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Microwave cavity perturbation measurements of complex dielectric permittivity and complex magnetic permeability

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Microwave Cavity Perturbation Measurements of Complex Dielectric
Permittivity and Complex Magnetic Permeability
A thesis
submitted by
Mi Lin
In partial fulfillment of the requirements
for the degree of
Master of Science
In
Electrical Engineering
TUFTS UNIVERSITY
May 2006
© Mi Lin
ADVISER:
Professor Mohammed Nurul Afsar
UMI Number: 1433639
UMI Microform 1433639
Copyright 2006 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, MI 48106-1346
Acknowledgements
The author would like to thank Professor Mohammed N. Afsar and Dr. Young Wang for
their guidance during the graduate study at Tufts University. The author would also like
to thank Mr. Paul Dee for his courteous helps in the design and construction of various
mechanical parts for the cavity resonators. Finally, the author would like to thank his
parents for their supports in study, in life and in everything else.
ii
Abstract
Measurement of complex permittivity and permeability of dielectric and magnetic
material plays an important role in microwave technology. Knowing properties of
material is the first step in all kind of design when the material is used. This thesis has
examined the rectangular cavity perturbation method as it used for material complex
permittivity and complex permeability measurement. A wide range of material’s
permittivity and permeability have been successfully determined based on the rectangular
cavity perturbation technique. Also, a modified rectangular cavity perturbation technique
has been introduced and a new set of formulas have been derived in accommodating with
a special measurement case namely when the material sample under measurement has
insufficient sample length comparing to the height of cavity resonator. These newly
derived formulas are more general than the standard forms and have been proved to be
accurate in yielding consistent results by measuring different samples with different
lengths.
iii
Contents
Chapter 1
Introduction
1
Chapter 2
Cavity Perturbation Technique
5
Chapter 3
Experimental Results
13
Chapter 4
The New Cavity Perturbation Technique
31
Chapter 5
Conclusion
40
Measured Resonant Frequencies for Each Designed Rectangular
Cavity Resonator
42
B
The General TRL Calibration Technique and Debye’s Equation
45
C
Calculation of Ne Due to Image Effect
53
Appendices
A
Publications
55
Bibliographies
56
iv
List of Tables
Table 2.1
The dimension of each designed rectangular cavity resonator.
Table 2.2
The operating modes for each designed rectangular cavity resonator.
Table 3.1
Theoretical resonant frequencies, measured resonant frequencies of empty
cavity and measured resonant frequencies of the cavity loaded with empty
capillary tube.
Table 3.2
Measured real and imaginary parts of complex permittivity of six chemical
liquids.
Table 4.1
Comparison of complex permittivity calculated by using (4.5a-4.5b) and
(4.10a-4.10b) with Ne at frequency 4.5463 GHz.
Table B.1
A discrete measured S-Parameters data retrieved from VNA.
Table B.2
Debye’s parameters for chlorobenzene from literature.
v
List of Figures
Figure 1
The vector diagram of complex permittivity.
Figure 2.1
Geometry of a rectangular waveguide.
Figure 2.2a
Side view of rectangular waveguide for cavity resonator.
Figure 2.2b
Front view of rectangular waveguide for cavity resonator.
Figure 2.2c
The coupling plate for rectangular cavity resonator.
Figure 2.3
A rectangular cavity resonator for a particular frequency band.
Figure 2.4
Changed in resonant frequency and Q-factor of a cavity resonator due to a
sample insertion.
Figure 3.1
Liquid samples measurement setup.
Figure 3.2
Measured resonant frequencies of empty rectangular cavity.
Figure 3.3
Complex permittivity of cyclohexane measured using cavity perturbation
technique and the waveguide transmission line technique at the X-band.
Figure 3.4
Complex permittivity of chlorobenzene measured using cavity
perturbation technique and calculated data by the Debye equation at the Xband.
Figure 3.5
Complex permittivity of ethyl alcohol measured using cavity perturbation
technique at the X-band.
Figure 3.6
Complex permittivity of 1-4dioxane measured using cavity perturbation
technique at the X-band.
Figure 3.7
Complex permittivity of benzene measured using cavity perturbation
technique at the X-band.
Figure 3.8
Complex permittivity of acetone measured using cavity perturbation
technique at the X-band.
Figure 3.9
Dielectric samples measurement setup.
Figure 3.10
Measured complex permittivity of boron nitride.
Figure 3.11
Measured complex permittivity of magnesium fluoride.
vi
Figure 3.12
Ferrite samples measurement setup.
Figure 3.13
Measured real parts of permittivity of 99.9% pure YIG, gadoliniumsubstituted YIG and aluminum-substituted YIG.
Figure 3.14
Measured complex permeability of 99.9% pure YIG.
Figure 3.15
Measured complex permittivity of nickel ferrite.
Figure 3.16
Measured complex permeability of nickel ferrite.
Figure 3.17
Measured complex permeability of strontium ferrite composite sample #1
Figure 3.18
Measured complex permeability of strontium ferrite composite sample #2
Figure 4.1
Polarization of the material sample with the external applied electric field
As a result, a depolarizing field of magnitude.
Figure 4.2
Variations of Ne for the normalized sample length to the height of the
cavity.
Figure 4.3
The new cavity measurement set up.
Figure A.1
Measured resonant frequencies for H band rectangular waveguide cavity
resonator.
Figure A.2
Measured resonant frequencies for C band rectangular waveguide cavity
resonator.
Figure A.3
Measured resonant frequencies for X band rectangular waveguide cavity
resonator.
Figure A.4
Measured resonant frequencies for Ku band rectangular waveguide cavity
resonator.
Figure A.5
Measured resonant frequencies for K band rectangular waveguide cavity
resonator.
Figure B.1
Block diagram of a network analyzer measurement of a two port device.
Figure B.2
Block diagram for Thru connection.
Figure B.3
Signal flow graph for Thru connection.
Figure B.4
Block diagram for Reflect connection.
vii
Figure B. 5
Signal flow graph for Reflect connection.
Figure B.6
Block diagram for Line connection.
Figure B. 7
Signal flow graph for Line connection.
Figure B.8
Resonant frequency generated from discrete data lists in table B.1.
Figure B. 9
Calculation of resonant frequency and Q factor for the discrete data lists in
table B.1.
Figure C.1
Image effect due to metallic ground plates.
viii
Chapter 1
Introduction
Dielectric and Magnetic Properties of Matter
Propagation of electromagnetic waves in a media or in a material is totally
described mathematically by Maxwell’s equations:
∇× E = −
∂B
∂t
(1.1a)
∇× H = −
∂D
+J
∂t
(1.1b)
∇•D = ρ
(1.1c)
∇•B = 0
(1.1d)
where
J is the current density
E is the electric field intensity
D is the electric flux density
H is the magnetic intensity field
B is the magnetic flux density
p is the charge density
The unique solution to a particular circumstance is decided by the boundary
conditions and the electromagnetic properties of the material. The permittivity (ε),
permeability (µ) and conductivity (σ) are commonly referred as the electromagnetic
properties of a material. They relate the six parameters of Maxwell equation in following
1
way:
D =ε ⋅E
(1.2a)
B = µ⋅H
(1.2b)
J =σ ⋅E
(1.2c)
At high frequencies, especially in microwave frequency and frequency beyond the
effect of conductivity is ignorable. This is true for most non-conductive material.
Therefore, the mainly interested electromagnetic properties of a material to be studied in
this thesis in high frequency are the permittivity and permeability.
Permittivity which is a complex quantity describes the interaction of a material
with an electric field. Complex permittivity is also called complex dielectric constant.
The notation for the complex permittivity is given as follow:
ε = ε ∗ = ε 0ε r
(1.3a)
ε r = ε r' − jε r''
(1.3b)
where
ε0 =
1
× 10 −9 , ε 0 is the free space permittivity and has a unit of F/m.
36π
The real part of complex permittivity (εr') is a measure of how much energy from
an external electric field is stored in a material. The imaginary part of permittivity (εr '')
is called the loss factor and is a measure of how dissipative or lossy a material is to an
external electric field. The ratio of the imaginary part of complex permittivity to the real
part of complex permittivity is called loss tangent ( tan δ). A simple vector diagram of
complex permittivity is illustrated in figure 1.
2
Figure1. The vector diagram of complex permittivity
Permeability which is also a complex quantity describes the interaction of a
material with a magnetic field.
Under an applied external static field the complex
magnetic permeability is well known as non-diagonalized tensor:
(1.4)
where µ ∗ , µ z∗ and k ∗ are all complex numbers. µ ∗ is the principal direction transverse
component of the magnetic permeability. µ z∗ is the parallel component. k ∗ is the offdiagonal transverse component. With the absent of external applied static fields, the
complex magnetic permittivity reduces to:
µ = µ = µ ∗ = µ0 µr
(1.5a)
µ r = µ r' − jµ r''
(1.5b)
where
µ 0 = 4π × 10 −7 , µ 0 is the free space permeability and has a unit of H/m.
