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Doctoral Thesis
Passive microwave circuit elements analyzed with thefrequency-domain TLM method
Author(s): Hesselbarth, Jan
Publication Date: 2002
Permanent Link: https://doi.org/10.3929/ethz-a-004315993
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ETH Library
Diss. ETH no. 14544
Passive microwave circuit elements analyzed with the
frequency-domain TLM method
εεεεr
JAN HESSELBARTH
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
2002
Diss. ETH no. 14544
Passive microwave circuit elements analyzed with the
frequency-domain TLM method
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
for the degree of Doctor of Technical Sciences
presented by
JAN HESSELBARTH Dipl.-Ing., Dresden University of Science and Technology (TU)
born April 13, 1970 citizen of Germany
accepted on the recommendation of Prof. Dr. Rüdiger Vahldieck, examiner
Prof. Ke Wu, PhD, co-examiner
2002
Acknowledgements I would like to thank cordially my supervisor Prof. Rüdiger Vahldieck
for his help and continuous assistance during the last years. I was particularly impressed with his confidence and his ability to recognize situations when to direct me and when to let me go my own way.
I would also like to thank Prof. Ke Wu, who accepted to serve as co-examiner and reviewed this work as well.
Special thanks are to Dr. Pascal Leuchtmann, who never hesitated to enter discussions on various aspects of electromagnetics, which helped me a lot to become familiar with the miraculous Maxwell theory.
I am also in debt to my colleagues in the Field Theory Group for all the fruitful discussions and the warm and creative atmosphere.
for Benjamin
1
Contents
Zusammenfassung ................................................................................. 3
Abstract .................................................................................................. 5
1. Introduction ..................................................................................... 7
2. Formulation of the Frequency-Domain TLM Method
2.1. General Aspects ..................................................................... 11 2.2. Derivation of the FDTLM Node ............................................ 13 2.3. Dispersion Relations of the FDTLM Nodes .......................... 28 2.4. Incorporation of Boundary Conditions .................................. 39 2.5. Conclusions ........................................................................... 44
3. Algorithms for the Application of the FDTLM Method ............ 45
3.1. Cavity Resonance Frequency ................................................ 46 3.2. Waveguide Mode Cutoff Frequency ..................................... 50 3.3. Waveguide Eigenmode Problem ........................................... 51 3.4. Scattering Parameters of 3D Structures ................................. 58 3.5. Implementation Aspects ........................................................ 65 3.6. Conclusions ........................................................................... 72
4. Application of the FDTLM Method to Passive Microwave Circuit Elements ............................ 73
4.1. Cavity Resonance Frequencies and the Non-physical Solutions ............................................. 74 4.2. Rectangular Waveguide Cutoff Frequencies and the Influence of Mesh Grading ....................................... 77
2
4.3. Non-physical Solutions in the Waveguide Eigenmode Algorithm ........................................ 81 4.4. Scattering Parameters of a Waveguide Filter and Accuracy Analysis .......................................................... 85 4.5. Higher Order Mode Scattering Parameter Extraction of a Waveguide Bend ........................................... 90 4.6. Resonance Frequencies of High-Permittivity Dielectric Resonators ............................................................ 95 4.7 Microstrip Patch Antenna on Corrugated Substrate ............ 100
5. General Conclusions and Outlook ............................................. 107
6. References .................................................................................... 113
Publications of the Author Related to this Thesis .......................... 117
Curriculum vitae ............................................................................... 119
3
Zusammenfassung
In der vorliegenden Arbeit wird die Transmission-Line Matrix Methode im Frequenzbereich (FDTLM Methode) für die Untersuchung passiver, linearer Mikrowellenbauelemente (wie beispielsweise Resonatoren, Wellenleiter und Mehrtorschaltungen) verwendet.
Zuerst wird die FDTLM Methode in ihren verschiedenen Varianten in einer einheitlichen Art und Weise aus den Maxwellschen Gleichungen hergeleitet. Die verschiedenen Diskretisierungsvarianten der FDTLM Methode werden hinsichtlich ihrer numerischen Genauigkeit (Dispersion) verglichen. In mehreren Fällen können analytische Ausdrücke für die Dispersion angegeben werden, die einen besonders klaren Vergleich gestatten.
In einem zweiten Teil werden algebraische Verfahren entwickelt, die es erlauben, praktische Probleme (beispielsweise die Berechnung von Resonanzfrequenzen, Eigenmoden und Streuparametern) mit der FDTLM Methode zu lösen. Dabei wird insbesondere die numerische Effizienz der verschiedenen Algorithmen im Hinblick auf ihre Anwendung auf realistische, numerisch grosse Problemstellungen untersucht.
Im letzten Teil wird die FDTLM Methode zur Bearbeitung einiger repräsentativer Problemstellungen verwendet. Hier werden zudem verschiedentlich auftretende nicht-physikalische Lösungen diskutiert. Die FDTLM Methode wird hinsichtlich der Genauigkeit der berechneten Lösungen sowie des dazu notwendigen numerischen Aufwandes untersucht.
Es stellt sich heraus, dass die FDTLM Methode im Vergleich mit anderen Frequenzbereichsverfahren (etwa der Methode der finiten Elemente) sowohl Vor-, als auch Nachteile besitzt und sie diesen Methoden durchaus ebenbürtig ist.
