Transmission-line modelling (TLM) for the simulation of circuits and fields by Professor C. Christopoulos The uidespread introduction in recent years ofpoweijiul and inexpensive cowlputer workstations in industry and education has pvovided a new impetus to the development and appliration o f numerical simulation as an analysis and design tool. Incteasingly, lager parts ofthe desi@ process are simulation based, thus leading fo lowev costs and vight-jrst-time designs. Moreover, the development ofthe digital computer has led to a re-examination oj-how numerical models ofphysical systems aye constructed and operated. Signijicant advances have been made in model construction and application and this process, spurred by vapid technological development, is set to continue. In this article a modern and ejicient simulation technique known as the transmission-line modelling (TLV) method is described and its application to electromaxnetics and other areas is illustrated.

The purpose of simulation

t is unfortunate that in niodern industrial societies the term ‘simulation’ in its common use has negative associations. It tends to describe a substitute for the real thing and soniething of a lower quality

at that. It is connectred with thing and activities distinct from real life, lacking in cubstance and foreign to human feelings and experiences. Phrases such as ‘simulated pleasure’ and ‘simulated concern’ make one somewhat embarrassed to be associated with ‘simulation’. Although in scientific circles the situation is not by any means as bad-we do not ay yet practice simulation behind closed doors-it is undeniably true that many practitioners in science and engineering have a natural mistrust of siniulation. Responsibility for this must also be shared by those who make exaggerated clairns for siniulation and fail to point out its inherent limitations. Let me, therefore, say from the start that I do not regard simulation as a substitute for experiment. It is. rather, coniplenientary-another tool, which together with expcrinientation can, if used intelligently, help in understanding the behaviour of complex systems and assist in advanced design. ‘Those who choose to ignore numerical <inidation are

therefore depriving themselves of a useful tool. Central to simulation is the construction ofa model,

which more often than not is a numerical model residing in the memory and procedures of a computrr. The model is not the real thing, nor should it be. If it were the real thing, it would lose much ofits clarity and flexibility and would, therefore, be of limited use in understanding and predicting. Once this has been accepted, it should immediately become apparent that a model always has limitations. A lot of the bad press for simulation is due to insufficient emphasis being placed on this fact by those who create models and those who use them.

The numerical model, its implementation in a computer, the solution algorithrn and associated software, its input and output, in short the entire process of simulation is subject to simplifications and errors. These must be made explicit as far as possible and the physical ideas associated with the entire process must be clear to the user and as close to his or her way of thinking as possible so that the model can be used intelligently It would be a retrogradc step if the desire to make a nunierical model ‘wer friendly’ masks its liniitations and permits the user to drift into a false sense of confidence.

ENGINEERING SCIFNCE AND EDUCATION IOUKNAL APRIL 1W.l

81

3 1

AY

- Ax

a

/3

b

3

1

C

Fig. 1 representation and (c) its schematic representation

(a) A shunt TLM node, (b) its transmission-line

Following this health warning it is worth pointing out some of the undoubted advantages of simulation based on numerical models. A numerical model offers something unique, namely the ability to vary parameters almost independently of each other in a strictly controlled environment, with instant access to responses anywhere in the system, uriaffccted by mearurenient interference. No experiment can match this flexibility and, although some of the ways that a modeller uses the model could not he reproduced in practice, it neverthclcss offers an ideal medium for ‘numerical experimentation’ which helps in establishing trends and identifying important parameters. Quantitative results of high accuracy can also be obtained, but this should not be regarded as the only, or in fact the most important, function ofmodels. The purpose of this article is to describe simulation using numerical models based on the transmission-line modelling (TLM) method.

