Chapter

Provisional 14
chapter

Materials Characterization
Materials Characterization Using
Using Microwave
Microwave
Waveguide Systems
Waveguide Systems

Kok Yeow
Kok Yeow You
You

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http://dx.doi.org/10.5772/66230

Abstract
This chapter reviews the application and characterization of material that uses the
microwave waveguide systems. For macroscopic characterization, three properties of
the material are often tested: complex permittivity, complex permeability and conduc-
tivity. Based on the experimental setup and sub-principle of measurements, microwave
measurement techniques can be categorized into either resonant technique or nonresonant
technique. In this chapter, calibration procedures for non-resonant technique are
described. The aperture of open-ended coaxial waveguide has been calibrated using
Open-Short-Load procedures. On the other hand, the apertures of rectangular waveguides
have been calibrated by using Short-Offset-Offset Short procedures and Through-Reflect-
Line calibration kits. Besides, the extraction process of complex permittivity and complex
permeability of the material which use the waveguide systems is discussed. For one-port
measurement, direct and inverse solutions have been utilized to derive complex permit-
tivity and complex permeability from measured reflection coefficient. For two-port mea-
surement, in general, the material filled in the waveguide has been conventional practice to
measure the reflection coefficient and the transmission coefficient by using Nicholson-
Ross-Weir (NRW) routines and convert these measurements to relative permittivity, εr
and relative permeability, μr. In addition, this chapter also presents the calculation of
dielectric properties based on the difference in the phase shifts for the measured transmis-
sion coefficients between the air and the material.

Keywords: microwave waveguides, relative permittivity, relative permeability, con-

ductivity, resonant methods, nonresonant methods, materials characterization

1. Introduction

For macroscopic material characteristic investigations, three properties of the material are
often measured: relative permittivity εr, relative permeability μr and conductivity σ. Normally,
many microwave measurements only focus on the properties of relative permittivity, εr rather

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342 Microwave Systems and Applications

than the permeability μr and the conductivity σ. Recently, there has been an increased interest
in the determination of dielectric properties of materials at microwave frequencies range. This
is because those properties were played the important roles in the construction of high-fre-
quency electronic components, the superconducting material properties, the quality of printed
circuit board (PCB) substrate, the efficiency of microwave absorption materials, metamaterial
characterizations and the performance of dielectric antenna design. Based on the setup and
sub-principle of measurements, measurement techniques can be categorized into either reso-
nant methods or nonresonant methods. In practice, the prime considerations in measuring the
dielectric properties of the materials are the thickness required of the material, the size of the
waveguide, limitations of the operating frequency and the accuracy of the measurements.
For material characterizing using resonant methods, a resonator is filled with a material or
sample as shown in Figure 1 [1, 2]. This produces a resonance frequency shift and also a
broadening of the resonance curve compared to the resonator without filled with any sample.
From measurements of shifting resonance frequency, the properties of the sample can then be
characterized. The particular resonance frequency for the resonator without filled with the
sample is depends on its shape and dimensions. The resonance measurement techniques are
good choices for determining low-loss tangent, tanδ values for the low-loss sample, but such
techniques cannot be used for the measurement of swept frequency.

Figure 1. The resonator cavity filled with sample under test [1, 2].

A free-space and transmission/reflection measurement techniques are grouped in the category

of nonresonant methods. The free-space technique is a far-field measurement, and a horn
antenna is used as the radiator as shown in Figure 2 [3–5]. The free-space method is suitable
for the measurement for thin film sample with high temperature because horn radiators do not
come into direct contact with the sample, and thus, the RF circuits of the instrument are safer
from heat damage. However, this method provides a less precise measurement because the
sensing field is highly dispersed. Furthermore, the distance between the sample surface and
the horn aperture is difficult to gauge precisely. The coaxial, circular or rectangular wave-
guides are implemented in transmission/reflection measurement techniques which are directly
 Materials Characterization Using Microwave Waveguide Systems 343
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in contact with the sample. Although various measurement techniques are available to be
used, when choosing the appropriate technique, some other factors are required to be consid-
ered in the selection of technique, such as accuracy, cost, samples shape and operating fre-
quency. This chapter is focused only on coaxial and rectangular waveguides.

