj
-7
-s
-4
10
'3 10
10 ”
conductivity
Flgura 52.
1
10
(S /m )
Variation of rod lapadanca, R*, with
conductivity.
161
0.005 -n
3
o.ooo 4
-
0.00
_Q
X
-0.015
-
0.020
4
10
'*
10
' 4.
10
"*
10
‘ *
10
”
1
10
conductivity (S/m)
Flgura 53.
Variation of rod lapadanca,
conductivity.
with
10
*
10 5
162
Flgurs 54.
Simplified circuit for low loss, thin rods.
163
95.
The rod la lncludad by adding an additional shunt
raactanca with sarlas loss.
Tha ssrlss loss can bs
convsrtsd to an squlvalsnt shunt loss whan tha quality
factor, Q, of tha circuit la high.
According to [70], tha
new raslstanca la:
R.
X*
" TT
(138)
Ha can now study tha cavity In teras of lapadanca: R, L and
C, or by tha cavity paraaatars: rasonant frequency, Q and
coupling coefficient.
Tha reflection coefficient can be
deteralned froa tha Input adalttance of tha cavity.
(139)
Tha circuit eleaents are related to tha cavity paraaatars as
follows:
z
o
(140a)
(140b)
(140c)
where Qo represents tha quality factor for tha unloaded
cavity with saaple present.
Figure 55.
Cavity equivalent circuit.
165
The input admittance of tha cavity axpraaaad in teraa
of cavity paraaetere la: [71]
and the reflection coefficient is [5]
IP I* -
1°
(0
- »)* ♦
(142)
+ !)■ ♦ ( 2 Q„Sgy
Y - Y
where p ■ y8 4. y'a
0
1n
The normalized absorbed power is
* »n« - 1 -
♦6
IPI1 (#
«■ IT
(143)
( * .« )
Maximum absorption occurs at resonance when the coupling
coefficient is unity (critical coupling).
Note that this
model is valid for cylindrical cavities as well, since it is
a general expression for resonance.
Using (136a-c), the absorbed power is expressed in
terms of circuit parameters:
166
4n»Va
1
-
|p |*
(144)
-
When the cavity la tunad to lta rasonant fraquancy, wa
hava
1
-
|p|*
(145)
-
Since tha raalatanca la lnvaraaly proportional to tha
conductivity of tha saaple wa hava
(146a)
where f(v) la tha noraallzed teaparatura dependent
conductivity function and £ la an arbitrary conatant.
Then,
Coupling to the cavity can be controlled by changing
the conatant £.
Since f(0) - 1 , a value of £ ■ 1 defines
critical coupling at v ■ 0.
If f(0) la aonotonlcally
increasing with teaparatura, the cavity will always be over­
167
coupled.
If we chooee 5 leee then one,
the cevlty will
becoee critically coupled when the eaaple reachee a certain
teaparatura defined by f(v) - 1 /5 .
Below thle teaparatura
the cavity la undercoupled, and above thle teaparatura It le
overcoupled.
The effect of cavity coupling on theraal stability can
be seen by aodlfylng (63).
2( c,v
♦ Cj ((v+ 1)4 - 1 ))
He have
2(c,v
♦ Cj ((v+ 1 )* - 1))
f(v)[l - 1* 1-1
(1
, ~
Sf(v))z
,147)
This expression Is valid when the following conditions
are satisfied: 1 ) fields Inside the saaple are unlfora,
there la no theraally induced skin effect; 2 )the saaple Is
thin enough to slapllfy (131); 3) cavity Q Is greater than
10, allowing for the circuit transforaatlon; 4) the cavity
Is aalntalned at resonance.
Figure 56(a) Is a plot of noraallzed teaperature vs.
power when the cavity Is critically coupled at rooa
teaperature, where f(v) ■ exp(.78v), c, ■ .002 and cz ■
.00018.
Figure 56(b) Is the power vs. teaperature plot for
the cylinder In free space.
Is aonotonlc.
He note that heating In 53(a)
However, It requires extreae power levels to
Increase teaperature In the high teaperature region, because
coupling Is weak.
These results aatch experlaental
168
8.00
-|
tem perature
6.00
4.00 -
2.00
■
0.00
0.000
0.010
0.020
0.030
0.040
0.050
power
Figure 96.
Normalized steady state surface teaperature vs.
noraallzed applied power for cylindrical rod In
rectangular applicator with f(v) * exp(.78v),
c, ■ .0 0 2 , c, ■ .00016, a) critical coupling
at 300K, b) free apace cylinder.
169
observations, when a small Iris Is used to begin heating a
low loss alumina sample.
The sample cannot be raised to the
sintering temperature with this fixed iris, because the
available power is limited.
Figure 97(a) is a plot of the other extreme, where the
coupling coefficient is chosen for critical coupling at
2000*C, somewhat above the sintering teaperature.
Figure
57(b) is the reference plot of the free space cylinder.
As
sxpscted, low tsmpsrature hsatlng requires large power
levels.
As tsmpsrature rises, coupling Improves,
accelsratlng the instability.
In tha frse space cylinder,
instability occurs nsar a noraallzed temperature of 4,
while for the undercoupled cavity instability occurs much
sarller, nsar a noraallzed teaperature of 1.4.
observation confirms this effect.
Experimental
When a large iris is
chosen, high power is rsqulred for initial heating and high
temperatures are impossible to control.
An intermediate iris may be selected, as shown in Fig.
58(a), where critical coupling occurs at 1000°C.
Figure 58(b) is the free space cylinder as defined earlier.
The compromise iris shows soms difficulty in initial
hsatlng, but aonotonlc behavior at high temperatures.
Intsrasdlats tsapsraturss ars difficult, but not impossible
to control.
Experimental observations using alumina
rods showed that thsrs is an optimum iris size.
It must be
170
8 .0 0 -i
tem perature
6.00
-
4.00
2.00
0 .0 0 - f - r
0.000
0.020
0.040
0.060
power
Flgura 57.
Noraallzad staady stats surftea taaparatura vs.
noraallzad appllsd powar for cylindrical rod In
rsctangular applicator with f(v) ■ sxp(.78v),
c, ■ .0 0 2 , ct ■ .00018, a) critical coupling
at 2300K, b) frss apaca cyllndar.
171
8 .0 0 -i
tem perature
6.00
-
4.00 -
2.00
0.00 -Hr0 .000
0.010
0.020
0.030
0.040
power
Figure 58.
Noraallzed steady stats surfacs tsapsraturs vs.
noraallzad applied power for cylindrical rod In
rectangular applicator with f(v) ■ exp(.78v),
c, ■ .0 0 2 , c, ■ .00018, a) critical coupling
at 1300K, b) free epace cylinder.
172
large enough to peralt high teaperature heating at
reaeonable power levels, but saall enough to provide high
teaperature stability.
A good guess would place the
critical coupling aldway between rooa teaperature and the
sintering teaperature.
Initial heating Is enhanced by
Increasing coupling, while
theraal runaway at high
teaperatures Is controlled
by decreasing coupling.
An adjustable lrle Is valuable In dynaalcally controlling
coupling.
However, for low loss saaples, like alualna,
a fixed circular Iris Is necessary to reduce arcing.
Figure 59(a-c) shows the coupling as a function of
teaperature for the three Irises used In Figs. 56-58.
Calculations show that the
Q of the cavity Is 1000when the
conductivity Is l.E-5 S/a,
for a .5 ca dlaaeter rod. The
drops with teaperature as the conductivity rises.
corresponds to a conductivity of .001 S/a.
Q
A Q of 10
Using the
exponential law, this corresponds to a noraallzed
teaperature of 5.9, or 2070 K.
Experlaental observations
show that tuning Is required during low teaeperature heating
of low loss saaples As the teaperature rises, the resonant
frequency decreases.
At high teaperatures, the Q Is
sufficiently dlalnlshed to spread the resonant spectrua,
thus cavity tuning Is not required.
This chapter shows the laportance of lrle eelectlon on
etable heating.
It aay be that the Iris which aaxlalzes
173
power
1.00
0.80
absorbed
0.60
0.20
0.00
0.00
2.00
4.00
6.00
8.00
10.00
tem perature
Figure 59.
Percent power absorbed ve. noraallzed
teaparatura for different coupling arrageaente,
a)solld, 300K, b )daeh, 1300K, c)dot, 2300K.
174
efficient energy transfer say not be the beet choice for
stable heating.
41
CHAPTER
V
MATERIAL CHARACTERIZATION
Knowledge of the teaperature dependence of electrical
conductivity la important in determining the thermal
atablllty of the syetem.
In this chapter we dlacuas methods
of material characterization.
First we consider low
temperature measurements (0 - 200°C) using the Automatic
Material Characterization Equipment (AMCE).
A discussion of
measurement range and sources of error is included.
We next
consider high temperature characterization methods.
Of
particular interest is an in situ characterization method
derived from Chapter III, where measurements of applied
power and surface temperature are used to determine
conductivity.
A. Automatic Material Characterization Equipment
A scalar network analyzer has been developed to measure
permittivity and conductivity over a wide frequency range
(2-lSGHz) with temperature control from (0 - 200°C) . [27,28]
It is useful in determlng initial values for materials to be
sintered.
We describe the apparatus and discuss measurement
limitations arising from Maxwell's Equations, poorly fitting
saaples and component imperfections.
178
176
i_i
The Apptrtttt
A simplified block diagram la presented In Fig. 60.
A
alcrowsve source, HP8390B Sweep Oscillator and accesorles,
sampled by the Incident alcrowave bridge, Narda Model 5082,
excites a saaple-f1 1 led segaent of coaxial line enclosed In
a temperature-controlled chamber.
A second bridge Is used
to sample the reflected signal, while the transmitted signal
Is measured directly.
The saaple Is padded to minimize
undeslred multiple relfectlons.
The use of an Internally
terminated alcrowave swlth enables the computer to
selectively measure the transmitted or reflected signals.
The source Is modulated at 1 kHz and the detector diodes
fsed directly to narrow-band amplifiers.
The computer
measures the signals at each of the detectors and controls
frequency and temperature.
The saaple holder la an ordinary
APC-7 connector which can been modified by undercutting the
Inner and outer conductors to support the samples.
shown In Fig. 61.
It Is
Care Is needed to prepare tightly fitting
saaples with parallel front and back surfaces.
Tsmpsrature Is controlled using a two thermocouple
method.
The oven controller uses an lntsrnal set point (set
by the computer) and a direct thermocouple feedback to
achieve a steady state oven temperature.
A second
thermocouple Is ussd by ths coaputsr to monitor the oven
teaperature.
The computer can alter the set point to
m ic ro w a v e
SOURCE
(U M t SO WAVE)
M l
M l
■**/“
NARROW IAND
AM RUFIERS
(IkNtl
OVM
Flour# 60.
Block diagram of Automatic Hatarial
Charactarization Equlpaant (AHCE)
178
ALL DIMENSIONS
Figure 61.
ARE IN MILLIMETERS
AMCK Sample holder, ehowlng a) end view and b)
croea-eectlonal view of the modified APC-7
connector.
179
enhance the heating rate.
L
Theory
The reflection and transmission coefficients are
derived fros a total wave analysis [54] and are given In
terse of complex permittivity
P m
.... (1 - £r
i*) sinhYd
.
(€r*
1)sinhYd ♦ 2j€^Tcoshvd
(£,* ♦ l)sinhYd ♦ 2ji7TcoshYd
(148ft)
(148b)
where free space permeability is assumed.
An Impedance change in the coaxial line is produced by
undercutting the sample holder, thus modifying (148a,b).
The corrected expressions are
- e-~) mlnhyA
(€r* ♦ kMsinhYd
-------
2k,|i/coshYd
2kJ?7
((,* ♦ k*)sinhYd ♦ 2kJ*7Tco*hYd
where
(149b)
iso
k
r H "
The numerator la tha Impedance of the undercut eectlon.
In the ecalar network analyzer, we measure the
magnitudes of the reflection and transmission coefficients.
Consequently, the complex permittivity must be derived
numerically.
Initially we utilized the Newton-Raphaon
method [47] to analyze the data at each frequency.
However,
small measurement errors can produce wild variations In the
calculated results.
It Is preferred therefore to use the
large frequency range of the Instrument to smooth the data.
Flrat, the measured reflection and transmission coefficients
are plotted as a function of frequency.
He then calculate
the reflection and transmission of an Ideal material with
assumed complex permlttltlvlty and superimpose this data
onto the plot of the measured data.
A good fit Is
determined visually.
b. Results
He present data obtained from measurements of teflon,
Stycast H1K, and silicon at room temperature from 2-18 GHz.
These results demonstrate the ability of the Instrument to
measure the constitutive parameters of low and high
181
permittivity, and low to medium conductivity eamples In the
coaxial line frequency band.
Typical published data for teflon Indicates that at
microwave frequencies, the dielectric constant la 2.08 and
the conductivity la .0004 S/cm [72].
Data taken on a 4.56
mm teflon sample showed an average dielectric constant of
2.11 over the range 4-14 (3Hz.
The transmission coefficient
was very large, on the order of 0.5dB, Indicating low loss.
The haIf-wavelength resonance of this sample occurs at 22.9
GHz which Is beyond the range of the Instrument.
The
quarter-wavelength anti-resonance occurs at 11.45 GHz and
corresponds to the region of high uncertainty In dielectric
constant.
Next we consider a high permittivity, low loss sample.
He used Stycast H1K, with a nominal permittivity of 16.
The
curved surfaces of the sample were coated with Heraeus 5450
conducting paste (a silver-filled polymer) to fill gap
regions between the sample and the walls of the holder, as
substantial error can be Introduced by poorly fitting
samples.
Figure 62 shows the measured reflection and
transmission coefficient of a 4.43 mm sample In dB as a
function of frsqusncy.
Two resonance points are observed,
the first at 8.48 GHz and the second at 16.93 GHz
representing the half-wavelength and one-wavelength
resonance points, respectively.
Calculated from the half-
182
••
••
o/
-10
0
1
i
o
I
-15
20
20
FREQUENCY* GH*
figura 62.
Raflactlon and transmission v s . fraquancy for
Styeaat H1K, o - la raflactlon data, □ - la
transmission data.
163
wavelength reeonant point, the dielectric conetant of the
Material la 16.0.
