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Electromagnetic Invariant and its Application for Non-Destructive Testing of Moisture Content Petro Zakharchenko,Anatoliy Syrtsov,Borys Nevzlin East-Ukrainian State University kv. Molodezhniy, 20A 91034 Lugansk UKRAINE E-mail: bnevz@vugu.lugansk.ua , Mykhaylo Zagirnyak E-mail: mzagirn@vugu.lugansk.ua, Yuriy Dyachenko E-mail: dyach@vugu.lugansk.ua Contact

Abstract

Paper is deal with the problems of developing of non-destructive instant monitoring of moisture of granular materials, such as grain, soil, sand, milled ores of minerals, building materials, grinded coal with applying of high-frquency method. The expression for electromagnetic invariant linking values of four- and two-element network of equivalent circuit is obtained. On the basis of the found formulae the algorithm of calculation of granular material moisture with increased precision is developed.
Index Terms-Moisture, high-frequency method, granular material (GM), equivalent circuit.

I. INTRODUCTION

The prompt determination of moisture without destruction of sample is necessary in agriculture, civil engineering, during mining and processing of minerals. The traditional high-frequency method of non-destructive measuring of moisture consisting in the analysis of electrical properties of instrument transducer (IT) with GM has exhausted opportunities of increasing of accuracy. The pronounced difference in electrophysical properties of material of GM particles and interparticle space does not allow to define moisture by traditional way - on integral performances of GM volume with error less than 0,5%. For decreasing error it is necessary to determine GM local parameters, on the basis of which the corrected calculation of moisture is possible.
For this purpose the authors use representation of GM as equivalent circuit relevant to division of material volume on space, occupied by particles and interparticle space. The calculation of such equivalent circuit is possible on the basis of measuring IT parameters with GM on two and more frequencies. With the help of the theory of electromagnetic invariant the analytical expressions for calculation of this equivalent circuit are obtained.

II. DETERMINATION OF ELECTROMAGNETIC INVARIANT

Suppose that the imagine discrete medium, including magnetic, dielectric and conductive components placed in the primary transducer consists of identical particles, e.g., at the cubic package is as shown in Fig.1(a).
For each of such particles two zones can be separated, which differ in physical state and the values of electromagnetic parameters corresponding to it: the body zone of the particle I and contact zone II [1]. For each zone there can be offered equivalent electrical circuit, containing R, L, C elements [(Fig.1(b)], which measured by, e.g., nonlinear diode bridge [2] and reflect physical properties (magnetic and dielectric permittivity, conductance) and concentration of the components located in the zones. If the field frequency w in the transducer allows the particle inductances L₁, L₂ to be neglected, then its equivalent circuit is transformed into four-element two-terminal network [Fig.1(c)]. Local parameters C₁, C₂, g₁, g₂ of this circuit enters the integral parameters C and g [Fig.1(d)], common for the whole suppose.

Fig 1: (a) Particles in the primary transducer; (b) particle equivalent circuit; (c) four element two-terminal network with local parameters; (d) two-terminal network with integral parameters.


In Fig. 1(d) values of integral parameters are

(1)

where g₁=1/R₁, g₂=1/R₂ - the active conductance of the resistors R₁, R₂;
w - electric field angular frequency.

Having taken the derivatives from C and g by w, there can be found

	(2)
From (2) it is evident that
	(3)

that is in four-element two-terminal network the partial derivative from the conductance of one type by the conductance of another type (reactive by active and other way round) equals to the ratio of sums of conductances of corresponding types.
It is to be noted, that the second derivative and all subsequent ones have just the same expression (3), that is,

(4)

Thus, the expression (4) represents invariant not only by frequency in the frequency-dependable circuit, but order of the derivative..

Fig 2: (a) Particle division into parallel zones ; (b) equivalent circuit with local parameters; (c) equivalent circuit with integral parameters.

Analogically with R-C network we find (Fig.2)

(5)

Having taken the derivatives from L and R by w, their ratio is determined.

(6)

All this gives the following determination of the electromagnetic invariant: partial derivative from the parameter of one type of four-element two-terminal network by the parameter of the other type equals to minus ratio of the sum of parameters of corresponding types. In the case of degeneration of four-element two-terminal network into two-element one, partial derivative is changed by the ratio of certain integrals.

