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Technology » "Microwave Systems and Applications", book edited by Sotirios K. Goudos, ISBN 978-953-51-2868-7, Print ISBN 978-953-51-2867-0, Published: January 11, 2017 under CC BY 3.0 license. © The Author(s).

Chapter 14

Materials Characterization Using Microwave Waveguide System

By Kok Yeow You
DOI: 10.5772/66230

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Overview

The resonator cavity filled with sample under test [1, 2].
Figure 1. The resonator cavity filled with sample under test [1, 2].
Free-space measurement setup for dielectric measurement of the thin sample [3–5].
Figure 2. Free-space measurement setup for dielectric measurement of the thin sample [3–5].
(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 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.
(a) WR510 waveguide-to-coaxial adapter and (b) WR 90, WR 75 and WR 62 waveguide-to-coaxial adapters [9].
Figure 4. (a) WR510 waveguide-to-coaxial adapter and (b) WR 90, WR 75 and WR 62 waveguide-to-coaxial adapters [9].
Open-port measurements using the (a), (b) coaxial probe and the (c), (d) rectangular waveguide.
Figure 5. Open-port measurements using the (a), (b) coaxial probe and the (c), (d) rectangular waveguide.
Two-port measurements using the (a) rectangular waveguide and the (b) coaxial transmission line.
Figure 6. Two-port measurements using the (a) rectangular waveguide and the (b) coaxial transmission line.
Cross-sectional and front views for the dimensions (in millimeter) of the (a) coaxial probe and the (b) rectangular waveguide.
Figure 7. Cross-sectional and front views for the dimensions (in millimeter) of the (a) coaxial probe and the (b) rectangular waveguide.
The experiment setup for the (a) coaxial cavity and the (b) rectangular waveguide cavity.
Figure 8. The experiment setup for the (a) coaxial cavity and the (b) rectangular waveguide cavity.
Error network and finite coaxial line: (a) terminated by air; (b) terminated by a sample under test.
Figure 9. Error network and finite coaxial line: (a) terminated by air; (b) terminated by a sample under test.
Finite coaxial line: (a) terminated by free space; (b) shorted by a metal plate; (c) immersed in water.
Figure 10. Finite coaxial line: (a) terminated by free space; (b) shorted by a metal plate; (c) immersed in water.
(a) Ku-band and X-band waveguide adjustable sliding shorter. (b) Connection between sliding short with waveguide-to-coaxial adapter.
Figure 11. (a) Ku-band and X-band waveguide adjustable sliding shorter. (b) Connection between sliding short with waveguide-to-coaxial adapter.
Calibration procedures of the aperture rectangular waveguide using an adjustable shorter. (a) Step 1; (b) Step 2; (c) Step 3.
Figure 12. Calibration procedures of the aperture rectangular waveguide using an adjustable shorter. (a) Step 1; (b) Step 2; (c) Step 3.
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.
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.
Comparison of real part, Re(Y/Yo), and imaginary part, Im(Y/Yo), of the normalized admittance for air.
Figure 14. Comparison of real part, Re(Y/Yo), and imaginary part, Im(Y/Yo), of the normalized admittance for air.
through-short-line (TRL) calibration procedures and its network errors. (a) Through connection; (b) Reflect connection; (c) Line connection.
Figure 15. through-short-line (TRL) calibration procedures and its network errors. (a) Through connection; (b) Reflect connection; (c) Line connection.
Variations in relative dielectric constant, εr with the thickness layer of (a) acrylic, (b) RT/duroid 5880 substrate and (c) FR4, respectively.
Figure 16. Variations in relative dielectric constant, εr with the thickness layer of (a) acrylic, (b) RT/duroid 5880 substrate and (c) FR4, respectively.

Materials Characterization Using Microwave Waveguide Systems

Kok Yeow You
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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 conductivity. 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 permittivity and complex permeability from measured reflection coefficient. For two-port measurement, 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 transmission coefficients between the air and the material.

