2. Phase-shift method

touched or not), destructive or nondestructive (depends on whether sample can be destroyed or not), narrowband and wideband (depends on frequency range), etc. Since each method has its own advantages and limitations, the selection of the appropriate one depends on a particular application, required accuracy, sample, and other factors. There are a number of commercially available holders, kits, and probes that operate on different principles and allow measurement of the dielectric constant of the material in different forms on different temperatures and frequency ranges [8, 11]. However, the most of them are designed to be connected with expensive instruments such as network analyzers, LCR meters, or impedance

One of the commonly used methods suitable for the material characterization in a wide frequency range, from around 10 MHz to 75 GHz, is the TL method. The TL method includes both measurements of the reflection and/or transmission characteristic [3–10]. This method can be used for characterization of permittivity as well as permeability of hard solid materials with medium losses. High-loss materials can be also characterized using this method, if the sample is kept relatively thin. The TL holders are usually made of a coaxial, a waveguide, or a microstrip line section. However, the specific design of the holder or specific multilayered configurations can be used for characterization of powders, liquid, or gases. Usually, the method requires initial sample preparation to fit into the section of the TL, typically the waveguide or the coaxial line. For accurate permittivity measurement, the sample has to be exposed to the maximal electric field, and therefore the position of

the sample is very important. A typical measurement configuration of this method consists of the TL section with a sample placed inside, a vector network analyzer (VNA) used to measure the two ports complex scattering parameters (S-parameters), and a software that converts the measured S-parameters to the complex permittivity or permeability. In addition, the TL method requires initial

The measurement of the phase shift of the transmitted signal represents relatively fast and simple method for determination of the dielectric properties of the material. It is characterized by fast time response, and in comparison with other methods, it is less sensitive to the noise [10, 12]. Furthermore, this method allows characterization at a single frequency which simplifies the development of a supporting electronic, allows easy integration with sensor element, and allows realization of low-cost in-field sensing devices. Therefore, it found application in the realization of different types of sensors such as soil moisture sensors [13], microfluidic sensor for detention fluid mixture concentration [14], etc.

In this chapter, the phase-shift method will be explained on the example of a microstrip line configuration, and the permittivity of the materials will be determined by measuring the phase shift of the transmitted signal. Theoretical background of the phase-shift method and mathematical equations for determination of real and imaginary part of complex permittivity based on the phase of the transmitted signal will be presented in Section 2. The unknown permittivity of material will be determined for several measurement configurations in Section 3.

Potential of the phase-shift method will be demonstrated through several applications in the characterization of an unknown dielectric constant in multilayered structure, a soil moisture sensor, and sensor for determination of fluid properties in microfluidic channel. Advance techniques for increasing the sensitivity of the phase-shift measurement will be presented in Section 4, while the simple in-field detection device for determination of the permittivity based on the phase measurement will be presented in Section 5. The conclusions are given in

calibration with various terminations before the measurement.

analyzers.

Electromagnetic Materials and Devices

Section 6.

70

#### 2.1 Determination of real part of the dielectric constant

Phase-shift method is based on the measurement of the phase shift of a sinusoidal signal that propagates along a transmission line.

Phase shift Δφ is defined by velocity and frequency of the propagating signal as well as physical properties of the transmission line:

$$
\Delta \rho = \frac{a L\_{\rm TL}}{v\_p},
\tag{1}
$$

where ω is the angular frequency, vp is the phase velocity, and LTL is the length of transmission line.

In order to determine a phase velocity of electromagnetic wave, we will start with the expression for imaginary part of the complex propagation constant for lossy medium [15]:

$$\beta = \frac{\alpha \sqrt{\mu \varepsilon}}{\sqrt{2}} \sqrt{\mathbf{1} + \sqrt{\mathbf{1} + \frac{\sigma^2}{\alpha^2 \varepsilon^2}}} \tag{2}$$

where μ, ε, and σ are the real parts of the permeability, permittivity, and electrical conductivity of the medium through which the signal is propagating, respectively. If the imaginary part of the complex propagation constant is known, the phase velocity can be determined as

$$\upsilon\_p = \frac{\alpha}{\beta} = \frac{\sqrt{2}}{\sqrt{\mu \varepsilon}} \frac{1}{\sqrt{1 + \sqrt{1 + \frac{\sigma^2}{a^2 \varepsilon^2}}}}. \tag{3}$$

Based on Eq. (3), it can be seen that phase velocity is dominantly influenced by permittivity, permeability and signal frequency, and then electrical conductivity.