The real part of complex permeability (μr') is a measure of how much energy from
3
an external magnetic field is stored in a material. The imaginary part of permittivity (μr '')
is a measure of how loss of material when subject to an external magnetic field.
Review of Measurement Techniques in Microwave Frequency Ranges
Measurement of complex permittivity and permeability of dielectric and
magnetic material plays an important role in microwave technology. Knowing properties
of material is the first step in all kind of design when the material is used. Quite a few
techniques such as cavity perturbation [1], open-ended coaxial probes [2], free space [3],
waveguide transmission line [4] and dispersive fourier transform spectrometer [5] have
been developed for material permittivity and permeability measurement. Nevertheless,
the cavity perturbation method has a reputation as one of most simple and accurate
techniques for measuring dielectric and magnetic properties of material over years.
The earliest treatment of cavity perturbation theory was given by Bethe and
Schwinger [6]. Following them, many researchers have been studied and improved cavity
perturbation technique. Up to nowadays, many theories related to it have been proposed
and experimentally verified. The cavity perturbation technique used to carry out the
measurement tasks as described in chapter 3 is mainly follow K. T. Matthew’s method
[1]. Then, a new cavity perturbation technique derived from K. T. Matthew’s method [1]
will be presented in chapter 4.
4
Chapter 2
Cavity Perturbation Technique
Rectangular Waveguide
The geometry of a rectangular waveguide is shown in figure 2.1. The rectangular
waveguide can propagate TE and TM modes of electromagnetic waves, but not TEM
waves. Rectangular waveguide have cutoff frequencies below that electromagnetic wave
the propagation is not possible. The cutoff frequency for a particular mode in rectangular
waveguide is determined by the following equation:
( f c ) mn =
1
2π uε
2
⎛ mπ ⎞ ⎛ n π ⎞
⎟
⎟ +⎜
⎜
⎝ a ⎠ ⎝ b ⎠
2
(2.1)
So, the corresponding wavelength is then:
(λc ) mn =
2
2
⎛m⎞ ⎛n⎞
⎜ ⎟ +⎜ ⎟
⎝ a ⎠ ⎝b⎠
2
(2.2)
Figure 2.1 Geometry of a rectangular waveguide.
5
The TE10 mode is the dominant mode (lowest cutoff frequency) of a rectangular
waveguide with a >b.
Design of Rectangular Cavity Resonators
The cavity resonator is formed by using a section of rectangular copper
waveguide with both ends covered by a pair of flat copper coupling plates. Figure 2.2a2.2c illustrates the structure of a designed rectangular cavity resonator. To cover the
measurement frequency from 4.5 to 26.5 GHz which composes H, C, X, Ku and K bands,
five different dimension pieces of waveguide are used, namely WR 187, WR 137, WR 90,
WR 62 and WR 42. In order to get the sample in and out of the cavity without
disassembling it, a narrow non-radiated slot was opened at the top surface of the cavity
wall. Table 2.1 lists the dimension of each designed rectangular cavity resonator. Table
2.2 lists the operating modes and corresponding frequencies for each designed band of
rectangular cavity resonators. The narrow opened slot has negligible effect on changing
the geometrical configuration of electromagnetic fields inside of the cavity. Figure 2.3
shows a complete constructed rectangular waveguide resonator.
Figure 2.2a Side view of rectangular waveguide for cavity resonator.
6
Figure 2.2b Front view of rectangular waveguide for cavity resonator.
Figure 2.2c The coupling plate for rectangular cavity resonator.
Figure 2.3 A rectangular cavity resonator for a particular frequency band.
7
The resonant frequency of rectangular cavity resonator for the TEmnL or TMmnL
mode is determined
f mnL =
C
2π u r ε r
2
2
⎛ mπ ⎞ ⎛ nπ ⎞ ⎛ Lπ ⎞
⎜
⎟ +⎜
⎟ +⎜
⎟
⎝ a ⎠ ⎝ b ⎠ ⎝ d ⎠
2
(2.3)
For the resonator shown in figure 2.3 with b < a < d, the lowest resonant frequency will
be the TE101 mode, corresponding to the TE10 dominant waveguide mode. Table 2.1 lists
the dimension of each designed rectangular cavity resonator. And, table 2.2 lists the
operating modes and the corresponding resonant frequencies for each designed
rectangular cavity resonator. The measured resonant frequencies for each designed
rectangular cavity resonator can be found in appendix A.
Table 2.1 The dimension of each designed rectangular cavity resonator.
Waveguide
b
(mm)
a
(mm)
d
(mm)
WR 187
WR 137
WR 90
WR 62
WR 42
22.16
15.84
10.12
8.04
4.38
47.56
34.91
22.80
15.78
10.69
91.43
91.32
89.45
90.01
79.91
Coupling
hole
diameter
(mm)
7.56
5.47
4.50
3.95
2.80
Opened
slot
width
(mm)
3.34
3.25
3.10
2.15
1.75
Volume
(mm^3)
Recommend
frequency
range (GHz)
Cutoff
frequency(fc)
(GHz)
96360.78
50497.62
20639.34
11419.68
3742.28
3.95-5.85
5.85-8.20
8.20-12.40
12.4-18.0
18.0-26.5
3.15
4.30
6.557
9.486
14.05
8
Table 2.2 The operating modes and the corresponding resonant frequencies for each
designed rectangular cavity resonator.
m
n
1
1
0
0
1
1
0
0
1
1
1
1
0
0
0
0
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
L
WR 187
2
3
WR 137
3
4
WR 90
3
4
5
6
WR 62
5
6
7
8
9
WR 42
6
7
8
9
10
11
12
fmnL (GHz)
λg (mm)
4.551
5.826
91.43
60.95
6.538
7.851
60.88
45.66
8.282
9.395
10.658
12.021
59.63
44.73
35.78
29.82
12.641
13.796
15.048
16.374
17.757
36.00
30.00
25.72
22.50
20.00
17.990
19.221
20.550
21.960
23.434
24.964
26.537
26.64
22.83
19.97
17.76
15.98
14.53
13.315
Measurement Theory
When a small material sample is inserted in a resonant cavity, it will cause the
complex frequency shift. The amount of complex frequency shift is given by Waldron [7]
as
−
δω
=
ω
(εr −1)εo ∫ E ⋅ Eo*dν + (µr −1)µo ∫ H ⋅ Ho*dν
Vs
Vs
∫ Do ⋅ E + Bo ⋅ H dv
*
o
*
o
(2.4)
Vc
where δω is the complex frequency shift; Do, Eo, Bo and Ho are the unperturbed cavity
fields, and E and H are the fields in the interior of the sample. εr = εr′ - j εr′′ and µr = µr′ -
9
jµr′′.
Vc and Vs are the volumes of the cavity and the sample, respectively.
The
fundamental idea of cavity perturbation is that the change in the overall geometrical
configuration of the electromagnetic fields upon the introduction of a material sample
should be small. This indicates that percentage change in the real part of resonant
frequency must be small [1]. Therefore, the constraint on equation (2.4) is that δω << ω.
For a sample placed at the electric field maximum position, equation (2.4) can be
simplified to
δω
−
ω
(ε
r
=
− 1)
∫
E ⋅E
*
o max
E
⋅E
dv
dν
(2.5)
Vs
2
∫
o
o
Vc
For a sample placed at the magnetic field maximum position, equation (2.4) can be
simplified to
δω
−
=
ω
(µ
r
− 1) ∫ H ⋅ H
*
o max
dν
Vs
2
∫
H
o
⋅H
o
(2.6)
dv
Vc
For a rectangular cavity operated at TE10L mode, the total fields can be written as
Lπz
d
(2.7)
=
− jE o max
πx
Lπz
sin
cos
Z TE
a
d
(2.8)
=
− j π E o max
L πz
πx
cos
sin
Kηa
a
d
(2.9)
E
y
= E
H
x
H
z
o max
sin
πx
a
sin
where a is the broader dimension of waveguide, d is the length of cavity, ZTE = (Kη/β), is
wave impedance of transverse electromagnetic fields, η is the intrinsic impedance of the
material filling the waveguide. For air filling, η equals 377 Ω. K and β are the
10
wavenumber and propagation constant of waveguide. And, the complex frequency shift
can separate into real and imaginary parts as [1]
δω
ω
where
≈
δ f
f
and
⎡
δfo
fs
⎡ 1 ⎤
+ jδ ⎢
⎥
⎣ 2Q o ⎦
f
=
o
s
1
δ ⎢
⎣ 2Q
0
⎤
⎥ =
⎦
s
−
f
f
(2.10)
o
s
1 ⎡ 1
1 ⎤
−
⎢
⎥
2 ⎣ Q s
Q 0 ⎦
fo and Qo are the resonance frequency and the Q-factor of the cavity without sample
inserted, respectively; fs and Qs are the quantities with the sample inserted.
