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Open-ended sensors for microwave nondestructive evaluation of layered composite media

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Order N um ber 9417610
Open-ended sensors for microwave nondestructive evaluation of
layered composite media
Bakhtiari, Sasan, Ph.D.
Colorado State University, 1992
UMI
3 00 N . Z eeb R d.
Ann Arbor, MI 48106
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DISSERTATION
OPEN-ENDED SENSORS FOR MICROWAVE NONDESTRUCTIVE
EVALUATION OF LAYERED COMPOSITE MEDIA
Submitted by
Sasan Bakhtiari
Department of Electrical Engineering
In partial fulfillment of the requirements
for the degree of Doctor of Philosophy
Colorado State University
Fort Collins, Colorado
Fall 1992
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COLORADO STATE UNIVERSITY
November 5, 1992
WE HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER
OUR SUPERVISION BY SASAN BAKHTIARI ENTITLED OPEN-ENDED SENSORS
FOR MICROWAVE NONDESTRUCTIVE EVALUATION OF LAYERED COMPOSITE
MEDIA BE ACCEPTED AS FULFILLING IN PART REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY.
Committee on Graduate Work
\V\ - XI, .
Adviser
Department Head
ii
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ABSTRACT OF DISSERTATION
OPEN-ENDED SENSORS FOR MICROWAVE NONDESTRUCTIVE
EVALUATION OF LAYERED COMPOSITE MEDIA
The need to monitor the composition, integrity, and maintenance of materials,
structures and equipments gave rise to the development of the field of NonDestructive
Testing and Examination (NDT &E). With the prevalent use of metals due to their
strength, early NDT techniques were established to deal mostly with metallic media. The
advent of new lightweight and exceptionally strong dielectric composite materials, replacing
metals in many applications, has diversified the field of NDT & E tremendously. The low
conductivity associated with these dielectric materials such as plastics, ceramics and carbon
fiber composites has rendered many traditional and conventional NDT techniques
ineffective. The ability of microwaves to penetrate inside dielectric media and their
sensitivity to minute structural and dimensional variations within the medium, coupled with
availability of large bandwidths are of great significance. Microwave techniques have
reemerged recently not only to fill many gaps left vacant by traditional NDT & E
techniques, but also to further expose new alleys in this expanding and demanding field.
Near-field interaction of electromagnetic waves with different material media is a
complex process. Without a thorough understanding of the nature of reflection, attenuation
and scattering mechanisms involved, even the most careful measurements may render
themselves useless. The objective of this work is to take a forward step toward filling
some of the existing gaps for concise theoretical formulations and practical implementation
in application to microwave NDT & E of layered dielectric composite media. Models
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pertaining to analysis of electromagnetic radiation into layered, generally lossy dielectric
composite media from two prevalent sensors, namely, an open-ended rectangular
waveguide and an open-ended coaxial line are introduced. Variational formulation is
evoked in conjunction with Fourier transform boundary matching technique for calculation
of the terminating aperture admittance of these two open-ended sensors radiating into
stratified dielectric media. The integrity of the numerical solutions are examined both
theoretically and experimentally.
The applications addressed here pertain to measurement and monitoring of
thicknesses variations, dielectric properties and disbonds/delaminations in layered dielectric
composite media. Results of a versatile technique are presented for measurement of
thickness and dielectric properties of materials (e.g. coatings) on top of conducting plates
using an open-ended rectangular sensor. The accuracies achieved are far better than what is
normally associated to microwave measurements at relatively low frequencies. Swept
frequency measurement outcomes which are also in line with the theoretical observations
indicate that optimization of measurement parameters may allow for accurate detection and
evaluation of thickness variations of dielectric slabs in the range of few microns and better
than 5% and 10% accuracy in estimation of permittivity and loss factor. Both the numerical
and experimental results in application to detection and evaluation of disbonds and
thickness variations within layered dielectric composite media clearly indicate the ability of
microwave techniques to resolve minute dimensional and dielectric property variations.
Results of theoretical expositions have been presented in application to examination
of layered dielectric composite media using an open-ended coaxial transmission line sensor.
Numerical result of the formulation presented here which are also in good agreement with
available general numerical models indicate the capability of this sensor to resolve detailed
dimensional variations within thin layered dielectric composite media. It is puiposed that
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incorporation of finite thickness of material media into the formulation may allow for
utilization of a more versatile calibration method. Detailed measurements using an openenede coaxial sensors are purposed for future endeavors.
Sasan Bakhtiari
Department of Electrical Engineering
Colorado State University
Fort Collins, Colorado 80523
Fall 1992
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ACKNOWLEDGMENTS
I would like to offer my sincere appreciation to those who supported me during this
work. Foremost, I am particularly grateful for the guidance and encouragement given by
my advisor Dr. R. Zoughi. I would also like to express my gratitude to Dr. D. Lile, Dr.
M. R. Azimi, and Dr. T. VonderHaar for serving on my committee. I wish to thank in
particular my colleagues Dr. S. Ganchev and Mr. N. Qaddoumi for their assistance
throughout this endeavor.
Finally, I would like to thank my family for their support and guidance in every
aspect of my life.
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TABLE OF CONTENTS
C H APTER I
I n t r o d u c t i o n .............................................................................................................. 1
C H APTER I I ......................................................................................................................4
A Survey o f Prom inent ND E Techniques........................................................... 4
11.1 Conventional NDE Methods............................................................................5
11.2
M icrowave NDE M ethods..................................................................... 10
11.3 Reference Table o f Prevalent NDT&E Methods..............................................14
11.4 Scope and Motivation o f This W ork................................................................17
11.5 Summary and Remarks.................................................................................... 18
C H A P TE R I I I
Theoretical Analysis o f Radiation From Rectangular W aveguide
In to
L a yered C om posite M edia........................................................................ 19
III.I Variational Formulation o f Aperture Admittance.........................................20
I I 1.2 Conductor Backed Lossy Dielectric Slab...................................................... 27
111.2.1 Theoretical Formulation.................................................................. 27
111.2.2
Verification o f The Theoretical Results.................................. 33
111.3 Radiation From Rectangular Waveguide into Stratified Composite
M e d ia ..............................................................................................................38
III.3.1 Theoretical Formulation.................................................................. 38
HI.3.2 Termination of Layered Media into a Loss-Less Half-Space
46
111.4 Summary and Remarks...................................................................................49
C H A P T E R IV
R ectangular Waveguide Sensor fo r Exam ination o f Layered
D ielectric Composite M ed ia .................................................................................. 50
IV .1
Conductor-Backed Lossy Dielectric Slab.............................................50
IV .1.1 Experimental Results vs. Theoretical Observations................. 51
IV2 Practical Implications..................................................................................... 56
IV.2.1 Estimation o f Dielectric Properties and Thickness........................ 56
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IV.2.2
Estimation o f Coating Thickness...........................................60
TV.2.3 Frequency Analysis For Sensitivity Maximization...................61
IV2.3.1 Measurement Apparatus................................................. 62
TV.23.2 Verification and Analysis o f the Results...................... <52
IV.3
Disbonds and Thickness Variations in Layered composite media.............69
IV3.1 Layered Media Terminated into an Infinite Half-Space............. 71
IV.3.2 Disbond in a Layered Media Terminated into a Conducting
Sheet.................................................................................................78
Summaty and Remarks................................................................................ 82
IV.4
C H A P TE R V
Coaxial Transm ission Line Sensor fo r Exam ination o f Layered
C o m p o site
M e d ia ................................................................................................. 84
V.l Analysis o f Radiation From Coaxial Transmission Line into Stratified
Composite M edia............................................................................................ 85
V.1.1 Theoretical Formulation................................................................... 86
V.l.2 Explicit Solution fo r a Two-Layer M edia........................................93
V.2 Verification o f the Numerical Results and Some Observations................... 96
V 3 Practical Implications..................................................................................... 105
V.4 Summaty and Remarks.................................................................................. 107
C H A P TE R V I
C onclusions and R e m a r k s .................................................................................... 109
B I B L I O G R A P H Y .......................................................................................................113
A P P E N D IX A
E x p lic it solution f o r Three-Layer M ed ia .................................................... 119
A P P E N D IX B
A nalysis o f Singularities ....................................................................................... 123
A p p en d ix C
C o m p u te r
C o d e s....................................................................................................125
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LIST OF TABLES
2.1
Comparison of several widely noted NDT&E techniques........................................ 1
3.1
Comparison of the results of Equation (3.42) with those using Lewin's
Infinite half-Space [Lew.51].......................................................................................33
4.1
Thickness measurement result at 10 GHz for e'r = 7.25 and tan 8 = 0.103.........60
4.2
Thickness measurement result at 10 GHz for e'r = 12.6 and tan8 = 0.19.............60
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LIST OF FIGURES
3.1
Aperture of arbitrary cross section in a perfectly conducting ground plane of
infinite area radiating into an infinite half-space..................................................... 21
3.2
Rectangular waveguide radiating into a conductor-backed slab of lossy
dielectric material......................................................................................................28
3.3a
Normalized conductance as a function of slab thickness for three different
lossy dielectric materials..........................................................................................36
3.3b
Normalized susceptance as a function of slab thickness for three different
lossy dielectric materials. Legend same as Fig.3.3a...........................................36
3.4a
Normalized conductance as a function of slab thickness for three different
frequencies. e'r = 5.0, tan8 = 0.1.......................................................................... 37
3.4b
Normalized susceptance as a function of slab thickness for three different
frequencies. Legend same as Fig. 3.4a................................................................. 37
3.5a
Cross section of a rectangular waveguide radiating into a layered media
terminated into an infinite half-space.......................................................................38
3.5b
Cross section of a rectangular waveguide radiating into a layered media
terminated into a conducting sheet............................................................... 39
3.6
Cross sectional view of a layered media terminating into an infinite half­
space of loss-less dielectric m aterial.............................................................46
3.7a
Effect of infinitesimal loss in the terminating medium on numerically
calculated value of conductance. d2 = 7.55m m ,^ , = 8.4, tanS = 0.107............ 48
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3.7b
Effect of infinitesimal loss in the terminating medium on numerically
calculated value of susceptance. Legend same as Fig. 3.7a.................................. 48
4. la
Comparison of the theoretical and measurement results of VSWR as a
function of slab thickness for / = 10 GHz, e'r = 2.59,tand = 0.007.................... 53
4.1
b
Comparison of the theoretical and measurement results of phase as a
function of slab thickness f o r /= 10 GHz, e'r = 2.59,tand = 0.007.
Legend same as Fig. 4.1a.........................................................................................53
4.2a
Comparison of the theoretical and measurement results of VSWR as a
function of slab thickness f o r / = 10 GHz, e'r = 7.25, tand = 0.103.............. 54
4.2b
Comparison of theoretical and measurement results of phase as a function of
slab thickness f o r /= 10 GHz, e ’r = 7.25, tand = 0.103. Legend same as
Fig.
4 .2 a ............................................................................................................... 54
4.3a
Comparison of theoretical and measurement results of VSWR as a function
of slab thickness for f = 10 GHz, e \ = 12.6, tand - 0.19.................................... 55
4.3
b
Comparison of the theoretical and measurement results of phase as a
function of slab thickness for f = 10 GHz, e'r = 7.25, tand = 0.103.
Legend same as Fig. 4.3a.........................................................................................55
4.4a
Conductance as a function of dielectric properties for a slab of thickness
d = 2 mm................................................................................................................... 58
4.4b
Susceptance as a function of dielectric properties for a slab of thickness
d = 2 mm. Legend same as Fig. 4.4a..................................................................... 58
4.5a
Conductance as a function of dielectric properties for a slab of thickness
d
4.5b
= 10 m m ........................................................................................................59
Susceptance as a function of dielectric properties for a slab of thickness
d = 10 mm. Legend same as Fig.4.5a.................................................................... 59
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4.6
Apparatus for swept frequency measurements............................................... 63
4.7
Comparison between the theoretical and measured phase for a lossy
dielectric sheet with d = 2.08 mm, and e \ = 12.4 - j 2.4.................................64
4.8
Comparison between the theoretical and measured phase for a lossy
dielectric sheet with d = 12.8 mm, and e'r = 12.4 - j 2.4................................ 64
4.9
Comparison between the theoretical and measured phase for a low-loss
TBC sample with er = 20 - j 0.02 for three different thicknesses.....................66
4.10
Comparison between the theoretical and measured phase for a low-loss
TBC sample with d = 2.0 mm for three different permittivities.......................67
4.11
Comparison between the theoretical and measured phase for two lossy
dielectric sheets with er = 12.4 - j 2.4......................................................... 68
4.12
Phase difference due to variations of lossy dielectric sheets of Figure 4.11,
with ^ =2.18 mm and d2 = 2.08 mm with er = 12.4 - j 2.4................................. 69
4.13
Measurement apparatus for examination of layered media.................................... 70
4.14
Cross sectional view of a layered media with a disbond............................... 71
4 .15a
Comparison of the numerical and experimental results for VSWR as a
function of air gap dn for three free-space dielectric constants of e'r = 1.0,
and tand = 0.0, lx lO '3, and lXlO'5. d2 = 7.55 mm,£'f, = 8.4,
tand = 0.107............................................................................................................. 73
4.15b
Comparison of the numerical and experimental results for phase as a
function of air gap d,, for three free-space dielectric constants. Legend
same as Fig. 4.15a................................................................................................... 73
4.16a
Comparison of the numerical and experimental results at 10 and 24 GHz for
VSWR as a function of slab thickness for an arbitrary air gap of
d, = 2 mm................................................................................................................. 76
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4 .16b Comparison of the numerical and experimental results at 10 and 24 GHz for
phase as a function of slab thickness for an arbitrary air gap of d, = 2 mm.
Legend same as Fig. 4.16a......................................................................... 76
4 .17a Comparison of the numerical and experimental results at 10 and 24 GHz for
VSWR as a function of slab thickness for arbitrary air gap of d, = 5 mm
77
4 .17b Comparison of the numerical and experimental results at 10 and 24 GHz for
phase as a function of slab thickness for arbitraiy air gap of d, = 5 mm.
Legend same as Fig. 4.17a..........................................................................77
4.18 Comparison of the numerical and experimental results for VSWR and phase
as a function of disbond thickness, d3 , for dl = 5 mm, d2 = 5.15 mm,
d4 = 7.55 mm.......................................................................................................... 78
4.19
Disbond in a layered media terminated into a conducting sheet..............................79
4.20
Comparison of the numerical and experimental results for VSWR and phase
as a function of d n for d2 = 7.55 mm, d3 = 0.............................................. 80
4.21
Comparison of the numerical and experimental results for V’SVW? and phase
as a function of d3, for.d, = 5 mm, d2 = 7.55 mm.................................................81
4.22
Difference between the numerical results of Figure 4.20 with the case where
a small disbond of d3 = 0.1 mm is also present........................................... 81
5.1a
Coaxial transmission line of inner conductor radius a and outer conductor
radius b opening onto a perfectly conducting infinite flange..................................86
5.1b
Cross section of a coaxial line radiating into a layered media terminated into
an infinite half-space................................................................................................ 87
5.1c
Cross section of a coaxial line radiating into a layered media terminated into
a perfectly conducting sheet.....................................................................................87
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5.2a
Cross section of a coaxial line radiating into a two-layer media with layer 2
being an infinite half-space....................................................................................... 94
5.2b Cross section of a coaxial line radiating into a two-layer media terminated
into a conducting sheet................................................................................... 94
5.3a Return loss versus slab thickness normalized with respect to outer radius of
the coaxial line, for r = bla, e'n = 10, en = 1 (air) and/ = 5 GHz......................... 100
5.3b Phase versus slab thickness normalized with respect to outer radius of the
coaxial line, for r = bla, e'n = 10, er2 = 1 (air) and/ = 5 GHz.............................100
5.4a Return loss versus slab thickness normalized with respect to outer radius of
the coaxial line, for r = bla, £'n = 10 a n d /= 5 GHz. Conductor backed......... 101
5.4b Phase versus slab thickness normalized with respect to outer radius of the
coaxial line, for r = bla, e'ri = 10 a n d /= 5 GHz. Conductor backed................. 101
5.5a
Return loss versus frequency f for different thicknesses normalized with
respect to outer radius of the coaxial line. T = d,l(ar), r = bla, e'n = 10,
tand = 0.1, er: = 1 (air)............................................................................................ 102
5.5b Phase versus frequency f for different thicknesses normalized with respect
to outer radius of the coaxial line. T = clj(ar), r = bla, £'n = 10, ran5=
0.1, £,.,
5.6a
= 1 (air)............................................................................................ 102
Return loss versus frequency f for different thicknesses normalized with
respect to outer radius of the coaxial line. T = d,l(ar), r = b/a, £'n = 10,
ta n 8
5.6b
Conductor backed.................................................................... 103
Phase versus frequency f for different thicknesses normalized with respect
to outer radius of the coaxial line. T = d,/(ar), r = bla, £'n = 10,
ta n 8
5.7a
= 0.1.
= 0.1.
Conductor backed.................................................................... 103
Return loss versus frequency for different disbonds d2, £r = 1 (air),
£',., = 10, tand = 0.01, d] = 0.5 mm, a = 1.18 mm, b = 3.62 mm................... 104
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5.7b
Phase versus frequency for different disbonds d2, en = 1 (air), e'n = 10,
tand = 0.01, d j = 0.5 mm, a = 1.18 mm, b = 3.62 mm................................. 104
5.8
Block diagram showing the measurement apparatus transfer function and
sources of error.......................................................................................................... 105
5.9
Simplification of transfer function using signal flow graphs............................106
B.1
Path of integration along the real axis in the complex domain.................................124
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LIST OF SYMBOLS
CHAPTER III
a
Large transverse dimensions of rectangular waveguide.
a'
normalized waveguide dimension,
a'~ k0a
A„
field coefficient related to the /-component of vector potential, where 1 denotes
4>or ¥ ' component and ± represents positive or negative travelling waves.
normalized field coefficients,
K
=ko K
b
small transverse dimensions of rectangular waveguide.
b'
normalized waveguide dimension,
b'= k0b
B
susceptance.
bs
normalized susceptance,
bs = —
Y
io
dn
thickness of layern.
en
transverse electric vector mode functions (bold or with a bar).
£r0
relative free-space permittivity.
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ern
relative complex permittivity of 11th layer.
G
conductance.
gs
normalized conductance,
G
8S
y
*0
hn
transverse magnetic vector mode functions (bold or with a bar).
k0
free-space propagation constant,
, _ ----- 1--------_ 2 k
Aq
kn
complex propagation of the nth layer,
bn
K x, y ,z
=
~
-y jk x + k y
+ A\?
normalized components of the complex propagation constant,
kx vKx . , r
—
0
n
vector potential (bold or with a bar) with c£>and f'.
R
complex reflection coefficient,
R = JV * =
i+ys
VSWR
voltage standing wave ratio,
VSWR =
i-r
Y
Admittance, Y = G + j B .
y0
free-space characteristic admittance,
S l
xvn
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Yn
normalized admittance (with respect to free-space characteristic admittance),
Yn
characteristic admittance of the n* waveguide mode.
Y0
characteristic admittance of fundamental mode,
Y0 = yo-\jerG ~
ys
j
’ £rc = 1 for air - filled waveguide
normalized admittance,
•u = —
Y
y s = 8 s + jbs
zn
total thickness of the layered medium up to interface n,
zn ~
i=J
zn
normalized thickness,
n
z n ~ ko
2-^/
i=J
*
complex conjugate notation
CHAPTER V
(Not written if the notation is the same as chapter III)
a
radius of inner conductor of coaxial transmission line.
b
radius of outer conductor of coaxial transmission line.
radial and angular electric field components in layer n.
Hankel transform of the E-field components.
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£rc
complex dielectric constant of the coax filling.
H n*
radial and angular magnetic field components in layer n.
Hankel transform of the H-field components.
Jn(x)
Bessel functions of first kind and order n.
kc
complex propagation constant inside the coax,
= ko-\j£ rc
cut-off wave number of higher order modes, also eigenvalues of radial
eigenfunction Rm.
N n(x)
Neumann functions of order n.
P*
complex conjugate of power flow (arrow on top denotes the direction of
propagation).
n„
vector potential (or Hertz potential) of magnetic type in layer n.
Hfi
Hankel transform of vector potential.
radial wave number, with axial component kZn,
Rm
radial function with eigenvalues Am,
K , = N m[j,(Xmp)Y0(Xma ) - J 0(Xma)Y,(Xmp)]
RL
return loss,
RL = 201ogf-p
ratio of the outer to inner conductor radius of coax,
b
a
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T
normalized thickness,
T=
b
Ym
admittance of the m * transmission line E-mode,
Yc
characteristic admittance of coaxial line,
Ye ~ y 0- f c
Yn(x)
Bessel functions of second kind and order n.
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1
C H APTER I
In tro d u ctio n
Nondestructive measurement methods utilizing microwave radiation for the purpose of
material examination, either in a contact or a non-contact manner, began years ago and have
been applied to various media
[Alt.63] [A/c.SS] [As.81] [Bah.82] [Bel.90] [Dec.74] [Gho.89] [Lun.72]
[Nyf.89] [Rud.74] [Tan.76] [Tiu.75] [Ven.86] [Zou.90],
NonDestructive Testing and Examination
(NDT & E) has been synonymous with such conventional methods as ultrasound, eddy
currents and X- or Gamma-ray imaging techniques. With prevalent use of metals for their
strength, early NDE techniques were originally established to deal in large with
examination of the integrity of metallic media. Consequently, utilization of Microwaves for
examination of such media received limited attention. At microwave frequencies the depth
of penetration into highly conductive materials is at best a few microns allowing for
examination of surface features only
[Bah.82] [Dal.73] [Hnt.70] [Kor.88].
It has not been until
recently that Microwave NDT methods have been mentioned alongside the more
conventional techniques.
The advent of new dielectric composite materials, replacing metals in many
applications, has diversified the implementation of NDT techniques. Dielectric composite
materials such as plastics, ceramics, and carbon fibers have revolutionized many facets of
engineering and industrial applications. With the increasing applications of these new
lightweight, durable, and strong composite materials many traditional NDT methods have
inherently rendered themselves ineffective. Penetration of microwaves inside dielectric
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2
media and their sensitivity to microscopic and macroscopic changes in material medium,
coupled with availability of relatively large bandwidths are of great significance.
Microwave techniques have reemerged not only to fill many gaps left vacant by these
traditional methods, but also to further expose new alleys in the expanding and demanding
field of NDT&E. For instance, unlike microwave techniques, eddy current methods have
experienced great difficulties in dealing with materials with low conductivity (i.e. lossy
dielectrics).
Moreover, the inherent limitations such as the contact nature of many
methods, susceptibility to a material surface condition and material density requirements
associated with techniques such as ultrasonic can not be disregarded.
Furthermore,
complexity and high cost of such methods as radiation imaging techniques have limited
their use for general applications. Microwave NDT methods offer significant advantages in
inspection of such materials which their composition and low conductivity limit the
usefulness of many traditional NDT techniques.
The use of microwaves in the past few decades has played a prominent role in
development and evolution of new and important applications. In the field of medicine
microwaves have been utilized for tomography, heating of tissue and examination of
biological state and chemical composition of organs [Ath.82b]
[Bur. 80} [ K m . 83a]
[Nik.89]
[Sea.89] [S/h.50]. In other applications, microwaves have been employed for commercial and
industrial heating purposes, either as household microwave ovens and dryers or as large
and high-power industrial microwave chambers for material curing.
Theoretically, characteristics of microwave radiation from various aperture antennas
have been studied in detail
[Gar.83-85].
[Col.60] [Gal.64a.b-65] [H ar.6l] [Lev.51] [Lew.5 1] [Ma.51] [Vil.65]
These radiators serve as standard sensors in most microwave NDT systems.
Theoretical analysis always plays an important role for a thorough understanding of the
interaction of electromagnetic fields with different material media. Since the interaction of
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3
microwave energy with composite media is a complex process, such expositions are
important since they may predict useful practical applications. Numerical simulations,
using either approximate or rigorous formulations play a fundamental role in these
analyses.
The use of microwave devices and components as versatile and accurate tools for NDT
applications has come about as a direct consequence of advances in the fields of microwave
semiconductor and integrated circuit (MIC) design and computer technology. Availability
of microwave solid-state components have made the assembly of low power, and versatile
microwave circuits and systems possible. Phenomenal advances in the microelectronic and
computer technology have brought upon powerful and inexpensive tools for acquisition
(i.e. A/D converters), storage and processing (i.e. microcomputers) of data which had
always posed great obstacles in development and implementation of practical on-line
applications.
The objective of this work is to show theoretical as well as practical aspects of several
microwave NDT methods utilizing two different sensors. In the following chapter, a
survey of several conventional NDT methods are presented along with a brief account of
each technique and its applicability. In the succeeding chapters numerical electromagnetic
analysis of the aperture admittance of two transmission line sensors radiating into multi­
layered composite media are presented. Experimental results using laboratory apparatuses
are also presented in verification the theoretical formulations.
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4
CHAPTER II
A Survey o f Prominent NDE Techniques
The need to monitor the composition, integrity and maintenance of materials,
structures, equipments and product quality control gave rise to the development of the field
of NDE. Initially, these techniques were developed to examine in large the condition of
engineering structures which were prominently metallic in nature. With the expansion of
applications due to the advent of many new dielectric composite materials the field of
NDT&E has tremendously diversified.
Numerous NDE techniques have evolved,
standardized, and tailored specifically to tackle specific problems.
In any NDT&E application, the effective testing method is chosen as a function of the
material structure, failure mechanisms and the operating environment. Most often it is
necessary to employ a combination of different methods to qualitatively examine and
quantitatively estimate different testing parameters. Therefore, a priori knowledge of the
building components of the material structure and the deterioration processes are of vital
importance in choosing the appropriate NDT method. The terms NDE - nondestructive
examination, and NDT - nondestructive testing, are distinguished according to the
standards set by the Danish Maintenance Society where the former refers to condition
monitoring and the later to manufacturing process control methods [flov.89].
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5
ll.l
Conventional NDE Methods
Due to the vast diversity of NDT techniques only a selected number of prominent
methods pertaining to applications such as detection and evaluation of cracks, material
condition, coating and material thickness will be briefly mentioned in the following review.
These techniques include, acoustic emission, dye penetrant examination, eddy current
testing, endoscopy, isotope techniques, laser distance and shearography measurements,
pulsed video thermography, computerized tomography, pattern recognition, ultrasonics,
microwaves, and radiography using X- or Gamma radiation. Microwave techniques will
be examined in more detail in a separate subsection.
Acoustic emission (AE) technique is a contact method (i.e. the sensor is in contact
with the material medium under test) which consists of detection of acoustical emissions
from material under stress by means of placing high frequency microphones on the clean
surface of the material. The presence of such changes in the material as formation of
cracks, fractures, deformation and dislocation causes energy release which propagates in
the material as elastic vibrations. Thus, the ability of a material to transmit acoustic waves
determines the applicability of AE technique. Since AE method can only detect changes in
material structure, any previously present defects may not be detected unless the defect size
increases under load. For this reason AE technique is mostly used as a supplement to other
NDE techniques which may first be used to detect any defective area.
Dye penetrant examination is also a contact NDE technique for determination of
defects such as cracks, fractures, leaks, and in some cases porosities in such materials as
metals, ceramics, and plastics. In this method a dyed or fluorescent penetrating liquid is
applied to the clean surface of the material. After allowing the penetrant to set for a period
of time and removing its excess, the developing liquid is applied and dried off. The
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6
developer draws the penetrant out of the defect and after a period of time indications of the
defect appear on the developer which may be observed directly or under ultraviolet light.
This is an inexpensive NDE method which requires careful preparation of material surface
and subsequent cleaning. This technique is rendered useless for examination of internally
embedded flaws.
Eddy current testing is an electromagnetic technique which may be used in a contact or
non-contact manner to test cracks, corrosion, fractures, erosion, leaks, and thickness of
electrically conducting materials. In this method an alternating current is applied to the test
coil of a probe, inducing a primary magnetic field in the vicinity of the coil which in turn
produces eddy currents in the test object. These eddy currents in the test material induce a
secondary magnetic field opposite to that present in the primary field which may be
recorded by means of an oscilloscope or galvanometer attached to the test coil. Once the
recording equipment is calibrated at the normal state of the test object (i.e. unhindered
currents), any change in the level of currents due to material inhomogeneity may be
detected as a result of change of fields in the primary coil. The eddy current measurements
are dependent on the electrical and magnetic properties of the test object as well as the
distance between the coil and the object. This method is quite versatile and can be quickly
implemented. Unless commercially available microprocessor based equipment are to be
purchased, the components needed to set up an eddy current measurement apparatus are
relatively inexpensive . Inherent to electromagnetic fields, the operating frequency
determines the depth of penetration inside the test object (i.e. lower frequency for higher
depth of penetration). The detection and evaluation of defects are done by means of
calibration with respect to known and well defined defects such as cracks. One of the
greatest limitation of this method is that it is mainly applicable to non-magnetic materials.
However, for certain application to magnetic media possibility of artificial magnetic
saturation of the material using a direct current has been suggested. This method is mainly
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applicable to conducting materials with reasonably smooth surfaces and trained inspectors
are often required to carry out the inspection task.
