References

1 CHO, K.: ‘A framework for alternate queueing: Towards traffic management by PC-UNIX based routers’. Proc. of USENIX ATC, 1998, pp. 247-257

2 KUZNETSOV, A.: ‘Linux packet scheduler’. Linux Kernel v2.1.124 neusched, 1998

3 http://www.netperf.orghetperf/NetperfPage.html 4 ROSCH, w.L.: ‘Hardware bible’ (SAMS Publishing, 1997) 5 MAZIDI, M., and MAZIDI, J.: ‘The 80x86 IBM PC and compatible

computers’ (Prentice Hall, 1998), Vols. I and I1

Non-interactive and distributive property of dielectrics in mixture

K.P. Thakur, K.J. Cresswell, M. Bogosanovich and W.S. Holmes

A new property of dielectrics in mixtures which is non-interactive and distributive has been defined. The Taylor expansion of this new property and subsequent use of the Pad6 approximation has generated the parametric expression for the new property which has been used to develop a dielectric mixture model that works better than some alternatives.

The determination of dielectric permittivity for mixtures of known composition is becoming of increasing importance for applications in diagnostics, microwave sensing and in processing various diel- tric materials [l, 21. In this Letter we present a more generalised approach towards the behaviour of the dielectrics in mixtures. We consider an unknown property P of the dielectric material that is non-interactive and distributive WID) in a mixture of dielectrics. Hence for a mixture of two dielectrics, we have

where v, and v, are the volume fractions of components 1 and 2, respectively, and P, P,, and P, represent the NID property of the mixture and its components 1, and 2, respectively. Furthermore, assume that the NID property P is a continuous function of per- mittivity of the material and its derivatives exist in a given inter- val, i.e.

p = f ( E ) (2) We now expand eqn. 2 into a Taylor series around E = E,,, the per- mittivity of free space

where the primes onflE) denote the derivative with respect to per- mittivity. After rearranging eqn. 3, we obtain

We now define a constant corresponding to E = E,,

f r ( & o )

f ( & O ) ff=-

which on substitution in eqn. 4 generates

(5)

Eqn. 7 is an exact representation of the NID property of the mate- rial which can be used in eqns. 1 and 8. However, our knowledge of the functionflE) and its derivatives with respect to E at E = E,, is quite limited at this stage and hence we need to use an approxima- tion technique. The Pad6 approximation has proven to be a very useful tool in several areas of theoretical physics [3, 41. Considera- tion of the second-order Pad6 approximation [3 - 51 which implies y, = 1/2!, y, = 1/3!, ... , yn = l/(n+l)!, .. etc, in eqn. 7 leads to

(10) pB = e 4 ~ - ~ o )

The property PB of dielectric mixtures is a new dimensionless property that is non-interactive and distributive. We further assume that the parameter a defined around E = E,,, stays the same for individual phases and their mixture. We obtain from eqns. 1, 8 and 10

+ v2eae2 (11) = w l e a E l

Here a is a constant indicating the measure of interaction among individual phases in the mixture and is unique for a given system. In principle, a can take any real value. However, in prac- tice, the values of a vary in a narrow band (both positive and neg- ative) around a = 0 depending on the interaction between the individual phases of the mixture. If there is no interaction between the individual phases in the mixture, the value of a tends to zero and eqn. 11 reduces to

6, = V I E 1 + VZEZ

which is a linear model for a mixture of two phases in a non-inter- active and distributive system which has been used to predict the dielectric properties of solid foods with some success [6]. To verify the validity of eqn. 11 and compare the results with the linear model we have taken the experimental data of Kraszewski [ 11 for a mixture of carbon tetrachloride (E’ = 2.228) and glass (E’ = 4.594). The results obtained according to eqn. 11 with a = 4.225 were in good agreement with the experimental results. We have further carried out similar comparisons for a variety of two-phase and three-phase systems for which data are available. Fig. 1 presents the experimental and theoretical (a = 4.36) results of the dielec- tric constant of the mixture of air and lentil grains having 8% moisture content. We present another comparison of theoretical

(12)

volume fraction of lentil m

Fig. 1 Comparison of theoretical and experimental values of dielectric constant of lentil and air mixture

0 3.95GHz (experiment) model

or

PB = 1 + C Y ( & - - E ~ ) + ~ ~ Q ~ ( € - - E ~ ) ~ + ~ . *+7nn-1an(&-&o)n+. . . (7)

(eqn. 11) and experimental data for a condensed milk and water system for 0.6GHz (Fig. 2), where a = 0.012 has been used in eqn. 11. Experimental data in Figs. 1 and 2 Were obtained bY using a waveguide cell and a vector network analyser.

Fig. 2 also presents the results of a widely used model [7], which fails to account for the present experimental data of dielectric con- stant of the condensed milk and water mixture (Fig. 2). However,

where P B is a dimensionless variable and the constants

are dimensionless

(8 ) P PB = -

f (EO)

ELECTRONICS LETTERS 8th July 7999 Vol. 35 No. 14 1143

References

80 -W

4 2 70

8 5 ti c 60

.Y 50

ii 40 5

30

_ _ 0 0.2 0.4 0.6 0.8 1 .o

fraction of water, v/v

1182121 Fig. 2 Comparison of theoretical and experimental values of dielectric constant of condensed milk and water mixture at 0.6 GHz