If the
perturbation condition is satisfied, we can assume E=Eo, H=Ho. After performing
integration and rearranging, equation (2.4) becomes [1]
ε r' − 1 =
(fo
⎛ 1
− fs )Vc
2 fs
Vs
1 ⎞ V
⎟⎟ c
−
ε r'' = ⎜⎜
⎝ Q s Q o ⎠ 4V s
(2.11a)
(2.11b)
Equation (2.6) becomes [1]
⎛ λ2g + 4a 2 ⎞ ( f o − f s ) Vc
⎟
u −1 = ⎜
⎜ 8a 2 ⎟ f s Vs
⎝
⎠
(2.12a)
⎛ λ 2g + 4a 2
u r'' = ⎜
⎜ 16a 2
⎝
(2.12b)
'
r
⎞⎛ 1
1
⎟⎜
−
⎟⎜⎝ Qs Qo
⎠
⎞ Vc
⎟⎟
⎠ Vs
where λg = 2d/L, is the guided wavelength and L=1,2,3,… . When volumes and resonant
frequencies and quality factors are measured, complex permittivity and complex
permeability can be calculated analytically. Figure 2.4 demonstrates the resonant
frequency and Q-factor changed due to insertion of a material sample into an empty
11
cavity resonator.
Figure 2.4 Changed in resonant frequency and Q-factor of a cavity resonator due to a
sample insertion.
12
Chapter 3
Experimental Results
To calibrate the set up, a custom built thru-reflect-line (TRL) calibration
technique is applied to it. The TRL calibration kit consists of a flat ground plate used as
short circuit standard and a piece of waveguide with a known waveguide length used as
the line standard. The length of line is calculated using the following equation:
λ gn =
2 ( λ gl × λ gh )
( λ gl + λ gh )
(3.1)
where λ gl and λ gh are wavelengths of the low frequency end and the high frequency end
at the corresponding frequency band. The general TRL calibration technique given by
David M. Pozar [8] can be found in appendix B. Since at each designed band waveguide
cavity resonator has multiple resonant frequencies, TRL calibration was re-used in each
resonant frequency range, and the measurement was separately performed in each
resonant frequency range.
The source of microwave signal and the scattering parameters measurement
equipment is an Agilent Vector Network Analyzer model 8510C. The VNA allows
measurement at discrete frequency point only, and offer several sets of measurement
points. 201 points set is chosen to carry out for all measurements. Since for cavity
perturbation measurements of complex permittivity and complex permeability are based
on the accurate calculation of resonant frequency and Q factor, a curve fitting technique
is applied to the discrete measured data. The curve fitting technique will improve
13
resonant frequency and Q factor calculation for a discrete measured data. An example of
a discrete measured data and calculation of resonant frequency and Q factor using curve
fitting are illustrated in appendix B. This chapter presents complex permittivity and
complex permeability measured results of a wide range of materials. These materials are
categorized as chemical liquids, dielectric solids, ferrites and custom made ferrite powerepoxy composite ceramics.
14
Liquids
The study of complex permittivity of chemical liquids at microwave frequencies
becomes more important than ever as the tendency of microwave technologies applied to
biomedical and biochemical engineering. For example, microwave dielectric heating is
rapidly becoming an established procedure in synthetic chemistry due to its several
advantages over conventional heating for chemical conversions [9]. Breast cancer
detection utilizing microwave imaging technique is one of rapid growing research areas
in biomedical application. One of motivations of developing microwave imaging
technique for breast cancer detection is the significant contrast in the dielectric properties
at microwave frequencies of normal and malignant breast tissue [10].
This section presents measurement results of complex permittivity of a number of
different chemical liquids at the X-band. Some comparisons of measured results with
published data measured using the waveguide transmission line technique and with
calculated data by the Debye equation [11] were made. Figure 3.1 shows the
measurement setup. The length of rectangular cavity is 89.45 mm. The waveguide cavity
operates at the TE103-6 modes, which correspond to the theoretical resonant frequencies of
f103-6 at 8.2863 GHz, 9.3880 GHz, 10.6823 GHz, and 12.0547 GHz.
Figure 3.1 Liquid samples measurement setup.
15
Figure 3.2 shows the measured resonant frequencies of empty waveguide cavity. The
measured resonant frequencies are at 8.2860 GHz, 9.3860 GHz, 10.6620 GHz, and
12.0260 GHz.
0
-10
S21 (dB)
-20
-30
-40
-50
-60
-70
-80
-90
8.0
9.0
10.0
11.0
12.0
f (GHz)
Figure 3.2 Measured resonant frequencies of empty rectangular cavity.
A capillary tube with negligible loss tangent was chosen as liquid sample holder.
However, ft and Qt should replace the fo and Qo in equation (2.11a) and (2.11b) for the
calculation of εr′ and εr′′ in this measurement, where ft and Qt are the resonance frequency
and Q-factor of cavity loaded with an empty capillary tube, respectively. Table 3.1 lists
and compares the theoretical resonant frequencies, measured resonant frequencies of
empty cavity and measured resonant frequencies of the cavity loaded with empty
capillary tube.
It is seen that the capillary tube has little effect on the resonant
frequencies.
16
Table 3.1 Theoretical resonant frequencies, measured resonant frequencies of empty
cavity and measured resonant frequencies of the cavity loaded with empty capillary tube.
Theoretical
resonant
frequencies (GHz)
Measured empty
cavity resonant
frequencies (GHz)
Measured empty cavity
resonant frequencies with
capillary tube (GHz)
8.2863
9.3880
10.6823
12.0547
8.2860
9.3860
10.6620
12.0260
8.2466
9.3450
10.6000
11.9510
Measured resonant
frequency different in
percentage (with and
without capillary tube)
0.476%
0.437%
0.582%
0.624%
The measurements were performed at room temperature of 22 0 C. The capillary
tube containing the liquid sample was placed at electric field maximum position for each
measurement. Figure 3.3 illustrates measured complex permittivity of cyclohexane using
cavity perturbation technique compared with published data measured by the waveguide
transmission line technique at the X-band. Figure 3.4 illustrates measured complex
permittivity of chlorobenzene using cavity perturbation technique, and comparing with
calculated data by the Debye equation. An illustration of using Debye’s equation for
calculation of chemical liquid can be found in appendix B. They all show quite well
match, especially for the imaginary part. Cavity perturbation technique measured the real
part of complex permittivity of cyclohexane and chlorobenzene are slightly large than
published data measured by the-waveguide transmission line technique and calculated
data using Debye’s equation. These consistent differentiations are possible due to the
effect of capillary tube in the cavity since the capillary tube also shows some degree of
dielectric constant in the X- band. Therefore, capillary tube, the sample holder’s effect
should be studied and taken into account for more accurate cavity perturbation technique
measurements. Figures 3.5-3.8 show the measured complex permittivity for ethyl alcohol,
1-4 dioxane, benzene and acetone, respectively. Table 3.2 lists the measured real and
imaginary parts of complex permittivity of all the six chemical liquids.
17
Table 3.2 Measured real and imaginary parts of complex permittivity of six chemical
liquids.
Frequency
(GHz)
8.2362
9.3345
10.5877
11.9380
Frequency
(GHz)
8.2340
9.3320
10.5860
11.9350
Frequency
(GHz)
8.2342
9.3317
10.5852
11.9352
Cyclohexane
ε'
2.21
2.18
2.24
2.22
Ethyl Alcohol
ε'
4.87
4.78
4.65
4.73
Benzene
ε'
2.47
2.49
2.48
2.46
ε''
0.00026
0.0031
0.011
0.00072
ε''
2.30
2.35
2.00
1.97
ε''
0.0067
0.013
0.0063
0.0012
Frequency
(GHz)
8.2165
9.3145
10.5660
11.8945
Frequency
(GHz)
8.2352
9.3321
10.5860
11.9355
Frequency
(GHz)
8.1725
9.2665
10.5415
11.8785
Chlorobenzene
ε'
4.92
4.83
4.79
4.57
1-4 Dioxane
ε'
2.34
2.45
2.41
2.44
Acetone
ε'
18.15
16.91
15.72
16.43
ε''
1.38
1.43
1.58
1.56
ε''
0.014
0.022
0.029
0.020
ε''
1.83
1.92
2.42
2.99
3
2.5
2
real (cavity technique)
1.5
imaginary (cavity technique)
1
real (in-waveguide technique)
imaginary (in-waveguide technique)
0.5
0
8
9
10
f (GHz)
11
12
Figure 3.3 Complex permittivity of cyclohexane measured using cavity perturbation
technique and the waveguide transmission line technique at the X-band.