Endoscopy is an optical technique for visual inspection of hard-to-reach surface
defects. The endoscopes may be stiff (borescopes) or flexible (fiberscopes).
The
borescope uses a halogen lamp and the fiberscope uses a cold light generator and optical
fiber as means of illumination of the object surface. The image is transferred from the
objective to ocular by means of optical system in the former device and optical fibers in the
later. The flexible fiberscope may be adjustable in length. Video endoscopes are also
available which operate based on the principle that a small electronic sensor detects and
transmits the recorded signal into a color TV monitor. Such devices may be operated with
minimal training for detection of visible defects.
Laser distance and shearography measurements are non-contact methods which have
recently been marketed for dimensional measurements and NDE of composites. Laser
distance measuring apparatus uses an optoelectronic instrument by means of triangulation
principle for measurement of distance to the surface of an object. The measurement system
usually consists of a laser, camera and a computer for data collection and processing.
Laser apparatuses may be used for fast and accurate dimensional measurements for variety
of materials given that the subject reflects infrared radiation. Some typical applications are
profile and thickness measurements in fabrication of wire and plates, and determination of
flatness and thickness in metal processing area. For thickness measurement, accessibility
to both sides of the object is required. The beam must be totally shielded to protect from
striking the eyes if output power greater than 2.5 mW is used. New shearography systems
incorporate a fiber optic laser, an image processing apparatus and a monitor for NDT of
composites, and surface examination of metals. Such systems may also be used for
thermal expansion studies and load deformation analysis. Commercial units available in
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today’s market are relatively expensive and a trained operator is required for evaluation of
images.
Pulsed video thermography (PVT) is a non-contact NDE thermal method which is
particularly effective in examination of fiber-reinforced composites, honeycomb structures,
metal-on-metal glued constructions and coatings. Moreover, PVT is particularly applied to
detection of delaminations, density variations, thickness variations, bonding defects, and
defects in composite materials. The principle of operation is based on illuminating the test
object surface with brief heat impulses causing transient temperature changes due to heat
impulse diffusion. Temperature changes which are strongly dependent on the presence of
defects in the sample may be detected via an infrared camera and may be displayed on a
monitor.
Optical pattern recognition uses a camera and a computer to store at full video speed a
digitized image of an analog signal from an object under test. A typical image of 512x512
pixels is stored in the image storage area of the computer with each pixel containing gray
level information between 0 to 255 for each point. The computer may then address each
memory cell and find appropriate transitions, check distances, and identify and check
contours in each image. The most common limitation of these systems which is decisive in
performing measurements is their ability in precise illumination of important details of the
test object. Background illumination in some cases may be the remedy to this problem.
The optical pattern recognition devices are relatively expensive and the ability of the
operator for optimization of system for the best illumination is of vital importance.
Ultrasonic examinations commonly use a pulse-echo method by transmitting a short,
high-frequency pulse of mechanical oscillations into a material and receive the echo of the
reflected signal from the defects or the opposite surface of the material. With the
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propagation speed of the sound waves within a certain homogeneous material being
constant, the time interval between the transmission and reception of a pulse renders
information about distance. The main areas of ultrasonic application are the steel industry,
for thickness measurement and detection of defects due to corrosion and erosion, for
medical purposes, and to a lesser extent for examination of concrete. Various ultrasonic
equipments are available ranging from battery-powered thickness gauges to expensive
automated P-scans mainly used in nuclear power plant industries.
Some practical
considerations are the need for reasonably smooth surface of the subject, maintaining of a
good probe contact and often the need for water as the coupling medium. As in many other
NDE methods, proper practical training of the operator is very important.
Isotope techniques use radioactive isotopes and measuring instruments to measure
radioactivity from the isotopes in variety of applications. These techniques are usually
categorized as tracer and radiometry methods. In tracer technique one takes advantage of
the fact that various isotopes of the same element have identical chemical properties and
follow the liquid, gas or solid material in a system containing the radioactive isotope. The
radiometry technique uses X-radiation for illumination of the object and a radiation detector
and measuring instrument to create an image of the distribution of radiation. The former
technique is mainly used to detect leaks from hidden pipe installations and the later allows
measurement of thickness and density or localization of defects such as inhomogeneities
and cavities. Unlike radiography which uses radiation-sensitive film to create an image,
radiometry measures the absorption and scattering of radiation quantitatively in a region of
material medium. The measuring instrument may be a Geiger counter or other sensitive
detectors for counting the individual number of pulses. The radiation levels are very low in
tracer methods, however, a shielded radiation source is always used in radiation methods.
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Radiography is an NDE technique in which an object is irradiated by X-rays or Gamma
radiation. The degree of penetration inside the test object, which depends on its thickness,
composition and wavelength of radiation, may then be recorded on a exposed radiationsensitive film. The fact that different materials absorb radiation to different extents, the
attenuation can reveal information about the material composition. The areas of usage may
vary from examination of light metals and reinforced concretes to plastics and composite
materials. Defects as small as 10 pm may be located in metal structures and composite
materials. The main limitation on usage of radiography, aside from the ability to penetrate
inside the material, is the need of access to both sides of the material this method.
Consequently, this technique is usually applicable in condition monitoring stage.
Computerized tomography (CT) using using X-ray, nuclear magnetic resonance to produce
cross-sectional images of the interior of a sample is a well known technology. CT devices
use ionizing radiation in a fan-shaped flat emission from a source to scan an object in cross
sectional patterns. Due to safety hazards associated with radiography methods a qualified
operator with proper experience, training and knowledge of the equipment is always
required. CT scanning equipments are very expensive. A typical scanner available in
todays market may cost between 0.5 to 1 million dollars.
II.2
M icrowave NDE Methods
Microwave nondestructive testing and examination is the term given to the process of
examination of materials using electromagnetic radiation at microwave frequencies.
Microwave frequency spectrum is considered to approximately cover the range of 300 MHz
to 300 GHz. This frequency range corresponds to free-space wavelengths between 1 m to
1 mm, with the small wavelengths below 1 cm commonly referred to as millimeter waves.
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11
In terms of their interaction with material media in the form of reflection, attenuation
and scattering, microwaves possess certain similarities with ultrasonic waves, eddy
currents, and X-rays, with the last two also using a part of electromagnetic spectrum.
However, there are substantial differences that distinguish microwaves from these other
elastic waves, induction and radiation sources. For instance, unlike eddy currents or Xrays which may penetrate inside a metallic object or an elastic wave that can propagate
through a metallic media, extremely small depths of penetration at even low microwave
frequencies may allow only surface features to be examined. On the contrary, while
microwaves may easily penetrate inside relatively low density, low-loss dielectric materials,
such media often present great difficulties when examined by means of eddy current or
ultrasonic techniques. Moreover, polarizability of microwaves similar to that of light,
distinguishes it from techniques employing ionizing radiation which interact with material
medium on atomic level only.
Ability of microwaves to penetrate inside dielectric materials such as ceramics, synthetic
rubber, plastics and dielectric composites can render information about material
dimensions, flaws (e.g. cracks and disbonds), composition, porosity, moisture content,
state of cure, etc. Furthermore, the polarizability of microwaves provides information
about orientation of embedded features and defects. Microwave NDT approaches unlike
many other techniques, have the capabilities to operate without requirements such as
transducer contact, special coupling medium, smoothness of contact surface, access to both
side of the material and etc. Such features are of utmost importance for real life NDT
applications.
Technological advances in the fields of solid states and microwave integrated circuit
technology has given access to many off-the-shelf sensors and components which unlike
their predecessors are compact, reliable, and inexpensive. Furthermore, great advances in
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microprocessor industry has made it possible to perform fast, on-line, and real time data
acquisition, storage and processing.
Most microwave NDE devices measure the modulus and/or the phase of the complex
reflection coefficient as means of deducing information about a test object. This may be
done either with transmission or reflection measurement setups. In transmission type
measurements the transmitter and receiver are located in opposite sides of the object and the
attenuation and phase shift of the incident wave passing through the object provides
information about the thickness and the physical state of the object.
In reflection
measurements both transmitter and receiver, which may also be one and the same unit, are
located on the same side of the test object. In this case either the two-way attenuation and
phase-shift of the transmitted signal (far-field measurement) or changes in the standing
wave pattern (e.g. near-field measurement) between the transmitter and test material may be
used to deduce information about the material under test.
Based on general application one may categorize microwave NDT&E measurements as
capable of:
- dimensional measurements,
- flaw detection and evaluation,
- material property and composition measurements and evaluation.
Dimensional measurements generally refer to thickness measurement of dielectric
sheets or metal plates as well as that of dielectric composite coatings on top of conducting
surfaces or measurement of variation of layers within laminated composite structures. For
thickness measurement of metallic sheets two transmitter/receivers are located on both sides
of the object. Combining the two signals and the knowledge of precise distance between
the two transducers provides information about the sheet thickness [As/i.S/] [Dal.73] [Bah.82]
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13
[Nyf.89],
Thickness measurement of dielectric plates and composite coatings on the other
hand may be done using either transmission or reflection techniques
[C hu.79] [Zou.90c],
[Bak.92] [Bak.93] [Bah.82]
In the later scenario, the two way propagation through the material,
caused by reflection at the interface of two dissimilar regions, renders information about the
material under test. For either technique the limitations are dictated by the thickness and
dielectric properties of the test material and the strength of the transmitted signal.
Flaws generally in the form of cracks, disbonds, delamination or voids may be
detected within layered composite materials or coatings on top of conducting plates
[B ah .78] [Lun.72] [Nyf.89] [Tha.82]
[Ven.86]
[Zou.90a] [Zou.90b] [Zou .91].
[Bak.90]
This is due to the
interaction of microwaves in the form of scattering, reflection and attenuation at
discontinuities and nonuniformities within the medium. Such changes, in terms of either
attenuation and/or phase shift experienced by the signal due to nonuniformities, may be
detected and evaluated.
Also, polarization properties of microwaves may render
information about orientation of reinforcements such as carbon fiber or metallic inclusions
within dielectric structures [Bah.78]
[Nyf.89] [Tiu.75].
only be used to detect surface features
In metallic structures, microwaves may
[Bah.82] [Dal.73] [Hm.70] [Kor.88] [Nyf.89\.
Material property and composition have also been examined using microwave
radiation.
As pointed out earlier, microwaves interact on macroscopic as well as
microscopic level with the material medium. The later may take place either in the form of
interaction with conduction electrons or with molecular dipoles (rotational processes) which
in general results in the attenuation of the wave. Such properties of microwaves have been
long in use mostly for accurate destructive measurements employing such technique as
partially filled waveguides, cavity resonator, and microstrip patches. Features such as
porosity, moisture content, dielectric property, and state of cure in non-metals have been
examined both in a contact and non-contact manner. Microwave NDE methods may also
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14
be coupled with holographic imaging techniques for detailed examination of composite
media.
Most often, the same microwave NDE apparatus may be used to measure different
parameters either individually or collectively. Simultaneous measurement of several
parameters is generally referred to as multiparameter measurement. Lets say that M is a
measured quantity which is a function of parameters a, A and y. Function M ,M = / (a,
fi, y), may render information about any of the three parameters a, p, and y, given that the
other two parameters are known. For instance, M may refer to the measured complex
reflection coefficient which is a function of material thickness, dielectric constant and loss
tangent. Moreover, a minimum of three independent measurements of M i = f) (a, p, y),
M 2 = f 2 (a >A Y)->M 3
three parameters.
(a, p, y) may be used to deduce information about any of these
Multivariable regression technique is one way of dealing with
multiparameter measurement situations [Kle.84], Signal processing techniques such as
neural networks may also be implemented in such situations. The speed and capacity of the
computer controller is the only factor determining the number of parameters and operations
to be handled.
I 1.3
Reference Table o f Prevalent NDT&E Methods
Table 1.1 is presented as a quick reference of the prevalent NDT methods reviewed in
the previous two sections [S/w.84], The table contains a short summary of each technique
along with its application to three different media. Although the table is by no means
complete, but as a quick reference it conveys the idea about general capabilities and typical
resolutions normally associated with each of these methods. In fact, it is the intention of
this research endeavor to demonstrate that the extent of capabilities of microwave NDE
methods stretch much beyond what is listed in this table.
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Reproduced with permission
I
Table 2.1 :
Comparison of several widely noted NDT&E techniques.
of the copyright owner. Further reproduction
Application to m aterials
Technique
Advantages/Disadvantages
M etals
N on-M etals
Adhesion
Radiography
prohibited without perm ission
X- and Gamma
Embedded flaws above about
1% o f the to tal equivalent
thickness o f m aterial.
In se n sitiv e to crack in g
unless aligned accurately with
beam .
Can be used to measure stress
and atom ic lattices
F lash ra d io g ra p h y for
m oving objects.
Neutron
C an re v e a l low d e n sity
m aterials w hen screened by
h ig h
d e n s ity
m e t a ll ic
structures.
Proton
M easu res sm a ll th ick n e ss
As for metals
changes and picks out edges
o f parts o f a structure_________
Crack detection.
A s for m etals, although a
R eso lu tio n o f voids less lo w e r fre q u e n c y m ay be
th a n
fo r
r a d i o g r a p h y , required for acoustically lossy
a lth o u g h
im p r o v e d
by m aterials.
im aging system s.
E ffective for d etec tio n o f
p o ro sity w h ich w o u ld be
_____
invisible to X -rays.
Surface breaking defects
As for m etals if m aterial is
n o n -ab so rb an t.
Ultrasonics
Liquid Penetrant
As for metals
G enerally sim ilar to X -rays
u nless in the p re s e n c e o f
m etal structures.
C an only detect gaps in
adhesive which gives a 1% or
g re a te r c h a n g e in to ta l
specim en thickness.
Great penetration depth into
dense media.
Superior contrast o f Xcompared to Gamma.
R elatively expensive.
N ot portable.
Sheilding from hazardous
radiation required.
May detect gaps in thick
adhesive layers m ore readily
than X-rays.
Access to nuclear reactor or
accelerator normally required.
Excessive scatter limits
specimen thickness to about
30m m .
P o te n tia lly c a p a b le o f
d etecting sm all ch an g es in
adhesive thickness.
W ill detect separations o f
below 1 micron.
In som e cases ultrasonic
signals can be related to bond
stren g th .
Cyclotron source.norm ally
required.
No application
Mostly some form o f
acoustic coupling necessary.
A ir coupling for non-m etals
som etim es possible.
Surface conditions of
specimens such as undressed
welds can cause problems
Inexpensive.
Careful preparation and
subsequent cleaning-needed.
M
U\
Reproduced with permission
Eddy Current
of the copyright owner. Further reproduction prohibited without p erm ission .
A coustic Emission
O ptical Techniques
Thermal Imaging
Magnetic Particle
M icrowave Techniques
Surface breaking defects and
C an m e a s u re co a tin g
some near-surface defects.
thickness on m etal substrate
No application
M aterial changes involving
Can detect defects in carbon.
conductivity and perm eability
may be measured.
Real tim e detection o f crack
Can detect grow th o f defects
grow th co rro sio n and other in more brittle materials.
C o u ld d e te c t g ro w in g
events g iving rise to stress
F iber co m p o sites copious d isbond.
waves.
em itters.
Location o f growing defects
p o ssib le
Sim ple view ing reveals some
Can reveal unbonds if these
surface defects.
As for metals
alter the stress pattern under
In te rfa c e te c h n iq u e s,
load in the specimen.
h o lo g ra p h y , an d sp e c k le
techniques can measure a wide
ran g e o f stra in s, su rface
e rro rs , an d re v e a l m any
defects.
surface temperature can reveal
As for m etals and generally
C an d etect u nbonds by
thinning o f walls in structures more effective as the thermal v a ria tio n s in c o n d u c tiv ity
such as furnaces and defects in im age is less d isp ersed by th ro u g h a sp ecim en w hich
electrical equipm ent.
thermal conductivity.
alters the surface tem perature
d istrib u tio n .
E ffe c tiv e fo r su rfa c e
b re a k in g and n ea r-su rfa ce
No application
No application
defects in magnetic materials.
Can measure Surface defects,
D im ensional and dielectric
M easure disbon ds, voids,
and slight surface roughness.
property variations if material and sligh t diem entional and
M easuere orientation and size not too thick and lossy.
dielectric property variations.
o f su rfa c e d e fe c ts u sin g
Can detect internal defects in
p o larizario n .
non-conductors and m easure
C heck strip thickness by burning ra te s by reflection
reflection from each side
from the reaction zone.
Measure movement.
H o lo g ra p h ic im a g in g
p o ss ib le .
P o ro sity d e te c ta b le by
scatter.
Easily autom ated.
N o n -co n tact m e a su re m e n ts
p o ssib le.
Significance o f defects in
metals andem issions for nonmetals difficult to quantify .
Non-contact.
Precise illum ination o f
im portant details is
necessary.
N on-contact
Relatively inexpensive
Non-contact.
R elatively inexpensive.
17
11.4
Scope and Motivation o f This Work
The strength and light nature of new dielectric composite materials has led to their
widespread use in many engineering and industrial applications. Parallel to this, there is a
vital and growing need for nondestructive testing and examination of these low
conductivity materials. Microwave NDE methods have shown great potential in dealing
with such material media. This was discussed in detail earlier and several reasons were
offered pertaining to this fact. The objective of this work is to take a step forward towards
filling the existing gap for concise theoretical formulation and practical implementation in
application to microwave NDT&E of layered dielectric composite media.
Interaction of electromagnetic waves with different material media is a complex
process. Without a thorough understanding of the nature of reflection, attenuation and
scattering mechanisms involved, most careful measurements would inherently render
themselves useless. Electromagnetic models pertaining to the analysis of radiation into
layered dielectric composite media from two prevalent sensors are introduced in this
endeavor. These models serve as principal tools for general exposition of the theoretical
aspects, and subsequently help in predicting the outcomes and analyzing their significance.
Variational formulation is evoked for calculation of admittance of apertures radiating into a
layered composite medium backed or unbacked by a conducting sheet. The resulting
integral equations are then solved using numerical methods. To verify the integrity of the
numerical formulations and to further exploit the possibilities open to the field of
microwave NDE, different practical apparatuses will be inU'oduced. These are mainly in
application to measurement and monitoring of thickness, dielectric constant, and flaws
such as disbonds and delaminations in layered dielectric composite media.
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18
II.5
Summary and Remarks
A survey of prominent NDE techniques was conducted in this chapter with a brief
explanation of each method. Table 2.1 provided a quick reference of the summary of
several prominent NDE techniques in application to flaw detection, dimensional
measurement and examination for structural integrity of different material media. The table
summarized the applicability of each technique to three categories of material media along
with generally expected resolutions associated with each technique. Although applications
of microwave NDE techniques, as the focus of this work, have only been barely noted in
this table, nevertheless they convey information about general mechanisms involved in each
technique along with its capabilities and limitations. Microwave NDE methods were
discussed in a separate section and several reasons were offered as to why they render
themselves useful in application to examination of new dielectric composite materials.
In the following chapters analysis of radiation from two open-ended sensors will be
presented. Initially, theoretical electromagnetic foundation based on variational formulation
in application to analysis of admittance of aperture antennas will be given. This will be
followed by expanding the theory to take into account different layered dielectric composite
geometries in front of the waveguide. Different measurement apparatuses using rectangular
waveguide sensors will also be introduced both to verify the numerical results and also
demonstrate the practicality of the work. Theoretical formulation for the terminating
aperture admittance of an open-ended coaxial transmission line radiating into layered
composite media will be presented in Chapter 5.
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19
CHAPTER H I
Theoretical Analysis o f Radiation From Rectangular Waveguides Into
Layered Composite M edia
Rectangular waveguides are probably the most commonly used transducers in
microwave NDE applications. As standard part of many microwave devices, they are
commercially available in variety of dimensions covering most of microwave and millimeter
wavelength spectra. As measurement sensors, one of their prominent features is that for
the dominant TE10mode of propagation, the field throughout the guide may be conveniently
monitored with minimum perturbation to the field distribution inside. This fact is taken
advantage of in slotted lines where a slot cut axially in the center of the guide is used for
insertion of a probe to accurately monitor variations of the standing wave in the guide.
Moreover, due to their simple geometry, the analysis pertaining to radiation from these
aperture antennas are relatively straight forward.
Much work has been done on rigorous theoretical analysis of radiation from rectangular
waveguides aimed at diverse range of applications. Most of the earlier literature have
addressed the problem of plasma covered aperture antennas
[C ro .6 7 -6 8 ].
[G al.64a.b-65) [Vil.65] [Com.64]
More recent near-field analysis of rectangular waveguides have been in
conjunction with such applications as measurement of dielectric properties of materials and
field interactions with biological tissues [7>«.S5]
[Jam .77] [D e c.7 4 ] [M ac.SO ] [N ik.89].
Considering typical measurement resolutions and accuracies in near-field examination of
generally lossy composite media, rigorous analytical solutions generally offer little practical
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20
advantage when taking into account the necessary computation time and difficulties
involved in replicating the results by means of well-controlled experiments.
The foundation of the theoretical implementation adopted here is parallel with the work
done by Compton in application to radiation from plasma covered aperture antennas
[C om .64].
The formulation is expanded and modified for application to microwave
nondestructive examination of layered lossy dielectric composite media. Variational
formulation is initially evoked to construct a stationary expression for the aperture
admittance of general cylindrical waveguides of arbitrary cross section radiating into an
infinite half-space. This fundamental result is subsequently used for analysis of radiation
into multi-layered lossy dielectric composite media. In the medium external to the
waveguide, Fourier transform boundary matching technique is applied to solve for the
appropriate set of field components pertaining to that specific geometry. This solution once
coupled with the admittance expression will render the appropriate aperture response. The
first case addressed here is that of a single layer of generally lossy dielectric material backed
by a conducting plate
[Bak.93).
The solution is consequently expanded to include N-layer
stratified composite media terminated by a conducting sheet or an infinite half-space. Some
important observations regarding the numerical results will be discussed. The numerical
results are inspected both by modification of geometry and comparison with other
theoretical formulations. A more thorough verification of the integrity of the results are
presented in the succeeding chapter in conjunction with practical applications of the work.
IIL I
Variational Formulation o f Aperture Admittance
Variational formulation is next implemented to construct a general expression for the
admittance of a cylindrical waveguide of arbitrary cross section opening onto a perfectly
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21
conducting ground plane, radiating into an infinite half-space. The geometry of the
problem is depicted in Figure 3.1.
Figure 3.1: Aperture of arbitrary cross section in a perfectly conducting ground plane of
infinite area radiating into an infinite half-space.
The waveguide geometry exhibits cross sectional symmetry orthogonal to the direction
of propagation, z. Analogous to the field components E and H in the guide, one may
introduce the transverse mode functions e(x,y) and h(x,y), along with the mode currents
and voltages V(z) and I(z), respectively
[Col.66] [H a r.6 l).
The following simple relations
would then hold among these functions
E (x,y,z) = e(x,y)V(z)
(3.1)
H (x,y,z) = h(x,y)I(z)
where the vector mode functions are subject to the orthogonal and normalization conditions
over the waveguide cross section and the transverse aperture plane as follows
e; = hi x a.
(3.2a)
hi = as x e i
(3.2b)
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22
H?i.ej dS = llhi.hj d S = \ 01 ! * {
S
S
I
\* j
s
(3.2c)
J
(3.2d)
s
Each mode vector is orthogonal to all the other mode vectors. This may be easily verified
with the aid of Green's first and second identities and the divergence theorem [Har.6l].
With the properties of orthogonal vector mode functions introduced, one can proceed to
construct the aperture admittance for the geometry of Figure 3.1 by evoking a variational
formulation method. The electric and magnetic fields in the waveguide may be written as a
summation of infinite number of discrete modes. With the dominant TEW mode being
incident on the aperture, these fields can be constructed in terms of the vector mode
functions as
E (x, y, 0) = V: e0(x,y) + l V r en (x, y)
(3.3a)
n~o
H(x,y,0) = Y0Vigh0(x ,y )- I YnVr hn(x,y)
{3.3b)
n=o
where Vin and V,.n are the magnitude of nlh incident and reflected modes respectively and
Yn is the admittance of the nth mode. The normalized aperture admittance in teims of these
mode voltages may be written as
Y
Yo
Vi
\r
1 + jo_
w
where Y0 is the characteristic admittance of the fundamental mode. The above expression
may be rearranged as
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23
v = ——
ro(vi . ~ vr.)
N
Y
------22 . = —
V*:Si + vr'ft
D
(3
{
5
)
J
where, for the ease of manipulations, N and D are chosen to denote the numerator and
denominator of the admittance function respectively and will be evaluated separately. To
construct D, both sides of (3.3a) are dot producted by e0 and integrated over the aperture
area as
| | E(x,y,o).e0dxdy = V0fje0.e0dxdy + || I Vr en.e0dxdy
S
S
(3.6)
S n= o
which with the aid of (3.2a) simplifies to
\
+\
= jjE(x,y,0).e0(x,y)dxdy
(3.7)
S
This represents the denominator of the admittance function. To construct N, (3.3b) is first
rearranged by taking the fundamental mode out of the summation and rewriting the
equation as
Yo{yio ~ vr„)^>(-V,y) = H (.v,y,0 ) + I Y„Vr J^(x,v)
(3.8)
n=l
which in simplified notation may be written as
Nh0 (x,y) = W
(3.9)
Subsequently, to extract N from this equation the aperture electric field is cross producted
by both sides of (3.9) and the z-component is integrated over the aperture surface resulting
in
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24
JjA /[£(jr,y,0)x/io(jc,y)].a. = J][£(x,y,fl) x W (x,y)].a: dxdy
{3.10)
Substituting (3a) in the left hand side of the above equation and using the orthogonal
properties of the vector mode functions in the following manner
J J * [ e (x, y, 0) x h0 (x, y)].a.
(3.11)
5
= NH J [vioe0(x,y) x ^0(jr,y)] +
I vren(x>y)xh>i(x’y)
\.d~dxd\
n=o
- '/v/ l | ^ / 0U > > ')-[^ U \y )x a.]+ £ Vrnen(x,y)[h„(x,y)xa: ]^dxdy
n=o
= N (v< . + \ )
results in the above simplified relation and leads to the following form of (3.10)
!S[EU,y.O)xWU.y)].u:dxdy = r„(v,a - V , J v ^ + V , ' )
(3.12)
The complete expression for W on the other hand may be simply expanded by once again
dot producting both sides of (3.3a) by an orthogonal vector mode function em and
integrating over the aperture cross section, resulting in
Vr =}] E(x,y,0).en(x,y)dxdy
s
(3.13)
Once the above is substituted back into (3.8) it results in the following
VTCv,y) = H(x,y,0)+ I Y„fin(x,y)jjE(Tj,^0).en(T],^)drid^
(3.14)
To be able to incorporate the above results in the admittance expression, (3.5) can be
manipulated to take the following foim
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25
(3.15)
which can be constructed using the results of (3.7) and (3.12). Consequently, substituting
the former equation for the denominator and the later for numerator respectively, renders
the variational form of the complex admittance function which can be expressed as
j|[£(jt,y,0) xW(x,y,0)].a,dxdy
Y = G +j B - —
(3.16)
jjE(x,y,0).eo(x,y)dxdy
S
where W is given by (3.14). The conductance G and susceptance B represent the real and
imaginary parts of the admittance respectively. The above expression is stationary,
meaning that small variations of the approximate aperture electric field distribution about its
exact value would not effect the determined parameter Y. A typical stationary expression
may be recognized from its general form of containing the square of the trial function in
both the numerator and denominator. This in effect implies that, for a reasonable estimate
of this function the resulting calculated parameter will not deviate form its actual value.
Proof of stationarity of such an admittance expression of (3.16) has been established by
several authors
[Sax.68] [Har.61] [Com.64 ].
In brief, this is shown by initially taking the trial
function, here the aperture electric field, as a sum of the actual field distribution plus a
variation 8E about this exact value. In variational symbolism this can be expressed as
E, (x, y, 0) = Ea (x, y, 0) + 8E (.v, y, 0)
(3.17)
with the subscripts t and a denoting the trial and actual values, respectively. Rewriting Y in
(3.16) in accordance with the above notation gives
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26
jj Et (x,y,0) x Wt (x,y,0).a,dxdy
(3.18)
where D t and Wt are
D( = D a + SD = jj Ea(x,y,0).e0(x,y)dxdy
(3.19)
S
+jj 8E(x,y,0).e0(x,y)dxdy
S
Wa = Wt + 8W
(3.20)
The proof of stationarity of (3.16) comes from the fact that the first variation of Y written
as
(3.21)
D% jjSE x Wa.a.dxdy + jjE a x 8W ,a: dxdy - 2Da8Djj Ea x Wa.d. dxdy
S
S
is equal to zero. This result can be deduced by showing that the first two terms of the
numerator of (3.21) are in fact equal and additionally, their sum within the bracket is equal
to the last term in the numerator. The demonstration of this may be done using the fact that
E a and H a, and by superposition 8E and 8H, satisfy both Maxwell's equations and the farfield radiation conditions in the region outside the aperture.