Using the foraula for the Q at reeonance,
the conductivity of the saaple la .005 S/ca.
The eaooth
curve euperlapoeed on the data polnte le calculated froa a
eaaple of dielectric conetant of 16 and conductivity of .005
S/ca.
Good agreeaent le eeen over the entire frequency
band.
As a final evaluation we perforaed aeasureaents on
silicon.
used.
A p-type, (111) oriented single crystal saaple was
DC aeasureaents were aade by the four point probe
aethod, and a DC conductivity of .017 S/ca was Measured.
Silicon has a peralttlvlty [73] of 11.8.
The contacting
surfaces of the saaple were coated with conducting paste.
Figure 63 shows reflection and transalsslon coefficients In
dB plotted versus frequency; thickness is 4.50 aa.
this case Is noticeably low.
The Q In
At the point of resonance, the
data Indicates a peralttlvlty of 11.8 and a conductivity of
.02 S/ca.
However, the curve for a peralttlvlty of 11.8 and
conductivity of .017 S/ca seeas to fit the overall data
better, and Is superlaposed on the graph.
It should be
noted that due to the low Q, the calculated conductivity
based on resonance Is only an estlaate of the true
conductivity.
184
*•
Og
oo
-
46
-
-10
-15
•20
-20
2
4
C
I
10
12
14
FREQUENCY - CHi
Vlgura 63.
Raflactlon and transmission vs. frsqusncy for
silicon, o - 0.017 S/ca. 0 - la rsflsctlon
data, □ - la transalsalon data.
18S
2.
Measurement Range and Sou r c e of Error
Three factors contribute to measurement Inaccuracies.
First, there Is a fundamental limitation Imposed by
Maxwell's Equations —
when conduction or displacement
current dominates, measurement of the other quantity Is
obscured.
Second, small gaps between the sample and walls
of the coaxial line produce errors In high permittivity and
high conductivity materials.
He describe a theoretical
model for the gap and determine that gaps must be eliminated
to acquire useful data.
Third, Imperfect components
Introduce multiple reflections and degrade the measurements.
A signal flow graph analysis la used to describe these
errors.
ib
Llmltatlonm Eased on Maxwell'a Equations
In this section we determine the physical limits on
accurate measurements and provide criteria for the selection
of a suitable saaple thickness.
He use Maxwell's Equations
to develop the theoretical boundaries on the measurement of
conductivity and permittivity.
know that In the steady state,
From these equations, we
186
7XH - 0 E ♦ Jw(,(0E
(150a)
We note that when the displacement current dominates there
la high uncertainty In the conduction current, and
conversely, when the conduction current doalnates, the
displacement current Is uncertain.
By applying this concept
to the present systea, the physical limitations on the range
of measurable material parametere may be discerned.
Theee
conditions guide selection of proper sample thickness.
In terms of complex permittivity,
(150a) becomes:
7XH - Jw£0tr'E -
(150b)
Assuming fixed dielectric constant, the prior Inequalities
can be expressed In terms of two limiting cases:
*nd
-» 0
“* 00 •
The first limit is obtained by either decreasing the
conductivity or Increasing the frequency.
Since the maximum
frequency Is restricted to the microwave frequency range,
only the diminishing conductivity limit will be considered.
From (148) we see that:
187
to*
\p\
V
1 *ln * *
- --------------B-------------- ,
to*
Ir I - £
(13la,b)
where
D - 1 ♦ ' S ; / 1'
^JT)
and thus,
|P|2 ♦ |t|* - 1
(152)
The second Halting condition Is aet by either
decreasing the frequency or by increasing the conductivity.
We consider each case separately.
In the low frequency
llalt the following results are obtained.
ton p - ------------------------------------------------ (153a)
#-•0
Om T - t
^
,153b)
and thus,
p +
t
■ 1.
(154)
188
Keeping the frequency fixed end taking the H a l t as the
epnductlvlty goes to Infinity, we obtain, for large
conductivity:
Ir | - NI T
•
- N-lT
(155a)
|t | - 0,
(155b,c)
In the Unit,
|p| - l, £*t
and thus,
iPl
♦ |r| - 1
(156)
Equation (156) can be used to describe the relationship
between reflection and transmission In either the low
frequency or the high conductivity case.
Equations
(152) and (156)describe the boundaries that
restrict the measurenent range of the Instrument.
He note
that (151)-(152) describe a region of high uncertainty In
conductivity and (153)-(156) describe regions of high
uncertainty in dielectric constant.
These limits are shown
In Fig. 64, where (152) and (156) are plotted as a function
189
1.0
04
1.0
♦
€,• 12
6 • 0.3mm
0.5
10’*
10*'
CONDUCTIVITY - S/cm
Figure 64.
Su b of aagnitudas and s u b of squares of
Bagnltudes of reflection and tranoBlaalon va.
conductivity.
190
of conductivity for a fixed frequency.
In the low
conductivity region, the sue of the equaree approachee
unity, (192).
In order to eatlafy thle Halt,
the
reflection and transalsslon euat be Independent of
conductivity, (191), and thle lepllee that dlaplaceeent
current dominates the conduction current and there la
uncertainty In the conductivity.
conductivity region, the
(196).
bub
Llkewlae, In the high
of aagnltudes approachee unity,
Thle U n i t le net only when the reflection and
tranemlealon are Independent of dielectric constant, (193)
and (199), and lnplles that conduction current donlnates the
dlsplacenent current and thus high uncertainty appears in
the dielectric constant.
Therefore we conclude that
accurate meaaureaents of both dielectric constant and
conductivity are confined to a Halted range of
conductivities.
These results are then used to deteralne the optlaua
thickness.
He view the saaple In three reglaes: the
aultlply reflecting saaple, (191)-(192), the high ineertlon
loss saaple, (199)-(196) and the electrically thin saaple,
(193)-(194), In each region, we will see that there Is a
different criteria for selecting the proper saaple
thlckneee.
In the aultlply reflecting caee, where the conductivity
Is sufficiently saall, and the saaple is sufficiently long,
191
aultlple reflections between the front and back surface of
the saaple produce resonant and antl-resonant behavior.
This resonant behavior can be predicted fros the sinusoidal
nature of (148) when the peralttlvlty Is purely real.
At
resonance (the half wavelength saaple) the reflection Is a
alnlaua and the transalsslon reaches a aaxlaua.
Anti-
resonance Is observed when the reflection Is a aaxlaua and
the transalsslon Is a alnlaua (the quarter wavelength
saaple).
The haIf-wavelength region Is ths aost sensitive
to changes In dielectric constant, whlls the quarter
wavelength region Is least sensitive.
The saaple thickness
required to observe resonance at a particular frequency Is:
d - j fjf.
(157)
and we note that an Increase of the dielectric constant of
the aaterlal, results In a lower resonant frequency.
In
Fig. 65, where reflection and transalsslon coefficients are
plotted as a function of dielectric constant at a fixed
frequency, conductivity and thickness, the sensitivity of
the resonant and antl-resonant regions Is deaonstrated.
To
aeasure dielectric constant aost accuratsly, the halfwavelength thickness should be chosen.
As ths dsslred
frequsncy Is decreased, this thickness would be Increased,
Halted only by the physical length of the eaaple holder.
Note that the resonant behavior of the aaterlal
REFLECTION
COEFFICIENT
MAGNITUDE - \ p\
192
DIELECTRIC CONSTANT
Vlgura 6 8 .
Raflactlon coafflclant vs. dlslsctric constant
for flxad fraqusncy and conductivity.
193
provides an altarnata way of measuring tha alactrlcal
propartlas of tha saapla.
Tha aaapla can now ba vlawad aa a
tranaalaslon cavity whara tha raaonant fraquancy la ralatad
to the dielectric constant and tha bandwidth or Q, la
related to tha conductivity (Appendix I).
Tha unloaded Q
for any homogeneous cavity, accounting for dielectric loasas
only, la [74]:
Q -
€
(158)
This Independent eolutlon can ba uaad to give tha first
guess for tha nuaerlcal analysis.
As tha conductivity Increases, resonant behavior
diminishes.
Whan tha conductivity is sufficiently large,
resonance disappears entirely and this defines the onset of
the high Insertion loss regime.
Here, the determination of
conductivity is very dependent upon the thickness of the
sample.
The maximum measurable conductivity Is determined
by the Insertion loss of the sample, the power available
from the source, and the noise level of the detector.
This
Is Illustrated In Fig. 6 6 , where the magnitudes of the
transmission coefficients of a thick saaple (a) and a thin
sample (b) are shown as a function of conductivity.
For the
thick sample (a), at a critical conductivity, 0c* the
transmitted signal would not be observed because It Is below
the noise level of the detector.
Thinning the sample, (b)
Flgur*
TRANSMISSION COEFFICIENT MAGNITUDE
66.
|t|- 46
Transmission
8
N
o
o
vs. conductivity
o
o
z
o
c
o
oI
M
•s.
194
for s) d • 4.56ss,
O.
199
brings ths signal Into obssrvatlon.
However, for a saapls
of lower conductivity, still In ths high lnssrtlon loss
rsglae, ths thicker saapls Is a bsttsr cholcs, bscauss ths
transalsslon cosfflclsnt Is aors ssnsltlvs to changss In
conductivity.
This Is sasn by coaparlng ths slopss of ths
transalsslon cosfflclsnts at conductlvltlss below oe.
This Is also sssn In (155) where
ms
nots that for high
conductlvltlss:
log |t | ~ -oid
(159)
Therefore, the thinner saaple Is aore useful for aeasurlng
high conductivities, as long as we are above the noise
threshold, while the thicker saaple Is better for aeasurlng
low conductivities.
A saaple Is teraed to be In the thin saaple region when
Its thickness Is less than a quarter-wavelength.
As the
frequency Is lowered, or as the thickness of the saaple
becoaes a saaller fraction of a wavelength, we approach a
Halting condition— (153) and (154), where the reflection
and transalsslon are deteralned solely by the conductivity.
This result Is Illustrated In Figs. 67(a) and (b) where the
reflection and transalsslon of saaples of given thickness
and peralttlvlty but different conductivities, ars plotted
versus frequsncy.
At very low frequencies, the reflection
$4
o
%4
>
o
c
•
s
»
S
TRANSMISSION COEFFICIENT MAGNITUDE-lT l
O®
2 1 10
TRANSMISSION COEFFICIENT M A G N lT U O E -|r|
u
%*
m
> ■
O
e s
o 01
• H
• «
■H
■ •
a
£ o
>o
z
w
3
o
m
m a
■o ■
e o
3s
M
c
oH
•H o
*•
«/>
b»
O
\d
•
u
• •
H
%* o
•
M •
m
O
I- aonilNOVN IN3QliJ3O0 NOI133U3U
O
n
O
l^l-aonilNOVN 1N3ICMJJ300 NOU03U3U
«
b
and transalsslon bscoas constant In both flgurss.
To
asasurs peralttlvlty, however, the aultlply reflecting
region of Fig 67(a) can be aoved to the left by Increasing
the thickness of the saaple The desired thickness of the
saaple can be estlaated by (157).
However, If the
conductivity Is so high that no aultlply reflecting region
can be observed, Fig. 67(b), Increasing the thickness would
only weakly laprove the aeasureaent of peralttlvlty.
The
highest frequency at which aeasureaents can be aade Is
deteralned by the onset of higher order nodes.
Higher order
nodes can propagate In a aaaple-f1 1 led region of line at
lower frequencies than In an enpty line.
However, we have
found no evidence of excitation of higher order aodes as
long as the saaple front and back surfaces are sufficiently
plane parallel.
Therefore It Is satisfactory to deternlns
ths upper bound on aeasureaent frequency fron the cutoff
frequency of the enpty line.
To suaaarlze our findings on optlaua saaple thickness,
we outline the following procedure.
Prepare a saaple of
nodest thickness and take a aeasureaent.
If weak resonance
or anti-resonance Is observed, an approxlaatlon of the
dielectric constant can be aade.
If the transalsslon drops
below the noise level of the detector at the higher
frequencies, a thinner saaple aust be used to deteralne the
conductivity.
If the reflection and transalsslon are
198
constant at the lower frequencies, the saaple is too thin.
These results can then be used to prepare a new saaple whose
thickness Is aore appropriate for characterization.
In
general, the dielectric constant Is aost accurately found by
considering a saaple whose thickness produces resonance at
the desired frequency.
Accurate conductivity Is found by
choosing thicknesses bounded by aaxlaua aeasurable insertion
loss on the high end and Insensitivity on the low end.
Note
that the analysis of optlaua saaple thickness Is based on
physical principles and thus can be applied to a vector
lnstruaent as well.
k:__ Q»p Effects
Another source of Inaccuracies In aeasurlng aaterlal
paraaeters In a coaxial line systea is the presence of
narrow gaps between the saaple and the aetal walls.[29] We
have found large discrepancies bstween neaaured and known
values for aaterlals that should be within aeasureaent
range.
We first review prior work on this topic, presenting
the capacitance aodel.
We develop a aore rigorous analysis
using the transverse resonance nethod, and finally we
consider the effect of filling the gap with a lossy
aatsrlal.
The results show aarked laprovsaent over a
calculable range.
199
The contact problem has been coneldered in the
literature, for various waveguide geometries
[79,76,77,78,79] and In coaxial line at fixed frequency
[80.81].
There are three basic approaches to this problem.
The first reduces the gap by applying pressure to the saaple
hoder, forcing the walls Into contact with the saaple
[75.80.81] A second technique Is to use a TE0l-mode
circular waveguide, where the electric field Is small near
the waveguide walls.[77] The final method Is to derive a
correction formula for the gap so that aeasureaents may be
adjusted to yield the proper result.
The aost elementary
approximation Is a perturbation analysis.[82] Next Is the
capacitance model, [83] and the aost advanced uses an
expansion of modes near the Interface.[84] An Illustration
of the gap effect Is shown In Pig. 6 8 , where observed
peralttlvlty is plotted vs.
gap thicknesses.
true peralttlvlty for different
The consensus Is that a correction formula
Is only valid for very small gaps.
A large gap obscures the
measurement and must be eliminated.
A pressure method Is Impractical In a coaxial geometry,
since It Is difficult to apply outward pressure on the Inner
conductor.
The second method, using a ?E01 circular
waveguide, Is not viable at low frequencies because the
physical dimensions are large.