18
6
5
4
Real (cavity technique)
Imaginary (cavity technique)
3
real (Debye)
2
imaginary (Debye)
1
0
8
9
10
11
12
f (GHz)
Figure 3.4 Complex permittivity of chlorobenzene measured using cavity perturbation
technique and calculated data by the Debye equation at the X-band.
8
7
6
5
4
3
2
1
0
real
imaginary
8
9
10
f (GHz)
11
12
Figure 3.5 Complex permittivity of ethyl alcohol measured using cavity perturbation
technique at the X-band.
19
3
2.5
2
1.5
Real
1
Imaginary
0.5
0
8
9
10
f (GHz)
11
12
Figure 3.6 Complex permittivity of 1-4dioxane measured using cavity perturbation
technique at the X-band.
3
2.5
2
Real
Imaginary
1.5
1
0.5
0
8
9
10
f (GHz)
11
12
Figure 3.7 Complex permittivity of benzene measured using cavity perturbation
technique at the X-band.
20
20
17.5
15
12.5
10
7.5
5
2.5
0
Real
Imaginary
8
9
10
f (GHz)
11
12
Figure 3.8 Complex permittivity of acetone measured using cavity perturbation technique
at the X-band.
21
Dielectric Solids
This section presents the cavity perturbation measurement results of complex
permittivity of boron nitride and magnesium fluoride. The measurement was performed
at C, X, Ku, and K four frequency bands ranges from 4.5 to 26.5 GHz.
Boron nitride has very high thermal conductivity, good thermal shock resistance
and high electrical resistance. It is widely used in electronics such as: heat sinks,
substrates coil forms or microcircuit packaging. Magnesium fluoride is commonly used
for UV windows, lenses and polarizers due to its optical properties. However, not much
information of complex permittivity about them can be found at microwave and
millimeter wave range. Figure 3.9 shows the measurement setup.
Figure 3.9 Dielectric samples measurement setup.
Figure 3.10-3.11 demonstrates measured complex permittivity of boron nitride
(Grade XP with parallel orientation) and magnesium fluoride. The measurements show
that both boron nitride and magnesium fluoride are very low loss dielectric materials.
Boron nitride has real part of permittivity about 4, and magnesium fluoride has real part
of permittivity about 5 in the frequency tested.
22
Permittivity
5
4
3
2
1
0
-1
real
imaginary
5
10
15
20
Frequency (GHz)
25
30
Permittivity
Figure 3.10 Measured complex permittivity of boron nitride.
7
6
5
4
3
2
1
0
-1
real
imaginary
5
10
15
20
Frequency (GHz)
25
30
Figure 3.11 Measured complex permittivity of magnesium fluoride.
23
Ferrites
Ferrite materials have found a diverse range of applications in modern microwave
and telecommunication systems. Ferrite materials such as nickel ferrite and yttrium iron
garnet ceramics with relative high permittivity, low loss and anisotropic properties are
often used in microwave filter, resonator, circulator or phase shifter designs. In order to
achieve the best design, the accurate measurement of their dielectric and magnetic
properties becomes important. This section presents measurement results of complex
permittivity and permeability of three types of YIG specimen and one nickel ferrite
sample
All of the samples have been measured in demagnetized state. The YIGs under
measurement are 99.9% pure YIG (4πMs = 1800 ±5% Gauss), Aluminum-Substituted
YIG (4πMs = 1000 ±5% Gauss) and Gadolinium-Substituted YIG (4πMs = 1200 ±5%
Gauss). The nickel ferrite sample is a commercially available isotropic soft nickel ferrite
material provided by TransTech, Adamstown, MD. (With the manufacturer data: 4πMs =
5000 ±5% Gauss, ∆H ≤200 Oe at 9.4 GHz). Figure 3.12 shows the measurement setup.
Figure 3.12 Ferrite samples measurement setup.
Figure 3.13 demonstrates measured real parts of permittivity of three types of YIG.
24
Measurements show real parts of permittivity of 99.9% pure YIG, gadolinium-substituted
YIG and aluminum-substituted YIG graduate increase with frequency. The imaginary
parts of permittivity values for YIG specimen are extremely low, and therefore can’t be
measured due to the limited sensitivity of the cavity resonator.
Real part of permittivity
18
17
16
15
14
YIG
Gd-YIG
Al-YIG
13
12
2
7
12
17
Frequency (GHz )
22
27
Figure 3.13 Measured real parts of permittivity of 99.9% pure YIG, gadoliniumsubstituted YIG and aluminum-substituted YIG.
Figure 3.14 illustrates that the 99.9% pure YIG has a magnetic resonance near 5
GHz. No magnetic resonances were found for Aluminum-Substituted YIG and
Gadolinium-Substituted YIG in the frequency range of testing. Real parts of permeability
of them are all around 1. Since pure and substituted versions of YIG are all soft ferrites,
the natural magnetic resonances should occur well below 30 GHz. The measurement
results demonstrated in this paper show well agreement with that.
25
Complex Permeability
1.2
1
0.8
0.6
real
imaginary
0.4
0.2
0
3
5
7
9
11
Frequency (GHz)
13
15
Figure 3.14 Measured complex permeability of 99.9% pure YIG.
Figure 3.15 shows the complex permittivity of nickel ferrite. Measurements show
that nickel ferrite has real part of permittivity around 12.5 and slowly increases with
frequency. Figure 3.16 shows the complex permeability of nickel ferrite. The figure
clearly shows magnetic resonance of nickel ferrite occurs around 13 GHz.
Complex Permittivity
20
15
10
real
imaginary
5
0
2
7
12
17
Frequency (GHz )
22
27
Figure 3.15 Measured complex permittivity of nickel ferrite.
26
Complex Permeability
1.4
0.9
real
0.4
imaginary
-0.1
6
8
10 12 14 16 18 20 22 24 26
Frequency (GHz )
Figure 3.16 Measured complex permeability of nickel ferrite.
27
Strontium Ferrite Powder Composite Ceramics
Magnetic composite materials are also inspired the interests of researchers over years.
The composite materials have several advantages, such as composite materials are easy to
prepare in the desired shape and to machine by molding.
Also by varying the
compositions in a composite material, the real and imaginary parts of complex dielectric
permittivity and complex magnetic permeability properties, as well as the position of
resonance frequency vary accordingly.
This section presents complex permittivity and permeability measurement results of
two custom made strontium ferrite powder-epoxy composite ceramic samples. The two
strontium ferrite composite ceramic samples with slightly different chemical composition
and specific gravity were prepared at Tufts University. The strontium ferrite samples
were prepared by mixing strontium ferrite powder with epoxy resin and a very small
amount of peroxide hardener. The mixtures were prepared homogeneously and baked in
an oven. These specimens were then flattened and polished and cut into specific shapes to
accommodate different measurement techniques. The first strontium ferrite composite
ceramic sample (Sample #1) has powder and epoxy resin ratio of 1: 2.3, and 1.9 g/cm for
specific gravity. The second composite ceramic sample (Sample #2) has a slightly higher
concentration of strontium ferrite powder as it compared to sample #1. It has powder and
epoxy resin ratio of 1: 2.0, and 1.8 g/cm for specific gravity.
28
4.5
Complex Permittivity
4
3.5
3
real
2.5
imaginary
2
1.5
1
0.5
0
0
5
10
15
20
25
30
Frequency (GHz)
Figure 3.17 Measured complex permeability of strontium ferrite composite sample #1
4.5
Complexer Permittivity
4
3.5
3
2.5
real
2
imaginary
1.5
1
0.5
0
-0.5
0
5
10
15
20
25
30
Frequency (GHz)
Figure 3.18 Measured complex permeability of strontium ferrite composite sample #2
Figures 3.17-3.18 shows complex permittivity measurement results for strontium
ferrite sample #1 and sample # 2. Measurements show that strontium ferrite sample #1
and sample #2 has real part of permittivity about 3.6 and 4.2 respectively in these
frequency region. These real parts of permittivity values are higher than the permittivity
values of original strontium ferrite powder in the frequency range tested. The strontium
ferrite powder has the average real permittivity 2.65 from 8 to 26.5 GHz [18]. In
29
previous measurements, Afsar et al show that when the concentration of strontium ferrite
powder in the mixture is increased, the absorption value at the resonance frequency
increases. Measurements also show that the imaginary part of permittivity values for
both samples gradually increase as frequency increases. For the strontium ferrite, it is
expected to have the magnetic resonance to appear around 56GHz. It is therefore the real
part of permeability values for both samples are expected to be about unity in these
frequency regions. The present measurements also reveal similar values to original
strontium ferrite powder values and that are all around unity.