Figure 2. Free-space measurement setup for dielectric measurement of the thin sample [3–5].

2. Microwave measurement using coaxial and rectangular waveguides

2.1. Coaxial and rectangular waveguides

There are various sizes of the coaxial probes and rectangular waveguides, which are depen-
dent on the operating frequency and its application. The coaxial probe is a waveguide
consisting of inner and outer conductors, with radii a and b, respectively, as shown in Figure 3.
On the other hand, the rectangular waveguide is a rectangular metal pipe with width, b and
height, a, which guides high-frequency electromagnetic waves from one place to another
without significant loss in intensity. There are several commercial rectangular waveguides,
such as WR510, WR90, WR75 and WR62 waveguides, which covering a broad measurement
range for L-band, X-band and Ku-band, respectively, as shown in Figure 4. Generally, the
material characterization using waveguide discontinuity methods can be categorized into
one-port and two-port measurements. The measurements assume that only the dominant
transverse electric, TE10 mode propagates in the rectangular waveguide. On the other hand,
only transverse electromagnetic mode (TEM) is assumed to be propagated in the coaxial line
waveguide.
344 Microwave Systems and Applications

Figure 3. (a) Keysight dielectric probe kit with inner radius of outer conductor, b = 1.5 mm and radius of inner,
a = 0.33 mm. (b) Customized small coaxial probe with b = 0.33 mm and a = 0.1 mm [6]. (c) RG402 and RG 405 semi-rigid
coaxial probe [7]. (d) SMA stub coaxial probe with b = 2.05 mm and a = 0.65 mm [8]. (e) Customized large coaxial probe
with b = 24 mm and a = 7.5 mm.

Figure 4. (a) WR510 waveguide-to-coaxial adapter and (b) WR 90, WR 75 and WR 62 waveguide-to-coaxial adapters [9].

2.2. Measurements principles

The one-port measurement is based on the principle that a reflected signal (reflection coeffi-
cient, S11) through the waveguide, which end aperture is contacting firmly with the material
under test (sample), will obtain the desired information about the material as shown in
Figure 5. The main advantage of using one-port reflection technique is that the method is the
simplest, broadband, nondestructive way to measure the dielectric properties of a material.
However, one-port measurement is suitable only for measuring the relative permittivity, εr, of
the dielectric material (nonmagnetic material, μr = 1). This is due to insufficient information to
predict the permeability, μr, if only obtained the measured reflection coefficient, S11 without
transmission coefficient, S21.
For Figure 5a and c measurements, the sample is considered infinite, as long as the sample
thickness d is greater than the radius of the outer conductor b. However, the radiation, or
sensing area, for an aperture rectangular waveguide is much greater than that of a coaxial
probe. For instance, the WR90 waveguide has a radiation distance up to 20 cm in the air from
the aperture waveguide. Hence, the sample under test must be much thicker when the rectan-
gular waveguide is utilized in the measurement. Besides, the coaxial probe and rectangular
waveguide are also capable of testing the thin film sample as shown in Figure 5b and d. The
measurements required that the thin sample is backed by a metallic plate.
 Materials Characterization Using Microwave Waveguide Systems 345
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Figure 5. Open-port measurements using the (a), (b) coaxial probe and the (c), (d) rectangular waveguide.

Figure 6. Two-port measurements using the (a) rectangular waveguide and the (b) coaxial transmission line.

The two-port measurement uses both reflection and transmission methods. Here, the material
under test is placed between waveguide transmission lines or segments of the coaxial line as
shown in Figure 6. The two-port measurement using coaxial or rectangular waveguides
became popular due to the convenient formulations derived by Nicholson and Ross [10] in
346 Microwave Systems and Applications

1970, who introduced a broadband determination of the complex relative permittivity, εr and
permeability, μr of materials from reflection and transmission coefficients (S11 and S21). For
measurements in Figure 6, the sample must be solid and carefully machined with parallel
interfaces, and must perfectly fill in the whole cross section of the coaxial line or waveguide
transmission line. The main advantage of using two-port Nicholson-Ross-Weir (NRW) tech-
nique [10, 11] is that the both relative permittivity, εr and relative permeability, μr of the sample
can be predicted simultaneously. When using NRW method for thin samples, the thickness of
the sample must be less than λ/4.