The main advantage of the phase-shift method lies on the fact that on the frequencies high enough, the influence of conductivity can be neglected:

$$\frac{\sigma^2}{\alpha^2 \,\,\sigma^2} \ll 1;\tag{4}$$

therefore, expression for velocity on high frequencies can be reduced to

$$v\_p = \frac{1}{\sqrt{\mu\varepsilon}},\tag{5}$$

where phase velocity is determined by permeability and permittivity only. This is especially important for soil moisture sensor which will be discussed later since most of the materials in the soil are diamagnetic or paramagnetic. If we assume that electromagnetic wave propagates through a nonmagnetic medium, that is, the magnetic permeability is equal to <sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup><sup>π</sup> <sup>∙</sup> <sup>10</sup>�<sup>7</sup> H/m, the phase velocity and the phase shift are dependent on dielectric permittivity only:

$$
\Delta \rho = a \mathcal{L}\_{\rm TL} \sqrt{\mu\_0 \varepsilon} = a \mathcal{L}\_{\rm TL} \sqrt{\mu\_0 \varepsilon\_0 \varepsilon\_r} = \frac{a \mathcal{L}\_{\rm TL}}{\varepsilon\_0} \sqrt{\varepsilon\_r} \tag{6}
$$

where c<sup>0</sup> is the speed of the light in the vacuum and ε<sup>r</sup> is a relative dielectric constant of the medium.

Unlike the coaxial line or the waveguide, the microstrip structure is particularly interesting for a simple and low-cost fabrication of compact sensors based on dielectric permittivity change which can be easily integrated with supporting electronics. In this way, standard printed circuit board technology used for microstrip manufacturing offers fabrication of a complete solution of the sensor in a single substrate. Therefore, the concept of the phase-shift measurement will be explained on the example of microstrip line. In the microstrip, the influence of the change in permittivity is reflected in effective permittivity which is determined by permittivity of a medium above the microstrip line and permittivity of the dielectric substrate, Figure 1.

Effective permittivity of the microstrip line shown in Figure 1 can be expressed as

$$
\varepsilon\_{\ell\overline{\mathcal{V}}} = \frac{\varepsilon\_{\mathfrak{s}} + \varepsilon\_{m}}{2} + \frac{\varepsilon\_{\mathfrak{s}} - \varepsilon\_{m}}{2} \frac{1}{\sqrt{1 + 12\frac{h}{w}}},\tag{7}
$$

and capacitance per unit length, C<sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81790

while the phase-shift range is

shift range can be reduced to

of the phase shift.

measured frequency range.

[18–22].

73

signal that propagates along microstrip can be defined as [17]

Δφ ¼ ωLTL

<sup>Δ</sup><sup>Φ</sup> <sup>¼</sup> <sup>Δ</sup>φmax � <sup>Δ</sup>φmin <sup>¼</sup> <sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

relative permittivities are equal to 1, and effective permittivity

<sup>Δ</sup><sup>Φ</sup> <sup>¼</sup> <sup>ω</sup> ffiffiffiffiffiffiffiffiffi

LC<sup>0</sup> p ffiffiffiffiffiffiffiffiffiffiffiffiffi

2.2 Determination of the imaginary part of the dielectric constant

Based on Eq. (13), it can be concluded that the phase-shift range can be increased by performing the measurements on higher frequencies or by increasing the total microstrip line inductance or capacitance. Therefore, the phase-shift range can be optimized to the capabilities of the supporting electronics for measurement

To validate the proposed method for the characterization of the material properties in wider frequency range, we compare the results of the dielectric constant calculated using proposed method with a predetermined value of dielectric constant, Figure 2. For the comparison, the microstrip transmission line was used where the unknown medium was placed above the microstrip line and dielectric substrate, as shown in Figure 1. The calculated dielectric constant was extracted from the phase shift of the transited signal using described procedure. The proposed result reveals that the phase-shift detection method provides high accuracy with the relative errors lower than 0.5% for the real parts of the dielectric constant in the

From the previous analysis, we demonstrate how the real part of the complex permittivity can be determined from the phase shift of the transmitted signal. However, the imaginary part of the complex permittivity cannot be directly calculated from the phase shift. It can be estimated using Kramers-Kronig (K-K) relation

The real and imaginary part of the complex dielectric constant is correlated with

