A COMPLEX PERMITTIVITY AND PERMEABILITY MEASUREMENT SYSTEM
FOR ELEVATED TEMPERATURES
Semiannual Status Report
July - December, 1989
Grant No. NAG 3-972
Submitted to:
NASA Lewis Research Center
Attn: Mr. Carl A. Stearns
21000 Brookpark Road
Mail Stop 106-1
Cleveland, Ohio 44135
Submitted by:
Georgia Institute of Technology
Georgia Tech Research Institute
Electronics and Computer Systems Laboratory
Electromagnetic Effectiveness Division
Atlanta, Georgia 30332-0800
Principal Investigator: Paul Friederich
Contractinq throuqh:
Georgia Tech Research Corporation
Centennial Research Building
Atlanta, Georgia 30332-0420
TABLE OF CONTENTS
Io
II.
A.
B.
C.
D.
III.
IV.
Appendix
Introduction .............................. 1
Project Progress ......................... 1
Background .............................. 1
Cavity Design ........................... 4
Room Temperature Measurements ........... 5
Error Analysis ......................... 46
Financial Status ....................... 87
References .............................. 87
A ................................... 88
List of Figures
Figure 1 Diagram of rectangular waveguide measurement
cavity.....................................................p 3
Figures 2-19 Permitivitty data versus frequency with
average and standard deviation.
................................................ pp 8-25
Figures 20-37 Permeability data versus frequency with
average and standard deviation.
............................................... pp 28-45
Figures 38-55 Permitivitty data versus frequency with error
bars.
............................................... pp 51-68
Figures 56-73 Permeability data versus frequency with error
bars.
............................................... pp 69-86
I. Introduction
This research is being supported by Grant No. NAG 3-972
from NASA Lewis Research Center (LeRC). The technical
monitor is Mr. C. A. Stearns of the Environmental Durability
Branch. The goals of this research program are threefold:
I) To fully develop a method to measure the permittivity and
permeability of special materials as a function of frequency
in the range of 2.6 to 18 GHz, and of temperature in the
range of 25 to ii00 ° C; 2) To assist LeRC in setting up an
in-house system for the measurement of high-temperature
permittivity and permeability; 3) To measure the complex
permittivity and permeability of special materials as a
function of frequency and temperature to demonstrate the
capability of the method.
II. Project Progress
A. Background
The method chosen for characterizing the sponsor-
furnished materials is based on standards issued by the
American Society for Testing and Materials (ASTM), standards
D 2520-86 [I] (Complex Permittivity of Solid Electrical
Insulating Materials at Microwave Frequencies and
Temperatures to 1650°C) and F 131-70 [2] (Complex Dielectric
Constant of Nonmetallic Magnetic Materials at Microwave
Frequencies). This method relies on perturbation of a
resonant cavity with a small volume of sample material.
Different field configurations in the cavity can be used to
separate electric and magnetic effects. Moore, et al. [3]
presented a detailed explanation of this technique at the
December, 1987 International Symposium on Infrared and MMW
Technology, with particular emphasis on applications with
anisotropic ferrites. An updated version of that paper hasbeen accepted for publication in the American Institute of
Physics (AIP) Review of Scientific Instruments.
Figure 1 illustrates the physical configuration of the
waveguide cavity and sample. The cavity consists of a
section of rectangular waveguide terminated at each end witha vertical slot iris. In the center of one wall is a small
hole through which the sample is introduced. For
permittivity measurements the hole is in the center of thebroad wall, and an odd resonance mode (i.e., TEI0n, with n
odd) is used. The sample is thus located at a point of
maximum electric field, and for small sample volumes the
field is nearly constant over the sample region. Similarly,
for permeability measurements the sample hole is located inthe middle of the narrow wall and an even resonance mode is
used. Thus, the sample is at a point of maximum magneticfield.
Typically, the sample is contained in a small bore
quartz tube. Such tubes have been used with powderedsamples, fiber samples, and thin ceramic rods. A
calibration measurement for such a sample would include
measurement of the cavity with an empty quartz tube in
place, so that perturbation effects could be solely
attributed to the sample.
2
Cavity
tr---1 _/_--, _\_, 1/J-_ Ouertz Tube:tt_ --Couptin9 con±_inin9
_/_ Iris 3emp[e
/Weveguide
Figure I. Diagram showing configuration of rectangular
waveguide cavity used for measurements. The illustrated
orientation of the sample would be used with odd modes for
permittivity measurements. For permeability measurements,
the sample would be inserted parallel to the broad wall.
