- [Book] the TLM Method in Electromagnetics (C. Christopoulos 2006)

8/2/2019 - [Book] the TLM Method in Electromagnetics (C. Christopoulos 2006)

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58 THE TRANSMISSION-LINE MODELING (TLM) METHOD IN ELECTROMAGNETICS

If we choose in (4.88) the line admittance as

yL =t

2(4.89

then we get,a2=

a3

= z1, i.e., a one time step delay but with opposite polarity. We n

that the choices in (4.87) and (4.89) imply that we must enforce the equality

yL =t

2 =

t (4.90

Hence, the velocity of propagation on the TLs is

u TL 2

t = 21

= 2c (4.91and the line admittance is

yL = t 2 = 12 2 2c = 1 2

(4.92

Equations (4.91) and (4.92) are in agreement with (4.27) and (4.28) obtained inconventional derivation of the 2D TLM node. To return back to the derivation of the scatterimatrix we note that for moden =2 we are not going to get a simple delay [factor of 2 in (4.64 To simplify the algorithm we assume that the impedance for this mode is the same as for n =1, to obtaina4 z1. This is an approximation in line with the discussion following(4.72). We are now in a position to evaluate all the elements of the scattering matrix (4. will evaluate as an example elementsS 11 and S 21.

S 11 =14

(a1 +2a2 +a4) = z114

(1 2 1) = 12

z1

S 12 =14

(a1 a4) =12

z1

Similar expressions are obtained for the remaining elements to give the scattering m

S =12

1 1 1 11 1 1 11 1

1 1

1 1 1 1

(4.93

Fortunately, (4.93) is the same expression as obtained from the conventional TLM mulations shown in (4.24)! It all makes sense, reaching (4.93) through the modal expansithe elds is an elegant methodological derivation revealing the sinews of the EM eld athe TLM model.


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TWO-DIMENSIONAL TLM MODELS 59

Finally, it is easy to show that in our 2D shunt TLM node each mode sees an open-circuitor short-circuit TL segment as implied by the discussion following Eq. (4.65). From Fig. 4.14,assuming that only moden =0 incident voltage pulses are applied at the four ports, we obtainafter deriving the Thevenin equivalent circuit and applying the parallel generator theorem that

the total voltage is

V = yL2V i 1 + yL2V i 2 + yL2V i 3 + yL2V i 4

4 yL(4.94)

where from the rst row of Table 4.1,V i 1 = V i 2 =V i 3 = V i 4 =1/ 2. Substituting in (4.94) weobtain that V =1. Hence the reected voltage on each of the four ports is the same and equal tothe total voltage minus the incident voltage, i.e., 1/2. This means that this mode sees effectivelyan open circuit as the reected voltage is equal to the incident voltage. Applying (4.94) anfollowing the same procedure for the remaining three modes (the rows of Table 4.1) we show

in each case that the total voltage is equal to zero and therefore from (3.21) that the reected voltage is equal to the incident voltage but with the opposite polarity, i.e., these modes seeffectively a short circuit as already indicated.

To summarize this section, I have shown that mapping the four incident port voltages tofour modes in the cylindrical expansion of the eld allows a systematic procedure for calculatinthe reected modal components and thus for obtaining the reected port voltages. Through thisprocedure we recover exactly the same TLM parameters and algorithm as from the standardderivation outlined in Sections 4.2 and 4.3. This elegant derivation would have been only justhata theoretical oddityif it did not give us an insight into how we may address some

complex practical problems. This you will see in the following section.

4.6 EMBEDDING A THIN WIRE IN A 2D TLM MESHI haveshown how a 2Dmeshcan be constructed to model EM elds inan inhomogeneous spaceand in the presence of conducting boundaries. However, most practical engineering problemsmake further demands on the modeler. The most obvious one is the modeling of wires. Howcan we model a wire in a 2D mesh? I will deal rst with the case where the wire radius a islarger than the mesh resolution . The situation is as depicted in Fig. 4.17(a). Clearly, thecross-section of the wire canbe mapped onto several nodes by inserting shortcircuits at the node

ports nearest to the wire perimeter. This is shown in Fig. 4.147(b) where we see that the smoothperimeter of the wire is approximated by stair-cased boundary. All we need to do is to tell thalgorithm to recognize during the connection phase the short circuits present at the nodesindicated. We see that we have a grainy approximation to the smooth outline of the wire This is referred to as stair-casing error and the ner the resolution of the mesh the smalleit becomes. As the mesh resolution gets coarser the approximation of the wire cross-section


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a

(a)

(b)

2

3

( x, y, z)2

3

( x + 1, y, z)2

3

( x 1, y, z)2

3

( x, y 1, z)2

3

( x, y + 1, z)

(c)

111

1

1

4

4

4

4

FIGURE 4.17: Mapping the round wire cross-section on a Cartesian mesh (a), stair-case approximati(b), and representing a wire cross-section by a single node with short circuits at its extremities (c)

becomes poorer. The case whena = /2 is depicted in Fig. 4.17(c). All we need to do iforce the nodes around node (x, y, z) to recognize the short circuits shown at the connect

phase, i.e.,k+1V

i 1 (x, y +1, z) = kV r 1 (x, y +1, z)

k+1V i 2 (x +1, y, z) = kV r 2 (x +1, y, z)

k+1V i 3 (x, y 1, z) = kV r 3 (x, y 1, z)

k+1V i 4 (x 1, y, z) = kV r 4 (x 1, y, z)

(4.95


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We must recognize, however, that the approximation to the wire radius obtained in this way is only a rough one. As a rule of thumb, depending on the application, we need a cluster about 44 nodes to describe with some accuracy the wire cross-section.

However, it gets worse whena < /2.Whatarewegoingtodothen?Theobviousoption

of decreasing to make it smaller than the wire radius is available to us but at a substantiacomputational cost which in most cases is prohibitive.To illustrate this point, consider a problemin free space where the highest frequency of interest is 1 GHz. At this frequency the wavelengthis 30 cm so an acceptable choice of space discretization is one tenth, i.e., 3 cm. If however wneed to model a wire of diameter equal to 3 mm we must increase the spatial resolution in eacdimension by at least a factor of ten to achieve a rough description as shown in Fig. 4.17(c). I2D this represents an increase in storage by a factor of 100 and in 3D by a factor of 1000! Thstorage costs scale up very rapidly and becomes unrealistic very quickly. Similar adverse scaltakes place as regards the required run time. In order that we advance the computation for the

same total period of time we reason that the reduction of by a factor of ten means that t isalso reduced by the same factor therefore for the same total problem observation time we wineed ten times more time steps. In 2D we need 100 times more calculations per time step andten times more time steps, i.e., run time scales up by a factor of 1000. Matters look even morunfavorable in 3D!

The purpose of this discussion is to convince you that trying to accommodate ne featuressuch as thin wires by increasing mesh resolution (the brute force way) leads rapidly to unmanageable computations. This is an example of what is referred to as amultiscale problem. Wehave in the same problem features that are electrically large (where resolution of the order/ 10

is adequate) and electrically small (where resolution of / 10 is insufcient). We need new waysof thinking to deal with multiscale problems if we are going to use computational resources ian intelligent and efcient way! The purpose of this section is to introduce ideas and techniquethat can be used for this purpose. The real challenge is to solve multiscale problems in 3D, bu we can illustrate better the basic concept in 2D with the minimum of mathematical complexityA review of multiscale techniques at a more advanced level than in this text may be found in [19].

Two possible approaches are available to us. First,mesh distortionwhereby the mesh isrened locally (only around the ne feature) thus saving in computational overhead in otheareas of the problem where a ne mesh is not required. This is shown in Fig. 4.18(a). This

appears to be an attractive option but there are problems. Consider in this gure the interfacebetween the ne and coarse mesh regions. It is obvious that one-to-one correspondence atthis interface has been lost. A coarse node has several ne node neighbors. Hence at thconnection phase some means must be worked out of sharing and combining pulses. Similarlysynchronism has been lost as the time step in the ne mesh is different compared to the coarsmesh. Some form of space and time ltering is required at the interface between regions o


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FIGURE 4.18: A small diameter wire represented in a multigrid mesh (a) and in a hybrid or varmesh (b)

different resolution. Schemes are available to do this but they are difcult to operate andlead to losses or instability. Such schemes are referred to asmultigrid schemes and they a

the subjects of continuing research interest. Mesh distortion may be implemented as shin Fig. 4.18(b). Here, a ne mesh region is generated around the ne feature but it is n well localized as in Fig. 4.18(a). This mesh is known as a hybrid or variable mesh and hobvious advantage that one-to-one spatial correspondence has been reestablished. Synchronmay also be achieved by a judicious use of stubs but at a reduced time step. We see herthe shape of each node is a general cuboid and this allows us to t better to various fepresent in the problem. This distortion is not achieved without a price. The use of stubs rein a higher dispersion compared to a regular mesh (one with cubes as the basic cell) andmust be taken to control errors. More details for the hybrid mesh may be found in [4] an

multigrid and other multiscale techniques in [19]. The second option is to keep a regular coarse mesh but toembed local solutionsinto thmesh that represent the local EM behavior of the ne feature. We focus on this approhere. It follows the ideas developed in Section 4.5. To illustrate more clearly this discussaddress the specic problem of a thin wire of radiusa placed centrally at a 2D node as shownFig. 4.19. This region of space containing the wire interacts with the rest of the problem throu


