Measurement setup

The measurement setup was proposed to reach high accuracy and suppress any uncertainties. We used dielectric probe SPEAG DAK-12, which is able to perform measurements in the frequency range of 4–3000 MHz [8]. The dielectric probe was fed vial coaxial cable from the handheld network-vector analyzer Keysight N9913A FieldFox being connected to PC with software for the calculation of dielectric parameters developed by SPEAG (Switzerland). The whole medium under test (MUT) and the probe were immersed in a container filled by distilled water tempered to the exact temperature of the measurement. To avoid cable and probe movements, the container and MUT were placed on a lifting table.

The measurement setup for the temperature-dependent spectroscopy of dielectric parameters is shown in Fig. 1(a). Figure 1(b) depicts the water container with the dielectric probe and two tissue mimicking-phantoms. In the control sample two temperature probes were inserted.

Fig. 1. Measurement setup for the measurement of temperature dependence of tissue mimicking phantoms (a) and the photo of the water container with the coaxial probe and MUT (b).

Temperature-dependent calibration

In our preliminary measurements, the probe calibration was realized by a three-term calibration (open, short and matching liquid). As a reference liquid, the 0.1 M saline solution was used. The calibration was performed only at laboratory temperature (20°C).

According to our initial results from [Reference Ley, Fiser, Merunka, Vrba, Sachs and Helbig6], the difference between the measured temperature dependence of distilled water and the Ellison model [Reference Ellison9] can be reduced by performing the temperature calibration. To do this, the probe, short, and reference liquid were preheated to the temperature during which the calibration was performed. The temperature calibration was done at seven temperatures in the range of 20–50°C at the 5°C intervals.

Statistical evaluation of measurement

The measurement was performed 10 times at each considered temperature. From that data the mean value together with the standard deviation (uncertainty type A – u _A) was calculated. From the data sheet [8] provided by the manufacturer of the dielectric probe the probe uncertainty at each frequency band was taken (uncertainty type B – u _B). Both types of error are considered as statistically independent from each other. The measured results in this paper are presented with the combined standard uncertainty (uncertainty type C – u _C) calculated according to the following equation (1), which is presented in the graphs.

(1)$$u_C = \sqrt {u_A^2 + u_B^2 } $$

Measurement procedure and accuracy

Figure 1 illustrates a measurement setup for the temperature-dependent spectroscopy of the muscle phantoms. The temperature-dependent calibration was performed before the start of the measurement. Two identical tissue-mimicking phantoms were inserted into the heated water. In the control sample two temperature probes were inserted (one in the center, second to the edge of the phantom). The probe was immersed in water and attached to the MUT. The water in the container was heated to the temperature of measurement and the temperature of the controlling sample was observed. When the difference between temperature (T ₁ and T ₂) was less than 0.2°C, the phantom was taken as uniformly heated. The calibration for this temperature was loaded and the measurement was performed.

For the accuracy evaluation, the dielectric parameters of the distilled water were measured in the temperature range of 25–45°C. The complex relative permittivity is defined as follows:

(2)$$\varepsilon ^\ast \lpar {\,f\comma \;T} \rpar = {\varepsilon }^{\prime}\lpar {\,f\comma \;T} \rpar -i{\varepsilon }^{\prime \prime}\lpar {\,f\comma \;T} \rpar $$

where ɛ* is the complex permittivity, ɛ′ is the relative permittivity, ɛ″ represents the dielectric loss, f is the frequency, and T is the temperature. The dielectric loss can be transformed into the effective conductivity σ according to the following formula:

(3)$$\sigma \lpar {\,f\comma \;T} \rpar = 2\pi f\varepsilon _0{\varepsilon }^{\prime \prime}\lpar {\,f\comma \;T} \rpar $$

where ɛ ₀ is the permittivity of free space.

In Fig. 2 the results of the measurement (relative permittivity ɛ′(f, T) and effective conductivity σ(f, T)) are presented (solid lines). This measurement was performed after the temperature-dependent calibration. The dashed lines are presenting the Ellison model [Reference Ellison9].

Fig. 2. Measured temperature-dependent relative permittivity (a)* and effective conductivity (b) with combined standard uncertainty u _C (equation (1)) of distilled water (solid lines) in the frequency range of 150–3000 MHz after temperature-dependent calibration compared to the model from Ellison marked as E – (dashed lines).

The relative permittivity and effective conductivity are decreasing with the increasing temperature. This corresponds to the Ellison model. The measured relative permittivity shows good correlation with the data from the Ellison model in higher frequencies. For the frequencies below 1 GHz, the deviation is increasing. This can be caused by the used calibration material for the short (copper), which is according to the probe manual intended for the frequencies above 500 MHz [8].