Throughout the rest of this work, the outcome of the fundamental results presented
above, namely formulation of a stationary form of the terminating aperture admittance, will
be used in the analysis of radiation from a waveguide into different composite media.
Although multi-mode functions may easily be incorporated into the formulation to define
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27
the approximation for aperture E-field distribution, only the dominant mode is taken into
account here. It will be demonstrated that for the cases considered here the standard
practice of dominant mode assumption is in fact quite valid.
HI.2
Conductor Backed Lossy Dielectric Slab
The first problem addressed here is that of a slab of generally lossy dielectric medium
backed by a conducting plate placed in contact with a flange-mounted rectangular
waveguide. A Fourier transform boundary matching technique is used to solve for the
fields external to the waveguide. Enforcing continuity of the fields over the aperture cross
section results in the solution of exterior fields. This result will be coupled into the
appropriate form of the stationary expression constructed in the previous section to
calculate the terminating aperture admittance. The theoretical results presented here are
based on the dominant mode assumption for the waveguide aperture field distribution.
Validity of this commonly practiced assumption has led to the construction of a simple
integral solution which is fast converging for generally lossy dielectric slabs, and may
easily be implemented for real time applications [Bak.93\.
H l.2.1
Theoretical Formulation
Figure 3.2 depicts the cross section of a rectangular waveguide radiating into a
homogeneous dielectric sheet which is terminated by a perfectly conducting plate. The
large and small transverse dimensions of the waveguide are denoted by a and b.
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28
^o'£ I
^
Waveguide
s:00
*1
a
y.
L.
*
t
a.
z=0
z =d
Figure 3.2: Rectangular waveguide radiating into a conductor-backed slab of lossy
dielectric material.
Considering the dominant TE,0 mode incident on the aperture, the field distribution at
this cross section may be given by
E v(x,y,0) = e0(x,y) =
2
free)
— cos —
'ab
\ a
(x,y) e Aperture
(3.22)
{x,y) g Aperture
where, according to (3.2c) it is chosen such that
2
i J |^ U ,.v ) |" ^ y = —
£
bi'2
j
j
a b y = - b l2 x = - a l2
itn x
cos- — dxdy = 1
V
(3.23)
U
By evoking vector potential formulation the fields in region 1 (dielectric slab) can be
constructed in the following manner. Fields in this region satisfy the wave equation,
V 2n + k j J j = 0
(3.24)
where k, is the complex propagation constant of the slab region. The vector potential /7
satisfies the field conditions for 0 < z < d and can be decomposed into two orthogonal
components
n = <Pax +
v
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(3.25)
29
The field components satisfying Maxwell's equations may then be expanded in the
following manner
_
— dV
E (x,y ,z) = - V x I J = -rf-a x oz x l z Qy+
dy
dx
a.
(3.26)
H (x,y, z) = —!—\kjn + VV •77]
jW o 1
1
d'4>
j( 0 /i0 11 dx
r d 20
(3.27)
J
d *F
.2 ,
ax
dxdy
d 2¥
, 2|„ V
~ B x B y ~ 3p '
p
f d 2<P | d 2 ^
+
dxdz
dydz
a.
Accordingly, general solution for the field components in region 1 satisfying the wave
equation may be presented as a set of integrations over the entire mode domain, namely
JJ
E x (x,y,z) = —
\ - j L A y e jk:,~ + jk. A y e jk:,z
(2 n) J J J
v
-> *
(3.28)
xe~jkxXe ]kyX dkxdkx
Ey, (x,y,z) =
] j \jk.t A%e Jk:‘Z- j L
I- *") —00—00
xe
= yd-f i f
(2 n r
J
(3.29)
jk**e Jk>y dkxdkx
j(on0
(3.30)
kxky
A ^ e Jk:‘z + A y e jk:‘: >e~jk*xe Jky>' dkx dkx
iW o
u (x,y,z)
(
\
Hyt
7
7J 7j \1^
= ——j
(2 n y _
kxkv
~ kv) A y e jkz,~ + A y e >kz,L
J(°Mc
A&e jk:r + A(pejk:>~ >e Jk*xe Jkyy dkx dky
jW o
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(3.31)
30
where kt is the complex propagation constant inside the slab-like material medium with
(3.32 a)
which is chosen such that
(3.32b)
Next, by using Fourier properties of the field solutions given by (3.28) through (3.31)
and applying the appropriate boundary conditions the unknown field coefficients
T will
be determined. Taking inverse transform of (3.27) at the aperture cross section, z - 0 , and
knowing that the x-component of the dominant mode for the aperture E-field is equal to
zero, results in the following simple relation
(3.33)
Further, taking the inverse transform of (3.29) at z = 0 and substituting for the aperture Efield given by (3.23), one can write
rjkxx e~ jJkyy
> dxdv
(3.34)
where the resultant g is introduced to simplify the following manipulations. Similarly, at
z = d interface taking the inverse transform of (3.27) and (3.28) and forcing the boundary
conditions, namely vanishing x and y components of the electric field, results in the
following relations
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31
A+
..I .
0
{ 3 . 35 )
- j 2k, d\
G{kx,ky)
(3.56)
Parseval's theorem is then used for the two sets of Fourier transform pairs {f),F j} and
{f2,F2h where F denotes the transform of/, to construct the numerator of the admittance
function given by (3.16). According to this theorem
J \fi{ x ,y ) f 2 {x,y)dxdy
(3.37)
—o o — oo
1
(2 jc)
J ]F 1(kx ,ky)F2\ k x.l:x)d kxcik,
Taking the following sets of Fourier transform pairs
fj( x ,y ) s Hx (x ,y ,0 )<=> Fj(kx ,ky) = —^ — [(kj -
)(i4j + A# )]
(3.38)
f 2 (x,y) = Eyi (x ,y , 0) <=> F2*(kx,ky) = { /k ( a J - A*)}
and subsequently putting the above back into (3.16) results in an integration over the entire
mode domain for the aperture admittance which can be written as
Y = G + jB
1
(3.39)
oo oo I
/ s t e - * ? ) 2-40 +
jc w 0 {2 x y -co-
jg (k x ,ky}
k,
g[kx,ky)jdkxdkx
where Q(kx,kx) was given by Equation (3.34).
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32
The above integration must be evaluated over two infinite ranges. This can be reduced to a
double integral over a single infinite limit by applying a polar coordinate transformation.
Normalizing the above expression with respect to free-space propagation constant k0 using
a' = k0a , b '- k0b , d'= k0d
(3.40a)
Kx = J - ’ Ky = J - ’ Kj = J - = { £rl - ( Kx + Ky)
(3.40 b)
and subsequently applying a polar transformation of the form
kx = $1 cos 0
(3.41a)
Ky = 3£.sin0
(3.41b)
^ = V v -^ T
(3.41c)
the aperture admittance, normalized with respect to guide admittance, can finally be written
as
(3.42)
ys = 8 s+jl>s
J i g(x.,o} (eri - %} cos
(2 n Y
I - | £ ' *.=00=0
2
oj
J
J
x ‘J ld 6 d‘J l
where
I a \ ■ f^'^-Sin#^
(a'%. cosd
Jcos
----j 2 a- ( 4 x ) s m [ -----
g(n.,o) = .
b'
(3.43)
(!£sin 0 )[;r2 - ( a ^ c o s t f ) 2]
g { ^ 0 ) e jK‘D
iK jS l^ K jd ')
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(3.44)
33
It should be mentioned that for a lossless dielectric slab singularities occur when integrating
(3.42). One can overcome this problem by integrating over a contour around singular
points [Col.60]. This case has not been dealt with in this work since only generally lossy
dielectric materials are of interest here.
111.2,2
Verification o f The Theoretical Results
To gain a better insight into the problem, and understand the implications of the
admittance formulation presented here, the numerical results of some test cases are
presented next. As preliminary verification, for the special case of termination into an
infinite half-space (i.e. very long electrical length), the results of the theory are initially
compared with those obtained by Lewin's infinite half-space formulation
[Lew.51}.
A more
detailed verification of the numerical results is presented in the next chapter. Table 3.1
shows the computed admittance results for three lossy dielectrics having different dielectric
characteristics at 10 GHz. The dielectric values for this comparison were chosen to
represent materials such as plexiglass and two types of synthetic rubber materials. Clearly,
there is good agreement between the two approaches, considering that the values calculated
from (3.42) are based on finite number of integration points (a Gauss quadrature method
was used).
Table 3.1:
Comparison of the results of Equation (3.42) with those using Lewin's
infinite half-space [Lew.51}.
r ...............
~
e
tan o
y s (Equation 3.42)
2 .6
1
x
0 .0 1
2.074 + j0.555
7.0
1
0 .1
3 .4 4 1 +j0.062
3.432 +j0.062
10.5
!
0.238
4 .1 6 6 -j0.169
4.166- jO. 166
ys (Lewin)
i
2.068 + j0.556
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34
The effect of slab dielectric properties, frequency, and thickness variation on the
aperture admittance are investigated next. The wavelength inside the dielectric medium can
be written as
7
■FZ
.
K
_ (ta n 5 }
8
+
0
j(tan # )4J
C3.45)
for lossless dielectricrs
Approx. for low -loss dielectricrs
where £ \m is the permittivity of the medium and
0
represents the sum of all higher order
terms. The approximation used in (3.45) is valid for most generally lossy dielectric media
examined in the microwave frequency range. For instance, even for a lossy dielectric
material with tan# of around
0 .1
the contribution of the second term inside the bracket is
approximately two orders of magnitude smaller than the first term. This expression also
indicates that assuming the dielectric properties of a material remains constant within a
frequency band, the frequency and permittivity variations would present similar effect on
the electrical length (2nd/Xm) of the medium. Meaning, increase in frequency leading to
smaller free-space wavelength consequently increases the electrical length of the medium
and vise versa. Similarly, increase in permittivity also decreases the electrical length of the
medium.
Variations of normalized conductance, gs, and susceptance, bs, as a function of slab
thickness over a free-space electrical distance of 1 X0 for three lossy dielectric samples at
10
GHz are shown in Figures 3.3a and 3.3b. The slabs have permittivity, e'n equal to 10, 5
and 5 respectively and corresponding loss tangents, tanS, of 0.1, 0.1 and 0.01. These
values were intentionally chosen so that the first two slabs have the same tanS with
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35
different permittivities. The second and third samples differ in their losses (i.e. £"r) only.
It can be observed that in both figures the first sample having a dielectric constant twice
larger than the other two has a shorter period of oscillation which is expected due to its
longer electrical length.
Consequently, the second and third slabs display similar
periodicity and eventually a faster damping response for the higher loss sample. The real
and imaginary parts of the admittance pertain to the radiation and storage of energy at the
aperture. Also, the oscillation between negative and positive values of the sign of
susceptance, bs, can be interpreted as the aperture being terminated by a capacitive or
inductive load for different sample thicknesses.
Figures 3.4a and 3.4b show variations of conductance and susceptance as a function of
slab thickness for three different frequencies in C, X, and A'-band. The dielectric constant
of the slab was chosen to be e'r = 5.0 and tan 8 = 0.1. Once again, similar behavior for gs
and bs can be observed for this case pointing to the change in effective electrical thickness
of the material. So, as explained earlier, both the operating frequency and dielectric
properties of the material play similar roles. The results of Figures 3.4a and 3.4b simply
demonstrate that as the frequency increases the depth of penetration decreases and
consequently the electrical length of the medium quickly reaches that of an infinite halfspace. This is true in practice given that the dielectric properties of the medium remains
relatively constant over the frequency range.
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36
16
= 10.0 , land = 0.1
14
= 5.0 . tanS = 0.1
©C
= 5.0 . tanS = 0.01
12
§
<o
10
8
6
£
:§
4
2
0
0.5
1.5
2.5
d (cm)
Figure 3.3a : Normalized conductance as a function of slab thickness for three different
lossy dielectric materials.
g
-20
-30
0
0.5
1.5
2
2.5
3
d (cm )
Figure 3.3b : Normalized susceptance as a function of slab thickness for three different
lossy dielectric materials. Legend same as Fig. 3.3a.
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37
• ■ X J- '
14
• —
1
/ = S.O GHz
- • e- - /= 10.0 GHz
/ = 2 4 . 0 G Hz
10
■i
<3
•«
s
0
0.5
1
1. 5
2
2.5
d (cm)
Figure 3 .4 a : Normalized conductance as a function of slab thickness for three different
frequencies.
10
-o
a
-10
-15
a
-20
-25
-30
0
0.5
1.5
2
2.5
d (cm )
Figure 3.4b : Normalized susceptance as a function of slab thickness for three different
frequencies. Legend same as Fig. 3.4a.
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38
I I I .3
R adiation F rom R ectangular W aveguide into S tra tified C om posite
M edia
The theoretical formulation presented earlier is next expanded to treat the general
problem of aperture admittance radiating into a stratified dielectric media. Fourier
transform boundary matching procedure is once again used to construct a complete set of
solutions for the external, z > 0 , medium. Consequently, the solutions are used in the
stationary admittance expression to achieve the desired solution. The versatility of such a
model for near-field in situ interrogation of stratified composite media arises from the fact
that it allows addressing non-contact as well as contact type measurements.
II I .3.1
Theoretical Form ulation
TE and TM field components are expanded in each layer in terms of Fourier integrals.
Subsequently, appropriate boundary conditions across boundary interfaces are enforced to
solve for the unknown field coefficients in each medium. Figures 3.5a and 3.5b depict the
cross section of an open-ended waveguide radiating into a layered medium which is
terminated into an infinite half-space and a perfectly conducting sheet respectively. Each
layer is assumed to be homogeneous and nonmagnetic with relative complex dielectric
constant ern.
£rN -l
Waveguide \
b'
-'n
L,
- =0
Figure 3.5a : Cross section of a rectangular waveguide radiating into a layered media
terminated into an infinite half-space.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
£n
CrN
(J—oo
Waveguide \
b
L
“N- I
z„=0
~N
Figure 3.5b : Cross section of a rectangular waveguide radiating into a layered media
terminated into a conducting sheet.
By evoking vector potential (or magnetic Hertz potential) formulation, the fields in each
region can be constructed in the following manner. In each layer, denoted by layer number
n, fields must satisfy the source-free wave equation
v 2n „ + k 2n n = o
(3.46)
and the vector potential can be represented in terms of its scalar components by
(3.47)
n n =
Subsequently, the field components in each region in terms of the vector potential can be
written as
En(x,y,z) = -V x Fln
(3.48)
Hn(x, v,z) = — — \k 2nT in + VV • n n]
JW o1
(3.49)
J
General solutions of (3.46) for the components of the vector potential in each bounded
layer are
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where n denotes the layer number. Referring to Figure 3.4a, for the case of termination
into an infinite half-space where the /Vth layer is unbounded in positive z-direction only
forward travelling waves exist. For this region the scalar potentials take the following form
A4
(2n)2 L L
.
-jk* z
<?
'
J
-/V
-j(kxx+kyy)
dkxdkx
(3.51)
\*:\
The field components of (3.48) and (3.49) can be written in terms of the scalar potentials as
1
E " ^ a
x
*
'
f d 2* n
d2V
JW o v ~ d ? - + ^
E.n = I
dy
dx
j m Q dxdz
ar
kn0n
1
, H ?= j m Q 3xdy
,2
2
dy2
'
(3.52)
a,
” ”
dydz )
General solutions for the electric and magnetic field components in each layer, in
accordance with (3.50), may be expressed in terms of integration over the entire mode
space as
//•
oo
oo
+A+ e 2k:"~ ±A~ e2k:"~
{1:1
C:}
x e - H ^ . y ) dkKdkt
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.53)
41
7
^ ( W
{;}
oo
oo
1
1
) =7T 3
7
A„2
—
( 2 tt) 2 _ „ o _ o o ;< y /i0
KxKy a +
IS)
4+
-/:2
fch
e A'- ' + <4
(S)
e
-
0.54)
C;}
£- ' a -: + /T
IS)
where kn is the complex propagation constant in each layer, and
(3.55a)
*2. = 4 kl - k* - k.
which is chosen such that
RejfcZn| > o , Im|A:Zn} < o
(3.55b)
and similarly, in accordance with (3.51) the field components in region N for the case of
termination into an infinite half-space take the following form
(3.56)
J
H i{;}
a (w
oo
oo
1
) = 7(2Tn y7 -Iao-aoja
I — ii0
•..2 _ ka
kN
{;}
y\ J
A+ - k xk ,A +
i:;l
• is )
0 .57 )
Next, the following normalized (with respect to k0) parameters are introduced into the
formulation
zn - ka 'L dj
i=l
kx v
K* . y = - r =
, a' = k0a
' K ,< ~
, b'= k0b
Ai
~i~ ~ \ £ r„
(3.58)
x
~ Kx ~
Y
Ky
fat
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
where n = 1,2,...N, d, is the thickness of the ith layer, and index I refers to
or <t>
components. Taking inverse transform of both components of (3.53) at the aperture cross
section, z = 0 , results in the following relations between the field coefficients in layer
1
(3.59)
4n
2a'
jK i\b '
Kxa
-cos
sin
k ^ j :2
G(KX’Ky )
(3.60)
JKi
Forcing the appropriate boundary conditions, namely continuity of the transverse field
components across each interface and vanishing tangential E-field over the conducting
surface expressed as
= z„) = E’ *‘(x.y,z = :„)
E”
for 1 < m < N -1
(3.61a)
\
form = N
K(*.y.z = * . ) «
(3.61b)
EfM{x,y,z = zH) = 0
(3.61c)
results in a set of relations among the field coefficients in the entire medium. It must be
noted that according to geometries of Figures 3.5a and 3.5b, the last boundary condition
stated by (3.61c) is only realizable for the case of Figure 3.5b where N boundary interfaces
exist. Consequently, forcing boundary conditions of (3.61a) and (3.61b) results in the
following set of relations between the field coefficients of adjacent layers
jq+
e JKmzm
Ci }
L
m'
(
(3.62)
\<Pm /
=
5 ”±!
.
te ', 1
te l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Consequently, when the medium is terminated into an infinite half-space the above
expressions take the following form at the (AM) 1*1 (last) interface
0 ~ j KNZN —
J3+
12
"
N -l
(3.64)
I’M
W -,J
)
er —k
.
K
it
{;}
+A+
ft )
q
J*NZN-I
(3.65)
121J
£,
- K
M
IyJ
jw
w -j
ft:;)
ft::)
jw
For a layered media terminating into a conducting sheet, the vanishing tangential E-field
components over the conductor surface renders an additional set of relations between the
field coefficients in the N^ 1 layer
53
(2 1
—p j^ KNzn a +
(3.66)
(2 1
Solving a simultaneous 4N-2 system of equations for the N-layered case terminated into an
infinite half-space results in the sought for field coefficients. Similarly, a system of 4N
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44
equations is needed for the N-layer medium terminated into a conducting sheet. In matrix
form the system of equations may be presented as
m-
C U \ " Cln
(3.67)
X
C n l '" C n n
nxn
nxl
r
J
m
nxl
where the Cnxn and $ nxj represent complex coefficient matrices relating the field
coefficients in the entire medium arising from (3.62) to (3.65). A simple matrix inversion
would then render the solutions for the field coefficients which can be written as
- 1
a *
Cl l \ " C l n
=
n • *h
X
C n l " ‘ C-nn
nxl
;
=5*
*
<3*
nXn
(3.68)
nxl
Finally, with the denominator of admittance expression of (3.16) equal to unity as a
result of normalization of the aperture field distribution, the numerator can be written as
oo
oo
| ([E (x ,y , 0 ) x H ( x ,y , 0 )]-az dxdy
(3.69)
—oo —oo
= - J ]Ell(x,y,o)H lx (x,y,o)dxdy
and choosing the Fourier transform pairs similar to that of (3.38), application of Parseval’s
theorem allows construction of the admittance expression. Once the field components in
layer 1 are calculated according to (3.68) and substituted into (3.69), they result in the
following admittance expression
oo
yn=
(2 k ) - U
oo
L . +M ^ )1
J
- 2
K
x K vA
.
y
I
K ‘
X (j(tC x. , K v j c l K v (j!Cv
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3-70)
45
which is normalized with respect to the free-space characteristic admittance. By applying a
polar coordinate transformation of the following form
kx =
cos 6 , Ky = 3(.sin 6
(3.71)
having a transformation Jacobian of the form
dKx 9 k x
3=
d(* X’Ky)
d(m,Q)
d%. d d
dK y dtcy
dm
(3.72)
de
Admittance expression of (3.70), which involves two infinite integrals, may be evaluated
over a single infinite integral only. Finally, normalization with respect to the guide
admittance renders the desired form of aperture admittance as
J
ys = gs + jbs =
X
j
(3.73)
it
7 jk
!R.-0«-o(
-
(ff,cosfl)2] | 2 A % 1 + J^
\
1
-'.0 ). - 2 ^ 2 sin 0 c o s0 J? ^ [
K1 J
J
xg(‘X.,d)fHcied3l
where Kj and Cj{$L,6 ) are given by (3.41c) and (3.43) respectively.
A lossless stratified medium is not the focus of this work, however, for such a case one
encounters singularities in (3.73). This can be remedied by integrating on a contour around
the singular points of the integrand located on the real axis while taking into account the
residues. In application to a composite medium which contains at least one generally lossy
layer, which is the case addressed here, the poles of the integrand move off of the real axis
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
and the integrand becomes smoother. Ultimately, this allows quick and efficient numerical
integration schemes such as a Gauss quadrature method (used here) to be applied.
II 1.3.2
Termination o f Layered Media into a Lossless Half-Space
To examine the validity and limitations of the numerical results presented above,
initially the the results of the two scenarios of layered media terminating into an infinite
half-space and into a conducting sheet were compared with each other. The geometry
configuration of the problem inferred from that of Figure 3.5a is shown in Figure 3.6. The
specific solution pertaining to this case as well as that of the three-layer medium terminated
into a conducting sheet are presented in appendix A.
Waveguide
Infmle
Half-Space
0
z= d
d, +d-
Figure 3.6 : Cross sectional view of a layered media terminating into an infinite half-space
of lossless dielectric material.
It was observed that for the case of layered media terminated into an infinite half-space
of lossless dielectric (e.g. free-space) deviations would occur for introducing negligible
amount of loss to the dielectric constant of the terminating infinite half-space medium.
Such inconsistencies did not occur when a conducting sheet was placed in the back of the
dielectric sample or the terminating half-space media was a lossy material. This, in effect,
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47
ruled out the possibility of potentially incorrect assumption of dominant mode aperture field
distribution (this would have influenced both cases, whereas now only one case is strongly
influenced). Figures 3.7a and 3.7b show variations of conductance and susceptance for a
slab of thickness d2 = 7.55 mm, e'r = 8.4 and tand = 0.107 over a distance of A</2 from
the aperture for three theoretical runs with free-space permittivities of
1
and tanS of 0 ,
lx lO '3, and lx lO '5. At first glance, for all practical purposes, loss tangents of this
magnitude once compared with that of the sample seem to be negligible (lossless
assumption of free-space permittivity is a common practice in theoretical formulations).
However, this can lead to significant error in the numerical calculations for a medium
terminated into an infinite half-space of air. This is verified by the fact that as the loss
tangent is reduced by two orders of magnitude (from lxlO "3 to lx lO '5) the theoretical
variations of the conductance and susceptance become insignificant and the solution
converges quickly so long as an infinitesimal amount loss is introduced to the free-space
dielectric constant.
This is a significant finding which shows the sensitivity of numerical
techniques to these types of issues. This means that one must be careful when comparing
the results of such numerical approaches, when they are normalized to the free-space
properties (e0 or k0), to experimental results. As it will be demonstrated in the following
chapter, much better agreement between the measurement and numerical results is obtained
when infinitesimal but finite amount of loss is associated with the free-space dielectric
constant. This is a valid observation due to the fact that in every non-ideal situation the air
always contains a certain percentage of moisture leading to losses in the medium.
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48
5
—tan 5 = 0.0
land = 1.0x10
4
tan S = 1.0x10
8
§
3
<o
2
"5
I
1
'
Jgyo^-c-o-e-fl-o-fr
0
0
2
4
6
8
10
12
14
16
A ir G ap. d { (m m )
Figure 3 .7 a : Effect of infinitesimal loss in the terminating medium on numerically
calculated value of conductance.
1. 5
1
0.5
Oi
8
CO
.a.
-0. 5
-1
-1. 5
0
2
4
6
8
A ir Gap, d
10
12
14
16
(m m )
Figure 3 .7 b : Effect of infinitesimal loss in the terminating medium on numerically
calculated value of susceptance.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IH .4
Sum m ary and Rem arks
Theoretical analysis of radiation from a rectangular waveguide into layered dielectric
composite media was presented in this chapter. Initially, variational formulation was
evoked to come up with a stationary expression for the terminating aperture admittance of
general cylindrical waveguides with arbitrary cross section. With application of Fourier
transform boundary matching technique solution was first given for the terminating
admittance of a conductor-backed slab. Upon modification of the geometry, the numerical
results of this case were verified against that of Lewin for an infinite half-space medium in
front of the waveguide. Formulation was then expanded to take into account general Nlayer media terminated into an infinite half-space or a perfectly conducting sheet. Some
important observations were addressed in regard to termination of the layered media into an
infinite half-space of lossless material.
In the following chapter, several measurement results will be presented and compared
with the numerical results of this chapter. These would serve both as further verification of
the theory and also for demonstration of practical implications of the work.
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50
C H APTER IV
R ectangular Waveguide Sensor fo r Exam ination o f Layered Dielectric
Composite M edia
In this part of the work, measurement results are presented both to test to a greater
extent the integrity of the theoretical formulations presented in the previous chapter, and to
further expose their practical implications. Several cases in connection with estimation of
thickness, dielectric properties, and detection and evaluation of thickness variations in
layered dielectric composite media are presented. Explicit references are given to such
cases as dielectric composite coatings on top of conducting plates and disbonds in layered
dielectric composite media. Furthermore, sensitivity analysis using swept frequency
measurements are presented to demonstrate the significance of the theoretical expositions in
optimizing the measurement sensitivity and resolution. For most cases pertaining to
thickness variations in layered media experimental measurements are performed using a
precise mechanical device built specifically for this purpose. As standard practice, a square
flange with sides greater than 1X0 is used to approximate an infinite ground plane
[Cro.67].
The validity of dominant mode assumption for the waveguide, based on standard practice is
confirmed by numerous experimental results presented throughout this part of the work
which show good agreement with the theory.
IV . 1
C onductor-B acked Lossy Dielectric Slab
Practical implications pertaining to theoretical modeling for the conductor-backed
dielectric slab of Figure 3.2 are discussed next. The theory is used to estimate such
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51
parameters as dielectric properties and thickness of the slab-like media backed by a
conducting sheet. Several measurement results in conjunction with the numerical outcomes
are presented in application to estimation of coating thickness and dielectric properties on
top of a conducting sheet. Results of an automated swept frequency measurement is also
presented for this scenario which display the importance of numerical simulations for
optimization of parameters for maximizing measurement sensitivity and resolution.
For measurement purposes, the normalized aperture conductance gs = R e{yJ and
susceptance bs = Im {yJ can be measured via the measurement of the VSWR =
(J+r)/(l-D
and standing wave null displacement with respect to the null location of a short circuit. The
complex reflection coefficient, R =Tei<!>is related to the normalized complex admittance ys,
by R = (l-y s)/(l+ y s). Thus, the magnitude, T, and phase, <2>, of R can be measured
directly via VSWR and shift in the location of the null of the standing wave with respect to
a short circuited waveguide
IV . 1.1
[Gan.93\.
E xperim ental R esults vs. Theoretical Observations
To experimentally verify the theoretical results of the previous chapter for the
admittance expression of (3.42), VSW7? and phase of the reflection coefficient as a
function of slab thickness were initially compared with a series of experimental results.
The experiments were performed at the frequency of 10 GHz (X-band) for three different
dielectric samples. These samples consisted of a low-loss and two higher loss dielectric
materials. Dielectric properties of these samples were measured using a short circuited
waveguide method
[Alt.63].
Figures 4.1a and 4.1b show these results for thickness variations for a slab of
Plexiglass with e'r = 2.59, tanS = 0.007. Similarly, Figures 4.2a and 4.2b show similar
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52
results for a slab of synthetic rubber with e'r = 7.25, tanS = 0.103, and finally Figures 4.3a
and 4.3b for a slab of similar type material with e'r = 12.6, tanS = 0.19, respectively. For
all cases the theoretical and experimental results follow each other closely. As discussed
earlier, with the increase in electrical length of the dielectric sample as a result of increase in
loss tangent, both VSWR and phase show greater number of periods. Moreover, as
expected, the material with smaller loss tangent displays slower VSWR damping. On the
contrary, the response for the slab having the highest loss tangent, shown in Figure 4.3a
and 4.3b, a much faster damping behavior is observed which quickly reaches the electrical
length close to that of an infinite half-space.
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53
T heory
o
30
Meas.
20
0
0.5
1
1. 5
2
d (cm)
Figure 4.1a:
Comparison of the theoretical and measurement results of VSWR as a
function of slab thickness for f = 10 GHz, e'r = 2.59, tanS = 0.007.