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a

= /2

4

3

2

1

FIGURE 4.19: Representing a ne wire (radiusa ) in a 2D node

the four port voltages. If we are able to present to incident voltages on these ports an admittance, which correctly represents the relationship betweenH and E in this region of space (includingthe impact of the wire), then we can obtain the reected voltages and implement connection tothe rest of the problem. The presence of the thin wire makes its impact through the modicationof the admittances compared to the case of free space. The best way of implementing this ito follow the same procedure as for the free space case (Section 4.5) by analyzing port voltaginto modes, reect each mode using the appropriate admittance and then combine the reectedmodes to obtain the reected mode voltages. The formal steps are the same as for the free spaccase the only differences are in the detailsthe nature of each mode and its admittance. Notethat in this way, the spatial resolution of the mesh ( ) is independent of the size of the wire(radius a). We can get away with quite a coarse mesh (a ) and still get an accurate wire

description. This substantially reduces computational demands. It is for these high stakes that we are playing in this section.

Our starting point is Eq.(4.41)whichshows theeld expansionin cylindricalcoordinates.In the case of free space the recognition of the singularity atr =0 meant that Dn had to bezero thus giving Eq. (4.42). In the present case of a wire at the center of the node the constanDn must be chosen so that the tangential electric eld on the surface of the wirer =a is zero,i.e.,

E z(a , )

=+

n=B

ne j n [ J

n(k

c a)

+D

n N

n(k

c a)]

=0 (4.96)

Hence,

Dn = J n(kc a) N n(kc a)

(4.97)


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Substituting (4.97) into (4.41) we obtain the following expression for the electricaround a wire of radiusr

E z(r , ) =+

n

=

Bne jn J n(kc r ) J n(kc a) N n(kc a)

N n(kc r ) (4.98

The expression for the magnetic eld is obtained from (4.43) which is repeated herconvenience.

H (r , ) =1

j 0 E z(r , )

r (4.99

The calculation now proceeds as for free space by identifying the modes and corresponadmittances. Starting with moden =0 we have by dividing (4.98) by (4.99)

E zH |n=0 = j 0

J 0(kc r ) N 0(kc a) J 0(kc a) N 0(kc r ) N 0(kc a) d

dr J 0(kc r )

J 0(kc a) d

dr N 0(kc r )

(4.100

We evaluate the numerator and denominator separately using small argument expanfor the Bessel functions (Appendix 2).

num = 1 (kc r )2

42

lnkc a2 + 1

(kc a)2

42

lnkc r 2 +

2

lnkc a2 +

2

lnkc r 2 +

2

lnar

where we have neglected terms (kc r )2, (kc a)2 relative to 1. Similarly, for the denominator obtain

denom2

lnkc a2 +

k2c r 2

2

1r

Hence, the impedance for moden =0 is obtained from (4.100) evaluated at the nodedge r = .

E z0

H z0 |r

= =j 0

lna

(kc )2

2ln kc a

2 + 1j 0 ln

a(4.101

A similar procedure is applied for the remaining modes to obtain

E znH n |z=

j 0n

2n a2n2n +an

, n =0 (4.102


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s/c

Z

Z Z

Z

Z s

FIGURE 4.20: Structure of a shunt 2D node representing a cell containing a ne wire

We will now devise a circuit which presents to moden = 0 the impedance given by (4.101) and to the other modes the impedance given by (4.102). The topology of the circuit isshown in Fig. 4.20 where the link lines have a characteristic impedanceZ and the short-circuitstub an impedanceZ s . Using the modal structure in Table 4.1 it can be shown that for modesn =0 the voltage at the center of the node is equal to zero. Hence, these modes see effectivelyshort-circuited TL segment of characteristic impedanceZ and length . We demand that theinput impedance of this segment is equal to the value given by (4.102).

Z IN = j Z tan (k TL ) = j 0

n

2n a2n2n +a2n

, n =0 (4.103) where the propagation constant is givenk TL = 00/ 2. Substituting and approximatingtan(.) we obtain

Z = Z TL1n

2n

a2n

2n +a2n , n =0 (4.104) where Z TL = 2 0/ 0 is the link line impedance of the nodes representing free space inthe absence of an embedded wire. This choice of Z ensures that moden =1 sees the correctimpedance. Mode n = 2, as before, is not presented with exactly the correct impedance. Wemust now ensure that moden = 0 sees the correct impedance. This is the only mode that is


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s/cZ Z s

Z IN0 Z IN s

FIGURE 4.21: Circuit seen by moden =0 (see Fig. 4.20)

affected by the stub! The equivalent circuit seen by this mode is shown in Fig. 4.21. Fromcascade of the two TL segments we obtain

Z INs

=j Z s tan (k TL )

Z IN0 =Z I Ns

Z + j tan (k TL )1 + j

Z I NsZ

tan (k TL )

Combining these two equations and demanding thatZ IN0 is equal to the expression(4.101) we obtain the required stub characteristic impedance,

Z s = Z TL ln a Z (4.105

Therefore, the presence of a wire inside a node is recognized by altering the paramof the node as shown in Fig. (4.20) and Eqs. (4.104) and (4.105). Pulses coming from thespace region, upon encountering the boundary of the node containing the wire, experiendiscontinuity (coming from an impedanceZ TL to an impedanceZ ). Scattering in the wire nois also different because of the presence of the stub. The radius of the wire can be much smthat the radius of the node (a ). In this way the size of the computation is not dictaby the size of the wire. Yet its impact is much more accurately included in the model. Claapproaches to this problem are based on the formulation given in [20]. Here the local soluembedded into the model is based on the quasi-static elds around the wire, i.e., the magnand electric elds near a wire carrying currentI and charge Q are given by

H (r ) = I 2 r , E (r ) = Q 2 r (4.106Oneshouldcontrast Eq. (4.106) with (4.98)and(4.99). In themodal expansiontechniq

(MET) adopted here we take account of four modes (in 2D), while in the classical appr


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that of only one. We therefore get a greater accuracy and we are also able to locate the wiranywhere inside the node (not just centrally). This is achieved by exploiting the addition theo-rems for Bessel functions. Further details on the MET and how it is applied for offset wires andmulticonductor systems (more than one wire inside a node) may be found in [2123]. Feature

other than wires may also be embedded inside a node such as dielectric rods, etc., using a similaapproach [24].


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69

C H A P T E R 5

An Unstructured 2D TLM Model

The development of TLM models we have described so far is based on highly structured meshes which are aligned along Cartesian coordinates. It is possible to develop meshes based on cylindrical or spherical coordinates to suit problems with a particular symmetry andthese areavailablein TLM [25, 26]. However, whatever the coordinate system chosen, these remain highly struc-tured meshes. This makes it easy to establish a mesh and because of the high level of regularit

(most nodes have exactly the same parameters) programing and tracking various quantities isimple. It isnevertheless true that a structured mesh isnot ideally suited to describingboundaries which do not coincide with coordinate axes, e.g., curved boundaries in a Cartesian mesh. This isillustrated in Fig. 5.1(a) where we see that a curved boundary is approximated by a stair-caseboundary resulting under certain circumstances in stair-casing errors [27]. In contrast, if insteadof a rectangle as the basic nodal element we employ a triangle as shown in Fig. 5.1(b) thenmuch improved description of the boundary is achieved. We now have, however, a collection otriangles of different shapes and sizes forming an unstructured mesh. The disadvantage is thateach node is potentially different thus requiring storage of a substantial amount of node-based

information. In addition, control of the time step to maintain synchronism across the mesh isnow not a trivial matter. The accuracy of the results is strongly dependent on the quality othe unstructured mesh. All these difculties are familiar to researchers of the Finite ElementMethod where traditionally unstructured meshes are employed. As is common in modeling, anunstructured mesh solves some problems and creates others and it is up to the modeler to reacha balanced judgement as to where and how an unstructured mesh should be employed.

We describe below the theoretical development of 2D TLM unstructured models in thetime-domain.

5.1 A TRIANGULAR MESH IN TLM We focus on a shunt node for TM waves shown schematically in Fig. 5.2 [28]. The gure showsthree ports connecting the node to its neighbors all originating from the center of the node. Thefull geometrical description is contained in the distancesi and angles i shown. The electriceld E z is perpendicular to the plane of the paper.