3
B. Cavity Design
Drawings of a waveguide cavity for use at X band have
been furnished separately to LeRC. Cavities for other bands
are similar except for size. Key features of the cavity
design include the location of sample holes as explained
above; the material from which the assembly is fabricated;
the length of the cavity; and the location of the joint
between pieces. The assembly is fabricated from Hastelloy,
an alloy of nickel developed to withstand temperatures in
excess of 1200 ° C. Cavity lengths are designed to support
three modes of the form TEI0n, so that each cavity will have
either two odd modes and one even, or two even modes and one
odd. It is expected that the complete system will include
two cavities, one of each type, in each band. Those
cavities with two odd modes can be joined at a seam through
the narrow walls, while those cavities with two even modes
can be joined at a seam through the broad walls. Location
of the seam in the narrow wall will minimize its effect on
permittivity measurements, while a seam in the broad wall is
best for permeability measurements. Table I shows possible
cavity lengths for each waveguide band, along with the in-
band resonant modes which would be expected for each length.
The width and height of each cavity are assumed to be the
dimensions of the standard rectangular waveguide for each
band, i.e., WR-187 for C band, WR-137 for Xn band, WR-90 for
X band, and WR-62 for Ku band.
4
TABLE I
RESONANT MODE VS FREQUENCY (GHZ) FOR VARIOUS LENGTHS
Mode: TEl02 TEl03 TEl04 TEl05
C band
6.6" 4.1393 4.7676 5.47046
4.8" 3.9980 4.8520 5.8415
Xn band
4.3" 5.9543
5.5"
X band
3.3" 8.4722
3.8"
Ku band
2.1" 12.692
2.6"
TEl06
6.9741 8.0988
6.0764 6.8763 7.7426
9.7039 11.0882
9.0325 10.1633 11.3939
14.710 16.954
13.132 14.792 16.598
C. Room Temperature Measurements
The third goal of this program is to demonstrate the
capabilities of this method by applying it to special
samples provided by NASA Lewis. Eight samples were sent in
the initial batch, five of which arrived intact. The other
three were broken and mixed together. We were unable to
distinguish the pieces by composition and reunite them into
measurable samples, so they will be put aside and returned
to NASA after the other samples are measured. These samples
have been labelled Batch I. Two other batches have also
been received, under the designations 60-1015 and 60-1520.
They will be referred to as batches 2 and 3, respectively.
All unbroken samples in the three batches have been measured
at room temperature.
5
Room temperature measurements were performed in
waveguide cavities which were either made of copper (C andXn bands) or gold-plated (X and Ku bands). These materials
provide a higher conductivity and, consequently, betterquality factor in the cavities than the nickel which is
required for higher temperatures. A higher quality factormakes smaller changes distinguishable, and thus makes themeasurements more sensitive.
Results from the room temperature measurements are
presented two different ways. One set of plots presents thedata with average and standard deviation values
superimposed; the other set of plots contains error bars
about each measured point. Each sample was measured twice
in each of the four frequency bands: C (4-6 GHz), Xn (6-8
GHz), X (8.4-12.4 GHz), and Ku (12.4-18.0 GHz). Each plotcontains data from one sample, and each line segment on the
plot represents one set of measurements in one cavity. Most
of the plots thus contain eight line segments (four bands
times two measurements). Within the cavity for eachfrequency band, typically eight different resonant modes
were obtainable, four odd modes and four even modes. For
complex permittivity measurements, each sample was inserted
parallel to the E-field in the cavity and the four odd modes
used; for complex permeability measurements each sample was
inserted perpendicular to the E-field in the cavity
(parallel to the broad wall of the waveguide) and the four
even modes used. Thus each line segment on a plot willcontain four data points corresponding to four different
resonant frequencies. (Permittivity results and
permeability results are presented in separate sets of
plots.) The resonant frequencies at the edges of the bands
between cavities sometimes overlapped. This is because some
measured resonances were outside the customary limits of thewaveguide band. All measured resonances were well above
cutoff and below the cutoff frequency for the next highermode, however.
6
Each sample was measured two times in each band. When
the sample was long enough, the two measurements were
performed at different locations along the length of the
sample. In several cases the sample was not long enough toextend through the entire cavity. Because the volume of
sample inside the cavity was indeterminate, it was notpossible to perform meaningful dielectric calculations in
those cases. Thus, measurements of samples 1-2 in the first
batch; samples 1-5 of the second batch; and samples 1-3 of
the third batch, at even modes (parallel to the broad wall)in the C-band cavity are not included. Also, samples 1-2 ofthe third batch and sample 4 of the second batch are not
characterized at even modes in the Xn band cavity.Figures 2-6 are plots of the calculated dielectric
constant and loss tangent of the five surviving samples inbatch i; Figures 7-11 show calculated dielectric constants
and loss tangents for the five samples in batch 2; andFigures 12-19 show the dielectric constants and loss
tangents for the eight samples of batch 3. All plots are
versus frequency, with the different line segmentsrepresenting individual measurements in a single cavity, as
explained above. The three dotted lines superimposed oneach plot represent an average value with one standard
deviation on either side. The average was taken over all
frequency values and all samples in a batch. (Thus theaverage and standard deviation lines are the same on all
plots of a given batch.) The average was taken across allfrequencies because normal dielectric behavior of ceramic
materials is not frequency dependent in the microwave
regions. These averages are compiled in Table 2.
