Printer Friendly

Particle swarm optimization based dielectric characterization of inhomogeneous media.

1 INTRODUCTION

Dielectric properties characterization of materials plays an important role in a wide range of applications. It is used for imaging, geometric profiling, quality control, and in many other processes. Their applications are based on the environmental operating conditions, the nature of the dielectric material to be physically and electrically characterized, the frequency of interest and the required accuracy [1]; which accuracy is strongly depending on the validity of the model used to represent the circuit for reliable calculation or estimation of the permittivity from the electrical measurements, e.g. impedance or admittance. Several permittivity measurement techniques have been developed. Among them the most effective ones are Resonant circuit [2], coaxial sample holder [3] transmission-line [4], resonant cavity [5, 6], and free-space techniques [7]. Those later, we are interested in, present more advantageous features since they are contact less, non disruptive and do not require a complex sample holding preparation. In a free-space transmission technique, a sample is placed between a transmitting and receiving antennas, and accurate characterization of the permittivity over a wide range of frequencies can be achieved through signals attenuation and phase shift measurements. In most systems, the accuracy of the determined complex permittivity depends mainly on the performance of the measuring system and the validity of the equations used for the calculation. Nevertheless, although their applicability, all these methods share the same lacks. They involve expensive equipments such as Network Analyzer and even necessitate a special sample holder and preparation system; furthermore when complex media are considered, mathematical analysis of the propagation properties, analytical or numerical should it be, may usually be not conductible. In such situations, an interesting alternative is to use Meta heuristics based optimization approaches.

Here, one proposes to deal with the characterization problem at hand by using Evolutionary Computation (EC) based algorithms which have been proven to be robust optimization tools. They were successfully used in a wide range of industrial, management and economic complex engineering problems such as optic fiber and indoor Radio communication networks design, engine shape design and aerial traffic management to name a few. Their success, due to their robustness, comes from the fact that they do not necessitate physical prior knowledge on the process to be optimized. Only general heuristics inspired from natural evolution laws are involved to capture the system behavioral parameters from environmental data (inputs and constraints). Among those optimization algorithms, the most used ones are Simulated Annealing [8], Genetic Algorithms (GAs) [9] and Tabu Search [10]. But although their effectiveness, their evolution models present a high implementation and computational complexities which are essentially due to the binary representation used for parameters encoding, e.g. in GAs, and hence are more adapted to combinatorial than continuous problems. To avoid theses drawbacks for continuous problems a genetic algorithm variant where real genes were considered instead of discrete ones were investigated [11]. But since binary encoding is not used, the schemata theory as stated by Holland can not still hold, and thus convergence of the optimization process is not guarantee. Recently, an emerging optimization tool, called Particle Swarm Optimizers (PSO) has been shown to be efficient to real optimization problems and have been introduced to solve EM problems by J. Robinson et al. [12]. These methods involve social behavior and cognitive knowledge of individuals or particles within a swarm such as for birds, fishes, bees and bacteria. Such a class of optimization tools was introduced by J. Kennedy and R. Eberhart [13] as an alternative to GAs and where the searched parameters are naturally manipulated as real numbers. Since their apparition, PSOs were successfully applied to hard optimization problems such as neural networks training for highly irregular environments [13]. Their major advantage is that they make full abstraction of genotypic attributes of the searched parameters; only phenotypic properties such as positions and velocities are considered and hence, since no bit-levels operations are involved, both computation and implementation complexities are considerably reduced, yielding a promising class of robust and efficient optimization tools.

In the present paper, the problem of free space dielectric characterization for inhomogeneous media based on the TLM-3D propagation model is considered. It is first formulated as a minimization problem involving a non-quadratic function of the unknown parameters. A PSO based solution is developed to determine the dielectric permittivity of a multilayered media sample, which correspond to the global minimum of the non-linear function describing the propagation process. This paper is organized as follows. Section 2 is devoted to the description of the measurement process and the reformulation of the characterization problem as an optimization issue. In section 3, a brief overview on PSOs motivation, metaphor and computing model is presented. In section 4 the simulation model retained for the propagation channel EM behavior prediction is briefly described, while in section 5, the proposed PSO solution is applied to the dielectric permittivity characterization problem of commonly used building material. Finally, concluding remarks and a reference list end the paper.