4
5
Abstract
The Frequency-Domain Transmission-Line Matrix (FDTLM) method is applied to the analysis of passive, linear microwave circuit elements, such as resonators, waveguides, and multiport devices.
First, the FDTLM method is derived from Maxwell’s equations in a consistent way and the accuracy of the discretization scheme is evaluated. In several cases, the accuracy of the discretization is described in an analytical form, giving more insight into the very nature of the FDTLM method.
Second, appropriate algebraic procedures are presented to solve the various problems at hand (resonances, waveguide modes, scattering parameters). The mathematical formulations are optimized in order to allow for the efficient treatment of large, real-world problems.
Finally, the overall applicability of the FDTLM method is evaluated and benchmarked using various case studies. The appearance of non-physical solutions is discussed, and the tradeoffs between accuracy and numerical cost are investigated.
It turns out that the FDTLM approach is competitive with other frequency-domain methods, such as the finite-difference method or the finite-element method.
6
7
1 Introduction
The idea to model electromagnetic wave phenomena using a network of transmission lines arose in the 1940’s. The propagation of voltage and current on a transmission line, described by the telegrapher’s equations, was found to be similar to the propagation of a plane wave in space, governed by Maxwell’s equations. G. KRON [27], [48] described how a waveguide cavity was modeled by a transmission line network, which was actually built and voltages were physically measured, allowing to conclude about the electromagnetic fields in the original problem.
This approach fall into oblivion until the 1970’s, where digital computers became widely available. P. JOHNS and R.L. BEURLE [25] are recognized having formed the name “TLM”, pushed the theory, developed a number of application examples, and brought it all to a wide attention.
Although development on the FDTD (finite-difference time-domain) method [49] started at about the same time as the work on TLM method, the latter had some serious drawbacks, namely the need of more computational power and memory, as well as the lack of a stringent bijective mapping between the real world (Maxwell’s equations) and the model (TLM). The more intuitive approach of TLM was rousing for some insiders, but most research and manpower was focused on FDTD.
It was not until the development of the so-called symmetrical condensed node (SCN) in 1987 [26], that TLM was able to catch up. It turned out that the SCN does not have any equivalent transmission line network, thus loosing the heuristic beauty inherent in the early TLM approaches. Instead, the SCN can be derived from Maxwell’s equations by finite differencing. By doing so, the resulting system of equations is largely over-determined. The SCN was found to have a smaller dispersion level as FDTD and allows for simpler boundary definitions than FDTD does. The required computer resources (arithmetic power as well as memory) are, however, about twice as high as for FDTD. The computational overhead of TLM is avoided in the so-called ATLM (alternating TLM) method which came up in 1995 [1], [38], [39] and was shown to have about the same computational requirements as FDTD.
In fact, there is evidence that the differences between TLM and FDTD are as large (or as small) as the differences between the different formulations of TLM themselves [8], [14], [28], [29]. The FDTD method, however, can be considered more mature than the TLM method because
8
during the last 30 years, much more research was dedicated to FDTD than to TLM in electromagnetics.
The applications of the TLM method in electromagnetics, from the beginning to our days, are more related to (free-space-) scattering and EMC problems than to guided wave and circuit problems. This can be explained by the fact that scattering and EMC problems can only be investigated by field solvers, whereas microwave circuits and guided wave problems could usually simplified until being approached as RLC circuits. This has several implications. The measurement equipment for microwave circuits and guided wave components has a much higher accuracy than scattering and EMC measurement setup. Thus, any numerical method for the modeling of microwave circuits and guided wave components must be very accurate. In addition, scattering and EMC problems have usually weaker resonances and higher loss (such as radiation in free space). As a result, few applications of TLM and FDTD are in the microwave circuits and guided wave area. The long-term stability and accuracy of these time-domain methods, needed to evaluate resonant, low-loss problems, are not well understood. Furthermore, time-domain methods are challenged by frequency-dependent material properties and modal excitations. A frequency-domain method will have much less problems in these regards.
The principles of the frequency-domain TLM (FDTLM) method were published first in 1992 [21], [23]. It turned out quickly that the new approach has significant advantages compared to time-domain TLM for some classes of problems. However, FDTLM still has to prove to be competitive with other general three-dimensional frequency-domain methods.
Yet another, though related, frequency-domain approach is the so-called method of Minimal Autonomous Blocks (MAB). The idea of this method is to solve Maxwell’s equations in a parallelepiped cell by expanding the fields in six orthogonal series of TEM waves. Each brick is then represented by a scattering matrix which can be connected to those of neighboring bricks, thus generating the scattering matrix of the computational domain. Matrix algorithms were developed [33] and the method was applied, e.g., to waveguide scattering problems [31] including gyromagnetic [32] and chiral [34] media. Note that although this method is similar to a large extend to the FDTLM method, except that the transmission line equivalence is not used, it employs an original notation and the TLM similarity was never mentioned in the literature. Furthermore, the method as a whole was described in the russian literature [33] only, but a few exceptions exist [31], [32], [34].
9
In this thesis, some developments are detailed which aim to make FDTLM more reliable and open to a wider range of problems. In order to do this, a consistent derivation of the discretization scheme from Maxwell’s equations allows to evaluate boundary conditions and mesh grading effects. Then, various algorithms suited to different classes of problems, such as resonance frequencies, eigenmode problems, and scattering parameters of multi-port structures, are presented. These algorithms are evaluated regarding accuracy and numerical cost. Finally, real world examples are benchmarked against measurements and solutions from FEM (finite element method) software.