7
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Frequency (GHz)
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ii iiiii iiiiiiiiiiii
I I I I I I I I
2 4 6 8 10 12 14 16
Frequency (GHz)
18
Figure 2. Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dottedlines).
D 12
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I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
0.018
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I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 3. Data from two measurements at each of four frequency bands is
plotted vs frequency. (solid lines) The average value over all fivesamples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
9
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4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 4. Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
i0
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Batch 1 / Sample 5
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
0.018
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4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 5. Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines. The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
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Batch 1 / Sample 8
I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
0.018
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I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 6. Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
12
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t I I I I I I t
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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I I I I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 7. Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
13
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I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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2 4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 8. Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
14
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I i I I I I i I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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0
iiiiiiiii................................................................... iiiiiiiiiii
I I I t I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 9. Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
15
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Batch 2 / Sample 4
I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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2 4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 10. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
16
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I I i I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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I I I I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
Figure II. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all five
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
17
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I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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0
4 6 8 10 12 14 16
Frequency (GHz)
18
Figure 12. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all eight
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
18
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iiiiii iiiiiiiiii6 : ..............................................................................................................................................
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6 I I t I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
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I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 13. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all eight
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
19
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I I I I I I I I
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Frequency (GHz)
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I I I t I I I t
4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 14. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all eight
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
20
Die
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Batch 3 / Sample 4
I t I I I I t I
4 6 8 10 12 14 16 18
Frequency (GHz)
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I I t I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 15. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all eight
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
21
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t t I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 16. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all eight
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
22
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I I I t t I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0,02
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o I f i I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 17. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all eight
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
23
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I I I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
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iiiiiiiiii iiiiiiiiiii
I I t t I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 18. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all eight
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dottedlines).
24
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Batch 3 / Sample 8
I I I I I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
0.02
0.018
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S0.012
T0.01
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iiiiiiiii .................................................................................................iiiiiiiii
I [ I I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
Figure 19. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all eight
samples, sixteen frequency points, and two measurements is superimposed
along with one standard deviation above and below the average (dotted
lines).
25
TABLE 2
AVERAGE DIELECTRIC VALUES FOR EACH SAMPLE BATCH
Batch
Average Standard
Dielectric Deviation
Average
Loss Tangent
Standard
Deviation
Constant
1 9.72 .73 .0088 .0022
2 8.58 1.05 .0065 .0023
3 8.55 .58 .0073 .0023
In Figures 20-24, the real and imaginary parts of the
complex magnetic permeability of the samples of batch 1 are
plotted versus frequency. Figures 25-29 represent the
results of measurements of batch 2 samples, and Figures 30-
37 are from the samples of batch 3. For the real and
imaginary permeability plots, average and standard deviation
values are again represented with dotted lines. This time,
however, the average is taken only over the samples of a
given batch at a particular frequency. Since typical
magnetic properties often show frequency dependence in the
microwave regions, each mode was averaged separately. Some
of these plots do not contain line segments from the C band
cavity, or in a few instances from the Xn band cavity. As
explained above, this is because the samples in these cases
were shorter than the broad-wall dimension of the waveguide.
One characteristic worth noting in the plots is a
sometimes wide disparity between multiple measurements of
the same sample. This is most likely due to inhomogeneity
in the samples. As noted above, these measurements were
performed on different sections of each sample rod whenever
possible. In addition, since more of each sample was
present in the larger cavity volumes of the lower frequency
bands, measurements at the lower bands have the effect of
averaging over more of the sample material. For this
26
reason, higher frequency cavities will be more sensitive to
inhomogeneities in the samples when different regions of the
sample are measured. Unfortunately, the same section was
not always measured in all four bands. Thus, traces from
band to band do not always represent the same section of
material; hence they would not necessarily "line up".
Finally, for most of the permittivity measurements,
especially in the cavities at Xn, X, and Ku bands, a slight
upward slope with increasing frequency is exhibited by the
data curves. This is due to the finite volume of the
samples. A first order correction for sample volumes which
do not vanish has already been applied in the analysis
software, but residual effects manifest themselves as the
slope present in most of these plots. The "true" value
would be approximately in the middle of the segment from
each band.
27
1.4R
e1.3
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Batch 1 / Sample I
I I i I t I I I i
4 6 8 10 12 14 16 18 20
Frequency (GHz)
0.4I
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e
a 0.15b
i 0.1I
i0.05
t
Y0 I I I I I I I t I I
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
Figure 20. Data from two measurements at each of three frequency bandsis plotted vs frequency (solid lines). The average value over all fivesamples and two measurements at each frequency point is superimposedalong with one standard deviation above and below the average (dotted
lines).
