electromagnetic analysis

8/6/2019 Electromagnetic Analysis

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3.10 Small Coupling Approximation of Second OrderQuasi-Coupling 1503.11 General 3D Scattering Process Using Cell Notation. 1523.12 Co mplete Iterative Equations 1643.13 Contributions of Electric and M agnetic Fields to Total Energy 166Plane Wave Behavior3.14 Response of 2D Cell M atrix to Input Plane W ave 1683.15 R esponse of 2D Cell M atrix to Input W aves WithArbitrary Am plitudes 1783.16 Response of 3D Cell M atrix to Input Plane W ave 1793.17 Response of 3D Cell Matrix to Input W aves W ithArbitrary Am plitudes 183Appendices3 A. 1 3D Scattering Equations With Both Coplanar andAplanar Contributions 1853A.2 3D Scattering Coefficients W ith Bo th Coplanar and AplanarContributions 1873A .3 3D Scattering Coefficients in Term s of Circuit Param eters 189

. CORRECTIONS FOR PLANE WAV E AND ANISOTROPYEFFECTS 1944.1 Partition of TLM W aves into Com ponent W aves 1944.2 Scattering C orrections for 2 D Plane Waves: Plane W aveCorrelations Between Cells 1964.3 Changes to 2D Scattering Coefficients 203Corrections to Plane Wave Correlation4.4 Correlation of Waves in Adjoining Media With DifferingDielectric Constants 2064.5 Modification of Wave Correlation Adjacent a Conducting Boundary ... 207Decorrelation Processes4.6 De-Correlation Due to Sign Disparity of Plane and SymmetricWaves 2114.7 Minimal Solution Using Differing Decorrelation Factors toRemove Sign Disparities 222


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I. Introduction to Transmission Lines and TheirApplication to Electromagnetic Phenom enaMaurice Weiner

United Silicon Carbide, Inc.

In recent years, an exciting new branch of research activity has emerged, dealingwith extremely fast phenomena in semiconductors and gases. The introductionof high speed instrumentation and devices, with time scales often in the 1 to1000 picosecond range, has prom pted the investigation o f a variety o f fast ph enom ena, including the generation of electromagn etic pulses and light, pho toconductivity, avalanching , scattering, fast recom bination, and many other physicalprocesses. The research has been driven by several applications [1], [2]. Theseinclude ultra-wideband imaging and radar , as well as ultra-wideband comrnuni-cations(thus avoiding the use of wires or optical fibers). In addition, the availability of new, high speed instrumentation has provided researchers with avaluab le tool for learning the fundamental properties of materials. In all theaforementioned applications, a central feature is the generation of electromagnetic pulses with either a narrow pulse duration or a fast risetime(or both). Theshort time interval involved (in either the risetime or the pulse duration), insuresthat a wide frequency spectrum is produced, a property which is essential for thecited applications.

The understanding of fast phenomena and ultra-wideband electromagneticsources is made more complicated by the very fast risetimes and by the factthat the wavelength of the signals being produced are often smaller orcomparable to the characteristic length of the device or experimentalconfiguration under study. As a result the use of lumped circuit variables isinappropriate and we must use either transmission line variables or Maxwell'sequations directly.

/


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2 Electromagnetic Analysis Using Transm ission Line VariablesElectromagnetic signals with very short wavelength may be generated by a

sudden transition in the conductivity of the medium. Suppose, for example, anelectric field bias first is applied to the medium and that subsequently theconductivity of a portion of the medium is suddenly increased(for example, byphotoconductivity or avalanching). The sudden change in conductivity willgenerate electromagnetic pulses with very steep risetimes, thus producing shortwavelength signals. In cases where light is produced (for exam ple, when carriersrecombine in gallium arsenide), the wavelength naturally will be smaller or atleast comparable to the device size. In any event , the analyses often used todescribe devices and experimental configurations do not adequately address theshort wavelength signals which are generated, and subsequently dispersedthroughout the device and the surrounding space. One should not underestimatethe impo rtance of the electromagnetic energy dispersal(which includes lightsignals). Often the physics of underlying processes are misunderstood becausethe electromagnetic energy dispersal, which delivers the physics to the detector,is not taken into account properly, particularly for fast phenomena. It is hopedthe ensuing discussion will help to correct this deficiency and lead to a betterunderstanding of the dispersal of ultra-wideband electromagnetic signals andassociated phenomena.

In this volume we endeavor to describe fast electromagnetic phenomena,relying on iterative rate equations which use transmission line matrix(TLM)variables. As with comparable numerical techniques, such as the finitedifference method, the transmission line element must be made very small inorder to attain accuracy, and solutions at a given time step depend on aknow ledge of solutions at a previous time step. In terms of physicalinterpretation and intuition , however, the TLM method is far superior to that offinite differences or other similar numerical techniques. The physical appeal ofthe TLM method m ay be viewed, in a conceptual way, from the two basiccomponents which comprise the TLM matrix: the transmission lines and thenodes wh ich form the intersection of the lines, as noted in Fig. 1.1. W ith thismodel, we can conceptually separate the phy sics and energy dispersal of a givenprob lem in electromagn etics. Accordingly, the nodes represent the physics, andphysical processes(such as conductivity changes) are mapped onto the nodes,which then control the flow of energy in the lines. The other component, thetransmission lines, are responsible for the energy distribution and storage.The


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12 Electromagnetic Analysis Using Transm ission Line VariablesTable 1.1 exhibits the relationship between the wave equation using field

variables, Eq.(1.2), and that using circuit variables, Eq .(1.3). An impo rtant simplification occurs when we select a small transmission line element (or cell) oflength Al. It is useful to state the total capacitance , inductance, and cond uctanceassociated with the line element, wh ich we identify asC = C'Al = sA l (1.4a)L = L'Al =nAl (1.4b)G = G ' A l = a A l ( 1 . 4 c )

TABLE 1.1 COR RESPOND ENCE BETW EEN W AVEAND c m c u r r VARIABLES

ELECTROMAGNETICVARIABLE

FIELDPERMMrrrvTTYPERMEABILITY

E8

nCHARACTERISTIC {\iJz } mIMPEDANCEPROPAGATIONVELOCITYLOSSRELAXATIONTIME

{sYm

CT

{8/a}

c m c u r rVARIABLEVcL'

{L7C'}1/2

(L 'c ymG '

C 7 G '


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Introduction to Transm ission Lines 13The total resistance R of the element is thus R=l/G. The relaxation time, e/o\ isequal to the "RC" discharge time for the cell, as noted from Eq.(1.4). We alsoidentify the impedance of the line as

Z 0 = (n/e)1/2 = (L7C')1 /2 (1.5)At this point w e can quantify the selection o f Al. In order to obtain accuracy, weselect Al such that

A l v (e /a )= v(CVG') (1.6)where v = 0isy 1 /2 =(L'C')"1 /2 (1.7)Eqs(1.6)-(1.7) state that the transit time delay in cell Al, equal to Al/v, is muchsmaller than the RC time of the cell. An equivalent statement is that the lumpedresistance, R, of the element Al is much larger than the characteristic impedance,Z0, or

R Z 0 (1.8)By virtue of previous equations, L ', C may be combined into a lossless transmission element, Z 0, and the conductance may be combined into tw o resistors,R, located at the ends of the transmission line, as shown in Fig. 1.7, where R isgiven by

R=2/aAl= 2/G'Al (1.9)A two factor appears in Eq.(1.9) since each of the two resistors, R , may be considered in parallel. Another way to view the introduction of the two factor is thefollowing. By focusing an a single TLM line element, we ignore the adjoiningTLM elements, each with similar end resistors; since such adjoining resistors arein parallel, a two factor should be introduced when "extracting " a single element from the chain.


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Introduction to Transmission Lines 17cepts, two important flaws exist. For one thing the circuit is one dimensional andit does not take into account conductivity and electromagnetic spreading in thetransverse directions. The second flaw, again related to the one dimensional nature of the circuit, has to do with the lack of a stable solution during equilibrium, i.e., prior to the activation of the cells, when R or R i2 , R23, etc... are verylarge. Suppose each of the cells in Fig. 1.9, which are connected in series, is initially charged to a different voltage. The cells will discharge into one another,unless artificial means are taken to prevent such a discharge, such as the artificial insertion of a series switch. A self consistent way to preserve equilibrium,prior to activ ation, is to insert an orthogonal transmission line in series; this thenconverts the one dimensional circuit into a 2D one. A similar extension to 3Dalso preserves the equilibrium. Before proving these assertions, we briefly de scribe the 2D and 3D arrays. This will be followed by background discussion intransmission line theory, which will allow us to place our previous claims on afirmer fooling.

Fig. 1.10 shows the circuit matrix used to describe electromagnetic and conductivity spreading in two dim ensions. O ne way to v iew the circuit cell is to notethat there are four square cells, with each region constant in voltage(Vi,V2,V 3,and V 4) but generally differing in value from the neighboring cell . Thesquare regions ,therefore, may be considered as conductors. Separating the constant voltage regions are the transmission lines. In this case the line imped ances,Zo, are the same. Initially the lines will charged up to a voltage value equal tothe voltage difference between adjacent cells. Note that the node resistance R islocated at the hub of the matrix and actually consists of four identical resistors,R, each terminating one of the four TLM lines. It is worthw hile to realize thatany signal arriving at the node will be equally scattered among the four transmission lines(in the absence of any significant conductivity). This property issimilar to that in electromagnetics, in which each region of the wave front maybe regarded as a point source. A similar extension of the circuit may b e ma de tothree dimensions. In 3D , how ever, the iso-potential region s are cubical, andthere are eight cubes centered about each node point(Fig.l.ll). As mentionedpreviously, the nodal resistors contain the bulk of the physics, since these tim evarying elements represent photoconductivity, avalanching, recombination,charge transfer, and myriad other phenomena. Also, for 3D, there are two independent , orthogonal fields associated with each cube edge.


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20 Electromagnetic Analysis Using Transmission Line VariablesR Z 0 regions exist, and also in App.7A.4 where we discuss the fielddissipation in terms of the elementary TLM waves.1.5 Transmission Line Theory BackgroundThe discussion in the previous sections will be quantified and reconciled withtransmission line theory. Before proceeding to this goal, however, we present abrief overview. The literature on transmission line theory is quite extensive(see,e.g., Reference [4]). In this discussion we limit ourselves to only those relationships which are deemed necessary for describing the technique. Before continuing, it is useful to describe the normal electromagnetic modes in a single seetionof transmission line, not coupled to any other line elements, in which the terminating resistors are extremely large, i.e., an open circuit. The transmission line isbiased to the voltage difference V0 and is in an equilibrium state, as shown inFig. 1.12(a) The analysis proceeds by first choosing the correct set of normalmodes which describe the standing waves during the off-state, when the line isbiased to voltage V0. This is not difficult to obtain, since we know that the general solution to the wave equation (Eq.(1.2)), or the equivalent Eq.(1.3) . Discarding the conductivity term, the solutions are a pair of waves traveling in opposite directions with velocity, v. The simplest set of modes which satisfy theboundary conditions, during the off-state, are two waves each with constant amplitude, each equal to half the bias voltage, Vo/2. The two waves travel in opposite directions, and are designated +V and ~ V in Fig. 1.12. +V designates thewave traveling in the plus x direction while ~ V is the backward wave travelingin the negative x direction. We adopt the convention that the voltage waves , asindicated by the vertical arrows, point in the direction of increasing volt-age(potential). The direction of the electric field is of course opposite to that ofthe voltage wave.The voltage waves fill the entire transmission element and are constrainedby the open circuit at both ends, where the waves are reflected so that +V converts to " V at one end, and vice versa at the other end. The waves obey thesymmetry requirement and of course the waves superimpose to give the correctvoltage at all times and at all points in the line during equilibrium, i.e., V0= +V+- y = +(Vo/2) + " (Vo/2). Thus, the general solution for the voltage consists of


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Introduction to Transm ission Lines 21

+BIAS

OPEN CIRCUIT (a\ OPEN CTRCUTTFig 1.12(a) STATIC FIELD IN A SINGLE TLM ELEMENT, EXPRESSED INWAVE VARIABLES. +V AND "V ARE REFLECTED AT ENDS WITH NOCHANGE IN POLARITY OR AMPLITUDE.