The third approach, a
correction formula, is considered in more detail.
Observed
dielectric
c o n s ta n t
200
100.00
0.01 S/ cm
0.0
80.00
60.00
40.00
20.00
i.OO
True d ie le c tric c o n s ta n t
rigur* 6 8 .
Obssrvsd dialactrlc constant vs. trus dlslsctrlc
constant for an air gap of a).l ua, b)l ua,
c) 10ua, d )100 ua
1. Capacitance Modal
The distributed cepecltence model (DCM), first proposed
by Westphal [83], views the section of line containing the
saaple as an equivalent lnhoaogeneous coaxial transalsslon
line, Fig. 69a, where r, and r„ are the radii of the Inner
and outer conductors, and r,. and r„ are the Inner and
outer surfaces of the sample, Fig. 69b.
In the absence of a
gap, the Inductance and capacitance per unit length of a
saaple filled line Is:
(160a)
L ■
oi
■
L ■
& * ($ ? )
2nf
*
When the effect of the gap Is Included, the measured
capacitance Is:
(160b)
(160c)
202
COAXIAL LINE
Figur* «*.
O«o»«try of tho g»P-
203
«nd L, - *»(??) . L, - * t ( ^ )
. L, - L, - L,.
The equivalent clrculte corresponding to Figs. 69a and 69b
are shown In Figs. 70s and 70b.
The aeasured peralttlvlty, €., can be expressed In
teras of the saaple peralttlvlty, 6., and the gap
peralttlvlty, €,:
€t% -
re
rt
J
(1«2 )
L,€
♦ L,€
1 r«
1 r|
Separating (162) Into real and laaglnary parts, we have
£r. - £r £ L,
'
€ L,(l + tan*6 ) ♦ €r L {1 + tan*6 )
— ----- ;--- =-----“ v --------- H -- Z
♦ L2£rttan*6f)
£r.L tanfi (1 ♦ tan26.) ♦ er.L_tan6 (1 +• tan26.)
tanfi ------------------ H ------*—1-----;--- *—
"
« , . M l ♦ tan26.) ♦ £,,1^(1 «■ tan26t)
where tan6 ■
(163a)
(163b)
'
'
204
1•
Figure 70.
Equivalent circuit for tha gap ualng a
distributed capacitance model.
205
Note that for a lossless saspls, DCM is lndspsndant of
frequency.
In Pig. 68 the seasured permittivity Is plotted as a
function of sasple permittivity for different gap
thicknesses.
The presence of a gap places an upper limit on
the observed permittivity.
This limiting value Is:
- €r, ^
(164)
In Fig. 71, the measured conductivity Is plotted as a
function of sample conductivity.
0.01
Note that even a gap of
microns can cause erroneous measurements for samples
with conductivities greater than 100 S/cm.
Using (15a) and (15b), the observed permittivity and
dielectric constant and conductivity are plotted as a
function of frequency In Figs. 72 and 73 respectively.
The
permittivity of the sample Is 1 2 , and the conductivity
Is .01 S/cm. The gap does not Influence the measurement
above 1 QHz when It la less than 10 microns, since the loss
tangent decreases with frequency and the Influence of the
gap Is diminished.
At this time there Is no criteria to
establish the size of the gap.
In general, for tests over a
large temperature range, gaps increase with Increasing
temperature.
( S /c m )
206
12.0
0.0
0.1
gap
Observed
conductivity
2 GHz
. ....W
. ■ ....Ml
10 ■*
■ . ...uJ
.
1 0 * i p
True c o n d u c tiv ity ( S /c m )
Figure 71.
Oba.rved conductivity vs. true conductivity for
an air gap of a ).01 im, b).l iia, c)l iia, d )10
Hi, a )100 iia.
3
•t
w
Observed d ie le c tric c o n s ta n t
O
O’
• •
►* H
S
*1
TTfl
<
<Q O.
m
•a a
o •
Hi M
•
• O
— r*
•1
O HH O
a o
P o
5■ °
o
;s
~•-*srt
55
o*
-Ull
s i
207
Figure
73.
TW
Observed
Observed c o n d u c tiv ity ( S /c m )
conductivity
«P
v s . frequency
208
for an sir
209
T r rtirr
avw
N to n w c t Model
The accuracy of DCM for aaaplea with high loaa tangents
aay be questionable.
In fact, In the H a l t of Infinite
conductivity, the samples aay be viewed as a noncontacting
short circuit.
„The noncontacting short has been extensively
Investigated by Huggins In [8 8 ], where the gap regions are
aodeled as narrow transmission lines.
A model which Is
valid for both high and low conductivity samples Is needed.
The transverse resonance method (TRM) Is used to derive such
a model.[8 6 ]
The cross-section of the partially-fllled coaxial line
Is viewed as a radial transmission line.
TRM Is used to
determine the propagation constant for the dominant mode In
the lnhomogeneous coaxial line.
For simplicity, we consider
only a single gap, between the sample and the outer
conductor.
From a reference plane at the boundary between
the sample and the gap we establish an equivalent network of
two short-circuited radial transmission lines of different
characteristic admittances, Y k and Y2.
The geometry
and equivalent network are shown In Fig. 74.
Resonance in
the transverse plane Is determined by equating the two
admittances at the gap sample boundary.
equal, the system will propagate. [8 6 ]
When they are
210
k
«1
kca
Yi(r )
Ya (r )
EQUIVALENT NETWORK
Figure 74.
Equivalent network for Traneveree Reaonance
Model (TRM).
211
Ytct(k„r„,kelr() - Ylct(k.ir...kelr0)
where £
'
- ^ * £r| , ke, k..£
** r I
(168)
, keI «*
The redial cotangent function ct(x,y) la defined aa
*
ct(x,Y) "
j i (*>y q (Y>
- Yi(*) Ja(y)
J0 (x)Y0 (y) - Y0 (x)J0 (y) •
(166)
When the eaaple Is lossy, kel, ke2 and the radial cotangent
are conplex.
The usual separation relations apply.
k*
- y* « S*^£ *.i , it-1• - /
_<
el
C r*
«*
- 5* _•
C r*
*
(167a,b)
a
The propagation constant, y, Is
> -
Solving for the Measured permittivity, £/, using (17,19
and 2 0 ) we have
(168)
212
(169)
Proa this general expression, two special cases are
exaalned, low and high conductivity.
When sample
conductivity Is low, and the gap Is narrow, the
approximations for Bessel functions of small arguments are
valid.
As a consequence, (169) becomes
(170a)
where A ■
(170b)
When the gaps are sufficiently narrow, or the frequency Is
sufficiently low, A ■ 1.
When A - 1, (170a) reduces to DCM
dsscrlbed In ths previous section.
The second limiting case of TRM Is highly conducting
213
saaples. that art insulated froa the walls of the coaxial
line by a narrow gap.
It is necessary to first aodlfy the
equivalent network that represents the radial transalssion
line systea.
He note that if the conductivity of the saaple
is high enough, reflections froa a radial short circuit will
be unobservable because ofthe high saaple
a consequence, the saaple,
attenuation.
As
viewed froa the inner gap,
represents an unbounded, highly attenuating radial guide.
The saaple, viewed froa the outer gap,coapletely fills
inner region, that is, the
the
presence of the inner conductor
does not effect the calculation.
The result of this
aodlflcatlon is that the radial cotangent functions appear
in a aodlfled fora.
The corrected expression for an inner
gap is
(171a)
and for an outer gap
ct (k^r.. ,k^r, ) ■
(171b)
The result is that wave propagation occurs in the gap
region, while the signal is attenuated in the saaple.
If
the saaple conductivity is high enough, the gap regions can
214
be modeled as TSM llnss.
problem.
This Is ths non-contacting short
It Is lntsrsstlng also to nots that for largs
radii and narrow gaps, ths sxprssslon for ths propagation
constant is equivalent to that of a parallel plans guide
[87], since radial tangents becoae tangents in the H a l t of
e
large radii.
We compare the two aodels, TRM and DCM in Pigs. 75 and
76.
Figure 75 shows observed conductivity vs. true
conductivity as a function of frequency, where (a) is TRM
and (b) is DCM.
value of 12.
The peralttlvty is assuned to be a constant
There are two branches to the transverse
resonance solution, one of which is spurious. The other is
in close agreeaent with DCM.
Figure 76 plots observed
peralttlvlty vs. true peralttlvlty with (a) and (b) as
described above.
The conductivity is assumed to be a
constant value of 0.01 S/cm.
agrees with DCM.
Again, the correct branch
A computer program for TRM analysis is
contained in Appendix I .
TRM describes the effect of the gap regardless of
saaple conductivity.
In the low conductivity limit, and
when the gaps are narrow, DCM is valid.
ilLx
Tschniquas for Removing Gaps
We now discuss means of overcoming ths effect of the
gap, using DCM.
Accordingly, we note that in order to
215
in
«, -
0.1
12.0
9, m 0.0
•i - 1-0
,
outer gop “ SO
f • 2 GHz
.
i
•5
o
3
•d
tS
o
a
•d
a>
0.01
0.001 r
&
eapocitonce ^
model
\
v
O o.oooi
~
■
0.0001
Figure 75.
■ *■■■■«•!
0.001
0.01
0.1
■ ■■■!■<
1
■
. I
10
\..«1
•
100
True c o n d u c tiv ity ( S /c m )
Oboorvod conductivity v « . truo conductivity for
• ) TRM nodol, b)cop«cit*nco nodol.
Observed
conductivity
( S /c m )
216
0.1
0.01
copocttonc*
modal
0.001
o, - 0.01 S /c m
- 12.0
0.0001
o. - 0.0
S
■ 1-°
outer gap
0.01
Figure 76.
— SO pm
illll
0.1
1
10
fre q u e n c y (GHz)
100
Obttrvvd conductivity ve. frequency for a)TRM
nodel, b) capacitance aodal.
reduce the effect of the gap, the gap capacitance auet be
auch larger than the saaple capacitance, or In teras of
reactance,
(172)
This can be accoapllshed In two ways: the first aethod
seeks to reduce the gap width to zero, and hence 1/C# ■ 0.
In the coaxial systeas, this option Is open only to samples
such as liquids, powders or "soft" solids, where the saaple
can be presssed Into the saaple holder.
An alterantlve approach Is filling the gap with a known
aaterlal.
We assuae that the gap Is saall, that is,
(173)
If we assuae that the gap Is coaposed of a material
whose conductivity Is greater than that of the saaple, then
(163a) and (163b) are
Cf.L.
L.
r•
(174a)
tsn6a - tan6. ♦ tanfif
(174b)
for tan 6t >> 1 , tanim - tan&.
(174c)
218
So we ••• that whan tha conductivity of tha gap la graatar
than tha saaple aaaaurad valuaa thaoratlcally match trua
valuaa.
Two tachnlquas ara usad for filling tha gap.
First, a
conducting paata la palntad on tha curvad aurfacas of tha
aaapla.
Tha aaapla la lnaartad In the aaapla holder and tha
paata la allowed to dry.
Tha hlghaat maasurabla aaapla
conductivity la H al t ed by tha conductivity of tha paste.
Tha second technique for achieving a higher
conductivity gap entailed electroplating tha saaple.
Tachnlquas for metallizing plastics [8 8 ] are usad.
A thin
layer of silver Is deposited on the saaple surface using an
electroless method, than tha sample Is electroplated with
copper.
The details of the process are described In
Appendix K.
After electroplating Is complete, the saaple Is
sanded to remove excess metal froa the flat surfaces and to
smooth the curved surfaces for a snug fit Into the sample
holder.
If the saaple dimensions are the same as the standard
7aa coaxial line, and the electroplating Is thick enough to
fit the undercut saaple holder, we can eliminate the
Impedance step caused by the undercut.
In more accurate measurements.
This should result
219
l£i
intrlHntt 1
We M i au r «d the characteristics of thrse known saaplss
to deteralne study ths gap offsets.
Measureaents wers aads
on Stycast H1K, and on two silicon saaplss of asdlua and
high conductivity.
Ws consider results for both an air gap
and a filled gap.
The Stycast H1K aaterlal was 4.43aa thick with a
nonInal peralttlvlty of 16.
Figure 77 Is a graph with three
different plots of the aagnltude of the reflection
coefficient In dB as a function of frequency: (a) the
theoretical results (solid curve), (b) the experlaental
results for the filled gap (circled points), and (c) the
experlaental results with an air gap (dashed curve).
The
theoretical curve Is based on a dielectric constant of 16
and a conductivity of .005 S/ca, and the gap filling
aaterlal Is Heraeus 5450 conducting paste. Figure 77(a,b)
Is a reproduction of Fig. 62.
In coaparlson, the plot of
the reflection coefficient with the air gap Is characterized
by a coaparatively large aaount of Irregularities, and the
resonance Is shifted In frequency.
The half-wavelength
calculation using (157) yields a false peralttlvlty of 10.4.
The capacitance aodel indicates that an air gap of 10
alcrons would produce this shift In dielectric constant.
Figure 78(a) and (b) Is a reproduction of Fig. 63, and
Fig.
78(c) Is the erroneous data acquired In the presence
220
m a g n itu d e(p )
(dB)
0.00
-3.00
-
10.00
Stycast HiK
a » 0.003 S /c m
00
fre q u e n c y (GHz)
Plgurs 77.
Ths sffsct of air gaps on tha aaasuraaant of
Stycaat HIK; a) dash, aaasuraaant with gap,
b) solid, aaasuraaant with flllad gap.
221
m a g n itu d e (p )
(dB)
o.co
•V
20.00
silicon
l.OO
fre q u e n c y (GHz)
Vlgura 78.
Tha affsct of air gaps on tha aaasuraaant of
.017 S/ca silicon; a)dash, aaasuraaant with gap,
b) solid, aaasuraaant with fIliad gap.
222
of a gap.
Tha faulty data la again charactarlzad by a shift
in rasonanca to a highar fraquancy and by graatar
irragularitias in tha rasponaa.
For high conductivity samples, tha conductive pasta
cannot ba usad, because its porosity H al t s its
conductivity.
Instead, we auat electroplate tha saaple.
Meaaureaents ware perforaed on p-type silicon.
DC
aeasureaents of tha 3.6aa saaple showed a conductivity of
.28 S/ca.
A aetalllzed saaple was prepared and tha results
are shown in Fig. 79.