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(a) (b)

FIGURE 5.1: Curved boundary approximation using a Cartesian mesh (a) and a triangular mesh

The general approach, as before, is to decompose the electric eld into modal compone(three in this case as we have three degrees of freedom) and work out an admittance mrelating magnetic and electric elds. After that this admittance matrix is mapped onto a cito obtain the TLM model.

We express the elds in the vicinity of the node as the superposition of the rst modes of an expansion of the eld in cylindrical coordinates around the node center. gives

E z(r , ) = J 0(kr ) X c 0 + cos( ) J 1(kr )2 X c 1

k+ sin( ) J 1(kr )

2 X s1k

(5.1

j 2j 1

j 0 1

2

0

Port 1

Port 2

Port 0

q

FIGURE 5.2: The basic notation and structure of a triangular node


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AN UNSTRUCTURED 2D TLM MODEL 71

where the coefcientsX represent the modal mix of the total eld. Similarly, for the magneticeld we have

j 0 H (r , ) = E z(r , )

r (5.2)

The electric eld at the three ports may now be expressed in terms of the modal components X by applying (5.1) at each port. As an example, for port 1 we get

E z1 = J 0(k 1) X c 0 + cos(0) J 1(k 1)2 X c 1

k+ sin(0) J 1(k 1)

2 X s1k

1 (k 1)2

4 X c 0 +

k 12

2 X c 1k

+ 0 X s1

X c 0 + 1 X c 1

(5.3)

where we have used small argument approximations for the Bessel functions and neglecteterms of the order (k )2. Equation (5.3) gives the rst row of a matrixT e relating the electriceld to its modal components.

E z = T e X (5.4)

where

E z = [ E z1 E z2 E z0 ]T , X = [ X c 0 X c 1 X s1 ]T

T e =

1 1 01 2 cos(0) 2 sin(0)1 0 cos(0 + 1) 0 sin(0 + 1)

(5.5)

We express in a similar way the magnetic eld using (5.2)

j 0 H (r , ) = J 0(kr )

r X c 0 +

J 1(kr )r

cos( )2 X c 1

k+

J 1(kr )r

sin( )2 X s1

k(5.6)

The derivatives of the Bessel functions are then replaced by their small argument approximations (Appendix 2) to obtain the magnetic eld in port 1.

j 0 H 1 = k2 1

2X c 0 +

2 X c 1k

cos(0)k2

+2 X s1

ksin(0)

k2

j 0 1 H 1 = k2 21

2X c 0 + 1 X c 1

(5.7)


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The last expression is effectively the rst row of the matrixT h relating the magnetic e vector to its modal components.

j 0 D H = T h X (5.8

where the transformation matrix is

T h =

(k 1)2

2 10

(k 2)2

2 2cos(0) 2 sin(0)

(k 0)2

2 0cos(0 + 1) 0 sin(0 + 1)

(5.9

D =

1

2

0(5.10

From (5.4) we obtainX = T 1e E z and substituting in (5.8) for X we get

j 0 D H = T h T 1e

E z (5.11

Inverting the matrixT e and carrying out the manipulations in (5.11) we obtain an expresrelating the magnetic eld in the three ports to the electric eld, i.e., the elements oadmittance operator. This is written out in full below.

H 1H 1H 1

= j 02

2 0 1s1 1 0 1s2 2 1 1s02 0 2s1 2 0 1s2 2 2 1s02 0 0s1 0 0 1s2 2 0 1s0

2 0s1 + 1 0s2 + 2 1s0

E z 1 E z 2 E z 0

+ 1 j 0

0s2 + 2s0 0s2 2s0 0s1 0s1 + 1s0 1s0

2s

1

1s

2 1s

2+

2s

1

2 0s1 + 1 0s2 + 2 1s0

E z 1 E z 2 E z 0

(5.12

wheresi = sin(i ). Examining (5.12) we can see that the rst term on the RHS is capaciand the second inductive. This gives us some encouragement that an LC network maconstructed to present an admittance in accordance with (5.12). To investigate further we


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electric elds to voltages and magnetic elds to currents as shown below.

E z 1 E z 2

E z 0

V 1V 2

V 0

,

s1 0 00 s2 0

0 0 s0

H 1H 2

H 0

I 1 I 2

I 0

(5.13)

where is a constant to be determined. Substituting (5.13) into (5.12) we obtain an equationrelating port voltages and currents.

I 1 I 2 I 0

= j 02

2 0 1s1s1 1 0 1s2s1 2 1 1s0s12 0 2s1s2 2 0 1s2s2 2 2 1s0s22 0 0s1s0 0 0 1s2s0 2 0 1s0s0

2 0s1 + 1 0s2 + 2 1s0

V 1V 2V 0

+

j 0

0s2s1 + 2s0s1 0s2s1 2s0s1 0s1s2 0s1s2 + 1s0s2 1s0s2 2s1s0 1s2s0 1s2s0 + 2s1s0

2 0s1 + 1 0s2 + 2 1s0

V 1V 2V 0

(5.14)

A circuit topology with three ports such that the port currents and voltages are relatedas shown in (5.14) would give us the desired model. However, there is a problem. Circuitconsisting of inductors, capacitors, etc., arereciprocal [29]the admittance matrix is symmetric.Examining (5.14) we see that the inductive term is reciprocal but this is not the case for thecapacitive term. In order to get a physically realizable circuit we need to have symmetricmatrices. This requirement invites further examination of thematrices to seewhether reciprocity can be achieved. It is evident from (5.14) that the inductive term dominates at low frequencie(the domain of model validity). However, we cannot altogether neglect the capacitive termas in the case when all port voltages are the same the inductive term vanishes. This can bconrmed by direct substitution into the second term on the RHS withV 1 = V 2 = V 0. Underthese circumstances, the only term left is the capacitive one and it is equal to

I 1 I 2 I 0

= j 02

1s1 0 00 2s2 00 0 0s0

V 1V 2V 0

,


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V 1

V 0

V 2

C 0

C 1

C 2

L1

L2

L 0

s i i

0 i

S i

0

L i =

C i = 2

FIGURE 5.3: Network representation for a triangular-shaped cell

The expression we now need to map onto a circuit is given below.

I 1 I 2 I 0

j 02

1s

1 0 00 2s2 00 0 0s0

V 1V 2

V 0

+

j 0

0s2s1 + 2s0s1 0s2s1 2s0s1 0s1s2 0s1s2 + 1s0s2 1s0s2 2s1s0 1s2s0 1s2s0 + 2s1s0

2 0s1 + 1 0s2 + 2 1s0

V 1V 2V 0

(5.16

The circuit representing this admittance matrix is shownin Fig. 5.3. This can be conrmdirectly by applying nodal analysis in the circuit to relate voltages and currents. We notthe circuit parameters are given by

Li = 0 i

asi , C i =

0si i 2

(5.17

In order for these components to be positive (to ensure stability)si = sin(i ) must bpositive and hence all the angles i must be less than . This constraint is familiar to nelement practitioners where similar meshes are used and is known as Delaunay triangu

[30]. It maximizes the minimum angles of all the triangles in an effort to avoid sliver and thin) triangles. In FE work a sliver triangle whose two sides are almost identical indsolutions which are almost the same and hence a system of equations containing two alidentical equations. This gives ill-conditioned matrices making a numerical solution difand inaccurate. In unstructured TLM, the Delaunay condition ensures positive passive elemand hence stability.


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A further condition to be investigated is the continuity of electric and magnetic eldsacross node boundaries. We see from (5.13) that continuity of voltage at the boundary betweennodes ensures continuity of electric eld. However, continuity of current does not automaticallyensure continuity of H since I i = si H i . In order to get magnetic eld continuity we need

to enforce the condition that si is the same in adjacent nodes. A full discussion of this may be found in [28] and it leads to two possibilities. The rst is to dene the node center asthe center of gravity (COG) of the triangle dened by its ports. The centroid (COG) of atriangle is the point at which the medians of the triangle intersect. The second approach is todene node centers as the circumcenters (CCM) of the Delaunay triangulation. We discussfurther only the CCM with the denition of the node center shown in Fig. 5.4. From thisgure we obtain, 2Rsi = i where R is the circumradius of the triangle and i is the sideof the triangle opposite to the angle i . If we choose the constant to be equal to 2R foreach node, then this gives for each node,I i = asi H i = 2Rsi H i = i H i . Since the length i

is the same across the sides of adjacent nodes it follows that continuity of current also meancontinuity of magnetic eld. The ports for the CCM conguration lie halfway between the nodecenters as shown in Fig. 5.4. Unstructured meshes need careful examination to obtain optimumconditions especially as regards the permissible time step. For computational efciency reasonit is important to maximize the time step. This, in a time-domain method, means that thelink line length must not be allowed to become too small. Even in a Delaunay mesh there arcircumstances where the circumcenter is outside the triangle and may be very close (or evecoincide) to the circumcenter of another triangle. Thus, this means that the link line betweenthese two circumcenters becomes very short with very unfavorable implications for the tim

step. Such conditions must be detected and avoided. One approach is to merge such trianglesinto a quadrilateral. Details of these techniques may be found in [28].

j i

i

R

FIGURE 5.4: Node center dened as the circumcenter (inner small circle), centers of adjacent nodes(outer dotted small circles) and node interfaces (full small circles)


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Z L 0

Z L 2

Z L 1

Y S 0Y S 1

Y S 2

FIGURE 5.5: TLM representation of a triangular cell

The implementation of the circuit shown in Fig. 5.3 in TLM is straightforward. Ooption is to model all inductance in the link lines and any decit in capacitance is addedstubs as shown in Fig. 5.5. Scattering and connection are implemented following the sprinciples as for the rectangular shunt node.