2 DIELECTRIC CHARACTERIZATION PROBLEM STATEMENT

We are considering multilayered inhomogeneous media that can be characterized by the free space measurement process depicted in Figure 1.

[FIGURE 1 OMITTED]

2.1 Characterization system description

As depicted in figure 1, in free space techniques, the sample to be characterized, placed between the emitting and receiving antennas is irradiated by an electromagnetic filed produced by a microwave generator. The input ([e.sub.i], [h.sub.i]) and the output ([e.sub.o], [h.sub.o]) time series EM waves are captured with an appropriate sampling period [DELTA]t and recorded by a measurement unit.

[FIGURE 2 OMITTED]

The proposed characterization technique is based on the synoptic given if figure2. In this approach the behaviour of the propagation channel constituted by the media sample and the free space between the two antennas in response to radiating EM waves is assumed to be described by a dynamical nonlinear and unknown function F as given in equation (1).

y(t) = F(u(t),[phi](t -1),[theta])+[eta](t) (1)

where {y(t)} = {[e.sub.o](t),[h.sub.o](t)} , {u(t)} = {[e.sub.i],(t),[h.sub.i](t}},and ([PHI](t-1)} respectively denote the input, output and the regression sequence accounting, in the sense of dynamical systems, for past input and output of the characterization process and where [theta] designates a real vector containing the set of the unknown parameters to be estimated. The corresponding simulated outputs [??](t) which are predicted by an accurate simulation model are compared to the measured ones through an objective evaluation function f([??]) that indicates how good the channel behaviour prediction is, which information is used by the optimizer for the evolution based adjustments of the searched parameters.

The sequence {[eta](t)} represent the total additive output disturbance accounting for internal noise, which originates within the microwave components, and external one which is depending on the interaction between the physical properties of the propagation channel within which the electromagnetic waves travel and its environing space.

Remark. For measurement accuracy enhancement, a special attention must be paid to the choice of the radiating elements, the design of the sample holder, and the sample geometry and location between the two antennas.

2.2 Characterization Problem statement

For a given material sample constituted of [n.sub.L] rectangular layers, each one presenting a dielectric permittivity [[epsilon].sub.i], having a depth [d.sub.i] and the same width height w and h, and subject to the assumptions enumerated here after, one can adopt the approximate propagation model given in figure 3 which represent the propagation channel presented by the space environing the media sample form radiating antenna to receiving one.

[FIGURE 3 OMITTED]

Let {[e.sub.o](t)} {[h.sub.o](t)} by the measured electromagnetic field sequences induced by the input exciting ones {[e.sub.i](t)} {[h.sub.i](t)} of fixed magnitude Ei, Hi and frequency [f.sub.0]; and let {[[??].sub.o](t}} and {[[??].sub.o](t)} be their estimates. Then, under the assumption that:

* The plane wave is uniform and is normally incident on the flat surface of a homogenous material

* The sample has infinite extent laterally

* The function F(.) is assumed to be bounded for any bounded u and satisfy the zero equilibrium condition (F(0)=0).

* All modes of the dynamics are assumed to be sufficiently excited by the input sequence {u(t)}.

The characterization problem at hand can be stated as follows. Given a consistent input and output measurement data set, Find the set of dielectric permittivity estimates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that minimize the output electromagnetic prediction least mean square error LMSE([??] ,t,[eta](t)) defined by equation (2),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where Q and R are semi definite positive real square weighting matrices. Subject to,

[[epsilon].sub.dMin] [less than or equal to] [[epsilon].sub.d] [less than or equal to] [[epsilon].sub.dMax], [for all]d [member of] {1,...,[n.sub,D]} (3)

where [[epsilon].sub.dMin] and [[epsilon].sub.dMax] respectively denote the lower and upper limits of the dth dimension the permittivity search space

3 PARTICLE SWARM OPTIMIZATION

As stated in the introduction, among exciting optimizers PSOs are ones of the most adapted tools for continuous optimization problems (real parameters). They are life inspired population-based search algorithms that work on a set of dynamic solution candidates to the problem under optimization. Starting with an initial swarm containing a set of randomly generate particles, the algorithm involves stochastic movements to put the search direction toward the global parameters optima. Their computation model is relying on social behavior and cognitive knowledge of bio colonies. The learning metaphor is based on the share of common resources and the exchange of information detected during previous exploration processes. In the following subsections, a mathematical description of the swarm optimizer and its stability conditions, and subsequently its parameters tuning setting are presented.