2.3. Measurements setup

In this chapter, the dimensions of the used coaxial probe and the rectangular waveguide as
examples of the one-port measurement are shown in Figure 7a and b, respectively. The coaxial
probe is capable of measuring the reflection coefficients covered the frequency range between
0.5 and 7 GHz. On the other hand, the rectangular waveguide adapter covers frequency from
8.2 to 12.4 GHz. For two-port measurement, a 5 cm length of the coaxial and the rectangular
transmission lines is implemented. The experiment setup of the waveguides with an Agilent
E5071C vector network analyzer (VNA) is shown in Figure 8.

Figure 7. Cross-sectional and front views for the dimensions (in millimeter) of the (a) coaxial probe and the (b) rectangu-
lar waveguide.

Figure 8. The experiment setup for the (a) coaxial cavity and the (b) rectangular waveguide cavity.
 Materials Characterization Using Microwave Waveguide Systems 347
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3. Waveguide calibrations

3.1. One-port calibrations

3.1.1. Open standard calibration
The reflection coefficient S11a_sample of the sample at the probe aperture (at the BB′ plane)
should be measured as shown in Figure 9b [8]. However, during the measurement process,
only the reflection coefficient S11m_sample at the end of the coaxial line (at the AA′ plane) is
measured. The measured S11m_sample must be calibrated due to the reflection at the AA′ plane,
which is separated from the sample (at the BB′ plane) by a coaxial line. Thus, a de-embedding
process should be done to remove the effects of the coaxial line.

Figure 9. Error network and finite coaxial line: (a) terminated by air; (b) terminated by a sample under test.

In this subsection, a simple open standard calibration is introduced which requires the probe
aperture open to the air as shown in Figure 9a. Firstly, the S11m_air for the air is measured. Later,
the probe aperture is contacted with the sample under test, and its S11m_sample is measured as
shown in Figure 9b. The relationship between the S11m_air at the plane AA′ and S11a_air at the
probe aperture BB′ is expressed in a bilinear equation as:

S11m_air −e00
S11a_air ¼ (1)
e11 S11m_air þ e10 e01 −e00 e11

Similarly, the relationship between measured S11m_sample and S11a_sample is given as:

S11m_sample −e00
S11a_sample ¼ (2)
e11 S11m_sample þ e10 e01 −e00 e11

The e00, e11 and e10e01 are the unknown scattering parameters of the error network for the
coaxial line.
348 Microwave Systems and Applications

The e00 is the directivity error that causes the failure to receive the measured reflection signal
completely from the sample being tested at plane BB′. The e11 is the source matching error
due to the fact that the impedance of the aperture probe at plane BB′ is not exactly the
characteristic impedance (Zo = 50 Ω). The e10e01 is the frequency tracking imperfections (or
phase shift) between plane AA′ and sample test plane BB′. For this calibration, the e00 and e11
terms in Eqs. (1) and (2) are assumed to be vanished (e00 = e11 = 0). By dividing Eq. (2) into
Eq. (1), yields

S11a_sample S11m_sample
¼ (3)
S11a_air S11m_air

Once the S11m_air and S11m_sample are obtained, the actual reflection coefficient, S11a_sample, of the
sample at the probe aperture, BB′ can be found as:

S11a_air
S11a_sample ¼ · S11m_sample (4)
S11m_air

The standard values of the reflection coefficient, S11a_air in (4), can be calculated from Eq. (5)
that satisfying conditions: (DC < f < 24) GHz.

1−jðω=Y o ÞðCo f −1 þ C1 þ C2 f þ C3 f 2 Þ
S11a_air ¼ (5)
1 þ jðω=Y o ÞðCo f −1 þ C1 þ C2 f þ C3 f 2 Þ

Symbol ω = 2πf and Yo = [(2π)/ln(b/a)]√(εoεc/μoμr) are the angular frequency (in rad/s) and
characteristic admittance (in siemens), respectively. For instance, the complex values of the
Co, C1, C2 and C3 in (5) for Teflon-filled coaxial probe with 2a = 1.3 mm, 2b = 4.1 mm and εc =
2.06 are given as [12]:

Co ¼ 5:368082994761808 · 107 þj 2:320071598550666 · 107 ðF · HzÞ

C1 ¼ 3:002820660256831 · 1014 j 2:988971515445163 · 1016 ðFÞ

C2 ¼ 1:112989441958266 · 1025 þj 7:730261500907114 · 1026 ðF=HzÞ

C3 ¼ 3:140652268416283 · 1036 j 6:786433840933426 · 1036 ðF=Hz2 Þ

It should be noted that this simple calibration technique will not eliminate the standing wave
effects in the coaxial line.