5.2 APPLICATIONS OF THE TRIANGULAR MESHI have indicated at the beginning of this chapter the advantages and disadvantages of unstured meshes. The development in Section 5.1 will have made it clearer how and when t

a triangular mesh. There is a signicant overhead as each node is different and the qualthe mesh has a signicant impact on accuracy. Controlling the time step to avoid patholoconditions such as those arising when two circumcenters coincide or are very near to eachis imperative. Extensive stubbing may be required to maintain synchronism and this againto dispersion. Each Delaunay mesh will give a solution with a different error. It appears seto suggest that unstructured meshes should be used only when necessary. I can see two areas where they may be employed with a clear advantage.

First, in case of intricate or curved boundaries, where the avoidance of stair-casing is important, using triangular meshes may be the best option. But even then consideration

be given as to whether the entire space must be meshed with triangles instead of only areathe boundary to conform the mesh better to it.Second, a useful approach in many problems is to use a hybrid mesh where certain reg

are meshed with a coarse structured mesh, others with a ne structured mesh and some an unstructured triangular mesh. In this way an optimum mesh is employed for each partipart of the problem. The triangular mesh can be used to stitch together different mesh area


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FIGURE 5.6: Triangular elements (shown in thicker line) used to join twoCartesian meshesof differentspatial resolutions

shown schematically in Fig. 5.6. This technique is described further in [31]. Therefore, ratherthan an indiscriminate use of an unstructured mesh a more intelligent approach is to use itas a high-quality, specialist, expensive modeling medium. Used in this way it is an excellencomplement to other meshes in TLM.


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79

C H A P T E R 6

TLM in Three Dimensions

6.1 BASIC CONCEPTS IN 3D TLM We have traced the development of TLM models for structured and unstructured meshes in2D and we must now examine how these techniques are applied in three dimensions (3D).Naturally, 3D networks need to be devised to do the necessary mapping. The approach is verysimilar to that adopted in 2D but with extra complexity due to the higher dimensionality. I will

therefore avoid excessive detail that does not add much to the physical understanding and themodeling philosophy that was already described as part of the 2D work. I will focus insteain this section on how modeling is done in 3D and leave more sophisticated topics for thefollowing sections and for self-study of the available literature.

We start again with a cell of space with dimensions x y z and seek to representthe properties of the EM in this cell by analogy to the behavior of a node in a 3D networkOver the years, a number of nodes have been developed for this purpose but the most widelused structure at present is the Symmetrical Condensed Node (SCN) shown in Fig. 6.1 [32]. Itconsists of a network of interconnected lines such that on each face of the cell there correspon

two ports orthogonal to each other. In this way, any eld polarization can be accounted for. The port voltages are labeled by numbers 112 in the traditional labeling scheme or by thresubscripts, the rst indicating the direction of propagation (x, y, or z), the second indicating whether the segment is along the negative or positive coordinate axis (n or p), and the third thepolarization of the pulse (x, y, or z). A superscript may be added to indicate incident or reectedpulse (i or r ). In addition stubs may be added to account for different materials, noncubicalcell shapes, and losses. Rather than complicating matters from the start, we will examine rsthe simple cubic SCN representing an air-lled cell to get the basic scattering and connectionschemes. We will also postpone a discussion of the parameters of the node to map a particula

cell to another section as this is intricately connected with material properties, shape of the celand the use of stubs. As in the case of the 2D nodes we need to address scattering, connectionexcitation and output, and treatment of boundaries. However, unlike in the case of the 2Dnodes a simple equivalent circuit for the SCN cannot be derived and it is therefore necessary tobtain its scattering properties by resorting to more general principles. Examining carefully thstructure of the SCN in Fig. 6.1 you may identify series and shunt nodes as its constituent


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ynzV ynx

zpyV

zpyV

xnyV

xnzV

ypz

ypx V zny

znx

xpz

xpyV

node ( x, y, z)

V

V

V

V

V

FIGURE 6.1: The 3D Symmetrical Condensed Node (SCN) with two alternative port notations

Two such nodes are shown in Fig. 6.2 where for generality I have also indicated the characterimpedance of each line segment to be different. It would be a mistake to suppose thatV y mabe calculated from Fig. 6.2(a) as for the 2D shunt node and therefore thatV r xny =V y V i xny This expression is incorrect for the 3D node. I have taken care to indicate in Fig. 6.2 tconnection at the center of these two nodes is somehow implied but does not actually An equivalent voltageV y and current I z may be dened but they are not the actual voltage current on each line. We will return to this idea later on. If a simple circuit cannot be dehow the scattering matrix can be obtained? The answer lies in enforcing fundamental princ[32, 33, 34]. Specically, we demand

Conservation of electric charge. Conservation of magnetic ux. Electric eld continuity. Magnetic eld continuity.


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TLM IN THREE DIMENSIONS 81

V y11 V xpyV xny 3

V zpy 8

4 V zny x

y

z

11 V xpyV xny 3

V ypx

12

1

V ynx

(b)

(a)

I z

FIGURE 6.2: Shunt (a) and series (b) nodes extracted from Fig. 6.1 used to illustrate scattering in the3D SCN

Enforcement of each of these principles gives three equations relating incident and re-ected voltages and therefore 12 equations in total. This is enough to calculate the elements othe scattering matrix relating the 12 reected voltages components to the 12 incident compo-nents. We assume for simplicity that the total capacitance and inductance of each link line areC and L, respectively. We enforce each condition separately.


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Followingelectric chargeconservation lawfor they-component of theelectric eld showin Fig. 6.2(a) we equate the total incident charge to the total reected charge for the fourlink lines

C 2 V

i xny +V

i zny +V

i xpy+V

i zpy =

C 2 V

r xny +V

r zny +V

r xpy+V

r zpy (6.1

Other two equations are obtained in a similar number for thex-component anz-component of the electric eld. Conservation of magnetic ux h B d s =0 implies thathe total ux linked to all the lines is zero, i.e.,

n n = n Ln I n = n Ln I

i n I r n =0 (6.2

Expressing this condition for thez-component of the magnetic eld shown in Fig. 6.2 we get

L2 I

i xny I i ypx+ I i xpy+ I i ynx =

L2 I

r xny I r ypx+ I r xpy+ I r ynx (6.3

Substituting in (6.3) in terms of the voltage and recognizing thatI i =V i / Z , I r =V r / Z we obtain

V i xny V i ypx+V i xpy+V i ynx =V r xny +V r ypxV r xpyV r ynx (6.4Other two equations are obtained in a similar manner for thex-component an

y-component of the magnetic eld.Continuity of the electric eld implies that they-component whether calculated on th

z-directed or thex-directed lines in Fig. 6.2(a) must be the same, i.e.,

V zpy +V zny = V xpy+V xnyor, expressing the total voltages in terms of incident and reected voltages we get

V i zpy +V r zpy +V i zny +V r zny =V i xpy+V r xpy+V i xny +V r xny (6.5

Other two equations are obtained in a similar number for thex-component anz-component of the electric eld.