Tha circled points show tha aeasured
transmission coefficient vs. fraquancy.
Tha solid curves
show theoretical transmission given a dielectric constant of
11.8 and conductlvtles of .2, .28 and .3 S/ca.
Tha
reflection coefficient was too large to aeasure accurately.
Notice that the conductivity is sufficiently high to
suppress resonance behavior.
For still higher conductivities, loss tangent greater
than 1000, Fig. 71 shows that even an oxide layer on the
sample surface could Influence the measurement.
In this
case, a metallizing layer aust wet the surface of the
saaple.
The gap effect can be corrected by using a suitable
filling aaterlal.
Theoretical models accurately predict the
effect of the gap but are unable to provide data correction
because the exact gap width is unknown.
223
-2 5 .0 0
silicon
OQ - 3 0 .0 0
- 3 3 .0 0
0 .2 S /e m
••
•-* - 4 0 .0 0
0 .2 9 S /c m
0.2 S /c m
- 4 5 .0 0
f r e q u e n c y (GHz)
Figura 79.
Transaisslon vs. frsquancy for high loss
(oDC " *28 S/ca) silicon using an
•lsctroplatsd saapls. Plgurs shows thsorstlcal
curvss for .2, .25 and .3 S/ca.
224
c. Component Imperfections
The other major source of error originates In the
microwave systea.
It arises froa laperfectlons In the
coaxial line coaponents.
Me found that the Initial version
of the Instrument, without appropriate attenuators, produced
erratic results.
Through a slgnal-flow graph [89] analysis,
It was possible to Isolate the source of error to aultlple
reflections between the bridge and the saaple, and aultlple
reflections between the saaple and the diode that aeasures
the transaltted signal.
Additionally, It Is possible to
Identify alternate paths to the detector diodes that limit
the sensitivity of the experiment.
Other aeasureaent
errors, produced by diode non-linearity or an unleveled
source, can be curtailed by proper calibration.
Figure 80 Is a signal flow graph model of the system.
The microwave bridges are modeled as three port devices and
are assumed to be Identical for the simplified analysis to
follow; where Stl Is the Insertion loss, S3I Is the coupling
loss, and S3, Is the directivity.
Slt, S„ and S33
represent the return loss of the bridge at each
corresponding port.
Both the attenuators and the sample are
modeled as two ports with the terms for Insertion loss and
return lose being clearly displayed in Fig. 80.
coaponents are all assumed to be reciprocal.
1,2
These
The diodes,
and 3 are represented as one ports loading the proper
225
9dB
3dB
BRIDGE I
BRIDGE 2
ATTEN I
«»
a
SAMPLE
ATTEN 2
'•T
u
’7 7
5*9
Plgura 00.
Signal flow graph rapraaantatlon of AMCE.
OS
226
portions of ths graph.
In reality, dlodss 2 and 3 rsprsssnt
a slngls dlods that Is controlled by ths switch. (Fig. 60)
When ths transmission Is bslng saspled, ths reflection port
Is terminated In a matched load.
Conversely, when the
reflection Is being measured, the transmission port Is
matched.
The Important first order loops are:
^33^*01 * ^33^01'
®77^03 '
®«7^D3
(175)
These loops represent multiple reflections between the
sample and the left attenuator, the sample and the
reflection bridge through the attenuator, the Incident
bridge and Its detsctor diode, the reflection bridge and Its
detector diode, the right attenuator and the transmission
detector diode, the sample and the right attenuator, and,
the sample and the transmission detector diode through the
attenuator, respescltlvely.
The signal at a point in the
graph can be computed by the relationship [90]:
b,
ST "
P,[l - 2 L,(l) ♦ •••] ♦ P,[l - S 1,(1) +•••]
1 - 2 M l ) ♦ Z L(2) - Z L(3) ♦ -
(176)
227
Where P„ represents the poeslble paths to point b, and Ln
represents non-touching loops with respect to path Pn.
The
denominator Is the sua of all loops.
Por the system, we first calculate the reflected signal
produced by a short circuit as normalized by the Incident
signal.
(177)
The error ln the reference signal, br can be reduced by
comparing It with the signal produced by an open circuit.
The aultlple reflections produced by the open circuit are
180o out of phase with the signal produced by the short
circuit.
The effect of these aultlple reflections can be
reduced by forming the reference signal from an average of
the short-circuit reference and the open-clrcult reference.
228
where the remaining error terms arise froa aultlple
reflections between the detector and the bridge, and are
Independent of aultlple reflections between the termination
and the bridge.
The signal reflected froa the saaple can
then be measured with respect to this reference.
The
approximate expression for the reflection coefficient Is:
s,,s4g} ♦ s„r 02 ♦ s 33r0l J
C^
* ^
♦ c" «
<179>
The teras ln the expression are explained as follows:
The first tera contains the reflection from the sample and
errors due to aultlple reflections between the sample and
the attenuator, bridge and diode.
The remaining teras
represent alternate paths, the first Is reflection froa the
front attenuator and the second Is due to the directivity of
the microwave bridge.
The secondary paths set the
sensitivity H a l t of the reflected signal, and the aultlple
reflections produce fine structure ln the measured signal.
If the attenuator between the bridge and the saaple Is
oaltted, a different approximate expression for the
reflection coefficient Is obtained:
229
(180)
The positive effect of the attenuator Is seen ln reducing
the aultlple reflections between the saaple and the bridge.
The negative aspect of the Insertion of the attenuator Is
that provides an alternate signal path to the detector and
that It Increases the effect of the directivity of the
bridge on the aeasured signal.
The transaltted signal can be considered ln a similar
way.
In this case, the reference signal Is obtained by
noraallzlng the transalsslon through an eapty holder to the
Incident signal.
The approxlaate expression for the
transalsslon coefficient Is given as:
r ■
Note that all of the error teras are due to aultlple
reflections.
There are no alternate paths to the
230
transalsslon detector.
The effect of the ettenuetore ie
eeen by reaovlng both ettenuetore froa the circuit, the
treneeleelon coefficient le now given ee:
r ■
(182)
So the left ettenuetor is useful in reducing reflections
between the bridge end the saaple, while the right
ettenuetor deeps reflections between the seaple end the
trensaleslon detector.
The deleterious effect of the
ettenuetor on the trensaltted slgnel Is to decreese
sensitivity.
While the Insertion of epproprlete attenuation reduces
the Interaction between reflective alcrowave components, the
aeesured values of the reflection and trensaleslon
coefficient aegnltudee still appear with soae residual
quasi-perlodlc saell variation due to reaelnlng multiple
reflections end secondary paths In the systea.
Their effect
could be reduced further by aore sophisticated data
analysis.
The signal flow graph analysis has been useful in
Identifying sources of error caused by reflective alcrowave
coaponents.
Multiple reflections between the saaple and the
bridge were reduced by Interposing a 3 dB attenuator.
231
Reflections between the saaple and the transalsslon detector
were reduced by a 9 dB attenuator placed between the two'
coaponents.
The particular value of attenuation was chosen
to reduce Interactions between coaponents without degrading
the sensitivity of the experlaent too auch.
232
». High T— pgratiure Characterization
In addition to rooa taaparatura aaaauraaants, It la
Important to know aaapla propartlas as taaparatura
lncraasas.
We saak a aystea to aeasure coaplax
peralttlvltles froa rooa taaparatura to sintering
teaperatures, noalnally 2000°C.
The aysten should provide
data at 2430 MHz and use actual saaples for sintering.
In
this section we consider existing aethods of high
taaparatura characterization, and propose a new technique
baaed on theoretical findings In Chapter III.
High Teaperature AMCE
The aost logical alternative would be to adapt AMCE for
high taaparatura use.
AMCE is currently Halted to
operation below 200*C for the following reasons: 1)teflon
spacer In saaple holder chars; 2 )copper conductors becone
Increasingly lossy 3) differential expansion nay cause gap
effects to re-energe.
To alleviate these difficulties we aust a) replace the
teflon spacer with a refractory window naterlal such as
beryl11a or alualna.
A lower dielectric constant Is
preferred since the spacer will add nultlple reflections to
the systea.
The saaple holder should be evacuable to avoid
high taaparatura oxidation effects, b) the saaple holder
233
should bs fabricated out of aolybdanua or a slallar aatal.
c) tha saaple should ba nickel plated.
Electroless
deposition of nickel is described in [8 8 ].
Differential
expansion aay either cause the saaple to fracture or cause
gaps to reappear around the saaple.
A correction foraula
for gaps produced through differential, expansion Is
developed using a aodal analysis In [91].
A correction
foraula Is now possible since the gaps produced by
differential expansion can be calculated.
These changes
should lncrsase the operating range to 1000°C.
2.!
Other High Teaperature Methods
A survey of the literature reveals several Interesting
characterization aethods.
Westphal [92] coapares several cavity aethods for
dielectric constant and loss tangent aeasureaent up to
1200°C: a) A TMo,0 re-entrant cavity with a centrally
aounted cylindrical saaple Is used near 1 GHz, with
aeasurable loss tangents ranging
froa 9 x 10'* to
.02
b) A coaplstely filled TEltl cavity at 4 GHz
aeasures loss tangents between 4
x 10'* and .009.
transversely aounted saaple In a
TEllt re-entrant
c) A
cavity at 8 GHz aeasures loss tangents between .002 and 1.
In all cases silver or platlnua foil Is used to Bake contact
234
between the staple end the wells of the cevlty.
Neesureble
lose tengente ere lower then those obtslnsble by AMCE et
roos tespereture.
High tespereture charecterlzetlon et sllllseter
wevelengths Is performed using quasl-optlcal techniques.
[93,94] Results compere well with lower frequency
(alcroweve) measurements [99].
These aethods would require
huge samples et 2450 MHz.
A method using e TE0I cavity composed of graphite Is
used to measure silicon end aluminum oxides et 10 GHz, froa
0 to 2000*C [96].
The sample Is 4.1 cm In diameter.
The
conductivity of alumina et 2000°C Is given as .1 S/a.
3.
In Situ Characterization
While high temperature characterization may be of use
In determining the trend of complex permittivity with
temperature, the response during actual sintering would be
desirable.
Araneta describes an In situ characterization
method using the rectangular applicator.
From the Impedance
analysis, we can deduce complex permittivity froa reflection
coefficient data.[18]
This asthod assumes that the saaple Is heated
uniformly, and works best when losses In ths iris and shortcircuit are kept to a minimum.
Ultimately, accuracy depends
on the ability to measure the reflection coefficient and
235
this is llaltsd by ths purity of the aagnetron sourcs.
Ms proposs an laproveaent to this charactsrlzatlon
asasursasnt using two alcrowave sources.
A possible systea
configuration Is shown In Fig. 81.
A aagnetron source Is
used to heat the saaple as before.
Since the aagnetron Is
only transalttlng 50% of the tlae (120 Hz), a precision low
power source can be used to sweep the cavity during the off
cycle of the aagnetron.
The low power source Is protected
by a directional coupler and a suitable nuaber of Isolators.
The high voltage froa the aagnetron Is used to switch a
transalt/receive (T/R) tube to protect the detector froa
high power levels.
Acquired data could be stored In
coaputer aeaory and analyzed at the coapletlon of a run.
Lx
A Hew Method for High Teaperature Characterization
We describe a high teaperature characterization aethod
using the analysis of Chapter III on a spherical saaple
placed In a alcrowave applicator.
This aethod provides a
aeans of deteralnlng teaperature dependent electrical
conductivity by aeasurlng surface teaperature, Incident
power and reflected power, along with Icnowledge of heat
transfer constants.
The aeasureaent Is Independent of
applicator geoaetry.
Equation (63) relates power to steady state surface
teaperature for a thin planar slab.
A slallar expression
236
Figure 81.
Block diagram of In situ characterization aathod
using a tranaait/rscsivs tubs.
237
has bssn dsrlvsd for an Infinite, thin cyllndar (1 1 2 ) and a
snail sphsrs (152).
Ths dlffsrsncs in ths sxprssslons Is a
constant dstsrnlnsd by ths surfacs arsa to voluns ratio of
ths sanpls.
To dstsrnlns ths conductivity froa sxpsrlasntal data an
lnvsrss rslatlon,
(67), must bs ussd.
Accurats rssults
rsqulrs that ths tsapsraturs bs unlforn within ths saapls,
othsrwlss ths asasursasnt will ylsld
sobs
average
conductivity.
In ths slab geometry, a saapls aountsd In rsctangular
wavsgulds has a non-unifora slsctrlc field, producing
centralized heating.
Observations indicate that for
cylindrical rods, non-unlforalty Increases with temperature
and is aost pronounced In low theraal conductivity
naterlals.
Figure 82 Is the axial teaperature variation of
an alualna rod.
Figure 83 shows the axial teaperature
variation for an alualna rdd with 30% titanium carbide.
The
profile Is aore uniform In Fig. 83, the higher electrical and
theraal conductivity aaterlal.
These figures show the trend
In axial teaperature variation and are consistent with
visual observations— the pure alualna saaple has a shifted
aaxlaa and Is highly non-unlfora.
Note that absolute
temperatures may be in error due to occlusion of the
pyrometer.
(The pyrometer looks at the saaple through a
beyond cutoff tube.
If the cone of observation intersects
238
t e mp e r a t u r e
(C)
1800-<
H
1
1
r>
60 u
1400
-1
i t_ *
i i
neignt
Plgura 82.
0
1
f
9
\
(cm)
Axial taaparatura profllaa In a alntarad alualna
saapla for varloua powar lavala.
239
Temper at ure
(C)
1600-1
1400J
*
♦
1200
-
1 0 0 0 ? i i i i i i i i r | i i i i i » r r tT i i i » i r l r i |~i i » i » » r t -> i
-
2
-
1
0
Height
Plgura 83.
1
(cm)
2
top
Axial taaparatura proflias In a slntarad
alualna-TlC (30%) saapls for various powsr
1 avals.
240
the beyond cutoff tube, a aeaeureeent error Mill
Introduced.)
In addition to axial non-uniforalty,
cylindrical aaaplee are preferentially heated on the side
facing the source.
The sphere is aore likely to have a uniform field
variation due to its aore coapact geometry.
(Theoretically
a saaple can be Bade saall enough to eliminate any nonuniformity. )
Phenomenological experiments with a pill-
shaped (diameter ■ height) alualna saaple suspended in a
quartz tube confirm this hypothesis.