28
1.4R
e1.3
aI
1,2
P
e 1.1
r
m 1
e
8 0.9b
i 0.8I
i0.7
t
Yo.6
o
Batch 1 / Sample 2
I t I I I I I I t I
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
0.4I
m 0.35a
g 0.3
P
e 0.25
r
m 0.2
e
a o.15b
i o.1I
i 0.05t
Y 0
0
I t I I I I t I t l
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
Figure 21. Data from two measurements at each of three frequency bandsis plotted vs frequency (solid lines). The average value over all fivesamples and two measurements at each frequency point is superimposedalong with one standard deviation above and below the average (dottedlines).
29
14
R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a o.9b
i o.8I
i0.7
t
Y0.6
0
Batch 1 / Sample 4
%
',.;, .--
I I t I I I I t I I
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
0.4I
m0.35
a
g0.3
P
e 0.25
r
rn 0.2
e
a 0.15
b
i o.1I
i0.05
t
YO I I 1 I I I I I t I
0 2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
Figure 22. Data from two measurements at each of four frequency bands
is plotted vs frequency (solid lines). The average value over all five
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dotted
lines).
30
1.4R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a 0.9
b
i o.8I
i0.7
t
Y0.6
4 6 8 10 12 14 16 18 20
Frequency (GHz)
0.4I
m 0.35a
g o.3
P
e 0.25
r
m o.2
e
a 0.15
b
i 0.1I
i 0.05t
Y o I I t I I t I I I I
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
Figure 23. Data from two measurements at each of four frequency bands
ks plotted vs frequency (solid lines). The average value over all five
samples and two measurements at each frequency point is superimposedalong with one standard deviation above and below the average (dotted
lines).
31
14R
e 1.3aI
1.2
P
e 1,1
r
rn 1
e
a 0.9b
i o.8I
i 0.7t
Y o.6
Batch 1 / Sample 8
I t I I I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2o
0.4I
n30.35
a
g0.3
P
e 0.25
r
m 0.2
e
a 0.15b
i 0.1I
i0.05
t
Y0 I I t I I I I I t
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2o
Figure 24. Data from two measurements at each of four frequency bands
is plotted ve frequency (solid lines). The average value over all fivesamples and two measurements at each frequency point is superimposedalong with one standard deviation above and below the average (dottedlines).
32
1.4
R
e1.3
aI
1.2
P
e 1.1
r
m 1
e
a o.eb
i o.8I
i0.7
t
Y0.6
Batch 2 / Sample i
I I t t I I f I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
2O
0.4I
m0.35
a
g0.3
P
e 0.25
r
m o.2
e
a o.15b
i o.1I
i0.05
t
Y0
O 2 4 6 8 10 12 14 16 18
Frequency (GHz)
20
Figure 25. Data from two measurements at each of three frequency bandsis plotted vs frequency (solid lines). The average value over all five
samples and two measurements at each frequency point is superimposedalong with one standard deviation above and below the average (dottedlines).
33
14
R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a o.9b
i o.eI
i0.7
t
Y0.6
Batch 2 / Sample 2
I I I I I t t I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2O
0.4I
m0.35
a
g0.3
P
e 0.25
r
m 0.2
e
a 0.15b
i 0,1I
i0.05
t
Y0 I I I I I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2O
Figure 26. Data from two measurements at each of three frequency bands
is plotted vs frequency (solid lines). The average value over all five
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dotted
lines).
34
1.4
R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a 0.9
b
i o.8I
i0.7
t
Y0.6
0
Batch 2 / Sample 3
I I I _ I I I I t
2 4 6 8 10 12 14 16 18
Frequency (GHz)
i
2O
0.4I
m0.35
a
g0.3
P
e 0.25
r
m 0.2
e
a 0.15
b
i o.1I
i0.05
t
Y0
iiiiiii!! iiiiiiii......
I I I I I I t I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2O
Figure 27. Data from two measurements at each of three frequency bands
is plotted vs frequency (solid lines). The average value over all five
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dotted
lines).
35
14R
e1.3
a
I1.2
P
e 1.1
r
rn 1
e
a 0.9b
i 0.8I
i0.7
t
Y0.6
Batch 2 / Sample 4
I I f t t I I t I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
2o
0.4I
m0.35
a
g0.3
P
e 0.25
r
m o.2
e
a 0.15b
i 0.1I
i0.05
t
y0 I I I I I I t I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2o
Figure 28. Data from two measurements at each of two frequency bands isplotted vs frequency (solid lines). The average value over all fivesamples and two measurements at each frequency point is superimposedalong with one standard deviation above and below the average (dottedlines).
36
14R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a 0.9b
i o.8I
i0.7
t
Y0.6
Batch 2 / Sample 5
2 4 6 8 10 12 14 16 18
Frequency (GHz)
2O
0.4I
m0.35
a
g0.3
P
e 0.25
r
m o.2
e
a o.15b
i o.1I
i0.05
t
Y0
0
""-,., .