R i J *V t t f $ RL1 "V t 0 , the total load impedance seenby the forward w ave in A w ill be R L = Z 0 , since the parallel com bination of R andZ is simply Z 0, since R is very large. Thus the forward wave going from line Ato line B will be unimp eded(i.e., "matched " to B). Similarly, the backward wavein B will be matched to A. Similar comm ents apply to lines B and C. The net effect is that the equilibrium conditions in line B( or any other cell in the chain)remain the same, with the same forward and backward waves (and with equalamplitudes)as in the previous time step. The equilibrium also is preserved whenthe adjoining cell impedances are unequal. Thu s there is no net transfer of


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26 Electromagnetic Analysis Using Transm ission Line Variables

A B CFORW ARD WAVE IN B FOR (K+1)TH TIME STEP:+V Bk+ ' = T+V AK+B-V BkBACK W ARD WA VE IN B FOR (K+1)TH TIME STEP:

-V B k + 1 = T - V c K + B + V B kT=2R L /(Z 0+ R L ), B = ( R L - Z O ) / ( Z 0 + R L ) , RL=Z 0R/(Z 0+R)

FIG 1.14 WAVES IN A AND C CELLS CONTRIBUTE TO THEFIELD IN B DURING THE ENSUING K+l TIME STEP ACCORDINGTO THE ABOVE RELATIONS. LOSS AND IMPEDANCE IN EACHCELL ARE ASSUMED IDENTICAL.

energy from one cell to the other.It is also of interest to determ ine the decay of the various cells for a uniformID chain in which each cell is initially biased to V 0 and the nod e resistance R isfinite and uniform throughout the chain. Under these circumstances one may useEqs.(1.17)-(1.18) to determine the decay within each cell. For large R (relativeto the line impedance) the forward and backward waves decay by an amount ( 1 -Z/R) with each time step. Because the nod e resistance and line impedance areuniform, however, we may adopt the view that there is no net transfer of energyfrom one cell to another. The cell voltage in each cell declines, to be sure, butthe decline may be regarded as internal to the cell. The situation changes ofcourse when the adjacent node resistors differ, even when the initial bias voltageis uniform for all the cells. Under these circumstances the fields will redistribute


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Introduction to Transmission Lines 291.8 Reverse TLM IterationNum erical techniques involving partial differential equations often require timeelements of t-At as well as t+At, in order to obtain correct solutions. It shou ldcome as no surprise, therefore, that in order to compare the numerical methodswith the TLM m ethod, we need to take into account reverse as well as forwarditerations. The generic form of the two types of TLM iterations are shown inEqs.(1.32)- (1.33), applicable not only to ID but 2D and 3D as well.

FORWARD+V k+ 1 = Z [ S.C. ] + V k + I [ S . C . r V k (1.32)- V k + 1 = [ S.C. ] +V k + [ S.C. ] " Vk (1.32b)

REVERSEV " 1 = Z [ S.C. ] + V k + I [ S . C . r V k ] (1.33a)V 1 = I [ S.C. ] +V k + I [ S.C. ] _ Vk] (1.33b)

where [ S.C. ] are the scattering coefficients (evaluated during the kth step) anddie summation is over wave contributions from adjacent lines. Note the important fact that in the reverse iteration, the (k-l)th wave is determined from wavesexisting during the ensuing kth step, whereas the forward iteration relates V k+1to waves existing during the prior kth step. Although Eqs.(1.32) and (1.33) appear to be superficially die same, they are different. First of all, the scatteringcoefficients w ill differ , especially if losses are present. In addition, the two iterations will have different node locations for the forward and reverse iterations,which we d iscuss later.

It seems somew hat strange to consider reverse TLM iterations. Onenaturally asks the question w hether it is truly necessary exam ine such a topic.There are at least two reasons, however, why it is important to take into accountthe reverse iteration. The first is th a t, using such an iteration, it may be possibleto determine an earlier physical state based on the present state. The secondreason is that the reverse iteration provides us with additional information, and


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Introduction to Transmission Lines 37d*E(x,tydx

2=[ E(x+Ax,t)-2E(x,t)+E(x-Ax,t) ]/Ax

2+higher order terms (1.43)

where Ax is the difference in the x coordinate. Similar expansions of^Efotydt2and dE(x,t)/& yielda2E(x,t)/at2 = [ E(x,t+At)-2E(x,t)+E(x,t-At) J/At2 + higher order terms (1.44)and3E(x,t)/dt = [E(x, t+At) -E(x, t- At) ]/2 At + higher order terms (1.45)wh ere At is the difference in the time. Substituting E qs.(1.43)-(1.45) into thewave equation and solving for E(x,y,t+At) yields the iterative equation,E(x,t+At) = E (x ,t ) - E(x, t -At) + ?c2[E(x+Ax,t) + E(x-Ax, t) - 2E(x,t)]

-a[E(x ,t+At) - E(x,t-At)] (1.46)where X2 = ^At 2 / Ax2 (1.47a)a = Ato72e = ZJ2R (1.47b)Eq.(1.46) simplifies by setting A.=l , or Ax=vAt. The value for k is within theallowable range needed to insure stability for the finite difference solution. Stability is assured when the "numerical" velocity, Ax/At, is greater than or equalto the wave veloc ity v (Reference [5]). Since Ax/At is set equal to v, the finitedifference solution is automatically stable and Eq( 1.46) becomesE(x,t+At) = - E(x , t-At) +[E(x+A x,t) +E(x-Ax, t)] -cc[E(x,t+At)-E(x,t-At)] (1.48)In order to compare the above w ith the TLM iteration we convert field variablesto TLM voltage variables in the above, using the following correspondence:


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Introduction to Transm ission Lines 41

=(V2-V,)J

D

t< >

* ^ p - B I

AV C B D ^ V Z - V , )

' / ,

FIG. 1.18 BOUNDARY CONDITION AT THE 2D NODEIS VA=Vc+VB-VD, WHERE V A=VCBD=V2-V I , Vc-Va-V*V B=V 4-V 3, V D = V 4 - V 2 .

Since we assume the node region is small compared to the wavelengths beinggenerated, we consider the field to be conservative about the node. Thus, thevoltage path from 1 to 2 is equivalent to 1 to 3 followed by 3 to 4 and then 4 to2. W e therefore take as our boundary conditionVA = V B + V C -V D (1.52a)

It is important to note that the line voltages in the above represent the total voltage, i.e., the sum of the backw ard and forward voltage waves. A lso note thenegative sign for V D , which stems from the fact that the path displacem ent is inthe negative direction, and therefore the voltage wave is negative. The negativesign for VD is completely dependent on the fact that we have selected V A as our


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46 Electromagnetic Analysis Using Transm ission Line Variables+ V B = T + V A (1.60)

T m ay be obtained fromT = [2RL/(RL+Z0)] (Zo/RL) =1/2 (1.61)

where the (Zo/RL) is appended to T since the voltage transfer to B representsonly a portion of the voltage delivered to RL. Combining ,+ V B = ( l / 2 ) + V A (1.62a)

We should note that +V B =VB (the total field) since there is as yet no backwardwave in B. Similarly, the voltage transfer to lines C and D are- V c = ( l / 2 ) + V A (1.62b)+VD= -(1/2) +VA (1.62c)

Again we should note that ~Vc and +V D represent the total fields V c , VD inlines C and D. We should also note the minus sign in Eq.(1.62c) , since +VD isdirected in the negative x direction. The previous equations are in agreementwith the boundary condition, Eq.(1.56b), as expected, i.e., the total field in linesB,C,D is then (3/2)V. Although we have considered a solitary w ave in one of thelines, the situation does not fundamentally change when waves from the otherlines (C,B, or D) are simultaneously incident on the node. Under these circumstances the waves moving aw ay from the node , in each line, will not only consist of the reflected wave , but will receive contributions from the incidentwav es in the other lines, which add in linear fashion to the reflected wave, justas in the ID case.

In the previous discussion, we focused on a single wave launched in one ofthe lines. The behavior of the 2D nodes when multiple coherent waves exist inparallel lines, how ever, is an important issue. Referring to Fig. 1.20, we inquirewhether identical waves, launched simultaneously in lines R, S, T , etc., willultimately simulate a plane wave. In other wo rds, we ask whether the waves inR, S , T, transfer com pletely intact to lines D, E, F, without transversescattering or reflection. In Ch apters III and IV we exam ine in detail the question


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Introduction to Transm ission Lines 55In order to facilitate the comparison of the TLM result with the finite

difference method , we add Vi(t+A t)+V2(t+At) to Vi(t-A t) +V 2(t-At), giving us[Vi(tf At) +V 2(t+At)]+[V1(t- At)+V2(t- A t)] = [V,(t) +V 2(t)]+(l/2)[V 1L (t)+V 3U (t)+V 1R(t)-V 4U(t)]+[l/2) [V2L(t)-V3D(t)+V 2R(t)+V 4D(t)] (1.76)Eq.(1.76) is the three tier TL M iteration. Note that the equation does not dependon any forward or backw ard w aves, but only the sums of the two w aves in eachline. We also remind ourselves that the previous equations apply only to thehorizontal lines. In the following we enumerate the similar relations for thetransverse lines. Corresponding to Eqs.(1.68)-(1.73),

+ V 3 (t+ A t) =(1/2) " V 3 (t)+(l/2)[ + V 3 D (t)- V 2 L + V 3 (t)+ " V 2 (t)] (1.77)" V 3 (t+A t) = (1/2) + V 3 (t)+ (l/2)[ - V 3 U (t)+ V 1 L ( t ) - " V , (t)] (1.78)

V 3(t+ At) = + V 3 (t+ At) +~ V 3 (t + At) (1.79)+ V 4 (t+ At) =(1/2) " V 4 (t)+ (1/2) [+V4D (t) - + V 2 (t) +- V 2 R (t)] (1.80)