The work presented in this thesis contributes the following original ideas: 1. The scattering matrices of various FDTLM nodes are derived in a
consistent manner from Maxwell’s equations directly. The numerical accuracy for plane wave propagation (dispersion) is used to evaluate the different nodes, including non-cubic node shapes and lossy media. In some cases, analytical formulas for the dispersion errors are presented.
2. A matrix eigenvalue algorithm for the calculation of cavity resonance frequencies is developed and allows an efficient calculation even for large problems. An algorithm for the calculation of waveguide cutoff frequencies is presented. A well known algorithm aiming to the 2D eigenvalue analysis of waveguides, is numerically improved. The appearance of non-physical solutions during resonance frequency and eigenmode calculations is discussed and possibilities for their treatment are considered.
3. Algorithms for multi-mode multi-port excitation of microwave circuits are developed and benchmarked. Special consideration is given to the computational efficiency of the methods. The most efficient algorithm found for large problems is based on one (or a few), possibly iterative solution(s) of a linear equation system (that is, it avoids matrix inversions). Sparse matrix methods and iterative algorithms are evaluated.
4. Real-world microwave circuit problems are used to benchmark the FDTLM method.
10
11
2 Formulation of the Frequency-Domain TLM Method
2.1 General Aspects
The propagation of electromagnetic fields in space and time is governed by Maxwell’s equations (2.1), (2.2).
tDJH
∂∂+=curl (2.1)
tBE
∂∂−=curl (2.2)
In order to make these partial differential equations applicable to numerical analysis, several simplifying assumptions are usually made. The most important one during the course of this thesis is that the space of interest shall be piecewise homogeneous, and the size of geometrical details is comparable to the wavelength of the electromagnetic waves.
The second important simplification of Maxwell’s equations results from the assumption that all involved materials are fully described by linear, time–invariant, diagonal tensors of permittivity, permeability, and electrical conductivity such as shown in equations (2.3)-(2.5). In most practical cases, however, the materials will be considered isotropic.
EED
rz
ry
rx
⋅
⋅==
εε
εεε
000000
0 (2.3)
HHB
rz
ry
rx
⋅
⋅==
µµ
µµµ
000000
0 (2.4)
EEJ
z
y
x
⋅
==
σσ
σσ
000000
(2.5)
In what follows, the electromagnetic field described by (2.1)-(2.5) together with boundary and initial conditions, will be analyzed by the so-called Transmission Line Matrix (TLM) method. The problems to be
12
considered will be restricted to cartesian coordinates, although TLM analysis of electromagnetic fields in other coordinate systems is possible and may be useful [6]. Under these circumstances the TLM method discretizes the space of interest by a rectilinear grid (mesh of parallelepipeds or bricks). A typical example is shown in Fig. 2.1. Here, a H-plane, 90 degrees, compensated, rectangular waveguide bend is discretized by means of 71 FDTLM nodes. All boundaries are supposed to be conductors, with the exception of the two port boundaries. Each port is discretized by 8 nodes, resulting in reasonable accuracy for fundamental mode scattering parameters. This example is discussed further in chapter 4.5.
Fig. 2.1: H-plane bend in rectangular waveguide, discretized for FDTLM analysis
Furthermore, it should be noted that the TLM method itself represents a numerical tool which allows to solve various kinds of differential equations. For example, thermal diffusion problems, Laplace and Poisson equations were solved using the TLM approach (see [11] for an overview).
13
2.2 Derivation of the FDTLM Node
The Transmission Line Matrix (TLM) method in cartesian coordinates discretizes space in parallelepipeds (bricks) having their edges in parallel to the axes. In the following, such a parallelepiped being the smallest element of the mesh, will be named node cell or node. Twelve mutually orthogonal plane waves are considered in the node cell, travelling and being polarized in the directions of the coordinate axes. Any plane wave traveling trough the node cell in a given direction, being polarized with a given polarization, can be projected onto these 12 waves. The TLM philosophy is based on the fact that the governing equations are similar for a plane wave propagating in a given direction, and a TEM wave traveling along a transmission line in the same direction. Thus, the twelve plane waves can be represented by voltage (and current) waves traveling on TEM transmission lines which point in the directions of the coordinate axes.
The relative weight and interaction of the 12 orthogonal waves in a node cell are described by Maxwell’s equations. This process is described numerically by the so-called node scattering matrix. Note that there is a duality from the very beginning of TLM modeling between approaches in terms of electromagnetic fields and the terminology (and thinking) of electrical networks. The advantage of this dual approach is obvious if the electromagnetic field problem shall be solved by actually constructing an equivalent RLC network. However, a computer algorithm may profit as well, because an algorithm having a network equivalent is energy conserving and stable. The so-called Symmetrical Condensed Node (SCN) introduced in 1987 by Johns [26] does not have an equivalent electrical network, but the mixture between ”network terminology” and ”field terminology” remained to our days. The easiest way to accept this mixture is to recall the equivalence between current and voltage on a TEM transmission line and the propagation of a plane wave in a small section of space.
Note that the decay of an evanescent field does not has a transmission-line equivalent, therefore, the FDTLM method handles evanescent fields by a finite-difference approximation.