III. CALCULATION OF ELEMENTS OF GRANULAR MATERIAL EQUIVALENT CIRCUIT

For the definition of mass moisture is necessary calculation of electrical parameters of GM particles which characterize moisture in sample and calculation of electrical parameters of interparticle space describing mass and density of GM.
The electrical model represents moist GM on fixed frequency is dipole relevant to division of GM particle on conduction zone with capacity C₁ and active conductance g₁ and contact zone with capacity C₂ and active conductance g₂ (Fig.3) [3]. We accept that the particles have identical sizes and spherical shape, are packaged into cubic lattice, flowing of current on chains of particles independent [Fig.1(b)].
For transition from IT parameters to equivalent circuit in Fig.1(c) is used intermediate equivalent circuit of GM (Fig.3), relevant to equivalent circuit of inhomogeneous dielectric with polarization in case of one relaxation time.

Fig 3: Equivalent circuit of dielectric: C4 - capacity relevant to geometrical capacity and electronic polarization; g3, C3 - active conductance and capacity from dipole-relaxational polarization; g4 - through conductance.

For calculation of equivalent circuit in Fig.3 active conductance g_O1 and g_O2 and capacity C_O1 and C_O2 IT with GM on two frequencies of electric field is measured.
Value C₃ is discovered analytically solving the equation

(7)

where



The transition to the circuit in Fig.3 is executed under the following formulae:)

	(8)
	(9)
	(10)

In the further calculation we use value of invariantÕ:.(11)

(11)

Value g₂ is obtained analytically solving (12) with the account (11):.)

(12)

The final transition to equivalent circuit in Fig.1(c) is realized under the formulae (13)-(15):

	(13)
	(14)
	(15)

Thus, with the help of electromagnetic invariant the analytical expressions permitting to calculate physical properties of particles and interparticle space of GM on integrated parameters of GM volume are obtained.

IV. ALGORITHM OF CALCULATION OF MOISTURE OF GRANULAR MATERIAL

On the basis of results of measuring of GM electrical parameters under the obtained formulae the calculations of local electrical parameters were performed. The analysis of results of calculations shows that:
1. On electric field frequencies 5.1 and 17 MHz the dependence of particle body capacity from moisture has exponential character (Fig.4) and does not depend on GM density.
2. On electric field frequencies of 5.1 and 17 MHz the coefficients a and b of linear function, approximating dependence of logarithm of particle body capacity on moisture are obtained. The dependences of such coefficients from interparticle capacity for different densities of GM have linear character (Fig.5).
3. On electric field frequencies of 5.1 and 17 MHz capacity of interparticle space does not depends on moisture (Fig.6) and characterizes density and grain-size parameters of GM.

Fig 4: Dependence of logarithm of particle body capacity on moisture coal.
	Fig 5: Dependencies of coefficients a and b of linear function, approximating dependence of logarithm of particle body capacity on moisture from interparticle space capacity for different densities of coal.

Fig 6: Dependence of capacity of interparticle space on moisture.

V. CONCLUSIONS

Thus, on the basis of obtained results the development of algorithm of moisture measuring with increased precision is possible:
1. by results of measuring capacity and active conductance of IT with GM on two optimum frequencies under the formulae (15), (14) we obtain values of capacity of GM particle C₁ and interparticle space C₂;
2. coefficients a and b of calibration characteristic are individual for each material and its density can be calculate on value C₂;
3. on value C₁ moisture of material are calculated

(16)

The calculation of coal moisture on the above-stated algorithm has allowed to increase measurement accuracy of GM moisture in 3.7 times only at the expense of handling observed data, without addition of essential changes to device construction.

REFERENCES

1. B.Nevzlin, "Determination of electromagnetic invariant of mathematical model of the controlled medium with actively-reactive components", in Int. Conf. on Fundamentals of Electrotechnics and Circuit Theory, Gliwice-Ustron ,Poland,May1999,pp. 261-263.
2. B.Nevzlin, Yu.Dyachenko, D.Polovinka, "Calculation characteristics of nonlinear diode bridge of the direct-alternating current", in Symp. "Electromagnetic Phenomena in Nonlinear Circuits", Liége, Belgium, September 1998, pp. 230-232.
3. B.I.Nevzlin, Y.Y.Dyachenko, "Analysis of methods of calculation of electric field in granular material", in Int. Conf. on Mathematical Methods in Electromagnetic Theory, Kharkov, Ukraine, June 1998, pp. 234-236.

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