--....
t t I I I I I I t
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2O
Figure 29. Data from two measurements at each of three frequency bands
is plotted vs frequency (solid lines). The average value over all five
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dotted
lines).
37
1.4
R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a o.9b
i o.8I
i0.7
t
Y0.6
0
Batch 3 / Sample I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
2O
0.4I
m 0.35a
g 0.3
P
e 0.25
r
m 02
e
a 0.15
b
i o.1I
i o.o5t
Y o
" :C--#
I I I I I _ I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
20
Figure 30. Data from two measurements at each of two frequency bands is
plotted vs frequency (solid lines). The average value over all eight
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dotted
lines).
38
14
R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a 0.9
b
i 0.8I
i0.7
t
Y0.6
Batch 3 / Sample 2
I I i t I I t I I i
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
0.4I
m0.35
a
g0.3
P
e 0.25
r
m o.2
e
a 0.15
b
i o.1I
i0.05
t
Y0
.. ............. ...
I I t I t I I I I t
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
Figure 31. Data from two measurements at each of two frequency bands is
plotted vs frequency (solid lines). The average value over all eight
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dotted
lines).
39--4
14R
e1.3
aI
1.2
P
e 1.1
r
m 1
e
a o.sb
i o.8I
i0.7
t
Y0.6
Batch 3 / Sample 3
I I t I I I I I f
2 4 6 8 10 12 14 16 18
Frequency (GHz)
2o
0.4I
m0.35
a
g0.3
P
e 0.25
r
m o.2
e
a o.15b
i o.1I
i0.05
t
Yo
o
I I I I t I I I I
2 4 6 8 lO 12 14 16 18
Frequency (GHz)
I
2o
Figure 32. Data from two measurements at each of three frequency bandsis plotted vs frequency (solid lines). The average value over all eightsamples and two measurements at each frequency point is superimposedalong with one standard deviation above and below the average (dottedlines).
4O
1.4R
e1.3
aI
1.2
P
e 1.1
r
m 1
e
a 0.9b
i o.sI
i0.7
t
Y0.6
_/Batch 3 / Sample 4
I t I I I I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
I
20
0.4I
m0.35
a
g0.3
P
e 0.25
r
m 0.2
e
a o.15b
i o.1I
io.o5
t
Yo t I I I I t t I I
2 4 6 8 lO 12 14 16 18
FreQuency (GHz)
I
20
Figure 33. Data from two measurements at each of four frequency bandsis plotted vs frequency (solid lines). The average value over all eightsamples and two measurements at each frequency point is superimposedalong with one standard deviation above and below the average (dottedlines).
41
1.4R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a o.s
b
i o.8I
i0.7
t
Y0.6
Batch 3 / Sample 5
/
I t t I t I I I
4 6 8 10 12 14 16 18
Frequency (GHz)
I
2o
0.4I
m0.35
a
g0.3
P
e 0.25
r
m 0.2
e
a 0.15
b
i 0.1I
i0.05
t
Y0
.. ........................ _..
I I t I I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2o
Figure 34 Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all eight
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dotted
lines).
42
1.4R
e1.3
a
I1.2
P
e 1.1
r
rn 1
e
a 0.9
b
i o.sI
i0.7
t
Y0.6
Batch 3 / Sample 6
\
"\ .. • ",,
"-..
//
I I I t I I I I t
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
2o
0.4I
m0.35
a
g0.3
P
e 0.25
r
rn 0.2
e
a o.15
b
i o.1I
i0.05
t
Y0
,,.....- ................. ,..,.,.
".-.._.
.... ;;L---
I I I t I I I I I
2 4 6 8 10 12 14 16 18
Frequency (GHz)
I
20
Figure 35 Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all eight
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dottedlines).
43
14
R
e 1.3a
I1.2
P
e 1.1
r
m 1
e
a 0.9
b
i o.8I
i 0.7t
Y o.6
Batch 3 / Sample 7
t I I I I I I I I I
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
0.4I
m0.35
a
g0.3
P
e 0.25
r
m 0.2
e
a o.15b
i o.1I
i0.05
t
Y0
"'......
I I I I I I I I I I
2 4 6 8 10 12 14 16 18 20
Frequency (GHz)
Figure 36 Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all eight
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dotted
lines).
_ 44
1.4R
e1.3
a
I1.2
P
e 1.1
r
m 1
e
a o.g
b
i 0.8I
i0.7
t
Y0.6
Batch 3 / Sample 8
0 2 4 6 8 10 12 14 16 18
Frequency (GHz)
2o
0.4I
m0.35
a
g0.3
P
e o.2s
r
rn 0.2
e
a o.15
b
i o.1I
io.o5
t
Yo
"'-,.....
iiiiiii!iiiiii!!!!!!!!!iiiiiiii....