TLM: the physical basis

The deaire tlo relate new arid complex phenomena to more familiar and simpler ideas is natural to liumans and evidence of their capacity for advanced thought directed at problem solving. It is by this proccss that increasingly complex problems in science and engineering can he reduced to simpler problems which can be solved by rather routine manipulation. As long as ways are found to devise and implement such strategies there is no dmernihle limit to the level of complexity that can be handled. At the beginning of the century, mechanic^ w a s the most devclopcd system of physical ideas, complete enough to be used to ‘explain’ new phenomena. For this reason electrical problems were solved using ‘mechanical models’. It will plcase the members of this Institution to learn that electrical science is $0 well developed today that we are able to turn the tables and use electrical circuit elements and associated concepts to model other physical systems including mechanical systems. The details of this process will be described later after complex electrical systems have been modelled by using simpler electrical elements.

Inherent to dynamic change, as opposed to mere stagnation and decay, is the ability to store and cxchange energy in various ways. At least two different ways of energy rtoragc arc ncccsyary to make life interesting, and in the nlechanical world are provided by potential a n d kirictic energy. The associated concepts in the electrical world are storage in electric and Iriagrictic ficlds or, equivalently, in capacitors and inductors.

It appears therefore plausible that these two storage elmments, capacitors and inductors, together with the inevitable source and dissipation elements should he suficierit tu dcscribe all electrical prOcKSSKS and, by analogy, many other physical pheiiotnena besides. This claim would lack credibility if lumped components such as capacitorc arid inductors could riot bc used to

study electromagnetic fields since, as already mentioned, circuit and field concepts are intimately related. Work along these lines was done many years ago, notably by Kron’ and by Whinnery and Kamo’, but it did not find immedlate application. It w a s not unbl the development of the modern computer and the pioneering work of Johns and Beurle3 that these ideas were explored and sufficiently developed to lead to the sophisticated TLM models available today

A model of light propagation, first described by Huygens three hundred years ago, is based on a sequence of impulsive collisions whereby an ‘ether’ particle colliding with a cluster of s i d a r pamcles causes the particles in the. cluster to scatter in all directions, thus spreadmg the disturbance. Hence, a wavefront may be considered to be made up of a number of secondary radiators each giving rise to a spherical wavelet. Subsequent wavefronts are thus formed by a combination of such wavelets. Further work by Fresnel expanded further on this idea and, although severe dlfficulties persisted in defining the physical assumptions and conditions necessary to explain propagation in this manner, it nevertheless formed an attractivc and simple physical conception of propagation. A more detailed discussion of this idea may be found in References 4 and 5. The value of this formulation from the engineering point of view is that it suggests a model which is particularly simple for numerical computation of wave problems.

In TLM, electrical impulses are used to describe wave propagation. In order for these pulses to propagate without dlstortion at a specified speed and in a given direction a network of transmission lines is established as the modekng medium sustaining these pulses. Four intersecting line segments are shown in Fig. la, where the lumped inductance and capacitance are related to the properties of the medium @,E) and the size of space (Ax, Ay) modelled by this network.

Lumping together the energy storage elements introduces space discretisation, i.e. a finite space interval, which for the two-dimensional network shown in Fig. l a is normally chosen to be Ax = Ay = Al. The question arises what the length A1 should be. A useful rule of thumb is to choose A1 much smaller (say one tenth) than the smallest wavelength ofinterest. The network shown in Fig. l a may, in turn, be reduced to two intersecting transmission lines, shown in full in Fig. l b and schematically in Fig. IC. The two lines form a TLM two-dlmensional node and a cluster of such nodes joined together forms a mesh of lines on whch wave propagation is simulated as explained below.