" V 4 (t+A t) =(1/2) + V 4 (t)+(l/2)[ " V 4 U (t)+ V i (t)- - V , R (t)] (1.81)V 4(t+At) = + V 4 (t+At) + " V 4 (t+A t) (1.82)

As with Vi(t+A t) and V 2(t+ A t ) , we form the sum of V3(t+ A t) and V4(t+A t) :V3(t+A t) +V 4(t+ A t) = (1/2) "V3 (t)+(l/2)[ +V 3D (t)- +V 2L (t)+ " V 2 (t)]+ (1/2) +V 3 (t)+ (l/2)t "V 3U (t)+ +V 1L (t)- "V, (t)]+ (1/2) "V4 (t)+ (1/2) [+V4D (t) -+V 2 (t) +-V 2R (t)]+ (1/2) +V 4 (t)+(l/2)[ "V4U (t)+ +V, (t) - "V 1R (t)] (1.83)


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Notation and Mapping of Physical Properties 75

R(n-2) . ^ C R(n-1)Z(n-2) ^ * S Z(n-1)R(n) _ ^Z(n) ^ S R(+l*vWV*..)> ^

n-1 n n +1

FIG.2.3 NOTATION FOR ID COUPLED CELLS. BYCONVENTION THE nth NODE , ATTACHED TO THENTH CELL, IS IN THE DHUECTION OF INCREASING n(TO THE "RIGHT" OF THE Z(n) CELL).unless they are germ ane to the discussion. T he characteristic impedance of eachcell is labeled by Z(n ). The n label for the cell Z(n) is used to indicate not onlythe cell impedance, but is also used to locate(at least for identical cells) the location of the cell within the chain. In Fig.2.3 the cell impedances are assumed to b ethe same and thus cell lengths are identical. Later, as well as in subsequentCh apters, the labeling also will allow us to consider differing values of cell im-pedance(and different line lengths), which will be necessary when treatingnonuniform dielectrics, dispersion, and boundaries between differing dielectrics.No te that each cell shares two node resistors, R (n -l) and R(n). By conventionthe node resistor R(n), corresponding to the nth cell, is located in the d irection ofincreasing n (i.e., located on the "right hand side "of the cell) wh ile R(n-l) is thenode resistor located in the direction of decreasing n.Having outlined the ID notation we can proceed to the calculation of thenode resistance. There are alternate means for obtaining the effective node resistance all of which are more or less equivalent. One approach, already alludedto in Chapter 1, relies on first calculating the end resistors for each TLM element. As shown in Fig.2.4, we first focus on the nth and (n+ l)th isolated cells,for which the end resistors are (see Section 1.3, Eq.(1.9)),

R '(n ) = 2p(n)/Al (2.1a)R'(n+ 1) = 2p(n+ l)/Al (2.1b)


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86 Electromagnetic Analysis Using Transmission Line VariablesThe node resistance R(n,m) is the sought after result which relates R(n,m) to

the average conductivity in much the same manner as the ID relation, except forthe factor of two , which is now present since adjoining end resistors are nolonger combined. The result in Eq.(2.21) may seem premature since we havedone the averaging in only in the y direction , while ignoring the x direction. Wemust therefore consider the Zyx(n,m+1) and Zyx(n,m) lines as well and performa similar averaging. The reader will quickly verify, however, that the same average, given by Eq.(2.21) is obtained. An "intuitive" method for obtaining thesame expression for R(n,m ), which works for 2D , is as follows. The averageconductivity about the node, aAv(n,m), is first obtained, as seen in Fig.2.10where we use an auxiliary cell centered about the node. We then assume anequivalent circuit where a pair of parallel resistors , each equal to R(n,m ), is directed in , say, the y direction. The combined resistance, R(n,m)/2, is then setequal to the equivalent resistance of the center cell, or, R(n,m)/2 = l/Ala Av(n,m),which is identical to Eq.(2.21).

(n,m+l)Ii ..ji

(n,m)

(n+1

(+

,m+l)

*Sl,m)

AUXILIARY^ CELL

aAv(n,m)=[ a(n,m)+ a(n+l,m) +a(n,m+l)+ a(n+l,m+l)]/4R(n,m)=2/AlaAv(n,in)FIG.2.10 EQUIVALENT R(n,m) W ITHCONDUCTIVITY CENTERED AT NODE.


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90 Electromagn etic Analysis Using Transm ission Line VariablesPropa gation Plane Transm ission Lines Fields

xy Z xy(n,m,q) , Zy^a.,m,q) Vxyfan^q) ,Vyx^m,^yz ZyZ(n,m,q),Z zy(n ,m,q) Vyzfon^q), V ^ n ^ q )zx Z ^ n .m .q ), ZxzCn.m.q) V zx(n,m,q), VM(n,m,q)

(n,m,q) NODE

FIG. 2.13(a) ELEM ENTARY CELL OF TL MAT RIX.Z ly(n,m,q) R(n,m,q) Zy2(n,m,q) ZZI(n,m,q)

ZyxOMn.q) Z > , m , q ) "X ZK(n,m,q)FIG.2.13(b) PROJECTION OF 3D TLM M A T R K ONTO 2D GRIDS.


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104 Electromagnetic Analysis Using Transm ission Line Variablessignals. For treating node dispersion , e.g., we use two parallel matrices wh ichoccupy the same space, as shown in Fig.2.19, which assumes no velocity dispersion. Each of the 2D matrices represents a different frequency( the light andEM frequencies), but the nodes are assumed to coincide. As in the I D case ofFig. 2.17, the dispersive node resistors results in the exchange of signals andconductivity at the nod e locations. The 2D matrices also are adaptable to thenode coupling approach for differing velocities, and also for treating boundariesbetween differing propagation region s, as discussed in Chapter V.

EM MATRIXLIGHT MATRIX

LIGHT AND CONDUCTIVITYPRODUCED IN EM MATRIX ISDISTRIBUTED IN LIGHT MATRIXLIGHT MATRIX PRODUCESCONDUCTIVITY IN EM MATRIX

FIG. 2.19 2D INTERACTION OF LIGHT AND EM SIGNALSSHOWN AS PARALLEL MATRIX ARRAYS(NO VELOCITYDISPERSION).


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Notation and Mapping of Physical Properties 107

P(x) = P0EXP[-a{ nAl/lo -1/2} 2] (2.49)wh ere a is the spread factor o f the light pulse and nAl = x is the distance measured from the elecrode. The total semiconductor length lo is equal to N0A1 whereN 0 is the total num ber of cells between electrodes. Po is the constant(in time)light power falling on the center cell, where the light is a maximum, and thepow er level tapers away from the center. In addition, we assume the light signalattenuates as it enters the semiconductor. If we assume an exponential attenuation then the signal follows EXP{-mAl/ho} where ho is the absorption dep th ofthe light signal. Th us, the actual pow er deposited in the mth cell will follow anexponential decay in the particular cell. We can now calculate the energy depo sited in the (n,m,q) cell during the kth time step. W e simplify m atters som ewhatby assuming the semiconductor is dispersionless, so that the light signal velocity is identical to that of the electromagnetic signals in the transmission lines. Ifwe denote the energy deposited in the (n,m,q) cell by y(n,m,q), we have

y(n ,m,q) = PoEXP[-a{ nAl/l0 -1/2 }2]D(m)At (2.50)whereD(m ) = 0 if k m (2.52)The delay nature of D(m ) expressed by Eq.(2.51) is understandable, of course,since no conductivity will be produced until the light signal reaches the cell inquestion. Once k> m then the wave energy is deposited in the cells. Eq.(2.52) isthe difference in the decay factor , EXP{-mAl/ho}, at depths mAl and (m-l)Al .The difference, D(m) , is thus proportional to the energy deposited in the mthcell. If U is the photon energy then the number o f photons deposited in (n,m ,q) isy(n,m,q)/U. To simplify the effect of the light signal we assume there is no lateral scattering of the light as it is absorbed in the semiconductor We now relatethe num ber of carrier pairs pr od uc ed , during At, to the deposited photons. If the


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Notation and Mapping ofPhysical Properties 111

F IG . 2 . 2 1 D E P E N D E N C E O F N O D ER E S I S T A N C E ON C E L L I N D E X N U M B E Rn . D A S H E D L I N E P O R T I O N S T A K E IN T O

A C C O U N T E L E C T R O D E S I N T H EA V E R A G I N G O F R ( n , m , q ) .

~ 9 0 0f 8003 7 0 0 +5 600 +S 5 0 0 400 +

3 0 03 4 5 6 7CELL NUMBER n

8 9 10

REFERENCES1. S.M. Sze, Semiconductor Devices ,John Wiley and Sons, New

York, 1985


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Scattering Equations 1133.1 ID Scattering EquationsAs in previous discussions it is easiest, by way of illustration, to first considerthe scattering in the ID case. Tow ard this goal w e utilize Fig.3.1, borrowed fromChapter II, and seek the fields in the nth cell. We assume the delay time in eachcell is identical, but do not require the cell lengths to be the same(i.e., we allowdiffering cell impedances). The forward and backward waves in the n cell, during the kth time interval, are denoted by +V k(n) and ~Vk(n) , respectively.Similarly the same wave s in the (n -l) th and (n+1) th cells are

+V

k(n-1) , ~V

k(n-l)and +V k(n+1), ~V k(n+l) respectively.

Our goal is to determine the fields scattered into the nth cell during the(k+l)th time step, based on the fields existing in the nth, (n-l)th, and (n+l)thcells during the kth time step. W e first determine +V k+ 1(n), i.e. , the forwardwave in the nth cell during the (k+ l)th interval. This wave will be the sum oftwo waves , consisting of a transmitted wave from the (n-1) cell as well as thereflected backward wave in the nth cell. Thus+Vk+ 1(n) = Tk(n-1,1) +V k(n-1) + B k(n-1,2) 'V k(n) (3.1)

where T k(n-l , l )and B k(n-1,2) are the transmission and reflection coefficients,respectively, of +V k(n-1) and ~Vk(n) at the (n -l)th node. The additional argument of one or two , in the scattering coefficient, is adapted to denote the factthat waves incident on the node are in the forward or backward directions, respectively. The coefficients, by definition, are:T k(n- l , l ) = 2 RL lk(n- l ) / [RLl k(n -l ) +Z(n-1)] (3.2)Bk(n-1,2) = [R L2 k(n-l)-Z(n)]/ [RL 2k(n-l)+Z(n)] (3.3)

where RLlk (n-l) and RL2 k(n- l) are the load impedances seen by the forwardand backward waves , incident on the (n-1) node. These impedances are easilycalculated from


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11 6 Electromagnetic Analysis Using Transm ission Line Variables3.2 2D Scattering EquationsWith the 2D matrix (and particularly the 3D) the scattering equations will naturally become more involved. However, by continuing to use the notation introduced in the previous Chapter, we will be able to describe the scattering in avery compact manner. We proceed by viewing a portion of the matrix in the vicinity of the (n,m) cell, repeated in Fig.3.2. The (n,m) cell and the surroundingcells and lines, pertinent to the scattering process, are shown. Our aim will be todetermine the f ields in the Z ^ ^ m ) and Z ^ m ) , for these are the two transmission lines associated with the (n,m) cell. As before, we will express the fields attime t + At in terms o f fields (in the surround ing lines) at time t. In the ensu ing

Z ^ n - l j i n + l ) Z^(n,m+1)

Z^(n- l ,m+l)

( n - l , m )

Z (n - l .m - l ) ^ R (n - l , m - l ) < Z^n,m-1)

(n,m-l)Z,(n-l,m-l)

R(n,m-1)V S A A T

Z^(n+l ,m- l )

7~fn.m-n

FIG. 3.2 2D TLM NOTATION C ON VE NT ION .-Z^m ) ANDZ^i^m) ARE THE TWO LINES(SHADED) ASSOCIATEDWITH THE (n,m) CELL. R(n,m) IS AT THE INTERSECTIONOF Z^n.m) AND Z^n.m).