30
Chapter 4
The New Cavity Perturbation Technique
The fundamental cavity perturbation condition is that the change in the overall
geometric configuration of the electromagnetic fields inside the resonant cavity upon the
introduction of a material sample should be small [1]. In other words, it requires that the
amount of the complex frequency shift be small. In rectangular cavity perturbation
measurements, the measurement is also limited by the sample shape requirement. It is
required to have the length of the sample under measurement equal to the height of the
cavity resonator for the formulas to work out. However, very often fabricating a long and
thin sample becomes quite challenging task especially for some materials are hard and (or)
fragile in nature. Therefore, it is necessary to extend the cavity perturbation measurement
technique to include the case when the sample length is less than the height of the cavity
resonator. In this chapter, cavity perturbation formulas have been expanded to include
the situation mentioned above when the sample length is insufficiently long comparing to
the height of the cavity. In order to check the validity of these new equations, complex
permittivity values of different samples with different sample lengths have been
measured and compared.
Brief Review of the Cavity Perturbation Theory
The frequency shift due to the insertion of a small material sample into a
resonant cavity is given by Waldron [7], [12] as
31
δω
=−
ω
(ε
∗
r
)
(
)
− 1 ε o ∫ E ⋅ Eo*dν + µ r∗ − 1 µo ∫ H ⋅ H o*dν
Vs
Vs
∫ Do ⋅ E + Bo ⋅ H dv
*
o
*
o
(4.1)
Vc
Where Do, Eo, Bo and Ho are the unperturbed cavity field quantities, and E and H are the
fields in the interior of the sample. ε r∗ = εr′ - j εr′′ and µ r∗ = µr′ - jµr′′. Vc and Vs are the
volumes of the cavity and the sample, respectively. δω/ω is the amount of the complex
frequency shift that can be separated into real and imaginary parts as [1]
δω
ω
δ f
where
f
⎡
and
δfo
≈
fs
⎡ 1 ⎤
+ jδ ⎢
⎥
⎣ 2Q o ⎦
f
=
o
s
s
1
δ ⎢
⎣ 2Q
0
⎤
⎥ =
⎦
−
f
f
(4.2)
,
o
s
1 ⎡ 1
1 ⎤
−
⎥
⎢
2 ⎣ Q s
Q 0 ⎦
fo and Qo are the resonance frequency and the Q-factor of the cavity without sample
inserted, respectively; fs and Qs are the quantities with the sample inserted. For a sample
placed at the electric field maximum, equation (4.1) can be simplified to
δω
ω
(ε
= −
∗
r
− 1) ∫ E ⋅ E
*
o max
dν
Vs
2
∫
E
o
⋅E
o
dv
(4.3)
Vc
A rectangular cavity operates at TE10L mode, the total electric fields can be written as
E o = E o max sin
πx
a
sin
Lπz
d
(4.4)
where a is the broader dimension of waveguide, d is the length of cavity and L=1,2,3,… .
L is an integer.
32
If the perturbation condition is satisfied, and the length of the inserted material
sample equals the height of the cavity resonator, then it’s valid to assume that E = Eo, the
electric field in the interior of the sample equals the unperturbed field. Along with this
assumption, substituting (4.2) and (4.4) into (4.3), evaluating the equation, and then
separating and equal the real and imaginary parts of it, the equation (4.3) becomes [1]
(fo
− fs
2 fs
ε
'
r
−1 =
ε
''
r
⎛ 1
1
= ⎜⎜
−
Q o
⎝ Q s
)V c
Vs
⎞ Vc
⎟⎟
⎠ 4V s
(4.5a)
(4.5b)
Equations (4.5a)-(4.5b) are precise solutions for calculating the complex permittivity
when the perturbation condition is satisfied [13]. However, the accuracy of the equations
(4.5a)-(4.5b) are also affected by the length of the inserted material sample. In other
words, these equations require that the inserted sample length equals the height of the
cavity. If the amount of the complex frequency shift is still small but either the length of
the inserted material sample is shorter than the height of the cavity or the sample is only
partially filling the cavity height, then the validity of the previous assumption E = Eo is
not longer true. Subsequently, evaluating equations (4.5a)-(4.5b) will yield incorrect
complex permittivity results.
Assessment of Complex Permittivity of a Material with the Sample Length Less
Than the Height of the Cavity
When the inserted sample length is shorter than the height of the cavity resonator,
a new estimation of E, the electric field in the interior of the sample, must be carried out
and new equations for calculating complex permittivity must be derived. To evaluate E,
33
one additional assumption need to be made besides the fundamental perturbation
condition. Namely, the electromagnetic fields inside the resonant cavity are assumed to
be static fields. This assumption is most likely to be accurate here. So, when the sample
is inserted and placed at the electric field maximum position of the cavity, we have an
electrostatic situation. The sample will be partially polarized by the applied electric field,
E0. Therefore, the sample will behave like a dipole with a length equal to the length of the
sample as illustrated in Figure 4.1.
Figure 4.1 Polarization of the material sample with the external applied electric field
As a result, a depolarizing field of magnitude.
E
d
=
−
NP
ε
(4.6)
o
will point in the opposite direction of the applied electric field. Where N is the
depolarizing factor and P is the polarization magnitude. In a linear medium, the electric
polarization is linearly related to the applied electric field as
(
)
P = ε r∗ − 1 ε o ⋅ E o
(4.7)
Therefore, the electric field in the interior of the material sample is the total vector sum of
the applied field and the depolarizing field. That is to say, now
34
= E
E
+ E
o
(4.8)
d
By substituting (4.8) along with (4.4), (4.6) and (4.7) into (4.3) and then evaluate the
right hand-side equation, equation (4.3) becomes
δω
= −
ω
[1 − (ε
∗
r
) ]
− 1 N ( ε r∗ − 1) ∫ E o ⋅ E o* max d ν
Vs
2 ∫ E o ⋅ E o dv
(4.9)
Vc
Again substituting (4.2) in (4.9), evaluating the equation, and then separating and equal
the real and imaginary parts of it, equation (4.9) becomes
[
]
( f s − f o ) = {N (ε ' )2 − 2ε ' − ε '' + 1 + (− ε '
fs
r
r
r
r
) ]⎛⎜⎜ 2 VV
1⎛ 1
1 ⎞
⎟ = ε r'' − 2 N ε r' − 1 ε r''
⎜⎜
−
2 ⎝ Q s Qo ⎟⎠
[
)}
⎛ V ⎞
+ 1 ⎜⎜ 2 s ⎟⎟
⎝ Vc ⎠
(
⎝
s
c
⎞
⎟⎟
⎠
(4.10a)
(4.10b)
The value of the depolarizing factor N depends on the geometrical dimension of the
sample, and has been calculated by a number of researchers [14-16]. According to
Bozorth [14], for a prolate ellipsoid with semi-axis h, b and c, (h > b =c) that is being
polarized along the longer axis, N is given by
1
⎡
⎤
m + m2 −1 2 ⎥
1 ⎢ m
−1
ln
N= 2
1
⎥
m −1 ⎢ 2 12
2
m − m −1 2 ⎥⎦
⎢⎣2 m −1
(
) (
)
(
(
)
)
(4.11)
Where m = h/b. For cavity perturbation measurements of permittivity, the sample usually
is prepared in a rod or a cylinder rather than in prolate ellipsoid shape. With a small
volume of the sample size, a prolate ellipsoid can be substituted by a cylinder with the
same volume for the calculation of the N value. The error of this substitution will be very
insignificant [17].
Thus far, the value of N used in (4.10a) and (4.10b) to calculate complex
35
permittivity has been computed for the sample in the absence of the surrounding cavity
walls. The value of N will be slightly affected by the present of the cavity conducting
surfaces due to the image effect. By including the effect of the images of the dipole
sample to the cavity, following Parkash and Abhai [17], the new N value will now be
Ne = N
πh
2H
cot
πh
(4.12)
2H
Ne is the effective depolarizing factor. H is the height of the cavity and h is the length of
the inserted material sample. An illustration of how to calculate the image effect given by
Parkash and Abhai can be found on appendix C.
Figure 4.2 illustrates the calculated values of Ne versus the normalized sample
length to the height of the cavity for a cylinder substituted prolate ellipsoid with different
diameters. The diameters of the cylinders illustrated here are 1.1 mm, 1.25mm, 1.5mm.
As noticed, for all three cases Ne approach zero as h approaches H, and Ne equals zero
when h equals H. Notice that when Ne equals zero, equations (4.10a)-(4.10b) will be
reduced to the standard forms, (4.5a)-(4.5b).
