Int. J. Reasoning-based Intelligent Systems, Vol. 5, No. 1, 2013 73

Copyright © 2013 Inderscience Enterprises Ltd.

Cylindrical grid-based TLM model of a coaxially loaded cylindrical cavity

Tijana Dimitrijević*, Jugoslav Joković and Bratislav Milovanović Department of Telecommunications, Faculty of Electronic Engineering, University of Niš, Niš, Serbia Email: tijana.dimitrijevic@elfak.ni.ac.rs Email: jugoslav.jokovic@elfak.ni.ac.rs Email: bratislav.milovanovic@elfak.ni.ac.rs *Corresponding author

Abstract: This paper explores the effectiveness of the Transmission-Line Matrix (TLM) method based on the cylindrical grid for the purpose of an analysis of a cavity with a circular cross-section and coaxial load. To consider the advantages of the cylindrical over rectangular TLM grid, coaxial inner load with different dielectric properties and dimensions have been used. Obtained numerical results have been verified comparing with corresponding analytic and measured results.

Keywords: TLM method; cylindrical mesh; coaxially loaded cavity.

Reference to this paper should be made as follows: Dimitrijević, T., Joković, J. and Milovanović, B. (2013) ‘Cylindrical grid-based TLM Model of a coaxially loaded cylindrical cavity’, Int. J. Reasoning-based Intelligent Systems, Vol. 5, No. 1, pp.73–79.

Biographical notes: Tijana Dimitrijević is currently a Research and Teaching Assistant with the Department of Telecommunications, University of Niš, Serbia, and working towards the PhD at the same Faculty. She received her MSc degree from the Faculty of Electronic Engineering, University of Niš, Serbia, in 2007. Her main research interests include software developments and application of the numerical TLM method for the purpose of an investigation of microwave applicators and microstrip antennas.

Jugoslav Joković is currently a Research and Teaching Assistant at the Department of Telecommunications, University of Niš, Serbia. He received his PhD from the Faculty of Electronic Engineering, University of Niš, Serbia, in 2007. His current research interests include theoretical developments and experimental verification of the numerical TLM method and its application to EMC and microwave heating.

Bratislav Milovanović received the PhD from the Faculty of Electronic Engineering, University of Niš, Serbia 1979. From 1972 onwards he was promoted, at the Faculty of Electronic Engineering to all academic positions to full Professor (1990). He was the Head of Department of Telecommunications in period from 1994 to 2000 and 2004 to 2010 and the Dean of the Faculty of Electronic Engineering from 1994 to 1998. Also, he was the president of Serbia and Montenegro IEEE MTT Chapter from 2004 to 2008. His present research work is in the field of telecommunications, particularly in the field of microwave theory and techniques, computational electromagnetic and neural networks applications.

This paper is a revised and expanded version of a paper entitled ‘TLM cylindrical model of a coaxially loaded cylindrical cavity’ presented at the ‘Jubilee 10th International Conference on Telecommunications in Modern Satellite, Cable and Broadcasting Services (TELSIKS’2011)’, Niš, Serbia, 5–8 October 2011.

1 Introduction

A circular waveguide or cavity loaded with a concentric dielectric cylinder has been the subject of many investigations (Shohet and Moskowits, 1965; Metaxas, 1974; Balasundaram

and Kennedy, 1989; Umashankar and Taflove, 1993; Kuwano and Kokubun, 1997; Chan and Reader, 2000). Besides the microwave heating of materials (Balasundaram and Kennedy, 1989; Chan and Reader, 2000), this model has been applied to calculate the Specific Absorption Rate (SAR) of a human

74 T. Dimitrijević, J. Joković and B. Milovanović

body, where the latter was modelled as a multi-layered dielectric (Metaxas, 1974; Kuwano and Kokubun, 1997), in scattering (Umashankar and Taflove, 1993) and in microwave generated plasmas (Shohet and Moskowits, 1965). In microwave heating applications, the model has been used to study various materials ranging from alumina to curing polyimide thin films for high performance multi-chip modules (Lewis et al., 1995).

As there is no analytical solution in most cases of widely used partially loaded cavity, computational electromagnetic techniques emerge as an invaluable tool in the cavity design. Several numerical techniques are available for microwave heating studies; among them the finite difference time domain (FD-TD; Kunz and Luebbers, 1993) and Transmission-Line Matrix (TLM; Christopoulos, 1995), known as full-wave methods, are the most popular in the field (Choi and Hoefer, 1986; Desai et al., 1992; Liu et al., 1994). Also, the Finite Element Method (FEM) is found to be a reliable technique for microwave heating applications (Dibben and Metaxas, 1994).