Keywords: microwave waveguides, relative permittivity, relative permeability, conductivity, 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 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-frequency 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 resonant 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.

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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 [35]. 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 waveguides are implemented in transmission/reflection measurement techniques which are directly 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 considered in the selection of technique, such as accuracy, cost, samples shape and operating frequency. This chapter is focused only on coaxial and rectangular waveguides.

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Figure 2.

Free-space measurement setup for dielectric measurement of the thin sample [35].

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 dependent 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.

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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.

media/fig4.png

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 coefficient, 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.

media/fig5.png

Figure 5.

Open-port measurements using the (a), (b) coaxial probe and the (c), (d) rectangular waveguide.

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 rectangular 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.

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 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) technique [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.

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Figure 6.

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

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.

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Figure 7.

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

media/fig8.png

Figure 8.

The experiment setup for the (a) coaxial cavity and the (b) rectangular waveguide cavity.

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.

media/fig9.png

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:

S11a_air=S11m_aire00e11S11m_air+e10e01e00e11
(1)

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

S11a_sample=S11m_samplee00e11S11m_sample+e10e01e00e11
(2)

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

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_sampleS11a_air=S11m_sampleS11m_air
(3)

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_sample=S11a_airS11m_air×S11m_sample
(4)

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

S11a_air=1j(ω/Yo)(Cof1+C1+C2f+C3f2)1+j(ω/Yo)(Cof1+C1+C2f+C3f2)
(5)

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=5.368082994761808×10-7j2.320071598550666×10-7(F×Hz)
C1=3.002820660256831×10-14 - j 2.988971515445163×10-16(F)
C2=1.112989441958266×10-25 + j 7.730261500907114×10-26(F/Hz)
C3=3.140652268416283×10-36 - j 6.786433840933426×10-36(F/Hz2)

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 terminated 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 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.

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:

media/fig10.png

Figure 10.

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

e00=S11a_sS11a_wS11m_aΔw_s+S11a_oS11a_sS11m_wΔs_a+S11a_aS11a_wS11m_sΔa_wS11a_wS11a_sΔw_s+S11a_aS11a_sΔs_a+S11a_wS11a_aΔa_w
(6a)
e11=(S11a_aΔw_s+S11a_wΔs_a+S11a_sΔa_w)S11a_wS11a_sΔw_s+S11a_aS11a_sΔs_a+S11a_wS11a_aΔa_w
(6b)
e10e01=(e00×e11)+S11a_aS11m_aΔw_s+S11a_wS11m_wΔs_a+S11a_sS11m_sΔa_wS11a_wS11a_sΔw_s+S11a_aS11a_sΔs_a+S11a_wS11a_aΔa_w
(6c)

where

Δa_w=S11m_aS11m_w, Δs_a=S11m_sS11m_a, and Δw_s=S11m_wS11m_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 calibration procedures are shown in Figure 12.

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Figure 11.

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

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Figure 12.

Calibration procedures of the aperture rectangular waveguide using an adjustable shorter. (a) Step 1; (b) Step 2; (c) Step 3.

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:

    (S11m_2)(S11m_1)(S11m_3)(S11m_1)(S11m_3)(S11m_2)

  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.

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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:

e00=S11m_1S11m_2(e2γl11)S11m_2S11m_3(e2γ(l2l1)1)S11m_1S11m_3(e2γl1e2γ(l2l1))(e2γl11)(S11m_2S11m_3)(e2γ(l2l1)1)(S11m_2S11m_1)
(7a)
e11=e2γl1(S11m_2e00)+e00S11m_1S11m_1S11m_2
(7b)
e10e01=(e00S11m_1)(1+e11)
(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 techniques were validated by comparing normalized admittance, Y/Yo with the literature data [1522] 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 operational range of frequencies.

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Figure 14.

Comparison of real part, Re(Y/Yo), and imaginary part, Im(Y/Yo), of the normalized admittance for air.