3.1 PSO equations and parameters tuning

For each iteration of the optimization process, the swarm search is reoriented by pulling particles toward the previous [x.sub.pi,,d] and global [[cho].sub.Gi,d] locations previously discovered by any particle within the swam along the dth dimension, and this according to equations (4)-(5),

[v.sub.i,d](t +1) = w(t)*[v.sub.i,d](t)+[c.sub.p]*[phi](t)* ([x.sub.i,d](t))+[c.sub.g]*[phi](t)*(x.sub.Gi,d] (t)-[x.sub.i,d])) (4)

[x.sub.i,d](t+1) = [x.sub.i,d](t)+[v.sub.i,d](t+1) (5)

where [x.sub.i,d] and [v.sub.i,d] respectively denote the dth dimension of the ith particle position [x.sub.i] and velocity [v.sub.i]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and where w is the particles inertia, [c.sub.p] and [c.sub.g] stand for their stochastic back motion accelerations and [phi](t) is a time function generating random numbers uniformly distributed in the interval [0, 1]; the parameters and particles numbers are designated by [n.sub.D] and [N.sub.P] respectively. In addition to particle maximal moving velocity, the search process is subject to the intrinsic constraints imposed by the problem under optimization. Theses constraints are taken in account in the definition of the parameters feasible search domain; and like for all Evolutionary Algorithms, they are expressed as negative functions of the searched parameters (8).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Due to the dynamical nature of this computing model, its stability must be ensured. This can be straight forwardly released by inspecting its eigen values through its characteristic polynomial. The yielding stability conditions are used for the PSO parameters setting. The commonly used values for easy to use PSO are 0.7 for the inertia and 2 for acceleration. More interesting information about PSO can be found in [13].

3.2 The PSO Algorithm

The algorithm used for the purpose of this study is implemented by one moving method that updates particles velocities and moves them according to equations (4) and (5), an objective evaluation procedure and a selection one which is used to update the best previous and best global locations which should be considered as new attraction locations for the next generation. As described in figure 4, the algorithm begins in the initialization step, by creating Np uniformly sparse particles in the considered search space. After what, for each generation of the optimization process, the implemented methods described above are launched until the meeting of the optimization objective which consists in finding the location of the global optimum that minimizes the characterization error and this with respect to the constraints imposed by the problem under optimization.
Figure. 4. Pseudo code of a standard PSO

Create Np particles and randomize their position
uniformly in the search space;
While the target objective is not achieved {
  for each iteration of the evolution process {
    for each particle{
        Handle constraints, update objective and
               the previous best location
        Detect the global best particle position;
       Update velocities and move particles;
     }
   }
 }


The problem now is to choose a simulation model which should be used to predict the propagation channel behavior for a given input parameter [theta] and exciting sequence u. This will be purpose of the following section.

4 THE PROPAGATION CHANNEL SIMULATION MODEL

To predict the behavior of a propagation canal in response to exciting electromagnetic fields, one may use one of the several simulation models proposed in the literature, in occurrence the MoM, FEM, FDTD and TLM methods. For high frequencies, and in the case of inhomogeneous and complex media, the most used ones are the Finite-Differences Time-Domain (FDTD) and the transmission lines method (TLM). The latter is more advantageous in the sense that its nodal structure makes possible the use pulse propagation theory in a transmission line, involving mathematical model yielding exact solutions for the Maxwell equations. It is for these reasons that, in this study, we shall use the Symmetrical Condensed Node for TLM method (TLM-SCN), which topology is given in figure 5 and for which the computing model can by summarized as will be presented here after.

4.1 Electromagnetic Fields Computing

By considering the in incident waves at a given node modeling a space block of dimensions (u, v, w) in the (x, y, z) directions (figure 5) , and the scattering properties obtained from general energy and charge conservation principle [14], the electric and magnetic fields, can be in the dissipative and inhomogeneous media, described as follows.

[FIGURE 5 OMITTED]

For each node of the mesh, the output electromagnetic fields are calculated according to equations (8)-(13).