Consider a voltage pulse, one volt in magnitude, incident on port 1 as shown in Fig. 2a. Assuming that each line segment has a characteristic impedance Z i t is then easy to establish how this pulse is scattered at the node. The pulses scattered into ports 2, 3 and 4 are determined by the transmission coefficient 2(2/3)/ (Z+Z/3), which in this case is equal to 0.5. Sidarly, the pulse reflected back into port 1 is determined by the reflection coefficient (2/3-2)/(2/3+2) and is thus

I I

Fig. 2 (a) A pulse of 1 V incident on a node and (b) its reflection

equal to 4 . 5 . The incidence of a pulse of 1 V on port 1 results in the scattering of pulses of magnitudes 4 . 5 V, 0.5 V, 0.5 V and 0.5 V into ports 1 to 4, as shown in Fig. 2b. It is easy to check that energy is conserved and spreads isotropically around the node. The energy associated with the incident pulse is proportional to 1’ and it is equal to the energy associated with the four scattered pulses (4 .5)? + 3(0.5)’. The pulse scattered back from port 1 combines with the incident pulse to produce a 0.5 V pulse, thus maintaining field continuity around the node.

The scattered pulses become incident on adjacent ports at the next timestep and thus set up simdar scattering events. The energy associated with the original pulse spreads out by successive scattering from adjacent nodes, the overall wavefront being determined by the superposition of numerous simple events as shown in Fig. 2. The TLM algorithm, which is described in more detail in the next Section, may thus be viewed as a discrete implementation of Huygens principle. The physical picture associated with TLM is simple and elegant and thus allows the user to build complex models with confidence and the minimum of effort. The intellectual economy associated with the TLM formulation is matched by computational simplicity. Since the model is in fact a physical network (a cluster of transmission lines) stability is uncondltional. Compliance with the principle of energy conservation can be monitored at all times by comparing energy associated with incident and scattered pulses and any inaccuracies, such as those due to round-off errors, may thus be identified and corrected.

The TLM algorithm

In the last Section, a simple description of TLM was presented to illustrate its relationshp to Huygens’ ideas of wave propagation. This simple concept can be generalised and applied to the most general field propagation problems in three dimensions. The

ENGINEERING SCIENCE AND EDUCATION JOUKNAL APRIL 1994

83

Fig. 3 Thevenin equivalent circuit of a shunt node used to calculate the scattered voltages

scattering process in such cases rriay be determined in various ways. One approach, which is comrnordy used in two-dimensional problcrru;, is to obtain a Thkvenin equivalent circuit for each node arid then to apply Kirchhoffs laws. This approach provides a more formal way of showing the equivalence or isomorphism between the circuit equations for voltage and current and the field equations for electric and magnetic field.

For the two-dmiensional node shown in Fig. 1, known in the literature as a shunt node, there will be a t any time step k four pulses k VI' to k V4' incident onto the four ports. Looking from the centre of thc node towards the four ports uric sees the four incident pulses on lines of-impedance Z which can be reprcsented by the Thbvenin equivalent circuit shown in Fig. 3. The voltage at the node at timestep k, kV, is thus equal to

The scattered or reflected voltages are then obtained by recogniting that the total voltage is the sum ofincident

3

1

3

4

4

Fig. 4 Schematic diagram of a cluster of nodes used to explain the 'connection' process

and reflected voltage pulses. Thus, for port 1:

kK'= k V - k K ' (2)

Similar expressions apply for the other ports. Scattering is fully determined by cqns. 1 and 2. The pulses incident at the next timestep k + 1 are deterrnincd equally simply. This process is described as 'connection' and it is dependent on the topology ofthe nodes in the TLM problem. Using Fig. 4 as an example. it it easy to see that the incident pulse on port 1 of node (x,y) at timestep k + l is simply the pulse scattered into port 3 of node (x,y-l) at timestep k, i.e.

(3) b+l VI' (x,y) = bV3' (x,)+l)

h i l a r l y for port 4 of node (x,y):

k+l V+' (.y,y) = kV2' (x+l,y) (4)

Equivalence between field arid circuit quantities leads to the following equations:

A typical TLM algorithm is thus as follows:

Initialisation: Input problem parameters, initial conditions, etc.

Scattering: Use eqns. 1 and 2 on all nodes to detcrniiric all reflected voltages.

Connection: Use eqns. 3 and 4 to find the new incident voltages on all ports and nodes in the problem.