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

The through-short-line (TRL) calibration model [23] is used for two-port rectangular waveguide 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.

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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.

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).

[1000S12m_Thru00000000S11m_Thru1000S12m_Thru000000000S22m_Thru10000000S21m_Thru0100S22m_Thru000001S11m_Short10000000000000S12m_Short0S12m_Short000000S21m_Short00000000000001S22m_Short1S22m_Short000001000ejβlS12m_Line00000000ejβlS11m_Lineejβl000S12m_Line000000000ejβlS22m_Lineejβl0000000ejβlS21m_Line0100S22m_Line00000][e00e11Δxke33ke22kΔyk00000]=[S11m_Thru0S21m_Thru0S11m_Short0S21m_Short0S11m_Line0S21m_Line0]
(8)

where k = e10/e23, Δx = e00e11e10e01 and Δy = e22e33e32e23. 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:

S11a_sample={(S11m_samplee00e10e01)[1+(S22m_samplee33e23e32)e22]e22(S21m_sampleS21m_Thrue10e32)(S12m_sampleS12m_Thrue23e01)}D
(9a)
S21a_sample(S21m_sampleS21m_Thrue10e32)D
(9b)

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

D={[1+(S11m_samplee00e10e01)e11][1+(S22m_samplee33e23e32)e22](S21m_sampleS21m_Thrue10e32)(S12m_sampleS12m_Thrue23e01)e22e11}

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, a_sample through equation: a_sample = (1 S11a_sample)/(1 + S11a_sample). The predicted value of εr is obtained by minimizing the difference between the measured normalized admittance, a_sample and the quasi-static integral model, (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].

Sample casesOpen-ended coaxial probe
Semi-infinite space sample (Figure 5a)
εr=(YojωC)(1S11a_sample1+S11a_sample)
(10)
Thin sample backed by metal plate (Figure 5b)
εr=(YojωC)(1S11a_sample1+S11a_sample)(a1+a2ed/M+a3e2d/M)
(11)

Table 1.

Explicit formulations for open-ended coaxial probe.

Sample casesOpen-ended rectangular waveguide
Semi-infinite space sample (Figure 5c)
Y˜=j8baγo0a0b{(ax){D1(by)cosπyb+D2sinπyb}×exp(jk1x2+y2)x2+y2}dxdy
(12)
where D1=1b2(k124ππ4b2), D2=1πb(k124π+π4b2), and k1=2πfcεr
Thin sample backed by metal plate (Figure 5d)
Y˜=j8baγo0a0bχexp(jk1x2+y2)x2+y2dxdy+j16baγon=10a0bχexp(jk1x2+y2+4n2d2)x2+y2+4n2d2dxdy
(13)

where
χ=(ax){D1(by)cosπyb+D2sinπyb}

Table 2.

Integral admittance formulations for open-ended rectangular waveguide.

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

Conventionally, the complex εr = ε′rr˝ and the μr = μrr˝ 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 procedures. The important formulations of the NRW and the TPS methods are tabulated in Table 3.

Waveguide factorsExplicit equations
NRW method [1011]Coaxial: ζ=1
εr=jζ(1Γ1+Γ)(c2πfd)ln(1T)
(14a)

μr=j1ζ(1+Γ1Γ){c2πfdln(1T)}
(14b)
Waveguide: ζ=ko2(πb)2
TPS method [24]Coaxial: ξ=0 and γo=ko
εr=1ko2{(γo+ϕ21_airϕ21_sampled)2+ξα2}
(15a)
Waveguide: ξ=(πb)2
γo=ko2(πb)2
εr=2αko2(γo+ϕ21_airϕ21_sampled)
(15b)

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 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 16ac shows the predicted dielectric constant, εr of the samples using Eq. (115a) 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].

media/fig16.png

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 properties of materials accurately. Thus, in this chapter, some of the waveguide calibrations are described in detail.

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