## 1. Introduction

For macroscopic material characteristic investigations, three properties of the material are often measured: relative permittivity *ε*_{r}, relative permeability *μ*_{r} and conductivity *σ*. Normally, many microwave measurements only focus on the properties of relative permittivity, *ε*_{r} rather than the permeability *μ*_{r} 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, tan*δ* values for the low-loss sample, but such techniques cannot be used for the measurement of swept frequency.

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 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.

## 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, TE_{10} 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.

### 2.2. Measurements principles

The one-port measurement is based on the principle that a reflected signal (reflection coefficient, *S*_{11}) 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, *ε*_{r}, of the dielectric material (nonmagnetic material, *μ*_{r}* *= 1). This is due to insufficient information to predict the permeability, *μ*_{r}, if only obtained the measured reflection coefficient, *S*_{11} without transmission coefficient, *S*_{21}.

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, *ε*_{r }and permeability, *μ*_{r }of materials from reflection and transmission coefficients (*S*_{11} and *S*_{21}). 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, *ε*_{r} and relative permeability, *μ*_{r }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**.

## 3. Waveguide calibrations

### 3.1. One-port calibrations

#### 3.1.1. Open standard calibration

The reflection coefficient *S*_{11a_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 *S*_{11m_sample} at the end of the coaxial line (at the *AA′* plane) is measured. The measured *S*_{11m_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.

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 *S*_{11m_air }for the air is measured. Later, the probe aperture is contacted with the sample under test, and its *S*_{11m_sample} is measured as shown in **Figure 9b**. The relationship between the *S*_{11m_air} at the plane *AA′* and *S*_{11a_air} at the probe aperture *BB′* is expressed in a bilinear equation as:

Similarly, the relationship between measured *S*_{11m_sample} and *S*_{11a_sample} is given as:

The *e*_{00}, *e*_{11} and *e*_{10}*e*_{01} are the unknown scattering parameters of the error network for the coaxial line.

The *e*_{00} is the directivity error that causes the failure to receive the measured reflection signal completely from the sample being tested at plane *BB ′*. The

*e*

_{11}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 (

**′***Z*

_{o}= 50 Ω). The

*e*

_{10}

*e*

_{01}is the frequency tracking imperfections (or phase shift) between plane

*AA*and sample test plane

**′***BB*. For this calibration, the

**′***e*

_{00}and

*e*

_{11 }terms in Eqs. (1) and (2) are assumed to be vanished (

*e*

_{00}=

*e*

_{11}= 0). By dividing Eq. (2) into Eq. (1), yields

Once the *S*_{11m_air} and *S*_{11m_sample} are obtained, the actual reflection coefficient, *S*_{11a_sample}, of the sample at the probe aperture, *BB ′* can be found as:

The standard values of the reflection coefficient, *S*_{11a_air} in (4), can be calculated from Eq. (5) that satisfying conditions: (DC < *f* < 24) GHz.

Symbol ω = 2π*f* and *Y*_{o} = [(2*π*)/ln(*b*/*a*)]√(*ε*_{o}*ε*_{c}/*μ*_{o}*μ*_{r}) are the angular frequency (in rad/s) and characteristic admittance (in siemens), respectively. For instance, the complex values of the *C*_{o}, *C*_{1}, *C*_{2} and *C*_{3} in (5) for Teflon-filled coaxial probe with 2*a* = 1.3 mm, 2*b* = 4.1 mm and *ε*_{c}* *= 2.06 are given as [12]:

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 *S*_{11a_a}, *S*_{11a_s} and *S*_{11a_w} represent the known reflection coefficients for the open, short and load (water) standards which are terminated at the aperture plane *BB ′*, while

*S*

_{11m_a},

*S*

_{11m_s}and

*S*

_{11m_w}are the measured reflection coefficients for open, short and load standards at plane

*AA*. The

**′***S*

_{11m_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

*S*

_{11m_s}by terminating the probe aperture with a metal plate as shown in

**Figure 10b**. Finally, the

*S*

_{11m_w}is obtained by immersing the probe in water as shown in

**Figure 10c**.

Once the complex values of *S*_{11a_air}, *S*_{11a_short}, *S*_{11a_water}, *S*_{11m_a}, *S*_{11m_s} and *S*_{11m_w} are known, the three unknown complex coefficients (*e*_{00}, *e*_{11}, and *e*_{10}*e*_{01}) values in Eq. (2) can be found as:

(6a) |

(6c) |

where

#### 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**.

In this calibration method, the measured reflection coefficients for one shorted aperture and two different lengths, *l* of offset short are required. Let *S*_{11m_1}, *S*_{11m_2} and *S*_{11m_3} represent the known measured reflection coefficients at plane *AA′* for the shorted aperture and the two offset shorts at location *l*_{1} and *l*_{2} from the waveguide aperture, respectively. Before calibration, the selection of the appropriate offset short length, *l*_{1} and *l*_{2} will be an issue. The lengths of the offset shorts can be determined by conditions:

The three phase shift between the

*S*_{11m_1},*S*_{11m_2}and*S*_{11m_3 }must not be equal:The resolution degree between any three phase shift must be significant large (>100°) as shown in

**Figure 13**. In this work, the distance*l*_{1}and*l*_{2}for the offset shorts from the X-band waveguide aperture are equal to 0.007 m and 0.013 m, respectively.