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Scattering Equations 117discussion we suppress the superscript k, designating the time element, untilsuch situations when it is significant, as in the iterative equationsTak e for example the Zxy(n,m) line. Th e forward wave , +Vxy(n,m), duringthe (k+l)th interval, results from several waves converging on the R(n-l,m)node during the prior interval. First, there are the transfer of waves intoZxy(n,m) from three neighboring lines: Zxy(n-l,m), Zyx(n-l,m), and Zyx(n-l,m+l).Second, there is the reflection of the backward wave at R (n -l,m ) in Zxy(n,m).Similarly, during the (k+ l)th step, the forward wave in the Zyx(n,m) line,+Vyx(n,m), will be the result of waves scattering at the R(n,m-1) node. Thebackward waves, on the other hand, for both the Zxy(n,m) and Zyx(n,m) lines,will involve scattering at the R(n,m ) nod e.

One can see that the scattering coefficients(i.e., both the transfer and the reflection type)will have man y po ssible values, depending on the transmission linefrom which the wave em anates and line to which it is directed. It will be convenient to label these coefficients before proceeding further w ith the analysis.Since we are dealing with only the Zxy(n,m) and the Zyx(n,m) lines, we needonly consider the nodes bounding these lines, wh ich are, from Fig.3.2, (n,m), (n-l,m), and (n,m-l). Table 3.1 lists the 16 scattering coefficients associated withthe (n,m) cell, 12 of which are transfer type and 4 reflection type. For example,the first transfer coefficient listed is Txy(n-l,m,l). The subscripts follow that ofthe incident wave (or TLM line), indicating the propagation and field directionsrespectively. The first two arguments n-l,m identify the node while the third argument, 1, simply indicates, by definition, that the wave is being coupled fromthe Zxy(n-l5m) line to the Zxy(n,m) one. A s an another example, the third transfercoefficient listed , Tyx(n-l,m,3), has the same node, but the third argument, 3,denotes the transfer of the wav e from the Zy x(n-l,m+l) line to the Zxy(n,m) line.In the case of the reflection coefficients , the first one listed, B ^ (n -l ,m ,l ) refersto the n-l,m node and the argument 1, indicates the reflection from that nodetakes place for a backw ard w ave in the Zxy(n,m) line.(the labeling of the 2D reflection coefficient differs from that used in the ID notation). W e emphasizethat Table 3.1 represents the scattering into the lines Zxy(n,m) and Zyx(n,m), associated with the (n,m) cell. It will also be useful to consider the scattering aboutthe node (n,m) , i.e., the scattering of all waves convergent on a particular (n,m)


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122 Electromagn etic Analysis Using Transm ission Line Variableswh ere the expression in the second parenthesis represents the share of the voltage transfer to Zyxfom), i.e., the ratio of the voltage transfer to Zyxfom) to thatof the sum of voltage transfers across the three lines Zyx(n,m), Zyx(n,m+1), andZxy(n+l,m). Next we consider the reflection coefficient, Byx(n,m-1,3). The arguments n,m-l denote the (n,m-l) node, while the 3 subscript indicates the reflection is for the backward w ave in the Z ^ a . m ) line. Utilizing the usual expression for the reflection coefficient then g ivesIV (n ,m -l,3 ) = [RL4(n,m-l>-Zyx(n,m)]/ [RL4(n,m-l)+Z yx(n,m)] (3.19b)where R L4 (n,m -l) is given by Eq.(3.18), but we replace m w ith m-1. The complete listing of scattering coefficients is given in Table 3.3, which represents thescattering about the node. The node parameters are given in Eqs.(3.11)-(3.14)and (3.15)-(3.18). W hen used in the scattering equations, the proper index is inserted in both the scattering coefficients and the node param eters.

Having enumerated the scattering coefficients, we are now in a position towrite down the fields, at time t, in the lines Zxy(n,m), Zyx(n,m), in terms of thefields of the previous time step. Starting with the forward wave in Zxy(n,m),during the kth time step, the voltage is (we now use the k superscript to indicatethe time step)V ^ m ) = T ^ n - l ^ l ) ^ / ^ ( n - 1 ,m)-Tkyx(n-1 ,m ,2) V ^ n - L m )+T kyx(n-l ,m,3)-Vkyx(n-l ,m+l)+B kxy(n-l,m,l)-V kxy(n,m) (3.20)We now explicitly describe each term in Eq.(3.20). The first, T kxy(n-l,m,l)+Vkxy(n-l,m) represents that portion of the forward wave, in the Zxy(n-l,m) line,that is transferred to the Zxy(n,m) line.The second term, T kyx(n-l,m,2)+V kyx(n-l,m) represents the energy coupled from the forward wave in the vertical Zyx(n-l,m ) line to the Zxy(n,m) line, via the (n -l,m ) node. A negative sign is presenthere since the +Vkyx(n-l,m) wave couples to the Zxy(n,m) in the negative direction. The third , Tkyx(n-l,m,3) ~V kyx(n-l,m+l), corresponds to the energy transferof a backw ard wave in the vertical Zyx(n-l,m +l) line to the Zxy(n,m) line. Finally the term Bkxy(n-l,m, l)"Vkxy(ii,m) represen ts the reflection of the backw ard


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134 Electromagn etic Analysis U sing Transm ission Line Variables

T ABL E 3 .6 a. 3D COPL ANA R SCAT T E RING CO E FFICIE N T SAND WAV E VOL T A GE S FO R YZ AND Z X PL A NE SSPE CIA L CA SE:An, Am, Aq =0

XY PLANE YZ PLANE ZX PLANE-> X

V ^ n . m . q )V y ^ n ^ q )Tjy(n,in,q,s)Tyx(n,in,q,s8^0,01^,8)8^(0,01^,8)

- - - ->->-

V ^ n . m . q )Vjy(n,m,q)^ ( 0 , 1 0 ^ , 8 + 1 2 )T,y(o,m,q,s+12)8^(0,01^,8+4)Bzy(D,m,q,s+4)

V M (o,m,q)V H (o ,m,q)TM(o,m,q,s+24)T(fl,m,q,s+24)B(o,m,q,s+8)8^(0,01^,8+8)

GEN ERA L TRAN SFO RM ATIO N: Ao , Am, Aq =0 o r +1Vjy(n+An,m+Am, q+Aq) -Vyz(n+Aq,ni+An, q+Am) V(n+Am,m+Aq, q+An)Vyj(n+Am,m+Aq, q+An) -V ly(n+Aq,m+An, q+Am) V^n+Am .m+Aq, q+An)Tw(n+An,m+Am,q+Aq,s) -Tyz(n+Aq,m+An,q+Am,s+12) TM(n+Am,m+Aq, q+An,s+24)Tyl(n+An,m+Am,q+Aq,s)-T zy(n+Aq,m+An, q+Am,s+12) Tm(n+Am,m+Aq, q+An,s+24)Bjy(n+An,m+Am, q+Aq,s)->Byj(n+Aq4n+An, q+Am,s+4) BM(n+Am,m+Aq, q+An^+8)Byl(n+An,m+Am, q+Aq,s)->Bzy(n+Aq,m+An, q+Am,s+4) Bjj,(n+Am^n+Aq, q+An^+8)


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Scattering Equations 137

+VYX (n,m+l,q) yU +H zx(n,m,q+l) c.INPUT +VXY(n+l,m,q) INPUT+Hxz(n,m,q)

+Hxz(n,m,q+1)

(a) (b)Vxy(n,m,q) H7A(n,m,q)

FIG.3.4COPLANAR SCATTERING OF +VxY(n,m,q) ANDASSOCIATED ^xzO^ rnq). INCORPORATION O F MAG NETIC FIELDIMPLIES SCATTERING NORMAL TO PROPAGATION PLANE.

3.5 Equivalent TIM C ircuit. Q uasi-Coupling EffectBased on the comments in the previous paragraph, it would appear that the scattering, including losses, is controlled by the equivalent nod e circuit in F ig.3.5which shows a wave +Vxy(n,m,q) incident on the (n,m,q) node. Th e circuit applies if line impedances surrounding the node obey certain symmetry conditions;for example, if the TLM lines surrounding the node have the same impedance.In the event the such symmetry conditions are lacking, however, then the incident +Vxy(n,m,q) wave will exhibit aplanar scattering not only to the z directionbut to the y direction as we ll. The equ ivalent circuit therefore m ust be revised.A first order revision of the circuit is shown in Fig.3.6 and is used to accountfor first order "quasi" coupling to the Z y ^ m ^ a n d Z yz ^m + ^q ) l ines .A quantitative description of qua si-cou pling , and the conditions for its existence, are


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142 Electromagnetic Analysis Using Transm ission Line Variables

VxY(n,in,q)(FIELD DIRECTION)FIG.3.7 APLANAR SCATTERING AT A 3D NODE. BOTH TH EDIRECTIONS OF PROPAGATION (SINGLE ARROW) AND FIELD(DOUBLE ARROW ) ARE SHOW N. QUASI-COUPLING MAY OCCUR ATA AND D, i.e., AT Z vz(ii,m,q) AND ZYz(n,m+l,q).

separate calculation must be performed for this type coupling. If the line impedances surrounding a node are identical, one can surmise that the quasi couplingvanishes due to symm etry.To determine first order coupling we examine the TLM lines in Fig.3.8,using simplified notation . Lines Zi and Z 2 in Fig.3.8 belong to the Rlyz(n.,m,q)and R3yz(n,m,q) branches in Fig.3.6, while Z 2 and Z 4 belong to the R2 zy(n,m,q)and R4zy(n,m,q) branches. We then allow for field +V, propagating perpendicular to the plane(em erging from the page) and with the field d irection as show n.