A distinct feature of the FDTLM method becomes obvious which contrasts to methods based on polynomial interpolation of the electromagnetic fields (such as finite difference or finite element methods): Since the propagation of both a plane wave and a TEM wave is governed by the same equations, a FDTLM node can model (under some
14
circumstances) the propagation of a plane wave exactly, that is, independent of the size of the node. This is even true for lossy media.
If the 12 orthogonal waves are projected into voltages and currents travelling on two-wire transmission lines, the typical picture of an SCN [26] is obtained (Fig. 2.2).
∆∆∆∆z
V+xy
∆∆∆∆xx
y
z
V+xz
V–xy
V–xz
V+yxV+yz
V–zy
V–zx
V+zx
V+zy
V–yxV–yz
∆∆∆∆y
Fig. 2.2: Symmetrical condensed node (SCN) schematic
Consider now the example of a y-polarized plane wave traveling in –z-direction. This wave is represented by current and voltage propagating on a parallel-plate line, generating the same field pattern between the plates as a plane wave would have (Fig. 2.3).
Then, both plane wave and TEM wave have the same velocity if the distributed capacitance, Czy, and inductance, Lzy, are given by
yxC ryzy ∆
∆= εε0 (2.6)
xyL rxzy ∆
∆= µµ0 (2.7)
15
perfect electric conductor
Ey
Hx
SV+zy
I+zy
perfect electric conductor
perf
ect m
agne
tic c
ondu
ctor
perf
ect m
agne
tic c
ondu
ctor
Fig. 2.3: Parallel plate line model of a stub of the SCN (propagation direction -z, y-polarization)
The power flow through the plane z = const. of both the plane wave and the TEM wave is identical if
xHI xzy ∆=+ (2.8)
yEV yzy ∆=+ (2.9)
In general, the following relations hold for {i,j,k} ∈ {x,y,z}, (i≠j≠k)
jkC rjij ∆
∆= εε0 (2.10)
kjL rkij ∆
∆= µµ0 (2.11)
ijijrkrjij CL
v 11
00
==µεµε
(2.12)
rj
rk
ij
ij
ijij k
jCL
YZ
εεµµ
0
01∆∆=== (2.13)
The field component Ey stems from y–polarized waves traveling in either x or z directions (transmission lines ±xy and ±zy). The total capacitance of the node in direction y is then defined as
16
stubyzyxyry
totaly CzCxC
yzxC +∆+∆=
∆∆∆= εε0 (2.14)
The last term, Cstub, represents an excess capacitance added in the center of the node. In order to keep time synchronism in time-domain TLM algorithms, this capacitance is not added as a simple shunt capacitor, but by means of an open-circuited transmission line (”stub”) which adds capacitance after the appropriate time delay (that is, one time step later).
The magnetic field component Hx is related to the transmission lines ±zy and ±yz, thus defining the total inductance in x-direction as
stubxyzzyrx
totalx LyLzL
xzyL +∆+∆=
∆∆∆= µµ0 (2.15)
Also here, an excess inductance can be added using a short-circuited transmission line (”stub”) in the node center.
Applying a similar procedure to all field components results in
stubxzxyxrx
totalx CzCyC
xzyC +∆+∆=
∆∆∆= εε0 (2.16)
stubyzyxyry
totaly CzCxC
yzxC +∆+∆=
∆∆∆= εε0 (2.17)
stubzyzxzrz
totalz CyCxC
zyxC +∆+∆=
∆∆∆= εε0 (2.18)
stubxyzzyrx
totalx LyLzL
xzyL +∆+∆=
∆∆∆= µµ0 (2.19)
stubyzxxzry
totaly LzLxL
yzxL +∆+∆=
∆∆∆= µµ0 (2.20)
stubzyxxyrz
totalz LyLxL
zyxL +∆+∆=
∆∆∆= µµ0 (2.21)
These six equations (2.16)-(2.21) represent a basic set of necessary conditions to be satisfied by any SCN. The 18 unknowns in the above equations are to be determined by an additional set of 12 constraints in a way that the node scattering matrix represents Maxwell’s equations as good as possible.
In time-domain TLM, for example, one set of constrains follows from the requirement that time-synchronism of all pulses at the node cell
17
boundaries has to be kept. That is, the delays for all pulses traveling from the node center to the node cell boundary (or vice versa) must be equal. Thus, 6 additional constraints are given by
zyzxyxyzxzxy vz
vz
vy
vy
vx
vx ∆=∆=∆=∆=∆=∆ (2.22)
The remaining 6 constraints can be set in various ways. If the characteristic impedances of all link lines are set to the impedance of the background medium, the scattering matrix of the standard SCN [26] can be found. On the other hand, if all stubs are eliminated (Ci
stub = 0, Listub = 0),
the scattering matrix is that of the so-called symmetrical supercondensed node (SSCN) [46], [47].
In frequency-domain TLM, synchronism of all pulses (such as given by equ. (2.22) for time-domain TLM) is not necessary. However, waves travelling in the same direction but having orthogonal polarizations must maintain the same velocity (3 constraints)
ikij vv = (2.23)
Furthermore, stubs can be avoided completely (6 constraints). Thus, there are 3 degrees of freedom left allowing for different definitions of link-line impedances and propagation constants.