Electromagnetically based numerical TLM time-domain method has been previously used to investigate an influence of different electromagnetic (EM) and geometric properties of dielectric materials used as a load on resonant frequencies in microwave applicators, based on different cavity structures (Milovanović et al., 2001; Milovanović and Doncov, 2002; Joković et al., 2006a; Joković et al., 2006b).

Structures of rectangular geometry can be relatively simply modelled by the network of TLM cubic-shaped nodes in a Cartesian grid (x, y, z). For that reason, some additional compact models (wire, slot, losses…) were implemented in the rectangular TLM mesh. There are problems, however, such as those with cylindrical or spherical symmetry, where rectangular nodes are not convenient for modelling of boundaries.

If a rectangular mesh is used for modelling of these structures a curved boundary would have to be described in a step-wise fashion (Joković et al., 2006a; Joković et al., 2006b). In these cases, boundaries would be presented approximately that, depending on a mesh resolution, might result in a deviation of resonant frequency values as well as in excitation of unwanted modes. Since a numerical error, caused by describing the boundaries in this way, depends on the mesh resolution, it could be reduced by applying the rectangular TLM mesh of higher resolution around cavity walls, which, on the other hand, increases duration of a simulation. For those problems, it would be more appropriate to use nodes that model directly non-cubic shaped blocks, that is, to carry out the modelling in a non-Cartesian grid (Al-Mukhtar and Sitch, 1981). One of the advantages of using grids other than Cartesian is that, in certain cases, it is possible to describe problem boundaries more accurately. Another advantage of a curvilinear mesh is that a saving in computer storage may be achieved by exploiting problem symmetry.

When modelling of a coaxially loaded cylindrical cavity is concerned (Figure 1), besides accurate modelling of cavity boundaries, it is necessary to enable accurate modelling of load properties represented by a dielectric constant εr. In order to achieve time synchronisation in the mesh, the TLM node

used for describing the load must be r times less than nodes dimension in the rest of the cavity filled with air. Obviously, the mesh resolution inside the load would be much higher than the resolution in the air. If the coaxially loaded cylindrical cavity is modelled by the non-uniform rectangular TLM mesh, the medium outside the load (the air) would have to be modelled with nodes of different dimensions, as it is shown in Figure 2a. As consequence, numerical results might not converge, especially in the case when the dielectric constant of a load is much greater than the dielectric constant of the air. This problem can be easily solved by using the cylindrical TLM grid (Figure 2b) with coordinates (φ, r, z) that provides uniformity of the mesh within the same medium.

Figure 1 A cylindrical cavity with a coaxial load

h

z

y

x

rl

a

�r

�0

Figure 2 (a) Rectangular TLM mesh, (b) Cylindrical TLM mesh

2a2rl

�r

�0

(a)

2a2rl

�r

�0

(b)

Cylindrical grid-based TLM model 75

In this paper, an efficiency of the TLM method based on the cylindrical grid for accurate modelling of a cylindrical cavity loaded with a concentric dielectric cylinder has been verified comparing the numerical results with analytic and measured results. Furthermore, obtained results have been compared with the corresponding results reached by the rectangular grid-based TLM method to show the advantages of using the cylindrical rather than rectangular TLM model, especially in cases of modelling of cylindrical cavities loaded with material of much greater dielectric constant than the dielectric constant of the air. Since only analysis of a load effect on resonant modes is of interest, an impulse excitation was used to excite a particular field component. However, if the compact wire model in a cylindrical TLM grid is used (Dimitrijević et al., 2012), there would be no limit for modelling of wire elements used for mode establishing and detection in practice.

2 TLM cylindrical model

In the conventional TLM time-domain method, EM field strength in three dimensions, for a specified mode of oscillation in a metallic cavity, is modelled by filling the field space with a network of link lines and exciting a particular field component through incident voltage pulses on appropriate lines. An efficient computational algorithm of scattering properties, based on enforcing continuity of the electric and magnetic fields and conservation of charge and magnetic flux (Trenkic, 1995), is implemented to speed up the simulation process. EM properties of different mediums in the cavity are modelled by using a network of interconnected nodes, a typical structure known as the Symmetrical Condensed Node (SCN; Trenkic, 1995). Each node describes a portion of the medium shaped like a cubic (Cartesian rectangular mesh) or a slice (non-Cartesian cylindrical mesh) depending on the coordinate system applied. Additional stubs may be incorporated into the TLM network to account for inhomogeneous materials and/or electric and magnetic losses.