Continuity of the magnetic eld implies that thez-component must be the same whethecalculated from thex-directed or y-directed lines in Fig. 6.2(b), i.e.,

I xpy I xny = I ynx I ypx


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or, expressing the currents in terms of incident and reected voltages we get

V i xpyV r xpy(V i xny V r xny) = V i ynxV r ynx(V i ypxV r ypx) (6.6)Other two equations are obtained in a similar manner for thex-component and

y-component of the magnetic eld.Equations (6.1), (6.46.6) and their equivalents for the remaining polarizations form a

system of 12 independent equations that give all the information necessary to derive scatterinfor the SCN

kV r =SkV i (6.7) where the scattering matrix for the SCN representing a cubic cell is

S =0.5

0 1 1 0 0 0 0 0 1 0 1 01 0 0 0 0 1 0 0 0 1 0 11 0 0 1 0 0 0 1 0 0 0 10 0 1 0 1 0 1 0 0 0 1 00 0 0 1 0 1 0 1 0 1 0 00 1 0 0 1 0 1 0 1 0 0 00 0 0 1 0 1 0 1 0 1 0 00 0 1 0 1 0 1 0 0 0 1 01 0 0 0 0 1 0 0 0 1 0 10 1 0 0 1 0 1 0 1 0 0 01 0 0 1 0 0 0 1 0 0 0 10 1 1 0 0 0 0 0 1 0 1 0

(6.8)

and the 12 vectors for incident and reected pulses follow the number labeling of Fig. 6.1 When programing scattering for computation we rarely use the matrix given in (6.8). Instead, itis more efcient to calculate the reected voltages directly from the following equations (usinthe alternative labeling).

V r ynx =12

V i znx +V i zpy +V i xny V i xpyV r ypx = 12 V

i znx +V i zpx +V i xpyV i xny

V r znx =12

V i ynx+V i ypx+V i xnz V i xpzV r zpx =

12

V i ynx+V i ypx+V i xpz V i xnz


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V r zny =12

V i xny +V i xpy+V i ynz V i ypzV r zpy =

12

V i xny +V i xpy+V i ypz V i ynzV r xny = 12 V i zny +V i zpy +V i ynxV i ypxV r xpy =

12

V i zny +V i zpy +V i ypxV i ynxV r xnz =

12

V i ynz +V i ypz +V i znx V i zpxV r xpz =

12

V i ynz +V i ypz +V i zpx V i znxV r ynz

=

1

2V i xnz

+V i xpz

+V i zny

V i zpy

V r ypz =12

V i xnz +V i xpz +V i zpy V i zny

(6.9

The scattering in Eqs. (6.8) or (6.9) is based on a cubical cell( x = y = z = ) where all the link lines have the same characteristic impedance and from (3.8) and

t = LC , Z 0 = L/ C . From (3.10) we also getC = t / Z 0, L = tZ 0. Capacitancand inductance must correspond to those of the cell representing the block of medium parameters , . For example, thex-directed capacitance of the cell must be represented byfour half lines with elds polarized in thex-direction, i.e.,

y z

x = =4t 2Z 0 =

2 t Z 0

(6.10

Similarly, the inductance represented by the four half lines must be

y z

x = =4t

2Z 0 =2 tZ 0 (6.11

Multiplying (6.10) by (6.11) to eliminateZ 0 we then obtained the required time steprepresent this block of medium

t = 2u (6.12 whereu =1/ . For a cell in free spaceu =c the speed of light.

We have described so far the basic scattering procedure in 3D TLM. Connection proceein exactly the same way as for the 2D nodes by exchanging pulses with neighbors. Compu


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procedures in 3D TLM are identical to those in 2D except for the fact that we now deal with12 rather than 4 pulses.

This basic introduction would be incomplete without a mention of how we deal withinhomogeneous materials and noncubical cells. We had a similar discussion in the previou

chapter regarding inhomogeneous media. Since the velocity of propagation changes, it appearthat we have two options. First, we may keep constant and thus we have one-to-onecorrespondence across material boundaries but then t will be different in different materialsand we will loose synchronism. Second, we may maintain synchronism but loose one-to-oncorrespondence. Neither of these alternatives is attractive. Instead, we follow a procedure wherlink lines represent a background medium (free space in most cases) and we add capacitive stuto account for r > 1 and inductive stubs for r > 1. In this way, we maintain both synchronismand one-to-one correspondence across material boundaries. The other desirable feature of beingable to alter the shape of a cell from a cube to a general cuboid shape is dealt exactly in the sam

way by introducing stubs to modelC and L correctly in the different directions. The only aspectto watch out for is that we keep stub parameters positive to ensure stability. The principles arstraightforward. If we wish to represent a material with r > 1 in a TLM model where thebackground material is free space, then in thex-direction the desired capacitance is made outof the capacitance of the link lines [see (6.10)] plus the capacitance of an open-circuit stub.

=2 t Z 0 +C ox (6.13)

Hence, substituting 0 =1/ (Z 0c ) in (6.13) we obtain

C ox =1Z 0

(r c 2 t ) (6.14)

Taking the round-trip time for the stub to be t we then obtain its characteristic admit-tance

Y ox =2C ox

t =2Z 0

r c t 2 =

1Z 0

4(r 1) (6.15)

where for stability the time step based on the material with the lowest dielectric permittivit

(free space) has been chosen. The presence of materials with different permeability is dealt with in a similar manner b

introducing inductive (short-circuit) stubs. The characteristic impedance of an inductive stubis then,

Z sx = Z 04( r 1) (6.16)


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Different stubs may be introduced in other directions to account for the anisotroproperties.

Losses may be introduced by inserting an appropriate conductance. For a medium a complex dielectric constant given by

= r 0 j = r 0 1 j

r 0(6.17

and conduction conductivity , the total effective electric conductivity may be dened as

e = + (6.18Equation (6.18) may be written in terms of the loss tangent for the material that in

porates in an approximate manner the conduction conductivity

=r 0(1

j tan ) (6.19

where tan = e / ( r 0). To account forx-directed electric conductivity we need to connto our TLM network a conductance given by

G x = e z y

x = e (6.20It is clear from the denition of e that it generally depends on frequency so its int

duction into the time-domain code can be done simply only if it is approximated by a srepresentative value. Special techniques will be presented in Section 6.4 to deal more rately with frequency-dependent features. Magnetic losses may be introduced by incorpora resistance into the TLM code as described in connection with (6.20),

Rx = mz y

x = m (6.21 where m is the magnetic resistivity ( m1). Naturally, the scattering equations need to derived again in the presence of stubs. Rather than doing this using the classical approachshow in the next section another, more efcient, way of implementing scattering in a 3D Tnode.

6.2 A SIMPLE AND ELEGANT SCATTERING PROCEDUREIN 3D TLMIn the previous section we have shown how scattering may be implemented in 3D TLM.now show how this process can be further streamlined and made more efcient. Our discuresumes from the paragraph in the previous section introducing the four principles on wscattering is based. We have already mentioned in connection with Fig. 6.2 that there i


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DC connectivity at the center of the SCN, but it may be possible to derive equivalent voltageand currents. There is some algebraic effort involved in developing this technique, but it i worthwhile as it gives a very compact formulation for scattering for the most general nod[34, 35]. I will illustrate the development by examining in detail conditions at the center o

the node for the y-component of the electric eld and thez-component of the magnetic eld.Conditions for E y are shown in Fig. 6.3(a). In addition to they-polarized link lines I haveintroduced a capacitive stub (admittanceY oy) and electric losses (conductanceG y). An average

x

y

z

I z 11 V xpyV xny 3

V ypx

12

1

V ynx

(b) R z

s/c

Z sz

V sz

V y11 V xpyV xny 3

V zpy 8

4 V zny

(a)

o/c

Y oy

G y

FIGURE 6.3: Same as in Fig. 6.2, but with open short-circuited stubs and electric/magnetic losses


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equivalent voltageV y may be obtained for thex-directed y-polarized line by imposing charbalance.

V xnyY xny +V xpyY xpyt

2 = V y Y xny +Y xpyt

2(6.22

In (6.22) the LHS is the total charge on the two half lines and the RHS is the same chargon the two lines assuming an average voltage. In terms of the line impedances the equiv voltage is obtained directly from this equation as

V y =V xnyZ xpy+V xpyZ xny

Z xpy+Z xny

=Z xpy

Z xny +Z xpyV i xny +V r xny +

Z xnyZ xny +Z xpy

V i xpy+V r xpy(6.23

where I have expressed the total voltages as the sum of incident and reected pulses.

In a similar fashion we dene an equivalent currentI z in Fig. 6.3(b) for thex-directed y-polarized line, where in addition to the link lines I have introduced an inductive (impedance Z sz) and also magnetic losses (resistanceRz). The average current is obtained imposing ux balance

I xpyZ xpy I xnyZ xnyt

2 = I z Z xny +Z xpyt

2(6.24

where on the LHS we have the total magnetic ux linked with this line and on the RHSsame ux associated with the average current. Substituting in (6.24) for the total currenterms of incident and reected voltages and solving for the equivalent current we get

I z =V i xpyV r xpyZ xny +Z xpy

V i xny V r xnyZ xny +Z xpy

(6.25

Multiplying (6.25) by Z xny adding to (6.24) and solving forV r xny we obtain

V r xny =V y + I zZ xny +V i xnyZ xny Z xpyZ xny +Z xpy

V i xpy2Z xny

Z xny +Z xpy(6.26

Note that if the two halves of thex-directed y-polarized line have the same characterisimpedance Z xy then (6.26) simplies to

V r xny =V y + I zZ xy V i xpy (6.27Similar expressions may be obtained for all other reected components. However, ne

(6.26) nor (6.27) can be used straightaway as we do not yet know V y and I z as a function incident pulses only [in (6.23) and (6.25) we have both incident and reected pulses oRHS]. In order to get these equivalent quantities in terms of incident pulses only, we n


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to take account of the whole node shown in Fig. 6.3not just thex-directed y-polarizedlines. To nd V y [Fig. 6.3(a)] we impose charge conservation and charge balance onx-directed y-polarized lines andz-directed y-polarized lines.

