Calibration of microwave antenna for porous materials moisture evaluation using frequency domain analysis

The article presents the possibility to use microwave contactless measuring system for evaluation of moisture of porous media, mainly building materials utilizing the frequency domain analysis. In the article the description of microwave sensing technique is presented and the methods of microwave data analysis described. In the second part of the manuscript, microwave readouts are presented and the apparent permittivity values estimated in the red brick samples depending on moisture. With the calculated data, calibration model is estimated and fitting quality evaluated. Using the achieved data and calculated values the potential of microwave sensing technique is evaluated and advantages and disadvantages presented.


Introduction
Water present inside the building barriers causes the construction material decomposition [1] and negatively influences the performance of the buildings both in the construction and health aspect [1][2][3].Removing of water is an important part in renovations of historical but also modern buildings and it requires to evaluate moisture status of the building partitions.Among techniques of detection, the indirect techniques are important because they enable quick and non-invasive moisture evaluation.One of this fast detection techniques is a microwave method.
Microwave moisture detection setup consists of the microwave Vector Network Analyzer (VNA) and wideband antenna which enables to evaluate the reflection coefficient and thus the complex impedance of the material under test.With this values information about material intrinsic impedance and complex permittivity can be evaluated [4].Complex permittivity is the measure of the material particles behaviour after applying the alternating external electric field, which forces the water and other polar molecules to rotate according to the direction of the field applied.This runs to the energy accumulation that could be released after the electric field is disabled [5].Complex electric permittivity consists of the real part that describes the amount of energy released in the alternating field (utilized in moisture evaluation) and the imaginary part that represents the loss of energy due to molecules rotating and ionic conductivity (dependent on material salinity) [6].

2
The complex value of the electric permittivity is often modelled by extended formula (1) or with the use of the Debye relaxation formula [7,8]: where: ε′r-real part of dielectric permittivity of medium at particular angular frequency (ω), ε″rimaginary part of dielectric permittivity of medium at particular angular frequency (ω) [-], i-imaginary unit (i 2 = −1), σ0-electrical conductivity [S/m], ε0-electric permittivity of vacuum [F/m], ω-angular frequency of the external electric field [1/s].The aim of this article is to develop a calibration model to determine red brick moisture.The model with utilize dielectric permittivity values determined using the microwave antenna with VNA and frequency signal analysis method.Together with the calibration model measuring uncertainty will be evaluated.

Materials
For the experiment a set of red ceramic bricks was used.Average bulk density of the samples equalled 1795 kg/m 3 .Maximum water content achieved by bricks equalled 14 mass% which was a saturation status of the selected material.The setup consisted of 15 bricks with that differed in moisture by 1 mass% from 0 to saturation.

Methods
The measurement was conducted in laboratory conditions, where the stand was surrounded by reflection free environment (microwave absorbers).The major components of the measuring setup were: • Agilent N5224A (VNA analyser) • double ridge pyramidal horn antenna.The measurement the antenna was placed 8 cm in front of surface of the sample (figure 1).

Figure 1. Measurement setup used in experiment
The experiment was conducted in laboratory conditions with constant temperature 20 ±1° C and air humidity 50 ±5%.The samples with various moisture were investigated separately by the VNA analyser equipped with double ridge pyramidal horn antenna.For the measurement the VNA analyzer was configured for reflection measurements on one port (the scattering parameter S11).The measurement was conducted in the frequency range from 1 GHz to 11 GHz.The results were achieved as the complex reflection coefficient S11 values (magnitude and angle).Those data was further used for frequency analysis.In parallel, the S11 values were automatically converted by the VNA analyser into the time form for time domain analyses to visualize a broadband signal reflected from material discontinuities along the direction of wave propagation.
Microwave readouts were recalculated using generic algorithms into relative permittivity values with the use of the frequency analysis method.This approach is to derive a mathematical model of propagation of electromagnetic wave in the material under test.The model takes into account the complex permittivity of the material and its physical dimensions.Additionally, the model was extended with the propagation of the wave through the transmit/receive antenna and on the free space distance between antenna aperture and the material surface.This technique allows calculating the reflection coefficient S11 and further compare it with the measured values.The calculated values of S11 for frequencies from range 1 to 11 GHz were optimised in order to obtain the best match with the measured values.The result of the optimization are the estimated values of the permittivity, which gives the bestmatch.
The achieved permittivity values were combined with the mass moisture evaluated gravimetrically in laboratory conditions.Having these dependencies, a polynomial regression model was developed and the calibration equation was established.To describe the quality of the model, R 2 (coefficient of determination), RSE (residual standard error) and RMSE (root mean standard error) were calculated, similarly to many articles about indirect techniques of moisture evaluation [9,10].
Additionally the uncertainty evaluation of the measurement was calculated for different levels of material moisture.Uncertainty evaluation was conducted according to GUM recommendations [11] and included combined standard uncertainty according to the following equation [11,12]: where: uc -combined standard uncertainty [mass%], S 2 -sample estimator of error variance, n -sample size, u -standard uncertainty of βi parameter expressed as its variance calculated from the calibration model, -partial derivative of  function with respect to the βi parameter, cov(βi βi) -covariance of both parameters calculated from calibrated model.
The expanded uncertainty was evaluated with the following formula: where: kp -coverage factor = 2.