1.4
1.2
Ne
1
Ne (b=1.1 mm)
Ne (b=1.25 mm)
Ne (b =1.5 mm)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
h/H
0.6
0.8
1
1.2
Figure 4.2 Variations of Ne for the normalized sample length to the height of the cavity.
36
Experimental Results and Discussion
The cavity resonator is formed by using a section of WR 187 rectangular copper
waveguide with both ends covered by a flat copper coupling plate. The length of the WR
187 waveguide is 91.52 mm and the diameter of the coupling hole is 5.52 mm. In order
to get the sample in and out of the cavity without disassembling it, a non-radiated slot
was opened at the top surface of the cavity wall. The width of the non-radiated open slot
is 3.48 mm.
The WR 187 waveguide cavity operates at the TE102 mode, which
correspond to the theoretical resonant frequency of f102 at 4.5488 GHz. The actual
measured resonant frequency of the empty cavity is 4.5463 GHz. The agreement between
the theoretical and measured values for the cavity resonant frequency is excellent. The
difference is only 0.055%. A small piece of rod-shaped styrofoam with both ends
attached by double-sided scotch tape is positioned firmly at the electric field maximum
point. The cylinder sample is inserted and placed at the center of the styrofoam by using a
tweeter with narrow-ended tips. The introduction of the styrofoam is found has very little
effect on the empty cavity’s resonant frequency and the Q-factor. But, when calculating
complex permittivity, the resonant frequency and the Q-factor measured with the
styrofoam are used as “empty” cavity quantities. Figure 4.3 shows the new cavity
measurement set up.
Figure 4.3 The new cavity measurement set up.
37
To calibrate the set up, a custom built thru-reflect-line (TRL) calibration
technique is applied to it. The TRL calibration kit consists of a short circuit and a piece
of waveguide with a known waveguide length. The measurement results obtained by
applying the custom designed TRL calibration have been proved to be very accurate by
testing a number of material complex permittivity and permeability for both cavity
perturbation and waveguide transmission line techniques [13] [18]. The resonance
frequency and the Q-factor used in calculation of complex permittivity are determined
from measured S-parameters from a Vector Network Analyzer (VNA).
The Vector
Network Analyzer used to carry out this experiment task allows measurement at discrete
frequency points only (Agilent model 8510 C with the 801 maximum measurement
points over a defined frequency range).
One of key factors for obtaining accurate
measurement results using cavity perturbation technique is the ability to precisely resolve
the resonant frequency and the Q-factor with and without a material sample inserted.
Therefore, a curve fitting technique is applied to the discrete measured S-parameters data
each time. The curve fitting technique will improve the accuracy in determination of the
resonant frequency and the Q-factor. The samples under test are Teflon, Magnesium
Fluoride and Plexiglas. Each sample was prepared with several different sample lengths.
The diameters of the same type sample with different sample lengths are made to be
identical. The diameter of Teflon and Magnesium Fluoride samples are 1.25 mm. The
diameter of Plexiglas samples are 1.26 mm.
38
Table 4.1 Comparison of complex permittivity calculated by using (4.5a-4.5b) and
(4.10a-4.10b) with Ne at frequency 4.5463 GHz. (Cavity height = 22.1 mm).
Teflon
22.1
15
10
5
εr′
2.08
2.06
2.02
1.86
εr′′
-----
From Eqs (10a-b)
with Ne
ε r′
εr′′
2.08
-2.07
-2.07
-2.04
--
Magnesium Fluoride
22.1
15
7
5.08
4.87
3.89
----
5.08
5.09
5.06
----
Plexiglas
4.1
1.81
0.10
2.51
0.22
Material sample
Length of
sample (mm)
From Eqs (5a-b)
The measured permittivity results are tabulated in Table 4.1. The table lists and
compares the permittivity results calculated using the standard form and the newly
derived equations. The imaginary parts of permittivity values for Teflon and Magnesium
Fluoride are extremely low, and therefore can’t be measured due to the limited sensitivity
of the cavity resonator. However, the real parts of permittivity values of Teflon and
Magnesium Fluoride with different together with Ne show much consistent and accurate
result than using the standard equations. Therefore, the newly derived equations should
be used for the calculation of complex permittivity of materials when the sample has
insufficient length comparing to the height of cavity resonator.
39
Chapter 5
Conclusion
This thesis has examined the rectangular cavity perturbation technique as it used
for material complex permittivity and complex permeability measurement. A wide variety
range of material’s permittivity and permeability have been successfully determined.
There are two main error sources associated with this permittivity and permeability
measurement task. First of all, the sample may only be placed at the maximum electric or
magnetic field positions with the accuracy of the experiment’s eyesight. As derived from
the cavity perturbation theory, the standard cavity perturbation measurement formulas
(equations 2.11(a-b) and 2.12 (a-b)) assume that a sample places at the maximum field
location. A sample not situated at the maximum field location under measurement causes
some degrees of error in obtaining the permittivity and permeability results. Another
measurement error source is related with calculating the resonant frequency and the Qfactor from the measured discrete S-Parameter points. Due the finite measurement
frequency points of a Vector Network Analyzer, a curve fitting technique is applied to the
measured S-Parameter data each time. Applied a curve fitting to the discrete measured SParameter data reduces the inaccuracy in resolving the resonant frequency and the Qfactor, but not totally eliminate the error associated with that. Finally, a modified
rectangular cavity perturbation technique has been introduced and a new set of formulas
have been derived in accommodating with a special measurement case namely when the
material sample under measurement has insufficient sample length comparing to the
40
height of cavity resonator. These newly derived formulas are more general than the
standard forms and have been proved to be accurate in yielding consistent results by
measuring different samples with different lengths.
41
Appendix A
Measured Resonant Frequencies for Each Designed Rectangular Cavity Resonator
-45.0
-55.0
-65.0
l S21 l dB
-75.0
-85.0
-95.0
-105.0
-115.0
-125.0
-135.0
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
f (GH)
Figure A.1 Measured resonant frequencies for H band rectangular waveguide cavity
resonator.
-40.0
-50.0
l S21 l dB
-60.0
-70.0
-80.0
-90.0
-100.0
6
6.2
6.4
6.6
6.8
7
7.2
7.4
7.6
7.8
8
f (GHZ)
Figure A.2 Measured resonant frequencies for C band rectangular waveguide cavity
resonator.
42
0
-10
-20
S21 (dB)
-30
-40
-50
-60
-70
-80
-90
8.0
9.0
10.0
11.0
12.0
f (GHz)
Figure A.3 Measured resonant frequencies for X band rectangular waveguide cavity
resonator.
-30.0
-40.0
l S21 l dB
-50.0
-60.0
-70.0
-80.0
-90.0
12.4
13.4
14.4
15.4
16.4
17.4
f (GHz)
Figure A.4 Measured resonant frequencies for Ku band rectangular waveguide cavity
resonator.
43
-30.0
-40.0
l S21 l dB
-50.0
-60.0
-70.0
-80.0
-90.0
-100.0
17.5
18.5
19.5
20.5
21.5
22.5
23.5
24.5
25.5
26.5
f (GHz)
Figure A.5 Measured resonant frequencies for K band rectangular waveguide cavity
resonator.
44
Appendix B
The General TRL Calibration Technique
The S parameters of DUT can be measured using the setup shown in figure B.1.
For a cascade network measurement, it’s convenient to convert S parameters
measurement to ABCD parameter measurement first and then convert the results back to
S parameters.
Figure B.1 Block diagram of a network analyzer measurement of a two port device.
Figure B.2-B.7 show general error characterization for TRL calibration technique.
Figure B.2 Block diagram for Thru connection.
45
Figure B.3 Signal flow graph for Thru connection.
Figure B.4 Block diagram for Reflect connection.
Figure B.5 Signal flow graph for Reflect connection.
Figure B.6 Block diagram for Line connection.
46
Figure B.7 Signal flow graph for Line connection.
The ABCD parameters for the DUT can be found as follow:
(B.1)
Table B.1 A discrete measured S-Parameters data retrieved from VNA.