200
15 0
100
50
c
-50
-100
-150
-200
0
0.5
1.5
2
d (cm )
Figure 4.1b: Comparison of the theoretical and measurement results of phase as a
function of slab thickness for f = 10 GHz, £’r = 2.59, ta n d = 0.007.
Legend same as Fig. 4.1a.
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54
25
o
Theory
Me as.
20
15
10
5
0
0
0.25
0.5
0.75
1
1.25
1.5
d (cm)
Figure 4 .2 a : Comparison of the theoretical and measurement results of VSWR as a
function of slab thickness for f = 10 GHz, e'r = 7.25, tand = 0.103.
200
15 0
100
50
-50
-100
-150
-200
0
0.25
0.5
0.75
1
1.25
1.5
d (cm)
Figure 4.2b : Comparison of theoretical and measurement results of phase as a function of
slab thickness for f = 10 GHz, £'r = 7.25, tan 8 = 0.103. Legend same as
Fig. 4.2a.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
30
— T heory
o Meets.
20
0
0 .2 5
0 .5
0 .7 5
1
1 .2 5
1 .5
d (cm )
Figure 4 .3 a : Comparison of theoretical and measurement results of VSWR as a function
of slab thickness for f = 10 GHz, £'r = 12.6, tand =0.19.
200
150
100
50
-5 0
-100
-1 5 0
-200
0
0 .2 5
0 .5
0 .7 5
1
1 .2 5
1 .5
d (cm )
Figure 4.3b : Comparison of the theoretical and measurement results of phase as a
function of slab thickness for f = 10 GHz, £’r = 7.25, tand = 0.103.
Legend same as Fig. 4.3a.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
IV .2
Practical Im plications
Accurate thickness measurement of dielectric slabs backed by a conducting plate is an
important issue in various industries. Several applicable examples are thickness of the
following materials over a conducting plate such as paint, thermal barrier coatings (used in
high temperature environments), adhesive layers, oxidation layers, bonding layers and
synthetic rubber layers in dry fuel rocket casings. Accurate dielectric property variation
monitoring of various ceramics and composite materials is also important. The dielectric
properties of a medium can be directly related to its important physical properties such as
moisture content, porosity, density, etc..
The aperture admittance of a waveguide terminated into a composite media at a given
frequency is a function of the relative dielectric properties of the slab (£,. = e'r - je "r,
tand = £"rle'r ) and its thickness d. Thus, admittance measurement may render thickness
information if the dielectric properties of the slab are known and vice versa by solving an
inverse problem. The slab thickness (or dielectric properties) can be determined from the
expression for admittance of an open-ended rectangular waveguide radiating into a
conductor-backed dielectric slab.
IV .2.1
Estim ation o f Dielectric Properties a nd Thickness
To calculate the thickness or dielectric properties of an unknown sample, conductance
gs and susceptance bs expressions may be treated separately in a root finding code in a
recursive manner
[Bak.92].
Starting with initial lower and upper bounds for thickness, gs
and bs are iterated alternately to find the root which is a close estimation of thickness or
dielectric properties of the sample. For both cases the root is found within some prescribed
range which can initially be estimated from plots of the variations of the parameters over
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57
some larger interval. This would also decrease the calculation time since a finer range can
be estimated in advance for the position of the root.
In general, for monitoring thickness or dielectric property variations, a priori
knowledge of these parameters is usually available. Figures 4.4 and 4.5 display typical
plots of gs and bs for two fixed slab thicknesses. Such plots over a wide range of dielectric
properties may used to come up with a finer initial range for estimation of the properties.
Here, for a known sample thickness the real and imaginary parts of the complex dielectric
constant are plotted over a large interval. Similar plots may be used to come up with an
appropriate initial estimate for the range of dielectric properties.
As stated earlier, as the loss tangent decreases, the integration of the admittance
expression requires that finer increments be evaluated due to slower convergence. This
simply means that the integrand approaches the singular behavior. However, as the loss
tangent increases, the integrand becomes smoother and the integration becomes more
manageable (faster convergence). Due to the fact that one of the main concerns in this type
of measurements is faster calculation speed, open interval integration methods which
commonly use convergence tests are rendered ineffective. Here, a Gauss quadrature
method with a fixed number of intervals was used to perform the integration. For such a
case it is vital to find, a priori, the necessary number of intervals for appropriate
convergence in the range of dielectric characteristics of the medium under test. Once an
upper bound on the necessary number of intervals is chosen the integrations can be
performed quite efficiently and accurately.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
2 .5
- - o* -
=
2.0
•a - e-r = 8.0
■a - e'r = 74.0
« — e' = 20.0
-cr
--
-Or -
0 .5
©-
-3
-2 .5
-2
-1.5
-1
-0 .5
0
£"
Figure 4 .4 a : Conductance as a function of dielectric properties for a slab of thickness
d = 2 mm.
2
1
0
1
9 -- - - - - -
2
-o
■o-
3
3
-2 .5
2
1.5
1
-0 .5
0
£"r
Figure 4.4b : Susceptance as a function of dielectric properties for a slab of thickness
d = 2 mm. Legend same as Fig. 4.4a.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
12
10
OC
=
8
— -® -
2.0
8.0
14.0
S§
20.0
6
N
<3
:§
2
0
3
-2.5
■2
1.5
1
- 0. 5
0
Figure 4 .5 a : Conductance as a function of dielectric properties for a slab of thickness
d = 10 mm.
-Cl
20
2
805
— ©-
3
2
1.5
1
e"
Figure 4.5b : Susceptance as a function of dielectric properties for a slab of thickness
d = 10 mm. Legend same as Fig.4.5a.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
IV.2.2
Estimation o f Coating Thickness
Thicknesses of several lossy dielectric slabs from two types of synthetic rubber at Xband were measured using the procedure outlined earlier. Different samples in sheet form
with known dielectric characteristics were chosen.
Using the measured dielectric
properties, the thicknesses of the samples were calculated using the above procedure. The
results of these experiments are shown in Tables 4.1 and 4.2. For both samples the
calculated thickness values were within three percent of their measured values. The first
column shows the physically measured values of slab thickness. The variations of the
calculated results in column three are indication of how close the root could be estimated in
the domain of both gs and by Better resolution can be achieved by choosing an optimum
frequency based on the material thickness and its dielectric properties.
Table 4.1: Thickness measurement result at 10 GHz for £'r = 7.25 and tand = 0.103.
T iZ f
3.91 ± 0 .0 4
5.92 ± 0.04
7.77 ± 0 .0 5
11.56 ± 0 .0 5
!
|
1
|
j
’ .<***■>
0.698 + j 1.433
10.030+ j l . 610
2.374 - j 1.212
5.570 +'j 1.850
Calculated
dim m )
j
1
„ a*
c
% Mean Error
3.9 ± 0.05
5.83 ± 0 .0 5
7.8 ± 0 .1
11.38 ± 0 .1 9
I
1
j
1
0.26
1.52
0.40
1.56
Table 4.2: Thickness measurement result at 10 GHz for e'r = 12.6 and tan 8 = 0.19.
7 (Z ?
3.63 ± 0.07
3.76 ± 0 .0 5
5.90 ± 0 .0 7
7.43 ± 0 .0 6
8.20 ± 0 .0 5
12.05 ± 0.06
13.0 ± 0 .0 5
1
1
|
j
1
1
I
1
y . <"**■>
3.67 + j4.25
4.85 + j4.83
2 .8 0 - j 1.69
3.08 + j 1.64
3.88 + j2.55
4.49 + j0.98
5.18 + j 1.17
Calculated
d(m m )
1
|
„
r
* Mean Error
3.59 ± 0.02
3.76 ± 0 .0 2
5.89 ± 0.06
7.45 ± 0.05
8.03 ± 0.05
12.20 ± 0 .1 0
12.64 ± 0.06
i
0.1
0.1
0.2
0.3
2.2
1.2
2.8
1
|
|
|
j
R eproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.
61
These experiments were performed to emphasize some important points that must be
taken into account in this type of measurements. To increase the sensitivity for thickness
measurements, the electrical length of the dielectric slab should not be such that it would
constitute an infinite half-space. This fact is clear from Figures 4.4a and b which show
more abrupt variations at smaller thicknesses.
Furthermore, an optimum operating
frequency must be chosen such that any possible dielectric constant variation would have
minimal effect on the thickness measurement due to sensitivity of the microwaves to
dielectric property variations.
IV .2.3
F requency A nalysis F or Sensitivity M axim ization
To illustrate the importance of frequency selection for maximum measurement
sensitivity (resolution), the results of several experiments are presented. The results
indicate that thickness variations in the order of a few microns or smaller can be detected at
relatively low microwave frequencies (around 10 GHz). Sensitivity to the detection of
dielectric property variations is also shown to be frequency dependent. Important practical
aspects of this technique are discussed in this section.
A swept frequency analysis provides for determining the most sensitive frequency (or
range of frequencies) at which small thickness variations can be detected. This can be
thought of as an optimization technique to achieve the best possible measurement accuracy.
It is an established fact that when the complex relative dielectric properties of slabs are to be
estimated, measurement confidence can be substantially increased if measured data from
various thicknesses of the same slab (multiple independent samples) are obtained. On the
other hand, if only one thickness of the slab is available, one can alternatively make the
measurements at several frequencies (swept frequency) to increase the measurement
confidence.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
IV .2.3.1
M easurem ent Apparatus
Figure 4.6 shows the measurement apparatus which consists of a dual-arm microwave
reflectometer. The oscillator generates the swept frequency signal which is then fed into a
waveguide Tee. The Tee splits the signal into two equal (phase and magnitude) signals
each entering one of the reflectometer arms. In the upper arm the signal is fed into an openended waveguide which is in contact with the conductor backed dielectric slab. The
reflected signal subsequently is coupled into the attached directional coupler and finally into
the test port of a microwave network analyzer. The other half of the input signal enters the
lower arm and is reflected off of a sliding short circuit. The reflected signal in this arm is
then fed into the reference port of a microwave network analyzer for comparison (phase
and magnitude) with the test signal. The sliding short in the reference arm allows
compensation for the extra length of the waveguide in the test arm over the entire range of
frequency sweep. Once the device is calibrated with respect to a short circuit at the test
port, any detected change in the measurement parameters would be as a result of changes in
the dielectric-conductor medium. The use of a computer controller via an A/D board allows
real time processing and display of the measurement results over the swept frequency
bandwidth.
IV .2.3.2
Verification and Analysis o f the R esults
To verify the consistency between the theory and measurement some initial experiments
were conducted. Figure 4.7 and 4.8 show the comparison of the phase of reflection
coefficient,
between the theoretical and experimental results for a lossy dielectric
samples (synthetic rubber er = 12.4 - j2.4) of thickness 2.08 mm and 4.28 mm at X-band
(8.2 -12.4 GHz) respectively. Generally, there is good agreement between the theory and
measurement for both cases. The small deviation of the measured points from those
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
predicted by the theory is attributed to the nonlinearities in the measurement apparatus,
slight variation in the dielectric properties of the material in the swept frequency range and
partially to the calibration process. Very good calibration can be achieved for a single
frequency, however, over a wide band of frequencies it becomes poorer. This problem
may be remedied by post processing of the measured data.
r
Computer
Controller
Processor
t
Sweep Oscillator
Network Analyzer
Dielectric Slab
RF Out
Tee
— nfc Matched Load
^ - v
\
\
\
/ \ / x' / S
\
A
\
-I
n.
V
/ s / N. / ^ / s / s / -v ✓« v _ X
Four Directional Couplers
X X X
4
---• •:••••x
- : n
Sliding Short
Figure 4.6 : Apparatus for swept frequency measurements.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
100
Theory
Meas.
50
e
-50
-100
10
10.2
10.6
10.4
10.8
11
F re q u e n c y (G H z)
Figure 4.7 : Comparison between the theoretical and measured phase for a lossy dielectric
sheet with d = 2.08 mm, and er = 12.4 - j 2.4.
200
Theory
Meas.
150
100
50
-50
-100
-150
-200
9.5
9.75
10
10.25
10.5
10.75
11
F req u en cy (G H z)
Figure 4.8 : Comparison between the theoretical and measured phase for a lossy dielectric
sheet with d = 12.8 mm, and er = 12.4 - j 2.4.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
To better understand the measurement resolution and sensitivity due to small variations
in the dielectric material thickness an experiment was conducted on a 2 mm coating of a
material commonly used as thermal barrier coating (TBC) in extremely high temperature
environments. This coating on top of a conducting plate may be considered in the category
of low-loss, high permittivity ceramics. The relative dielectric constant of this coating was
measured to be £r = 20 - j0.02 at 9 GHz [Bak.92] [Gan.92]. Typically the accuracies by which
the permittivity and loss factor are measured are around 5% and 10% respectively. The
thick solid line in Figure 4.9 represents the measured results from 8 to 9.5 GHz. The thin
solid line is the theoretical predication. Clearly the two results are in fairly good agreement.
The deviation of these two lines from each other is attributed to the factors stated earlier.
Subsequently, the theoretical results for a ±50 micrometer change (2.5% variation) in the
thickness of this coating were obtained and are shown as dashed lines. The results indicate
that at frequencies higher than 9.5 GHz the resolution is relatively poor. However, in the
range of 8 - 8.5, a 50 micrometer thickness change causes about 25° of phase change.
This translates to a resolution of about 2 microns per degree. It is important to note that <I>
can be measured accurately within one degree. Clearly, the choice of operating frequency
strongly affects the measurement resolution. From Figure 4.9 it is also evident that lower
than 8 GHz may render higher measurement sensitivity (the measurement apparatus here
was limited to X-band region). The most important conclusion that may be drawn from
this analysis is that resolutions in the few micron range can be achieved even at relatively
low microwave frequencies. Normally this type of resolutions are associated with the
millimeter wave frequency.
The effect of permittivity variations on the phase of the reflection coefficient was also
investigated. Again the same thermal barrier coating with its results depicted in Figure 4.9
was used. In Figure 4.10 the solid lines are the same as those described in Figure 4.9 and
the dashed lines show the effect of phase variation due to the 5% permittivity change. It is
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
clear from these results that such small changes in the permittivity of the dielectric coating
may easily be detected over the appropriate frequency range. The results display a large
phase difference (in average better than 50°) over the entire frequency range. Figure 4.10
shows excellent discrimination over the entire sweep. However, one can see that beyond
9.5 GHz all of the curves converge. This means that by arbitrarily choosing a frequency
(e.g. 10 GHz) for permittivity discrimination, one may not obtain the most sensitive
results. Whereas, an analysis such as the one shown in Figure 4.10 provides for the most
sensitive frequency of operation. Moreover, one can see that accuracies of better than the
conventional 5% for permittivity can be obtained using this technique.
100
M eas.
50
— Tlteo. d - 1.95 mm ~~
Theo. d = 2.05 mm
-50
-100
-1 5 0
8
8.2 5
8 .5
8 .7 5
9
9 .2 5
9 .5
F requency (G H z)
Figure 4.9 : Comparison between the theoretical and measured phase for a low-loss TBC
sample with er = 20 - j 0.02 for three different thicknesses.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
200
le a s .
150
Theo. cr = 20
-
100
-
' "
- j 0.02 ~
Theo. e = 21 - i 0.02
r
‘
Theo. C = 19 - j 0.02
r
1
'
50
-50
-100
-150
-200
8
8 .5
9
9 .5
F req u e n c y (G H z)
Figure 4.10 : Comparison between the theoretical and measured phase for a low-loss
TBC sample with d = 2.0 mm for three different permittivities.
Figure 4.11 shows the theoretical and experimental results for two samples of lossy
dielectric materials with er = 12.4 - j2.4 and thicknesses of 2.08 mm and 2.18 mm.
respectively. Once again the theory was used to deduce information about the appropriate
range of frequency in the X-band region which would result in maximum measurement
sensitivity.
The results show significant difference between the two phases, both
theoretical and experimental, due to 0.1 mm change in the material thickness. Calculation
of the phase difference between the two samples at around 10.3 GHz (about 150°) shows
variations of less than 1 micron per degree.
To analyze the resolution and sensitivity theoretically, the thicknesses of the two
samples were changed by 10 microns and their respective phase differences (with respect to
the actual thickness of each sample) were studied. Figure 4.12 shows both the theoretical
and experimental phase differences of samples in Figure 4.11 along with two theoretical
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
simulation results displaying the variations of phase difference due to 10 microns of change
in the thickness of each sample. There is a difference of about 30° for both samples due to
10 microns of thickness variation. This roughly translates to a change of less than 1
micron per three degrees. Keeping in mind that phase can be measured with about 1°
accuracy in the laboratory, this results indicates better than one micron measurement
resolution. Taking into account that these measurements are performed at A'-band, one can
see the tremendous potential of such swept microwave nondestructive method for accurate
measurements of minute changes in material thicknesses.
t o o -------
“
o
\X
o
.
-50 "
— —— Theo.
77R 't> . ci<i——2.0$
2.0$ mm
mm
--------
.
“— Theo. cl = 2.1$ mm
\
*'
_ \
O
o \\
° \\
\i
\
•
- 1 5 0 — 1— 1— 1— 1— 1— 1— ■— i— i
1 0 .2 5
’.O
\
\ °
\ vg>
-10° -
10
M eas. cl —
- 2.1$ mm
®
'
5 0
I
\°
0
© ffi
>
_ii
1 0 .5
®__©
i
>
ii
—o..O_o
II i■_■ i » « ■ ■
1 0 .7 5
11
F req u en cy (G H z)
Figure 4.11 : Comparison between the theoretical and measured phase for two lossy
dielectric sheets with er = 12.4 - j 2.4.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
150
Theo.
125
100
e
50
Ad
=
0.01 mm
25
10
1 0 .2 5
1 0 .5
1 0 .7 5
F req u en cy (G H z)
Figure 4.12 : Phase difference due to variations of lossy dielectric sheets of Figure 4.11,
with d] = 2.18 mm and d2 = 2.08 mm withe,. = 12.4 - j 2.4.
IV .3
Disbonds and Thickness Variations in Layered Composite M edia
The preceding results presented thus far, although nondestructive, pertain to contact
type measurements only. In many NDE applications measurements must be performed in a
non-contact manner. Furthermore, when examining dielectric composite media, most often
the geometry of the material is composed of layered stratified materials. Expansion of the
theoretical results of the previous chapter to general multi-layered media allows theoretical
modeling and examination of such cases. In this section experimental results are presented
for examination of dimensional variations in layered lossy dielectric composite media.
Non-contact, nondestructive evaluation (NDE) of disbonds, delaminations and minute
thickness variations in stratified composite materials is of great interest in many industrial
applications. Good examples would be thickness variations, disbonds and delamination
detection in ceramic, synthetic rubber, and honeycomb composite structures. In many
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
instances, the layered structure is backed by a conducting plate, or it is a dielectric coating
on top of a conducting sheet
Figure 4.13 depicts the apparatus utilized to conduct experimental measurements
described throughout the rest of this chapter. The mechanical setup seen on the right hand
side of the picture consisted of two sliding sample holders which allow precise adjustment
of the sample positions, via two micrometers, in front of the waveguide aperture. The
measurements presented here were performed at a frequency of 10 GHz using dielectric
sheets with permittivity of e ‘r = 8.4 and loss tangent tand= 0.107 (synthetic rubber
sheets).
The experimental results presented here are the average value of several
independent measurements with observed accuracies of better than 5 percent (deviation
between the upper and lower bound of measured data for the worst case scenario).
Figure 4.13 : Measurement apparatus for examination of layered media.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
71
IV.3.1
Layered Media Terminated into an In fin ite H alf-Space
To examine the validity and limitations of the numerical results of the previous chapter
in regard to layered media, initially variations of VSWR and phase were monitored as a
function of distance between the aperture and the sample (air gap) using a single slab with
= 8.4, tand = 0.107 terminated into free-space. The numerical results of this case in
terms of conductance and susceptance were discussed in Section III.3.2. It should be
noted that the geometry of all the following measurements pertaining to termination into an
infinite half-space are derived from Figure 4.14. For the case addressed here, the dielectric
sample thickness was chosen to be d 2 = 7.55 mm, with d3 and d4 set to be zero. This
models a three layer medium in front of the aperture.
Waveguide^ J
1
' j
{disbond )
Figure 4.14 : Cross sectional view of a layered media with a disbond.
The slab thickness was obtained from the measured value of 7.55 ± 0.03 mm, using a
micrometer, over the aperture area. The results of the measured VSWR for a range of air
gap values displayed significant difference when compared with the preliminary numerical
results.
Upon extensive measurement repetition and scrutiny of the apparatus, the
possibility of measurement inaccuracies as a cause of this difference was ruled out.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
Moreover, such inconsistencies did not occur when a conducting sheet was placed in the
back of the dielectric sample. This, in effect, also ruled out the possibility of potentially
incorrect assumption of dominant mode aperture field distribution (this would have
influenced both cases, whereas now only one case is strongly influenced).
Further
observations pointed to the invalid theoretical assumption of ideal lossless free-space
medium behind the slab. Figures 4.15a and 4.15b show the results of the measured
VSWR and phase over a distance of XJ2 from the aperture compared with three theoretical
runs with free-space permittivities of 1 and tand of 0, lxlO-3, and lx lfr5. At first glance,
for all practical purposes, these values of loss tangent once compared with that of the
sample seem to be negligible (lossless assumption of free-space permittivity is a common
practice in theoretical formulations). However, this can lead to significant error in the
numerical calculations for a medium terminated into an infinite half-space of air. This is
verified by the fact that as the loss tangent is reduced by two orders of magnitude (from
lxlO '3 to lxlO'5) the theoretical variations of the VSWR and phase become insignificant
and the solution converges quickly so long as an infinitesimal amount loss has been
introduced to the free-space dielectric constant. From Figures 4.15a and 4 .15b, it is clear
that much better agreement between the measurement and numerical results is obtained
when a finite amount of loss is associated with the free-space dielectric constant. This is a
significant finding which shows the sensitivity of numerical techniques to these types of
issues. This means that one must be careful when comparing the results of such numerical
approaches when normalized with respect to the free-space properties (eQ or kQ) to
experimental results. Figure 4.15b shows that the phase of the reflection coefficient varies
more for small air gap and displays much less variation at lager air gap values. This
suggests that depending on the measurement criteria one needs to decide on the appropriate
air gap size. For example a larger air gap may be used if small variations of the air gap are
not to effect the phase measurements drastically.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
14
►
0
2
4
6
8
10
12
tanS=0.0
tanS= 1.x10
tanS=IxlO
Meas.
14
16
A ir G a p . d ] (m m )
Figure 4 .1 5 a: Comparison of the numerical and experimental results for VSWR as a
function of air gap d}, for three free-space dielectric constants of e'r = 1.0,
and tanS = 0.0, 1x103, and lxlO '5.
200
150
100
50
o
-50
-100
-150
-200
0
2
4
6
8
10
12
14
16
A ir G a p , d l (m m )
Figure 4 .1 5 b : Comparison of the numerical and experimental results for phase as a
function of air gap d h for three free-space dielectric constants of £'r = 1.0,
and t a n 8 = 0.0, lxlO 3, and lxlO'5.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
Next, variations of VSWR and phase were investigated as a function of the slab
thickness at 10 and 24 GHz for a fixed arbitrary air gap distance dt . Once again Figure
4.14 illustrates the geometry for this case with d2 = 7.55 mm and d3 and d4 equal to zero.
Figures 4.16a-b and 4.17a-b show the numerical results at 24 GHz along with both the
numerical and experimental results at 10 GHz for different slab thicknesses. Figures 4.16a
and 4.16b show variations of VS1W? and phase respectively for an air gap of dj = 2 mm.
The numerical results of the phase variation at 24 GHz shows a change of around 120°
over the first 4.5 mm range of the slab thickness. This indicates that for this case thickness
variations as small as 0.04 mm (40 microns) may be detected since the measurement
apparatus used here may resolve a change of about one degree. Figures 4.17a and 4.17b
display the results for an air gap of dl = 5 mm. The measured phase at 10 GHz, which is
also predicted by the theory, shows significant variation over this thickness range. Due to
greater signal attenuation at larger air gaps and increase in the electrical length of the lossy
sample at higher frequencies, variations of VSWR and phase are expected to degrade as
shown for the 24 GHz results in Figure 4.16b. Such observations are made to indicate that
the measurement resolution (accuracy) may be maximized by using the theory to optimize
the air gap distance and the operating frequency based on the sample thickness and its
dielectric properties. Moreover, having such resolutions at X-band suggests that one may
not need to go to much higher frequencies to achieve high resolutions for examination of
the type of dielectric samples used here.
To examine the effect of disbond in a stratified composite material an experiment was
conducted in which the separation (disbond) between two slabs of the same dielectric
material was varied for a fixed air gap. From Figure 4.14 for this case d2 - 5.15 mm, d4 =
7.55 mm, and with both slabs having the same dielectric constant as before. Layers 1, 3,
and the infinite half-space in the back have free-space permittivity. Figure 4.18 shows the
numerical and experimental results for dj = 5 mm as d3 varies over a A ,/10 range. The
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
measurement outcomes, which are in good agreement with the theory, clearly indicate that
for this specific case, the phase of the reflection coefficient shows much more sensitivity to
the disbond dimension than VSWR. The phase drops around 15° over the first 0.5 mm
variation of disbond in the layered medium which translates to 33 microns per degree.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
10
— 24 G H z (Num .)
— 10 G H z (Num .)
8
10 G H z (M eas.) .
6
I
4
2
0
3
4
6
5
7
8
d. (mm)
Figure 4 .1 6 a : Comparison of the numerical and experimental results at 10 and 24 GHz
for VSWR as a function of slab thickness for an arbitrary air gap of
dj = 2 mm.
200
150
100
50
■& 0
-50
-100
-150
-200
3
4
5
6
7
8
dn (mm)
Figure 4.16b : Comparison of the numerical and experimental results at 10 and 24 GHz
for phase as a function of slab thickness for an arbitrary air gap of
dj =2 mm. Legend same as Fig. 4.16a.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
77
12
■ ■ -2 4 G H : (Num.)
10 G H : (Num.)
10
►
10 G H z (M eas.)
8
3
6
4
2
0
3
4
6
5
7
8
dn (mm)
Figure 4 .1 7 a: Comparison of the numerical and experimental results at 10 and 24 GHz
for VSWR as a function of slab thickness for an arbitrary air gap of
dj = 5 mm.
100
50
o
-50
-100
3
4
6
5
7
8
4, (mm)
Figure 4.17b : Comparison of the numerical and experimental results at 10 and 24 GHz
for phase as a function of slab thickness for an arbitrary air gap of
d] = 5 mm. Legend same as Fig. 4.17a.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
78
60
. . — P h a se (N u m .)
V SW ft
(N u m .)
P hase (M ea s.)
V S W /?
(M ea s.)
O
50
40
_
4 .5
V o
30
3 .5
10
2 .5
0
<v
-10
-20
0
Figure 4.18 :
IV.3.2
0 .5
1
1 .5
Disbond. d3 (mm)
2 .5
3
Comparison of the numerical and experimental results for VSWR and
phase as a function of disbond thickness, d 3 , for d j = 5 mm,
d2 = 5.15 mm, d4 = 7.55 mm.
D isbond in a
Layered Media Terminated into a C onducting Sheet
In this section the results of similar measurements are presented for a case of a disbond
in a layered medium backed by a conducting plate. Figure 4.19 depicts the cross sectional
geometry of the problem geometry. Once again, the apparatus of Figure 4.13 was used to
perform the measurements.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Waveguide
I
1
(disbond )
Figure 4.19 : Disbond in a layered media terminated into a conducting sheet.
A slab of the same material with thickness of d 2 = 7.55 mm was first placed flush
against a metal plate (d3 - 0), and dj varied over a X0/2 distance. The numerical and
experimental results for VSWR and phase are shown in Figure 4.20. Comparing VSWR
in Figure 4.20 with its counterpart in Figure 4.16a where the conductor in not present one
can clearly observe the effect of strong reflection from the surface of the metal plate causing
slower damping of the VSWR variation in Figure 4.20. Similarly, comparison between
phase of these two cases (Figures 4.15b and 4.20) shows the strong influence of the
conductor backing both in terms of a shift in position of minima change of slope.
Figure 4.21 displays variations of VSWR and phase as a function of disbond
thickness, d2, between the dielectric sheet and the metal plate for an air gap of dj = 5 mm.
Both VSVF/? and phase values, which are in close agreement with the theory, display
significant change over the A^/IO range. Comparing the phase shown in Figure 4.21 with
that of Figure 4.16b, it is clear that the presence of the conducting sheet in the back
enhances the measurement sensitivity. Such an effect is expected due to reflection at the
interface between the dielectric slab and the surface of conducting plate.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
To further stress the effect of air gap on the measurement resolution, the numerical
results of VSWR and phase of Figure 4.20 (disbond d3 = 0) were compared against a case
where a disbond of d3 = 0.1 mm exists between the dielectric sheet and the metal plate.
The results of this test are shown in Figure 4.22. The solid line displays the difference
between the magnitude of VSWR (in dB) and the dashed line is the phase difference
between the two cases. The results clearly indicate a resolvable difference as a result of a
0.1 mm disbond between the dielectric slab and the conductor surface. The maximum
difference of the VSWR and phase occurs at different air gaps. Such tests clearly indicate
that the theory may render important information on the air gap for the position of
maximum separation of either the VSWR or the phase for optimization of measurement
resolution and accuracy.