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Scattering Equations 153quasi scattering. If we wish, quasi-coupling may ignored entirely by setting thequasi- node parameters (to be described explicitly) equal to zero. As we havedone before, it is useful to define auxiliary node parameters consisting of theparallel combination of the line imped ance and the node resistance. T o illustratethe definitions, we focus on the (n,m,q) node and examine a wave +Vxy(n,m,q)incident on the node. For coplanar (xy plane) scattering the node parameterdefinitions remain the same as before and the relevant relationships are identicalto Eqs.(3.26a)-(3.26d) repeated here to give

Rlxyforr^q) = [R(n,m,q)Zxy(n,m,q)]/[R(n,m,q)+Zxy(n,m,q)] (3.71)R2yx(n,m,q) =[R(n,m,q)Z yx(n,m,q)]/[R(n,m,q)+Zyx(n,m,q)] (3.72)

R3xy(n,m,q) = [R(n,m,q)Z 3q,(n+l,m )q)]/IR(n,m,q)+Zxy(n+l,m,q)] (3.73)R4yx(n,m,q) =[R(n,m,q)Z yx(n,m+l,q)]/[R(n,m,q)+Zyx(n,m+l,q)] (3.74)

Because of aplanar effects the wave +Vxy(n,m,q) will scatter unto the yzplane with coupling to the lines Zzyfoir^q) , Zzy(n,m,q+1) , Zyzfom.q) andZ y ^ m + ^ q ) lines. Because of the aplanar and quasi scattering the values of thenode parameters will differ from the conventional type, following the circuit ineither Fig..3.6 or Fig.3.9 and the discussions in the previous Sections. In thefollowing it will be more useful and less confusing if we consider the cell notation of the node param eters for jus t the ha lf impedance portion s, i.e., for eitherZi/2, Z3/2 (first order) or if we use the second or higher order param eters. In thefollowing we always assume losses are present.

3.11(a): Qu asi-Node P arameters in Cell NotationWe first describe the first order quasi-node parameters in cell notation. In thefirst order approximation we continue to use Z 3/2 , Zi/2 for the coupling parameters , using Fig.3.6 as the circuit. The cell notation for the node parametersinvolved in quasi-coupling are as follows.


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Scattering Equations 167netic field energies in the space at any one mom ent. This allows us to determineif there are regions dom inated by either field, or whether a region contains significant contributions from both types of energy. In fact, we shall see in thefollowing Sections, wh ich describe plane waves incident on a cell matrix, thatcertain TLM lines will be dominated by magnetic energy while others will bedom inated by the electric field. W e know from previous comments that whenthe field is dominated by the electric field, usually the forward and backwardwaves are about equal,

+V = ~V (3.101)(For this Section, we drop the cell designation). Such is the case, e.g., whenstatic conditions prevail. On the hand, when the magnetic field dom inates, theforward and backward waves have opposite am plitudes, i.e.,

+V= -"V (3.102)We employ a straightforward technique for finding the relative contributions tothe total energy. W e first calculate the energy levels residing in the current andvoltage(i.e., in the magnetic and electric fields), designated respectively by Uhand U e. These quantities may be estimated from

U h = (1/2 )( +I+ I )2Z At = (l/2) (( +V/Z> ~V/Z))2Z At (3.103)U e = ( l /2 ) ( + V + " V ) 2 At/Z (3.104)

The total energy U t is given byU, = U h +U e = [ V /Z + V /Z]At (3.105)

using Eqs.(3.103) and (3.104). W e may interpret Eq.(3.105) to say that the totalenergy is the sum of the energies associated with the forward and backwardwaves. It is often convenient to define energy partition param eters


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Scattering Equa tions 169

+V kXY(n,m+l)

^ x Y ^ m )

(n,m)

V x v C n ^ - l ) WAVE FRONTFIG.3.10 PLANE WAVE CONDITIONS IN UNIFORM REGION:V x ^ m - l ) = +\*xY(n,m) = + V W I M H + I ) , ETC. . .V X Y (n , m ) = V^xvCn+l,);V'xyCn.m) =0 FOR n BEYOND FRONT

vanish and that the forward wave, expressed in the horizontal lines, be preserved as it m ove s from the nth cell to the (n + l)th cell for arbitrary values o f mand n. The pertinent relationships in cell notation for the plan e wave (2D)are,for a uniform region,V ^ m - l ) = " V ^ m ) = V ^ m + l ) , etc..V x y ( n , m ) = V ^ n + U m )+Vkxy(n,m) =0 (n beyond front)

(3.109)(3.110)(3 .1H)

Eq.(3.109) simply states the uniformity of the field in the y direction. The second relation (3.110) requires that the forward w ave in the horizontal lines re-


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178 Electromagnetic Analysis Using Transmission Line VariablesZxyOti.m) as purely electrical in nature and equal in magnitude to the magneticenergy, as expected for a pure transient plane wave.3.15 Response of 2D Cell M atrix to Input Waves With Arbitrary AmplitudesIn this Section we determine the conditions under which a limited portion of anon-uniform input wave behaves simulates the behavior of a plane wave, asseen by the TLM matrix. Unlike the previous plane wave analysis, suppose wenow allow the three forward waves , ^ ' ^ ( l . m - l ) , """ V^ ljm), and +V 1xy(l,m+1)to have arbitrary am plitudes, i.e., we assum e the w ave is in general non-uniform.As noted previously during the second time step the input waves fill up the longitudinal lines and partially transfer energy to Z^(2,m). At the end of the thirdtime step, the transfer of forward wave energy to Zxy(2,m) was m ore or lesscomplete, as noted by the fact that +V3xy(2,m) was unity and approximately remained so for the subsequent steps. We apply the same criterion to the casewhen the three input waves have arbitrary amplitudes, carrying out the analysisto the third time step by which tim e the trend is observable. W e then calculate+V3xy(2,m), using the same techniques described earlier, except for the differinginput amplitudes. We then obtain the forward wave, in terms of the amplitudesduring the first step,

+Y%(2,m)=(y2)+V\(l,m) + (l/4)[ +V \!y (l,m-1)++V 1,iy(l,m+1)] (3.130)We now determine what constraints are imposed on the inputs if we require

that the output during the third step satisfiesV ^ . n O a T v V l . m ) (3.131)

If we combine Eqs.(3.130) and (3.131) we obtain the resultV ^ ^ m ) = (l/2)[ V ^ l . m + 1 ) + V ^ L m - l ) (3.132)

or, using m ore general notation,


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Scattering Equations 185APP. 3A.1 3D SCATTERING EQUA TIONS : WITH

BOTH COPLANAR AND APLANAR CONTRIBUTIONSINTO UNIT CELL LINES ZYz(n,m,q), ZzY(n,iii,q) (yz PLANE).

+V k + 1iy(n ,m ,q)= T ^ n ^ - ^ q ^ ^ V ^ n . m - ^ q J - T ^ n ^ - ^ q ^ O f V ^ n ^ - l . q )+^"(11,111-1^,31) V ^ m - ^ q + l ) +Byzk(nm-l5q,5) Vyz(n , i i i ,q)+T nk(n,m-l ,q^2)+V k K (n,m-l,q)+T ]a k(n,iii-l,q,33)+V kzy{n+l,m-l,q)+T iy Q k(n,m-l,q,34)+V kxy(n,m-l,q)+T iy Q k(n,iii-l,q,35) V k iy(n+l ,m-l ,q)" V ^ ^ m i iM i ) = ^ ( 1 1, 0 1^ , 3 6) ^ ( n . m + l . q ) -^"(11,111,^37) V ^ n ^ q + l )

+^"(11,111^,38) + V \ y ( i i, m , q ) + 8^ (1 1,1 11 ^,6 ) ^ ( n . m . q )+T H k(n,m ,q^9) V U n ^ q ) + TH k(n+l,m ,q,40) V U n + l ^ q )+T iyQ k(ii,ni,q,41)+V kiy(ii,ni,q)+T xyQk(n+l,m,q,42)-Vkiy(ii+l,in,q)

V ^ n ^ q ) = -T yzk(n 5ni,q-l,43)

+V

kyi(n,m,q-l)T iy

k(n,m,q-l,44)

+V

kzy(n,m,q-l)+ ^1 1,1 11 ^-1 ,4 5) V ^ n . m . q - l ) +B zy k(n,m,q-1,7) V ^ i m q )+ T ^ k ( n , m , q - l , 4 6 ) V ^ n , r o , q - l ) + T ^ V m , q - l , 4 7 ) Vk v ( n + l , m , q - l )+T K Q k(n ,m ,q-l,48) V U n ^ q - l ) +T n Q k(n,m,q-l ,49)V k x z(n,m,q-l)

" V ^ ^ n , ^ ) = T y^ n^ q.S O ) V y ^ m . q ) +T zy k(n,in,q+l,51) V kiy(n,ni,q+1)-^"(11,111^,52) " V ^ n ^ + l . q ) +B zy k(n ,m ,q ,8) V ^ n . m . q )+T iy k(n,m,q,53) V ^ n . m . q ) +^"(11,111,^54) - \ \^ n + l , n M 0+T IZ Q k(ii,iii,q,55)+V ka(ii,m,q) +T3raQk(n,ni,q,56) V U n + l ^ q )


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192 Electromagnetic Analysis Using Transmission Line Variables

APP. 3A.3(CONT) NODE PARAMETERS (ALL THREE PLANES) WITHOUT QUASI-COUPLING.THE NODE PARAMETERS ARE REFERENCED TO THE (n,m,q) NODE.REPLACE n WITH n+1, n-1, ETC.. , AS APPROPRIATE, IN THE SCATTERINGEQUATIONS. TO SAVE SPACE WE HAVE OMITTED THE (n,m,q) ARGUMENT INTHE LOAD RESISTANCE PARAMETERS.

LOAD RESISTANCE PARAMETERS

RL1XY=[R4ZY*R2ZY *R*D1]/{R4ZY*R2ZY*R+D1 *R4ZY*R2ZY+D1 *R*R4ZY+D1 *R*R2ZY}Dl = R2YX +R3XY +R4YXR L 2 Y X = [ R 1 Z X * R 3 Z X * R * D 2 1 / { R 1 Z X * R 3 Z X * R + D 2 * R 1 Z X * R 3 Z X + D 2 * R * R 1 Z X + D 2 * R * R 3 Z X }

D2 = R IX Y +R 4 Y X + R 3 X YRL3XY= [R4ZY*R2ZY*R*D3]/ {R4ZY*R2zy*R+D3*R4ZYR2ZY+D3*R*R2ZY+D3*R*R4Zy}D3 = R2YX +RlXY+R4yXRL4YX=IR1ZX*R3ZX*R*D41/{R1ZX*R3ZX*R+D4*R1ZX*R3ZX+D4*R*R1ZX+D4*R*R3ZX}

D4= R l x y + R 2 Y X + R 3 X YRLlY?7=[R4Xz*R2xz*R*D5]/{R4xz*R2xz*R+D5*R4xz*R2xz+D5*R*R4xz+D5*R*R2xz}

D5= R2ZY+R3yz+R4ZY

RL2ZY=[RlXY*R3XY*R*D6]/{RlXY*R3XY*R+D6*RlXY*R3XY+D6*R*RlXY+D6*R*R3xYD6 = R1YZ +R4ZY + R 3 Y ZRL3YZT1 [R4xZ*R2xz*R*D7]/ {R4XZ*R2XZ*R+D7*R4XZR2XZ+D7*R*R2XZ+D7*R*R4XZ}