When cylindrical structures are concerned, a non-Cartesian cylindrical mesh in the coordinate system (φ, r, z) can be used for the modelling purpose. The coordinate system used and the port designations are shown in Figure 3.

Figure 3 A cylindrical SCN

�r

�z

x

yr

z

�

��

Vrpz

Vznr

Vrnz

Vzpr

Vzn�

Vpr�

Vpz�

Vnr�

Vnz�

Vzp�

Vrn�

Vrp�

Simulation proceeds exactly as for an SCN with stubs in a Cartesian grid (Kunz and Luebbers, 1993). The only modification involves the calculation of stub parameters where account must be taken of the details of the new geometry. Capacitance and inductance of the TLM node shown in Figure 3 can be expressed as follows:

0 0,r r r r

r z r zC L

r r

, (1)

0 0,r z r z

C Lr r

, (2)

0 0,z z z z

r r r rC L

z z

. (3)

Starting from above equations normalised admittances of capacitive stubs and impedances of inductive stubs of TLM nodes be expressed in a cylindrical grid by:

0 0

( ) ( )2 4, 2 4r r r r

z r z rY Z

u t r u t r

, (4)

0 0

2 4, 2 4( ) ( )

z r z rY Z

u t r u t r

, (5)

Cavity walls are modelled as external boundaries of an arbitrary reflection coefficient by terminating the link lines at the edge of the problem space with an appropriate load (Milovanović et al., 2009).

When modelling of cavities containing lossy loads is concerned, implementation of losses in the TLM model is carried out by introduction of stubs with losses in the nodes where scattering is going on. Stubs with losses may be considered as infinitely long transmission lines, or equivalently, as lines terminated with its characteristic impedance. They can be used to model either electric or magnetic losses. In case of the symmetrical condensed node, stubs with losses are directly implemented in the scattering procedure, including coupling with the corresponding EM field component.

If ek and mk represent effective electric and magnetic conductance, respectively, in k-direction, where k (φ, r, z), elements in the TLM node used for modelling of losses are defined as:

,ek ek mk mk

i j i jG R

k k

, (6)

where (Δi, Δj, Δk) (rΔφ, Δr, Δz). Starting from

* *,ek mkk o rk k rkj j

, (7)

(Kunz and Luebbers, 1993), it is possible to define a loss tangent at the appropriate frequency as:

0 0

tan , tan2 2

ek mkek mk

rk rkf f

(8)

Finally, corresponding equations for reflected total voltages and currents in k-direction have to be modified in case of modelling of mediums with losses (Kunz and Luebbers, 1993).

76 T. Dimitrijević, J. Joković and B. Milovanović

3 Numerical results

This section demonstrates capabilities and usefulness of the cylindrical TLM method to model a coaxially loaded cylindrical cavity. To point out advantages of the cylindrical over rectangular TLM model, cavities loaded with materials of different radius and dielectric constant have been considered (Figure 1).

Before the abilities and advantages of the cylindrical TLM method are discussed, it is important to verify its accuracy. This has been accomplished by using the method to model the cavity of circular cross-section loaded with a concentric dielectric cylinder for two different substrates. The cavity radius and height are a = 46.8 mm and h = 75 mm, respectively, while the radius of the load is rl = 6 mm, which is in accordance with dimensions of the cavity considered in the work of Chan and Reader (2000). Thus, the ratio between radius of the load and the radius of the cavity is rl/a = 0.128. Two different materials used as the load are as follows: the perspex rod, which has a dielectric constant εr = 2.6, and the load having a dielectric constant is εr = 10. Small loss, ε’’ = 0.0015 (tanδe = 0.00058 and tanδe = 0.00015, respectively), has been included in the model as well, but it has no great impact on results since it is not of significant value relative to that of the dielectric constant. Cylindrical TLM grids used for the modelling purpose have been chosen to be (φ r z) = (36 15 20) nodes and (φ r z) = (36 18 20) nodes, respectively. The medium of the cavity filled with air has been modelled by nodes of dimension in r direction ∆ra = 3.4 mm in both cases, whereas the node used for describing the load has been chosen to be r times less compared to nodes in the air, in order to achieve time synchronisation in the mesh. In that context, the load has been described by nodes of dimension ∆rl = 2.0 mm and ∆rl = 1.0 mm, respectively, for both considered cases. The EM field component Ez has been excited along the line (φ, r, z) = (1, 7, 1÷20) and (φ, r, z) = (1, 10, 1÷20), respectively, for either case. In this way, two resonant modes, TM010 and TM110, can be excited in the frequency range of interest (fmax = 5 GHz), as it is shown in Figure 4 for both analysed cases.