200
9
“— P h a s e
VSWR (Num.)
VSWR (Meas.)
( N u m .)
P h a se (M ea s.)
150
8
100
7
50
6
5
\o
2
-5 0
__
-100
-150
0
4
6
8
10
12
14
16
Air Gap, d (mm)
Figure 4.20 : Comparison of the numerical and experimental results for VSWR and phase
as a function of dj, for d 2 = 7.55 mm, d3 = 0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
200
• — P h a s e (N u m .)
150
>
O'
100
o '
P h a se ( M e a s .)
**
50
VSIVR (N u m .)
K S V W ? (M e a s .)
-50
-100
-150 -
0
-2 0 0
0
1
0 .5
1 .5
2 .5
D is b o n d , d 3 (m m )
Figure 4.21 : Comparison of the numerical and experimental results for F5W7? and phase
as a function of d3, for.d1 = 5 mm, d2 = 7.55 mm.
20
10
“a
<3
e
0 .5
-10
-0 .5
0
4
6
8
10
12
14
16
A ir G a p , d j (m m )
Figure 4.22 : Difference between the numerical results of Figure 4.20 with the case where
a small disbond of d3 = 0.1 mm is also present.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
IV .4
Sum m ary and Rem arks
Numerical and experimental results were presented in application to contact and noncontact examination of layered dielectric composite media backed or unbacked by a
conducting plate.
The integral solution developed in the previous chapter for the
terminating admittance of an open-ended rectangular waveguide radiating into a lossy
dielectric slab backed by a conducting plate was used in a recursive manner for estimation
of the thickness and dielectric properties. The thickness measurement results presented
here were for the worst case within three percent of the physically measured values. Some
important issues were pointed out in application of this technique for thickness
measurements. The results show this method to be a fast and reliable technique for
microwave nondestructive measurement of dielectric slabs or composite coatings backed by
conducting plates. For slab thickness variation detection, it is best to operate in the region
in which attenuation and phase experience more rapid fluctuations which can generally be
achieved by operating at an optimum frequency. This technique renders measurement
accuracies of better than 0.5% for dielectric slab thicknesses of less than 5 mm and no
worse than 3% for thicker slabs. For this simple technique these accuracies are better than
desired for most industrial applications (in Table 4.2 and for thicknesses less than 5 mm the
accuracies are better than 40 micro meters). It was pointed out that even better accuracies
may be obtained if the measurement parameters (specifically the operating frequency) are
optimally chosen for a given slab thickness and dielectric properties.
To obtain maximum measurement accuracy or resolution, the choice of the operating
frequency depends on the dielectric properties of the slab and its thickness. The results of
the swept frequency approach outlined here showed that at some frequencies measurement
accuracies of better than one micron are possible. Whereas, at a slightly different
frequency in the same microwave band the measurement accuracy is degraded drastically.
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83
The results of this study also indicated that detection of thickness variations in the order of
a few microns is quite possible. This is important as these measurements were conducted
in the 8.2 - 12.4 GHz region, and not in the millimeter wave region. Furthermore, the
sensitivity of small permittivity variations on the phase of the reflection coefficient was
demonstrated.
This approach also indicated that frequency selection for accurate
permittivity measurement is crucial. This can be accomplished by using a reliable
theoretical model such as the one described here.
Theoretical and experimental results were also presented in application to microwave
non-contact or in situ NDE of layered dielectric media terminating into either free-space or
a conducting sheet. The integrity of the numerical results of the earlier chapter were
examined by conducting several measurements using a precise mechanical device built
specifically to duplicate the geometry of the theoretical model. The measurements
presented here were performed at 10 GHz using generally lossy slab-like samples.
Variation of VSWR and phase of the reflection coefficient as a function of parameters such
as the air gap between the sample and the waveguide, and sample thickness were
investigated. Furthermore, numerical and experimental results were presented for very
small disbond thicknesses in a layered media. The experimental results showed good
agreement with the numerical outcomes. These results clearly indicate that high resolutions
may be achieved in examination of generally lossy dielectric samples without the need to
operate at very high frequencies. Moreover, they demonstrate the sensitivity of such a
versatile NDE technique as it applies to the interrogation of layered composite media, and
the importance of a fast and reliable numerical model as a tool to gain real time a priori
knowledge of the underlying process. It is concluded that the theoretical model can render
important information leading to optimization of such measurement parameters as position
of the sensor and operating frequency leading to better resolutions.
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84
CHAPTER V
Coaxial Transmission Line Sensor f o r Examination o f Layered Composite
M edia
The use of an open-ended coaxial line as a sensor for measurement of material
properties at microwave frequencies has received considerable attention
[Ath.82a] [M ar.87] [El.87] [S ea.89] [M is.90] [Sta.90] [Xu.91] [Zhe.91].
[Sn<.80] [M os.81]
Utilizing an open-ended
coaxial transmission line is very attractive because it offers many advantages such as wide
frequency band of operation and small aperture area. Unlike an open-ended waveguide, an
open-ended coaxial line provides operating frequencies from DC to the cut-off of the first
higher order mode. The cut-off frequency is directly dependent on the geometry of the
coaxial line. Open-ended transmission line methods are inherently nondestructive and offer
the possibility of making in situ measurements of material material properties. These
features have made an open-ended coaxial sensor a versatile tool in the contemporary
microwave biomedical and microwave engineering applications.
Several approaches are commonly sighted for modelling of the terminating admittance
of an open-ended coaxial line. The prevalent analytical procedures used in practice are
mostly limited to electrically small apertures or low operating frequencies allowing lumped
parameter approach or quasi-static approximations
[Sm.SO] [Mis.8 7 ].
Furthermore, they
generally pertain to the specific case of termination of the aperture into an infinite dielectric
half-space and do not address the problem of finite thickness media. The available results
pertaining to layered media
[And.86] [Fan.90 ].
More rigorous formulations have also been
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85
presented by taking into account the effect of higher order modes while considering an
infinite half-pace case
[Mos.81].
Such formulations require intense computations once the
geometry under consideration is different from the special case of infinite half-space. The
analysis presented here is in line with the work done by Swift [5h-/.67] which in turn stems
from the fundamental results developed by Levine and Papas
[Lev.51].
The formulation is
corrected and expanded to take into account general N-layer stratified media terminating
into an infinite half-space or a conducting sheet. Explicit solutions are given for the
specific cases of a two-layer medium. Accordingly, the numerical results presented here
also pertain to these two geometries. For the coaxial region only the fundamental TEM
mode is considered to propagate. Analysis of singularities for the case of lossless layered
media has also been briefly addressed in Appendix B. Although, it must be noted that
lossless layered media is not a practical representation in application to examination of
composite media. Verification of the theory is accomplished via comparison with readily
available solutions for the special case of termination into an infinite half-space medium.
V .l
Analysis o f Radiation From Coaxial Transmission Line
into
Stratified Composite Media
Aperture admittance of a coaxial transmission line with a perfectly conducting flange of
infinite area, opening into a layered dielectric composite media is formulated next. Similar
to the theoretical analysis presented in Chapter III, the results here pertain to two cases of
general N-layer media terminated into an infinite half-space and a conducting sheet
respectively. Fourier-Bessel integral transforms are employed to construct the field
solutions. Consecutively, an N+l region boundary value problem consisting of the
internal coaxial region plus the N-layered external media is solved. Continuity of the
power flow as a direct consequence of Poynting's theorem at the aperture cross section
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86
allows formulation of the terminating aperture admittance. Some practical issues are also
discussed.
V.1.1
Theoretical Formulation
Figures 5.1a shows the designated coordinate axis and geometry of a coaxial
transmission line with an infinite flange. Figures 5.1b and 5.1c depict the cross sectional
view of the line radiating into an N-layered media which is terminated by an infinite half­
space and a perfectly conducting sheet respectively.
Each layer is assumed to be
homogeneous and nonmagnetic with relative complex dielectric constant e) n.
Figure 5 .1 a : Coaxial transmission line of inner conductor radius a and outer conductor
radius b opening onto a perfectly conducting infinite flange.
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87
rN - l
Coax
» 0 -Cr
Z„=0
Figure 5.1b : Cross section of a coaxial line radiating into a layered media terminated into
an infinite half-space.
<7 = 00
Coax
W r,
z= 0
Figure 5 .1c:
“N - l
-N
Cross section of a coaxial line radiating into a layered media terminated into
a perfectly conducting sheet.
With the fundamental TEM mode incident on the aperture, the structure only supports
H ^ E p , and Ez field components with no angular, 0, dependancy. The external fields
may be constructed using an electric or magnetic Hertz potential having a single scalar
component as
nn(p,</),z) = n*(p,z)a.<P
(5.1)
where n denotes the layer number. The vector potential satisfies the source-free Helmholtz
wave equation in each region, written as
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88
v 2n n (p, 0 , z )+ k 2
*7ntnip,<p,z) = o
where
(5.2)
is the complex propagation constant of the nth layer. In terms of scalar component,
the above expression can be expanded using (5.1) as
d2
Id
d2
■)- +i — ——i— r + \ 2 - ^
dp
p dp dz 2 . n P 2
IJ*{p,z) = 0
(5.3)
J
Subsequently, the field components satisfying the the wave equation can be constructed
from the vector potential formulation as
En(P’Z) = — V x n n(p,<p,z)
(5.4)
>ant
ap +
£„ dz
Hn(p,z) =
—
a,
[kl + VV-1/7,, (p,0,z) = j(o n ! d $
J W 0£ n L
1
(5.5)
9
where en is the complex dielectric constant of the nth layer. The solution of the wave
equation (5.3) in cylindrical coordinates can be written in terms of Fredholm's integral
equation of first kind
n * (p ,z ) =
(H ,z)J , ( ‘Kp)d%.
(5.6 a)
0
and its Hankel integral transform
(P’z) = °]pTl*(p,z)J ,(<Xp)dp
(5.6b)
The notation - represents the transformation and Jj is the Bessel function of first kind and
order one. Substitution of (5.6a) into (5.3) using the derivatives of Bessel functions
results in
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89
Z n t( t,z ) jj( C p ) d t = o
(5.7)
which once multiplied by the orthogonal Bessel eigenfunctions and manipulated using the
Dirac delta Bessel integral representation
(5.8)
7
o
£
results in
dz
n i(K ,z ) = o
n
(5.9a)
where
(5.9b)
sc
Equation (5.9a) is a one dimensional representation of the wave equation in terms of the
Hankel transform of the scalar potential. Next, one may construct the solutions in the
external medium, z > 0 , in terms of standing waves for a bounded region and positively
travelling waves for the terminating semi-unbounded region as
( ^ , z) =
7 ■" + S l n
fl.Z (% J e
fli( ^ ,z ) = A Z W e
- jJk ~- n z
i
( $ S ) e ^ z"
~
(5.10a)
Region bounded
Region ubounded in
+z
dir.
(5.10b)
where
= Id ,
i=i
< n < N - 1
N
and- { / s S» l
(5.11)
N - th layer unbounded in + z dir.
N - th layer backed by conducting sheet
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90
Consequently, with the aid of (5.4) and (5.5) solutions for the transformed field
components may be constructed as
£?n (X ,z) =
(X)e~]kznZ - ^ ( 3 > A’"Z]
(5.12)
fn
x t(% .,z ) = ;< o [< (5 0 e “A »-' + - W ) A * ]
(5.13)
In accordance with (5.10b) the second terms inside the brackets in the above equations
vanish for the field components in layer N if the region is unbounded in +z-direction. It
must be noted that according to Figures (5.1b) and (5.1c), for an N-layer media, aside
from the aperture interface, N -l boundary interfaces exist when the medium is terminated
into an infinite half-space and N boundary interfaces when terminated into a conducting
sheet respectively. Next, application of the appropriate boundary conditions which may be
written as
For 0 < n < N - 1 ,
I
= Z/l) =
K ( S U
= z„ ) = <
= 2n)
(5.14a)
#( * , ; = z„ )
J
i pn (X ,z = zN) - 0
and,
(5.14b)
where n - 0 represents the internal coaxial region. The fields in this region, z < 0 and
a < p <b, can be expanded as a sum of the incident and reflected fields together with a
summation over all the higher order modes as
ES(p,z) =
[M a .5 l] [Gal.69) [Lev5 1 }
+Re~jk<z)+ i A mR J p ) e y'":
m=I
P
H*(p,z) =
P
V
- R e - Jk'z)+ h mAmRm(p)er"':
1 m=l
where
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(5.15)
(5.16)
91
k c -k o fc ,
Yc - y 0j r
7rn =
(5.17a)
Rc{ym} < 0
(5.17b)
where erc is the permittivity of the dielectric filling inside the coax, y 0 is the free-space
characteristic admittance and R is the complex reflection coefficient. R m represents the
radial eigenfunction whose eigenvalues A.m, which are also the cut-off wave numbers of the
m* higher order modes, may be calculated from the transcendental equation
J o { K a )M0{kmb) = N 0(Xma)J 0(Xmb)
where J 0 and N 0 are the Bessel and Neumann functions of order zero.
(5.18)
With the
assumption of dominant TEM mode propagating in the coax, the summation over the
higher order modes drops out of (5.15) and (5.16), consequently resulting in
Ep
0 (p,z) = ^ ( e jk' z + R e 'jk' z)
P v
'
(5.19)
H g(p,z) =
{5 2 0 )
p
'
- Re-* r-\
t
To construct a solution for the aperture E-field in terms of the internal field quantities,
one can take the integral Hankel transform of both sides of (5.19) at z = 0
1 e p ( p , z = 0)J,(Cp)pdp
o
=l^ -il + R M & pdp
n P
(5.21)
and with the left hand side term representing the Hankel transform of the aperture E-field,
the right hand side may then be evaluated over a < p <b, resulting in
= -A o (l + R)
}~
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(5.22)
92
Next, using the Poynting’s theorem continuity of power flow across the aperture is
enforced. In general, with the fields having no angular 6 dependancy, the complex
conjugate of power flow at a cross sectional area S can be written as
P* = ^ fjj£ * (p ,z ) x H (p,z)j.£i2pdpd0
*s
(5.23)
= 7t J {£p (p,z)J H ^(p,z)pdp
P=°
Subsequently, application of Parseval's theorem for the integral transforms
[Arf.85\
renders
an equivalent relation for the transform pairs of the field components, resulting in
J {Ep ( p ,z ) } V ( p ,z ) p d p = J {
p=o
£
P
&
,
z
)
}
*
(5. 24)
&=0
From the above results, the outward power flow from the aperture at z - 0 and a < p < b ,
using (5.19) and (5.20) can be evaluated as
P *=0 = n Y A A rfu + R f i l - R f J - )
\aj
(5.25)
where the arrow denotes the direction of flow. Similarly, (5.24) may be used to calculate
the complex conjugate of inward power flow from layer one. So, at the aperture cross
section one may construct the following
p :=0
=
=
0)}*Af(0i,z
=
ow n.
0
= 7r°j[^(3l,z =
o )}
\T(n.)i%(n.,z = o)]n.dn.
0
= n \ \iP ( ^ ,z =
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(5.26)
93
where in the above equation the transform of the H-field component in region 1 is replaced
with its equivalent expression containing the transform of aperture E-field.The function
^!X), relating
(!$L,z) and
(X ,z) results from enforcement of boundary conditions of
(5.14) in the layered media and at the aperture. Equating (5.24) and (5.25) and substituting
(5.22) for the transform of the aperture field as
7i\A0\2Yc( l- R ) ( l + R)* l n ^ j
(5.27)
= n\A0\2( l+ R ) \l + R) / 1
0
consequently allows the construction of the terminating aperture admittance. Rearranging
the above equation and using the normalization parameter £ = (!£. / k0), ultimately results in
y s = 8 s + jbs = ~ ^
1+ A
_
=
where as noted earlier,
(5.28)
£n
T
\
J
I
*
is the unknown function which is to be evaluated from the
enforcement of the boundary conditions.
V.1.2
E xplicit Solution fo r a Two-Layer Media
The procedure leading to construction of the explicit form of J(C,) for the case of N = 2
both when the media is terminated into an infinite half-space and a conducting sheet is
presented next.
The problem geometries are depicted in Figures 5.2a and 5.2b
respectively.
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94
Coax
Figure 5.2a : Cross section of a coaxial line radiating into a two-layer media with layer 2
being an infinite half-space.
■
i
$
d, '<<Figure 5.2b : Cross section of a coaxial line radiating into a two-layer media terminated
into a conducting sheet.
These two specific cases were chosen due to fact that they represent a broad class of
real life material geometries. Examination of finite size slabs, and disbonds in coatings on
top of conducting sheets are some typical examples. In view of Figures 5.2a and 5.2b, at
z = 0, (5.12), (5.13) and (5.14a) result in the following field coefficients in layer 1. In
terms of the Hankel transform of the aperture E-field these can be written as
VQ =
s [1 + K($L)]ejk:rl‘i pn {'H,z = 0 )
r
--------T
j 2 kZj [ K{X) cos kZizt +j sin kZ/zsj
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(5.29)
95
_/
s
<%J( =
£r \ l - K ( ^ ) } e 'jk:<:,^ ( X , z = 0 )
—L:________ i_________ __________
( 5 . 30 )
j 2 kZj [ k W c o s k . z 1 + j sin k . z ^
£ r ^ £r . ~ I —
£r L
'• V
1 kQ
where, k ($ J = —— - - — -----------/
\2
£ r: k z,
&
£ r A £r,
(5.31)
\ ko
which once substituted back into (5.13) renders the following result for the case of Figure
5.2a
(k0y0£n )[C(H, z) - jK(3QS(3jt z)]£P ( f t , z
Xf(*l,z) =
=
o)
(5.32)
k2[ [ir(£)cosA2j zj + jsin k 2izj]
where
C( X, z) = c o s k 2i ( z - Z j )
(5.33 a)
S(%.,z) = sink2i ( z - z j )
(5.33 b)
Similarly, for the case of Figure 5.2b the procedure results in the succeeding expression for
the transform of the magnetic field component in layer 1
- a
knvner ,EZ(,J i,z = o)
xf(lV z) = —
! - ■-
k-
------
(5 .3 4 )
“i
[C0K.,Z)- jK(X)S(X,z)] + e~jk* (2z2~z‘ \c(H,z) - jK(40S(3L,z)]}
|r A'2Z/[ic(^m^) +^ ,2 ) ] - r y^ (2z2'Z/)[jc(atm,z)-y5(^,z)]j
Consequently, substitution of (5.32) into (5.26) and following the procedure outlined
earlier results in the following expression for 7 (t), for the case of termination into an
infinite half-space
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96
l + jK'(C) tana(C)
(5.55a)
where
(5.35b)
a(0 =
(5.35 c)
Similarly, following the same procedure using (5.34) results in the succeeding expression
for
for the case of termination of two layer media into a conducting sheet
[1 + yV (Q tan a (Q ] + e jP(^ [ l + yy'(Q tan a(Q ]
where
(5.36 b)
and with td(£) and a (£) given by (5.35b) and (5.35c) respectively.
V.2
Verification o f the N um erical Results and Som e Observations
To examine the validity and significance of the formulation presented above a series of
numerical results for the two-layer media are presented next. The two geometries
considered are those of a dielectric slab terminated into an infinite free-space and a
conducting sheet respectively. In view of Figure 5.2b the latter represents the case were dj
or d 2 is set equal to zero. For verification purposes, results of these cases are compared
with the special case of aperture termination into a infinite half-space reported by [Lev.51].
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97
This is done by using an electrically thick enough slab of generally lossy dielectric material
over the aperture so as to result in negligible variations of the aperture response. For all
cases presented here the real part of the complex dielectric constant of the medium is
arbitrarily chosen to be e'r = 10. The value of loss tangent tand = e"rl£'r may vary between
0.01 to 1.0. These values are in the range of various synthetic rubber products and coating
type materials on top of conducting sheets. For examination of thickness dependency of
the response, the plots are given as a function of normalized slab thickness T = d,l(ar)
which is a function of normalized coax radial dimensional ratio r = b/a . Figure 5.3a and
5.3b display a series of graphs for the return loss (RL = 20 logHT) and phase <t>of the
complex reflection coefficient R = Tei® for the case of the aperture terminated into a slab of
finite thickness and backed with an infinite free-space medium. Referring to the geometry
of Figure 5.2a, the slab thickness dj is increased to ultimately reach that of an infinite half­
space of the dielectric material. As expected, the response for the material with higher loss
flatens out faster both for the return loss an phase. The crosses shown on the plots are the
results calculated using reference
[L ev.5l\
which are in good agreement with this theory.
Alternatively, setting dj = 0 directly renders the results of the reference for the case of
aperture termination into free-space media. Figures 5.4a and 5.4b show the same scenario
except with the infinite half-space of air being replaced with a conducting sheet. For the
two-layer aperture admittance formulation and depicted in Figure 5.2b this pertains to
d 2 —0 and finite dj. Once again, the results reach to that of infinite half-space of dielectric
media for a thick slab. Consequently, as the slab thickness goes to zero, the return loss
and phase reach the values for that of a short circuited coax.
Figures 5.5a and 5.5b show the return loss and phase as a function of frequency again
for the case of a single slab of finite thickness terminated into an infinite free-space
medium. The results are shown for a coax with r = 0.307 for several different normalized
slab thicknesses T = djl(ar) having e'r = 10 and tan 5 =0.1. The points marked with x and
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98
+ are calculated using infinite half-space formulation [Lev51] for the case of dielectric with
the same properties and free-space, respectively. For small thicknesses the curves follow
the trend of free-space results. However, the general trend of the plots show that as the
thickness and frequency increase, both the return loss and phase responses setde down
around the infinite dielectric half-space curve. At higher frequencies where the wavelength
is comparable with coax radial dimensions, the overall aperture admittance response will be
a contribution of both the fringing fields as well as radiating waves. For a fixed frequency,
the dependance on the finite slab thickness results in nonuniform spacing between the
individual curves.
Such theoretical expositions display the importance of a sound
numerical model to gain a priori knowledge leading to optimization of the parameters for
practical purposes.
Subsequently, Figures 5.6a and 5.6b show the return loss and phase for the case of
layer 1 being terminated into a conducting sheet with the other parameters kept the same as
previous case. Comparison of Figure 5.6a with is counterpart 5.5a displays a more
smooth response for this case which could be contributed to the two way attenuation
caused by the presence of the conducting plate in the back. Eventually, this effect at high
enough frequency gives rise to the phase transition for the thinnest slab of Figure 5.6b.
Once more, this figure shows that once slab electrical thickness gets large enough the
response becomes independent of slab thickness and similar to that of infinite half-space
media (marked with x on the graph).
Figures 5.7a and 5.7b depict the return loss and phase versus frequency for two-layer
conductor backed media. These figures attempt to demonstrate the potential of detecting a
disbonded layer in layered composites. In view of Figure 5.2b, each curve pertains to
different thickness of the second layer d 2 (i.e. different disbonds). The calculation
parameters for this case are a = 1.18 mm, b = 3.62 mm, e ’n = 10, tanS - 0.01, dj = 0.5
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
99
mm, and e'n = 1 (air disbond). To simulate a disbond, any increase in d 2 is compensated
by a decrease in dj, such that the total thickness d {+d2 of the two-layer medium remains
constant. It can be observed that for this case the phase response displays more sensitivity
to the presence of the disbond. Moreover, both return loss and phase display much greater
sensitivities to the disbond thickness variation at higher frequencies. These results clearly
demonstrate the capability of the coaxial sensor for detection and evaluation of small
disbonds in layered composite media.
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100
10
t o n 5 = 0.01
tan 5 = 0.1
8
tan 5 = 0.5
tan 5 = 1.0
6
4
2
0
0
1
2
3
4
5
d fa r ) 1
Figure 5 .3 a : Return loss versus slab thickness normalized with respect to outer radius of
the coaxial line, for r = b/a, e'r, = 10, er = 1 (air) a n d /= 5 GHz.
-6 5
— tan 5 = 0.1
-7 5
— tan 5 = 0.5
— t o n 5 = 1.0
-8 5
"
e
-■
-9 5
-1 0 5
-1 1 5
0
2
4
d /a r )'1
Figure 5 .3 b : Phase versus slab thickness normalized with respect to outer radius of the
coaxial line, for r - bla, e'n = 10, En = 1 (air) and/ = 5 GHz.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
12
10
tan 5 = 0.01
tan S = 0.1
8
tan S - 0.5
tan S = 1.0
1 6
S:
4
2
0
0
1
2
3
4
5
d j(a r) 1
Figure 5 .4 a : Return loss versus slab thickness normalized with respect to outer radius of
the coaxial line, for /• = b/a,
= 10 and/ = 5 GHz. Conductor backed.
-8 5
-9 5
-1 0 5
e
-1 1 5
tan S = 0.0 J
tan 5 - O . I
tan 5
-1 2 5 -
=
0.5
tan 5 = 1.0
-1 3 5
0
2
3
4
d t ! (a r)-J
Figure 5.4b : Phase versus slab thickness normalized with respect to outer radius of the
coaxial line, for r = b/a, e'n = 10 and/ = 5 GHz. Conductor backed.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
5 .5
4 .5
-
7
=
1.0
- T = 2.0
3 .5
7 = 4 .0
0 .5
- 0 .5
0
2
4
6
8
10
12
f(GHz)
Figure 5 .5 a : Return loss versus frequency for different thicknesses normalized with
respect to outer radius of the coaxial line. T = dftar), r = bla, e'rl = 10,
tan 8 = 0 .1 , en = 1 (air).
-20
-4 0
-6 0
^
-8 0
-100
-
T = 0.5
-120
~ T = 2.0
-1 4 0
— T = 4.0
-1 6 0
0
2
4
6
8
10
12
f(GHz)
Figure 5.5b:
Phase versus frequency for different thicknesses normalized with respect to
outer radius of the coaxial line. T = d,/(ar), r = bla, e 'rl = 10,
tanS = 0 . 1, er: = 1 (air).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10 3
7
■ ■ T = 0.125
6
T = 025
-
5
t
= i.o
T = 4.0
4
os
3
2
1
0
4
10
6
12
f(GHz)
Figure 5 .6 a : Return loss versus frequency for different thicknesses normalized with
respect to outer radius of the coaxial line. T = d,/(ar), r = bla, e'ri = 10,
tan 8 = 0.1. Conductor backed.
• --
-4 5
T = 0.125
T = 0.25
T = 1.0
-1 3 5
-1 8 0
0
2
6
4
8
10
12
f (G H z)
Figure 5.6b:
Phase versus frequency for different thicknesses normalized with respect to
outer radius of the coaxial line. T = d,l(ar), r - bla, e 'rI = 10,
tan5= 0.1. Conductor backed.
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104
d , - 0.01 mm
0.8
5
2.
S;
d.
=
0.05 mm
d*
=
0.2 mm
°-6
0 .4
0.2
0
2
4
6
8
10
12
f(G H z)
Figure 5.7a:
Return loss versus frequency for different disbonds d2, £n = 1 (air),
£'n = 10, tand= 0.01, dj = 0.5 mm, a = 1.18 mm, b = 3.62 mm.
-4 5
d_
—
-1 3 5
~
=
0.0 mm
- 0.01 mm
d~ - 0.05 mm
-1 8 0
0
2
4
6
8
10
12
f(GHz)
Figure 5.7b:
Phase versus frequency for different disbonds d2, e r2 = 1 (air), e'ri = 10,
tanS = 0 .01 , d] = 0.5 mm, a = 1.18 mm, b = 3.62 mm.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
105
V.3
Practical Im plications
For practical purposes, the measurement outcomes requires modifications to allow
comparison with the results of the theoretical formulation.
Unlike waveguide
measurements in which the standing wave may be monitored directly, several sources of
measurement error contribute significantly to coaxial line measurements. The disparities
between the measurement outcomes and theoretical expectations can be attributed to such
measurement errors as source match, directivity, and frequency tracking
[HPA.77],
Figure
5.8 shows a general block diagram for the S-parameter of a typical measurement. The
measured Smn related to the actual (expected) 5au for a reflectometer setup
Sources of
Measurement Error
Fig 5.8 :
Block diagram showing the measurement apparatus transfer function and
sources of error.
where 5oe represents the effective reflectometer directivity error which is a vector sum of
leakages and miscellaneous reflections at the measurement port. S$e is the effective source
match error which may be contributed to re-reflections due to mismatch between the test
port and the characteristic impedance of the line (e.g. 50 Ohms). This source of error leads
to variations of the magnitude and phase of the incident field which may be difficult to
eliminate. Finally, STe represents the tracking error causing variations in the frequency
response. The relationship between Smn and S°u through SDc„ S je and SSl, is commonly
referred to as the calibration of the reflectometer measurement setup. Inherently, better
matching may be achieved by incorporating larger number of explicit error sources into the
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106
calibration model at the expense of more rigorous and com putational intensive calibration
procedure.
The block diagram of Figure 5.8 may be simplified to render an explicit transfer
function of the measurement device relating the measured and expected results. Figure 5.9
depicts the S-parameters signal flow graph and its simplifications taking into account the
error sources.