3.1.2. Open-short-load (OSL) standard calibrations

In this subsection, the three-standard calibration is reviewed in which open, short and load
standards are used in the de-embedding process [13]. Let S11a_a, S11a_s and S11a_w represent the
known reflection coefficients for the open, short and load (water) standards which are termi-
nated at the aperture plane BB′, while S11m_a, S11m_s and S11m_w are the measured reflection
coefficients for open, short and load standards at plane AA′. The S11m_a is measured by leaving
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the open end of the probe in the air as shown in Figure 10a. Later, the measurement is repeated
to obtain the S11m_s by terminating the probe aperture with a metal plate as shown in Figure 10b.
Finally, the S11m_w is obtained by immersing the probe in water as shown in Figure 10c.

Figure 10. Finite coaxial line: (a) terminated by free space; (b) shorted by a metal plate; (c) immersed in water.

Once the complex values of S11a_air, S11a_short, S11a_water, S11m_a, S11m_s and S11m_w are known, the
three unknown complex coefficients (e00, e11, and e10e01) values in Eq. (2) can be found as:

S11a_s S11a_w S11m_a Δw_s þ S11a_o S11a_s S11m_w Δs_a þ S11a_a S11a_w S11m_s Δa_w
e00 ¼ (6a)
S11a_w S11a_s Δw_s þ S11a_a S11a_s Δs_a þ S11a_w S11a_a Δa_w

−ðS11a_a Δw_s þ S11a_w Δs_a þ S11a_s Δa_w Þ

e11 ¼ (6b)
S11a_w S11a_s Δw_s þ S11a_a S11a_s Δs_a þ S11a_w S11a_a Δa_w

S11a_a S11m_a Δw_s þ S11a_w S11m_w Δs_a þ S11a_s S11m_s Δa_w

e10 e01 ¼ ðe00 · e11 Þ þ (6c)
S11a_w S11a_s Δw_s þ S11a_a S11a_s Δs_a þ S11a_w S11a_a Δa_w

where

Δa_w ¼ S11m_a −S11m_w , Δs_a ¼ S11m_s −S11m_a , and Δw_s ¼ S11m_w −S11m_s

3.1.3. Short-offset-offset short (SOO) standard calibrations

The open-short-load (OSL) technique is rarely used in the one-port rectangular waveguide
calibration due to unavailable commercial open kit for the rectangular waveguide. In this
subsection, the short-offset-offset short (SOO) calibration [14, 15] is introduced for waveguide
calibration by using waveguide adjustable sliding shorts as shown in Figure 11. The calibra-
tion procedures are shown in Figure 12.
350 Microwave Systems and Applications

In this calibration method, the measured reflection coefficients for one shorted aperture and
two different lengths, l of offset short are required. Let S11m_1, S11m_2 and S11m_3 represent the
known measured reflection coefficients at plane AA′ for the shorted aperture and the two
offset shorts at location l1 and l2 from the waveguide aperture, respectively. Before calibration,
the selection of the appropriate offset short length, l1 and l2 will be an issue. The lengths of the
offset shorts can be determined by conditions:
1. The three phase shift between the S11m_1, S11m_2 and S11m_3 must not be equal:

∠ðS11m_2 Þ−∠ðS11m_1 Þ≠∠ðS11m_3 Þ−∠ðS11m_1 Þ≠∠ðS11m_3 Þ−∠ðS11m_2 Þ

2. The resolution degree between any three phase shift must be significant large (>100°) as
shown in Figure 13. In this work, the distance l1 and l2 for the offset shorts from the X-
band waveguide aperture are equal to 0.007 m and 0.013 m, respectively.