D7 = R2 ZY +R1 Y Z + R 4 Z YRL4ZY=[R1XY*R3XY*R*D8]/{R1XY*R3XY*R+D8*R1XY*R3XY+D8*R*R1XY+D8*R*R3XY}

1/8 Rly2 +R22Y " "" VZR L 1 Z X = ( R 4 Y X * R 2 Y X * R * D 9 ] / { R 4 Y X * R 2 Y X * R + D 9 * R 4 Y X * R 2 Y X + D 9 * R * R 4 Y X + D 9 * R * R 2 Y X }D 9 = R2XZ+R3ZX+R4XZR L 2 X Z = [ R 1 Y Z * R 3 Y Z * R * D 1 0 ] / { R 1 Y Z * R 3 X Y * R + D 1 0 * R 1 Y Z * R 3 Y Z + D 1 0 * R * R 1 Y Z

+ D 1 0 * R * R 3 Y Z }D10= R1ZX+R4XZ+R3ZX


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Corrections for Plane W aves 199are treated as having zero values for correlation purposes. Likewise, if +Vxy(n,m)is negative , any positive values of+V xy(n,m-1) and +Vxy(n,m+1) are assignedzero values when calculating the correlation. The same Categories and conditionsalso apply of course to the backward waves and the transverse waves.In the following w e use Category I as an illustrative examp le, with theforward wave +Vxy(n,m) considered p ositive. The first step is to divide the w avein each line into two identical waves, the sum of whose energies is equal to theoriginal energy. Using the partitioning given in Eqs.(4.4a)-(4.4b), the half energyeffective wave is denoted by +VxyiD(n,m) and

+VxyJD(n,m) = +V xy(n,m)/2 1,2 (4.7)with similar relationships for +Vxy jD(n,m+l) and +Vxy ;D(n,m-l). W e now invokea symm etry argument and state that one of the two waves, +VxyD(n,m) ,correlates with +V x yD (n,m +l) , while the other +Vxy,D(n,m) correlates to+Vxy(n,m-1) . Based on these correlations we must now determine whether eachof the +V xyD(n,m ) must be further partitioned . W e first look at the correlation of+Vxy,D(n,m+l) to +V xyD(a.m). Since category I has been selected, +V xy D(n,m+l) >VxyoO^m). There is no need, therefore, to partition the upper +VxyD(n,m) sincethe amplitude in Zxy(n,m+1) is more than sufficient to produce plane wavecorrelation. The upper +Vxy,D(n,m) is entirely of the p lane wave type( note that ifwe were to start out with Zxy(n,m+1) , however, a partitioning of Zxy(n,m+1)would then be necessary) .The "upper" plane wave component, designated by

Vxyp^n^m), is therefore"Vxypufom) = X . D f o m ) = X f o m ) /2 m (4.8a)

Thus there is no need for an "upper " symmetric component. Using a similarnotation for the symm etric com ponent,+V xySU(n,m)= 0 (4.8b)

Next we consider the correlation of +Vxy>D(n,m) to +V x y D(n,m-l), where+VxyjD(n,m) >+Vxy iD(n,m-l). Here a partitioning of+Vxy)D(n,m) is necessary since


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200 Electromagnetic Analysis Using Transmission Line Variablesthere is insufficient amplitude in +VxyD(n,m-l) to produce a complete plane wavecorrelation. We therefore set +Vxy,D(iMn) equal to

X . o f a r n ) = +V ^>D(n,m-l) + +A^(n,m) (4.9)where +Axy(n,m) is defined by Eq.(4.9). We must not forget to use Eqs.(4.4a )-(4.4b) to insure proper scaling. The new effective components for the lowercom ponents, with similar notation(replacing subscript U with L ) , are

X n t e m ) ^ V ^ D f o m - l ) +W mU(n,m)]V2 (4.10a)X s L f o m ) = f A ^ m ) X o f o m ) ] 1 7 2 (4.10b)

Eq.(4.10a) represents the plane wave contribution due to the correlation of the+Vxy(n,m) and +Vxy(n,m-1) waves. Eq.( 4.10b) is the "normal' wave which isscattered in all directions.(lines). In addition to Eqs.(4.10a)-(4.10b) we must alsoadd, in quadrature, the plane wave contribution resulting from the correlationwith +Vxy(n,m+1), which we have shown to be equal to +VxyD(n,m). The totalplane wave, designated by +Vxyp(n,m), relates in quadrature to +V xyPL(n,m) and+V xy D(n,m) b y

[ X p f o m ) ] 2 = T V ^ n . m ) ] 2 + [ X ^ m ) ] 2 (4.11)The portion of the wave which undergoes normal, symmetric, scattering isdesignated b y +Vxys(n,m), and is simply

^ xy sO u n) = X s L ^ m ) (4.12)

It is more useful to express +VxyP(n,m) and +Vxys(n,m) in terms of the originalwave amplitudes. Using Eqs.(4.7)-(4.10), +Vxyp(n,m) and +V xy S(n,m) thenbecome\ K n , m ) = (l/2)1 /2[+V xy(n,m-l)+V xy(n ,m)+ { X ^ m ) } 2 } m (4.13)


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Corrections for Plane W aves 203Similarly when +Vxy(n,m) is negative, +Vxy(n,m+1) and +Vxy(n,m-1) will vanishwhen they we positive. W e should also note that when +V xy(n,m) is negative theroles of+Vxy(n,m-1) and +Vxy(n,m+1) are reversed so that Eqs.(4.13) and (4.14)apply to Category II (instead of I) while Eqs.(4.17) and (4.18) apply to CategoryI, and so forth. Ap p.4.1 goes through the steps , parallel to the 2D development,for deriving the 3D planar and normal components for the various Categories.Throughout the previous discussion, we postulated that a plane wavecorrelates with an adjacent wave, with the correlation strength roughlyproportional to the product of the waves; however, any excess in the waveamplitude relative to the adjacent wave, causes that portion of the wave to scatterin all directions. As mentioned before, this assumption is supported byelementary quantum mechanical arguments, described in App.4A.2, where weshow that the quantum cross-coupling between adjacent regions is consistent withplane wave correlations.One may have noticed that throughout the discussion we have avoided thefollowing question concerning the plane wave analysis. The plane wavecorrelation process can only occur instantaneously if the wave in a particular line"kno w s" what the status is of the wave in the adjoining line. If we assume thetwo lines somehow "communicate" with each other, however, we return to ouroriginal problem, in which a signal delay occurs because of signals propagatingin the transverse lines. The only way to "resolve" this dilemma is throughintensive quantum mechanical considerations, where such questions are posed ina different manner. The issue is a matter for debate even to this day. Indeed, ourtreatment of wave correlations represents a classical description of phenomenamore appropriately described by quantum mechan ics. Need less to say, theissues just discussed becom e more important as the cell size is reduced. Furtherexamination of this topic will take us far beyond the scope o f the present subject.4.3 Changes to 2D Scattering CoefficientsHaving determined the effective wave amplitudes for a partitioned wave, our taskis now to modify the scattering coefficients. In particular, we must modify thescattering equations to account for the plane wave component of the wave. Thuswe need to change Eqs.(3.33)-(3.36) for coplanar scattering or Eq s.(3.97)-


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Corrections for Plane W aves 205

Tyx ,P(n,m,2) = 0 : (4.25){Z x y(n ,m)-Z x y(n+l ,m)=0}

since Zxy(n+l,m) and Zxyfom) are shorted out.In the case of 3D assume the wave +Vxy>P(n,m,q) is incident on the (n,m ,q)node. A total of six transverse lines must be decoupled by setting Zyx( n,m,q),Zyx(n,m+l,q), Z ^ r r ^ q ) , Z ^n .m .q +l ) , Z^ fam .q) , and Zyz(n,m+l,q) equal tozero. These zero values for the impedance lines are then substituted in the 3Dscattering coefficients.Once the scattering is completed each line will contain forward andbackward waves, each comprised of plane wave and symmetric components.Provided that the net plane wave and net symmetric components have the samesign, the two components will add in quadrature, resulting in the final field. Inthe event of a sign disparity between the two com ponents, a "decorrellation"process must be imp lem ente d, to be discussed later in the Chapter.A computer simulation of plane wave correlations in a light activated,semiconductor switch , with a parallel plate geom etry, is given in Chapter VII.As we shall see, there are very substantial differences, compared to the situationin which only symmetric scattering prevails.Finally, one should allow for the possibility that the node resistance (or anyother node parameter) is dependent on the wave amplitude. If so, is the nodedependence changed when the wave is partitioned into plane wave andsymmetric parts? Although second order effects are always possible, in this studywe assume the dependence of the node on the full wave amplitude is unchangedwhen the wave is partitioned. Thus, for example, the avalanche threshold at anode is assumed the sam e, regardless of the degree of partitioning.

Correct ions to P lan e W av e Corre lat ionThe plane wave correlation described in the previous Sections is satisfactoryprovided the neighboring TLM lines are uniform. W hen the bordering TLM linesdiffer, however, the previous correlation is not complete and we must consider


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206 Electromagnetic Analysis Using Transmission Line Variablesaugm enting the correlation appropriately. First we consider changes to thecorrelation when the adjoining cells have different dielectric constants.Correlation changes should not be surprising since the dielectric constant affectsthe wave energy and this in turn will influence the correlation. W e then go on todiscuss the modifications to the wave correlation required when the TL M wave isadjacent a conductor. This is an important modification because of theomnipresence of conductors, as well as the frequent application of guided wavesadjacent to a conductor(or for that matter, a dielectric). The topic of de-correlation of the waves, i.e., the conversion of plane waves into symmetricwaves, is discussed later in the Chapter.4.4 Correlation of Waves in Adjoining Media With Differing D ielectricConstantsIn the previous discussion we have tacitly assumed th at jh e correlation processtakes place between the amplitudes of adjoining waves propagating in the samedielectric media. With differing dielectric media , however, proper weight mustbe given to the energy residing in each wave, taking into account the dielectricconstant. A simple example will illustrate the point. If a wave in Zxyfom)correlates with another wave in Zxy ( n ,m + l) , situated in a very high impedanceregion , the correlation ignores the fact that the wave in Zxy(n,m+1) carries littleenergy and should therefore carry less weigh t. The simplest way to correct forthis oversight is to replace every amplitude +Vxy(n,m), located in Zxy(n,m), by+Vxy(n,m) /(Zxyfom))1'2, which takes into account the wave impedance. Theprevious correllation equations, such as Eqs.(4.13)-(4.14) and Eqs.(4.17)-(4.22),should be modified by the following replacements

^ ( r w m ) -+ X f o m ) / ( Z ^ m ) ) 1 7 2 (4.26)+N^n,m+\) -> X f o m + l ) / ( Z ^ m + l ) ) 1 ' 2 (4.27)

" V ^ m - l ) -> " V ^ m - l ) / ( Z ^ m - l ) ) 1 7 2 (4.28)


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Correc tions for Plane W aves 207as well as a similar set of transformations for +VxyP(n,m), +V xyP(n,m-l),+V xyP(n,m+l) and ^xysCn.m), +V xys(n,m-1), +VxyS(n,m+l). The criteria for the sixCategories of wave correlation, listed in Fig.4.2, are also modified accordingly.Thus Category I, for example, is: fVxyfom +l) / ( Z ^ m + l ) ) 1 ' 2 ] > fVxyOo.m)/ ( Z ^ m ) ) 1 ' 2 ] > r v ^ m - ^ / C Z ^ m - l ) ) 1 7 2 ] .