Output: Use eqns. 5 to 7 to calculate the field coniporients as required.

Repeat: Repeat this process as required, starting again from 'scattering' to obtain quantities at the next time step.

This hasic scheme with adaptations, such as the addition ofstubs to deal with inhomogeneous niaterialt and diffcrcnt reflection coefficients to deal with conductors and open boundaries, has been used to solve many engineering problems. An attractive software package which uses this algorithm to sirnulate field propagation in two dimensions is described in Reference 6.

Another structure, described as the 'series node', rilay be used to simulate field components E,, Ey, H,, or, alternatively, duality inay be exploited, as described in Reference 6, to associate these components with the shunt riodr.

ENGINEEKINC SCIENCE ANI) EDUCATION JOURNAL APRIL 1994

84

Many engineering problem? require simulation in three-diniensions and ‘I’LM modcls wcrc dcvclopcd for this purpose. The symmetrical condensed node (SCN) is the structure most commonly used for 3D simulation’. I t consists oftwelve ports, two for each co- ordinate direction, as shown schematically in Fig. 5. lt is difficult, and it does not appear profitable, to try to establish a network for thic node. Instead the connnon approach is to determine scattering by demanding that the scattering matrix is unitary (energy conservation) and to exploit other symmetry and conservation laws’. Mvrc rcccntly, a bimplcr and more elegant approach has been introduced in the determination of scattering which in addition leads to a simpler implementationH. It is based on applying at three sets of four ports the requirements of conservation of electric charge and magnetic flux linkage, and clcctric arid rriagictic field continuity. The computational benefits in implementing this algorithm are described in References 8 and 9.

Equivalence between circuit and field quantities for the SCN implies that E, is associated with the average x-directed voltage (ports 1,2,9,12). Similarly, H, is associated with ports driving current on the y-2 plane (ports 4,5,7,8), AU other field components are sitilllarly determined. The computational algorithm is identical to that for the 2U-node with obvious modifications to account for the new three-dimetisional node.

In two recent colloquia organised by the IEE, various aspects of the TLM method arid its application were explored”’.”. These include the development of TLM to its present state as one of the most versatile numerical sininlation methods, and its application to a whole variety of electromagnetic, thermal and acoustic problems.

Of particular significance in recent years has been progress in describing different parts of the problem by using a different spatial resolution. This becomes essential as more complex problems are tackled. An example of such a problem is the modelling of a fine wire inside a large box. The spatial resolution required to describe the wire is much higher than that for the rest ofthe space and it is inefficient to use the same fine space step throughout the problem. Although advances in coniputer hardware mcan that increasingly larger problems can be tackled in considerable detail, it is nevertheless true that computers powerful enough to tackle even simple realistic systems are not yet available. Hence efficiency in modelling and computation remains an important consideration.

Selected areas of tine resolution may be obtained in various ways as shown in Fig. 6. The standard uniform mesh is shown in Fig. 6a, where each node is either the symmetrical condensed node described earlier or an improved, more eficient structure, such as the hybrid SCN”.

Areas of finer resolution may be obtained as shown in Fig. 6b using nodes which describe noncubic shaped volumes of space. This mesh and the associated nodes are referred to as ‘variable’ or ‘graded’. Such nodes

+I2

7 1 I ,+.

+9 f - , ~

5

Fig. 5 Schematic diagram of a 12 port symmetrical condensed node.

contain stubs so that the material and geometrical properties of space are modelled correctly whilst synchronism and connectivity are niaintaincd. Changing the shape of nodes and/or material properties alters the velocity of propagation and hence propagation delays. The rcquircmcnt of synchronism is imposed so that all scattering events occur simultaneously, thus making computation siniplc and efficient. The conncctivity requirement conies about from the need to establish a simple one-to-one correspondence between ports on acfiacent nodcs. This can be sccn from Fig. 4 where, if the part of space centred at (x+l,y) is modelled by two nodes to increase resolution, it is unclear how the ‘connection’ process to port 4 of node (x,y) should be iniplemented. Thc variable mesh shown schrniatically in Fig. 6b overcomes these problems at the expense of using stubs at each node and of changing resolution throughout the problem. These features increase storage and also introduce numerical dispersion into the model.