K-K relation [18–20]. This relation is a direct consequence of the principle of

microstrip line with length LTL can be expressed as

vp <sup>¼</sup> <sup>1</sup> L0

Phase-Shift Transmission Line Method for Permittivity Measurement and Its Potential in Sensor…

ffiffiffiffiffiffiffiffiffi L0 <sup>C</sup><sup>0</sup> <sup>p</sup>

The capacitance can be written in the form of the vacuum capacitance C0, which represents the capacitance of the microstrip line when both substrate and medium

<sup>C</sup> <sup>¼</sup> <sup>C</sup>0εeff . : (12)

In addition, if we assume that the electromagnetic wave propagates through a nonmagnetic medium, the inductivity of the transmission line does not depend on the sample under test (Lmax ¼ Lmin ¼ L). Therefore, the expression for the phase-

<sup>ε</sup>eff max <sup>p</sup> � ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>¼</sup> <sup>ω</sup> ffiffiffiffiffiffi

LmaxCmax <sup>p</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

LminCmin � � <sup>p</sup> : (11)

<sup>ε</sup>eff min � � <sup>p</sup> : (13)

Combining Eqs. (1) and (9), phase shift of the signal propagating along

, on high frequencies, the phase velocity of the

<sup>C</sup><sup>0</sup> : (9)

LC <sup>p</sup> , (10)

where ε<sup>r</sup> and ε<sup>m</sup> are the permittivity of dielectric substrate and the medium above the line, respectively, h is the height of the substrate, and w is the width of the microstrip line [16].

It can be seen that a variation in the permittivity of the medium causes a variation in effective permittivity that results in the change of the phase velocity. The change in the phase velocity changes the phase shift of the signal. It is evident that phase shift is determined by the value of the permittivity of the medium above the microstrip line. Therefore, the real part of dielectric constant of unknown medium can be detected with simple measurement of the phase shift of the transmitted signal.

The range of the phase shift ΔΦ, is determined by the upper and lower values of the measured permittivity:

$$
\Delta\Phi = \Delta\rho\_{\text{max}} - \Delta\rho\_{\text{min}} = \alpha L\_{\text{TL}} \sqrt{\mu\_0 \varepsilon\_0} \left(\sqrt{\varepsilon\_{\text{eff}\_{\text{max}}}} - \sqrt{\varepsilon\_{\text{eff}\_{\text{min}}}}\right). \tag{8}
$$

This parameter needs to be adjusted to the supporting electronics that measure the phase shift. From Eq. (8) it can be seen that with appropriate choice of the operating frequency and the proper optimization of the geometrical parameters (mostly the length of the transmission line), the range of the phase shift can be optimized to the maximal measurable value.

As stated above, the phase shift depends on the properties of the transmission line. Since the microstrip line can be characterized by inductance per unit length, L<sup>0</sup> ,

Figure 1. Configuration of the microstrip line.

Phase-Shift Transmission Line Method for Permittivity Measurement and Its Potential in Sensor… DOI: http://dx.doi.org/10.5772/intechopen.81790

and capacitance per unit length, C<sup>0</sup> , on high frequencies, the phase velocity of the signal that propagates along microstrip can be defined as [17]

$$v\_p = \frac{1}{L'C'}.\tag{9}$$

Combining Eqs. (1) and (9), phase shift of the signal propagating along microstrip line with length LTL can be expressed as

$$
\Delta \rho = a \text{L}\_{\text{TL}} \sqrt{L' \text{C}'} = a \sqrt{L \text{C}}, \tag{10}
$$

while the phase-shift range is

where c<sup>0</sup> is the speed of the light in the vacuum and ε<sup>r</sup> is a relative dielectric

interesting for a simple and low-cost fabrication of compact sensors based on dielectric permittivity change which can be easily integrated with supporting electronics. In this way, standard printed circuit board technology used for microstrip manufacturing offers fabrication of a complete solution of the sensor in a single substrate. Therefore, the concept of the phase-shift measurement will be explained on the example of microstrip line. In the microstrip, the influence of the change in permittivity is reflected in effective permittivity which is determined by permittivity of a medium above the microstrip line and permittivity of the dielectric sub-

Unlike the coaxial line or the waveguide, the microstrip structure is particularly

Effective permittivity of the microstrip line shown in Figure 1 can be expressed as

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>12</sup> <sup>h</sup> w

<sup>q</sup> , (7)

εeff min � � <sup>p</sup> : (8)

