IEEE 2007 International Symposium on Microwave, Antenna, Propagation, and EMC Technologies For Wireless Communications

Analysis of Two-dimensional Fields

in the Inhomogeneous Media

Aixin Chen1 Aihong Chen2^Beijing University ofAeronautics and Astronautics, Beijing 10083, China;

2 Agricultural University of Hebei, Baoding 071001, China )

Abstract: An electrical tool is considered. By usingtwo-dimensional finite element method, the simulated

measurements are carried out in the inhomogeneousmedia. Several examples are given to demonstrate the

capability of this tool, and the influence of

resistivities of the media on measurement results.

Keywords: Finite element method, Simulation,Inhomogeneous media

1 Introduction

In many electromagnetic fields measurement

problems, the measurement tools and the measuredmedia sometimes have cylindrical symmetry. In thatcase, if we simulate the measurement by usingnumerical methods, the actual three-dimensional

problem can be simplified into a two-dimensionalmodel. The storage and the computed time then willbe greatly reduced [1].

In this paper, an electrical tool is applied to themeasurement of the surrounding complex media.Because the tool and the media both have cylindricalsymmetry, the simulation of measurement is carriedout by two-dimensional finite element method. Themeasurement principle and the analysis process are

introduced. And several simulated results are given to

show the measurement capability of the tool.

2 Description of the tool and

principle of measurement

The electrical tool consists of three electrodes

1-4244-1044-4/07/S25.00 ©2007 IEEE.

(Figure 1). It measures the resistivity of the media,and all the media have cylindrical symmetry. It is a

section of a three-dimensional media model in Figure1, medium 1 is one kind of fluid medium, and othermedia are solid media. During the measurement, thetool is placed in medium 1, and moved from thebottom to the top.

Medium 2

Medium 3

Medium 4

IAi

I

Medium 2

Medium 3

Medium 4

Medium 1

Figure 1 Measurement modal

In the measurement, the electrode Ao transmitscurrents, electrode A\ and A2 transmit the same

currents to shield the current of electrode Aq from

disperse. As the potential on electrode A0 is constant,the current on it is related to the resistivities of themedia close to the electrodes. When the resistivity ofthe media is changed, the current on each electrodewill change respectively. So record the current on A0,and transform it into resistivity. According to the

relationship of the current and resistivity, the apparentvalue of the media can be written as [2]

Ra=kU/I (1)where k is the transform coefficient of the electrode

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IEEE 2007 International Symposium on Microwave, Antenna, Propagation, and EMC Technologies For Wireless Communications

and related with the geometry of the tool, U is the

voltage on electrode A0, I is the current on electrodeA0, Ra is the resistivity recorded by electrode A0 whichrepresents the real value of the media close to theelectrodes.

3 Two-dimensional finite

element method

First of all, establish the energy functional as

below [2]

^M^if/*7 dz dpdz (2)

where u is the unknown, S is the solution region,and o is the conductivity of the medium. As infigure 1, in different medium, ^has different value.On the interfaces between two media, theseconditions satisfy

IA\ Vt ry

dn= CTo du2

dn(3)

where subscript 1 and 2 added to the end of u or o

stand for two conjoint media, and n is the normaldirection of the interface. On the boundary ofD, thepotential is w=0.

The solution region is discretized into a lot oftriangular elements. And the form of approximationfor u within an element is polynomialapproximation

u =al+a2p + a3z (A)Consider a typical triangular element, the

potential u\, u2, u3 at nodes 1, 2, and 3, respectively,are obtained using (4), and then it gives

=!», (5)

where N* are element shape functions. The

functional in an element is given by

F.-Ufa +¦du dpdz (6)

Assembling all the elements in the solutionregion, the energy associated with the assemblageof elements is

f(«)=2f«(«) (7)e=l

When the total energy in the solution region isminimum value, the partial derivatives of F(u) withrespect to each nodal value of the potential are zero,i.e.,

oui ~l duiwhere N is the total number of nodes in the mesh. Inmatrix form, it leads to

[JT] [.]-[»] (»)where \K\ is the global coefficient matrix, [b] is a

vector associated with the current on electrodes.Solving for [u] and substituting it into (1), Ra isobtained.

Rs

Rn Rt

Rs

Figure 2 3-layer horizontal media modal

4 Numerical results

Three examples are presented to demonstratethe capability of the tool and the influence ofresistivities of the media on measurement. The firstexample is a 3-layer horizontal media modal(Figure 2). In Figure 2, only half of the modal isshown, the resistivity of the fluid medium is Rm=\

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IEEE 2007 International Symposium on Microwave, Antenna, Propagation, and EMC Technologies For Wireless Communications

Q . m, and the resistivities of the solid medium are

Rs=2 & . m, and 7^=100 Q . m respectively. Thesimulated result by the finite element method isshown in Figure 3. At the location of middle solidmedium, the apparent resistivity Ra is lower than thereal resistivity Rt.

Depth

Figure 3 Simulated result of example 1

The second example is based on the firstexample. In this example, the resistivity of the fluidmedium is changed as Rm=2 & . m, and theresistivities of the solid medium keep the same

value as those in the first example. The simulatedresult is shown in Figure 4. In Figure 4, the dashline is the result of the first example for comparison.Compared with it, at the location of middle solidmedium, the apparent resistivity Ra in the secondexample is a little lower. This is because the fluidmedium is one kind of lossy medium, less current

flows into the middle solid medium.

o so"a

70

60

50

40

30

20

10

0/

Depth

Figure 4 Simulated result of example 2

The third example is also based on the firstexample. In this example, the resistivity Rs is

changed as 10 Q . m, and the other resistivitieskeep the same value as those in the first example.The simulated result is shown in Figure 5. In Figure5, the dash line is also the result of the first example.Compared with the first example, at the location ofmiddle solid media, the apparent resistivity Ra is a

little bigger. This is because after flowing across thefluid medium, less current flows into the upper andthe nether solid media.

Figure 5 Simulated result of example 3

AcknowledgmentThe work is supported by the Programme of

Introducing Talents of Discipline to Universities(No. B07009).

Reference

[l]Matthew N. O. Sadiku, Numerical Techniques in

Eletromagnetics (Second Edition), CRC Press, 2001

[2]A.X. Chen, Z.R Nie, Domain Decomposition Method

Applied to Three-dimensional Inhomogeneous Media

Imaging, Journal of Electronics and Information Technology,2003, 25(3), 383-388

[3]P.P. Silvester, R.L. Ferrari, Finite Elements for Electrical

Engineers, Cambridge, Cambridge University Press, 1990

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