In Fig. 3 the mean deviation δ of the measured temperature dependence of distilled water and model of Ellison is shown. The mean deviation is calculated according to the following equations:

(4)$$\delta {\varepsilon }^{\prime}\lpar {\,f\comma \;T} \rpar = \overline {{\varepsilon }^{\prime}} _{measured}\lpar {\,f\comma \;T} \rpar -{\varepsilon }^{\prime}_{Ellison}\lpar {\,f\comma \;T} \rpar $$

(5)$$\delta \sigma \lpar {\,f\comma \;T} \rpar = \bar{\sigma }_{measured}\lpar {\,f\comma \;T} \rpar -\sigma _{Ellison}\lpar {\,f\comma \;T} \rpar $$

where $\overline {{\varepsilon }^{\prime}} _{measured}$ is the measured mean value of the relative permittivity and $\bar{\sigma }_{measured}$ is the measured mean value of conductivity. The differences of the relative permittivity without temperature calibration (Fig. 3(a) – dashed lines) are increasing with the frequency and temperature. The deviation is approximately 2.5 units at the upper frequency. In the case of used temperature-dependent calibration, the deviation has decreased and is relatively similar for all measured temperatures. The deviation is ~0.7 units. Figure 3(b) is representing the deviations of the effective conductivity of our measurements and Ellison model. The solid lines show results with the temperature-dependent calibration and the dashed lines without temperature calibration. For the lower frequencies (below 500 MHz) the deviation is negligible for both cases. With the increasing frequency, the deviation is growing mainly for the measurement without temperature-dependent calibration.

Fig. 3. Differences between the Ellison model and measured temperature-dependent relative permittivity (a) and effective conductivity (b) depending on the calibration procedure. Dashed lines corresponding with the measurement without temperature dependent calibration and solid lines with temperature calibration.

The deviation of the measurement, where the temperature-dependent calibration was used, is fluctuating around zero. The deviation for this measurement is ~0.05 (S/m). It is obvious that the temperature dependent calibration reduced the deviation of the measurement. It is very convenient to perform the temperature-dependent calibration before the measurement. According to our opinion, the measurement equipment and the proposed measuring procedure ensure the sufficiently precise measurement of the temperature-dependent dielectric parameters of liquids and solids.

Fitting procedure

A frequency and temperature-dependent Cole–Cole model was determined by measured data. The model enables to simplify the results of measurements and calculate complex relative permittivity and effective conductivity at any frequency and temperature within the given range. The same principle was used to describe the biological tissues data, for example, in [Reference Gabriel, Gabriel and Corthout11].

The Cole-Cole model is fitted to the measured data to represent the interaction of the material with the electromagnetic field. The Debye form of the model was used for the materials with constant relaxation times. To model the dispersion accurately, the two pole Cole–Cole model was chosen and fitted. Dispersion behavior is actually present in a broader frequency range, more than it was expected, especially for liquid and solid dielectrics. For this reason, empirical index α was introduced in the original formula. Following two-pole Cole–Cole model is used in this study:

(6)$$\varepsilon ^\ast \lpar {\,f\comma \;T} \rpar = \;\varepsilon _\infty \lpar T \rpar + \displaystyle{{\Delta \varepsilon _1\lpar T \rpar } \over {1 + {\lpar {i\omega \tau_1\lpar T \rpar } \rpar }^{1-\alpha _1}}} + \;\displaystyle{{\Delta \varepsilon _2\lpar T \rpar } \over {1 + {\lpar {i\omega \tau_2\lpar T \rpar } \rpar }^{1-\alpha _2}}} + \;\displaystyle{{\sigma _s\lpar T \rpar } \over {i\omega \varepsilon _0}}$$

where Δɛ ₁, Δɛ ₂ are the dispersion magnitudes, ɛ _∞ represents the permittivity at very high frequencies (ideally infinite frequency), τ ₁, τ ₂ are the relaxation times (typically in ns or ps), σ _s is the static conductivity, and ω = 2πf is the angular frequency.

At first Levenberg–Marquardt algorithm was used to fit the two-pole Cole–Cole model to the measured data for each temperature. As there are many fitting parameters, indexes α ₁, α ₂ were fixed to the value 0.1 for simplification of calculations. The initial values were set according to Gabriel's parameters for muscle tissue. The fitting procedure is taken from [Reference Ley, Schilling, Fiser, Vrba, Sachs and Helbig7]. The solution was divided into the real and imaginary part of complex relative permittivity. To obtain the real part, we calculated parameters ɛ _{∞, fit}, Δɛ _{1, fit}, Δɛ _{2, fit}, τ _{1, fit}, τ _{2, fit} by minimizing the mean absolute error over the frequency range of 10 to 3 GHz. After this procedure, the static conductivity sigma was computed to fit imaginary part by the same method as the real part. To get the temperature dependence, a second-order polynomial function with coefficients A _n, B _n, C _n, was used to fit the temperature dependence of the Cole–Cole parameters.

