Determina Tion of Complex Dielectric Permittivity of Loss Materials at Microwa Ve Frequencies

The method of determination of complex dielectric permIttIvIty of loss materials at microwave frequencies (X-band) using measured amplitudes of reflection and transmission coefficients and numerical calculations is developed. Different numerical methods namely graphical, bisection (halving), newton, and secant, are applied in order to determine the permittivity of cement-based materials. Simulation time and errors of these methods are compared. It is shown that the fastest and most accurate method is the bisection (halving) method because it is a global method. . KeywordsComplex dielectric permittivity of loss materials, root-finding and graphical methods, cement-based materials, microwave measurement. Dielectric properties, usually referred to permittivity, are intrinsic properties that describe wave-matter interaction and, therefore, are dependent on contents of the material, for example: moisture content, density and temperature. Cement-based materials (cement paste, mortar, concrete etc.) are widely used in many structures of the construction industry. For that reason, the knowledge of dielectric properties of such materials is very important for both propagating related research, for example: microwave propagation modelling to develop indoor wireless communication systems [4,7], and quality and content of these materials against time. Free-space microwave techniques have been widely used for dielectric property measurements particularly since recent advances in microwave components and. instrumentation made them more convenient. Complex permittivity of samples can be determined from measured reflection and/or transmission coefficients using microwave techniques [2,4,5,8,9] and suitable numerical methods [3]. The graphical method, the first applied method, can be applied for the problems with more than one independent variable [2,5]. However, its simulation time is high and its precision is low. To reduce the simulation time and to increase the precision, bisection, newton, and secant methods [3] are tested. In this paper, the complex permittivity of loss specimens by using only the amplitudes of reflection and transmission coefficients at microwave frequencies (Xband) is determined. Then, the numerical methods used for the determination of the complex permittivities of loss samples are compared. Finally, an error analysis for both measurement system and numerical methods are presented. 2. THEORY A typical situation in the measurement of the permittivity of a sample using the free-space technique is shown in Figurel. ( ) E· d I ) Et ) ( Er SAMPLE II III IV Figure 1. Typical situation in the measurement of permittivity of a sample. The wave travels from the radiating. antenna to the receiving antenna through the two different media, air and sample. Reflection occurs at the. interface of the air-sample I, and multiple reflections occur between each sides of the sample. For loss materials, the expressions for reflection coefficient, r, and transmission coefficient, t, can be expressed as [1] r = 'i2 t = (1'i~)ej8 where k = 2n o A' o In the foregoing equations in (1) and (2), .,10 ' d, and £ are the wavelength in free-space, thickness and dielectric permittivity of the material, respectively. In experimental techniques, the amplitudes of reflection and transmission coefficients IrI and It 1 are measured in decibels, defined as T=-20logltl, R=-20loglrl· (3) In order to find the permittivity, equations in (1) should be solved together. To simplify the solution of the equations, let 1£ = a jb . After this manipulation, a nonlinear equation that has a dependence only on a can be found as f(a)=~2a(1-lrI4)-(1-lrI2)2 exp{- 2a(1+1: 12 ) -(1+a2)}-lt 1 a=o (1-lrl ) where = kod. The root of the function, a, can be computed using appropriate numerical methods. After finding the value of a, the value of b can be calculated by (1a)2 (1+ a)21r12 b= 1r1 -1 uetermination of Complex Dielectric Permittivity Of Loss Materials at Microwave Frequencies After all, the real and imaginary parts of the permittivity can then be found b using the following simple formulas £' = a2 b2 and £N = 2ab. (7) 3. MEASUREMENT SYSTEM Typical loss cement-based structures with different contents, and dimensions were prepared and used for investigation of permittivity values at X-band (8-12 GHz) by using free space method. There are two main parts of the proposed microwave measurement system: microwave and control parts, respectively. The microwave part of the measurement system is used to determine the amplitudes of incident, reflected and transmitted waves as three separate parameters, and the main purpose of the control circuit is to provide communication between the microwave experimental set-up and the compu.ter. The detailed information about microwave measurement set-up and control circuit set-up can be found in [5] and [6] respectively. 4. NUMERICAL CALCULATIONS WITH DIFFERENT METHODS 4.1. Determination of Permittivity Appropriate numerical methods are developed to determine the dielectric permittivities of loss specimens by using only the amplitudes of reflection and transmission coefficients, 11'1 and It I ' respectively. These are graphical method and rootfinding methods namely, bisection method, newton method and secant method. We firstly applied graphical method to determine the complex permittivity of samples. The chart for this method is shown in Figure 2. In this method, each prospective value of real and imaginary parts of the permittivity is put into equations in (1), then the results are matched with the input parameters. After that, constant value lines of reflection and transmission coefficients (CR and CT) for the values of the amplitudes of the reflection and transmission coefficients (Rmea and Tmea) are obtained. In Figures (4) and (5) are the constant value lines for Rmea and Tmea shown, respectively. In the calculation, the frequency f = 10.38 GHz and the sample thickness d = 150 mm. For loss media case, there is just one cross point between the lines CR and CT [2] ~sulting in the complex permittivity of loss sample. The cross point can found by utting the CR onto CT. The result is shown in Figure 6. To find the complex ~rmittivity, a permittivity finding algorithm is employed. The results of this algorithm ~given in Table 1. Secondly, bisection, newton, and secant methods are employed. The geI1eral chart each of these methods is shown in Figure 3. The purpose of all of these root-finding hods is to find a domain. Not only should the domain support a suitable region, but it should r~duce the time for determining the complex permittivity. For that reason, \pplied an algorithm for finding the best region that complies with the foregoing rements. The results of the determined complex permittivity values are 1arised in Table 1. As is clearly seen from Table I, the results obtained from the applications of each ~nt algorithm are in good agreement except that the graphical method has the tivity values slightly different from the other values. Initial value arrangement and the format of the program Initial value arrangement and the format of the program FINDING POINT Finding initial point(s) for analysis Set initial values of the program Index I = I, Index2 = 0, Accuracy, N 1, N2 Set initial values of the program Iteration, Index = 0. Func = -1 Root-Finding Algorithm(s) Figure 2. The ASM chart for graphical method. Figure 3. The general ASM chart for root-finding methods For all methods, the error degree value is put the same. It is observed that the bisection method produces results faster than the others. The bisection method is called global method because the starting point information need not be accurate at all. Besides, the newton and regula falsi methods are called local methods because their models and assumptions apply only near the solutions. Therefore, the simulation time for newton and secant methods is slightly greater than that of bisection method although they are faster than bisection method. As is seen from the Table 1 that the simulation time for graphical method is much longer than the simulation times of other methods. The reason is that the domain of graphical method is much broader than the domains of others. 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