The same cavity has also been modelled using the rectangular TLM method. When the cavity with a coaxial load of dielectric constant εr = 2.6 has been analysed, the TLM grid of resolution (x y z) = (52 52 38) nodes (∆x1 = ∆y1 = 1.2 mm in the load and ∆x2 = ∆y2 = 1.94 mm in the air) has been applied, whereas the cavity with a load having εr = 10 has been modelled using the TLM grid of resolution (x y z) = (61 61 38) nodes (∆x1 = ∆y1 = 0.63 mm in the load and ∆x2 = ∆y2 = 1.94 mm in the air). Opposed to the cylindrical TLM model where uniformity of the mesh within the same medium has been easily achieved, in a rectangular model some areas outside the load have been modelled with nodes of considerably different dimensions from one area to another, as it is shown in Figure 2b. Nevertheless, in the considered case, due to small dielectric constant and relatively small radius of the load

compared to the cavity radius (rl/a = 0.128), the rectangular TLM grid has enabled good modelling of the given cavity.

Figure 4 Resonant frequencies distribution in the coaxially loaded cavity for (a) εr = 2.6 and (b) εr = 10

(a)

(b)

Tables 1 and 2 compare numerical results obtained by cylindrical and rectangular grid-based TLM methods with measured data and those calculated analytically (Chan and Reader, 2000), for two different values of the dielectric constant. As one can observe, a good agreement between measured, analytic data and corresponding simulated results based on cylindrical and rectangular TLM method has been achieved, confirming validity of the TLM model of coaxially loaded cylindrical cavity in a cylindrical grid.

Table 1 Analytic (Chan et al., 2000), simulated and measured (Chan et al., 2000) data for cylindrical cavity with a coaxial load of εr = 2.6

Resonant frequencies (GHz)

Mode Analytic TLM_cyl TLM_rec Measured

TM010 2.332 2.331 2.353 2.3335

TM110 3.894 3.896 3.915 –

Cylindrical grid-based TLM model 77

Table 2 Analytic (Chan et al., 2000) and simulated data for cylindrical cavity with a coaxial load of εr = 10

Resonant frequencies (GHz)

Mode Analytic TLM_cyl TLM_rec

TM010 1.845 1.872 1.857

TM110 3.826 3.839 3.845

To demonstrate possibilities of the TLM method based on the cylindrical grid, modelling of the considered cavity loaded with materials of different dielectric constants has been carried out. Simulations have been done using not only cylindrical but also rectangular TLM grid, in order to emphasise the usefulness of the cylindrical TLM method. According to the obtained results, shown in Table 3, it can be concluded that both approaches give almost the same results, but the advantage of the cylindrical over rectangular TLM model, besides accurate modelling of boundaries, can be seen in the fact that rectangular TLM model is limited in its applications, that is, when the non-uniform rectangular TLM mesh is used, simulated results, considerably differs from the analytic data not allowing detection of modes, in cases where a dielectric constant of a load has significant value relative to that of the air. In the case considered here, TM010 mode cannot be detected when a material of εr = 25 is used as a load, whereas in case of a material of εr = 30 both modes cannot be detected in the frequency range of interest.

Table 3 Simulated data for the coaxially loaded cylindrical cavity for different dielectric constant values

Resonant frequencies (GHz)

Dielectric constant Mode TLM_cyl TLM_rec

TM010 1.615 1.625 εr = 15

TM110 3.729 3.750

TM010 1.465 1.448 εr = 20

TM110 3.613 3.363

TM010 1.353 1.340 εr = 25

TM110 3.463 –

TM010 1.253 – εr = 30

TM110 3.242 –

The reason for that can be found in different dimensions of nodes used for modelling of different mediums in the cavity. Generally, due to inhomogeneity of the medium inside the cavity, dimensions of nodes used for modelling of the load have to be r times less compared to nodes in the rest of the cavity filled with air. This means that in cases where a dielectric constant of a load is much greater than that of the air, the mesh of much higher resolution is required for modelling of the load. As a result, if a rectangular non-uniform TLM mesh is used to model a cylindrical structure, the specific areas outside the load would have to be modelled with nodes of considerably different dimensions from one area to another, as presented in Figure 2b. These differences become more significant when the ratio between radius of the load and the cavity radius is increased, and thus lead to results that do not converge. This problem could be overcome by using the rectangular mesh of higher resolution which, on one hand, results in increasing the