I I t I t I I I I
0 2 4 6 8 10 12 14 16 18
Frequency (GHz)
2o
Figure 37 Data from two measurements at each of four frequency bands is
plotted vs frequency (solid lines). The average value over all eight
samples and two measurements at each frequency point is superimposed
along with one standard deviation above and below the average (dottedlines).
45
D. Error Analysis
The same data points in Figures 2-37 are plotted again
in Figures 38-73 with error bars (and without the average
and standard deviation lines). This section will present
the analysis by which the error bars were obtained. The
equations by which the real and imaginary parts of the
permittivity are calculated simplify essentially to the
following:
fo - fs Vc Fr VrE r - 1 = --
fo 2 V s 2
Ei = I l l VcQs - Qo 4 V s
V r
4
Where
fo - fsF r =
fo
V C
V r =
V_
A Q = Qs - Qo
The real and imaginary parts of the permittivity are, of
course, represented by c r and Ci, respectively. The other
variables are:
46
V c
Vs
f0
fs
Q0
Os
Volume of the cavity
Volume of sample inside the cavity
Resonant frequency of the empty cavity
Resonant frequency with the sample present
Quality factor of the empty cavity
Quality factor with the sample present
The uncertainties in these quantities are related by
2
I rl+o 12 (72 ( E r ) -: (72 ( pr ) _Fr _Vr
(72 ( Er ) -- i ( (72 ( Fr ) Vr 2 + (72 (Vr) Pr 2 )2
and
I 1111 i _i i I_il 24(72(ci)= o2 A _ I i I + (72(Vr)
[_a(O)I _Vr
where the standard deviation in a sample population is taken
as the uncertainty. It is also necessary that the
measurement uncertainty in the volume term be independent of
the measurement uncertainty in the frequency term or quality
47
factor term. Then the uncertainty in these terms can be
determined from physical measurement considerations.
For the volume ratio term, the uncertainty is derived
from
2 2
= + (v_)
2
-- + (vc)Vs Us 2
The cavity volume contributes
02(Vc) = 02 (a)(bl) 2 + 02 (b)(at) 2 + 02 (_)(ab) 2
where the uncertainty in the transverse dimensions, a and b,
are the mechanical tolerances of the waveguide, and the
measurement uncertainty of 0.08 cm (0.03 in.) in the length
is used for _(_)
The volume of sample in the cavity is calculated for
each measurement using the appropriate transverse waveguide
dimension (broad wall for permeability measurements and
narrow wall for permittivity measurements) and the cross
sectional area of the individual sample. The sample cross
section is assumed to be uniform and is calculated using the
density values supplied with the samples after weighing each
sample and measuring its length. Then the sample volume
uncertainty is
o2(v_) = o2(x)(t,)2 + o2(t,)(x)2
48
where Is is the length of sample in the cavity and x is the
cross section area. Of particular concern is the effect of
the value used for the density of a sample material. The
precision of the density value supplied with the samples was
taken to be ±.01 g/cc. With this precision for the density,
the uncertainty in the sample volume is dominated by the
uncertainty in the measured length of the sample, which was
determined to be .08 cm (.03 in).
The frequency ratio term F r contributes to the
measurement uncertainty as follows:
2 2
2
+ t J
Here the variance terms are computed from the measurement
results. Specifically, the variance term for the empty
cavity resonance, f0, is calculated as the sample variance
for the population of empty cavity resonant frequency values
recorded at each mode with each batch of samples. The
variance term for the perturbed resonant frequency, fs, is
likewise calculated from the measurement population at each
mode for each batch of samples. Note that this means that
sample to sample variation within a batch, as well as
inhomogeneity within each sample, will contribute to the
measurement uncertainty, and this is accounted for in the
plotted error bars. Similarly, uncertainty in the quality
factor measurements is taken as the measurement variance at
each mode for each batch of samples, and contributes to the
overall measurement uncertainty through the Q term as
follows:
49
I 1A 1
II I lll !ii+
I'l--o2 I +O_o
Again, the effects of sample to sample variation within a
sample batch, and the effects of inhomogeneity within a
particular sample, have been accounted for in the error bars
for these terms.
Plots of the variations in the measured quantities at
each resonant mode have been included in the appendix as a
quick visual indication of the magnitude of sample-to-sample
variations. For these plots, the empty cavity quantities
have been plotted as deviations from the average value,
where the average is computed across all empty cavity
measurements at a given frequency mode. For the top plot on
each page the plotted quantity is a frequency deviation,
while the bottom plot shows variations in the quantity I/Q.
The dashed lines in each plot represent the change
associated with the introduction of a sample into the
cavity. Thus the solid lines indicate the magnitude of the
standard deviations used for f0 and I/Q 0 in the error bar
calculations, while the dashed lines show the relative
magnitude of the standard deviations of the quantities fs
and I/Q s .