Charge conservation (KCL) gives

Y xny V i xny V r xny +Y xpy V i xpyV r xpy +Y zny V i zny V r zny +Y zpy V i zpy V r zpy+Y oy V i oyV r oy G yV y =0

(6.28)

The last term in (6.28) represents the charge reected into the conductance representinglossesthere is no incident pulse (not a storage component).

Using (6.23), charge balance for thex-directed y-polarized lines gives

Y xnyV r xny +Y xpyV r xpy =V y Y xny +Y xpy Y xnyV i xny Y xpyV i xpy (6.29)

A similar expression is obtained by imposing charge balance on thez-directed y-polarizedlines.

Y znyV r zny +Y zpyV r zpy =V y Y zny +Y zpy Y znyV i zny Y zpyV i zpy (6.30)Substituting (6.29) and (6.30) into (6.28) and solving forV y we obtain

V y =2Y xnyV i xny +Y xpyV i xpy+Y znyV i zny +Y zpyV i zpy +Y oyV i oy

Y xny +Y xpy+Y zny +Y zpy +G y(6.31)

We now have the equivalent voltage as desired in terms of incident pulses only.

We now focus on the calculation of the equivalent current. Magnetic ux conservation(KVL) in Fig. 6.3(b) gives

V i xpy+V r xpy V i xny +V r xny + V i ynx+V r ynx V i ypx+V r ypx V i sz +V r sz I z Rz =0(6.32)

Using (6.24), ux balance for thex-directed y-polarized line gives

V r xpyV r xny =V i xpyV i xny I z Z xny +Z xpy (6.33)Similarly, ux balance for they-directed x-polarized line gives

V r ypxV r ynx =V i ypxV i ynx+ I z Z ynx+Z ypx (6.34)Substituting (6.33) and (6.34) into (6.32) and solving forI z we obtain

I z =2V i xpyV i xny +V i ynxV i ypxV i sz

Z xny +Z xpy+Z ynx+Z ypx+Z sz +Rz(6.35)


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This is as desired in terms of incident pulses only. We have now all we need to implscatteringwith (6.35) forI z and (6.30) forV y substituted into (6.26) or (6.27) for the simpcase we canobtain the reectedvoltageV r xny. Similar expressions can be derivedfor the remainreected voltage pulses. In compact form, the scattering equations for the most general

are

V r inj =V j I kZ inj V i ipj +hij (6.36V r ipj =V j I kZ ipj V i inj +hij (6.37

where the upper signs apply for(i , j , k){(x, y, z) , ( y, z, x) , (z, x, y)}and the lower signfor (i , j , k){(x, z, y), ( y, x, z), (z, y, x)}. The equivalent voltages and currents are given

V i =2Y kni V i kni +Y kpi V i kpi +Y jni V i jni +Y jpi V i jpi +Y oi V i oi

Y kni +Y kpi +Y jni +Y jpi +Y oi +G i (6.38

I i =2V i jpkV i jnk +V i kn j V i kp j V i si

Z jnk +Z jpk +Z kn j +Z kp j +Z si +Ri (6.39

where(i , j , k){(x, y, z) , ( y, z, x) , (z, x, y)}. The h-factors in (6.36) and (6.37) are given

hij =Z inj Z ipj Z inj +Z ipj

V i inj V i ipj (6.40

Note that for the common case of the same impedance for both halves of lines withsame direction and polarizationh-factors are equal to zero. The reected pulses into the st(open circuit and short circuit) are

V r oi =V i V i oi (6.41V r si = I i Z si +V i si (6.42

wherei {x, y, z}. The voltages reected into the components representing electric magnetic losses in thei -polarization areV i and I i Ri , respectively.

Thus, the scattering procedure is to calculate rst the three equivalent currents andthree equivalent voltages from (6.39) and (6.38), then theh-factors from (6.40) (if non-zer

and after that the reected pulses into the link lines (6.36) and (6.37), the stubs (6.41)(6.42).Scattering coefcients as components of the matrix in (6.8) may also be obtain

required (e.g. for dispersion analysis), for all nodes and are available in the literature [4Most TLM nodes in common use are not as general as assumed in this section therescattering is further simplied.


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6.3 PARAMETER CALCULATION AND CLASSIFICATIONOF TLM NODES

We have obtained the scattering matrix in (6.8) for a cubic node where all link lines havthe same characteristic impedance and another scattering procedure in the last section for a

General SCN (GSCN) node with stubs and losses and also potentially all half lines having adifferent characteristic impedance. You may wonder under what circumstances this GSCN may be employed and how all these parameters may be calculated. We will see that many differenchoices are possible resulting in a number of different nodes to model elds but the numericaproperties (dispersion, number of arithmetic operations, storage) differ. Firstly, we focus on thcase where the impedance of ani -directed j -polarized link line is the same on both sides of the node, i.e., Z inj = Z ipj = Z ij . Nodes where this equality does not hold are rare and only usedto obtain unequal arm lengths and adjust better near boundaries [34, 36]. For obtaining theparameters of any node we need to enforce the following conditions:

The total capacitance in each of the three coordinate directions must be equal to thecapacitance of the block of space represented by the cell.

The total inductance in each of the three coordinate directions must be equal to theinductance of the block of space represented by the cell.

Synchronism must be maintained (same propagation time on all link lines).

We have 18 unknowns: 6 link line capacitances, 6 link line inductances, 3 stub capacitances, and 3 stub inductances. The equation we obtain from the rst of the above conditions i

C yx y+C zx z +C ox = x y z

x(6.43)

The rst term in this equation is the capacitance in thex-polarization of they-directedline, the second term is for thez-directed line, and the third the contribution of thex-polarizedstub. The RHS is the cell capacitance assuming that the permittivity in thex-direction is x.Similar expressions may be written for the other two polarizations

C zy z +C xy x +C oy = yx z y

(6.44)

C xz x +C yz y+C oz = z y xz (6.45) We now impose the second condition

L yz y+Lzy z +Lsx = x y z

x(6.46)


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The rst term in this equation is they-directed z-polarized link line inductance, tsecond is for thez-directed y-polarized link line and the third for the contribution of tx-directed inductive stub. All three termscontribute to thex-directed magnetic eld. The RHis the inductance of the cell associated with thex-component of the magnetic eld. Simil

expressions are obtained for the remaining two components:

Lzx z +Lxz x +Lsy = yx z y

(6.47

Lxy x +L yx y+Lsz = z y x

z(6.48

The delay time along a line is given by (3.8) which applies to each of the six lintaking into account that we use per unit length quantities gives

t

=x C xyLxy t

=x C xzLxz t

= y C yzL yz

t = y C yxL yx t = z C zxLzx t = z C zyLzy (6.49Equations (6.43)(6.49) are 12 in total and we have 18 unknowns. We have thu

degrees of freedom and we can derive many different congurations by imposing additconstraints. This is the origin of the different nodes. It is not appropriate in this text to go the details of each node but a general classication will be given to assist the reader in msense of the various options.

We can put these equations into a compact form by using the synchronism equatioexpress all inductances and capacitances in terms of the characteristic impedances and atances

Z ij =Lij i

t Y ij =

C ij i t

Z sk =2Lsk

t Y ok =

2C 0kt

(6.50

Then Eqs. (6.43)(6.48) reduce to

Y ik +Y jk +Y ok2 = k

i j k t

(6.51

Z ij +Z j i +Z sk2 = k

i j k t

(6.52

If we take advantage of the six degrees of freedom to impose the condition that alink line impedances are equal toZ 0, the intrinsic impedance of the background medium (fspace in most cases,Z 0 = 0/ 0), we obtain thestub-loaded SCN [4, 32]. Equations (6.51


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and (6.52) become

2Y 0 +Y ok2 = k

i j k t

(6.53)

2Z 0 + Z sk2 = k i j k t (6.54) The open-circuit stub admittances and short-circuit stub impedances are then obtained

from

Y ok =2Y 0 rk

c t i j

k 2 (6.55)

Z sk =2Z 0 rkc t

i j k 2 (6.56)

where rk, rk are the relative permittivity and permeability in thek-direction, respectively.Clearly, for stability, the time step must be chosen such that the expressions in the bracketsin (6.55) and (6.56) are positive. This imposes a maximum permissible value fort . In freespace and for a cubic node we see that the maximum permissible time step is as expected fro(6.12). However, for a noncubic node the maximum time step is affected by the smallest nodadimension and also the aspect ratio of the node [34]. Indiscriminate grading of the cell tointroduce long and thin shapes may result in unacceptably small time steps.