Fig 5.9 : Simplification of transfer function using signal flow graphs.
where 1 in the return path is due to assumption of a lossless line. The S-parameters in the
above diagrams may be written as [Col.66\
(53T>
s - 1
S '- STeS
-
(
5
.
5
S
,
(5 . 3 9 )
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107
where S'" and Smu are equal. Manipulating (5.39) allows construction of an equivalent
expression in terms of Smu as
ffi=
(5.40)
^ ------SSe[SU “ ^DeJ + ^Te
knowing that Sn represents the reflection coefficient T o f the measurement port (i.e. openended coax), solution of a system of three equations results in a solution of the three
unknown S-parameters. Thus,
r.m _ c
i ? = — r i ,— ^ —
S s .[ r r - S o .\+ S t'
= ■= 1’2-3
V .4 D
The conventional method of calibration calls for measurement of the reflection
coefficient of three known loads to solve the above system of equations. Common practice
is to use a short circuit, an open circuit and a matched load (very hard to construct for open
ended coax) which is usually substituted by a known calibration liquid. As a result of
incorporation of finite thickness into the solution, the theoretical formulation presented here
could be used to improve upon the conventional calibration method by taking advantage of
several load thickness.
V.4
Sum m ary and Rem arks
Theoretical formulation of the aperture admittance of an open-ended coaxial
transmission line terminated by layered dielectric composite media was presented. Explicit
solutions were constructed for specific cases of termination into a two-layer media backed
or unbacked with a conducting sheet. Verification of the results was possible by modeling
the geometry to represent an infinite half-space termination of the aperture and comparing
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108
with the available numerical solutions. For all cases the result were in good agreement with
those of the other formulation. For practical implementation of the theory, a calibration
method based on an S-parameter solution of a reflectometer transfer function was discussed
by taking into account prominent sources of measurement errors. The equivalent error
sources discussed can lead to great disparities between the measurement and theoretical
results. The traditional calibration methods are rather cumbersome and require three
different loads. The numerical formulation discussed here, may be utilized for a more
convenient calibration method by taking advantage of the load thickness and consequently
do without the use of hard to acquire loads. Finally, It is worth noting that the formulation
for the terminating aperture admittance of the coaxial geometry presented here may also be
used for cylindrical waveguides by setting the inner radius of the coax to zero.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
109
CH APTER V I
C onclusions and Rem arks
In this work theoretical formulations and practical implications were addressed in
application to microwave NDT & E of layered dielectric composite media. With growing
applications of new lightweight and durable dielectric composites many traditional and
conventional NDT methods have rendered themselves ineffective. Although microwaves
have been utilized to examine material properties for quite some time, their application to
the field of NDT has been minimally addressed. Microwaves have recently reemerged to
fill many gaps left vacant by traditional NDT methods. In Chapter II, a survey of some
prominent NDT &E methods was presented.
Table 2.1 gave a quick reference
summarizing general applications and resolutions associated with these techniques. Upon
investigation of available microwave techniques and their implications, variety of reasons
were given as to why microwaves should be utilized more extensively in todays expanding
and demanding field of NDT.
In many cases dielectric composite materials consist of layered generally lossy dielectric
media. Examination of layered composite dielectric materials for flaws such as disbonds,
delaminations and dimensional inhomogeneities within layered media also pertain to this
class and can be modeled in this fashion. The interaction of electromagnetic fields with
such material media is a complex phenomenon. This in effect emphasizes on the
importance of reliable theoretical models for a priori analysis and investigation of problems.
In Chapter III analysis of radiation from rectangular waveguides into layered composite
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110
media was presented. Initially, variational formulation was evoked to construct a stationary
expression for the terminating admittance of a rectangular aperture in an infinite ground
plane.
Using Fourier transform boundary matching technique first the admittance
expression was formulated for the specific case of a single layer on top of a conducting
sheet. Upon preliminary verification of the numerical results, the formulation was then
expanded to general N-layer media backed or unbacked with a conducting sheet.
In Chapter IV extensive experimental results using laboratory apparatuses were
presented both to further investigate the integrity of the numerical results and to further
address some practical implications. Specific applications such as the measurement of
dielectric properties and thickness of coatings on top of conducting sheets were shown.
Furthermore, results of a swept frequency analysis were presented to stress on the
importance of a priori theoretical expositions for optimization of measurement parameters.
Results of this investigation showed the possibility of detection of dimensional variations in
the order of few microns which is not commonly associated with relatively low frequency
microwaves. Experimental results in conjunction with numerical outcomes were also
presented in application to detection disbonds and delaminations and monitoring of
thickness variations in layered media. Numerical codes for the two cases of a five layer
media with the last region being an infinite half-space and a three layer media backed with a
conducting sheet are supplied in appendices.
Due to small aperture size, wide frequency of operation and inherently nondestructive
nature of measurements open-ended coaxial transmission lines are utilized in a verity of
medical, biological and engineering applications. Parallel to the theoretical formulations of
Chapter III, in Chapter V the problem of coaxial transmission line opening in to a layered
dielectric composite media was addressed. This was in part in response to lack of concise
and versatile theoretical models in application to nondestructive examination of finite size
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Ill
media. In this part of the work only theoretical results are presented and experimental
verification and further analysis of the results are left for future endeavors.
The
formulation presented pertains to general N-layer media backed or unbacked with a
conducting sheet. The terminating aperture admittance of a coaxial line was formulated by
employing Hankel transforms and solving the wave equation in cylindrical coordinates.
For the layered external medium Fourier transform boundary matching technique was used
to construct the field solutions in this region. Continuity of the complex conjugate of
power flow across the aperture interface allowed for construction of the terminating
aperture admittance. Only the dominant TEM mode was considered for the internal coax
fields. This is a valid assumption as most coaxial transmission lines are used in the
frequency band below the cut-off frequency of first higher order mode. Preliminary
verification of the numerical results with another theoretical model for the special case of
termination in to an infinite half space have shown good agreements. As an extension of
the coaxial problem geometry, the formulation may also be applied to cylindrical
waveguides by allowing the inner radius of the coax to be equal to zero.
The theoretical formulations presented in this work for two prominent open-ended
sensors for examination of layered composite media may serve as building blocks for
future endeavors in the expanding field of microwave NDT. The formulations may be
further improved upon by incorporation of higher order modes into the solution for the case
of radiation into inhomogeneous layered media. For such scenarios where higher order
modes play a more prominent role, such improvements can utilize for a more accurate
assessments leading to further optimization of measurement parameters. Also, numerical
models may be incorporated to account for a more realistic representation of variations of
complex permittivity in swept frequency analysis over a wide frequency band. Models for
variable permittivity profiles are also useful in application to certain material media.
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112
Data processing techniques may serve as valuable tools in complex cases where the
measurement parameters such as disbonds/delaminations and thickness variations are
highly correlated. Multivariable regression models and neural networks are some of the
techniques which can be used both for theoretical expositions and on-line measurement
setups to aid in multiparameter measurement scenarios. A detailed study in this field is
definitely of great significance for future endeavors in the field of microwave NDT.
It is foreseen to have commercially viable sensors designed and built based on the
results presented here for dielectric slab and disbond thickness variation estimation. These
sensors are intended to be relatively simple in design and inexpensive. This fact is not only
considered as part of the future extension of this research, but also indicates the practical
aspects of this endeavor.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
113
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Slabs," Accepted for presentation at the 8th annual IEEE IMTC/91 Conference,
Atlanta, GA, May 14-16, 1991.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11 9
APPENDIX A
E xplicit solution fo r Three-Layer Media
The field coefficients for two specific three-layer geometries are presented here. The
first case is that of Figure 3.6 where the layered media is terminated into an infinite half­
space and the second case is for the configuration where the third region is terminated by a
perfectly conducting sheet. The system of equations are manipulated to come up with a
more efficient matrix solution. Instead of 4N and (4N-2) system of equations needed in
general to solve for the two cases, as discussed earlier, the simplified results are given in
terms of a system of four equations for both cases.
For the case of Figure 3.6 the manipulated C matrix elements of equation (3.68) can be
written in terms of the field coefficients of layer 2 as
C llC l2 C13C141
C21C22 C23C24
I"®;
x <B2
C31C32C33C34
f 4 l C 4 2 C 4 3 C 44 J 4>i4
(A.1)
l ‘B 4
where
j Sin(KjZj ) K]
£r i- K 2x
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04.2)
120
COS(fCyZy) K2
C12 -
£ r2
j s m ( K }Z j ) K j
Kx K y
Kx
COs(?CyZy) K2
C 13 ~
CA.3)
o~JK2: 1
+1
04.4)
js in (K jZ j) Kj
£n - K
Kx K y
Cj4
aj K2zl
en - k]
COS( K j Zj )
■
2
£r, ~ Kx
k2
J
oJK2--I
(A.5)
j s i n ( K j Z j ) Kj
COs(/CyZy) K 2
+1
C21 - -
o ~ J K2 z l
(A.6 )
j s i n ( K j Z j ) Kj
COS( K j Z j )
C 22
-
k2
1
o')K2zI
cos(xryZy) £ r, ~
C23 -
k2
Kx Ky
jsin(K jZ j)
c o s { k j Zj ) e r, -
C24 -
jsm (K jZj)
£ r3 ~ K
K2
Kx K y
Kj
C 31 ~
£q ~
C32 ~ ~
1+ *1
C41 ~ C3J
C 43 -
»
k] k2
Kx K y
K
Kj
[ £ r: ~ K l
,JK2Z1
Kx K y
e f2 - K\
o>K2:l
Kx K y
04.9)
(A. 10)
Kx K y
~
K
■Jj k 2z2
Kx K y
0-JK2--
{A. 11)
{A. 12)
aJ K2z2
(A. 13)
(A.14)
C 42 - C34
Kj
(A. 8 )
~ K
Kj
£r3 ~ K K
K x Ky
Kj
* x K2 | £ r2
Kx K y
C33 ~ 1 - * L
C 34
(A .7 )
j s i n ( K j Z j ) Kj
k:
~JKiZ->
K x Ky
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A. 15)
121
£ r}
C 44 ~ ~
~ Ky *2
KxKy
®y=«2= —
J «1
, £ r: - K
K3
j
(A 16)
KxKy
COS(*7-7/) J Kr-i
ysin(x'yZy)
(A /7 )
ib3 = b 4 = o
(A /5 )
Once the above coefficients are substituted into matrix equation (A.l), the field coefficients
of layer 1 may subsequently be calculated from
(A.19)
=
**
=
" V " 1"]
<**>
which may then be used in (3.73) for calculation of the terminating aperture admittance.
For the case of the three layer medium terminated into a perfectly conducting sheet the
complex coefficients are in part similar to the case presented above. The terms that are
different can be written as
cos[ k 3 ( z 3
C3l - ~
C32-
- z 2 )]
k,k,
y s in j^ --,)]
k3
g,: - k 2 x
0 ~ J k 2z 2
Kx Ky
(A. 21 )
CQS[ ^ ( z ? - z 2 )] erj - k 2x k 2 ^ £,: - k 2 x
qJ k 2-2
/sin fje^ Z j-Z j)]
c o s[x -j(z ,? - z 2 )]
C 33 ~
£n - K 2x K2
KxKy
Kl
^
K3
KxKy
~j K-iZi
_ysin[/ci (zi - z 2)] k 3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.22)
(A.23)
122
c ° s [ y i ( z j - z 2 )]
k
2
[ 1
J K 2~-2
C j4 - -
(.A.24)
js in [ K 3 (z3 - z 2 ) \ K 3
C4I
C33
’ C42
c o s[
k
3(z j -
(>1.25)
C34
e rj -
z2 )]
y *
k
2
k
3
e f2 -
y j
C43 = ~
y s in [ /c i (z i - z 2 )]
C O s[ K j (z 2 — Z2 )]
£ r? -
k
]
k
7
k
3
£ r, -
K.
C44 y s in [ K j( z j - z 2 )]
K xKy
,j~Jk2z:
(A. 26)
K x tcy
0JK2z 2
(A.27)
Kx Ky
Once again, following the same procedure as the earlier case would result in the aperture
admittance expression for the waveguide radiating into a thre-layer conductor backed
medium.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12 3
A p p en d ix B
A nalysis o f Singularities
With the dielectric properties of the medium being complex, the admittance expression
of (5.28) may be readily evaluated using appropriate numerical integration routines.
However, for the specific case of loss-less media singularities occur in this integration over
the real axis. To simplify the notation, this admittance expression may be written as
(B .l)
= \G (C )dC
0
(B .l)
where A* is a constant multiplier and A and B represent functions in the numerator and
denominator of the integrand and J represents the Bessel function term. The poles of G(£)
are the roots of D(£) which lie on the real axis along the path of integration. Consequently
D ( £) will vanish at £ = Q and G(£) becomes singular. Referring to (B.l), along with a
pole at £ = 0 and a branch point at £ = erl there exist a series of poles for £ > en . Thus, this
expression may be evaluated using residue theorem as [Arf.85]
y i„ = P r { jG (O d +
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(BJ)
124
where Pr represents the principal part of the integral and Res{ £,} is the residue evaluated at
pole Q. The position of the complex poles over the real axis along with the branch point
are depicted in Figure B. 1
lm(C}
Branch
Point
R e {£ }
Figure B .l : Path of integration along the real axis in the complex domain.
For an analytic function with a pole at £ = 0 the residue may be calculated as
[Arf.85]
(B.4)
o -i =
where a is the coefficient of the (£-£,) in the Laurent expansion of G(£) at C, = ^ and D '
represents the derivative of the denominator function. Equation (B.3) may then be
evaluated as
(B.5)
"(£)
i=l>
For a lossless media, poles may be interpreted as excitation of surface waves which is
the trapped energy within the bounded dielectric medium.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
125
A ppendix C
C om puter Codes
This appendix contains four fortran codes simulating the theoretical results presented in
Chapters III and V. These codes calculate the terminating aperture admittance of a
rectangular waveguide and coaxial transmission line radiating into a layered media and
subsequently write the outputs in terms of conductance and susceptance, and measurable
parameterssuch as reflection coefficient, phase and return loss. The first two codes
simulate the five-layer and three-layer conductor-backed cases of the rectangular waveguide
sensor depicted in Figures 4.14 and 4.19 respectively. The other two codes simulate the
geometries of Figures 5.2a and 5.2b for the coaxial transmission line sensor.
To increase the calculation speed, the Gauss Legendre numerical integration routine
[Pre.88] used in all the codes supplied here uses uniform segments and fixed number of
points over the integration interval. For integrations over infinite ranges care must be taken
to insure proper convergence.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
Program "S lT l.f
c
c
c
c W aveguide D im ensions (bw :large . awrsmull)
c T hese dim ensions should m atch w ith the chosen o p erating freq.
c
c
Calculates G & B o f a 5_layer medium fo r som e given range of
any o f the slab thicknesses.
c
c
c
c
c
c
** T his specific code varies the thickness o f Layer 111 (m iddle).
c
c
w ith slight m odification o f I/P data any other layer
may be varied, c
**E xternal slab-like m edium s have arbitrary diel. const.
c
aw =2.019
X-band ; fm in=8.2, fm ax=12.4. fc=6.56 (GHz)
bw®2.286
aw =1.016
K u(P)-band ; fm in® !2.4. fm ax=18.. fe= 9.49 (GHz)
e
c
c C -band t fm in=4.9. fm ax=7.05. fc=3.71 (GHz)
c
bw =4.039
c
**
S ingle ferq. calculation.
c
c
c
**
Internal subroutine used for solution o f the system o f 8*eq.'s.
c
c
c
c
**
**
Internal G L integration.
O /P files : gb.d (gs, bs)
c
c
c
aw=.79
c K-band ; fm in=18., fm ax=26.5, fc= 1 4 .\ (GHz)
c
bw =1.067
c
c
rlp.d (IRI, RL, Phase)
c
c
c
c
c
c
d o u b le co m p lex c f,e p rl,e p r2 ,e p r3 ,e p r4 ,e p r5
d o u b le com plex ro l,ro 2 ,ro 3 ,ro 4 ,ro 5 ,R ,Y
com plex F
bw =1.5ii
aw=.432
K a(R)-band ; fm in*26.5, fm ax=40, fc=21.1 (GHz)
b w = .7 U
aw=.356
x la m = 3 0 ./fre q
g la m = x la m /(s q rt(l .- (x la m /(2 .* b w ))* * 2 ))
x g l= g lam /x lam
x k 0 = 2 .* p i/x lam
ak»xkO*aw
bk=xkO*bw
do u b le p recision pi,tpi.F G ,F B ,R r,R i
d o u b le p recisio n alfm n .alfm x,betm n,bctm x.b etn m n .b ctm x n
d o u b le p recisio n xa(50).w a{50),xb(50),w b(50)
external G A U LE G .F.M A T IN V
com m on /dat 1/ p i,x k 0 .a k ,b k ,zk l,zk 2 .z k 3 ,zk 4
com m on /d a t2 / ro l,ro 2 ,ro 3 ,ro 4 ,ro 5
r o l = e p r l * * 0 .5
ro 2 = e p r2 * * 0 .5
o p en (u n it® l,filc= s,g _ b .d \sta tu s= s’unknow n')
r o 3 = e p r3 * * 0 .5
open(unU®2,file=T_rl_ph.d\stalus®,unknown,)
r o 4 ® e p r4 * * 0 .5
ro 5 = e p r5 * * 0 .5
p i® 4 .* a ta n (l.)
tp i= 2 .* p i
c
c
c
w rite(6,*) '** T his code varies the thickness o f layer III*
w rite(6,*) ’slight m odification o f input thickness data n eed ed ’
w rite(6,*) 'if variation o f any other la y er thickness is desired'
w ritc(6,*) * '
w ritc(6,*) 1 Freq. of operation (G H z )-? '
rcad (5 ,* ) freq
w rite(6,*) ' T hickness o f layer I (cm )* ?'
rcad (5 ,* ) d l
D river fo r G L integration
w rite(6,*) 'Lim its o f External integral M in. & M ax. = ?'
c
c
c
read (5 ,* ) b etm n .b etm x
b ctm n = 0 .
betm n® 10.
w rite ^ ,* ) 'No. o f segm ents = ?’
read(5.*> n se g
n seg= 50
w rite(6,*) ' C om plex P erm ittivity o f layer I = ?'
x e i® b e tm x -b e tm n
w ritc(6,*) * ( e 'r l - e " r l ) ’
re a d (5 ,* ) e p rl
s e g = x c i/n s e g
w rite(6,*) 'No. o f G auss pts. in each in terval(E xtem al) = ?’
w rite(6,*) * Thickness o f layer II (cm )* ?’
read (5 ,* ) d2
w rite(6,*) * Com plex Perm ittivity of layer II® ?’
w rite(6,*) ’ (e 'r2 - e " r2 ) *
rc a d (5 ,* ) cp r2
read (5 ,* ) ng p b
c
c
ngpb=50
c
w rite(6,*) 'L im its o f Internal integral (Fixed 0 - 2 p i)‘
alfm n=0.
alfm x=tpi
w rite(6,*) 'No. o f G auss pts. in each interval(Intem al) ®?'
rcad(5,*) ngpa
w rite(6,*) ’ M in. T hickness o f layer III (cm)® ?’
read (5 ,* ) d3m n
w rite(6,*) ’ M ax. T hickness o f layer III (cm)= ?’
rcad (5 ,* ) d3m x
w rite(6.*) * T hickness increm ent for layer III (cm) = ?’
read (5 .* ) d3in
c
w rite(6,*)
w rite(6,*)
re a d (5 ,* )
w rite(6,*)
c Calculation o f G auss pts. for the internal integral by calling
c subroutine 'G A IIL E G '
’ C om plex Perm ittivity of layer III* ?’
’ (e‘r3 - c " r 3 ) ’
cpr3
* Thickness o f layer IV (cm)= ? ’
n g p a= 5 0
call gauleg(alfm n,alfm x,xa.w a,ngpa)
read (5 ,* ) d4
w rite(6,*) ‘ C om plex P erm ittivity o f layer IV® T
w rite(6,*) ’ (e 'r4 - e " r4 ) *
re a d (5 ,* )
w rite(6,*)
w ritc(6,*)
re a d (5 ,* )
epr4
* C om plex P erm ittivity o f lay er V® ? ’
’ (e'r5 - e " r5 ) ‘
epr5
do 10 in * l.n m ax
in l = i n - l
d 3 = d 3 m n + in l * d 3in
z l® d l
n m a x * l+ n in t( (d3m x-d3m n)/d3in )
w rite ( 1,*) 'd l ^ .d d ^ p r H e p ^ .e p r B .e p r d .e p r S '
w rite ( 1.*) d l,d 2 .d 4 .e p rl.c p r 2 .e p r3 ,e p r 4 ,e p r5
z2 ® d l+ d 2
z 3 = d l+ d 2 + d 3
z 4 = d l+ d 2 + d 3 + d 4
z k l= x k ()* z l
e p s O = l.
zk2*xkO *z2
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
zk3= xk0*z3
zk4= xkO *z4
c
100 stop
x x r= 0 .
x x i= 0 .
end
do 110 ik » l,n s e g
c
c
ik l - i k - 1
c
Function ' F fo r calculation o f the integrand o f the
b e tm n n a b e tm n + ik l * scg
b e tm x n = b e tm n + ik * s e g
c
c
adm ittance expression,
com plex function F (alf.bct)
d o u b le precision p i.a lf,b et,et,si.ff
Calculation o f G auss pts. fo r each sub-interval o f external
integral by calling subroutine 'G A U L E G '
d o u b le co m p lex
c a ll g au lcg (b eim n n ,b etm x n ,x b ,w b ,n g p b )
x x in = 0 .
x x rn * 0 .
do 31 i= l,n g p b
do 32 j= l,n g p a
cf^FC xa(j),xb(i) )
FG = real( c f 1
F B sim ag l c f )
x x m = x x m + w b (i)* w a (j)* F G
32
x x in = x x in + w b (i)* w a(j)* F B
c o n tin u e
31
c o n tin u e
x x rsx x r+ x x rn
x x i= x x i+ x x in
110 continue
c
c
c
com m on /d a tl/ pi,xkO ta k ,b k ,z k l(zk 2 ,z k 3 .zk 4
co m m o n /d a l2 / r o l,ro 2 ,ro 3 ,ro 4 ,ro 5
c
e t^ b e t* d c o s(a lf)
si= b e t* d sin (a lf)
r l = ( r o l * r o l- c t* e t- s i* s i ) * * 0 .5
r2=s( r o 2 * r o 2 - e t* e t- s i* s i) * * 0 .5
r3 = :(ro 3 * r o 3 - e t* e t- s i* s i) ,,* 0 .5
r 4 s s ( ro 4 * ro 4 -e t* e t-s i* s i)* * 0 .5
r 5 = ( r o 5 * r o 5 - e t* e i- s i,ls i)* * 0 .5
G e = x x r* x g l
B e = x x i* x g l
f f= 4 .* p i* ( 2 .* b k /a k )* * 0 .5 * d s in te t* a k /2 .) * d c o s ( s i* b k /2 .)
1 /(et*(pi*pi-si*si*bk*bk))
y = c m p lx (G e ,B e )
R = ( l.- y ) /( l.+ y )
c t l l= c d c o s (rl * z k l )/( (0.,1 .)* c d s in (rl * z k l ) )
ref= c d a b s (R )
s w r = ( l.+ r e f ) /( l .- r c f )
r ld l= c d e x p { ( 0 .,l.) * r l * z k l )
r le = ( r o l * ro l-e t* e t)/(e t* s i)
a tl= 2 0 .* a lo g l0 (sw r)
R r= re a l(R )
e s l = e t * s i / ( r o l * r o l- s i* s i)
r 2 e = ( r o 2 * r o 2 - e t* c t) /t e t* s i)
R i= im ag (R )
r 2 s = { r o 2 * r o 2 - s i* s i) /( e t* s i)
p h i= a ta n fR i/R r)
r 2 1 s = ( r o 2 * r o 2 - s i* s i) /( r o l * r o l - s i * s i )
c
c
c
r o l,r o 2 ,r o 3 ,r o 4 ,r o 5 .r l,r 2 ,r 3 ,r 4 ,r 5
d o u b le co m p lex C tll,r le .r 2 e ,r 3 e .r 4 c ,r 5 e ,e s l,r 2 s .r 3 s .r 4 s .r 5 s
d o u b le co m p lex r ld l,r 2 d 2 ,r 2 ls ,r 2 d 2 s .r 2 d lp ,r 2 d ln ,r 2 d 2 1 ,r 2 d 2 1 s
d o u b le c o m p le x r3 d 3 ,r3 d 2 p ,r3 d 2 n ,r3 d 3 s ,r3 d 3 2 ,r3 d 3 2 s
d o u b le co m plex r4 d 3 p .r4 d 3 n .r4 d 4 p .r4 d 4 n
d o u b le com plex fil,f i2 .f i3 .s il,s i2 .s i3 .s l.f r d
double com plex fc .fl,f2 .f3
in trinsic d co s.d sin .cd co s.cd sin .cd ex p
p aram ete r (L D A = 8 ,N = 8 )
d o u b le co m p lex A (L D A .L D A ).B (N ).X (N ),d et
in teg er L L (L D A ).M M (L D A )
external M ATINV
if(R i.g t.0.a n d .R r.lt.0.) phi= phi+ pi
r 2 d 2 = e d e x p ( ( 0 .,l.) * r 2 * z k 2 )
r 2 d 2 s = c d c x p ( ( 0 .,2 .) * r 2 * z k 2 )
r2 d lp = c d e x p ( (0..1 J * r 2 * z k l )
r 2 d l n = cd c* x p ((0 ..-1. )* r2 * zk I >
if(R i.It.0 .a n d .R r.lt.0.) phi= phi-pi
if(R i.e q .0 .a n d .R r.n e .0 .) phi= pi
r2 d 2 1 = cd cx p ( (0..1 .)* r2 * ( z k 2 - /.k l) )
r2d 2 1 s= cd cx p ( ((> ..2.)*r2*(zk2-zk 1) )
p h i= p h i* I8 0 ./p i
p h asesphi
r 3 e = ( r o 3 * r o 3 - e t* e i) /i c t* s i)
r3 s = (ro 3 * ro 3 * s i* s i ) /( c t* s i)
C alculation o f appropriate quadrant fo r phase
r3d2 p = cd ex p ( (0 .,l.)* r3 * z k 2 )
r 3 d 2 n = c d e x p ( ( 0 .f- 1 .) * r3 * z k 2 )
c
c
c
conversion o f layer thickness from cm to mm
r 3 d 3 = c d e x p ( ( 0 .,l .)* r3 * z k 3 )
r3 d 3 s = c d e x p ( ( 0 .,2 .) * r 3 * z k 3 )
c
c
d lm m = d l* 1 0 .
d2m m =d2*10.
d 3 m m = d 3 * 10.
r3d32=cdexp{ (0 .,l.)* r3 * (z k 3 -z k 2 ) )
r3d 3 2 s= cd ex p ( (0 .,2 .)* r3 * (z k 3 * z k 2 ) )
r 4 e = ( r o 4 * r o 4 - c t* e t ) /( e t* s i)
c
d4m m =d4*10.
r 4 s = ( r o 4 * r o 4 - s i* s i )/( e t* s i)
r4 d 3 p = cd ex p ( ( 0 .,l.) * r 4 * z k 3
w ritef 1,71) d3m m ,G c,B e
r4 d 3 n * cd ex p ( (0 .f-I.)* r4 * z k 3 )
r 4 d 4 p = c d c x p ( ( 0 .,l. )* r4 * z k 4 )
r 4 d 4 n = c d e x p ( f 0 .,- l .)* r4 * z k 4 )
r 5 c —( ro 5 * r o 5 - e t* e t )/(e t* si>
r5 s= l r o 5 * ro 5 -s i* s i )/(e t* s i j
71
f o rm a t(3 f l0 .4 )
w rite (2 ,7 2 ) d 3 m m ,ref,rl,p h ase
72 f o r m a t(4 fl0 .4 )
c
w ritc(6,*) ’ '
c
c
c
)
w rite (6 ,* ) *d3(mm ), (G ,B)=*,d3m m ,G c,Be
w rite (6 ,* ) '(R .p h a se J^ '.rc f.p h a sc
w rite(6,*) ' '
frd s* ff* rld 1 /((()., 1.) * x k 0 * x k 0 *r 1)
c o n tin u e
c
A ( l,2 ) = r 2 d lp * ( ( C tl l* r 2 /r l ) * ( f 2 d 2 1 s - l .) + ( r 2 d 2 1 s + l .) )
A( 1 ,3 ) = r 3 d 2 p * r 2 d 2 1 * (c t 1 ] * r l e * r 2 / r l + r 2 e ) * ( r 3 /r 2 ) * ( l .- r 3 d 3 2 s )
c
A( I ,4 ) = r 3 d 2 p * r 2 d 2 1 * ( c t l 1 * r 2 /r l + 1. )* (r 3 /r 2 ) * ( r 3 d 3 2 s - l.)