5O
D 14
i
e13
I
e12
C
t
r 11
i
C 10
C 9
0
n 8
s
t7
a
n6
t
Batch 1 / Sample 1
I st Meas. -- 2nd Meas.
0
: : \ ' ' o'-- ..... ZZ
I t t I I 1 I
4 6 8 10 12 14 16
Frequency (GHz)
I
18
0.03
L 0.025
0
s
s 0.02
T o.o15
a
n
g O.Ol
e
n
t 0.005
0
_ 1st Meas. -- 2nd Meas.
-2 5 ,£2 -. oi ' _. o=_ ., ¢ G ' '.', "
;_ ,V il i i :v __ii1_ i - 0
0 o 0 G - o
I ! I t I I I
4 6 8 10 12 14 16
I
18
Figure 38. The data from Figure 2 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
51
D 14
i
e13
I
e
12C
tr 11
i
c lo
C 9
0
n 8
s
t7
a
n6
t
2
Batch 1 / Sample 2
m
<_ 1st Meas. 2nd Meas.
I
14B 10 12
Frequency (GHz)
16 1B
0.03
L 0.025
0
S
S 0.02
T 0.015
a
n
g 0.Ol
e
n
t 0.005
2 4 6 8
1st Meas. -- 2nd Meas.
-m
10 12 14 16 18
Figure 39. The data from Figure 3 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
52
D 14i
e13I
e12
C
tr 11ic 10
C 9
0
n 8s
t7
a
n6
t
Batch 1 / Sample 4
<_ 1st Meas. --" 2nd Meas.
4 6 8 10 12 14 16
Frequency (GHz)
I
18
0.03
L 0.025
o
s
S 0.02
T 0.015a
n
g O.Olen
t 0.oo5
<_ _ 1st Meas. m 2nd Meas.
ml=
- ©¢
o_
_ _,_, ;:
e- d i
2 4 6
¢i
!
0
, __ ¢:; =lm O
- 0<3 O, ',0 --
8 10 12 14 16 18
Figure 40. The data from Figure 4 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
53
D 14
i
e13
I
e12
C
t
r 11
i
C 10
C 9
o
n sS
t7
a
n6
t
Batch 1 / Sample 5
1st Meas. -- 2nd Meas.
t I i I I I I
4 6 8 10 12 14 16
Frequency (GHz)
I
18
0.03
L 0.025
o
s
s 0.o2
-r o.oi 5
a
n
g O.Ol
e
n
t o.oo5
©
1st Meas. -- 2nd Meas.
: ,!
I _ i 1 I I I
4 6 8 10 12 14
I
16
I
t8
Figure 41. The data from Figure 5 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
54
D 14
i
e13
I
812
c
t
r 11
i
c 10
C 9
0
n s
s
t7
a
n6
t
Batch 1 / Sample 8
1st Meas. -- 2nd Meas.
I I t I I I
4 6 8 10 12
Frequency (GHz)
14 16 18
0.03
L 0.025
0
s
s 0.02
T 0.015
a
t3
g 0.01
e
n
t 0.005
0
rap=
<>
<>
i 0 ,,,_._N i _
t¢ "0 '14 6 8
1st Meas. -- 2nd Meas.
_o
- $<> :5 -¢
I I I I I
10 12 14 16 18
Figure 42. The data from Figure 6 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
55
O 14
i
e13
I
e12
C
t
r 11
i
C 10
C 9
0
R 8
S
t7
a
ri6
t
Batch 2 / Sample 1
1st Meas. -- 2nd Meas.
Frequency (GHz)
I
18
0.03
L o.025
o
s
s 0.02
T o.o15
a
n
g O.Ol
e
n
t o.oo5
2
1st Meas. -- 2nd Meas.
I
18
Figure 43. The data from Figure 7 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
56
O 14
i
e13
I
e12
C
t
r 11
i
C 10
C 9
0
n 8
s
t7
a
n6
t
Batch 2 / Sample 2
1st Meas. -- 2nd Meas.
! !
I
4
606
¢
L , --__
t
i
n
h
I I I t
6 8 10 12 14 16
Frequency (GHz)
I
18
0.03
L 0.025
0
s
s 0.02
T 0.015
an
g O.Ole
n
t 0.005
1st Meas. _ 2nd Meas.
I
18
Figure _4. The data from Figure 8 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
57
D 14
i
e13
I
e12
c
t
r 11
i
c lO
C 9
0
n 8
S
t7
a
n6
t
Batch 2 / Sample 3
1 st Meas.
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4 6 8 10 12 14 16
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Figure 45. The data from Figure 9 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
58
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Figure 46. The data from Figure i0 is plotted here with error bars
around each point. The error bars represent a calculated uncertaintylevel of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
59
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Figure 47. The data from Figure ii is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
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Figure 48. The data from Figure 12 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
61
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Figure 49. The data from Figure 13 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
62
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I I t t I t t
4 6 8 10 12 14 16
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Figure 50. The data from Figure 14 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
63
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Figure 51. The data from Figure 15 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
64
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Figure 52. The data from Figure 16 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
65
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Figure 53. The data from Figure 17 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
66
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1 st Meas. -- 2nd Meas.