,

<sup>2</sup> <sup>þ</sup> <sup>ε</sup><sup>s</sup> � <sup>ε</sup><sup>m</sup> 2

where ε<sup>r</sup> and ε<sup>m</sup> are the permittivity of dielectric substrate and the medium above the line, respectively, h is the height of the substrate, and w is the width of

It can be seen that a variation in the permittivity of the medium causes a variation in effective permittivity that results in the change of the phase velocity. The change in the phase velocity changes the phase shift of the signal. It is evident that phase shift is determined by the value of the permittivity of the medium above the microstrip line. Therefore, the real part of dielectric constant of unknown medium can be detected with simple measurement of the phase shift of the trans-

The range of the phase shift ΔΦ, is determined by the upper and lower values of

This parameter needs to be adjusted to the supporting electronics that measure the phase shift. From Eq. (8) it can be seen that with appropriate choice of the operating frequency and the proper optimization of the geometrical parameters (mostly the length of the transmission line), the range of the phase shift can be

As stated above, the phase shift depends on the properties of the transmission line. Since the microstrip line can be characterized by inductance per unit length, L<sup>0</sup>

ffiffiffiffiffiffiffiffiffi μ0ε<sup>0</sup> p ffiffiffiffiffiffiffiffiffiffiffiffiffi

εeff max p � ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>ε</sup>eff <sup>¼</sup> <sup>ε</sup><sup>s</sup> <sup>þ</sup> <sup>ε</sup><sup>m</sup>

ΔΦ ¼ Δφmax � Δφmin ¼ ωLTL

optimized to the maximal measurable value.

constant of the medium.

Electromagnetic Materials and Devices

strate, Figure 1.

the microstrip line [16].

the measured permittivity:

mitted signal.

Figure 1.

72

Configuration of the microstrip line.

$$
\Delta\Phi = \Delta\rho\_{\text{max}} - \Delta\rho\_{\text{min}} = \alpha \left( \sqrt{L\_{\text{max}} C\_{\text{max}}} - \sqrt{L\_{\text{min}} C\_{\text{min}}} \right). \tag{11}
$$

The capacitance can be written in the form of the vacuum capacitance C0, which represents the capacitance of the microstrip line when both substrate and medium relative permittivities are equal to 1, and effective permittivity

$$\mathbf{C} = \mathbf{C}\_0 \mathbf{e}\_{\mathbf{f}\mathbf{f}}.\tag{12}$$

In addition, if we assume that the electromagnetic wave propagates through a nonmagnetic medium, the inductivity of the transmission line does not depend on the sample under test (Lmax ¼ Lmin ¼ L). Therefore, the expression for the phaseshift range can be reduced to

$$
\Delta\Phi = \alpha \sqrt{LC\_0} \left( \sqrt{\varepsilon\_{\text{eff}\_{\text{max}}}} - \sqrt{\varepsilon\_{\text{eff}\_{\text{min}}}} \right). \tag{13}
$$

Based on Eq. (13), it can be concluded that the phase-shift range can be increased by performing the measurements on higher frequencies or by increasing the total microstrip line inductance or capacitance. Therefore, the phase-shift range can be optimized to the capabilities of the supporting electronics for measurement of the phase shift.

To validate the proposed method for the characterization of the material properties in wider frequency range, we compare the results of the dielectric constant calculated using proposed method with a predetermined value of dielectric constant, Figure 2. For the comparison, the microstrip transmission line was used where the unknown medium was placed above the microstrip line and dielectric substrate, as shown in Figure 1. The calculated dielectric constant was extracted from the phase shift of the transited signal using described procedure. The proposed result reveals that the phase-shift detection method provides high accuracy with the relative errors lower than 0.5% for the real parts of the dielectric constant in the measured frequency range.

#### 2.2 Determination of the imaginary part of the dielectric constant

From the previous analysis, we demonstrate how the real part of the complex permittivity can be determined from the phase shift of the transmitted signal. However, the imaginary part of the complex permittivity cannot be directly calculated from the phase shift. It can be estimated using Kramers-Kronig (K-K) relation [18–22].

The real and imaginary part of the complex dielectric constant is correlated with K-K relation [18–20]. This relation is a direct consequence of the principle of

#### Figure 2.

Real part of the dielectric constant calculated using the phase-shift method compared with the predetermined value.

causality [21]. K-K relation describes a fundamental correlation between the real and imaginary part of the complex dielectric constant and allows us to retrieve imaginary part of the dielectric constant from the measured real part or vice versa.