One possibility is to set all link-line impedances equal to the impedance of the medium [21], [23]. For an isotropic, loss-free medium (εrx = εry = εrz = εr, µrx = µry = µrz = µr), one obtains
r
rmij ZZ
εεµµ
0
0== (2.24)
ikij CC = (2.25)
ikij LL = (2.26)
By substituting equations (2.24)-(2.26) into (2.16)-(2.21), it is readily found that
ijijikiji CL2ωγγγ −=== (2.27)
and
( )
∆
∆∆−∆∆+
∆∆= 22 x
zyyz
zykj m
xγ (2.28)
18
( )
∆
∆∆−∆∆+
∆∆= 22 y
zxxz
zxkj m
yγ (2.29)
( )
∆
∆∆−∆∆+
∆∆= 22 z
yxxy
yxkj m
zγ (2.30)
with km being the propagation constant of a plane wave in the medium,
rrmk µµεεω 00= (2.31)
Historically, this node definition has been the first proposed for the frequency-domain TLM method.
Another possibility is to set the propagation constants of all link-lines equal to half the propagation constant of a plane wave in the medium
rkrjm
ijjkj µµεεωγγ 0022
=== (2.32)
By substituting equations (2.10), (2.11), (2.32) into (2.16)-(2.21) and performing some manipulations, quadratic equations are obtained for the link-line impedances, having two identical solutions
rj
rk
ijij k
jY
Zεεµµ
0
01∆∆== (2.33)
This node is known in the literature as ”distributed node” or ”propagation constant node” [3], [24]. It provides much better numerical accuracy than the first node mentioned above.
It should be noted that for isotropic material and cubic node cell geometry, both nodes presented are identical.
The connection of the twelve link lines in the node center is described by a scattering matrix. This scattering matrix can be derived from Maxwell’s equations by finite differencing and averaging as will be shown now. Alternatively, a derivation using conservation of charge and magnetic flux, in combination with the continuity of the electric and magnetic fields is possible and gives identical results [45].
The scattering matrix [S] of the SCN relates incident and reflected voltages on the boundaries of the node cell, incbV and refbV , respectively, as
( ) [ ] ( )incbbrefb VSV ⋅= (2.34)
19
The procedure is as follows. First, field components tangential to a node boundary will be mapped onto incident and reflected voltages on the respective node boundary. Then, these voltages will be transformed to the node center. The same transformation is applied to the field components, assuming that they form a plane wave traveling in the direction of the respective transmission line. The center-transformed field components of opposite sides of the node are averaged in order to find the field values at the actual node center. Maxwell’s equations in a finite-difference form are stipulated at the node center. After some manipulation, the scattering matrix at the node center is obtained. It relates incident and reflected voltages on the link-lines at the center of the node. Finally, by shifting these voltages back to the node boundaries, the scattering matrix [ ]Sb according to equ. (2.34) will be found.
Note that each field component is described at four out of six node boundaries. Therefore, the averaging process in the node center is not unambiguous. There are two choices to take the average of field values on opposite sides. As a result of the ambiguity, spurious solutions arise, which will be discussed in chapters 4.1 and 4.2.
Consider first the following relations between incident and reflected voltages and currents on a transmission line and the total voltage and current
refij
incijij VVV ±±± += (2.35)
( )refij
incij
ijij VV
ZI ±±± −= 1 (2.36)
Applying these two equations and referring to Fig. 2.4, the mapping between tangential field components at the node cell boundaries and the incident and reflected voltages on the link-lines at the node boundaries is found as
( )refyx
bincyx
bx VV
xzyyxE ±± +
∆=∆± 1),
2,( 000 (2.37)
( )refzx
binczx
bx VV
xzzyxE ±± +
∆=∆± 1)
2,,( 000 (2.38)
( )refxy
bincxy
by VV
yzyxxE ±± +
∆=∆± 1),,
2( 000 (2.39)
20
( )refzy
binczy
by VV
yzzyxE ±± +
∆=∆± 1)
2,,( 000 (2.40)
( )refxz
bincxz
bz VV
zzyxxE ±± +
∆=∆± 1),,
2( 000 (2.41)
( )refyz
bincyz
bz VV
zzyyxE ±± +
∆=∆± 1),
2,( 000 (2.42)
( )incyz
brefyz
b
yzx VV
xZzyyxH ±± −
∆±=∆± 1),
2,( 000 (2.43)
( )refzy
binczy
b
zyx VV
xZzzyxH ±± −
∆±=∆± 1)