(7)$$\eqalign{\varepsilon _{\infty \comma fit}\lpar T \rpar = & \;A_1T^2 + B_1T + C_1 \cr \Delta \varepsilon _{1\comma \;fit}\lpar T \rpar = & \;A_2T^2 + B_2T + C_2 \cr \tau _{1\comma \;fit}\lpar T \rpar = & \;A_3T^2 + B_3T + C_3 \cr \Delta \varepsilon _{2\comma \;fit}\lpar T \rpar = & \;A_4T^2 + B_4T + C_4 \cr \tau _{2\comma \;fit}\lpar T \rpar = & \;A_5T^2 + B_5T + C_5 \cr \sigma _{s\comma \;fit}\lpar T \rpar = & \;A_6T^2 + B_6T + C_6} $$

The final temperature-dependent dielectric properties of the proposed phantom can be computed by the following formula:

(8)$$\eqalign{\varepsilon _{\,fit}^\ast \lpar {\,f\comma \;T} \rpar = & \;\varepsilon _{\infty \comma fit}\lpar T \rpar + \displaystyle{{\Delta \varepsilon _{1\comma fit}\lpar T \rpar } \over {1 + {\lpar {i\omega \tau_{1\comma fit}\lpar T \rpar } \rpar }^{1-\alpha _1}}} \cr & + \;\displaystyle{{\Delta \varepsilon _{2\comma fit}\lpar T \rpar } \over {1 + {\lpar {i\omega \tau_{2\comma fit}\lpar T \rpar } \rpar }^{1-\alpha _2}}} + \;\displaystyle{{\sigma _{s\comma fit}\lpar T \rpar } \over {i\omega \varepsilon _0}}} $$

The difference between the measured value and fitted value of the real part of complex relative permittivity and effective conductivity at a certain frequency and temperature indicates the quality of the fitting procedure.

(9)$$\delta {\varepsilon }^{\prime}_{measured\comma fit}\lpar {\,f\comma \;T} \rpar = \;{\varepsilon }^{\prime}_{measured}\lpar {\,f\comma \;T} \rpar -{\varepsilon }^{\prime}_{\,fit}\lpar {\,f\comma \;T} \rpar $$

Effective conductivity was firstly calculated by

(10)$$\sigma _{\,fit}\;\lpar {\,f\comma \;T} \rpar = \omega \;\varepsilon _0{\varepsilon }^{\prime \prime}_{\,fit}\;\lpar {\,f\comma \;T} \rpar $$

Then the difference is

(11)$$\delta \sigma _{measured\comma fit}\lpar {\,f\comma \;T} \rpar = \sigma _{measured}\lpar {\,f\comma \;T} \rpar -\sigma _{\,fit}\lpar {\,f\comma \;T} \rpar $$

Fitting parameters and temperature coefficients of the measured phantom are shown in following Table 2 along with Fig. 5, where the results of the fitted Cole–Cole model are presented.

Fig. 5. (a) Relative permittivity and (b) effective conductivity of tissue-mimicking phantom modeled from the fitted two-pole Cole–Cole model.

Table 2. Temperature coefficients of the two-pole Cole–Cole model of the muscle tissue phantom

Figure 5 shows the temperature dependence of the dielectric properties of the muscle tissue phantom based on the fitted parameters of the Cole–Cole model. The graphs from Fig. 5 are based on the temperature-dependent Cole–Cole model – equation (8). The modeled effective conductivity has an intersection at frequency ~2500 MHz. This is corresponding with the results from the measurement. The trend of the temperature dependence is consistent with the models presented by Ley [Reference Ley, Schilling, Fiser, Vrba, Sachs and Helbig7] or Gabriel [Reference Gabriel, Gabriel and Corthout11].

The efficiency of the fitting procedure is shown in Fig. 6. It representing the differences between measured and modeled data of the muscle tissue phantom. The differences between measured and modeled relative permittivity are lower than 0.7 in case of relative permittivity (A) and lower than 0.4 (S/m) in case of effective conductivity (B).

Fig. 6. Differences between measured and modeled data by the two-pole Cole–Cole model of (a) relative permittivity and (b) effective conductivity.