As expected the node described above is not the only choice. We may develop a Type Hybrid SCN (HSCN) where all inductance is modeled by the link lines and hence we do notneed inductive stubs. This implies that the impedance of link lines modeling different magneticeld components may be different. Details of this node may be found in [37, 38]. Alternativelya Type 2 HSCN may be developed where all capacitance is modelled by the link lines (ncapacitive stubs) [39]. Similar constraints apply as for the stub-loaded SCN as regards themaximum permissible time step. The type 1 HSCN is the node most often used in commercial TLM codes. There are several other nodes described in the literature, e.g., MSCN [40], ASCN[41]. I also mention nodes that are based on non-Cartesian meshes [25]. There are also nodessuitable for frequency-domain implementations of TLM [42, 43]; these publications can beconsulted if required.

6.4 FIELD OUTPUT IN 3D TLMElectric/magnetic elds, current/charge densities are obtained in a similar way as for the 2Dnode. As regards elds there are two possibilities: mapping at the cell center or at the ceboundaries.


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First we may dene elds at the center of the node from the equivalences

E i = V i

i (6.57

H i = I i

i (6.58

where the equivalent voltages and currents are obtained from (6.38) and (6.39). As an exfor the 12-port SCN (no stubs) in free space thex-component of the electric eld is

E x = V x

x = V i ynx+V i ypx+V i znx +V i zpx

2 = V 1 +V 12 +V 2 +V 9

2(6.59

where both port notations have been used. Similarly, thex-component of the magnetic eld

H x = I x

x = V i

zny +V i

ypz V i

ynz V i

zpy

2Z 0 = V 4 +V 7 V 5 V 82Z 0 (6.60

Similar expressions apply for the remaining eld components. For the stub-loaded S we have 18 voltagespulses (12 from the link lines and three each fromthe capacitive and industubs) that map to six eld components (three for electric eld and three for magnetic at the node center. Therefore there is no one-to-one correspondence. A one-to-one (bijectmapping may be established at the cell boundaries. With reference to Fig. 6.4 we are interin the elds at the RHS of the boundary between the two cells. We illustrate as an exaconditions on the x-directed y-polarized line. The total y-polarized voltage on the RHS the boundary at timek +1/ 2 is made out of the sum of the voltage reected from the nodthe right at timek and the voltage incident on the node at timek +1 (this originates from thpulse reected from the node on the left at timek). The two pulses meet at timek +1/ 2.

k+1/ 2V y = kV r xny +k+1V i xny (6.61Similarly, the current there is

k+1/ 2 I y =kV

r xny k+1V

i xny

Z xny(6.62

Thus the two pulseskV r xny, k+1V i

xny at the boundary of the cell to the right map toV y an I z, i.e., E y and H z. Similar expressions may be derived for all other eld components tangeto the cell boundaries.


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x-directed

y-polarized

line, Z xny

k

k + 1/2

time

V r xpy ( x)k V r xny ( x + 1 )k

V i xny ( x + 1 )k + 1

FIGURE 6.4: Schematic showing the propagation of pulses used for eld calculation at cellboundaries

6.5 MODELING OF GENERAL MATERIAL PROPERTIES IN TLMIn the previous section we mentioned briey how losses may be introduced in TLM modelsthrough conductance and resistance for electric and magnetic losses, respectively. However, i

is evident that in this way only a xed loss value may be introduced which is independent othe frequency. Many materials exhibit strong loss dependence on frequency and in additionanisotropy, chirality, and nonlinear behavior. In order to account for the entire range of materiaproperties more sophisticated material models need to be developed that go beyond the mereintroduction of xed loss components. The topic is a complex one but of great practical valueIt cannot be fully treated in the present text. I will give, however, a introduction to the relevantechniques so that the reader may nd it easier to access more advanced work in the literatur[4450].

I will illustrate the modeling philosophy by focusing on modelingfrequency-dependent

conductivity. One can view a frequency-dependent component as introducing a transfer function(TF) H (). Modeling this TF in a frequency-domain model is straightforward. However, ina TD model, where frequency does not appear as a variable, the TF cannot be accounted fo without rst converting to the discrete time-domain. The problem posed is similar to the onein electrical lter theory: given an analog lter (specied in the FD) derive an equivalent digitlter (specied in the discrete TD). There are formal methods in lter theory to do just this,


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E y

H z

V 3 V 11

x

y

z

FIGURE 6.5: The TLM node for the study of 1D propagation in a general medium

e.g., through the application of z-transforms and we have to do something similar. Since algorithm resulting on the completion of this process is in the discrete TD it can be direembedded into the TLM algorithm. To reduce complexity and thus illustrate better the esseof the approach I develop fully a 1D model that can be readily generalized for 3D mod[4750]. The network conguration is shown schematically in Fig. 6.5. We need to deriv1D eld equation and the corresponding network equation, establish equivalence and then sothe network in the discrete TD. The process is similar to that described in Section 2.2 but the important difference that now the electrical conductivity is a function of frequency. Findependent of frequency, the electric current density and the electric led are related siby J e = E . Frequency dependence of a quantity implies delays and that the time histoimportant. A simple multiplication in the frequency-domain is translated into a convolutiothe time-domain.

J e (x, t ) =t

0

e (x, ) E (x, t )d = e (x, t ) E (x, t ) (6.63


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With this complication, Faradays laws and Amperes laws in 1D reduce to

H z (x, t )

x = J efy(x, t ) + e (x, t ) E y(x, t ) +0 E y(x, t )

t (6.64)

E y(x, t )

x = 0H z (x, t )

t (6.65)

In (6.64) the term J efy is to account for the possible presence of free electric sources. Wenow seek to derive a circuit analog to these equations. Unlike the work we have done so fa where we have used dispersionless TLs to construct nodes, we must now recognize that th TLs may contain components that in addition to normal spatial dependence also exhibit timedependence. In the particular case of the eld problem described by Eqs. (6.64) and (6.65) wanticipate that resistive losses which depend on (x, t ) must be incorporated. We start with a TL

equivalent as shown in Fig. 6.6. In addition to the normalLC components we have a shuntbranch consisting of a resistanceG e (x, t ) to represent the losses and a current sourceI f y(x, t )to account for the possible presence of free sources. Applying KCL in this circuit gives

I z (x, t )

x = I 3 + I 11 = I f y(x, t ) +G e (x, t )V y(x, t ) +C V y(x, t )

t (6.66)

I will now demonstrate that (6.66) is analogous to (6.64). Dividing both sides of (6.66)by and setting G e (x, t )/ = e (x, t ) the medium electrical conductivity, andC / = 0, we get

I z (x, t )

x = I f y(x, t )

+ e (x, t )V y(x, t ) +0V y(x, t )

t (6.67)

We now dene the following analogies between eld and circuit parameters

E y(x, t ) = V y(x, t ) , H z(x, t ) =

I z(x, t ) (6.68)

V 3 V 11

I 3 I 11

Ge I fy

L L

C C

FIGURE 6.6: Parameters of the line representing a medium with frequency-dependent electrical con-ductivity


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and the free current density as

J efy(x, t ) = I f y(x, t )

( )2(6.69

Substituting (6.68) and (6.69) into (6.67) gives an equation identical to (6.64) thus showthat solution of the circuit in Fig. 6.6 will give us complete information about the correspondeld problem. We can also apply KVL and show in a similar way that we obtain the canalog of (6.65). Let us now focus on solving (6.67). To do this we rst go through a procnormalization to simplify thevariousterms in preparation forsolution.We introducenormalitime T and spaceX variables by expressing the relevant derivatives as

x =

1 X

, t =

1t

T

(6.70

We also represent the current by a new quantity i z, which has the dimensions of vol

dened by the following expression

I z( X , T )

= i z( X , T ) 1

0(6.71

Substituting (6.70) and (6.71) into (6.67) and multiplying both sides by 0 we get

i z ( X , T )