[E.sub.x] = 2([V.sup.i.sub.1] + [V.sup.i.sub.2] + [V.sup.i.sub.9] + [V.sup.i.sub.12] + [Y.sub.x] [V.sup.i.sub.13] / u(4 + [Y.sub.x] + [G.sub.x] (8)

[E.sub.y] = 2([V.sup.i.sub.31] + [V.sup.i.sub.4] + [V.sup.i.sub.8] + [V.sup.i.sub.11] + [Y.sub.y] [V.sup.i.sub.14] / u(4 + [Y.sub.y] + [G.sub.y] (9)

[E.sub.z] = 2([V.sup.i.sub.5] + [V.sup.i.sub.6] + [V.sup.i.sub.7] + [V.sup.i.sub.10] + [Y.sub.x] [V.sup.i.sub.17] / u(4 + [Y.sub.z] + [G.sub.z] (10)

[H.sub.x] = 2([V.sup.i.sub.4] - [V.sup.i.sub.5] - [V.sup.i.sub.7] - [V.sup.i.sub.8] - [V.sup.i.sub.16] / [Z.sub.0]u(4 + [Z.sub.x] + [R.sub.x] (11)

[H.sub.y] = 2([-V.sup.i.sub.2] - [V.sup.i.sub.6] - [V.sup.i.sub.9] - [V.sup.i.sub.10] - [V.sup.i.sub.17] / [Z.sub.0]v(4 + [Z.sub.y] + [R.sub.y] (12)

[H.sub.z] = 2([-V.sup.i.sub.3] - [V.sup.i.sub.1] - [V.sup.i.sub.11] - [V.sup.i.sub.12] - [V.sup.i.sub.18] / [Z.sub.0]w(4 + [Z.sub.z] + [R.sub.z] (13)

Where [V.sup.i.sub.n] (n=1,..,18 ) is the incident voltage on port n and Zo denote the air impedance

and where [G.sub.x], [G.sub.y], [G.sub.z], [R.sub.x], [R.sub.y], [R.sub.z], [Y.sub.x], [Y.sub.y],[Y.sub.z] and [Z.sub.x], [Z.sub.y], [Z.sub.z] denote the normalized sub admittances and sub impedances as defined in [14].

[Y.sub.x] = 2[[epsilon].sub.r]/c[DELTA]t vw/u -4 (14)

[Y.sub.y] = 2[[epsilon].sub.r]/c[DELTA]t uw/v -4 (15)

[Y.sub.z] = 2[[epsilon].sub.r]/c[DELTA]t vu/w -4 (16)

[Z.sub.x] = 2[[micro].sub.r]/c[DELTA]t vw/u -4 (17)

[Z.sub.y] = 2[[micro].sub.r]/c[DELTA]t uw/v -4 (18)

[Z.sub.z] = 2[[micro].sub.r/c[DELTA]t vu/w -4 (19)

[G.sub.x] = [[sigma].sub.e(wv)/u [Z.sub.0] (20)

[G.sub.y] = [[sigma].sub.e(uw)/v [Z.sub.0] (21)

[G.sub.z] = [[sigma].sub.e](uv)/w [Z.sub.0] (22)

[R.sub.x] = [[sigma].sub.m(vw)/[Z.sub.0]u (23)

[R.sub.y] = [[sigma].sub.m(uw)/[Z.sub.0]v (24)

[R.sub.z] = [[sigma].sub.m(uv)/[Z.sub.0]w (25)

and where [[epsilon].sub.r], [[mu].sub.r], [[sigma].sub.e] and [[sigma].sub.m] perceptively designate the relatives permittivity, permeability, electric and the magnetic conductivities; c is the light celerity in the air ([3.10.sup.8]m/s) and [DELTA]t is the discretization time step.