Once the *S*_{11m_1}, *S*_{11m_2}, *S*_{11m_3}, *l*_{1} and *l*_{2} are obtained, the three unknown complex coefficients (*e*_{00}, *e*_{11}, and *e*_{10}*e*_{01}) values in Eq. (2) can be found as:

(7a) |

The complex reflection coefficient, *S*_{11a_sample}, at the waveguide aperture which is open to the air was measured. Then, the measured *S*_{11a_sample} was converted to normalized admittance, *Y*/*Y*_{o} parameter by a formula: *Y*/*Y*_{o} = (1* *− *S*_{11a_sample})/(1 +* S*_{11a_sample}). The SOO calibration techniques were validated by comparing normalized admittance, *Y*/*Y*_{o} with the literature data [15–22] as shown in **Figure 14**. The real part, Re(*Y/Y*_{o}), and the imaginary part, Im(*Y/Y*_{o}), of admittance results were found to be in good agreement with literature data over the operational 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 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**.

The error coefficients (*e*_{00}, *e*_{11}, *e*_{10}*e*_{01}, *e*_{33}, *e*_{22}, *e*_{32}, and *e*_{23}) in **Figure 15** can be obtained by solving the matrix equation of Eq. (8).

(8) |

where *k *= *e*_{10}/*e*_{23}, Δ_{x} = *e*_{00}*e*_{11}–*e*_{10}*e*_{01 }and Δ_{y} = *e*_{22}*e*_{33}–*e*_{32}*e*_{23}. Once the *S*_{11m_sample}, *S*_{21m_sample}, *S*_{12m_sample} and *S*_{22m_sample }at plane *AA ′* and

*BB*for the sample under test are measured, the calibrated reflection coefficient,

**′***S*

_{11a_sample }at plane

*CC*and transmission coefficient,

**′***S*

_{21a_sample}at plane

*DD*can be calculated as:

**′**## 4. Material parameters extraction

### 4.1. Reflection measurements (one-port measurements)

There are two methods of determining sample parameters (*ε*_{r}, *μ*_{r} 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, *S*_{11a_sample}, while the inverse method is implemented rigorous integral admittance model to estimate the sample parameters(*ε*_{r}, *μ*_{r} or *σ*) using optimization procedures. For coaxial probe measurement cases, the explicit relationship between *ε*_{r} and *S*_{11a_sample }[8] is tabulated in **Table 1**. For rectangular waveguide cases, the measured *S*_{11a_sample} is transferred to normalized admittance, *Ỹ*_{a_sample }through equation: *Ỹ*_{a_sample} = (1* *− *S*_{11a_sample})/(1 +* S*_{11a_sample}). The predicted value of *ε*_{r }is obtained by minimizing the difference between the measured normalized admittance, *Ỹ*_{a_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 (*Y*_{o}, *C* and *γ*_{o}) and the coefficients (*a*_{1}, *a*_{2} and *a*_{3}) in Eqs. (10)–(13) can be found in [8, 9, 17].

Sample cases | Open-ended coaxial probe |
---|---|

Semi-infinite space sample (Figure 5a) | |

Thin sample backed by metal plate (Figure 5b) |

Sample cases | Open-ended rectangular waveguide |
---|---|

Semi-infinite space sample (Figure 5c) | where |

Thin sample backed by metal plate (Figure 5d) | where |

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

Conventionally, the complex *ε*_{r} = ε′_{r}–*jε*_{r}˝ and the *μ*_{r} = *μ*′_{r}–*jμ*_{r}˝ of the sample filled in the coaxial or rectangular waveguide are obtained by converting the calibrated reflection coefficient, *S*_{11a_sample} and the transmission coefficient, *S*_{21a_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 factors | Explicit equations | |
---|---|---|

NRW method [10–11] | Coaxial: | |

Waveguide: | ||

TPS method [24] | Coaxial: | |

Waveguide: |

where *k*_{o} = 2π*f*/*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 *ϕ*_{21_air} and *ϕ*_{21_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 16a**–**c** shows the predicted dielectric constant, *ε*_{r}** ′** 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].

## 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.