We observe no axial
variation, and heating appears to be uniform around the
circumference of the sample.
Therefore, it is proposed that
spherical saaples be used in characterization studies.
To aake an accurate measurement of the sample requires
that the absorbed power be known.
This requires a
measurement of incident power and reflected power.
It is
not necessary to acquire reflection coefficient data.
The
location of the sphere in the cavity does not affect the
calculation, since we assume that all of the power
dissipated in the cavity is dissipated in the sample.
However, low loss samples should be placed in a region of
maximum electric field to improve heating efficiency.
With a knowledge of theraal properties, saaple diameter
and surface teaperature, the electrical conductivity can be
deduced.
241
An uncertainty analysis rsvsals ths relative Importance
of certain measured values.
The error Is found froa (63)
and Is
Error ■ 1
(183)
where two supposedly Identical measurements are compared.
A 5* error In measured power or sample geometry
translates directly to a 5k error In conductivity.
A 5k
uncertainty In ck has a large effect on conductivity at low
temperatures.
Pig. 84.
It Is plotted as a function of temperature In
The result is anticipated, since radiation
dominates at high temperatures.
Plgure 85 plots the effect
on conductivity of a 5k error In cz as a function of
temperature.
As expected, cz has less effect at low
temperatures, where convection dominates.
A 5k error In surface temperature can produce up to a
20k error In conductivity as shown In Fig. 8 6 .
This is a
result of the fourth power variation In temperature.
The uncertainty analysis reveals that the surface
temperature measurement can produce the largest errors In
conductivity.
In a typical experiment, we know the
temperature within 50°C at 1500°C.
This would translate
242
A 00 -
T r\ r*
«*/ . sJW
ir 2.00
\
*^
(*
I
n
n r\r*
_»
I
..Jw -Ji
-J
0
0.00
2.00
4.00
6.00
8.00
10.00
tem oeroture
Plgurs 84.
Porcont orror In conductivity B M s u r m n t vs.
normalized taaparatura for a 5% error In c,.
243
5.00 -
error
4.00 -i
1.00
0.00
2.00
4.00
6.00
temperature
— ...
8 oo
1 0 .0 0
244
20.00 --i
16.00 -s
3
/
-t
*^ ^ ^ ^
o
•4
.
.VS-/ J
-1
3
-<
o
&
8 •nW nw
J
^
/
3
-i
t /
2/
A .00
3
3
n
3
-t
0 . 0 0
I
n
' i i r r 'r i
0.00
rr
i i
2.00
n
i i i i
i i i i i i
*00
11
i i i i i i t ) i i i i i i i i
6.00
8.00
i
10.00
tennDeroture
Flgur* 8 6 .
Psrcsnt srror in conductivity M u u r n t n t vs.
normalized tsmpsraturs for • 5% srror In v.
245
into an error of a little aore than 10% In conductivity.
While thle aethod aay not provide high accuracy
reaulte, it ehould provide ueeful conductivity data for
alcrowave alnterlng applications.
It has several advantages
over the aethods described earlier: A saaple does not need
to contact the cavity, so the "gap effect" is not a concern;
phasor detection of reflection coefficient is unnecessary,
only aagnltude is laportant, so source quality la not
critical; and saaples can be studied in an "real" sintering
envlronaent.
C H A P T M VI
C0WCL08I0W8 AMD RICOHHgyDATIOHS FOR f U T U W WOWC
This study has provldsd sddsd Insight Into ths
alcrowave hsatlng of aatsrlals.
Specifically, wc havs
dsaonstrated the laportance of teaperature dependent
electrical conductivity In deteralnlng the heating behavior
of the saaple.
Matheaatlcal analysis Indicates that when
electrical conductivity rises too rapidly, for electrically
thin saaples, the heating process Is uncontrollable above a
critical power level, and that there Is a aaxlnua attainable
steady state teaperature associated with this power level.
We have considered the heating process In a variety of
geoaetrles Including a seal-infinite space, a planar slab,
an Infinite cylinder and a sphere.
The result Is that
heating of a seal-infinite space is always aonotonlc because
the "skin effect" H a l t s field penetration at high
conductivities (teaperatures).
theraal Instability.
Thin saaples can exhibit
In fact, the aaxlaua teaperature Is
the saae for the slab, cylinder and sphere.
This "runaway"
behavior is consistent with laboratory observations.
To predict stability/instability In the heating of a
saaple the teaperature dependent electrical conductivity
246
247
aust be known.
Characterization aathods and their
llaltatlons are discussed for both rooa and high
teaperatures.
He find that to aeasure high loss tangent
aaterlals the saaple surfaces which contact the walls of the
saaple holder aust be electroplated to reaove any gaps.
A
high teaperature technique based on the relationship between
steady state teaperature and absorbed power Is also
described.
This study has answered a nuaber of questions
concerning the nature of anoaalous behavior observed In
experlaent.
Core aeltlng Is attributed to theraal gradients
due to low theraal conductivity In which the surface
teaperature Is cooler than the Interior. "Theraal runaway"
can occur above a critical power level when electrical
conductivity Is a strong function of teaperature.
Control
of cavity coupling can In soae cases restablllze the system.
Puture work Is needed to further our understanding of
the sintering process.
In the theoretical area, a coaplete
analysis of the slab Is needed.
He have developed a
solution for the case where the electric field distribution
Is nearly unifora and does not depend on teaperature.
At
high teaperatures, the skin effect reduces field
penetration, coapllcatlng the analysis.
For very high
conductivities, the saaple reseables the seal-lnflnlte
space.
Hork Is also needed In developing aore sophisticated
aodels for the radiant wall applicator and the "hot wall"
248
applicator.
Tha cavity analysis should bs sxpandsd to
lncludsd wall lossss.
Studios should bs sxpandsd to
consldsr teaperature gradlsnts In aultlpls dimensions, as
wall as sintering dynamics.
High tsapsraturs charactsrlzatlon squlpasnt Is badly
nssdsd.
As ws slntsr unlqusly prsparsd aatsrlals such as
alualna-TIC, at various concsntratlons of TIC, a knowlodge
of tsapsraturs dspsndsnt conductivity Is essential.
Such
squlpasnt aay help answer still unresolved problems, such
as: why do unslntsrsd saaples heat aore quickly than
sintered saaples?
Unslntsrsd saaples should have a
lower conductivity due to their greater porosity.
It Is
recommended that a characterization study be implemented to
classify aatsrlals as good/bad candidates for alcrowave
sintering based on their teaperature dependent electrical
conductivity.
Suitable dopants alght be used to change
teaperature dependent conductivity and thus enhance the
slnterablllty of certain low loss materials.
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APPENDIX A
ER99IMH T9B PQHPyiHW TRAIT?I HIT TPgjBAIgMLgRgrifcH Iff A
SEHI-INFIKITX 5PAC1 KITH CONSTANT CONDUCTIVITY
1 PLANE HAVE HEATING OF SEMI-INFINITE SPACE
' EXPLICIT METHOD SOLUTION OF HEAT EQUATION
I
DEFINT I ,J ,N
’ T STORES TEMP DATA, T 1 STORES ELECTRIC FIELD DATA
DIM T(2,1000),T1(1000)
OPEN "0",#1,"PROFIL.DAT"
INPUT "INTERVAL SPACING";D
INPUT "LENGTH";L
INPUT "TIME STEP";DT
N-INT(L/D+.5)
' SET CONSTANTS USING MKS UNITS
SO-.002
•ROOM TEMP ELEC CONDUCTIVITY
ER-10
'DIELECTRIC CONSTANT OF SPACE
PI-3.14159265
F-2.45E+9
'TEST FREQUENCY
W-2*PI*F
C-2.998E+8
'SPEED OF LIGHT
EO-8.854E-12
'FREE SPACE PERMITTIVITY
K-20
'THERMAL CONDUCTIVITY
'HEAT CAPACITY
CP-1000
RHO—3970
•DENSITY
'EMISSIVITY
EL-.6
'STEFAN-BOLTZMANN CONSTANT
S-5.67E-8
'CONVECTION CONSTANT
HH-10
'AMBIENT TEMPERATURE (KELVIN)
T00-300
R-DT/D* 2 *K/CP/RHO
PRINT R
'R MEASURES STABILITY < .025
PN-2*10*5
'INCIDENT POWER LEVEL
J-l
Jl-2
FOR 1-0 TO N
'SET SAMPLE TO AMBIENT TEMP
T (J ,I) - TOO: T (J 1 ,I) - TOO
NEXT I
FOR K2-1 TO 1000000
'COMPUTE INTERIOR FIELDS
IF K2 > 1 THEN 15
'FIRST PASS ONLY
FOR 1-1 TO N
X-SO/W/EO/ER
RT-W/C*SQR( ER)• (1+X' 2 )".25*SIN(-. 5*ATN (X) )
T1(I )-377*PN*EXP(2*RT*I*D)*S0
NEXT I
10 M-(l+X‘2)‘.25
'COMPUTE REFLECTION COEFF
257
258
MM«M*SQR(ER)
PH— 5*ATN(X)
NUM-SQR((MM* 2-1)* 2+4*MM“2*SIN(PH)“2)
DEN-MM* 2+1 + 2 *MM*COS(PH)
P-l-(HUM/DEN)‘2
'P IS ABSORBED POWER RATIO
PRINT p
15 T-T+DT
'INCREMENT TIME
PRINT "TIME";T;"SEC"
H1-S*EL*T(J,1)‘3+HH
'LINEARIZE HEAT TRANSPER
T(J,0)«T(J,2)-(2»H1/K)»T(J,1)•D+2/K*TOO*(HH+S*EL*TOO*3)*D
T(J,N)-T(J,N-2)-2*T(J,N-1) 'COMPUTE BOUNDARY VALUES
POR 1-1 TO N-l
T(J1,I)-R*(T(J,1+1)+T(J,1-1))+(l-2*R)*T(J.I)+
+ T1(I)*DT/CP/RHO*P
NEXT I
IP K2/500 > INT((K2+.5)/500) THEN 20
LPRINT "TIME ";T;"SEC"
'STORE RESULTS OCCASIONALLY
POR 1-1 TO 100
LPRINT USING "#####.#♦ ";T(J1,I) - TOO;
LPRINT #1, USING "###♦#.## ";T(J1,I) - TOO;
NEXT I
LPRINT
20 J-Jl
J1-3-J
NEXT K2
MTiwm p
EB99RAH T9 gQBPffll TBAWSIIWT TPff MATVEI PR9KM? IB MHIIHTIBITI tfAVI WTB IBfflBATVEI PI FIEPBfT-KISIRISAIr
CONDUCTIVITY
PLANE NAVE HEATING OF SEMI-INFINITE SPACE
EXPLICIT FINITE DIFFERENCE SOLUTION
OF HEAT AND WAVE EQUATIONS
T STORES TEMPERATURE DATA, T1 STORES ABSORBED POWER DATA
KR AND KI ARE COMPLEX PROP CONSTANTS IN EACH GRID REGION
EPR, EPI, ENR AND ENI ARE COMPLEX ♦ AND - TRAVELING WAVES
IN EACH GRID REGION.
DIM T (2,500),T1(500),XR(500),KI(500),EPR(500),EPI(500)
+ ENR(500),ENI(500)
OPEN "0 " ,# 1, "PROF.DAT"
INPUT "INTERVAL SPACING";D
INPUT "LENGTH";L
INPUT "TIME STEP";DT
'REM SET CONSTANTS IN MKS UNITS
N-INT(L/D+.5)
•ENDPOINT FOR SG(T) CALC
NH-N .
'FOR HIGH ATTENUATION
NN-N
SO-.002
EXPONENTIAL CONSTANT
Cl-.0026
SG-SO
ER-10
PI-3.14159265
F-2.45E+9
W»2*PI*F
C-2.99E+8
EO— 8 .8S4E-12
K-20
CP-1000
RH0-397C
EL-.6
S-5.67E-8
R-DT/D“2 *X/CP/RHO
H-10
T0-300
'R MEASURES STABILITY, <.25
PRINT R
PN-10'6
J-l
Jl-2
259
260
POR 1-0 TO N
'SET SAMPLE TO AMBIENT TEMP
T (J ,I) - TO : T(J1,I) - TO
NEXT I
POR K2-1 TO 1000000
'COMPUTE NORMAL PIELD ATTENUATION ON PIRST PASS
IP K2 -1 THEN 5
'IP DIPPERENTIAL CONDUCTIVITY IS HIGH, COMPUTE USING GOSUB
IP ABS(1 - EXP(Cl*T(J,1))/EXP(C1*T(J,2))) < .008 THEN 5
’IP CONSTANT CONDUCTIVITY SKIP APTER PIRST PASS
IP K2 > 1 AND Cl-0 THEN 18
GOSUB 40
GOTO IS
8
POR 1-1 TO N
SG«S0*EXP(C1*T(J,I))
X-SG/W/EO/ER
RT«W/C*SQR(ER)*(1+X*2)“.25»SIN(-.5»ATN(X))
Tl#(I)-377*PN*EXP(2*RT*I*D)*SG
NEXT I
10
'COMPUTE ABSORBED POWER RATIO ON PIRST PASS
SG-S0«EXP(C1*T(J,1))
M-(l-»-SG‘2) ‘ .28
A-SQR(ER)*M»C0S(-.3*ATN(SG))
B-SQR(ER)*M*SIN(-.8 *ATN(SG))
DD«(A+l)“2+B“2
NR-A‘2+B‘2-1
NI-2»B
P"l - (NR“2+NI"2 )/DD‘2
15
T-T+DT
'INCREMENT TIME
IF K2/100 > INT((K 2+ .5 )/ 100 ) THEN 16
PRINT "TIME";T;"SEC"
16
H1-S*EL*T(J,1)‘3+H
T( J ,0) *T{ J ,2)“ (2*D/K)*(T(J,1)*H1 - TO*(H
S*EL*T0‘3))
T (J ,N ) * T (J ,N - 2 )-2 * T (J ,N - l )
POR 1-1 TO N-l
T(J1,I)-R*(T(J,I+1)+T(J,I-1))+(l-2*R)*T(J,I)+
Tl#( I)*DT/CP/RHO*P
NEXT I
IP K2/500 > INT((K2+.5)/SOO) THEN 20
POR 1-1 TO N-l
LPRINT USING "#####.## ##.##### ";T(J1,I),
S0*EXP{C1*T( J1,1)) .