The incident voltage [V.sup.i.sub.n] on the port n is computed in the previous time iteration from the reflected voltage [V.sup.r.sub.n] according to connection rules and Higdon Absorbing Boundary Conditions (ABCs) being the reflected voltage [V.sup.r.sub.n] computed from incident voltage at iteration k-1 through the scattering matrix S. for more implementation details see [14].
Figure. 6. The TLM Simulation Algorithm

The TLM Simulation Algorithm
           Initialization;
           For each time iteration [t.sub.k] {
              Excite input node with the voltage source;
              Compute the output EM fields;
              Apply scattering operations;
              Connect adjacent nodes;
              Handle boundary conditions;
           }


5 NUMERICAL EXAMPLES

Many examples have been simulated to validate the accuracy of the proposed PSO based solution to dielectric permittivity characterization. In the present section, three study cases considering single layer and 2 multilayered inhomogeneous materials commonly used buildings construction are presented. This concerns (1) the Glass, (2) the Clay- Concrete cellular and (3) Brick- Concrete cellular combinations.

5.1 Operating Condition and PSO Parameters Setting

The search space is of dimension 2.54 cm the first case (a) and 2 for later cases (b and c). Knowing that relative permittivity is considered the lower boundaries for each dimension are set to 1.0, the upper one is set arbitrarily to +10.0 respectively. The simulated measurements of the recorded time series electromagnetic fields induced by a 2450Mz frequency radiating microwave wave of amplitude Em = 100 V/m , and Hm = 0.3 A/m, corrupted with an artificial white Gaussian noise of 20dB signal-to-noise ratio which simulates measurement and modeling errors, are reported on figure 7. At this effect, let us note that for a real characterization process assembling, since operating frequency is too much high to be digitalized by conventional Analog to Digital Converter (ADC) which is limited to 1 GSPS (see for e.g. Analog Device). In such situation, a classical method is to use analogical recorder measurement system that captures the input and output EM waves. The recorded sequences are then discretized (in differed time) using classical (Low frequency) ADCs. Since we are operating in microwave frequency ranges, magnetic and electric fields are linearly related, only electric field is considered. The function used to evaluate the optimization objective can then be defined as the characterization process prediction LMSE of the true and estimated output electromagnetic waves given by equation 26 which correspond to Q=1 and R=0 in equation 2.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Where [theta] is the unknown vector parameters concatenating the permittivity of the unknown multilayered media sample.

The PSO parameters settings are those given in table 1.

[FIGURE 7 OMITTED]

5.2 Optimization Results

For the retained study, the dielectric properties estimates as well as those corresponding to the true values (assumed to be unknowns) for the three cases are reported on tables 2. We have selected the case of Brick- Concrete cellular combination for example to show the details results, the evolution of the RMS error between the measured and estimated fields, with and without noise, are depicted in figures 8. Figures 9 give the predicted electric fields obtained by simulating the propagation with the founded permittivity estimates while their evolutions are given in figures 10; which results are validated through the examination of the output electric field prediction error autocorrelation functions given in figures 11 and which confirm that the predicted waves differ from the measured ones only by a zero mean white noise.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

5.3 Optimization Performances

As illustrated in figure 8 (noise free and noisy cases) the output prediction LMSE becomes negligible after approximately just twenty optimizing iterations; and as depicted in figures 10, the searched materials permittivity, are estimated with a good precision, which shows the accuracy and efficiency of the PSO-based dielectric characterization of multilayered inhomogeneous media.

6 CONCLUSION

This paper has focused on dielectric characterization of multilayered media using tools from evolutionary computation. An implementation of a standard Particle Swarm Optimizer was used to estimate the dielectric permittivity of each matter layer constituting the media sample. The electromagnetic levels used were chosen so that safety operating conditions were ensured. The characterization results presented in the previous section are quite satisfactory and show how such optimization algorithms are well suited to the dielectric characterization task. For further related investigations, one should examine the on-line characterization to deal with quality control in industrial and manufacturing processes. Also a special attention must be paid to their therapy capabilities by predicting the SAR induced by EM waves in some critical parts or organs of the human body.

References

[1.] Nelson, S.O., "Dielectric properties measuring techniques and applications", ASAE Paper No. 983067. St. Joseph, MI: ASAE, 1998.

[2.] Field, R.F., "Lumped circuits and dielectric measuring techniques", In Dielectric Materials and Applications, ed. A. von Hippel, 12-22. New York, NY: John Wiley and Sons, 1954.

[3.] Corcoran, P.T., Nelson, S.O., Stetson, L.E. and Schlaphoff, C.W., "Determining dielectric properties of grain and seed in the audio frequency range", Transactions of the ASAE 13(3): 348-351, 1970.