Real part
Imaginary part
l S21 l in dB
4.11E-05
1.44E-05
1.58E-05
1.14E-05
4.75E-05
3.53E-05
5.74E-06
2.35E-05
1.66E-05
3.12E-05
2.81E-05
2.71E-05
3.84E-05
2.50E-05
1.66E-05
3.79E-05
2.35E-05
3.09E-05
2.29E-05
3.59E-05
2.99E-05
2.63E-05
2.44E-05
8.24E-06
3.14E-05
3.37E-05
4.01E-05
4.85E-05
2.95E-05
3.52E-04
3.61E-04
3.72E-04
3.74E-04
3.76E-04
3.53E-04
3.81E-04
3.67E-04
3.64E-04
4.02E-04
3.92E-04
3.93E-04
3.83E-04
4.03E-04
3.80E-04
3.93E-04
4.06E-04
4.04E-04
4.25E-04
4.22E-04
4.09E-04
4.27E-04
4.21E-04
4.36E-04
4.29E-04
4.23E-04
4.31E-04
4.47E-04
4.39E-04
-69.021
-68.853
-68.588
-68.535
-68.419
-69.003
-68.377
-68.689
-68.768
-67.881
-68.119
-68.088
-68.293
-67.885
-68.402
-68.070
-67.808
-67.841
-67.428
-67.470
-67.736
-67.375
-67.492
-67.199
-67.323
-67.453
-67.271
-66.949
-67.140
Frequency point
(MHz)
8185.000
8185.375
8185.750
8186.125
8186.500
8186.875
8187.250
8187.625
8188.000
8188.375
8188.750
8189.125
8189.500
8189.875
8190.250
8190.625
8191.000
8191.375
8191.750
8192.125
8192.500
8192.875
8193.250
8193.625
8194.000
8194.375
8194.750
8195.125
8195.500
47
3.90E-05
1.92E-05
4.04E-05
3.38E-05
3.21E-05
4.90E-05
1.26E-05
4.17E-05
1.98E-05
3.90E-05
2.60E-05
4.28E-05
5.03E-05
3.82E-05
4.14E-05
4.20E-05
3.33E-05
4.08E-05
3.46E-05
3.07E-05
3.63E-05
6.23E-05
5.16E-05
5.11E-05
4.28E-05
3.23E-05
6.08E-05
5.12E-05
3.72E-05
3.15E-05
4.93E-05
6.66E-05
5.51E-05
7.27E-05
6.87E-05
6.47E-05
5.50E-05
4.42E-05
3.56E-05
5.92E-05
7.62E-05
6.00E-05
7.28E-05
6.53E-05
6.66E-05
6.98E-05
4.83E-05
5.42E-05
5.66E-05
7.68E-05
5.59E-05
6.74E-05
8.13E-05
4.42E-04
4.49E-04
4.60E-04
4.69E-04
4.74E-04
4.65E-04
4.68E-04
4.44E-04
4.79E-04
4.79E-04
4.84E-04
4.72E-04
4.97E-04
4.90E-04
5.00E-04
5.14E-04
5.26E-04
5.06E-04
5.26E-04
5.09E-04
5.34E-04
5.51E-04
5.17E-04
5.56E-04
5.46E-04
5.67E-04
5.40E-04
5.49E-04
5.62E-04
5.66E-04
5.87E-04
5.95E-04
5.51E-04
5.82E-04
5.80E-04
6.11E-04
6.46E-04
6.42E-04
6.38E-04
6.33E-04
6.17E-04
6.28E-04
6.47E-04
6.73E-04
6.55E-04
6.83E-04
6.87E-04
6.69E-04
6.87E-04
7.03E-04
7.30E-04
7.20E-04
7.38E-04
-67.065
-66.940
-66.716
-66.559
-66.457
-66.602
-66.588
-67.005
-66.388
-66.361
-66.282
-66.484
-66.037
-66.174
-65.998
-65.750
-65.561
-65.895
-65.565
-65.853
-65.431
-65.120
-65.695
-65.063
-65.231
-64.907
-65.305
-65.164
-64.989
-64.932
-64.590
-64.457
-65.132
-64.633
-64.666
-64.228
-63.769
-63.834
-63.886
-63.936
-64.130
-64.000
-63.724
-63.403
-63.631
-63.261
-63.241
-63.461
-63.233
-63.005
-62.709
-62.814
-62.582
8195.875
8196.250
8196.625
8197.000
8197.375
8197.750
8198.125
8198.500
8198.875
8199.250
8199.625
8200.000
8200.375
8200.750
8201.125
8201.500
8201.875
8202.250
8202.625
8203.000
8203.375
8203.750
8204.125
8204.500
8204.875
8205.250
8205.625
8206.000
8206.375
8206.750
8207.125
8207.500
8207.875
8208.250
8208.625
8209.000
8209.375
8209.750
8210.125
8210.500
8210.875
8211.250
8211.625
8212.000
8212.375
8212.750
8213.125
8213.500
8213.875
8214.250
8214.625
8215.000
8215.375
48
5.84E-05
5.63E-05
6.41E-05
7.04E-05
8.45E-05
8.21E-05
5.87E-05
9.81E-05
7.57E-05
7.69E-05
8.74E-05
8.14E-05
8.02E-05
1.00E-04
1.13E-04
8.34E-05
1.29E-04
1.02E-04
9.41E-05
1.25E-04
9.97E-05
1.03E-04
1.26E-04
1.04E-04
1.00E-04
1.27E-04
1.21E-04
1.39E-04
1.38E-04
1.49E-04
1.42E-04
1.37E-04
1.41E-04
1.52E-04
1.86E-04
1.54E-04
1.73E-04
1.77E-04
1.76E-04
1.88E-04
2.06E-04
2.08E-04
1.72E-04
2.54E-04
2.51E-04
2.58E-04
2.58E-04
2.89E-04
2.96E-04
2.93E-04
3.23E-04
3.40E-04
3.66E-04
7.19E-04
7.53E-04
7.43E-04
7.86E-04
7.58E-04
7.89E-04
7.88E-04
7.94E-04
8.25E-04
8.27E-04
8.46E-04
8.53E-04
8.74E-04
8.75E-04
8.81E-04
9.17E-04
9.05E-04
9.45E-04
9.69E-04
9.51E-04
1.00E-03
1.01E-03
1.00E-03
1.03E-03
1.05E-03
1.05E-03
1.05E-03
1.10E-03
1.13E-03
1.16E-03
1.16E-03
1.19E-03
1.20E-03
1.22E-03
1.25E-03
1.27E-03
1.31E-03
1.32E-03
1.36E-03
1.39E-03
1.43E-03
1.45E-03
1.49E-03
1.54E-03
1.56E-03
1.61E-03
1.66E-03
1.72E-03
1.76E-03
1.80E-03
1.84E-03
1.91E-03
1.98E-03
-62.839
-62.439
-62.550
-62.062
-62.354
-62.007
-62.041
-61.938
-61.632
-61.610
-61.404
-61.340
-61.131
-61.104
-61.026
-60.713
-60.780
-60.444
-60.230
-60.364
-59.959
-59.876
-59.921
-59.687
-59.561
-59.475
-59.524
-59.091
-58.868
-58.613
-58.612
-58.426
-58.389
-58.204
-57.957
-57.858
-57.567
-57.522
-57.247
-57.032
-56.809
-56.682
-56.463
-56.154
-56.044
-55.769
-55.498
-55.183
-54.977
-54.800
-54.581
-54.229
-53.916
8215.750
8216.125
8216.500
8216.875
8217.250
8217.625
8218.000
8218.375
8218.750
8219.125
8219.500
8219.875
8220.250
8220.625
8221.000
8221.375
8221.750
8222.125
8222.500
8222.875
8223.250
8223.625
8224.000
8224.375
8224.750
8225.125
8225.500
8225.875
8226.250
8226.625
8227.000
8227.375
8227.750
8228.125
8228.500
8228.875
8229.250
8229.625
8230.000
8230.375
8230.750
8231.125
8231.500
8231.875
8232.250
8232.625
8233.000
8233.375
8233.750
8234.125
8234.500
8234.875
8235.250
49
3.80E-04
4.08E-04
4.52E-04
4.82E-04
5.07E-04
5.37E-04
5.79E-04
6.28E-04
6.87E-04
7.64E-04
8.00E-04
8.87E-04
1.01E-03
1.11E-03
1.20E-03
1.38E-03
1.56E-03
1.77E-03
2.04E-03
2.41E-03
2.84E-03
3.40E-03
4.09E-03
5.02E-03
6.21E-03
7.76E-03
9.61E-03
1.17E-02
1.35E-02
1.43E-02
1.36E-02
1.18E-02
9.67E-03
7.68E-03
6.05E-03
4.75E-03
3.82E-03
3.09E-03
2.54E-03
2.09E-03
1.77E-03
1.48E-03
1.25E-03
1.12E-03
9.74E-04
8.44E-04
7.27E-04
6.45E-04
5.70E-04
5.39E-04
4.70E-04
4.30E-04
4.00E-04
2.03E-03
2.11E-03
2.17E-03
2.24E-03
2.36E-03
2.43E-03
2.55E-03
2.65E-03
2.76E-03
2.88E-03
3.03E-03
3.17E-03
3.34E-03
3.52E-03
3.74E-03
3.94E-03
4.18E-03
4.45E-03
4.72E-03
5.05E-03
5.40E-03
5.79E-03
6.15E-03
6.52E-03
6.76E-03
6.82E-03
6.37E-03
5.19E-03
2.92E-03
-1.32E-04
-3.34E-03
-5.66E-03
-6.96E-03
-7.41E-03
-7.37E-03
-7.04E-03
-6.65E-03
-6.19E-03
-5.77E-03
-5.38E-03
-5.02E-03
-4.70E-03
-4.41E-03
-4.14E-03
-3.90E-03
-3.69E-03
-3.47E-03
-3.32E-03
-3.18E-03
-3.04E-03
-2.89E-03
-2.76E-03
-2.64E-03
-53.683
-53.359
-53.105
-52.790
-52.338
-52.067
-51.661
-51.294
-50.923
-50.525
-50.084
-49.660
-49.138
-48.649
-48.126
-47.584
-47.000
-46.397
-45.776
-45.049
-44.295
-43.457
-42.628
-41.695
-40.747
-39.715
-38.764
-37.872
-37.217
-36.920
-37.084
-37.663
-38.480
-39.435
-40.415
-41.414
-42.307
-43.195
-44.007
-44.778
-45.476
-46.138
-46.779
-47.354
-47.913
-48.448
-48.995
-49.404
-49.803
-50.213
-50.660
-51.069
-51.465
8235.625
8236.000
8236.375
8236.750
8237.125
8237.500
8237.875