0 experiment eqn. 7 - eqn. 8

we find very good agreement between experimental and theoretical results according to eqn. 11 in Figs. 1 and 2. The main difference between Figs. 1 and 2 is that the curves are concave in one and convex in the other with respect to the concentration axis. This departure from linearity in either direction cannot be explained by any of the existing models for the dielectric mixtures. The NID property is a unique property of the system, which does not change with concentration. This property will always satisfy eqn. 1. Several published models can be derived on the basis of this property. For example, if we consider the function AE) = E in eqn. 1 and eqn. 2, we can obtain eqn. 12. For another function,f(~) =

eqns. 1 and 2 generate another equation [q. Consideration of a functionflE) = In(&) in eqns. 1 and 2 generate the mixture model

[71 _ _ _ -

E, = &;I €2 Further consideration of a refractive index function for the NID property,f(~) = in eqns. 1 and 2 generate the mixture model proposed by Kraszewski [ 11:

Eqn. 7 is an exact representation of the reduced NID property of the mixture of dielectrics and eqn. 10 is an approximate represen- tation of this property in an analytical form. The results generated by most of the existing models can be reproduced by eqns. 10 and 11 by taking an appropriate value of a. For example a = 0 in eqn. 11 yields eqn. 12. If a higher order Pad6 approximation is used [4], the logarithmic function for PE can be generated hence yielding eqn. 13. The dielectrics in a mixture interact with each other. However, there exists a property of the dielectric mixtures that does not interact with a similar property of another material and this property is distributive as well. This property is a continuous function of dielectric constant A&). The plots of dielectric constant of a mixture can be concave or convex with respect to the concen- tration axis depending on the nature of interaction among the individual phases that constitute the mixture. The new model can account for the dielectric constant of different types of mixtures where some existing equations fail. Hence the new model may still be far from perfect but it is better than some available alterna- tives.

Acknowledgments: The authors thank the New Zealand Foundation for Research Science and Technology for financial support.

0 IEE 1999 Electronics Letters Online No: 19990766 DOI: 10.1049/el:I9990766

K.P. Thakur, K.J. Cresswell, M. Bogosanovich and W.S. Holmes (Imaging and Sensing Team, Industrial Research Limited, 24 Balfour Road, Parnell, PO Box 2225, Auckland, New Zealand)

E-mail: k.thakur@irl.cri.nz

18 May 1999

KRASZEWSKI, A.: ‘Prediction of the dielectric properties of two phase mixture’, J. Microw. Power, 1977, 12, pp. 215-222 SUN, E., DATTA, A., and LOBO, s.: ‘Composition-based prediction of dielectric properties of foods’, J. Microw. Power Electromagn. Energy, 1995, 30, pp. 205-212 BAKER, G.A., Jr., and GAMMEL, Y.L.: ‘The Pade approximant in theoretical physics’ (Academic Press, New York, 1970) THAKUR, K.P., and DWARI, B.D.: ‘Behaviour of sodium chloride crystal on its P-V-T surface’, J. Phys. C. Solid State Phys., 1986, 19, pp. 3069-3081 PADE, H.: Thesis, Ann. Ecole Nor. 1892, 9, Suppl. 1 MUDGETT, R.E., and GOLDBLITH, D.I.c.: ‘Prediction of dielectric properties in solid foods of high moisture content at ultrahigh and microwave frequencies’, J. Food Proc. Perserv.. 1977, 1, pp. 119- 151 LOOYENGA, H.: ‘Dielectric constants of hetrogeneous mixtures’, Physica. 1965, 31, pp. 401-406

Preconditioning of electromagnetic integral equations using pre-defined wavelet packet basis

H. Deng and H. Ling

A preconditioner based on the pre-defined wavelet packet (F‘WP) basis is proposed to accelerate the convergence of iterative solvers for large-scale electromagnetic scattering problems. With the moment equations more efficiently represented using the wavelet packet bases, an effective block-diagonal preconditioner can be constructed. Simulation results show that the convergence rate for inlet-type scatterers can be si&icantly improved while maintaining a moderate computation cost for the preconditioning operation.

Introduction: The method of moments is widely used to solve elec- tromagnetic integral equations numerically. Direct solution of the moment equation requires O(W) operations, making such solu- tions prohibitively expensive for large-scale problems. Recently, we proposed a class of pre-defined wavelet packet (PWP) bases to efficiently represent the moment matrix in electrodynamic prob- lems [l, 21. We found that the moment matrix represented by the PWP basis has -qN3) significant elements for small-size prob- lems. For large-size problems, the number of si&icant elements approaches qMogN). Consequently, the complexity of solving the moment equation can be reduced when an iterative solver is employed. However, the convergence rate of the iterative solution process is strongly problem-dependent. For scattering geometries containing a large number of multiple interactions such as inlet ducts, the moment matrices are not well conditioned, and could result in an unusually slow convergence rate [3, 41. For such geometries, preconditioning is needed to improve the spectral properties of the moment matrix and accelerate convergence. Pre- conditioners such as incomplete factorisation may be effective, but require high complexity to implement [5]. In this Letter, we describe the construction of a preconditioner for accelerating the convergence rate of PWP-based moment equations. Since the moment equation represented by a wavelet basis is sparse and diagonally concentrated due to its vanishing moment and multi- scale properties, an effective preconditioner based on the block- diagonal inverse can be constructed. Our results show that the convergence rate for inlet-type scatterers can be significantly improved while maintaining moderate computational costs for the preconditioning operation.

Preconditioning using pre-defined wavelet packet basis: The pre- defined wavelet packet (PWP) is generated from a spectral decom- position tree that grows along the free-space wavenumber ko rather than the zero frequency in the conventional wavelet trans- form [l, 21. If the PWP basis transform matrix is m, the moment equation after the basis transformation is

[z]J = E

1144 ELECTRONICS LETTERS 8th July 1999 Vol. 35 No. 14