Another issue arises when we examine the correlation results at thedielectric-dielectric interface. Suppose we wish to obtain the correlation for theTLM wave belonging to the lower dielectric region (the larger cell) andpropagating parallel the interface. Normally this wave will be correlated with asingle wav e, in the high dielecric region, corresponding to the "nearest nod e" atthe interface(see Sections 5.3 and 5.4 in the next Chapter). Rather than consideronly a single cell, a better approxim ation takes into consideration the m ultitudeof high dielectric cells sharing the border with the larger low dielectric cell.

For exam ple, suppose at the interface the ratio of dielectric constants isnine. The cells in the low dielectric region will then be 3X larger than the theircounterparts in the high dielectric region, and therefore each large cell will sharea border with three of the smaller cells. We can improve the accuracy of thecorrelation of the wave in the larger cell if we perform the correlation with allthree waves, belonging to the smaller cells, rather than with only a singlewave(belonging to the nearest node). In performing the correlation we utilize theaverage of the fields in the three cells, bordering the large cell. During thecorrelation process we still utilize the modified relationships, Eqs.(4.26)-(4.28),discussed previously. The same averaging technique in the correlation may ofcourse be ex tended to arbitrary ratios for the d ielectric constant.4.5 Modification of Wave Correlation Adjacent a Conducting BoundaryThe need for this modification is most easily understood if we consider a planewave next to a conducting plane. If we assume the plane wave +Vxy(n,m) canpropagate next to the conducting zx plane, without loss of its plane waveproperties, then the previous wave correlation must be modified, as we shall see.Suppose the zx conducting plane is just beneath the Zxy(n,m) line. Since the w avein the Zxy(n,m-1) vanishes there is no plane wave correlation with that line, and a


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Corrections for Plane W aves 221is then obtained for each of these two cases. The larger value of a D is thenselected, which corresponds to the lesser degree of decorrelation needed toremove the sign disparity, i.e, the minimum Path whether it be I or II. In certaincases the only solution will be a D = 0 , in which case all the input plane wavefields will converted to symmetric fields.A simple example is discussed in order to render the concepts more clearly.Suppose we wish to eliminate a sign disparity in Zx y(n +l,m ). Th e de-correlationfactor ocD based on the planar scattering , must then be determined. W e assumefor the mom ent that no planar fields are present in the transverse lines and thusno decorrellation need be applied to these fields. Given these circumstances,how do we go about setting +Vk+1xyp(n+l,m) or +V k+1xyS(n+l,m) equal to zero,assuming a sign disparity is present. Several possib ilities exist. To simplify wefirst assume Path I prevails , and to make matters concrete, we also assum eB(n,m,l) is positive and ~Vkxyp(n+l,m) , +Vkxyp(n,m) are as well. We then take+V k+1xyS(n+l,m) to be neg ative, w hich thus makes +Vk+1xyP(n+l,m) positive. N owsince Path I applies, +V k+1xyP(n+ l,m) is to be eliminated. Because both incomingplane waves contribute to the reduction, we attach ao to both +Vkxyp(n,m) andVkxyp(n+l,m). In this case, in order to reduce the output plane wave componentto zero, we mu st likewise force the two planar inputs to zero, i. e. , a D = 0.During this reduction +V k+1xyS(n+l,m) may change in value but will remainnegative. Now consider a slightly different variation of the previous, in which allthe field magnitudes and polarities are unchanged but B (n ,m ,l) is negative. Inthis situation the decorrelation factor attaches to only one of the two inputs,namely, +V xyP(n,m). In this instance it may b e possib le to analytically determineccD from the scattering equation, Eq.(434)

D = -Bxypfom,!) -VxyKn+Mn) / T ^ ^ m J ) V ^ p f o m ) (4 .41a)Suppose the solution of the above does not satisfy 0 < a < l ? Then we proceed asbefore and com pletely decorrelate the two inputs, effectively forcing a D =0.We use the same example as before but now examine the possibility thatPath II applies. B (n ,m ,l) is once more negative and the field magnitudes andpolarities are as before. The strategy followed is then to reduce +V k+1xyS(n+I,m)


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Corrections for Plane W aves 227simultaneous presence of forward and backward plane waves witn me samepolarity in neighboring series TL M lines. This decorrelation is termed nonessential in the sense that it is used to simplify the decorrelation process, but isnot needed to obtain the final decorrelation. For this decorrelation, w e take a cluefrom equilibrium situations in which we know that at equilibrium, the forwardand backward waves are equal in amplitude and sign. In the process of achievingthe equilibrium state, we may regard the correlation between the two waves asindicative of the degree to which the system has achieved equilibrium. In order toinsure that the system tends toward equilibrium, however, we require acorresponding increase in the symmetric scattering , at the expense of the planewave scattering, i.e., we perform a decorrelation of the opposite waves As wewill, see this type decorrelation applies not only to equilibrium states, but tonon-equilibrium situations as well, i.e., time dependent situations in which planewave components always appear.

We can best illustrate the previous assertion by considering two oppositelydirected plane waves approaching one another with equal amplitudes, one in thepositive x direction and the other in the negative x d irection. Until the two w avesconverge , the waves proceed as normal plane waves. However, when the wavesborder one another, or even overlap, our interpretation of the two waves willchange. In fact the two waves converging on the (n,m) node, +Vxyp(n,m) and"Vxyp(n+l,m) may just as well be regarded as symmetric waves +VxyS(n,m) and~V kxys(n+l,m). Indeed it is very easy to show that, for these oppositely directedsymmetric waves, the net scattering to the transverse lines Zyx(n,m) andZyx(n,m+1) vanishes. We should also take note that in arriving at the zeroscattering to the transverse lines, we conside red opp osite waves in the Zxy(n,m)and Zxy(n+l,m) lines, rather than both waves in the Zxy(n,m). However, if thechanges in the x direction vary slowly, and the TLM length is correspondinglysmall, then we can compare forward and backward waves in the same Zxy(n,m)line.

In the previous discussion we saw that a pair of opposite plane waves, ofequal amplitude, are equivalent to symmetric waves so far as transversescattering is concerned. W hat about the reflected and transmitted waves inZxyfom) and Z*y(n+l,m)? Indeed it is easy to show , for example, that the


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Corrections for Plane W aves 259G s(n,m) = 1- [(U FP 1(n,m)+(UFp2(n,m)]/U TOT(n,m) (4.105)

On the other hand w e require the sym metric weight for each grid. Since the finalsymmetric field must be diluted by the number of grids, the simplest approach isto assign equal weight to each of the grids, and thus the weight for the Nth gridfor the symmetric p a rt , GsN(n,m), isGSN(n,m) = G s(n,m)/ N T (4.106)

We are now prepared to obtain the node resistance, averaged over the Nxgrids. The actual quantity to be averaged is (1 / R(n ,m)), i.e, the conductance, asmentioned previously. We denote the node resistance of the Nth grid byRN(n,m). The averaging then takes the formAV[(l/R(n,m)] = G 1(n,m)(l/RN1(n,m)) +G 2(n,m) (l/R N 2(n,m))+ ,NT I GsN(n,m) (l/R N(n,m)) (4.107)

The above represents the average for the node resistance, but other nodeproperties mentioned in Chapter II may also be averaged along the same lines.For pure plane waves the last term in the above vanishes, and conversely, forpure symmetric waves the first two terms vanish.4.22 Comp arison of Standard Num erical M ethods and TLM MethodsIncorporating TLM Correlations/Decorrelations and Grid OrientationWithout wave correlations and grid orientation effects w e should expect the TLMand numerical solutions of Maxwell's equations to yield identical simulationresults. With the introduction of wave correlations and grid orientation effects,however, we will start to see departures from the standard numerical techniques.The departures will be most evident in extremely fast electromagnetic pulses andin the description of the initial field profiles. These situations cannot possibly b edescribed by standard num erical methods without taking into account thesignificant effects of plane wave correlations and grid orientation. The resultshould be greater accuracy in the prediction of electromagnetic energy dispersal.


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Corrections for Plane W aves 265In the light of the above discussion, it is easier to see what we have done intreating plane wave correlations earlier in the Chapter. In this case, instead ofdistributing the overlap to both photons, we have arbitrarily concentrated theoverlap onto one of the two photons in the m+1 line. The other photon in them+1 line is then completely isolated and does not feel the overlap from the mthline. W e also emphasize though we have only considered the simplified case ofthree photons distributed in the m and m+1 lines, we may extend the discussion

to arbitrary amplitudes in the two lines, in which the number of photons in eachline is proportional to the amplitude. The zero order wav e functions, and theirassociated probability functions, possess the potential for a natural source ofcross-coupling between waves in adjacent regions(TLM lines). What w e have no taccomplished is to further mod el the nature of the coupling between waves inadjacent states. To do this a more detailed use of quantum mechanical methodsmust be employed.


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Boundary Conditions and Dispersion 267in impedance levels. A conducting surface also will prov ide total reflection butwith a resultant field inversion.Other boundary conditions relate to the input/output of the electromagneticsignal. Here we specify the cond itions under which electromagnetic energy isintroduced into or leaves the region o f interest, whether it be a closed device oran antenna. The difference between the input and output energies results in eitherthe dissipation or storage of energy in the region of interest. In the following wedescribe ways in which to simulate the boundary conditions, using the TLM cellmatrix and appropriate values of node resistors and transmission lineimpedances.5.1 Dielectric-Dielectric InterfaceThere are several choices for positioning the TLM boundary of a dielectric-dielectric interface, two of which are shown in Fig. 5.1. The side view shows thedielectric - dielectric interface with constants Si and e2 . The permeability jx isassumed to be the same in both regions. In contrast, we assume ei > e2 . Thepropagation velocities in each region satisfy

( v , / v 2 ) = [6 2/s,]1/2 (5.1)where vi and v2 are the velocities in regions 1 and 2 respectively. If we wishto retain the same time step in each region, At , then we are forced to adoptdifferent cell sizes in each region. Since the linear dimension of each cell isinversely proportional to the velocity, the transmission line lengths in eachregion, Alj andAl2 , satisfy

(A1,/A12) = [e 2/e,]1/2 (5.2)The particular choice of boundary dictates what value of characteristicimpedance is selected for the transmission lines at the interface and parallel to it.In Fig.5.1(a) the horizontal lines at the interface, Zxy(n,m), Z ^ n + l . m ) ,


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Boundary Conditions and Dispersion 287applications) cannot be used toward the existing prob lem. In these otherapplications both uniform and non-uniform grids are established throughout thespace. This software must then be adopted by attaching a node, cell, andsurrounding TLM lines to each of the grid points. In the final analysis, thedecision must then be made as to whether it is m ore expedient to adopt existingsoftware to establish the TLM m atrix, or whether it is easier to develop thenecessary TLM matrix from the outset.