2,,( 000 (2.44)
( )refxz
bincxz
b
xzy VV
yZzyxxH ±± −
∆±=∆± 1),,
2( 000 (2.45)
( )inczx
brefzx
b
zxy VV
yZzzyxH ±± −
∆±=∆± 1)
2,,( 000 (2.46)
( )incxy
brefxy
b
xyz VV
zZzyxxH ±± −
∆±=∆± 1),,
2( 000 (2.47)
( )refyx
bincyx
b
yxz VV
zZzyyxH ±± −
∆±=∆± 1),
2,( 000 (2.48)
Ey
I+xy ∼∼∼∼ –Hz
y x0 + ∆x/2x0 – ∆x/2
EyHzHz
I–xy ∼∼∼∼ Hz
V+xy∼∼∼∼ Ey
V–xy∼∼∼∼ Ey xz
Fig. 2.4: Boundary oriented field mapping
The incident and reflected voltages at the node center, inccV and refcV , respectively, are related to those at the node boundary, incbV and refbV , by
21
the following transformation, which represents a simple phase shift in loss-free medium
∆−⋅= ±± 2exp
iVV ijref
ijcref
ijb γ
(2.49)
∆−⋅= ±± 2exp
iVV ijinc
ijbinc
ijc γ
(2.50)
By applying these transformations, the field values at the node boundaries are also transformed towards the node center in the sense that they form a plane wave traveling in the direction of the respective transmission line. A field value directly at the node center is found by averaging two field values on opposite sides of the node center. In terms of incident and reflected voltages, it is found, for example, using equs. (2.49), (2.50) together with equ. (2.39) and equ. (2.47), respectively,
( )incxy
crefxy
cincxy
crefxy
cy VVVV
yzyxE −−++ +++
∆=
21),,( 000 (2.51)
( )incxy
crefxy
cincxy
crefxy
c
xyz VVVV
zZzyxH −−++ +−−
∆=
21),,( 000 (2.52)
Adding and subtracting these two equations yields incxy
czxyy
refxy
c VzyxHzZzyxEyV ∓−∆±∆=± ),,(),,( 000000 (2.53)
Applying the same procedure (that is, averaging, addition, subtraction), similar expressions are readily obtained for the other field components
incxz
cyxzz
refxz
c VzyxHyZzyxEzV ∓∓ −∆∆=± ),,(),,( 000000 (2.54)incyz
cxyzz
refyz
c VzyxHxZzyxEzV ∓−∆±∆=± ),,(),,( 000000 (2.55)incyx
czyxx
refyx
c VzyxHzZzyxExV ∓∓ −∆∆=± ),,(),,( 000000 (2.56)inczx
cyzxx
refzx
c VzyxHyZzyxExV ∓−∆±∆=± ),,(),,( 000000 (2.57)inczy
cxzyy
refzy
c VzyxHxZzyxEyV ∓∓ −∆∆=± ),,(),,( 000000 (2.58)
The averaged fields at the node center and those transformed towards the node center must fulfill Maxwell’s equations at the frequency fπω 2=
22
( )000 ,, zyxEjzH
yH
xxy
cz
c
εω=∂
∂−
∂∂ (2.59)
( )000 ,, zyxEjxH
zH
yyz
cx
c
εω=∂
∂−∂
∂ (2.60)
( )000 ,, zyxEjyH
xH
zzx
cy
c
εω=∂
∂−∂
∂ (2.61)
( )000 ,, zyxHjyE
zE
xxz
cy
c
µω=∂
∂−∂
∂ (2.62)
( )000 ,, zyxHjzE
xE
yyx
cz
c
µω=∂
∂−∂
∂ (2.63)
( )000 ,, zyxHjxE
yE
zzy
cx
c
µω=∂
∂−
∂∂ (2.64)
where the imaginary parts of ε and µ describe the losses. The differentials of the field components are replaced by differences. For example, Maxwell’s equation (2.59) reads, using equs. (2.46), (2.48) and accounting for the transformations (2.49), (2.50)
zx
refzx
cinczx
crefzx
cinczx
c
yx
refyx
cincyx
crefyx
cincyx
c
xx ZzyVVVV
ZzyVVVV
Ej∂∆
−+−+∆∂
−+−= −−++−−++εω (2.65)
On the right-hand side of equ. (2.65), the terms refyx
cV± and refzx
cV± can be substituted using equs. (2.56) and (2.57), respectively. One obtains
zx
inczx
cinczx
c
yx
incyx
cincyx
c
zxyx
xx Zzy
VVZzyVV
ZzyZzyxjEx
∂∆++
∆∂+
=
∂∆+
∆∂+
∆∆ −+−+11
2ωε (2.66)
It is assumed that the node cell shape, defined by zyx ∆∆∆ :: , has been kept constant while shifting the fields from the node boundary towards the node center, that is
ζ=∆∂=
∆∂=
∆∂
zz
yy
xx (2.67)
Under the assumption 0→ζ , equation (2.66) reads
23
( ) ( )zxyx
inczx
cinczx
czx
incyx
cincyx
cyx
x YYVVYVVY
zyxEx+
+++=∆ −+−+),,( 000 (2.68)
where the impedances are conveniently replaced by admittances. All other field components can be treated in a similar way, giving
( ) ( )kiji
incki
cincki
cki
incji
cincji
cji
i YYVVYVVY
zyxEi+
+++=∆ −+−+),,( 000 (2.69)
kjjk
incjk
cincjk
cinckj
cinckj
c
i ZZVVVV
zyxHi+
+−−=∆ −+−+),,( 000 (2.70)
If (2.69), (2.70) are substituted back into equs. (2.53)-(2.58), the scattering matrix of the node center is obtained
( ) [ ] ( )incccrefc VSV ⋅= (2.71)
By replacing the voltages at the node center with those at the node boundaries according to equs. (2.49), (2.50), one obtains
( ) [ ] ( ) [ ] [ ] [ ] ( )incbcincbbrefb VEXPSEXPVSV ⋅⋅⋅=⋅= (2.72)
where the voltage column vectors are given by ( )
( )Trefzy
brefzy
brefzx
brefzx
brefyx
brefyx
brefyz
brefyz
brefxz
brefxz
brefxy
brefxy
b
refb
VVVVVVVVVVVV
V