simulation time. On the other hand, resolution of a rectangular mesh is limited in cases of modelling of a cylindrical structure that includes accurate numerical investigation of a feed/monitoring probe influence on the resonant mode frequencies and level of the EM field inside the cavity, detected from S11 and S21 characteristics (Dimitrijević et al., 2012), since an implementation of the compact wire model into the rest of the TLM mesh demands that the ratio between dimensions of a wire conductor and dimensions of nodes through which a wire conductor passes is optimal.

The accuracy of the cylindrical TLM method has been confirmed on one more example involving the cylindrical cavity of radius a = 40 mm and height h = 75 mm with a coaxially placed load which has a radius rl = 10 mm and the dielectric constant εr = 25 (Chan and Reader, 2000). In this case, the ratio between radius of the load and the radius of the cavity is almost twice as high as the cavity that is considered previously and equals rl/a = 0.25. The cavity has been modelled using the cylindrical grid of resolution (φ r z) = (36 16 20) and an impulse excitation in the line (φ, r, z) = (1, 11, 1÷20). In this case, dimension of nodes in r direction are ∆rl = 1.0 mm and ∆r2 = 5.0 mm in the load and air, respectively. Corresponding simulated pattern, representing TM0n0 modes excited in the frequency range of interest, is shown in Figure 5. Table 4 demonstrates numerical results based on cylindrical TLM grid compared to the analytic ones. As can be seen, there is a good agreement between analytic and numerical data. It can be concluded that in case where the ratio between the load and cavity radius is significant, a cylindrical TLM model for chosen resolution, which is pretty much the same as for the much lower ratio rl/a, gives good results, in terms of detection of modes and resonant frequency values.

Table 4 Analytic (Chan et al., 2000) and simulated frequencies of the TM010 and TM020 modes for the coaxial load of εr = 25

Resonant frequencies (GHz) Mode

Analytic TLM_cyl

TM010 1.0189 1.0396

TM020 3.5702 3.5592

Figure 5 Resonant frequencies distribution in the cavity with a coaxial load of εr = 25

78 T. Dimitrijević, J. Joković and B. Milovanović

4 Conclusion

In this paper, a coaxially loaded cavity with circular cross-section has been investigated using the TLM method based on the cylindrical grid in order to present the advantages and possibilities of the method. Obtained numerical results are verified by comparison with corresponding analytic and measured values of resonant frequencies.

In general, compared to the rectangular TLM grid, which implies approximate modelling of cylindrical cavity external and internal boundaries since they do not coincide with the rectangular grid axes, a cylindrical TLM grid enables accurate boundaries modelling independently of a mesh resolution, because a numerical error caused by the step-wise approximation of boundaries in a rectangular grid is avoided.

Another advantage of using the cylindrical grid can be found when modelling of a partially loaded cavity is concerned, as it is the case of a cylindrical cavity loaded with a concentric dielectric cylinder. It has been shown that by fulfilling the demand of choosing the mesh resolution, in order to achieve the time synchronisation, TLM cylindrical model provides better results than rectangular TLM model, in terms of accuracy of determining resonant modes in the frequency range of interest. Namely, when the coaxially loaded cylindrical cavity is modelled by the non-uniform rectangular TLM mesh, the medium outside the load (the air) has to be modelled with nodes of considerably different dimensions from one area to another within the same medium. As a result, numerical results might not converge, especially in cases of coaxially loaded cavities where the dielectric constant of a load is much greater than the dielectric constant of the air and where the load radius has considerable value compared to the cavity radius. Thus, accurate results, in terms of detection of modes and resonant frequency values, can be achieved for lower resolution of a cylindrical grid compared to the corresponding rectangular grid resolution.

A solution for this problem could be using the rectangular TLM mesh of higher resolution. However, this solution demands more time for simulation, and, what is more important, cannot be applied for all cases involving a cylindrical cavity containing feed/monitoring probes, since in these cases dimensions of nodes through which the wire conductor passes cannot be too small. Thus, one more advantage of a cylindrical grid is that in real case of probe-coupled cavity a cylindrical grid enables modelling of probes with much greater radius compared to the TLM method based on a rectangular grid.

Acknowledgements

The work was supported by the project III-43012 of the Serbian Ministry of Education, Science and Technological Development.

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