A f I .I ) = r 2 d lp * {
10
( c t l 1 * r l e * r 2 / r l )*( 1 .-r2 d 2 1 s ) - r 2 e * ( l.+ r 2 d 2 1 s )
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
)
128
A ( I ,5 ) = r 4 d 3 p * ( r 4 / r 2 ) * ( c t l I * r le * r 2 / r l+ r 2 e ) * r 2 d 2 1 * r3 d 3 2
A ( l,6 ) = -
A ( 7 ,7 ) = - l .* r 4 d 4 p * ( r 4 / r 5 + l. )
A ( 7 ,8 ) = r 4 d 4 n * ( r 4 /r 5 - l .)
B ( 7 )= (0 .,0 .)
I .* r 4 d 3 n * ( r 4 /r 2 ) * ( c t l 1 * r le * r 2 /r l+ r 2 e ) * r 2 d 2 1 * r3 d 3 2
A ( l , 7 ) = - l . * r 4 d 3 p ’ ( r 4 / r 2 ) * ( c t l l * r 2 / r l + l.) * r 2 d 2 1 * r 3 d 3 2
A ( 1 ,8 ) = r 4 d 3 n * ( r 4 /r 2 ) * ( c tl 1 * r 2 /r l + l.) * r 2 d 2 1 * r3 d 3 2
B( 1 ) = ( 1 .- c t l 1 )*frd
A ( 8 ,l ) = ( 0 ..0 .)
c
c
A (2 ,l)= < r2 d lp * e s l* (
A ( 2 ,2 ) = r 2 d lp * (
( c t lI* r2 /rl) * (l.- r2 d 2 1 s ) - < l.+ r 2 d 2 1 s >
A ( 8 .2 ) = ( 0 .,0 .)
A ( 8 .3 ) = ( 0 .,0 .)
A ( 8 .4 ) = ( 0 .,0 .)
A ( 8 ,5 ) = A ( 7 ,7 )
)
( c t l I * r2 /rl) * (r2 d 2 1 s - l.)+ r 2 1 s * < r 2 d 2 1 s + l.)
)
A ( 8 ,6 ) = A ( 7 ,8 )
A ( 2 .3 ) - r 3 d 2 p * r 2 d 2 1 * ( r 3 / r 2 ) * e s l * ( c t n * r 2 / r I + l . ) * ( l . r3 d 3 2 s )
A ( 8 ,7 ) = r 4 d 4 p * ( r 4 s + r 5 s * r 4 /r 5 )
A (8,8)= > r4d4n*( r4 s - r 5 s * r 4 /r 5 I
B (8 )= (0 .,0 .)
A ( 2 ,4 ) = r 3 d 2 p * r 2 d 2 1 * ( r 3 /r 2 ) * ( c tll * r 2 /r l+ r 2 1 s ) * ( r 3 d 3 2 s - l.)
A ( 2 ,5 ) = r 4 d 3 p * ( r 4 / r 2 ) * e s l * ( c t l l * r 2 / r l + l.) * r 2 d 2 1 * r3 d 3 2
A ( 2 ,6 ) = -
c
c
c
I .* r 4 d 3 n * ( r 4 / r 2 ) * e s l * ( c l l l * r 2 / r l + l.) * r 2 d 2 1 * r 3 d 3 2
A ( 2 ,7 ) = - l .* r 4 d 3 p * ( r 4 / r 2 ) * ( c t l l * r 2 / r l + r 2 1 s ) * r 2 d 2 1 ‘ r3 d 3 2
A ( 2 ,8 ) = r 4 d 3 n * ( r 4 /r 2 ) * ( c tll * r 2 /r l+ r 2 1 s ) * r 2 d 2 1 * r 3 d 3 2
B (2 )= B (1 )
Soultion o f system o f equations
call MATDMV (X, B. A. LDA. 0, 1, LL, MM. LDA, del)
c
c
s l = ( 0 . , l . ) / ( 2 . * c d s i n ( r l * z k l >)
s i3 = X '(4 )* r3 d 3 s + (r4 /r3 ), (X (8 )* r4 d 3 n -X (7 )» r4 d 3 p )* r3 d 3
fi3 = X (3 )* r3 d 3 s+ < r4 /r3 )* (X (6 )* r4 d 3 n -X (5 )* r4 d 3 p )* r3 d 3
s i2 = X ( 2 ) * r 2 d 2 s + ( r 3 /r 2 ) * ( s i3 * r 3 d 2 n - X ( 4 ) * r 3 d 2 p ) * r 2 d 2
f i2 = X ( l) * r 2 d 2 s + ( r 3 /r 2 ) * ( f i3 * r 3 d 2 n - X ( 3 ) V 3 d 2 p ) * r 2 d 2
s il= s l * ( ( r 2 / r l ) * ( s i 2 * r 2 d l n - X ( 2 ) * r 2 d l p ) + f r d )
fiI= s l" (r2 /rl)* (fi2 * r2 d ln -X (l)* r2 d lp )
A ( 3 ,l )= 2 .* r2 d 2 * r2 e
A ( 3 ,2 ) = - 2 .* r 2 d 2
A (3 ,3 )= r3 d 2 p * ( ( r 2 e * r 3 /r 2 )* (r 3 d 3 2 s -l.)-r 3 e * ( r 3 d 3 2 s + l.)
)
A (3 ,4 )= r3 d 2 p * ( ( r 3 /r2 ) * (l.- r3 d 3 2 s ) + ( l.+ r3 d 3 2 s ) )
A ( 3 ,5 ) = - l .* r 4 d 3 p * ( r 4 / r 3 ) * ( r 2 e * r 3 /r 2 - r 3 e ) * r 3 d 3 2
A ( 3 ,6 ) = r 4 d 3 n * ( r 4 /r 3 ) * ( r 2 e * r 3 /r 2 - r 3 e ) * r 3 d 3 2
A ( 3 ,7 ) = r 4 d 3 p * ( r 4 /r 2 - r 4 / r 3 ) * r 3 d 3 2
A ( 3 ,8 ) = - l .* r 4 d 3 n * ( r 4 / r 2 - r 4 /r 3 ) * r 3 d 3 2
B ( 3 ) = ( 0 .,0 .)
c
c
fc= ( 0 .,- l.) 'f f /( 2 .* p i) * * 2
f 1= < r o l* r o l- s i* s i) * ( 2 .* ( x k 0 * x k ( > * s ilM O ..l.) * f f /r l)
c
c
f2 = 2 .* e l* s i* (x k 0 * x k 0 * fil)
A (4 ,1 )= A < 3 ,2 >
A ( 4 ,2 ) = 2 .* r 2 d 2 * r 2 s
f3 = fc* (fl-f2 > * b et
F = f3
A (4 .3 )= r3 d 2 p * (
(r3 /r2 )* ( l.- r3 d 3 2 s )+ (l.+ r3 d 3 2 s )
A (4 ,4 )= r3 d 2 p * (
A ( 4 ,5 ) = A ( 3 ,7 )
A ( 4 .6 ) = A ( 3 ,8 )
( r2 s * r3 /r2 )* (r3 d 3 2 s -l.)-r3 s * (r3 d 3 2 s + I.>
A ( 4 .7 ) = - l .* r 4 d 3 p * ( r 4 / r 3 > * ( r 2 s * r 3 /r 2 - r 3 s P r 3 d 3 2
A ( 4 ,8 ) = r 4 d 3 n * ( r 4 /r 3 ) * ( r 2 s * r 3 /r 2 - r 3 s ) * r 3 d 3 2
B ( 4 ) = ( 0 ..0 .)
c
c
A < 5 ,1 > = (0 .,0 .)
A ( 5 ,2 ) = ( 0 .,0 .)
A ( 5 .3 ) = 2 .* r 3 d 3 * r 3 e
A ( 5 ,4 ) = - 2 .* r 3 d 3
A (5 ,5 )= -1 .* r 4 d 3 p ,'( r 3 e * r 4 /r 3 + r4 e)
A ( 5 ,6 ) = r 4 d 3 n * ( r 3 e * r 4 / r 3 - r 4 c )
A ( 5 ,7 ) = r 4 d 3 p * ( l . + r4/r3>
A ( 5 ,8 ) = r 4 d 3 n * ( l .- r 4 / r 3 l
B (5 )=( 0. ,0 .)
)
)
re tu rn
end
ccceccccccccccccccccecccccccccccccccu:eccceecoccoceccccccccccccccccccccccc
c
"GAULEG’'
c
c Given the low er and u p p er lim its o f integration X I & X 2, and
c
c given N . this routine returns arrays X & W o f lenght N , containing c
c
c
th e abscissas and w eights o f the G auss-L egendre N -point
quadrature form ula, [from N um. Reeip. book)
c
c
ceececccccccccccceccccececcccccccececceeeeececeeececcccccccccccccccccccccc
c
SUBRO U TIN E GAULEG(X1.X2.X.W .N>
IM PLICIT DOUBLE PRECISION (A -II.O-Z)
D O U BLE PRECISION X(N).W (N)
c
c
c
Increase if you don’t h av e this Boating precision,
PARAM ETER <EPS=1.D-14>
M =(N + l)/2
A ( 6 ,l ) = ( 0 .,0 .)
A ( 6 ,2 M O ..O .)
X M =0.5D0*(X2+X1)
XL=0.5D0*(X2-X1)
DO 12 I=1,M
Z=CO S(3.141592654D O *(I-.25D(I)/(N +.5DO ))
A ( 6 ,3 ) = A ( 5 .4 )
A (6 ,4 ) = 2 .* r3 d 3 * r 3 s
A ( 6 ,5 ) = A ( 5 .7 )
A ( 6 .6 ) = A ( 5 ,8 )
A ( 6 .7 ) - - l.* r 4 d 3 p * ( r 3 s * r 4 / r 3 + r 4 s )
A (6 ,8 ) = r4 d 3 n * ( r 3 s * r 4 /r 3 - r 4 s )
B ( 6 ) = ( 0 .,0 .)
c
e Starling with the above approxim ation to the Ith root, we enter the
e loop o f refinem ent by N ew ton's m ethod.
1
CONTINUE
P i = 1.DO
P2=0.D(I
DO II J= !.N
A ( 7 ,l) = ( 0 .,0 .)
P3=P2
A ( 7 ,2 l= ( 0 .,0 .)
A (7 .3 ) = ( 0 .,0 .)
P2=PI
A ( 7 ,4 ) = ( 0 ..0 .)
A (7 ,5 ) = r 4 d 4 p * ( r4 e + r 5 c * r 4 /r 5 )
A ( 7 ,6 ) = s r 4 d 4 n * ( r 4 e - r 5 c * r 4 /r 5 )
11
c
c
P l= ((2 .D 0 * J-I.D 0 > * Z * P 2 -lM .D (» * P 3 )/J
CONTINUE
P I is now the desired Legendre polynom ial.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
W e next com pute
P P. its
45
c derivative, by a standard relation involving also P2, the polynom ial
c
o f o n e lo w er order.
if(i-k ) 5 0 ,5 5 ,5 0
c
50
55
60
z(iJc)=z(i,k)/(continue
do 65 i= l ji
do 65 ja l j i
if(i-k) 60,64,60
iftj-k) 62,64,6
62
64
z(i,j)=z(i,k>*
continue
PP=N*<Z*P1 -P 2)/(Z *Z -1.DO)
Z l* Z
2 -Z 1 -P 1 /P P
IF(A B S(Z*Z l).G T.EPS)G O T O 1
X (l)=X M -X L*Z
X (N+1-I)=XM +XL*Z
W lI)=2.D 0*X L /((l.D 0-Z *Z )*P P *P P )
W(N+1-I)=*W(I)
12
65
C O N TIN U E
RETU RN
END
70
75
c
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
" M ATIV ••
c
c S ubroutine M ATIV inverts a Com plex square m atrix and solves a c
c system o f sym ultaneous lin ear equations.
c
c U pon entry to M ATIV, Z (U ) denotes the O riginal matrix.
c
c O n exixt from M ATIV, Z(I,J) contains the Inverse matrix.
e
c V (I) denotes the right hand collum n o f the m atrix equation.
c
c T h e solution is stored in C(I).
c
c L (I) & M (I) arc integer work arrays.
c
100
105
108
c
c
110
120
125
c If IW R=1 . M ATIV w ill print th e solution C(I).
c
c
c If 112-1 , M ATIV will Invert th e matrix and M ultiply by V(I),
c
c If 112=2 , M ATIV will skip inversion and sim ply multiply
c
Z (I,J) by V (I).
c
c
19
20
25
30
35
z(j,i)=hold
j=m (k)
iffj-k) 100,100.125
continue
z(J,i)=hold
go to 100
150
cmx=0.0
do 220 i= ljie q
s=(0.0,0.0)
210
220
do 210 j= l.n e q
s=s+zli,j)*v(j)
sabs=cdabs(s)
iftsabs.gt.cm x) cinx
c(i)=s
if (iw r.le.0) go to 250
write(6,5)
do 240 i= l ,neq
l(k)=k
m (k)=k
if(cdabs(bigz)-cdabs(z(i,j))) 15,19.19
bigz=z(i,j)
l(k)=i
m(k)=j
k= k-l
if(k) 150,150,105
i=l(k)
if(i-k) 120,120,108
do 110 j= l,n
130
do 80 k = l,n
10
15
continue
k=n
do 130 i= l,n
sub ro u tin e m atinv(c,v,z.idm ,iw r,iI2,l,m ,ncq,d et)
d o u b le com plex c(idm ).v(idm ),s
bigz=z(k,k)
do 20 j= k ji
do 20 i=k,n
continue
d e t= d e t* b ig z
h o ld = z(k ,i)
z (k ,i)= -z (j,i)
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
2
5
if(j-k ) 7 0 ,7 5 ,7 0
z(kj)=z(fc,j)/big,
hoU =z(jJt)
z(jjc)=*z{j,i)
c
c If IW R=0 , M ATIV will not print the solution C(I>.
continue
do 75 j = l j t
z(k,k)=1.0/bigz
80
c D E T denotes the double com plex determ inant o f Z.
c IDM denotes the dim ension o f C, V, Z. L & M.
c N EQ denotes the actual size o f the m atrix Z.
d ouble com plex z(idm .idm ),bigz,hold.det
d im ension l(idm ),m (idm )
form at! Ix .i5 ,fl0 .3 ,fl 5.7410.1)
form at (lhO)
n=neq
if ( il2 .n e .l) go to 150
d c t= (l.,0 .)
continue
do 55 i = l j i
s=c(i>
sabs=cdabs<s)
sn=sabs/cm x
240
250
w rite(6.2) i.sn.sabs
writc(6,5)
return
end
coniinue
continue
j=l(k>
if(j-k) 35,35,25
continue
d o 30 i—I ,n
hold= -zlk,i)
z(k,i )=z(j,i)
z(j,i)=hold
i=m (k)
if(i-k) 45,45.38
38
continue
do 40 j= l,n
hold=-z(jJc)
z(j.k )-z (j,i)
40
zC j.i)=hold
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
Program “31cT.h’'
c
c
c
c
c
c C alculates gs,bsJR I,R L ,P hase(R )
for a 3_layer m edium backed
c by a conductor for som e given range o f thickness
fo r any layer.
c
c
c
c
c
c
c
e
c
** E xternal slab-like m edium s have arbitrary die!, const.
c
c
** S ingle ferq. calculation.
** T hickness o f any o f th e 3 layers m ay be changed.
c
c
c
c
c
c
** Internal m atrix inversion subroutine used fo r solution of the
system o f 4-eq.*s .
* • Internal G auss Legendre integration.
** O /P files : gbc.d fgs, bs)
c
c
c
c
c
rlpc.d (IRI, R.L., Phase)
c K -band ; fmin®18., fm ax=26.5. fc®14.1 (GHz)
K a(R)-band ; fmin==26.5, fm ax=40, fc® 2 l.l (GHz)
b w = .7 ll
aw®. 356
x la m = 3 0 ./frc q
g la m = x la m /(s q rt( l.- ( x la m /( 2 .* b w ) ) * * 2 ) )
xgl® glam /xlam
x k 0 = 2 .* p i/x la m
ak®xk0*aw
b k = xk0*bw
c
c
bw® 1.067
aw®.432
r o l = e p r l * * 0 .5
r o 2 ® e p r2 * * 0 .5
ro 3 ® e p r3 * * 0 .5
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
d o u b le co m p lex
co m p lex F
c f,e p rl,e p r2 .e p r3 ,ro l.ro 2 ,ro 3 ,R ,Y
c
c
do u b le p recision pi,tpi,F G ,F B ,R r,R i
do u b le p re c isio n alfm n .alfm x .b ctm n .b ctm x .b eim n n .b etm x n
do u b le p re c isio n x a(5 0),w a(50),xb(50),w b(5 0 )
external G A U L E G ,F .M A T IN V
D river fo r G L integration
w rite(6,*) 'Lim its o f External integral M in. & M ax. =?'
re a d (5 ,* ) b ctm n .b etm x
w rite (6 ,* ) 'No. o f segm ents = ?’
r e a d (5,*) nseg
x e is b c tm x - b e tm n
s e g s x e i/ n s e g
w rite(6,*) ‘No. o f G auss pts. in each interval(E xtem al) = ?'
com m on /d a ta / p i,x k 0 ,a k ,b k ,z k l,z k 2 ,z k 3 ,ro l.ro 2 .ro 3
open( unit® 1,file® ‘g b c .d ‘,status® 'unknow n‘)
o p e n (u n it= 2 ,filc ® 'rIp c .d \sta tu s ® ’unknow n')
read(5,*> ngpb
p i= 4 .* a ta n ( l.)
c
tp i= 2 .* p i
c
w rite(6,*) ‘L im its o f In tern al integral (Fixed 0 - 2p i)‘
w rite(6,*)
w rite(6,*)
w rite(6,*)
w ritc(6,*)
alfm n = 0.
'* • T h is code varies the thickness o f layer I’
'slight m odification o f input thickness data needed'
’if variation o f any other layer thickness is desired'
' '
w rite(6,*) * F rcq. o f operation (GHz)®?’
read (5 ,* ) freq
w rite(6,*) * M in. Thickness of layer 1 (cm)® ?'
read (5 ,* ) d lm n
alfm x=tpi
w rite(6.*) 'No. o f G auss pis. in each interval (Internal) ®?‘
read (5 ,* ) ngpa
c C alculation o f G auss pts. fo r the internal integral by calling
c
w rite(6.*) ‘ M ax. Thickness of layer I tcm )= T
read(5,*> d lm x
su broutine 'G A U L E G '
call gauleg(alfm n.alfm x.xa.w a.ngpa)
w rite(6,*) ' T hickness increm ent for layer 1 (cm ) ® T
rcad (5 ,* ) d lin
w rite(6,*) ' C om plex P erm ittivity o f layer I = ?'
do 10 in = l.n m ax
in l= in -l
w ritef6,*) ' ( e 'r l - e " r l ) '
re a d (5 ,* ) e p rl
d l= d lm n + in l* d lin
z l= d l
w rite(6,*) ' T hickness o f layer II (cm)® ?'
read (5 ,* ) d2
w rite(6,*)
w rite(6,*)
re a d (5 ,* )
w rite(6,*)
rc a d (5 .* )
w rite(6,*)
w rite(6.*)
re a d (5 ,* )
’ C om plex Pcrm itiivity o f layer II = ?'
’ (e 'r2 - e " r2 ) '
ep r2
' T hickness of layer III |cm )= ?'
d3
' Com plex Perm ittivity of layer III®
’ (e 'r 3 - e " r3 ) '
epr3
n m a x = l+ n in t(
(d lm x - d lm n ) /d lin
z 2 = d l+ d 2
z 3 = d l+ d 2 + d 3
z k l= x k 0 * z l
z k 2 = x k 0 * /2
zkJ= xk0*z3
c
x x r= 0 .
xx i= 0 .
do 110 ik = I.n seg
ik l = i k - l
)
b e tm n n = b e tm n + ik l *seg
b e tm x n = b c tm n + ik * sc g
e p s O = l.
c W aveguide D im ensions (bw darge . aw:smali)
c C alculation o f G auss pts. fo r each sub-interval o f external
c T h e se dim ensions should match w ith the chosen operating freq.
c
c C -band ; fm in=4.9, fm ax=7.05, fc®3.71 (GHz)
c
b w = 4.039
c
aw «2.019
c X -band ; fmin®8.2, fm ax=12.4, fc=6.56 (GHz)
bw = 2.286
c
c
c
a w ® l.0 I6
K u(P)-band : fm in=12.4, fm ax= lS .. fc®9.49 (GHz)
bw®1.58
aw =.79
integral by calling subroutine 'G A U L E G '
c a ll g a u le g (b ctm n n ,b ctm x n ,x b ,w b ,n g p b )
xx in = 0 .
xxrn® 0.
do 31 i= l,n g p b
do 32 j® I,ngpa
cf=F( xa(j).x b (i) )
FG=real( c f )
FB®imag( c f )
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
32
31
110
x x m » x x m + w b (i)* w a (j)* ‘FG
r l = ( r o l * r o l- e t* e t- s i* s i) * * 0 .5
xxm ® xxin+ w b(i)*w a(j)*F B
c o n tin u e
r 3 = ( r o 3 * r o 3 -e t* e t- s i* s i) * * 0 .5
c o n tin u e
x x r= x x r+ x x rn
ff= 4 .* p i* ( 2 .* b k /a k )* * 0 .5 ,‘d s in ( c i* a k /2 .)* d c o s (s i* b k /2 .)
1 A ct*(pi*pi-si*si*bk*bk))
r 2 = (ro 2 * ro 2 -e t * e t-si^ s i )* *0.5
x x i» x x i+ x x in
continue
r ld l= c d e x p ( l0 ..1 .) * r l* z k l )
r2 d lp = cd e x p ( (0 .,l.> * r2 * z k l )
c
c
r2 d ln = cd c x p (
c
( 0 .,- l.) V 2 * z k l
)
c t l l = c d c o s ( r l * z k l )/( ( 0 .,l.) * c d s i n ( r l * z k l)
r e l = ( r o l * r o l * e t* c t) /(e t* s i)
G e » x x r* x g l
B e® x x i* x g l
y = c m p lx (G e ,B c )
R « * (l.-Y )/(1 .+ Y )
ref= c d a b s(R )
)
r c 2 = ( r o 2 * r o 2 - e t* e i) /( e t* s i)
r e 3 = ( r o 3 * r o 3 - e t* e t) /( e t* s i)
s w r * ( l .+ r e f ) / ( l.- r c f )
frk= <ff*rldI/( < 0 .,U * rl* x k 0 * x k (> )
A (1,1 ) ^ - r 2 d l n * ( c t l 1 * r e l * r 2 /r l + r e 2 )
A ( l , 2 ) » r 2 d l p * ( c t 11 * r e l * r 2 /r I -re2>
A ( l, 3 l » r 2 d l n * ( c t 11 * r 2 /r l + 1.)
rl= «20.*alogl0(sw r)
R r= rc a l(R )
A ( 1 ,4 ) » - r 2 d lp * ( c t 11 * r 2 /r l -1 . >
B(1 ) = f r k * ( I . - c t l l »
R i= im ag (R )
p h i= a ta n (R i/R r)
r s l= c t* s i/( ro l * ro l - s i * s i )
c
c
c
r 2 Is = (ro 2 * ,r o 2 -s i* s iJ /( r o ! * r o l- s i* s i>
A (2,1 ) = - r 2 d ln * r s I * (c tl 1 * r 2 /r l + l .)
C alculation o f appropriate quadrant fo r phase
A ( 2 ,2 ) = r 2 d l p * r s l * ( c t l 1 * r 2 /r l -1 .)
if(R i.g t.0 .an d .R r.lt.0 .) phi= phi+ pi
A (2 ,3 )s r2 d ln * (c tll* r2 /rl+ r2 ls )
if(R i.lt.0 .a n d .R r.h .0 .) phi= phi-pi
A ( 2 ,4 ) = - r 2 d l p * ( c t 11 * r 2 / r l - r 2 Is)
if(R i.e q .0 .a n d .R r.n e .0 .) phi= pi
B (2 )= B (1 )
p h i= p h i* l8 0 ./p i
p h asesphi
r2 d 2p= cdexp( ( 0 .,I .) l,'r2 * zk 2 )
r2d 2 n = cd ex p ( (0 .,-l.l* r2 '* z k 2 1
c l2 = c d c o s (r3 * tz k 3 -z k 2 ))
c t3 = ( 0 .,l.) * c d s in ( r 3 * ( z k 3 - z k 2 |)
ct2 3 = c t2 /ct3
c
c
c
conversion o f layer thickness from cm to mm
d lm m = d l* 1 0 .
d 2 m m = d 2 * 10.
d3m m =d3*10.
c
c
A ( 3 ,l ) = r 2 d 2 n * ( r e 2 - r c 3 * c t2 3 * r 2 /r 3 )
A ( 3 ,2 ) = r 2 d 2 p * ( r e 2 + r e 3 * 'e t2 3 * r2 /r 3 )
A ( 3 ,3 ) = r 2 d 2 n * ( c i 2 3 * r 2 /r 3 - 1.)
w rite( 1,71 > d lm m .G e.B e
71
fo rm a t(3 fl0 .4 )
A ( 3 ,4 ) = - r 2 d 2 p * ( c l2 3 * r 2 /r 3 + I .)
B (3 )= 0 .
72
c
w rite (2 ,7 2 ) d lm m .re f.rl.p h a se
f o r m a t(4 fl0 .4 )
w rite(6,*) ' *
r s 2 = ( r o 2 * r o 2 - s i* s i) /( e t* s i)
r s 3 = ( r o 3 * ro 3 -s i* s i)/(e t* s i>
c
c
w rite (6 ,* ) 'd l( m m ), (G ,B )= ',d lm m ,G e,B e
w rite (6 ,* ) '(R .p h a sc )* ',re f,p h a se
A ( 4 ,l )= A (3 ,3 )
A ( 4 ,2 ) = A ( 3 ,4 )
c
w rite(6,*)
A ( 4 ,3 ) = r 2 d 2 n * ( r s 2 - r s 3 * c t2 3 * r 2 /r 3 )
A ( 4 ,4 ) = r 2 d 2 p * ( r s 2 + rs 3 * c t2 3 * r 2 /r 3 )
B (4 )= 0 .
10
’ *
c o n tin u e
c
c
100
c
c
stop
end
Soultion o f system o f equations
call M ATINV (X. B. A . LDA. 0, 1. LL. M M , LDA. del)
s r d l = ( 0 ..1 .) /( 2 .* c d s in ( r l * z k l ))
c
sslp = x k O * x k O * srd l*( ( r 2 /rl)* (r2 d ln * X (3 )-r2 d lp * X < 4 )) + frk
c Function ' F for calculation o f the integrand of the
c adm ittance expression,
c
sp lp = x k O * x k O * s rd l * ( r 2 /r l ) * ( r 2 d ln * X ( I ) - r 2 d lp * X ( 2 ) )
fc= (0 ..-l.)* ff/(2 .* p i)* * 2
com plex function F (alf.bct)
f 1—( ro l * r o l- s i* s i) * ( 2 .* s s lp - ( 0 .,l . ) * f f / r l )
f2 = 2 .* e t* s i* s p lp
double precision p i.a lf.b et.et.si.ff
d o u b le com plex ro 1,ro 2 ,ro 3 ,rl ,r2 ,r3 ,r l d 1,r2d 1p.r2d 1n
d o u b le co m p lex frk ,rs l,rs 2 ,rs3 ,r2 1 s ,r2 d 2 p ,r2 d 2 n .c t2 ,c t3
d o u b le com plex r e l,r e 2 ,r e 3 ,s r d l,c tl l,c t2 3
d o u b le com plex s s lp ,s p lp ,f c ,fl,f 2 ,f3
in trin sic dcos.dsin.cdcos.cdsin.cdcxp
p a ram ete r (L D A =4,N = 4)
d o u b le com plex A (L D A ,L D A ),B (N ).X (N ),dct
in te g e r LL (LD A ),M M (LD A )
external M ATINV
)
f3 = fc* (fl-f2 > * b ct
F = f3
r e tu r n
end
eccccccccccccccccccccccccccccccccccccccccccccctccccccccccccccccccccccccccc
c
'G A U LH G "
c
com m on /d a ta / p i,x k 0 .a k ,b k ,z k l,z k 2 .z k 3 .ro l.ro 2 .ro 3
c
c
c
Given the low er and up p er lim its o f integration X I & X 2, and
given N , this routine returns arrays X & W o f lenght N , containing
the abscissas and w eights o f the G auss-L egendre N -point
c
c
c
c ts b e t* d c o s (a lf)
e
quadrature form ula, (from N um. Recip. book)
c
s i* b e t* d sin (a lf)
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
do 80 k = l,n
SUBRO U TIN E GAULEG(X1.X2.X.W ,N)
IM PLICIT D O U BLE PRECISION (A-H.O-Z)
Uk)=k
m(k>=k
DOUBLE PRECISION X(N),W (N)
bigz=z(kJO
do 20 j=kut
c
c
Increase if you don't have this floating precision,
do 20 i= k ji
c
10
PA RA M ETE R (EPS=1.D -14)
M = (N + l)/2
15
X M =0.5D 0*(X2+X 1)
X L=0.5D 0*(X 2-X I)
D O 12I= 1,M
Z=COS(3.141592654DO*(I-.25DO)/{N+.5DO)>
19
20
c
c
S tarting w ith th e above approxim ation to the 1th root, we enter the
25
c loop o f refinem ent by N ew ton’s m ethod.