Figure 11. (a) Ku-band and X-band waveguide adjustable sliding shorter. (b) Connection between sliding short with
waveguide-to-coaxial adapter.

Figure 12. Calibration procedures of the aperture rectangular waveguide using an adjustable shorter. (a) Step 1; (b) Step
2; (c) Step 3
 Materials Characterization Using Microwave Waveguide Systems 351
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Figure 13. The three phase shift of the measured reflection coefficients for the shorted aperture and two offset shorts with
l1 = 0.7 cm and l2 = 1.3 cm, respectively.

Once the S11m_1, S11m_2, S11m_3, l1 and l2 are obtained, the three unknown complex coefficients
(e00, e11, and e10e01) values in Eq. (2) can be found as:



S11m_1 S11m_2 ðe−2γl1 −1Þ−S11m_2 S11m_3 e2γðl2 −l1 Þ −1 −S11m_1 S11m_3 e−2γl1 −e2γðl2 −l1 Þ
e00 ¼
 (7a)
ðe−2γl1 −1ÞðS11m_2 −S11m_3 Þ− e2γðl2 −l1 Þ −1 ðS11m_2 −S11m_1 Þ

e2γl1 ðS11m_2 −e00 Þ þ e00 −S11m_1

e11 ¼ (7b)
S11m_1 −S11m_2

e10 e01 ¼ ðe00 −S11m_1 Þð1 þ e11 Þ (7c)

The complex reflection coefficient, S11a_sample, at the waveguide aperture which is open to the
air was measured. Then, the measured S11a_sample was converted to normalized admittance, Y/
Yo parameter by a formula: Y/Yo = (1 − S11a_sample)/(1 + S11a_sample). The SOO calibration tech-
niques were validated by comparing normalized admittance, Y/Yo with the literature data [15–
22] as shown in Figure 14. The real part, Re(Y/Yo), and the imaginary part, Im(Y/Yo), of
admittance results were found to be in good agreement with literature data over the opera-
tional range of frequencies.

3.2. Two-port calibrations [through-reflect-line (TRL)]

The through-short-line (TRL) calibration model [23] is used for two-port rectangular wave-
guide measurement. The TRL technique requires three standards, which are through, short and
line measurements at the CC′ and DD′ planes so-called reference planes (at the front surface of
the sample under test) as shown in Figure 15.
The error coefficients (e00, e11, e10e01, e33, e22, e32, and e23) in Figure 15 can be obtained by solving
the matrix equation of Eq. (8).
352 Microwave Systems and Applications

Figure 14. Comparison of real part, Re(Y/Yo), and imaginary part, Im(Y/Yo), of the normalized admittance for air.
2 32 3 2 3
1 0 0 0 S12m_Thru 0 0 0 0 0 0 0 e00 S11m_Thru
60 S11m_Thru −1 0 0 0 −S12m_Thru 0 0 0 0 07 6 7 6 0 7
6 76 e11 7 6 7
60 0 0 0 S22m_Thru −1 0 0 0 0 0 07 6 Δx 7 6 S21m_Thru 7
6 76 7 6 7
60 S21m_Thru 0 1 0 0 −S22m_Thru 0 0 0 0 07 6 7 6 0 7
6 76 ke33 7 6 7
61 −S11m_Short 1 0 0 0 0 0 0 0 0 7
0 76 ke22 7 6 S11m_Short 7
76 6
6 7
60 0 0 0 −S12m_Short 0 −S12m_Short 0 0 0 0 07 6 7 6 0 7
6 76 kΔy 7 ¼ 6 7
60 −S21m_Short 0 0 0 0 0 0 0 0 0 7
0 76 k 7 6 S21m_Short 7
76 6
6 7
60 0 0 1 −S22m_Short 1 −S22m_Short 0 0 0 0 07 6 7 6 7
6 76 0 7 6 0 7
61 0 0 0 e−jβl S12m_Line 0 0 0 0 0 0 07 6 0 7 6 S11m_Line 7
6 76 7 6 7
60 ejβl S11m_Line ejβl 0 0 0 −S12m_Line 0 0 0 0 07 6 7 6 7
6 76 0 7 6 0 7
40 0 0 0 e−jβl S22m_Line −e−jβl 0 0 0 0 0 0 54 0 5 4 S21m_Line 5
0 ejβl S21m_Line 0 1 0 0 −S22m_Line 0 0 0 0 0 0 0
(8)

where k = e10/e23, Δx = e00e11–e10e01 and Δy = e22e33–e32e23. Once the S11m_sample, S21m_sample,