An alternative approach, which is described as a multigrid mesh, is shown in Fig. 6c. Here the fine resolution is achieved at specified parts ofspace without using stubs. In principle, resolution may be increased by any integer factor but in practice factors larger than

rn

I = b C

Fig. 6 (a) uniform; (6) graded; (c) multigrid

Different space discretisation schemes:

ENGINEERING SCIENCE AND EDUCATION JOURNAL APKlL lYY4

85

eight have not been used. Neither synchronism nor connectivity requirements are met for this arrange- ment. However, procedures have beeri established to exchange pulses across the fine-coarse mesh interface without increasing significantly complexity or errorsli.

Fine features can thus be modelled without incurring prohibitive computational costs. Irrespective of how fine a resolution is used, any surface which is not perpenmcular to one of the co-ordmate axes is modelled by a staircase approximation and this can sometimes be a &sadvantage. Thic difficulty can be overcome if the nodal dimensions can be trimmed to represent a slightly longer or shorter node compared with the space lengths Al. Such a development has been described recently“. Special nodes have also been developed to describe thin wires as integral parts of a TLM node as described in the paper by Johns et d . in Reference 11. Losses may be modelled in TLM by introducing stubs matched at their terminations so that they extract energy from the field. Electric and magnetic losses can thus be individually defined for all the nodes in the problem. Since the TLM method described here operates in the time-domain, special techniques are necessary to introduce frequency- dependent components into the computation. This can be done either by constructing a network having an impedance with the desired frequency dependence, which then is modelled in the time-domain”. or by introducing a wave digital filter description of the frequency dependent parameterI6. A recent innovation is the formulation of TLM in the frequency-domain to tackle problems more suited to this approach”.

Applications of TLM

TLM has been applied widely across thr entire spectrum of electromagnetics. Its versanlity and its close

connection to circuits which are familiar to the electrical engineer mean that it can be used as a natural modelling medium for electroniagnetics. Electro- magnetic compatibility (EMC), where the interaction between fields arid circuits is paramount, has been a particularly fruitful application area. Many problems have been tacklecl, such as field-to-wire coupling inside conducting enclosures, penetration through apertures, determination of the resonant structure inside screened rooms and cavities, coupling and crosstalk in printed circuit boards, the interaction of lightning with structures and many others.

An example of a TLM simulation of a complex radntor is shown in Fig. 7. The plots show the power profile from an equipment box, typical of thosc used for rack-mounted equipment, when there is a small gap between the lid and the rest of the box. The radiated power at two different frequencies is shown in the proximity of the box. Further examples of this work and comparisons hetwren TLM sirnulatioris a n d measurements may be found in Reference 18 and in the paper by T h f f y er al. in K

TLM has been applied to the modelling o f micro- wave circuits to describe in a self-consistent manner the nature of dxontinuitiec and interconnects. Some of this work is desribed in the papers by A-Basha and Snowden. and by Seager r f al. in Refererice 1 1. Morc details of applications in this area may be found in the work reported by HoefeP.”’.

Other applicationc in electromagnetics include work in antannas”’ and radar cross-section”,

The application of TLM has not beeri liriiited to electromagnetics alone. Suitable choice of parameters leads to a niodcl describing diffkion type effects, such as thermal conduction and mass transfer. The pioricering work in this area was done by P B. Johns and progress has continued to this day In the

08

03

-02

4 7

-1 2

E s

Fig. 7 Radiation pattern from an equipment cabinet obtained by TLM simulation at (a) 350 MHz and (b) 773 MHz

ENGINEEKING SCIENCE AND EDUCATION JOURNAL APRII 1994

86

M1

hour the workpiece from the start of the heating cycle to the establishment of steady-state conditions”.