Other Boundary Condi t ions5.8 Dielectric - Open Circuit InterfaceOn occasion, we may wish to constrain or direct electromagnetic energy within aparticular region, by mean s of an extremely high permeability constant materialat a given interface. Such an interface may correspond to an actual representationof the experimental facts, where an extremely large positive mismatch exists; orelse the interface may correspond to a mathematical simplification of a radiationproblem in which we impose the reflection of electromagnetic energy by m eansof an "open circuit". This is in contrast to the field reversal experienced byarbitrarily placing a conducting region at the interface. Since the relatively highimpedance of the transmission line often arises from a large permeability, n, wemust include this in the determination of the cell sizes. Thus, Eq .(5.2) becomes

(AWAfe) = [ Eaua/s^f2 (5.10)In the limit o f open circuit impedances, i.e., very large jo., the num ber of cells inthe slow region will grow very large, thus adding an unnecessary degree ofcomplexity. A simpler way to simulate an open circuit is to maintain the samecell sizes on both sides of the interface, but to assign extremely largecharacteristic values to Z(n,m,q) in the open circuit region , as indicated in Fig.5.14. The im plication of having the same cell size is that the dielectric constant


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Boundary Conditions and Dispersion 303calculate the displacement of charge(including loss) within atomic systems, fromwhich the dispersive polarization follows. From this we obtain the dispersion ofthe complex inductive capacity. Once we obtain the proper dispersion relation,the complex inductive capacity may be used to obtain K, which may becom pared with Eq .(5.16) to obtain the dispersive relationships for K and a.Using he localized field app roa ch[l], we can obtain the polarization P,which is now dispersive, as a function of the macrosco pic field E. Thus

P(ro)=a280E/(coR

2-o)

2-jcog) (5.17)

adopting the notation in [1]. a2 is proportional to the number of oscillators perunit volume, each with resonant frequency co0, g is the dissipative term, and CORis related to co0 by co2R = co20 -(l/3)a2 . The phenomenological constants COR ,g ,and a are determined from experiment. If K ' is considered to be a complexspecific inductive capacity, thenP = [ K ' - 1 ] E E (5.18)

Comparison of Eqs.(5.17) and (5.18) then providesK' = l+{a2/[co2R-co2-jcog]} (5.19)

In terms of the com plex propagation constantK = ct+jp = (CO/C)K'1/2 = (co/c)[l+a2/(co2R - co2 -jog)]172 (5.20)

We then separate the terms inside the radical into real and imaginary parts, andassume that the dissipative term g is much smaller than co and (coR -co). Withthese assumptions,K = a + j p = (co/c)[l+{a2/(co2R-co2)}+ja2cog/(co2R-co2)2]1/2 (5.21)


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Boundary Conditions and Dispersion 309velocity) lose their usual meaning. In this discussion, we assume co is still farenough from resonance, such that the phase velocity is still meaningful.

Incorporat ion of Dispers ion into TLM Formulat ion5.20 Dispersion ApproximationsFrom the previous discussion we may infer that dispersion introduces anothercomplication, related to the non-uniformity of the cell field, into the iterationprocess. Without dispersion the wave energy travels exactly one cell length for agiven time step. By adding dispersion, how ever, certain Fourier com ponents maytravel only a fraction of a cell length w hile others m ay travel several cell lengths.Indeed, in the previous discussion, we saw that for "normal" dispersion, whereV(CD )< v(0) at the lower frequencies, the wave components will travel a celllength or less. For the " ano malous" case, on the other hand, v(co)>v(0) as coapproaches COR , and thus some components will travel more than one celllength. In the ensuing discussion we adop t a matrix utilizing the cell lengthdetermined by the "zero frequency " velocity , v(0 ), so that Al = v(0)At. W ethen continue the approximation by confining ourselves to two Categories ofdispersion. In both categories we will tacitly assume the max imum velocityexcursion , Av0 is always much less than v{0). AvQ is assumed to be adispersionless constant. To simplify matters we also assume in the following tha tthe neighboring cells are identical, so that v(k) is the sam e in each cell. Differingcells will of course modify the calculation, but not alter the basic technique. InCategory I the velocity is always less than or equal to v(0), so the individualwav e com ponents never travel more than one cell length during the time step At.W e express the velocity limits in terms o f both co and the propagation constant k:

CATEGORY 1v(0)- Av0 < v(co)


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Boundary Conditions and Dispersion 321conservative one, which is to recalculate the spectrum after each time step,provided such a procedure is practical in terms of computer capacity.In the previous we confined the discussion to the Category I forward wa veand in particular to the transm ission o f the forward wave in the n-1 cell to the nthcell. There are two other contributions of course. One is the forward wave in thenth cell during the kth time step. The bulk of this wave will have left the cellonce the time step is completed. Since the velocity is slightly less than v(0) ,however, a small portion of the wave will remain in the nth cell. The calculationfollows the same course as before except that now we first obtain the FourierTransform of the forward wave in the nth cell for the kth step. We then obtainthe contributions which remain in the nth cell and add these to the previouscontributions. The final contribution originates from the reflection of thebackward w ave at the (n-1) node. The reflected wave is related to the backw ardone by

~A(n, k) = B(n-1, k) + A(n,k) (5.64)With the use of numerical techniques, k is of course replaced by kj. The initialcalculation differs here in that we first obtain the Fourier Transform of thebackward wav e. We then imagine the backward wave, with amplitude modifiedby Eq .(5.64), flowing , in the forward direction, into the nth cell at the n-1 node.For this wave the bulk of the contributions will remain in the nth cell. Thesecontributions are then added to those of the previous waves discussed.We may reduce the amount of computer computation, at least for thereflected wave. W e do this by examining the (n-1) node and combining thetransmitted and reflected waves with identical wavenu mbers. We can employ th istechnique provided w e use the same kc cutoff in each cell , and therefore thesame set of kj By this technique, the transmitted and reflected waves arecombined into a single forward wave in the nth cell. This enables us to employanalytic expressions while reducing the amount of numerical computation. Thisprocedure works, of course , only wh en the adjacent cells are uniform and thuseach wave in the two adjoining cells has the same velocity dispersion.

Let us denote the kth transmitted wave in the nth cell by +AxR(n,k) and isrelated to the forward w ave in the (n -l)th cell by Eq.(5.55). The forward wave


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322 Electromagn etic Analysis Using Transm ission Line Variablesresulting from the reflected wave is denoted by +AREF(n,k) and is given byEq .(5.64). The sum of these two waves is determined by sinusoidal addition , asopposed to the usual arithmetic add ition when the waves are uniform throughoutthe cell. Denoting the sum by +A s(n,k ), the result is

(+As(n,k))2=(+ATO(n,k) )2+(+AREF(n,k))2+2+ATR(n !k)+AREF(n,k)cos(A(p(k)) (5.65)

where A(p(k) is the phase angle difference between the two waves, i.e., Acp(k)=tp(k,n)-cp(k.n-l).Unfortunately we are unable to perform a similar addition for the forwardwave originally in the nth cell during the kth time step. This is because we cannever match the waves with the same k num ber. Neither the transmitted orreflected waves catch up with the original forward wave, all of which have thesame velocity for the same k(or kj) value. Waves with differing values of k mayof course catch up with one another, leading to significant changes in the fieldprofile.

5.23 Initial Conditions With Dispersion PresentThe most com mon initial condition is that in which the field in a particular cellis bo th static and uniform so that both the forward and backw ard wav es arelikewise uniform ,i.e., rectangular in shape. As an exam ple, therefore, we seekthe initial Fourier Transform in each of the cells belonging to a cell chain as inFig.5.21 or 5.22. It is relatively easy to obtain each o f these transforms if we firstobtain the transform of the auxiliary cell, centered about x=0, with total celllength Al. Assume for the mom ent, therefore, that this cell is isolated from itsneighbors and that the field profile is rectangular shaped, i.e., the field is Vo for-Al 12 < x


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Boundary Conditions and Dispersion 32 7~V(0,x). = (Vy7i)J0*[sin(kAiy2)/k][cos{kx -k(q(k,t) Al +kv(k)t}]dk-Al /2< x


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334 Electromagnetic Ana lysis Using Transm ission Line Variablesdisadvantage in using the same matrix is that the time step being used is usuallyso small that very little happens, when analyzing slower phenomena during thatstep. In the case of drift, for ex ample, the only change that occurs is near the cellboundary. One way of dealing with this disparity is to evaluate the slowerphen om ena less frequently, instead of after each step. Indeed we will see laterthat if we use the same electromagnetic cell size to examine transportphenomena, then it is appropriate to use a slower "sampling " speed.

Before bringing the carrier description under the umbrella of thetransmission line matrix model, we will first discuss the discharge of neighboringcells, resulting from the potential difference between them. The discharge, i.e.,the transfer of charge, occurs via the node resistors connecting the cells. We willsee later that the description of this discharge is consistent with the introductionof carrier drift into the TLM cell matrix m odel.6.1 Charge Transfer Between CellsW ithin the TLM mode l there exists iso-potential cells which are separated bytransmission lines, which represent differences in potential between cells andwhich also account for the conveyance of electromagnetic energy. The nodalresistors simulate the conductivity of the medium and provides the m eans for thecells to discharge into one another. Our scattering equations automatically takeinto account any changes in potential difference between cells, resulting fromeither a change in the wave status, or from charge tra nsfe re e., current) .Equivalently this allows us to track the evolution of net charge on each cell, aswe shall show in a mo ment.

A natural question which arises is what happens if the resistivity returns toits formerly large value, characteristic of equilibrium. Suppose, for example, thelight activation process in a semiconductor, which produces conductivity, ceasesat time t = ti . We further assume an exponential "recovery" of the noderesistance R (n,m,q). Thus,R(n,m,q) = [R(n,m ,q)]M lEXP((t-t1)/x) (6.1)

where {R(n,m,q)}M i is the node resistance value at t = ti, and x is the "recoveryconstant". For times (t-ti) large compared to x one might expect Eq.(6.1) to rein-


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Cell Discharge and Transport Integration 351

6.9 RecombinationIn this Section we include recombination effects in the iteration. As is wellknown, there are a large num ber of recombination m echanisms, many o f whichoccur simultaneously in semiconductors. Typically the time scales involved inthe recombination process will vary over a wide range but are usually muchlonger than the electromagnetic delay time. For illustrative purposes we select asingle quite common mechanism.For concreteness, we assume recombination of carriers is achieved via theexistence o f a single energy level in the midgap region. The midgap energy levelserves as an indirect means for carrier recombination, i.e., these deep level sitesachieve the recombination by a two step process: first an electron is capturedfollowed by the capture of a hole. The capture and emission rates, involved in therecombination process, are assumed to differ for holes and electrons and holes,and to be field dependent as well. First we set forth the following defin itions[l]Nx(n,m,q) = Num ber of recombination sites in (n,m,q) cellT(n,m,q) = Number of recombination sites filled with electrons in (n,m,q) cell/>r(n,m,q) = Nu mber of empty recombination sites in (n,m,q) celland which satisfy

N T(n,m,q)= nT(n,m,q) + />r(n,m,q) (6.44)

With these definitions we are able to write down the rate equation for electronsd/dt)REcoM= enT(n,m,q)-cn/>r(n,m,q)(n,m,q) (6.45)

en is the emission coefficient representing the transitio

electromagnetic analysis

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