S12m_sample and S22m_sample at plane AA′ and BB′ for the sample under test are measured, the
calibrated reflection coefficient, S11a_sample at plane CC′ and transmission coefficient, S21a_sample
at plane DD′ can be calculated as:
8     9
> S11m_sample −e00 S22m_sample −e33 >
>
< 1þ e22 >
=
 e10 e01 e23 e32 
> −e22 S21m_sample −S21m_Thru
>
:
S12m_sample −S12m_Thru >>
;
e10 e32 e23 e01
S11a_sample ¼ (9a)
D


S21m_sample −S21m_Thru
e10 e32
S21a_sample ≅ (9b)
D

The denominator, D in (9a) and (9b) is given as:

 Materials Characterization Using Microwave Waveguide Systems 353
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Figure 15. through-short-line (TRL) calibration procedures and its network errors. (a) Through connection; (b) Reflect
connection; (c) Line connection.
354 Microwave Systems and Applications

8      9
>
> S11m_sample −e00 S22m_sample −e33 >
>
< 1 þ e11 1 þ e22 > >
=
e10 e01 e23 e32
D¼   
>
> S21m_sample −S21m_Thru S12m_sample −S12m_Thru >
>
:− e22 e11 >
>
;
e10 e32 e23 e01

4. Material parameters extraction

4.1. Reflection measurements (one-port measurements)

There are two methods of determining sample parameters (εr, μr or σ), which are the direct
method and the inverse method. The direct method involves the explicit model to predict the
sample under test based on the measured reflection coefficient, S11a_sample, while the inverse
method is implemented rigorous integral admittance model to estimate the sample parameters
(εr, μr or σ) using optimization procedures. For coaxial probe measurement cases, the explicit
relationship between εr and S11a_sample [8] is tabulated in Table 1. For rectangular waveguide
cases, the measured S11a_sample is transferred to normalized admittance, Ỹa_sample through equa-
tion: Ỹa_sample = (1 − S11a_sample)/(1 + S11a_sample). The predicted value of εr is obtained by mini-
mizing the difference between the measured normalized admittance, Ỹa_sample and the quasi-
static integral model, Ỹ (in Table 2) [9, 17] by referring to particular objective function. The
procedures of direct method are more straightforward than the inverse method. The detail
descriptions of the parameters (Yo, C and γo) and the coefficients (a1, a2 and a3) in Eqs. (10)–(13)
can be found in [8, 9, 17].

4.2. Reflection/transmission measurements (two-port measurements)

Conventionally, the complex εr = ε′r–jεr˝ and the μr = μ′r–jμr˝ of the sample filled in the coaxial or
rectangular waveguide are obtained by converting the calibrated reflection coefficient,
S11a_sample and the transmission coefficient, S21a_sample by using Nicholson-Ross-Weir (NRW)
routines [10, 11]. In this section, another alternative method, namely transmission phase shift
(TPS) method [24], is reviewed. The TPS method is a calibration-independent and material
position-invariant technique, which can reduce the complexity of the de-embedding proce-
dures. The important formulations of the NRW and the TPS methods are tabulated in Table 3.

Sample cases Open-ended coaxial probe

  
Semi-infinite space sample (Figure 5a) Yo 1−S11a_sample
εr ¼ (10)
jωC 1 þ S11a_sample

  
Yo 1−S11a_sample
Thin sample backed by metal plate (Figure 5b) εr ¼ ða1 þ a2 e−d=M þ a3 e−2d=M Þ (11)
jωC 1 þ S11a_sample

Table 1. Explicit formulations for open-ended coaxial probe.