Ez(1:130. 1:60.16), nVlmlHr

the designer to ascertain susceptibility risks and decide 011 optinium cable routing. A single computa-

An example of such a calculation is shown in Fig. 8. This Figure shows a cross-section through a microwave oven. The microwave source (MI) is used to launch microwave energy which penetrates into the alumina ceramic

300

10.0

0.0 njl -

tional run provides similar information over a wide frequency range. The sensi- tivity ofthe field distribution to the presence of passen- gers, other loads, and neighbouring vehicles can also be established using the TLM model.

block shown in the iruddle. Dark shaded areas indicate

ENGINEERING SCIENCE AND EDUCATION JOUlINAL APKlL 1994

87

’’0 Although TLM is

primarily used as a field

algorithm are also increasingly exploited to solve circuit problems and coupled equations representing a variety of physical systems. The general approach is to uce discrete transmission-line models of inductors, capacitors arid trarisforrricr elements or, cyuivalcntly, of differential, integral and coupling terms in simultaneous equations. This approach offers the advantages of a thoroughly physical approach to thc solution procedure, an engineering interpretation of modelling errors which now appcar as stray components, and outstanding stability even for stiff systems. Further details of this technique, described as a discrete TLM transform method, may be found in References 23 and 24.

Discussion and conclusions

Transmission-line modelling (TLM) is a powerful and versatile method for solving circuit arid ficld problems. It is ideally tuned to the way of thinking of the applied scientist and engineer. It has been successfully applied to all aspects of electromagnetic research and detign and also in thermal, acoustic and coupled problems. As a time-domain method it gives complete information on the evolution of phenomena and, if properly applied, produccs rcsults over a wide frequency range. Thc simplicity and inherent parallelism of the TLM algorithm make it certain that it will benefit from new parallel computer architectures presently coming on stream. This aspect is further explored in the paper by Vasta and Pompei presented in Reference 11. Over two hundred research publications related to TLM are currently in the literature, and the method and its applications are vigorously pursued by sevcrd rcscarch teams both in the UK and abroad. Many industrial laboratories use TLM-based software and the impact of the method in industry and education is set to increase.

Acknowledgments

I thank Dr. J. L. Herring and Mr. V Trenkic for help with computations.

References

1 KRON, G.: 'Equivalent circuit of the field equations of Maxwell', Prof, IRE, 1944, 32, pp.289-299

2 WHINNERY, J. R., and RAMO, S.: 'A new approach to the solution of high-frrqucncy problrins', Pror. IRE, 1944, 32, pp.284-288

3 JOHNS, P B., and BEURLE, R. L.: 'Numerical solunon of 2-dmiensional scattering problenis using a transniission- line matrix', Pro[, IEE, 1071, 118, pp.1203-1208

4 JOHNS, P. B.: 'A new niatherilatical model to describe the physics of propagtioii', Rridic t; Elenroii. E I ! ~ . , 1974, 44, pp.657-666

5 HOEFER, W. J. R.: 'Huygens and the coiiiputcr-a powcriul alliance in nunirrical electroiilagrietics', Pror. IEEE, 1991, 79, pp.1459-1471

h HOEFER. W J. R., and SO. P, F! M.: 'The rlrctroninpetic wave simulator' (Wilcy, 1991)

7 JOHNS, f! U,: 'A symmrtrical condrnsed node for the 'I'LM mrthod', IEEE ' I h s . , 1987, MTT-35, pp.37(!-377

8 HERRING, J. L., and CHKISTOPOUI.OS, C.: 'I>cvclopnicnt~ in thc TLM nicthod'. Proc. 8th ACES, NPS Moritrrey, USA, 16th-20th Mdrch 1992, pp.523-530