+−+−+−+−+−+−
=
,,,,,,,,,,,
( )( )Tinc
zybinc
zybinc
zxbinc
zxbinc
yxbinc
yxbinc
yzbinc
yzbinc
xzbinc
xzbinc
xybinc
xyb
incb
VVVVVVVVVVVV
V
+−+−+−+−+−+−
=
,,,,,,,,,,,
and the shift matrix writes
24
e 2–γxy∆x
e 2–γxy∆x
e 2–γxz∆x
e 2–γxz∆x
e 2–γyz∆y
e 2–γyz∆y
e 2–γyx∆y
e 2–γyx∆y
e 2–γzx∆z
e 2–γzx∆z
e 2–γzy∆z
e 2–γzy∆z
0 0 00 0 0
00 0 0 0 0
0
0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0
0 0 0
0 0 0
0
00 0 0 0 0
0 0 0 0 0
0 000 0 0
00000
0 00
00 0 0 0
0
000
0 0
[ EXP ] =0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
00
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
and the scattering matrix at the node center is given as
25
0 0 0 0
0 0 0 0
0
0 0 0 0
0 0 0
0
0 0
0
0
0 0 0 0
0 0 0 0
00 0
00
0 0
00
0 0
[ cS ] =0
0
0
0
0
0
0
0
0
0
0
0
00
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
axy
axy
axz
axz
ayz
ayx
ayz
ayx
azx
azx
azy
azy
cxz
cxz
cxy
cxy
cyz
cyz
cyx
cyx
czx
czx
czy
czy
bxy0
bxy bxy
bxy
bxz
bxz bxz
bxz
byz
byz byz
byz
byx
byx byx
byx
bzx
bzx bzx
bzx
bzy
bzy bzy
bzy
dxy
–dxy dxy
–dxy
dxz
–dxz dxz
–dxz
dyz
–dyz dyz
–dyz
dzy
–dzy dzy
–dzy
dzx
–dzx dzx
–dzx
dyx
–dyx dyx
–dyx
with their elements given by
ij
ji
kj
ijij
ZZ
ZZa
+−
+=
1
1
1
1
ij
kjij
ZZb
+=
1
1
11
1
1
1 −+
++
=
ij
ji
kj
ijij
ZZ
ZZc
ij
jiij
ZZ
d+
=1
1
In case of a cubic, isotropic node cell, the above parameters simplify to aij = cij = 0 and bij = dij = ½. This is also the case if all link-line impedances are set equal.
In the lossless case (permittivities and permeabilities are all real), the above scattering matrix becomes unitary, that is [S] –1 = [S] T.
Note that in the derivation presented here, only averaging of the field values together with finite differencing of Maxwell’s equations was
26
applied. Identical formulas are obtained by decomposing the node in series and shunt circuits and using equivalent voltages and currents. The latter approach, however, suffers from a lack of foundation due to the fact that an equivalent network model of the SCN does not exists.
It should be noted that the above equations (2.53)-(2.58), (2.69), (2.70) are quite similar to those obtained for the (stubless) time-domain SSCN [47], if the so-called ”equivalent voltages Vi” and ”equivalent currents Ii”, respectively, are defined as Vi = ∆i Ei and Ii = –∆i Hi.
It is possible to derive a FDTLM node without shifting the voltages and field values towards the node center and back. Then, the finite differencing will take place accross the entire node dimensions. Because of 1=ζ in (2.67), equations (2.69), (2.70) read
( ) ( )
ikjjYY
VVYVVYzyxEi
ikiji
incki
bincki
bki
incji
bincji
bji
i
∆∆∆++
+++=∆ −+−+
2
),,( 000 εω (2.73)
ikjjZZ
VVVVzyxHi
ikjjk
incjk
bincjk
binckj
binckj
b
i
∆∆∆++
+−−=∆ −+−+
2
),,( 000 µω (2.74)
Since there are no link-lines, their impedances cancel each other out
ikjZjVVVV
zyxEii
incki
bincki
bincji
bincji
b
i
∆∆∆+
+++=∆ −+−+
22
),,(0
000 εω (2.75)
( )
ikjYj
VVVVYzyxHi
i
incjk
bincjk
binckj
binckj
b
i
∆∆∆+
+−−=∆ −+−+
22
),,(0
0000 µω (2.76)
The node scattering matrix [ ]Sb can be found by substituting (2.75) and (2.76) back into (2.53)-(2.58). This node scattering matrix was derived first in [22], but the derivation was not related to other, previously known discretization schemes shown above.
In the coming chapters of this thesis, all link-lines voltages are those at the boundaries of the node cell. Similarly, the node scattering matrix refers always to the one at the node boundary. For simplicity, the superscript b (for ‘boundary’) will be omitted.
27
The node developed last (equs. (2.75), (2.76)) differs from the two other nodes shown before in that it does not uses link-lines. As a result, the fields are linearly approximated across a mesh cell (much like in a finite-difference frequency-domain method) and therefore, all nodes have to be small compared to the wavelength. The time-domain equivalent of this node is the stub-loaded standard node having all link-line impedances set to the intrinsic impedance of the background medium [47]. The propagation characteristics of this time-domain node (numerical dispersion) are not very good. Similar results are to be expected for the frequency-domain case.