PRINT #1. USING "#♦##♦.## ##.##### ";T (J1,I )i
♦ S0*EXP(C1*T(J1 ,1)),
NEXT I
LPRINT
20
J-Jl
J1-3-J
NEXT K2
' SG(T) CALCULATION
40 TD# - S0*EXP(C1*T(J,N))/W/EO/ER
N-NN
' SET FIELDS TO TRAVELING AT DEEP INTERIOR POINT
KR#(N ) - -H/C*SQR(ER)•(1+TD#“2)“.25*SIN(-.5*ATN(TD#))
KI#(N) - -W/C*SQR(ER)*U+TD#‘2) ‘ .25*COS(-.5*ATN(TD#))
KR#(0) - 0
KI#(0) - -H/C
EPR#(N ) - EXP(KR#(N)*N*D)*COS(KI#(N)*N*D)
EPI#(N) - -EXP(KR#(N)*N*D)*SIN(KI#(N)*N*D)
ENR#(N ) - 0
ENI#(N) - 0
FOR II - N-l TO 0 STEP -1
SG# ■ SO*EXP(Cl*T(J,11))
TD# - SG#/H/EO/ER
IF 11 « 0 THEN 45
KR#(11) - -W/C*SQR(ER)*(1+TD# ‘2)‘.25*SIN(-.5 •ATN(TD#))
KI#(11) - -H/C*SQR(ER)*(1+TD# ‘2)‘.25*COS(-.5 *ATN(TD#))
KR#(11)
Ql#
Q2#
KI#(11)
Q3#
KR#(I1+1)
KI#(I1+1)
Q4#
Q5#
EPR#(I1+ 1 )
EPI#(I1+1)
Q 6#
Q7#
ENR#(I1+1)
Q 8#
ENI#(11+1)
AR#
Ql#-Q3#
AI#
Q2#-Q4#
BR#
Ql#+Q3#
BI#
Q2#+Q4#
CR#
-Ql#-Q3#
Cl#
-Q2#-Q4#
DR#
-Ql#+Q3#
DI#
-Q2#+Q4#
HR#
EXP(AR#*D)*COS(AI#*D)
HI#
EXP(AR#*D)*SIN(AI#*D)
XR#
EXP(BR#*D)*COS(BI#*D)
XI#
EXP(BR#*D)*SIN(BI#*D)
YR#
EXP(CR#*D)*COS(CI#*D)
YI#
EXP(CR#*D)•SIN(CI#*D)
ZR#
EXP(DR#*D)*COS(DI#*D)
ZI#
EXP(DR#*D)•SIN(DI#*D)
MR#
Q5#«(BR#*HR# - BI#*HI#) - Q 6#*(BI#*HR# + BR#*HI#
MI#
Q5#*(BI#*HR# + BR#*HI#) + Q 6#*(BR#*HR# - BI#*WI#
NR#
Q7#*(AR#*XR# - AI#*XI#) - Q 8#*(AI#*XR# + AR#*XI#
NI#
Q7#*(AI#*XR# + AR#*XI#) + Q 8#«(AR#*XR# - AI#*XI#
QR#
Q5#*(DR#*YR# - DI#*YI#) - Q 8#*(DI#*YR# + DR#*YI#
Q5#*(DI#*YR# + DR#*YI#) + Q 6#*(DR#*YR# - DI#*YI#
Ql#
RR#
Q7#*(CR#*ZR# - CI#*ZI#) - Q 8#*(CI#*ZR# + CR#*ZI#
RI#
Q7#*(CI#*ZR# + CR#*ZI#) + Q 8#*(CR#*ZR# - CI#*ZI#
2*(Q1#“2 + Q2#‘2)
DD#
262
EPR#(II) - (Q1#*(MR#+NR#) + Q2#*(MI#+NI#))/DD#
EPI#( II) - (Ql#* (MI#+NI#) - Q2#* (MR#+NR#) )/DD#
ENR#(II) - (-Ql#*(QR#+RR#) ♦ Q2#*(QI#+RI#))/DD#
ENI#(I1) - (-Q1#*(QI#+RI#) - Q2#*{QR#*RR#))/DD#
Rl# - EPR#(II)
R2# - EPI#(II)
R3# - ENR#(II)
R4# - ENI#(II)
IP II - 0 THEN 50
PI# - EXP(-Q1#*I1*D)
P2# - EXP(Q1#*I1*D)
P3# - C0S(Q2#*I1*D)
P4# « SIN(Q2#*I1*D)
SR# - R1#*P1#*P3# + R2#*P1#*P4# ♦ R3#*P2#*P3# + R4#*P2#*P4#
SI# » R2#*P1#*P3# - R1#*P1#*P4# + R4#*P2#*P3# ♦
♦ R3#*P2#*P4#
Tl#(II) - 377*PN*(SR#“2 + SI#‘2)*SG#
NEXT II
50
P - 1 - (R3‘2 ♦ R4“2)/(Rl“2 + R2‘2)
'ABSORBED POWER
CON - SR‘2 ♦ SI‘2
POR II - 1 TO N-l
Tl#(II) - T1#(I1)/C0N*EXP(-2*KR#(1)*D)
'CHECK POR HIGH ATTENUATION
IP Tl#(Il)/Tl#(l)<lE-3 THEN NN-I1: GOTO 60
NEXT II
N-NH
RETURN
60
POR 12 - II TO N-l
Tl#(12) - 0
'RENORMALIZE POWER MATRIX
NEXT 12
N-NH
GOTO 55
APPKWDIX C
WQW-DIMKKSIOWAL ANALYSIS
He scale the heat and wave equations
dlaenslonal fora.
to a non-
This peralts analysis In terms of non-
dlaenslonal parameters.
The solution we present Is general
and can be applied to a variety of geometries.
We begin
with the heat equation:
K7,*T - pcpT, - Ao(T) |E(X) |*
(C-la)
and tcT„ + h(T - T0) ♦ •CJT 4 - T04) - 0
(C-lb)
where the left hand side of (C-la) Is a one-dimensional
Laplaclan.
The boundary condition (C-lb) Is expressed In
terms of a normal derivative.
This representation notation
Is used so that the scaling can be applied to rectangular,
cylindrical and spherical geometries.
He scale the length In terms of some convenient
physical value.
This might be the half-thickness of a
planar alab, the radius of a cylinder, or, In the case of
the semi-infinite solid, the wavelength in the material:
- f dx - JdX
(C-2a)
The time Is scaled accordingly,
263
264
( C-2b)
and the temperature, using (32) Is
T - T0
u - — s— 1
*o
(C-2c)
Substituting (C-2) Into (C-l) we have
7,‘u u„
♦
(u)
(C-3a)
T0’( (u+1)4 - 1) - 0
(C-3b)
He define the following constants
_ _ hL
c. " ir » cf ■
s(.L ,
o„Lz
k T0 , X ■
1E01
(C-4a,b,c)
The heat equation becomes
7 m*u - ut - -Xf(u)
(C-5a)
un ♦ c,u + Cjttu+l)4 - 1) - 0
(C-5b)
For the wave equation, we scale the length to the
wavelength In the material at t ■ 0.
geometry,
We have In a planar
265
“** * ®*<1 -
0
r
“ 0
(C-6a)
*x ♦
3YoE - 2 JY0
(C-6b)
and,
x ■ 6X , dx -6dX
(C-6c)
The non-dlmenslonal wave equation becoaes
E«, ♦ (1 - Jc3f(u))K- 0
♦ JB ■ 2J
wh.re c, - 5 ^ -
(C-7a)
(C-7b)
APPENDIX P
DAMPED HARMONIC OSCILLATOR
We show the difficulty that arises
of the perturbedharaonlc oscillator.
In alinearexpansion
We thenderive
an
approximate solution uolng the method of multiple
scales.[52]
The pertinent equation Is:
u’ + (u' ♦ u ■ 0
(D-la)
or u’ ♦ u ■
(D-lb)
-€u".
where € << 1 .
We look for an approximate solution using a Taylor series
expansion:
e
es
u - Z'Ut)*"
(D-2)
n*0
Taking the first two terms,
u 'v u0 + (u,
(D-3)
Substituting (D-3) Into (D-lb), we have
u0 - e'*
(D-4a)
and u^ ♦ ui ■ -is11
(D-4b)
266
267
The solution of (D-4b) Is
u, - - £te'*
(D-5)
Combining (D-5) with (D-4a), yields
u ~ e" - tte''
(D-6 )
For large values of t, u, > u0, asking the approximation
Invalid.
However, when t Is small, the solution looks like
.n
u 'v e “ e 2
(D-7)
So, we look for a solution In
We express u In terms of
u ■ v(t,u,€) , where
ti
this fora.
two paramters,
■ ft
(D-8 )
Substituting (D-8 ) Into (D-la) we have
v tt + 2£vtn ♦ €2vw + €vt ♦ e2v„ ♦ v - 0
Rewriting (D-9a), eliminating
have
(D-9a)
second orderterms In c, we
26 8
v,f ♦ v - -€ (2V,, ♦ v.)
(D-9b)
We now look for a solution using s ssrlss expansion In t
and T):
v ~ v0(t,T») ♦ €v,(t.D) ♦ ...
(D-10)
The solution of (D-9b) for the first two terns Is as
follows:
Vo.it ♦ v0 « 0 , v0 - Ao(Tl)e(,t)
v .,.» ♦ v. “ -M2A.* ♦ A0)e(,t) , v, - - j (2A0
(D-lla)
+ A0)te"
He define the constant tern of v, to be zero.
(D-llb)
This
yields
.u
Ao - a0e *
(D-12a)
. ii
and v0 - SgS 'e"
which Is equivalent to (D-7).
(D-12b)
APPENDIX 1
THERMAL EQOILIBRIOM OF SEHI-IltTIMTE SOLID
The following derivation deaonstrates the conaervatlon
of power at eteady atate —
•eal-lnflnlte solid.
theraal equilibrium, for the
The heat and wave equations as defined
In (31a-d) are used.
We fora the conjugate wave equation as follows:
E.’. ♦ V
'(1 ♦
- 0 , x > 0
" - 2 JV0 , x - 0 .
(E-la)
(E-lb)
We aultlply (31c) by E* and (E-la) by E and take their
difference, which yields
(E-2)
The right hand side of (E-2) is proportional to the right
hand side of (31a).
We Bake the substitution and Integrate
the equations over the length of the solid:
270
/ (*'I„ - BE/Jdx o
f
0
*< T) | K ( x ) | ’dx
(E-3)
o
T„dx - - O0nopo f f (T) I E(x) I
o
*dx
(E-4)
The left hand aide of (E-4) le Integrated and, from (31b)
and (33b) we have
T- L “ T-l0 “ " h(T “ To> "
f
f (T) | E(x) | *dx -
~ V)
[h(T - T0) ♦ s€JT« - T04)]
<E-3)
(E-6 )
O
Subetltutlng (E-6 ) Into (E-2) we
f
now have
(e'E„- EE^Jdx - J—p^[h(T - T0) ♦ 9 i j r - T/)]
(E-7)
0
He Integrate the left hand side of (E-7) and evaluate the
solution at the boundaries, using (31d), (33a) and (E-lb):
271
(e'E. -
e \ )|~
- 2J#m(E*K.)
(E-8 )
| K(0) | * - 2jy0E* , E( 0 ) - 1 ♦ p
S'K. - Jy0[2(l ♦ p') - (1 ♦ p)(l ♦ p’)] -
(E-9a)
jy0 (1- IpIM
(E-9b)
Substituting (E-9b) Into (E-7), we have
P0d
- IP I* ) - h(T - T0) ♦ s€JT 4 - T04).
This Is
a
statement of thermal equilibrium.
(E-10)
The Incident
power, less the reflected power Is equal to the heat lost at
the surface.
I
APPENDIX F
PROGRAM FOR COHPUTIIfQ TRANSIENT TEMPERATURE PROFILES IK A
PLAPAff SfcAP
1 PLANE WAVE HEATING OF PLANAR SLAB
DEFDBL A-Z
DIM T( 2,500),T1(500)
OPEN "0",#1,"SLBPRO.DATINPUT -INTERVAL SPACING-;D
INPUT "LENGTH";L
INPUT "TIME STEP";DT
N-INT(L/D+ .3)
SO-.002
'DEFINE CONSTANTS, SEE APPENDIX A
Cl-.0026
SG-SO
ER-10
PI-3.14159265
F-2.45E+9
W-2*PI*F
C-2.99E+8
EO-8.854E-12
K-20
CP-1000
RHO-3970
EL-.6
S-5.67E-8
R-DT/D”2*K/CP/RH0
H-10
T0-300
PN-10‘6
J-l
Jl-2
FOR 1-0 TO N
'SET SAMPLE TO AMBIENT TEMP
T(J,I) - TO : T(J1,I) - TO
NEXT I
FOR K2-1 TO 1000000
5
FOR 1-1 TO N
'ASSUME CONSTANT E-FIELDS
SG-S0*EXP(C1*T(J,I))
X-SG/W/EO/ER
Tl#(I)-377*PN*SG
NEXT I
M-(1+SG*2)'.23
'CALCULATE % ABSORBED POWER
A-SQR(ER)*M*COS(-.5•ATN(SG))
B—SQR(ER)*M*SIN(-.5*ATN(SG))
DD-(A+1)*2+B*2
NR-A" 2"fB“2-1
272
15
20
273
NI-2*B
P«1-(MR‘2+NI ‘2 )/DD “2
T-T+DT
•INCREMENT TIME
IP K2/100 > INT((K2+.5)/100) THEN 15
PRINT "TIME";T;"SEC"
H1"S*EL*T(J,1)*3+H
'TWO BOUNDARIES, HI, H2.