[4.] Nelson, S.O., Stetson, L.E., and Schlaphoff, C.W., "A general computer program for precise calculation of dielectric properties from short-circuited wave-guide measurements", IEEE Transactions of Instrumentation and Measurement 23(4): 455-460, 1974.

[5.] Sucher, M. and Fox, J., "Handbook of Microwave Measurements", New York, NY: Polytechnic Press Institute, 1963.

[6.] de Loor, G.P. and Meijboom, F.W., "The dielectric constant of foods and other materials with high water contents at microwave frequencies", Journal of Food Technology 1: 313-322, 1966.

[7.] Trabelsi, S., Kraszewski, A.W., and Nelson, S.O., "A new density-independent function for microwave moisture content determination in particulate materials", In Proceedings of IEEE Instrumentation and Measurement Technology Conference, 1997.

[8.] Kirkpatrick, S. and Gelatt, C. D., and Vecchi, M. P., "Optimization by Simulated Annealing", Science, Vol 220, Number 4598, pages 671-680, 1983

[9.] Holland, J. H., "Adaptation in Natural and Artificial Systems", University of Michigan Press, Ann Arbor, 1975.

[10.] Glover, F., "Tabu search - Part I. ORSA", Journal on Computing. 1(3): 190-206, 1989.

[11.] Bosworth, J., Foo, N., and Zeigler, B.P., "Comparison of genetic algorithms with conjugate gradient methods", NASA-CR-2093; NTIS. N72-30526, 1972.

[12.] Robinson, J., Sinton, S., and Rahmat-Samii, Y., "Particle Swarm, Genetic Algorithm, and their Hybrids: Optimization of a Profiled Corrugated Horn Antenna", Proc. Of IEEE International Symposium on Antennas & Propagation. San Antonio, Texas. June, 2002.

[13.] Kennedy, J. and Eberhart, R. C., "Particle swarm optimization", Proceedings of IEEE International Conference on Neural Networks, Piscataway, NJ. pp. 1942-1948, 1995

[14.] Johns, P. B., A "symmetrical condensed node for the TLM method", IEEE transaction on microwave theory and techniques, vol. MTT-35. No. 4 April 198

S. Mouna *, H. Ammor *, and M.Khalladi **

* Communications and Electronic Laboratory, Mohammadia Engineering Institute, Av. Ibn Sina. P.Box 765, Rabat -MOROCCO

** Faculty of Science Abdelmalek Essaadi University P. O. Box 2121 Tetuan 93000 Morocco Smouna7@yahoo.fr
Table 1 PSO parameters tuning

Symbol            parameter                value

[n.sub.D]         Particle dimension       2
[N.sub.p]         Particles number         100
[w.sub.i]         Inertia                  0.7
[c.sub.i]         Max back motion factor   2.0
[p.sub.r]         Repulsion probability    0.1
[absolute value   Max Velocity             0.5
  of [V.sub.i]
  (t).sub.max]]

Table 2 Estimates and true values of the media dielectric
permittivity

Study case                                Symbol    Permittivity

Glass (a)                       [epsilon]           Unique layer
                                [[epsilon].sub.1]   First layer
Clay-Concrete cellular (b)      [[epsilon].sub.2]   Second Layer
                                [[epsilon].sub.1]   First layer
Brick-Concrete cellular (c)     [[epsilon].sub.2]   Second Layer

Study case                      Estimate   True

Glass (a)                       1.581      1.58
                                1.943      1.94
Clay-Concrete cellular (b)      1.498      1.50
                                2.946      2.940
Brick-Concrete cellular (c)     1.505      1.500
COPYRIGHT 2011 Springfield Publishing Corporation
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2011 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Mouna, S.; Ammor, H.; Khalladi, M.
Publication:International Journal of Emerging Sciences
Article Type:Report
Geographic Code:6MORO
Date:Dec 1, 2011
Words:4270
Previous Article:Characterization of variable high voltage D.C. power supply designed and constructed for low pressure gas discharge.
Next Article:Integrating pollution remediation potentials of green plants in Herbarium system.
Topics:

Terms of use | Privacy policy | Copyright © 2021 Farlex, Inc. | Feedback | For webmasters |