8238.250
8238.625
8239.000
8239.375
8239.750
8240.125
8240.500
8240.875
8241.250
8241.625
8242.000
8242.375
8242.750
8243.125
8243.500
8243.875
8244.250
8244.625
8245.000
8245.375
8245.750
8246.125
8246.500
8246.875
8247.250
8247.625
8248.000
8248.375
8248.750
8249.125
8249.500
8249.875
8250.250
8250.625
8251.000
8251.375
8251.750
8252.125
8252.500
8252.875
8253.250
8253.625
8254.000
8254.375
8254.750
8255.125
50
3.35E-04
3.25E-04
2.71E-04
2.58E-04
2.49E-04
2.18E-04
2.13E-04
1.94E-04
1.94E-04
1.69E-04
1.57E-04
1.46E-04
1.42E-04
-2.55E-03
-2.44E-03
-2.37E-03
-2.30E-03
-2.20E-03
-2.14E-03
-2.06E-03
-2.02E-03
-1.94E-03
-1.89E-03
-1.83E-03
-1.78E-03
-1.73E-03
-51.807
-52.165
-52.463
-52.726
-53.098
-53.363
-53.689
-53.862
-54.215
-54.421
-54.711
-54.968
-55.188
8255.500
8255.875
8256.250
8256.625
8257.000
8257.375
8257.750
8258.125
8258.500
8258.875
8259.250
8259.625
8260.000
fo of empty cavity with capillary tube
- 30. 0
8185
- 35. 0
8195
8205
8215
8225
8235
8245
8255
- 40. 0
l S21 l dB
- 45. 0
- 50. 0
- 55. 0
- 60. 0
- 65. 0
- 70. 0
- 75. 0
f ( GHz)
Figure B.8 Resonant frequency generated from discrete data lists in table B.1.
51
Figure B.9 Calculation of resonant frequency and Q factor for the discrete data lists in
table B.1.
Debye’s Equation
Debye’s equation is used to carry out the computation of complex permittivity of
materials as follows:
ε = ε∞ +
εs − ε∞
1 + jωτ
(B.2)
where
ε is the complex permittivity
εs is the static permittivity
ε∞ is the permittivity at the infinite frequency
τ is the relaxation time
Table B.2. Debye’s parameters for chlorobenzene from literature
Materials
ε∞
εs
τ (pico-sec)
Chlorobenzene
2.3
5.54[19]
10.3[19]
52
Appendix C
Calculation of Ne Due to Image Effect
Figure C.1 Image effect due to the metallic ground plates.
Let A= the area of cross section of sample, 2h = length of sample and 2H = height of
cavity resonator. The total depolarizing electric field due to the polarization of the sample
and its images can be written as
(C.1)
where
53
(C.2)
is the magnitude of the polarization of the first image A1 B1. Use the standard result
(C.3)
lead to
(C.4)
Therefore, Ne can be defined as:
(C.5)
54
Publications
[1]
Mi Lin, Yong Wang, and Mohammed N. Afsar, “Precision measurement of
complex permittivity and permeability by microwave cavity perturbation
technique”, presented at The Joint 30th International Conference on Infrared and
Millimeter Waves and 13th International Conference on Terahertz Electronics,
Williamsburg, Virginia, September, 2005.
[2]
Mi Lin and Mohammed N. Afsar, “Measurement of dielectric and magnetic
characteristics of nickel ferrite and strontium ferrite composite from 4.5 to 26.5
GHz frequency range”, accepted by IEEE IMTC 2006 – IEEE Instrumentation
and Measurement Technology Conference, Sorrento, Italy, April, 2006.
[3]
Mi Lin and Mohammed N. Afsar, “Cavity perturbation measurement of dielectric
and magnetic properties of ferrite materials in microwave frequency range”,
accepted by IEEE INTERMAG 2006 – IEEE International Magnetics Conference,
San Diego, California, May, 2006.
[4]
Mi Lin and Mohammed N. Afsar, “A new cavity perturbation technique for
accurate measurement of dielectric parameters”, accepted by IEEE MTT-S
International Microwave Symposium, San Francisco, California, June, 2006.
55
Bibliographies
[1]
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[2]
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open-ended coaxial probes, Meas.Sci.Technol. 7, 1996, pp. 1164-1173.
[3]
Rene Grignon, Mohammed N. Afsar, Yong Wang, and Saquib Butt, “Microwave
broadband free-space complex dielectric permittivity measurements on low loss
solids,” IEEE Instrumentation and Measurement Technology Conference,
Colorado, USA, pp. 865-870, May 2003.
[4]
Yong Wang and Mohammed N. Afsar, Measurement of complex permittivity of
liquids using waveguide techniques, Progress In Electromagnetics Research,
PIER 42, 131-142, 2003.
[5]
Mohammed N. Afsar and Kenneth J. Button “Millimeter-wave Dielectric
Measurement of Materials”, Proceeding of the IEEE, vol.73, no.1, January 1985.
[6]
H.A. Bethe and J. Schwinger, NRDC report D1-117, Cornell University, Ithaca,
NY, 1943.
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vol. 107C, p. 272, 1960.
[8]
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Michael P. Mingos, “Dielectric parameters relevant to microwave dielectric
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Elise C. Fear, Xu Li, Susan C. Hagness, and Maria A. Stuchly, “Confocal
Microwave Imaging for Breast Cancer Detection: Localization of Tumors in
Three Dimensions,” IEEE Trans. Biomed. Eng., vol. 49, no.8, pp812-822, Aug.
2002.
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Fabienne Duhamel, Isabelle Huynen, and Andre Vander Vorst, “Measurements of
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Nostrand, 1970, ch. VI, pp. 292-318.
[13]
Mi Lin, Yong Wang, and Mohammed N. Afsar, “Precision measurement of
complex permittivity and permeability by microwave cavity perturbation
technique,” presented at 30th Int.Conf. Infrared and Millimeter Waves, Virginia
USA, 2005
[14]
R. M. Bozorth and D. M. Chapin, “Demagnetizing factors of rods,” J. Appl. Phy.,
vol. 13, pp. 320-326, May.1942.
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J.A. Osborn, “Demagnetizing factors of the general ellipsoid,” phy. Rev., vol.67,
no.11, pp. 351-357, Jun. 1945.
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A. Sihvola, P.Y.Oijala, S.Jarvenpaa, and J.Avelin, “Polarizabilities of platonic
solids,” IEEE Trans. Antennas Propagat., vol. 52, no.9, Sept. 2004.
[17]
A. Parkash, J.K.Vaid, and A. Mansingh, “Measurement of dielectric parameters at
microwave frequencies by cavity-perturbation technique,” IEEE Trans.
Microwave Theory Tech., vol.MTT-27, no.9, Sept. 1979.
[18]
Adil Bahadoor, Yong Wang, and Mohammed N. Afsar, “Complex permittivity and
permeability of barium and strontium ferrite powders in X, Ku, and K-band
frequency ranges,” J. Appl. Phys. Vol.97,10F105, May. 2005.
[19]
V.P. Pawar and S.C. Mehrota, “Dielectric relaxation study of liquids having
chloro- group with associated liquids,” Journal of solution chemistry, Vol. 31, July
2002.
57
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