9 NAYLOK, f!, and AITSAI)I, K.: 'Simple method for dstsrniining 3-11 TLM nodal scattering in non5calar problems', Elerivoti. Letr.. 1992, 28, pp.23.53-2354

10 Colloquiun~ o n 'Transmission-lint. niatrix modelling- TLM', organised by the IEE, Savoy Place, London, 18th October 1991, Digest No. 1991/157

11 Colloquiuni on 'Devrlopments in transmission-line modclling', r)rgdniscd by thc IEE, Nottirrghdrlr University, 2nd April 1993, Digest No. 1993/156

12 SCAKAMUZZA, R . A., and LOWERY, A. J.: 'A hybrid symmrtrical condcnsrd node for the TLM mcthod', ELrtron. L ~ r t . , 1090, 26, pp.107-1949

13 HERRING, J. L., arid CHRISTOP'OULOS. C.: 'Mnlti- grid transm~ssion~line modelling method for solving clectroinagneric field problem5', E/errron. l m . , I991, 27, pp.1794-1795

14 GERMAN, F. J.: 'Infiriitc\irirally adjustable boundaries 111

synimetricd condensed node TLM s~mulations'. Yth Annual Rrv. of Prog. in Applied Computational Elrctro- magnetics. Montrrey, USA, 22nd-26th March 1993. pp.482190

15 CHKISTOPOULOS, C.: 'Propagation ofsurges abovr the corona threshold in a line with a lossy rarth return', hii. J. Ginip. niid Mafheiiintin in Elrrtriral and Elvrtmirii EIX. . COMPEL, 1985, 4, pp.91 102

16 DAWSON, J. E: 'Keprrsenting ferrite absorbing tiles as frequency dependent boundarm in TLM', Elecrroii. Lett., 1993,29, pp.791-792

17 JOHNS, D. I?, and CHRISTOPOULOS. C.: 'Lossy diclrctric arid thin lossy filni modelc for 31) steady-ctatr TLM', E/ertroi~. Lett., 1993, 29, pp.348-349

18 HERRING, J. L., and CI IRISTOPOULOS, C.: 'Trans- mission-line modelling in elcctromag~ctic cor~~patibility stud&', 1iii.J. &'iim. :Miiile//iq~, 1991, 4, pp.143-152

19 ITOH, T. (Ed.): 'Numerical techniques for niicro\vavr and millimeter-wave passive strncti~res' (Wiky 1'189)

20 NDAGIJIMANA, E, SAGUET, I?, and BOUTHINON, M.: 'Tapered dot antcnna andlysis with 311-TLM method', Ehtroii. Lett., 1990, 26, pp.408-470

21 GERMAN, E J., GOTFIARD, G. I<., RIGGS, I-. S., and GOGGANS, P M.: 'The calculation ofradar cm-section using the TLM method', IN. J. A'II~II. MoiMiig, 1989, 2, pp.267-287

22 TRENKIC, V., CHKISTOPOULOS, C., and BINNER. J. G. F!: 'The application of the transmission-line modclling (TLM) nirthod in combined thermal and clectmrnagnetic prohlmis'. Proc. 8th Irit. Cont: 011 Numerical Methods in Thernidl Problems, Swansea, 11 th-16th July 1993

23 HUI, S. Y R. , and CHRISTOPOULOS, C.: '1)iscrrtt. transforin for solving intcgro-diffrreritial equations 111

digital computers', IEE Prm 24 HUI, S. Y. K., and C H K

transform for solving non-linrx circuits 2nd cquatiom'. IEE Pmr.-A% 1992. 139. pp.321-328

D IEE. 1994

Pnifrssor Chrisropoulos is with the Ilepartment of Electrical and Electmnic Engineering, Thc Univrrsity of Nottingham, Univrrsity Park, Notringham NG7 2RD. UK.

kNGINEkRING SCIENCE AN13 EDUCATION JOURNAL APRIL 1994

88

~~ -.