Results and discussion
In figure 2 there are presented readouts of the real part of permittivity achieved by microwave signal time domain (a) and frequency (b) analysis.The results are compared with gravimetric water evaluation and second order polynomial regression models have been evaluated.

Figure 2. Dependence between mass moisture and real part of relative permittivity estimated from microwave measurements
Relation between both types of data has been described mathematically using the second order polynomial regression models, which is frequently applied for this type of data [9,12].The general formula of dependence is the following: where: w -mass moisture estimated by polynomial model [mass%],  -relative permittivity [-], a,b,cpolynomial model estimators.
The values of regression model estimators and other parameters of the formula (4) are presented in table 1.Both graphical dependence presented in figure 2 and the model parameters in table 1 confirm the possibility to evaluate material moisture using microwave antenna signal analysed by the frequency method."a" estimator significance level p value is less than 0.01 which means that is statistically significant and the polynomial dependence is suitable to describe examined phenomenon and better than other types of fit, linear for example.Coefficient of determination (R 2 ) is the indicator that describes the quality of the model of the investigated phenomenon.Its value higher from 0.9 confirms that the developed model represents the dependence at satisfactory level, in this case R 2 value equals 0.96 which confirms that moisture estimation is very good in this case.Additionally for the achieved data both RSE and RMSE were calculated.RSE value equals 0.986 mass% and the RMSE value equals 0.882 mass%.Both discussed values are below 1 mass% which is satisfactory and below the values given in many reports about material moisture evaluation using dielectric techniques [12][13][14].
Additionally in figure 3 there is presented the dependence between mass moisture estimation by microwave antenna and gravimetric measurement in laboratory conditions.The dependence between microwave detection and gravimetrical evaluation is clearly linear which is visible in the diagram presented in figure 3. Dots visible in the diagram show moisture of the same samples evaluated using microwave antenna and the gravimetric measurement.Blue line shows the linear dependence between readouts achieved using both methods.It should be noticed that dependence is satisfactory due to the values of slope and y-intercept value.In case of the slope estimator, its value equals about 0.96 which confirms good linear dependence between data achieved using both techniques.Also y-intercept is close to zero value which means that microwave technique is not significantly overestimating or underestimating the readouts.
The detailed uncertainty analysis covers both standard uncertainty and expanded uncertainty that have been calculated for each value of moisture of the material.The results of calculation are presented in figure 4. It can be noticed that both standard and expanded uncertainty depend on material moisture.The lowest values of uncertainty are observed for the intermediated states of material moisture between 3 and 9 mass% where expanded uncertainty value reaches about 0.7 mass%.In case of dry material the expanded uncertainty value exceeds 1.2 and 1.4 mass% in case of saturated.It should be mentioned here that estimated values of uncertainty are smaller than the the uncertainties presented in the literature, for example 1.1 and 1.3 vol% [13] or 0.4-1.8vol% [14].Expanded uncertainty values are comparable to other research, anyway it must be emphasised that in many cited articles volumetric moisture is estimated instead of mass moisture.

Conclusions
With the data achieved and calibration model developed the following conclusions can be made:  Microwave technique has significant potential for non-destructive quick moisture evaluation of the building partitions. Calibration formula applying the polynomial fit gives the satisfactory quality of moisture evaluation which confirms coefficient of determination R 2 equal 0.96. Both RSE and RMSE errors are below 1 mass% which is satisfactory value for moisture evaluation using indirect techniques. Expanded uncertainty equals 0.6 mass% in the intermediate material moisture which is better comparing to the other techniques of indirect moisture evaluation.

Figure 3 .
Figure 3. Dependence between moisture readouts by microwave antenna system and material moisture evaluated gravimetrically in laboratory conditions

Figure 4 .
Figure 4. Standard und expanded uncertainty depending on material moisture