H2-S*EL*T(J,N-1)*3+H
T (J ,0) *»T(J ,2)-(2*D/K)•(T (J ,1)*H1-T0*(H ♦ S*BL*T0'3))
T(J,N)»T(J,N-2)-(2*D/K)*(T(J,N-l)*H2-T0*(H ♦
♦ S*EL*T0“3 ))
POR 1-1 TO N-l
T(J1,I)-R*(T(J,I+1)+T(J,I-1))+
♦ (1-2*R)*T(J,I)+T1#(I)*DT/CP/RHO*P
NEXT I
IP K2/500 > INT{(K2+.5)/500) THEN 20
LPRINT "TIME ";T;"SEC"
POR 1-1 TO N-l
LPRINT USING "#####.## ##.##### ";T(J1,I),
♦ SO*EXP(Cl*T(J1,I)),
PRINT #1, T(J 1 ,I )
NEXT I
LPRINT
J-Jl
J1-3-J
NEXT K2
tfp m i i i a
PROGRAM FOR COMPUTING STEADY STAT1 TEMPERATURE PROFILIS IN A
PLAWAR 9 L U JtfM 9 A lE H E R IV A b-lSU Q ST IlfQ — tfElEQD
In Chapter III, Section C.2.b.l, a "shooting" aethod is
described to determine the relationship between steady state
temperature and applied power.
The program listed below
uses the Runge-Kutta method to solve a system of linear
differential equations to determine the Incident power level
required to obtain a prescribed steady state temperature
profile.
The program generates a curve relating Incident
power to steady state temperature, and Interior temperature
profiles at each steady state temperature.
The program Is
written In Turbo-Basic and MKS units are used.
rem
rea SHOOTING METHOD PROGRAM USING RUNGE KUTTA
rem
dim k(4,4), y (500,4)
rem DEFINE CONSTANTS
'error margin for convergence
zerr* .00001
'convection constant
hl- 10
'thickness of the slab
11-4.00000IE-03
'thermal conductivity
k- 2 0
'Stefan-Boltzmann Constant
s-5.67E-08
'emlsslvity
el- . 6
'ambient temperature
tO—300
'non-dimensional constants
cl-hl*ll/k
c2-e*el*ll/k*t0‘3
'elec. cond. constant
kk - .78
'currantly exp(kx)
'grid spacing for calculation
h - .02
'number of linear diff eqs.
n2-4
'number of points
nl-99
'file for power v. temp data
open "o",#1 ,"shootp.dat"
274
275
l-cl*.l
'first guess for norm pwr, 1 .
rem
rs» M i n loop, cslculsts power (1) for eech new temp(a)
res
for a - .1 to 10 step .1
'loop with norm, temp, a.
y(0 ,l)-a
<s«t initial values
y(0,2 )b c 1*s +c 2*((a+1)~4 - l)
y(0,3)-0
y(0,4)-0
Sub:
GOSUB Runge
d b y(nl,2)+cl*y(nl,l)+c2*((y(nl,l)+l)‘4 “ 1 )
17 ABS(d)<zsrr THEN 640
'test for convergence
dp-y(nl ,4)+cl*y(nl,3)+4»c2*(Y(nl,1)+l)*3*y(nl,3)
lBl-d/dp
'use Newton's Method to comp
GOTO Sub
'next point
print #l,L,a
'store power, temp data
b$ b str$(int(a*10+.05))
a$ b "shot"+ald$(b$,2 ,len(b9 )-1 )+".dat"
open "o",#2 ,a9
for 1b 0 TO nl
'store temp profile
print # 2 ,y(1 ,1 ),i
next 1
close #2
next a
STOP
Runge:
'first point
xbo
for Jb Q
yi
y2
y3
y4
i-i
GOSUB
yi
y2
ys
y*
i-2
GOSUB
yi
y2
ys
y*
1-3
GOSUB
yi
y2
to nl
y(J.i)
y( J. 2 )
y( J* 3)
y(J.*)
oopl
yl ♦
y2 ♦
y3 +
y4 ♦
.5*k(1,1)
.5*k(l,2)
.5*k( 1,3)
.5*k(1 ,4)
oopl
yl ♦
y2 ♦
y3 ♦
y4 ♦
.5*k( 2,1)
.5*k(2 ,2 )
.5*k(2,3)
.5*k(2 ,4)
oopl
yl ♦ k{3,l)
y 2 ♦ k(3,2 )
276
y3 » y3 '♦ k(3,3)
y4 - y4 ♦ k(3,4)
1-4
OOSUB loopl
for 1 - 1 to 4
Y(J*1.1) - y(d.i) ♦ (k(l,i)+2*k(2,i)+2*k(3,i) +k (4,1) )/6
next 1
x-x-fh
'next point
next j
return
loopl:
k(l,l)
k(l, 2 )
k(1,3)
k(l,4)
return
-
h*y 2
'define functions here
-h*l*exp(kk*yl)
h*y4
-h* (l*kk*exp(kk*yl)*y3 ♦ exp(kk*yl))
ATfBTPIX B
H0D13 IW A PIELECTRICALLY LOADED, CIRCULAR CAVITY
Presented below Is s derivation of the fields and nodes
within a dielectrically loaded cylindrical cavity.
The
cylindrical dielectric Is hoaogeneous, lossless, and
centsred In the cavity.
The cavity geometry Is shown In
Pig. 65.
He will describe the fields as a summation of trial
wave functions for TE and TM waves. [31]
For TM waves,
(H-la)
(H-lb)
(H-2a,b)
(H-2c,d)
H
- 0
and for TE waves,
277
(H-2e,f)
278
ZZ
to
Hour. »T.
a.o«trv Ot • U P '1 lB T,‘" c«wity-
279
E
- 0
H. ■ 3 uii Laz* + *'J *
(H-«e,f)
where S(x) Is e Bessel function.
The seperetlon relations are:
kj, ♦k* - w*€,U,
(H-5a)
kj, ♦k* - ^*€,11,
(H-5b)
In the present cavity, k, ■
q*ir*
so this becomes,
«.»*
k;, ♦-^r ’ ;* ‘r
(»-«•)
where q la an Integer representing the axial order
resonance.
I
H .. I
H*. I
We now define the boundary conditions:
- E•2
I,...
A
I ....
T? B
* H- 1 ....
* *.*
^
D
BJc
♦ <3D^s
281
The following amplifications prove useful:
(H-8 a,b)
(H-8 c,d)
F, - K* <*,•>
The solution can be represented In aatrlx fora where the
eigenvalues of the solution can be found by setting the
determinant to zero.
k»
2!■
¥7P*
k n
I
k n
u)U7ip«
282
the determinant Is:
[K
€,»,»;]
2
[ K, u.r.F, - k,, u,FzF, ]
k*n2 r k5. - k?. 1 *
- A ?
p ' p< p> p< - 0
For n ■ 0 this separates Into TE and TM modes:
TE:
[ k,, u,F,F, - „ U,F,F,]
- 0
TM:
[ k,, e.FlF, - k,, £,F,F, ] - 0
(H-lOa)
(H-lOb)
The fields In the TM,,,*, cavity are easily discovered.
We look for a solution of the form
3B„(kp) - a0 J„(kp) ♦ Y„ (kp)
(H-ll)
Fields must be finite at the origin, so we require that Yn ■
0 In region I.
Furthermore, since the tangential electric
field Is zero at the cavity walls, we can write Ft and F,.
F, - Jn(k,,a) , F, - -
* YJKi*)
(H-12a,b)
283
Setting n ■ 0, and using (H-l), (H-2) and (H-lOb), the
fields given In (131) are obtained.
To determine the fields In the "TE,,," cavity, we must
use (H-9).
The solution Is a hybrid aode, for n > 0.
Again
we assume fields of the form (H-ll), and require the same
boundary restraints.
F »
"
F z
The constants 7,, 7,, 7, and 7, are:
■
(H-13a,b)
Y„ (k.,b)
" - jt ixlb)
(H-13C)
Using (H-3), (H-4), and (H-7) along with (H-13), we can
describe the fields within the sample:
■ A Jn{k,,p) cos ne cos
(H-l4a)
(H-14b)
284
w
-XdV *
u>€, dpdz
.
1
P d#
■ t * ¥ i^ 7 ..J . t k . , * )
_
E**
"
E
_
^
i
[
- T SJ .(* '.,P ) ]
« •
»• • ! "
d V ‘. a**'
~5*5i♦
-fr
~a¥ J n (k,,P) ♦ B k^jjl^.p) ] aln n* sin SJ5 (H-lSb)
- -4- I -^L ♦ k* I 0;*'
“T
■
Laz*
J
" i^7
Jn(k»iP) coa n* coa ^
***• " £ l£~ ♦ dr; 1 ^ -
H*> ■ ■ ^
y&z
H.* “
He now aolve for conatanta.
(H-15c)
(H-i5d)
(H-15e)
(H-15f)
Fro* (H-7a) and (H-7b), we have
285
Substituting Into (H-16a,b) Into (H-7d) we obtain:
B * * ? iS^S [
] F.p. /
[
Equation (H-15) can be simplified —
E.. " A, [ kM J„-, - n(Bi/
r,r, - k.. r,r: ]
<h-i7)
(H-15a-c) become
2) J„ ] cos n* »ln *Sr
(H-lBa)
r
n(B. -fl)
1
OTTZ
E*i ■ Ai [ Ki*x Jn-.-----v --- Jn ] *in nb sin 3^-
(H-18b)
E .» ■ A . “ $ r k .*i J r> c o s
( H- 1 8 C )
n* cos
wh.r. A, . * 3*
Jinrl
n
B‘
m
n
r
«
r—
—
1
J
F lE «
[ k " r;»,
The lowest order mode, HE
F.p;)
is obtained by setting n
286
■ 1. The electric fields Inside the saeple ere listed In
(136).
MTHfPIX-1
PROGRAM TO STUDY OAF EFFECTS 03IWO TOT METHOD Of
TRAWSVRRS1 RESONANCE
REM
REM TRANSVERSE RESONANCE METHOD FOR GAP EFFECT
3 REM
4 REM SET CONSTANTS
10 LET PI-3.1415926538#
20 LET C0-3E+10
25 LET BO-8.854E-14
DIMENSIONS OF COAX, SAMPLE
30 LET A-.152
40 LET C-.35
45 LET B-.345
46 INPUT "DIELECTRIC CONSTANT AND CONDUCTIVITY OF INNER
+ REGION "'El SI
47 LET E2-l! S2-0
'AIR GAP IN OUTER REGION
50 LET F-2
'FREQUENCY OF OPERATION IN GHZ
51 PRINT F
60 LET W«2*PI»F*lE+09
70 LET T1-S1/W/E0/E1: T2-S2/W/EO/E2
72 REM
74 REM COMPUTE CAPACITANCE MODEL SOLUTION FOR COMPARISON
75 REM
76 LET L1-L0G(C/B): L2-L0G(B/A): L3-L0G(C/A)
77 LET EM-(E2*E1‘2*L1»L3*(1 + Tl*2) ♦ E2‘2*B1*L2*L3*(1 +
♦ T2‘2) )/( (L1*E1 ♦ L2*E2)*2 ♦ (L1*B1*T1 ♦ L2*E2*T2)‘2)
78 LET TM-(E1*L1*T2*(1 + T1‘2) ♦ E2*L2*T1*(1 ♦
+ T2 “2))/(El*Ll*(l ♦ T1‘2) ♦ E2»L2*(1 ♦ T2‘2))
79 LET SM«TM*EM*EO*W
00 LET A4-W*SM/E0/C0‘2
85 LET B4-W‘2/C0‘2*EM
86 LET ZR-SQR(.5*B4+.5*SQR(B4*2+A4“2 ))
87 LET ZI-A4/2/ZR
90 PRINT "EM ";EM;" SM ";SM;" ZR ";ZR;" ZI ";ZI
95 INPUT G1,G2
96 LET ZR-G1: LET ZI-G2
100 LET A1#-W»S1/E0/C0‘2-2*ZR*ZI
110 LET A2#-W*S2/EO/CO‘2-2*ZR»ZI
120 LET B1#-W‘2»E1/C0‘2+ZI‘2-ZR*2
130 LET B2#-W“2*E2/C0* 2+Z1‘2-ZR“2
140 LET VI#— .5*B1#+.5*SQR(A1#*A1#+B1#*B1#)
145 LET V2#-Al#*Al#/4/Vl#
150 LET Wl#— .5*B2#+.5*SQR(A2#»A2#+B2#*B2#)
155 LET W2#-A2#*A2#/4/Wl#
160 LET J3-SQR(V2#)
1
2
288
165
170
175
180
185
199
200
205
210
220
230
231
240
250
251
260
270
280
281
290
300
301
310
320
330
331
340
350
351
360
365
370
380
385
390
392
400
410
411
420
430
435
440
441
450
460
461
470
480
485
LET J4-SQR(V1#)
LET K3— SQR(W2#)
LET K4«SQR(W1#)
PRINT J3"2“J4"2-B1#,K3“2-K4“2-B2#
PRINT 2*J3*J4-A1#,2*K3*K4-A2#
REM COMPUTE COMPLEX BESSEL FUNCTIONS
LET N*0
IF J3-0 THEN LET TH«PI/2*SGN(J4): GOTO 220
LET TH-ATN(J4/J3)
LET R-SQR(J3‘2+J4‘2)*A
GOSUB 1000
REM JO(KCIA)
LET Ql-SR#: Q2-SI#
GOSUB 1500
REM YO(KCIA)
LET Q3-SR#: Q4-SI#
LET R-R*B/A
GOSUB 1000
REM JO(KCIB)
LET Q5-SR#: Q 6 -SI#
GOSUB 1500
REM YO(KCIB)
LET Q7-SR#: Q 8 -SI#
LET N-l
GOSUB 1000
REM Jl(KClB)
LET Ll-SR#: L2-SI#
GOSUB 1500
REM Yl(KClB)
LET L3-SR#: L4-SI#
IF K3-0 THEN LET TH-PI/2*SGN(K4): GOTO 380
LET TH-ATN(K4/K3)
LET RaSQR(K3“2+K4‘2)*B
IF T2>1 THEN 410
GOSUB 1000
REM J1(KC2B)
LET L5-SR#: L 6 -SI#
GOSUB 1500
REM Y1(KC2B)
LET L7-SR#: L8 -SI#
LET N-0
IF T2>1 THEN 460
GOSUB 1000
REM JO(KC2B)
LET Rl-SR#: R2-SI#
GOSUB 1500
REM YO(KC2B)
LET R3-SR#: R4-SI#
LET R»R*C/B
IF T2<1 THEN 490
289
486
487
490
491
500
510
511
520
530
531
540
541
550
560
561
570
571
580
590
600
601
610
611
620
630
631
640
641
650
660
670
680
690
700
710
720
730
740
750
760
770
780
790
800
805
810
820
825
LET R5-1: R6-0: R7-0: R8-0
GOTO S30
GOSUB 1000
REM JO(KC2C)
LET R5-SR#: R 6 -SI#
GOSUB 1500
REM YO(KC2C)
LET R7-SR#: R 8 -SI#
REM COMPUTE CYLINDRICAL COTANGENTS
REM J1(KC1B)*Y0(KC1A)
LET DR-L1*Q3-L2*Q4: DI-LI*Q4+L2*Q3
