a complex permittivity and permeability measurement …

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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s

t

a

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t I t I I I I I t

4 6 8 10 12 14

Frequency (GHz)

16 18

0.02

0.018

L o.o16

o

s o.o14

so.o12

TO,Ol

a

n o.oo8

g

e o.oo6

n

t 0.004

0.002

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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c

t 10

r

i

C 9

C

0 8

n

S

t 7

a

n6

t

Batch i / Sample 2

I I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

L o.o16

o

S 0.014

S

0.012

T0.01

a

n 0.008

ge 0.006

n0.004

t

0.002

I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)


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

D 12

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c

t lO

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i

C 9

C

O 8

n

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a

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Batch I / Sample 4

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

L 0.016

O

S 0.014

S

0.012

T0.01

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0.002

4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

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a

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Batch 1 / Sample 5

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

L 0.016

0

S 0.014

S0.012

T0.01

a

n 0.008

ge o.oo6

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0.002

4 6 8 10 12 14 16 18

Frequency (GHz)


plotted vs frequency (solid lines. The average value over all five



lines).

ii

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C

0 8

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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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S 0.014

S0.012

T0.01

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I I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

12

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0 8

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a

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Batch 2 / Sample i

t I I I I I I t

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

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S0.012

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2 4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

13

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0 8

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a

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Batch 2 / Sample 2

I I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

L O.Ol 6

o

S 0.014

S0.012

T0.01

a

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n

t 0.004

0.002

2 4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

14

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C

0 8

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t 7

a

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Batch 2 / Sample 3

I i I I I I i I

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

L o.o160

S 0.014

S0..012

T0.01

a

n 0.008

ge 0.006

n

t 0.004

0.002

0

iiiiiiiii................................................................... iiiiiiiiiii

I I I t I I I I

2 4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

15

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0

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S

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a

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12

11

10

9

6

2

Batch 2 / Sample 4

I I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

L o.o160

S 0.014

S

0.012

T0.01

a

n o.oos

ge 0.o06

n

t o.oo4

0.002

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



lines).

16

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C

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Batch 2 / Sample 5

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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S

0.012

T0.01

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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




lines).

17

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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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0

S 0.014

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0.012

T0.01

a

n o.oo8

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4 6 8 10 12 14 16

Frequency (GHz)

18


is plotted vs frequency (solid lines). The average value over all eight



lines).

18

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n

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iiiiii iiiiiiiiii6 : ..............................................................................................................................................

7

6 I I t I I I I

2 4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

L 0.016

0

S 0.014

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0.012

T0.01

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I I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

19

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C

0

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Batch 3 / Sample 3

I I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

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L o.o160

S 0.014

S

0.012

TO.01

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I I I t I I I t

4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

20

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12

11

10

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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)

0.02

0.018

L 0.016

0

S 0.014

S

0.012

T0.01

a

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e 0.006

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I I t I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

21

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Batch 3 / Sample 5

t t I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

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S 0.014

S

0.012

T0.01

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I I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

22

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0 8

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Batch 3 / Sample 6

I I I t t I I I

4 6 8 10 12 14 16 18

Frequency (GHz)

0,02

0.018

L 0.016

0

S 0.014

S

0.012

T0.01

a

n 0.008g

e 0.006

n

t 0.004

0.002

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2 4 6 8 10 12 14 16 18

Frequency (GHz)





lines).

23

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c 9

C

0 8

n

s

t 7

a

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Batch 3 / Sample 7

I I I I I I I I

4 6 8 10 12 14 16 18

Frequency (GHz)

0.02

0.018

L 0.0160

s 0.014

s0.012

T0.01

a

n 0.008

g

e 0.006

n

t 0.004

0.002

iiiiiiiiii iiiiiiiiiii

I I t t I I I I

2 4 6 8 10 12 14 16 18

Frequency (GHz)





24

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0

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S

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a

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12

11

10

9

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

L 0.016

O

S 0.014

S0.012

T0.01

a

n 0.008ge 0.006

n

t 0.004

0.002

iiiiiiiii .................................................................................................iiiiiiiii

I [ I I I I I I

2 4 6 8 10 12 14 16 18

Frequency (GHz)





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

aI

1.2

P

e 1.1

r

m 1

e

a o.sb

i o.8I

i0.7

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Y0.6

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

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 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

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r

m 1

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8 0.9b

i 0.8I

i0.7

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Yo.6

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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

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a

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P

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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

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P

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a 0.15

b

i o.1I

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YO I I 1 I I I I I t I

0 2 4 6 8 10 12 14 16 18 20

Frequency (GHz)



samples and two measurements at each frequency point is superimposed


lines).

30

1.4R

e1.3

a

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P

e 1.1

r

m 1

e

a 0.9

b

i o.8I

i0.7

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4 6 8 10 12 14 16 18 20

Frequency (GHz)

0.4I

m 0.35a

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P

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r

m o.2

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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)


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


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




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





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





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



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




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

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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

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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




lines).

42

1.4R

e1.3

a

I1.2

P

e 1.1

r

rn 1

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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

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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





43

14

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I1.2

P

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r

m 1

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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

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I I I I I I I I I I

2 4 6 8 10 12 14 16 18 20

Frequency (GHz)





lines).

_ 44

1.4R

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a

I1.2

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r

m 1

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a o.g

b

i 0.8I

i0.7

t

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Batch 3 / Sample 8

0 2 4 6 8 10 12 14 16 18

Frequency (GHz)

2o

0.4I

m0.35

a

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P

e o.2s

r

rn 0.2

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a o.15

b

i o.1I

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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





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 .

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4 6 8 10 12 14 16

Frequency (GHz)

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4 6 8 10 12 14 16

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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

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4 6 8 10 12 14 16

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Frequency (GHz)

I

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0.03

L o.025

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T o.o15

a

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2


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6 8 10 12 14 16

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0

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s 0.02

T 0.015

an

g O.Ole

n

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1st Meas. _ 2nd Meas.

I

18

Figure _4. The data from Figure 8 is plotted here with error bars




57

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Batch 2 / Sample 3

1 st Meas.

i

I I I I I

4 6 8 10 12 14 16

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Frequency (GHz)

0.03

L 0.025

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s o.o2

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1St Meas. -- 2nd Meas.

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4 6 8 10 12 14 16

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58

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1st Meas. <_

2nd Meas. _ O _ =i, i; i! :i! :

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1 ,

I I I I

2 4 6 8 10

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12 14 16

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0.O3

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0

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T o.o15

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1 st Meas. --2nd Meas.

I

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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




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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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21313. 13

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Time

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A-12

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A-13

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Time

100. 0

Empty Covity R_sononcm # 2 in C bond

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A-31

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Time

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a complex permittivity and permeability measurement …

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