1
CO NTINU E
P3=P2
P2=P1
30
35
z(j,i)=hold
i=ni(k)
if<i-k> 45.45,38
38
continue
do 4 0 j= l,n
Pl=((2.D 0*J*l.D O )*Z*P2-(J-l.D O )*P3)/J
CONTINUE
c P I is now the desired Legendre polynom ial.
P P . its
c
c
c
W e next com pute
40
45
derivative, by a standard relation involving also P2, the polynom ial
o f one low er order,
50
55
Z1=Z
Z= Z1-P1/PP
1F( A B S lZ -Z l ).G T.EPS)G O T O 1
X (I)=X M -XL*Z
60
X(N+l-I)=XM+-XL*Z
62
64
W (I)= 1D 0*X L /(<I.D 0-Z *Z )*PP *P P )
W (N +1-I)=W (I)
65
CONTINUE
RETURN
END
70
75
c
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
" M ATIV ”
c S ubroutine M ATIV inverts a Com plex square m atrix and solves a c
c system o f sym ullancous lin ear equations,
c
c
Upon entry to M A TIV , Z(1,J) denotes the O riginal matrix,
O n cxixt from M A TIV , Z(I,J) contains the Inverse matrix,
80
100
105
c V (I) denotes th e right hand collumn o f the m atrix equation,
c T h e solution is stored in C(I).
c
108
L (l) 8c M(I) are integer work arrays,
c
If 1WR=1 , M ATIV w ill print the solution C (l).
c
c
c
c
If IW R=0
, M ATIV will not print the solution C(I).
c
If 112=1 ,
M ATIV w ill Invert the matrix and M ultiply by V (l). c
If 112=2 ,
MATIV will skip inversion and sim ply multiply c
Z (U )b y V (I).
c
c
continue
do 75 j= Ia i
if(j-k) 7 0 ,7 5 .7 0
z(k,j»=z(k,j)/bigz
continue
d e t= d e t* b ig z
z(k,k)= L 0/bigz
continue
k=n
k= k-l
if(k) 150,150.105
i=llk)
ifti-k) 120.120.108
do 110 j= l.»
z(j,i)=ho!d
j=m (k)
if(j-k) 100.100,125
continue
do 130 i= l,n
h o ld = z(k ,i)
z (k ,i)= -z (j,i)
130
z(j,i)=hold
150
subroutine m atinv(c,v,z.idm ,iw r,il2,l,m ,neq.d ct)
double com plex c(idm ),v(idm ),s
go to 100
cm x=0.0
do 220 i= I,n cq
s=(0.0,0.0)
210
fo m iat(lx ,i5 4 ‘10.3,fl5.7J'10.1)
fomiaUIhO)
if(il2 .n e .l) go to 150
d ct= (l.,0 .)
z(i,j)=z(i,k)*z(k,j)+z(i,j)
continue
125
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
n=neq
do 65 i= lji
do 65 j= I.n
if(i-k> 60.64,60
if(j-k) 62.64.62
z (j.k )= 'Z (j,i)
110
120
c N EQ denotes the actual size o f the m atrix Z.
2
5
z(iJO =z(i,k)/(-bigz)
continue
h o !d = z(j,k )
c D E T denotes the double com plex determ inant o f Z.
c ID M denotes th e dim ension of C, V, Z, L 8c M.
double com plex z(idni.idm ),bigz,hold.dct
dim ension K idm ).m (idm )
hoId=-z(j,k)
.i)
z(j,i)=ho!d
continue
do 55 i= lj t
if(i-k ) 50.5 5 ,5 0
PP=N *(Z*P1 -P 2)/(Z *Z -1.DO)
12
if(j-k) 35,35.25
continue
do 30 i= l,n
hold=-z(k,i)
z(k,i)=z(j,i)
P1=1.D0
P2=0.D0
DO II J - l .N
U
c
if(cdabs(bigz)-cdabs(z(i,j») 15,19,19
bigz=z(i,j)
l{k)=i
m(k)=j
continue
continue
j= l(k)
do 210 j= I,n cq
s=s+z(i,j)*v(j)
sabs=cdabs(s)
220
if(sabs.gt.cm x) cm x=sabs
c(i)=s
if (iw r.le.0) go to 250
writc(6,5)
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
133
do 240 i= l jie q
s=c(i)
sabs=cdabs(s)
240
250
sn=sabs/cm x
w rite(6,2) i,sn,sabs
w rite(6,5)
return
end
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cccccccccccccccccccccccccccccccccctxcccccccccccccccccccccccccccccccccccc
c
Program "C O A X 2L T .f
c
c
c
c Calcultcs *gs, bs, 1RI, RL, Phase o f a
C oaxial T x . line opening
c into a fin ite slab backed by an infinte
half-space o f another c
c m aterial both having arbitrary dielectric properties.
c
c
c
c
c
c
**
**
**
•*
L ayer 1 has finite thickness.
E xternal m edium s have arbitrary diel. const.
S ingle ferq. calculation.
var. thickness for the slab.
c
c
c
c
c
c
c
c
c
•*
**
**
**
Internal G auss-L egendre N -point form ula used.
Internal Bessel function calculation.
a » R adius o f Inner conductor
b = R adius o f outer conductor
c
c
c
c
c
** O /P files: gb.d (gs.bs)
c
c
D riv er fo r G A U LE G 2D
c
c
c
Lim its o f Integral
writc{6,*) 'Lim its o f Integral M in & M ax*?'
re a d (5 ,* ) b e tm n .b e tm x
b e tm n * 0 .
b e tm x = 5 0 .
w rite(6,*) ' N o. o f segm ents =?'
c
c
rcad (5 ,* ) n seg
nseg = 2 0 0
c
x e i* b e tm x - b e tm n
s e g s x e i/n se g
c
rlp.d (IRI.RL,Phase)
w rite(6,*) ‘no. o f G auss pts. in each sub-interval?’
read (5 ,* ) ngpb
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
c
c
c
ngpb*50
co m p lex F
do 111 ii* lm m a x
d o u b le com plex c f,e p rl,e p rI,e p r2 ,x le ,R ,Y
d o u b le p recision pi,xb(50),w b(50)
i i l= ii- l
d w = d m in + iil* d in
z k l= x k 0 * d w
d o u b le p recisio n ref,b etm n ,b ctm x ,b ctm n n ,b etm x n
ex tern al G A U LE G .B E SSJO .F
zle * d w /x le
com m on /d a tl/ p i.a k .b k .zk l
co m m on /d a t2 / e p rl,e p rl,e p r2
x x r* 0 .
xx i= 0 .
o p e n t u n i t s l . f i l c a 'g b . d '. s t a t u s a 'u n k n o w n ’)
do 10 ik = I.n seg
ik l = i k - l
o p c n ( u n it* 2 ,f ile * ‘rlp .d ',s ta tu s* 'u n k n o w n ')
b e tm n n * b e tm n + ik l *seg
p i= 4 .* a ta n (l.)
w rite(6,*)
w ritc(6,*)
read (5 ,* )
w rite(6,*)
read(5,*>
b e tm x n * b e tm n + ik * s e g
’** T his code varies the thickness o f layer I'
'M ax. T hicbness o f m aterial (cm )*?'
dm ax
'M in. Thickness o f m aterial (cm )*?'
dm in
c C alculation o f G auss pts. fo r each sub-interval
c
by calling subroutine 'G A U L E G '
ca ll g a u le g (b e tm n n .b c tm x n .x b .w b .n g p b )
w rite(6,*) T h ick n ess Increm ent (cm )*?'
rcad (5 ,* ) din
x x in * 0 .
n m a x * l+ n in t( (d m a x -d m in )/d in )
w rite(6,*) ‘ Freq. o f operation (GHz) * ?'
read (5 ,* ) freq
do 31 i* l,n g p b
cf*F ( xb(i) )
fg * re a l(c f)
fb = tm ag (cf)
x x m * x x rn + w b (i)* fg
x x rn * 0 .
w rite(6,*) 'Rel. Diel. Char, o f S lab (C om plex)* ?'
w rite (6 ,* ) 1 ( e 'r l . - e " r l ) ’
re a d (5 ,* ) e p rl
w rite(6,*) ‘Rel. Diel. Char, o f Infinite half-space (C om plex)* ?'
w rite(6,*) ' (c r2 ,-e " r2 )'
read (5 ,* ) epr2
31
x x in = x x in + w b (i)* fb
c o n tin u e
x x r= x x r+ x x rn
x x i= x x i+ x x in
10 c o n tin u e
e p s O = l.
c
Coax D im ensions (cm);
c a = In n er conductor Radius . b * O uter conductor Radius
w rite(6,*) 'R adius of Inner conductor V
rcad(5,*) a r
c
c
G e= xxr
B e= x x i
= ?'
Y = c m p ix (G e,B e)
R = ( l.- Y ) /( l.+ Y )
re f= c d a b s(R )
w ritc(6,*) 'R adius o f O uter conductor 'b ' * ?'
read (5 ,* ) b r
a r= .1 1 8
b r= .3 6 2
w rite(6,*) 'Complex Perm, o f coax filling * ?'
read (5 ,* ) ep rl
c
s w r = ( l.+ r e f ) /( l .- r e f )
a ttd b = 2 0 .* a lo g I0 (s w r)
rl= 2 0 .* d lo g l()( l ./ r c f )
R r= re a l(R )
R i= im ag (R )
c F or Teflon filling
c
e p rl* ( 2 .0 7 ,0 .)
c
x la m * 3 0 ./fre q
c
c
p h a s = a ta n (R i/R r)
C alculation o f appropriate quadrant fo r phase
x k 0 = 2 .* p i/x la m
ak * x k O * ar
if(R i.gt.O .and.R r.lt.0 ) ph as* p h as+ p i
if(R i.It.O .and.R r.lt.O ) p h as* p h as-p i
bk*xkO *br
x le = x la m /( e p r l * *0.5)
if(R i.eq.O .and.R r.ne.O ) p h as= pi
p h a s e = p h a s * 1 8 0 ./p i
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
135
c
c
conversion o f layer thickness from cm to mm
Z=C O S(3.141592654D O *(l-.25D O)/(N+.5D O))
c
d lm m * d w * 1 0 .
71
c S tartin g with the above approxim ation to the Ilh root, w e en ter the
c loop o f refinem ent by N ew ton’s m ethod.
I
CONTINUE
PI=1.D 0
P2=0.D0
w rite (l,7 1 ) d lm m .G c .B e
f o r m a t( 3 fl0 .4 )
w rite (2 ,7 2 ) d lm m ,re f,rl,p h a se
72
c
f o rm a t(4 f l0 .4 )
w rite(6,*) ' '
c
w rile (6 ,* ) 'd l(m m ),
c
c
w rite (6 ,* ) '(R ,p h ase)= ',ref,p h a se
w rite(6,*) ' ’
111
DO 11 J=1,N
P3=P2
(G ,B )= \d lm n i,G e ,B e
P2=P1
P l= ((2.D O *J-l.D O )*Z *P2-(J-l.D O )*P3)/J
CONTINUE
II
c
c P I is now the desired Legendre polynom ial.
its
continue
c lo s e (l)
c lo s e (2 )
W e next com pute PP,
c derivative, by a standard relation involving also P 2, the polynom ial
c o f one lo w er order,
c
100
stop
P P=N *(Z*PI -P2 )/(Z *Z-1 .DO)
end
Z1=Z
Z= Z1-P1/PP
c
IF(A B 5(Z -Z I l.G T.EPS )GO T O I
c
Function * F fo r calculation o f the integrand o f the
X(I)=XM -XL*Z
c
c
adm ittance expression,
X(N+1-I)=XM +XL*Z
W (I)=;2.D 0*XL/((1.D ()-Z*Z)*PP*PP)
W (N +1-I)=W (I)
com plex fu nction F (bet)
d o u b le p recisio n p i.bct.rj
12
d o u b le co m p lex s q c r l,s q e r 2 ,e rl2 ,e r2 1 b ,m ,r d ,c f
d o u b le co m p lex e p rl,e p rl,e p r2 ,z k s,a rg s,a rg c ,a rg t
intrinsic cdsin.cdcos
ex tern al BESSJO
com m on / d a tl/ p i.a k .b k .zk l
com m on /d a t2 / e p rl,e p rl ,epr2
CONTINUE
RETURN
END
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c Function "BESSJO’*
c
c C alculates Bessel functions o f 1st kind, o rd er 0.
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
e t= b e t* b k
si= b e t* a k
z k s = z k l* ( (e p rl-b e t* b e t)* * 0 .5
args=cdsin< zks )
argc=cdcos( zks )
FUN CTIO N BESSJO(X)
REA L*8 Y ,P1,P 2.P 3,P 4,P5,Q 1,Q 2,Q 3,Q 4,Q 5,R 1,R 2,R 3,R 4,R 5.R 6,
*
S1.S2.S3.S4.S5.S6
D ATA P l,P 2 ,P 3 ,P 4 ,P 5 /I.D 0 .-.1 0 9 8 6 2 8 6 2 7 D -2 ,.2 7 3 4 5 1 0 4 0 7 D -4 .
)
a rg t= a rg s /a rg c
*
-.20733706391> 5,.20938872l ID -6 /. Q 1,Q 2,Q 3,Q 4,Q 5/.1 5 6 2 4 9 9 9 9 5 D -
s q e r l =( e p r l - b e t *bct>* *0.5
s q c r 2 = 4 c p r2 -b e t* b c t)* * 0 .5
rj= (l.D 0 /b e t)* ( ( B E S S J0(ct)-B E S S J0(si) )**2 )
e rl2 = e p rl/e p r2
e r2 I b = ( (e p r2 -b et* b et)* * 0 .5
)/( { cp rl-b et* b et)* * 0 .5
* 1.
*
. 1430488765 D -3,-.6 9 1 114765 ID -5 ,.7 6 2 1095161D -6,.9 3 4 9 4 5 1 5 2 D - 7 /
)
D ATA R 1 .R 2.R 3.R 4.R 5.R 6/57568490574.D 0,13 3 6 2 5 9 0 3 5 4 .DO.6 5 1 6 1 9 6 4 0 .7D
r n = l .D 0 + ( 0 .,l .) * c r l 2 * e r 2 l b * a r g i
rd=( e rl2 * sq e r2 ) + ( (0 .,l.)* s q e rl* a rg t )
cf= ( (l.D 0 * e p r l) /(
F = l .* c f
*0,
tep rl* * 0 .5 )* alo g (b k /ak ) > )*rj*rn/rd
*
r e tu r n
end
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
"GAULEG'*
c G iven the low er and upper lim its o f integration X I & X 2, and
c given N , this routine returns arrays X & W o f lenght N , containing
c the abscissas and w eights o f the G auss-L egendre N-point
c
quadrature form ula, (from Num. Recip. book)
cccccccccccccccccccccccccccccccccccceccccocccccccceccceeececcceccoocccec
c
SU BRO U TIN E G A U LEG (X 1X 2,X .W ,N )
IM PL IC IT D O U BLE PRECISION (A-H.O-Z)
D O U B LE PRECISION X(N).W (N)
c
c
c
Increase if you don’t have this floating precision,
c
-1 1 2 1 4 4 2 4 .18D 0.77392.3 3 0 17DO.-184.9052456DO/,
S 1 .S 2 .S 3 .S 4 .S 5 .S 6 /5 7 5 6 8 4 9 0 4 1 I.D 0 .I0 2 9 5 3 2 9 8 5 .D 0 ,
9494680.718D 0.59272.64853D O .267.8532712D 0.1.D 0/
IF(A BS(X ).LT.8.)TH EN
Y=X**2
B ESSJ0=(R 1+Y *(R 2+Y *(R 3+Y *{R 4+Y *(R 5+Y *R 6)))))
*
/(S1+Y *(S2+Y *(S3+Y *(S4+Y *(S5+Y *S6)))))
ELSE
c
c
AX=ABS(X)
Z=8./AX
c
c
Y=Z**2
X X =A X -.785398164
B E S S J0 = S Q R T (.6 3 6 6 1 9 7 7 2 /A X )* (C O S (X X )* (P I+ Y * (P 2 + Y * (P 3 + Y * (P
4+Y
*P5))))-Z*S1N (X X )*(Q 1+Y *(Q 2+Y *(Q 3+Y *(Q 4+Y *Q 5)))))
ENDIF
RETURN
END
PA RA M ETE R (EPS=1.D-14>
M =(N + I)/2
X M =0.5D 0*(X 2+X 1)
X L=0.5D 0*(X 2-X 1)
D O 12 1=1,M
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
x le -xlam /leprl ••0.5)
CCCCOCCCCCCCeCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCOCOCCCCCCCCCCCCCCC
c
c
c
Program "C O A X 2LC T .f'
Calcultcs *g$, bs, IRt, RL, Phase o f a Coaxial
c
into a 2-layer dielectric m edium backed by a C onducting sheet,
Tx. line opening
c
e
c
c
c
c
c F inite dielectric lay er thicknesses.
c
c
** E xternal m edium s have arbitrary diel. const.
c
c
c
** S ingle fcrq. calculation.
c
c
c
c
c
c
c
c
c
** var. thickness fo r th e slab.
** Internal G auss-L egendre N -point form ula used.
** Internal B essel function calculation.
** a » R adius
o f Inner conductor
•* b a Radius
o f outer conductor
** O /P files: gbc.d (gs.bs)
rlpc.d (IRI.RL.Phase)
D river fo r G A U LE G 2D
c
c
c
c
c
c
c
Limits o f Integral
w rite(6,*) ‘Lim its o f Integral Min & M ax*?'
re a d (5 ,* )
b etm n = 0 .
c
c
b e tm x s lO .
w rite(6,*) * N o. o f seg m en ts =?'
read (5 ,* ) n seg
n se g = 2 0 0
x e i= b c tm x -b c tm n
s e g = x e i/n s e g
c
c
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
writc(6.*» 'no. o f G auss pis. in each interval?*
read (5 ,* ) ng p b
co m p lex F
d o u b le com plex c f,e p rl,e p rl,e p r2 txle,R ,Y
d o u b le p recisio n p i,x b (50),w b(50)
c
c
d o u b le p re c isio n ref.b etm n .b etm x .b etm n n .b etm x n
ex te rn a l G A U LE G ,B E SSJO .F
c
ngpb=50
do 111 ii^ ljim a x
co m m on /d a t l / p i.a k ,b k ,z k l,z k 2
co m m on /d a t2 / e p rl,e p rl tepr2
ii l= ii- l
d l= d lm n + iil* d lin
zl - d l
z 2 = d l+ d 2
o p e n ( u n it= l,file = 'g b c .d ',s ta tu s = 'u n k n o w n ')
o p e n (u n it= 2 ,file = ’rlp c .d ',s ta tu s s'u n k n o w n ')
z k l= x k 0 * z l
z k 2 = x k 0 * z2
p i= 4 .* a ta n (l.)
z le s d w /x le
x x r= 0 .
xxi=0.
w ritc(6,*) ’** T his code varies the thickness o f layer P
w rile(6,*) * '
do 10 ik = l.n se g
ik 1—ik - 1
w ritc(6,*) * Freq. o f operation (G Hz) = ?'
rcad (5 ,* ) freq
w rile(6,*) ‘M ax. T hichness o f laycr-I (cm )*?'
rcad (5 ,* ) d lm x
w rite(6.*) 'M in. T hickness of layer-1 (cm )=?'
re a d (5 ,* ) d lm n
w ritc(6 ,* ) T h ic k n e s s Increm ent (cm )* ?’
rcad (5 ,* ) din
b c tm n n s b e lm n + ik l * seg
b e tm x n s ;b e tm n + ik * s e g
c Calculation o f G auss pts. fo r each sub-interval
c
by calling subroutine 'G A U L E G '
n m a x a l + n i n t ( ( d l m x - d l m n ) / d l in)
w rite(6,*) 'Rel. D iel. Char, o f laycr-I (Com plcx)= ?'
w rite (6 ,* ) ' ( e T l , - e " r l ) '
re a d (5 ,* ) e p rl
ca ll g au leg (b ctm n n ,b etm x n ,x b ,w b ,n g p b )
x x in = 0 .
x x rn = 0 .
do 31 i= l,n g p b
cf=F( xb(i) )
fg = real(cf)
tb= im ag(cf)
w rite(6,*) T h ick n ess o f laycr-II <cm)=?'
read tS ,* ) d2
w rite(6,*) 'Rel. D id . Char, o f laycr-II (Complex )= ?'
w rite (6 ,* ) ’ ( c 'r 2 ,- e " r 2 ) ‘
re a d (5 ,* ) epr2
x x rn = x x rn + w b (i)* fg
x x in » x x in + w b (i)* fb
e p s O = l.
c
b etm n .b etm x
31
Coax D im ensions (cm);
c a = In n er conductor Radius , b = O uter conductor Radius
c o n tin u e
x x r= x x r+ x x rn
xx i= x x i+ x x in
10 c o n tin u e
w ritc(6,*) 'Radius o f Inner conductor 'a ' = T
re a d (5 ,* ) a r
c
c
c
c
w ritc(6,*) 'Radius o f O uter conductor 'b* = ?'
re a d !5 ,* ) br
a r= .1 1 8
b r= .3 6 2
w rite(6,*) 'Com plex Perm, o f coax filling = ?'
w rite (6 ,* ) ' (c‘ rl.-e*'rl)'
read (5 ,* ) eprl
G e= x x r
B e= x x i
Y = c m p lx (G e,B c)
R = ( l.- Y ) /( 1 .+ Y )
rc fs c d a b s (R )
sw r= ( l.+ r c f ) /( 1.-re f)
c
c For T eflon filling
c
e p rl= (2 .0 7 ,0 .)
a ttd b = 2 0 .* a lo g l0 (sw r)
r l= 2 0 .* d lo g l0 ( l./ r e f )
R r= reaI(R )
R i= im ag(R )
p h a s= a ta n (R i/R r)
x la m = 3 0 ./fre q
c
x k 0 = 2 .* p i/x la m
ak= xkO *ar
c C alculation o f appropriate quadrant fo r phase
c
bk=xkO *br
if(R i.gt.O .and.R r.lt.0) phas= phas+pi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
137
if(R i.lt.0.and.R r.lt.O ) phas= phas-pi
IM PLICIT D O U BLE PR EC ISIO N IA-H.O-Z)
if(R i.cq.O .and.R r.ne.O ) phas=pi
D O U BLE PREC ISIO N X(N).W (N )
p h a s e = p h a s * 1 8 0 ./p i
c
c
c
c Increase if you d o n 't have this floating precision,
c
conversion o f layer thickness from cm to m m
c
PARAM ETER (EPS=1.D -14)
d lm m = d w * 1 0 .
M =(N + l)/2
XM =0.5D 0*(X 2+X 1)
XL=0.5D0*(X2-X1)
DO 12 1=1 ,M
dw m m =z2*I0.
71
w ritc (l,7 1 ) d lm m .G e .B e
f o rn ia t(3 f l0 .4 )
72
w rite (2 ,7 2 ) d lm m .re f.rl.p h a se
f o rm a t( 4 fl0 .4 )
c
w rite(6,*) ' '
c
c
c
w rite (6 ,* ) 'd l(m m ) , (G ,B )= '.d lm m ,G e,B c
w rite (6 .* ) '(R ,p h a s e )= ’,ref,p h ase
w rite(6,*) ' '
111
100
Z=C O S(3.14I592654D O *(I-,25DO )/(N +.5D O ))
c
c Starting with the above approxim ation to th e Ith root, we enter the
c loop o f refinem ent by N ew ton's m ethod.
I
CONTINUE
P1=»1.D0
P2=O.DO
DO 11 J=1,N
continue
P3=P2
c lo s e d )
c lo se d )
P2=P1
P1=((2.D0*,J-1.D 0)*Z *P2-(J-1.IX ))*P3)/J
CONTINUE
II
c
stop
end
c PI is now the desired Legendre polynom ial.
its
c
c derivative, by a standard relation involving also P2, the polynom ial
c o f one low er order,
c
P P = N * (Z * P l-P 2 )/(Z * Z -l.D 0 )
c Function ' F for calculation o f the integrand o f the
c ad m ittan ce expression,
c
c o m p lex fu nction FlbeU
d o u b le precision p i.b e t.rj
Z1=Z
Z=Z1-P1/PP
d o u b le com plex s q e rl ,sq er2 ,crk ,cf
d o u b le com plex rrrl ,m 2 ,m 3 ,m ,rd l,rd 2 ,r d
IF(A BS<Z-Zl).G T.EPS)G O T O 1
X(I)=XM -XL*Z
d o u b le com plex e p rl,e p rl,e p r2 ,z lk ,z 2 k ,a rg s ,a rg c ,a rg t
X(N+1-I)=XM +XL*Z
W (I)= 2.D 0*X L /((L D 0-Z *Z )*P P *PP )
W fN + l-I)=W (I)
intrinsic cdsin.cdcos
e x te rn a l BESSJO
com m on / d a tl/ p i,a k ,b k ,z k l,z k 2
co m m o n /d a t2 / e p rl,e p rl,e p r2
12
e t= b e t* b k
s i= b et* a k
CONTINUE
RETURN
EN D
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c Function "BESSJO"
c
s q e r l = ( e p r l- b e t * b e t) * * 0 .5
s q e r 2 = ( e p r 2 - b e t* b c t) * * 0 .5
z lk = z k l* s q c rl
z 2 k = (0 .,-2 .)* z k 2 * s q e r2
args= cdsin( z l k )
c
Calculates Bessel functions o f 1st kind, order 0.
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
FUNCTION BESSJO(X)
argc= cdcos( z l k )
a rg t= a rg s /a rg c
REA L*8 Y.P1 ,P 2.P 3,P 4.P 5,Q 1,Q 2,Q 3,Q 4,Q 5,R 1,R 2,R 3,R 4,R 5.R 6.
S1.S2.S3.S4.S5.S6
*
rj= U .D 0/bet )*( ( BESSJO iet)-BESSJO (si) )**2 >
DATA PI ,P2,P3,P 4,P 5/1 .DO.-. 10 9 8 6 28627D -2..2734510407D -4,
*
-.2073370639D -5,.2093 8 8 7 2 1 1 D -6/, Q 1.Q 2.Q 3.Q 4.Q 5/.1 5 6 2 4 9 9 9 9 5 D -
e r k = ( e p r l/e p r 2 ) * ( s q e r 2 /s q e r l)
r n l = I .D 0 + ( 0 .,l .)* c rk * a rg t
rn2 = l.D 0 - ( 0 .,l . )" c rk * a rg t
m 3= cdexp( z2k )
* 1.
*
.14 3 0 4 8 8 7 6 5 D -3 .-.6 9 1 1147651D -5,.7 6 2 1 0 9 5 161D-6,.9 3 4 9 4 5 1 5 2 0 - 7 /
r d l= erk + (0 .,l.)* a rg t
rd2= erk - (0..1.)*argt
DATA R1 .R 2.R 3.R 4.R 5.R 6/57568490574.D 0,13 3 6 2 5 9 0 3 5 4 .D 0 .6 5 1 6 1 9 6 4 0 .7 D
*0.
r n = m l + (rn 2 * rn 3 )
rd = s q c rl* (rd l- (rd 2 * m 3 ))
c f—( (l.D 0 * e p r l)/(
F = l .* c f
r e tu r n
end
W e next com pute PP.
(ep rl* * 0 .5 )* alo g (b k /ak ) ) )*rj*rn/rd
*
-11214424.18D 0.77392.33017D 0.-184.9052456D 0/,
SI.S2.S 3.S 4.S 5.S 6 /5 7 5 6 8 4 9 0 4 1 1 .D 0 .I0 2 9 5 3 2 9 8 5 .D 0 ,
*
9494680.718D 0.59272.64853D 0.267.8532712D 0.1.D 0/
IF( ABS(X J.LT.8. )TUEN
Y=X**2
B ESSJ0=(R 1+Y *<R2+Y *(R 3+Y *(R 4+Y *(R 5+Y *R 6)))))
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccca'cccccccc
c
"GAULEG"
c
c
Given the low er and upper lim its o f integration X I
& X 2, and c
c
given N, this routine returns arrays X & W o f lenght N , containingc
c
c
the ab scissas and w eights o f th e G auss-Legendre N-point
quadrature form ula, [from N um . Rccip. book]
cccccecccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
SU BRO U TIN E G A U LE G (X l JC2.X.W.N)
c
c
*
/JSI+Y *(S2+Y *(S3+Y *(S4+Y *(S5+Y *S6)))))
ELSE
AX=ABS(X)
Z=8./AX
Y=Z**2
X X =A X -.785398164
B ESS J0 = S Q R T (.6 3 6 6 1 9 7 7 2 /A X )*(C O S( X X )*t P I+ Y * (P 2 + Y * (P 3 + Y * (P
4+Y
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
*P5»))-Z*SIN(XX)*(Q1+Y*(Q2+Y*(Q3+Y*(Q4+Y*Q5)))))
E N D IF
RETURN
EN D
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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