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Figure 54. The data from Figure 18 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
67
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Figure 55. The data from Figure 19 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
68
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Figure 56. The data from Figure 20 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
69
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Figure 57. The data from Figure 21 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
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Figure 58. The data from Figure 22 is plotted here with error barsaround each point. The error bars represent a calculated uncertaintylevel of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
71
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Figure 59. The data from Figure 23 is plotted here with error bars
around each point. The error bars represent a calculated uncertaintylevel of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
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Figure 60. The data from Figure 24 is plotted here with error bars
around each point. The error bars represent a calculated uncertaintylevel of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
73
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Figure 61. The data from Figure 25 is plotted here with error bars
around each point. The error bars represent a calculated uncertaintylevel of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
74
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Figure 62. The data from Figure 26 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
75
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Figure 63. The data from Figure 27 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
76
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Figure 64. The data from Figure 28 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
77
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Figure 65. The data from Figure 29 is plotted here with error barsaround each point. The error bars represent a calculated uncertaintylevel of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
78
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Figure 66. The data from Figure 30 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
79
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8 10 12 14 16 18
Frequency (GHz)
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Figure 67. The data from Figure 31 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
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Frequency (GHz)
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Figure 68. The data from Figure 32 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
81
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Figure 69. The data from Figure 33 is plotted here with error bars
around each point. The error bars represent a calculated uncertaintylevel of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
82
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Figure 70. The data from Figure 34 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
83
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Figure 71. The data from Figure 35 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
84
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Frequency (GHz)
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0
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0 0 - _.i ¢ --_"_ 1st Meas. _ ' i
-- 2nd Meas. <_ :
-- 0t t I I I i t
n
2 4 6 8 10 12 14 16
I
18
Figure 72. The data from Figure 36 is plotted here with error bars
around each point. The error bars represent a calculated uncertaintylevel of one standard deviation as explained in the text. They includethe effect of variations from sample to sample within the same batch.
85
14
R
e1.3
a
I1.2
Pe 1.1r
rn 1
ea 0.9
b
i o.8Ii
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\ :vk, _ 1st Meas.( ©
i ii _ . -: "\ ii , <_
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i I t t tA
4 6 8 10 12 14 16
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18
0.4
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a
g 0.3
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0
n
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0 = ¢ =¢
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0 1st Meas. : " <_ "_"
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1
2 4 6 8 10 12 14
¢
t
16
I
18
Figure 73. The data from Figure 37 is plotted here with error bars
around each point. The error bars represent a calculated uncertainty
level of one standard deviation as explained in the text. They include
the effect of variations from sample to sample within the same batch.
86
III• Financial Status
Of the total grant of $100,030, expenditures through
December 31 have totalled $32,640, or approximately 33%.
.
•
References
American Society for Testing and Materials standard
D 2520-86, "Complex Permittivity of Solid Electrical
Insulating Materials at Microwave Frequencies and
Temperatures to 1650°C. ''
American Society for Testing and Materials standard
F 131-70, "Complex Dielectric Constant of Nonmetallic
Magnetic Materials at Microwave Frequencies."
Moore, et al., "Second Order Correction in Cavity
Constitutive Parameter Measurements with Application to
Anisotropic Ferrites," International Symposium on
Infrared and MMW Technology, Orlando, Florida, December,
1987.
87
APPENDIX A
The plots in this appendix give an indication of the
repeatability of the measurements. A solid line represents
a quantity in the unperturbed cavity for the mode indicated
in the plot title. In all of the plots, a solid line
represents the deviation from the average value of the empty
cavity resonant frequency, f0, or the empty cavity quality
factor, as represented by the quantity I/Q 0. Thus, the
smoothness of the solid lines is an indication of the
repeatability of empty cavity measurements, and consequently
of the calculated quantities. The dashed line represents
perturbation caused in a given mode by the presence of a
sample. For odd modes, the difference between the solid
line and dashed line in the frequency plot would be the
quantity used to calculate the dielectric constant. The
difference in the quality factor plot would be used to
calculate the imaginary part of the permittivity. The loss
tangent is just the ratio of imaginary and real
permittivity. Similarly, the difference between solid and
dashed lines in the frequency plot for even modes is used to
calculate the real part of the permeability, while the
difference in the quality factor plot is used to calculate
the imaginary part. The smoother a line is, the less
variation one would see in calculated parameters. The three
sets of lines in each plot represent results from the three
different batches of samples. The abscissa is meant to
indicate the chronological order of the measurements rather
than any fixed time increment, with one unit representing
one measurement.
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