X = I f y( X , T ) 0 + e ( X , T )V y( X , T ) 0 +00 t

V y( X , T )T

The rst term on the RHS of this equation is represented by a quantity i f y (dimensionof volts) dened as

i f y( X , T ) = J efy( X , T ) ( )2 0 (6.72 The second term is equal tog e ( X , T )V y( X , T ) where we have dened a normaliz

conductivity as

g e ( X , T ) = e ( X , T ) 0 (6.73Recognizing that / t =c results in the coefcient of the derivative in the third te

being equal to one. Hence, the circuit equation reduces to the following form

i z ( X ,

T ) X = i f y( X , T ) + g e ( X , T )V y( X , T ) +

V y( X ,

T ) T (6.74

The circuit in Fig. 6.6 can be redrawn in terms of the normalized quantities in (6.74observing that theLC components at the two ends of the segment can be expressed as twosegments of characteristic impedances0 and since we have normalized all other impedanto this value the normalized impedance of each TL is equal to one. This is shown in Fig. 6


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1 g e i fyV 3 V 11

+

g 2V 3 e i fy

1 1

V y(b)

+

Z =1/2

g e( X , T )

i fy( X , T )V y( X , T )

(c)

(a)

+i 2V 11

i

V 3i V 11

i+

FIGURE 6.7: TL representation in normalized form of the problem in Fig. 6.6 (a), and Theveninrepresentations (b) and (c)

with the corresponding Thevenin equivalent circuit in Fig. 6.7(b). Replacing the two parallebranches with voltage sources by their Thevenin equivalent we obtain the circuit shown inFig. 6.7(c). Summing the three currents on the branches to zero (KCL)

V y( X , T ) [V i 3 ( X , T ) +V i 11( X , T )]1/ 2 + g e ( X , T )V y( X , T ) +i f y( X , T ) =0

and rearranging we obtain

2[V i 3 ( X , T ) +V i 11( X , T )] i f y( X , T ) =2V y( X , T ) + g e ( X , T )V y( X , T ) (6.75) This is (6.74) in terms of the incident voltage pulses. The LHS of this equation is

essentially the excitation of this segment at timeT (incident pulses from the left and fromthe right and any free sources present within the segment) and it is convenient to express thi


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source term in terms of a single quantity.

2V r y ( X , T ) =2[V i 3 ( X , T ) +V i 11( X , T )] i f y( X , T ) (6.76Substituting (6.76) into (6.75) we obtain

2V r y ( X , T ) =2V y( X , T ) + g e ( X , T )V y( X , T ) (6.77Equation (6.77) must now be solved to obtain the total voltageV y( X , T ) at this segmen

Note that this is notstraightforward, as theunknown appears also insidetheconvolution integ(second term on the RHS). Transforming this equation in the discrete time-domain using z-operator gives

2V r y ( X , z) =2V y( X , z) + g e ( X , z)V y( X , z) (6.78 where in the z-domain the convolution becomes a simple multiplication. We manipulateexpression further by putting the frequency-dependent termg e ( X , z) in the form

(1 +z1) g e ( X , z) = g e 0( X ) +z1 [ g e 1( X ) + g e ( X , z)] (6.79 What this does is to make the frequency-dependent part of the conductivity to dep

on the eld at the previous time step (delay operatorz1 in front of the brackets on the RHS This gives an explicit algorithm. Multiplying (6.78) by (1+z1), substituting from (6.79) ansolving forV y ( X , z) after some algebraic manipulation

V y( X , z)

=T e 2V r y ( X , z)

+z1S ey( X , z) (6.80

whereT e = (2 + g e 0)1 is a frequency-independent term, and

S ey( X , z) =2V r y ( X , z) +e ( X )V y ( X , z) g e ( X , z)V y( X , z) (6.81 The frequency independent terme ( X ) = (2 + g e 1( X )). We note from (6.80) that th

total voltage at any one instant depends on the current excitation (rst term on the RHS)contributions from theprevioustime step (second term on the RHS). The quantity S ey( X , z which represents this stored information is called the main accumulator. It is given by (and includes information from the previous excitation, total voltage and also the recent hiof the material [the third term in (6.81)]. Since we do not know at this stage the details omaterial we cannot give substance to the third term so we introduce yet another accumu(let us call it the conduction accumulator)S ec , where

S ec ( X , z) = g e ( X , z) V y ( X , z) (6.82


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V y y

r

z 1

S ey

V

T e

k e

+

++

+

+

S ec

2

g e (z )

FIGURE 6.8: Flow diagram in the z-domain to implement scattering

Equations(6.80)(6.82) areembodied into thegraph showninFig6.8(essentiallya digitallter algorithm). This is the formal solution to our problems of determining the total voltageV y in terms of the excitation. No further progress can be made without an exact specicationfor the frequency-dependent term g e ( X , z). Depending on the form of this function (andit may be a complicated function of z) we may end up needing to produce yet another lterto replace the box labeled g e ( X , z) in Fig. 6.8. Also note that if the frequency dependencefor the material is in the conductivity only the scheme in Fig. 6.8 will work for uswe onl

need to work out in each case the details of this box. I will do this for one particular materiaunmagnetized plasma [47], where the frequency-dependent conductivity has the form

g e (s) = g ec

1 +s c (6.83)

For simplicity, I suppress hereafter showing space dependence. In (6.83)s is the Laplace variable. We transform the conductivity into the z-domain by applying the bilinear transfor-mation [18].

s 2t 1 z1

1 +z1 (6.84)Substituting fors into (6.83) we obtain after some algebra

g e (z) =K 1 1 +z1

1 z1a1(6.85)


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where the two constants are given by

K 1 = g ec

2 c

t +1

, a1 =2 c

t 12 c

t +1

(6.86

Multiplying (6.85) by 1 +z1 we obtain after some algebra

1 +z1 g e (z) = K 1 +z1b0 +z1b11 z1a1

(6.87

where the two new constants are

b0 = K 1 (2 +a1) , b1 = K 1 (6.88Comparing (6.87) with the standard form in (6.79) we obtain

g e 0 = K 1, g e 1 =0, g e (z) =b0 +z1b11 z1a1

, e = 2 (6.89 We now have the functional form of the material properties for the magnetized pla

and we can thus proceed to design an algorithm to calculate the conduction accumulat(6.82).

S ec (z) = b0 +z1b11 z1a1

V y (z) (6.90

We dene a state variableX 0(z) such that

S ec (z) = (b0 +z1b1) X 0 (z) (6.91Substituting (6.91) into (6.90) we obtain

X 0 (z) = z1a1 X 0 (z) +V y (z) (6.92 This equation allows us to calculate the state variable at the current time step as a funct

of its value at the previous time step and the current value of the total voltage. Equations (and (6.92) may be implemented by the ow graph shown in Fig. 6.9. We can conrm thatransfer function of this lter is as desired by checking directly or by applying Masons ru

TF =F 1 +F 21 L1

(6.93

whereF i and Li are the forward and loop gains. For the scheme in Fig. 6.9 these gains areF 1 =b0, F 2 = z1b1, L1 = z1a1. Substituting these gains into (6.93) we obtain (6.90


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V y

z 1

b0

b1a1

X 0

+

+

S ec

FIGURE 6.9: Structure of the shaded box in Fig. 6.8 for the particular example of an unmagnetizedplasma

expected. Figure 6.9 replaces box g e (z) in Fig. 6.8. There is one delay element in Figs. 6.8 and6.9 introduces another one, hence two quantities need to be remembered (stored) to advancethe calculation. In addition the incident voltages from ports 3 and 11 are required.

The complete algorithm for this particular material is as follows:

Obtain excitation at the current time step from (6.76). Calculate thetotalvoltageV y( X , T )at the current timestepfrom(6.80)using the current

value of the excitation and the main accumulator value obtained from the previous timstep.

Calculate the current value of the state variable using its previous value and the curren value of the total voltage using (6.92).

Calculate the current value of the conduction accumulator from (6.91). Calculate the current value of the main accumulator from (6.81). Calculate the reected voltages from

V r 3 ( X , T ) =V y( X , T ) V i 3 ( X , T ) , V r 11( X , T ) =V y( X , T ) V i 11( X , T ) (6.94) Implement connections to the adjacent segments and repeat the process given above

for the next time step.


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How may we generalize these ideas? First, we may consider introducing a full mothe magnetic properties that requires the incorporation of series components to supplemenshunt components in Fig. 6.6. Second, we may extend the model to 3D to take full accof the spatial structure of the 3D node. The principles for doing this work are similar to t

explained so far but the algebra is more complex and thus it is not included here. In geterms we need to update Eq. (6.76) to the form

F r = RT 1 V i 0.5V f (6.95 where F r represents the node excitation consisting of the incident voltage pulses (rst termthe RHS), and any free sources (second term on the RHS). The matrix RT 1 represents the TLMprocess and the vectorV f includes electric and magnetic sources. Equation (6.78) can nowput in the general form

F

=t (z)

F r (6.96

where F is the vector representing electric and magnetic elds andt (z) is a matrix representithe transmission properties of the material. In the presence of series components, theequations in (6.94) must be modied to account for the total voltage at each port that depend on th

Documents

- [Book] the TLM Method in Electromagnetics (C. Christopoulos 2006)