 Materials Characterization Using Microwave Waveguide Systems 355
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Sample cases Open-ended rectangular waveguide

ða ðb ( n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )
Semi-infinite space
~ ¼ j8b πy πyo exp ð−jk1 x2 þ y2 Þ
sample (Figure 5c) Y ða−xÞ D1 ðb−yÞ cos þ D2 sin · pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy
aγo 0 0 b b x2 þ y2
(12)


2 
k21 π k1 π 2πf pffiffiffiffi
where D1 ¼ b12 ; D2 ¼ πb
4π − 4b2
1
4π þ 4b2 ; and k1 ¼ c εr
Thin sample backed ða ðb pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
~ ¼ j8b exp ð−jk1 x2 þ y2 Þ
by metal plate Y χ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy
(Figure 5d) aγo 0 0 x2 þ y2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða ðb
exp ð−jk1 x2 þ y2 þ 4n2 d2 Þ
j16b ∞
þ χ ∑ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxdy (13)
aγo n¼1
0 0 x2 þ y2 þ 4n2 d2
πy

where χ ¼ ða−xÞ D1 ðb−yÞ cos πy

b þ D2 sin b

Table 2. Integral admittance formulations for open-ended rectangular waveguide.

Waveguide factors Explicit equations

    
NRW method Coaxial: ζ ¼ 1 1−Γ c 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εr ¼ jζ ln (14a)
[10–11] 2ffi 1þΓ 2πf d T
Waveguide: ζ ¼ k2o − πb
   
1 1þΓ c 1
μr ¼ j ln (14b)
ζ 1−Γ 2πf d T

( 2 )
TPS method [24] Coaxial: ξ ¼ 0 and γo ¼ ko 1 φ21_air −φ21_sample
ε′ r ¼ γo þ þ ξ−α2 (15a)
k2o d

 
Waveguide: 2α φ21_air −φ21_sample
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi ε″ r ¼ γo þ (15b)
ξ ¼ πb γo ¼ k2o − πb k2o d

Table 3. Explicit formulations for reflection/transmission measurements.

where ko = 2πf/c is the propagation constant of free space (c = 2.99792458 ms−1); b (in meter) are
the width of the aperture of the waveguide, respectively; d(in meter) is the thickness of the
sample. The expressions for parameters Γ and T in Eqs. (14a, b) can be found in [10, 24]. The
φ21_air and φ21_sample in Eqs. (15a, b) are the measured phase shift of the transmission coefficient
356 Microwave Systems and Applications

in the air (without sample) and the sample, respectively. On the other hand, symbol α (in
nepers/meter) is the dielectric attenuation constant for the sample.
You et al. [24] have been mentioned that the uncertainty of the permittivity measurement is
high for the low-loss thin sample by using TPS method due to the decreasing of the sensitivity
for the transmitted wave through the thin sample, especially for transmitted waves that have
longer wavelengths. However, the literature [24] did not discuss how the thickness of the thin
sample may affect the uncertainty of measurement using TPS technique in quantitative. From
this reasons, the TPS method is reexamined in this section. Various thicknesses of acrylic, FR4
and RT/duroid 5880 substrate samples were placed in the X-band rectangular waveguide and
measured for validation. Figure 16a–c shows the predicted dielectric constant, εr′ of the
samples using Eq. (15a) at 8.494, 10.006 and 11.497 GHz, respectively. Clearly, the TPS method
is capable of providing a stable and accurate measurement for operating frequency in X-band
range when the thicknesses of the samples have exceeded 2 cm [25].

Figure 16. Variations in relative dielectric constant, εr with the thickness layer of (a) acrylic, (b) RT/duroid 5880 substrate
and (c) FR4, respectively.

5. Conclusion

The brief background of the microwave waveguide techniques for materials characterization is
reviewed and summarized. Not only that the measurement methods play an important role,
the calibration process is crucial as well. However, most of the literatures have ignored the
description of calibration. Measurement without calibration certainly cannot predict the prop-
erties of materials accurately. Thus, in this chapter, some of the waveguide calibrations are
described in detail.
 Materials Characterization Using Microwave Waveguide Systems 357
http://dx.doi.org/10.5772/66230

Author details

Kok Yeow You

Address all correspondence to: kyyou@fke.utm.my

Department of Communication Engineering, Faculty of Electrical Engineering, Universiti

Teknologi Malaysia, Johor, Malaysia

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