REM Y1(KC1B)*J0(KC1A)
LET ER-L3*Q1-L4*Q2: EI-L3*Q2+L4*Q1
LET MR-DR-ER: MI-DI-EI
REM JO(KC2B)YO(KC2C)
LET DR-R1*R7-R2*R8: DI-R1*R8+R2*R7
REM YO(KC2B)*JO(KC2C)
LET ER-R3*R5-R4*R6: EI-R3*R6+R4*R5
LET NR-DR-ER: NI-DI-EI
LET N1-NR*MR-NI*MI : N2-NR*MI+NI *MR
REM JO(KCIB)*Y0(KC1A)
LET DR-Q5*Q3-Q6*Q4: DI«Q5*Q4+Q6*Q3
REM YO(KCIB)*J0(KC1A)
LET ER-Q7*Q1-Q8*Q2: EI«Q7*Q2+Q8*Q1
LET MR-DR-ER: MI-DI-EI
REM J1(KC2B)*Y0(KC2C)
LET DR-L5*R7-L6*R8: DI-L5*R8+L6*R7
REM Y1(KC2B)*J0(KC2C)
LET ER-L7*R5-L8*R6: EI-L7»R6+L8»R5
LET NR-DR-ER: NI-DI-EI
LET D1»NR*MR-NI*MI: D2-NR*MI+NI*MR
LET D-Dl“2+D2*2
LET HR— (N1*D1+N2*D2)/D: HI-(N2*D1-N1*D2)/D
REM COMPUTE COMPLEX PROP CONSTANT
LET D-E2"2+(S2/W/E0)“2
LET NR- (El *E2-fSl *S2 / (W*EO)“2 )/D : NI-(El *S2-E2*S1)/D/W/EO
LET N1-NR*HR-NI*HI: N2-NR*HI+NI»HR
LET AR-N1*2-N2“2: AI-2*N1*N2
LET D-{1-AR)*2+AI“2
LET ER-E2*AR+S2*AI/W/E0: EI-E2»AI-S2*AR/W/E0
LET MR-E1-ER: MI-S1/W/EO-EI
LET PR-(MR*{1-AR)-MI*AI)/D:PI-{MR*AI+MI*(1-AR))/D
LET A3-W“2*PI/C0*2
LET B3-W“2*PR/C0“2
LET XI— .5*B3-»-.5*SQR(A3*24-B3‘2)
LET X2-A3‘2/4/Xl
LET KR-SQR(X2)
LET KI-SQR(Xl)
830 PRINT "REAL GUESS ";ZR;M REAL CALC";KR
840 PRINT "IMAG GUESS ";ZI;" IMAG CALC";KI
850 LET EN«CO‘2/W‘2*(KR‘2-KI‘2)
860 LET SN«CO“2/W*EO*2*KR*KI
870 PRINT "EN ";EN;" SN " ;SN
875 PRINT "K3 ";K3;" K4 ";K4;" J3 ";J3;" J4 ";J4
880 PRINT "B1 ";B1#;" B2 ";B2#;" B3 ";B3
890 PRINT "T1 ";T1;" T2 ”;T2
920 GOTO 90
990 END
1000 REMCOMPUTE COMPLEX BESSEL PUNCTION OF 1ST KIND
1010 REMN-ORDER OF J
1020 REM R-MAG OF ARG, TH-PHASE OF ARG
1025 REM RETURNS SR# AND SI#
1030 LET SR#-0
1035 LET SI#-0
1040 FOR M-0 TO 1000
1041 GOSUB 1050
1042 GOTO 1165
1050 REM COMPUTE FACTORIALS
1060 LET Fl#-0
1080 IF M-0 THEN 1120
1090 FOR U-M TO 1 STEP -1
1100 LET F1#-F1#+L0G(U )
1110 NEXT U
1120 LET F2#-F1#
1130 IF N-0 THEN 1161
1135 LET F2#-0
1140 FOR U-M+N TO 1 STEP -1
1150 LET F2#-P2#+L0G(U )
1160 NEXT U
1161 RETURN
1165 LET S#—SR#
1166 LET T#-SI#
1169 LET Pl#-(-l)‘M*EXP(M*L0G(R/2)-Fl#)
1170 LET P2#-P1#*EXP(M*L0G(R/2)-F2#): P#- P2#*(R/2)‘N
1171 LET SR#-SR#+P#*C0S((2*M+N)*TH)
1172 LET SI#-SI#+P#*SIN((2*M+N)*TH)
1175 IF ABS(S#-SR#)<.0001 AND ABS{T#-SI#)<.0001 AND M>0
+
THEN 1200
1190 NEXT M
1200 RETURN
1500 REM COMPUTE COMPLEX BESSEL FUNCTION OF 2ND KIND
1510 REM N-ORDER OF Y
1520 REM R-MAG OF ARG, TH-PHASE OF ARG
1525 REM RETURN SR# AND SI#
1530 GOSUB 1000
1535 LET JR#-SR#
291
1537 LET JI#-SI#
1540 LET GAMMA -.577215665#
1545 LET P-LOQ(R/2)+GAMMA
1547 LET YR#—2/PI*(P*JR#-TH*JI#)
1548 LET YI#-2/PI* (P*JIF+TH*JR#)
1550 LET TR#-0
1551 LET TI#-0
1555 IP N-0 THEN 1800
1560 FOR M-0 TO N-l
1570 LET Fl#-1
1580 LET F2#-l
1585 IF M-0 THEN 1620
1590 FOR U-M TO 1 STEP -1
1600 LET F1#-F1#*U
1610 NEXT U
1620 IF N-M-1-0 THEN 1660
1630 FOR U-N-M-l TO 1 STEP -1
1640 LET F2#-F2#*U
1650 NEXT U
1660 LET P#—F2#/F1#*(R/2)"(2*M-N)
1661 LET TR#-TR#+P#*COS ((2 *M-N)*TH)
1662 LET TI#-TI#+P#*SIN((2*M-N)*TH)
1665 NEXT M
1666 LET SR#-0
1667 LET SI#-0
1670 FOR M-0 TO 1000
1660 LET Kl#-0
1690 LET K2#-0
1700 FOR K-l TO M+N
1701 LET K3#-l/K
1705 IF K>M THEN 1720
1710 LET K1#-K1#+K3#
1720 LET K2#-K2#+K3#
1730 NEXT X
1740 GOSUB 1050
1741 LET F1#-EXP(F1#)
1742 LET F2#-EXP(F2#)
1745 LET S#-SR#
1746 LET T#—SI#
1749 LET Pl#-(-l)*M*(Kl#+K2#)*(R/2)‘M/Fl#
1750 LET P2#-P1#*(R/2)'M/F2#: P#-P2#*(R/2)“N
1751 LET SR#»SR#+P#*COS((2*M+N)*TH)
1752 LET SI#-SI#+P#*SIN((2*M+N)*TH)
1760 IF ABS(SR#-S#)<.0001 AND ABS(SI#-T#)<.0001 THEN 1780
1770 NEXT M
1780 LET SR#-YR#-TR#/PI-SR#/PI
1785 LET SI#-YI#-TI#/PI-SI#/PI
1790 RETURN
292
1800
1801
1809
1810
1815
1820
1830
1840
1850
1860
1870
1880
1881
1890
1891
1895
1896
1900
1920
1940
1945
1990
LET SR#«0: SI8-0
FOR M«1 TO 1000
LET Kl#-0
FOR X-l TO M
LET K3#-l/K
LET X1#-E1#+K3#
NEXT K
LET Fl#-1
FOR U-M TO 1 STEP -1
LET F1#-F1#*U
NEXT U
LET S#-SR#
LET T#-SI#
LET Pl#-(-l)‘(M+l)»Kl#«(R/2)*M/Fl#
LET P#-Pl#*(R/2)*M/F1#
LET SR#-SR#4P#*C0S(2»M*TH)
LET SI#-SI#+P#«SIN(2*M*TH)
IF ABS(S#-SR#)<.0001 AND ABS(T#-SI#)<.0001 THEN 1940
NEXT M
LET SR#-2/PI*SR#+YR#
LET SI#-2/PI*SI#+YI#
RETURN
AffBTPIE J
8AMPL1 RKS0WAWC1 Ilf COAXIAL LIHI
The quality factor, or Q of a resonating saaple can be
determined from the measured reflection coefficient and la a
meana of determining the conductivityof samples.
be shown ualng(148).
We note
This can
thatfor lowconductivities:
~ jr ( i -
,
J
'
1 )
At the resonant frequency, w r, from (157),
~
j*
(> -
<J-*>
The reflection coefficient Is a minimum at the resonant
frequency.
When the reflected power Is doubled, at
frequencies w r + Aw and w r + A w :
yd ~ j, (i ,
* it?) ' )
[l -
( J - 3 )
Substituting into (148),
293
294
p(wf t Aw)
(£' *
( W ,
4
+ *&[*■ ■ J2 w r°£0(r)
| p(wr ± Aw) | *
- I > - »>• + O f r ) 1] [ m [ ; I M ' ]
18«. ♦ 8 JTl«, ♦ 1 ) :rfer '- o 'r
We know that
2 | p(wr) | * ■
| p(wr t Aw) | 2
(J-6)
and thus
W
W £«£
- -4 ^
Which la by definition [74] the unloaded Q.
(J-7)
AEPIWPIX K
1LICTRQPLATIWQ PROCESS
To fora an electroplated layer of copper on the non­
conducting samples used In aaterlal characterization we
follow a four step process outlined In [8 8 ].
Plrst the
surface Is "roughened" using an etchant solution; next a
"sensitizing" solution Is used to prepare the sample surface
to receive a metallic film:; a thin film of silver Is then
deposited; and finally copper Is electroplated onto the
sample using the thin silver film as a conducting surface.
After the electroplating process Is complete, the sample Is
sanded to remove metallization from the flat surfaces and to
fit the sample holder.
The chemical solution used to etch the sample Is
100 ml Sulfuric acid
15 g Potassium dlchromate
50 ml Hater
It Is recommended that the sample be Immersed In this
solution for 2 minutes.
solutions are available.
For delicate samples other etching
It Is Important to keep the sample
free of grease and other contaminants.
To prepare the sample to receive the metallizing film,
a sensitizing solution Is needed In order to Improve surface
adhesion, producing a more even deposition.
295
296
1 g Stannous chloride
4 al Hydrochloric acid
100 al Water
After lnerslng the saaple for 1 to 2 alnutes In this
aqueous solution, with agitation, the saaple should be
rinsed In distilled water.
A silver flla can now be deposited on tbe saaple.
The
procedure Is to prepare two solutions a "slivering" solution
and a "reducing solution."
When these two solutions are
alxed sliver will preclpate onto the saaple foralng a thin
aetalllc layer.
This Is the "alrror” process.
Silvering Solution
6 g
Silver nitrate
100 al Water
6 al Aaaonlua hydroxide (28k)
Reducing Solution
6.5 al Poraaldehyde (40%)
100 al Water
When the reaction is coaplete, an oha aeter can be used to
verify the presence of a conductive flla.
The saaple should
be dryed overnight.
Copper aay now be electroplated on the saaple.
It Is
recoaaended that a "flashing" bath, containing weak acid, be
used to fora a thin copper layer to protect the silver flla
froa daaage.
The saaple can then be laaersed in the
“plating" bath to build up to the required thickness.
Two
297
electrodes arc Immersed Into tha solution, one connected to
the saaple, and one a sheet of pure copper.
The electrodes
are connected to a battery capable of producing about 1 aap.
Plashing bath (pH - 3)
40
Plating bath
200
15
- 100 g/1 Copper sulfate
2.5 g/1 Free sulfuric acid
(pH ■ 1)
-300 g/1 Copper sulfate
- 40 g/1 Pree sulfuric acid
We add a small quantity of gun arable to the solution to
Inhibit the formation of dendrites and to aid the formation
of a smooth copper layer.
Techniques for depositing nickel and other metals onto
non-conducting surfaces are also given In [81].
h i a
Name:
David Qray Hattara
Date of Birth:
Place of Birth:
November 25, 1960
Bvanaton, IL
Educational Background
Master of Science In Electrical Engineering
Northwestern University
Evanston, IL 60208
Project:
"An Automatic Scalar Network Analyzer for
Microwave Materials Characterization"
Graduated: June 1985
Bachelor of Science in Electrical Engineering
Northwestern University
Evanston, IL 60208
Graduated: June 1983
Publications
1.
D.G. Watters and M.E. Brodwln, "Automatic Material
Characterization at Microwave Frequencies," IEEE
Instrumentation and Measurement Technology Confsrence,
April 27 -29, 1987, Boston, MA, p 247.
2.
D.G. Watters, M.E. Brodwln and G.A. Krlegamann, "Dynamic
Tsmeperature Profiles for a Uniformly Illuminated Planar
Surface," Material Research Society 1988 Spring Meeting.
Reno, NV, April 4-8, 1988, M2.2.
3.
D.G. Watters and M.E. Brodwln, "Automatic Material
Characterization at Microwave Frequencies," IEEE Trans
on Instrumentation and Measurement, IM-37, June 1988, p
280-285.
4.
D.G. Watters, M.E. Brodwln and G.A. Krlegsmann, "Dynamic
Temperature Profiles for a Uniformly Illuminated Planar
Surface," 23rd International Microwave Power Symposium.
Ottawa, Canada, August 29-31, 1988, p 45-47.
5.
G.A. Krlegsmann, M.E. Brodwln and D.G. Watters,
"Microwave Heating of a Ceramic Half-Space," Tech Rept
8729. Dept of Engr Science and Appl Math, Northwestern
University, Evanston, IL, Aug 1988.
298
3 5556 019 165 778
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