The imaginary part of the complex dielectric constant can be calculated as

$$\epsilon''(\boldsymbol{\alpha}) = -\frac{2\boldsymbol{\alpha}}{\pi} \not\!\!\!\!\!\rho \int\_0^\infty \frac{\epsilon'(\boldsymbol{\Omega}) - 1}{\boldsymbol{\Omega}^2 - \boldsymbol{\alpha}^2} d\boldsymbol{\Omega},\tag{14}$$

agreement with the predetermined value of imaginary parts. However, the relative error is about 9% since the accuracy of Eq. (14) depends on the measured frequency range. It can be noted that the range from 0 to ∞ in the integral in Eq. (14) is not achievable in practice. Therefore, the limited range of frequencies affects the accu-

Phase-Shift Transmission Line Method for Permittivity Measurement and Its Potential in Sensor…

In a previous section, we presented how the real and imaginary parts of the dielectric constant of the conventional microstrip configuration can be determined using the phase-shift method based on the phase of the transmitted signal. The same method can be applied for nonhomogeneous substrate such as multilay-

microstrip multilayered substrate configurations interesting for the realization of different sensor topologies. The phase-shift method will be used for the calculation of the effective dielectric constant and determination of the real part of dielectric constant of individual layers. Similarly, the imaginary part of the complex permittivity can be determined from real part using procedure described in

If we assume that the microstrip transmission line is realized on a multilayered substrate consisting of N layers with different dielectric constants, the effective dielectric constant, εs, of the multilayer substrate with N layers can be calculated

> <sup>i</sup>¼<sup>1</sup> di j j ∑<sup>N</sup> i¼1 di εi

where N is the number of the layers, ε<sup>i</sup> is the dielectric constant of the i-th layer,

where K and K<sup>0</sup> = K(ki) are the complete elliptical integrals of the first kind [24]

cosh <sup>π</sup><sup>w</sup> 4∑<sup>N</sup> <sup>i</sup>¼<sup>1</sup>hi

where w is the width of the microstrip line and hi is the thickness of the i-th layer. If all geometrical parameters are known as well as the dielectric constants of all layers except of one arbitrary layer, this unknown dielectric constant can be determined based on effective dielectric constant of the multilayered substrate. Previously, the effective dielectric constant has to be determined from the phase

If the parameters of several layers are unknown, the unknown values of the dielectric constants can be found by solving system of equations. This procedure requires several measurements with different sets of geometrical parameters, typically the length of the microstrip line. In that case, the number of the essential measurements required for the determination of all parameter depends on the

<sup>⋯</sup> � K kð Þ<sup>1</sup> K0 ð Þ k<sup>1</sup>

, (15)

, (17)

, (16)

<sup>ε</sup><sup>s</sup> <sup>¼</sup> <sup>∑</sup><sup>N</sup>

and di is a coefficient which can be calculated using the following equation:

ki <sup>¼</sup> <sup>1</sup>

� K kð Þ <sup>i</sup>�<sup>1</sup> K0 ð Þ ki�<sup>1</sup>

di <sup>¼</sup> K kð Þ<sup>i</sup> K0 ð Þ ki

racy of the results and causes the error that can be observed in Figure 3.

ered or heterogeneous substrate. In this section we will analyze different

3. Multilayered substrate configurations

DOI: http://dx.doi.org/10.5772/intechopen.81790

Section 2.2.

using [14, 23]

and ki is

75

shift of the transmitted signal.

number of the unknown materials.

where ω is the angular frequency, ε<sup>0</sup> is the frequency-dependent real part of the dielectric constant, and ℘ is the Cauchy principal value [22].

The imaginary part of the complex dielectric constants retrieved using K-K relation from the real part of the dielectric constant (Figure 2) measured by phaseshift method is shown in Figure 3. The calculated imaginary part shows a good

#### Figure 3.

Imaginary part of the complex dielectric constant retrieved by K-K relation from real part of the dielectric constant measured by phase-shift method.

Phase-Shift Transmission Line Method for Permittivity Measurement and Its Potential in Sensor… DOI: http://dx.doi.org/10.5772/intechopen.81790

agreement with the predetermined value of imaginary parts. However, the relative error is about 9% since the accuracy of Eq. (14) depends on the measured frequency range. It can be noted that the range from 0 to ∞ in the integral in Eq. (14) is not achievable in practice. Therefore, the limited range of frequencies affects the accuracy of the results and causes the error that can be observed in Figure 3.