A comparison of the respective numerical accuracy of the three aforementioned nodes will be given in the following chapter.
28
2.3 Dispersion Relations of the FDTLM Nodes
A plane wave propagating through a homogeneous medium obeys the dispersion relation k = ω / v (wavenumber k, frequency ω). Here, the phase velocity v depends on the material properties ε, µ, and σ. In the finite-difference approximation of Maxwell’s equations, the numerical wavenumber knum will usually differ from the physical wavenumber k, a phenomenon described as numerical dispersion. Errors introduced due to numerical dispersion may depend on grid spacing (both size and aspect ratio of a node cell), on mesh grading (variation of mesh shape over space), on the direction of wave propagation through the grid, and finally on the accuracy of the discretization scheme used to approximate Maxwell’s equations locally. Even if numerical dispersion errors are small, they may sum up in electrically large structures and/or become unacceptable in the modeling of high-Q (narrowband) problems.
A common method to quantify dispersion errors is to model the propagation of a plane wave in homogeneous medium with a regular (that is, periodic) mesh. Then, the investigation can be performed using one mesh cell only. Note that evanescent waves are not considered. Furthermore, accuracy degradation effects introduced by neighboring nodes of different shape (mesh grading effects) cannot be evaluated.
The homogeneous medium is considered to be isotropic and characterized by its (complex) permittivity and permeability. The plane wave is characterized by the wavenumber and the direction of propagation. All field components are known at the boundaries of the node cell. By using the mapping between fields and voltages on the link lines (equs. (2.37)-(2.48)), a matrix [T] can be found such that
( ) [ ] ( )refinc VTV ⋅= (2.77)
The mesh cell is described by the node scattering matrix
( ) [ ] ( )incref VSV ⋅= (2.78)
Both matrices [T] and [S] depend on the wavenumber (or frequency). Given a wavenumber k defining matrix [T], equs. (2.77) and (2.78) will hold both only if the wavenumber knum defining matrix [S] is somewhat different from k, that is, knum ≠ k. The wavenumber knum is found from
[ ] [ ] [ ]{ } 0)()(det =−⋅ EkTkS num (2.79)
with [E] the unit matrix. The error can be computed as
29
%100⋅−=kkkerror num (2.80)
This error will depend on the direction of the plane wave in space as well as on the size of the node (compared to the wavelength), on the shape of the node cell, and on the material properties. The subject of the following investigation is to find out how the numerical dispersion error depends on these parameters. It is to be expected that some of the three discretization schemes (presented in chapter 2.2) perform better than others.
First, the matrix [T] representing the plane wave will be derived. The coordinates are shown in Fig. 2.5. A plane wave is given by
( )rkjEE −⋅= exp0 (2.81)
( )EkH ×⋅=µε (2.82)
⋅=
=
θϕθϕθ
cossinsincossin
kkkk
k
z
y
x
(2.83)
There are six points of interest such as (see Fig. 2.5)
∆−=
00
2/
1
xr (2.84)
Then, the following formulas are found for the field components (in fact, this is simply the projection of the wave vector onto the coordinate axes)
( ) ( ) ( )ϕθ cossinexp12 xkjrErE ∆−⋅= (2.85)
( ) ( ) ( )ϕθ cossinexp12 xkjrHrH ∆−⋅= (2.86)
( ) ( ) ( )ϕθ sinsinexp34 ykjrErE ∆−⋅= (2.87)
( ) ( ) ( )ϕθ sinsinexp34 ykjrHrH ∆−⋅= (2.88)
( ) ( ) ( )θcosexp56 zkjrErE ∆−⋅= (2.89)
( ) ( ) ( )θcosexp56 zkjrHrH ∆−⋅= (2.90)
By applying the mapping between fields and link-line voltages on the boundaries of the node cell, the matrix [T] is found as
30
z
r2 =+ ∆∆∆∆x/2
y
r1 =– ∆∆∆∆x/2
r3 =– ∆∆∆∆y/2
r4 =+ ∆∆∆∆y/2
r5 =– ∆∆∆∆z/2
r6 =+ ∆∆∆∆z/2
x
x
y
z
ϕϕϕϕ
θθθθ
Fig. 2.5: Coordinate system for plane wave propagation through a periodic mesh of FDTLM node cells
e+jkx∆x
0 0 00 0 0
0 0 0 0 0
0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0 0 0
0 0 0
0 0 0
0
0 0 0 0 0
0 0 0 0 0
000 0 0
0000
00
00 0 0
00
0 0
[ T ] =0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
00
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0
0
0
0
e–jkx∆x
e–jkx∆x
e+jkx∆x
e–jky∆y
e+jky∆y
e–jky∆y
e+jky∆y
e–jkz∆z
e+jkz∆z
e–jkz∆z
e+jkz∆z
31
By solving (2.79), the error due to numerical dispersion can be evaluated. The different discretization schemes will be abbreviated as CAN (TLM node with fixed link-line impedances according to (2.24)-(2.31) [21], [23]), PCN (TLM node with fixed link-line propagation constants; see (2.32)-(2.33) [3], [24]), and FDN (finite-difference node according to (2.75)-(2.76) [22]).
Cubic node cells of various size (loss-free material) For cubic node cells, CAN and PCN are identical. The numerical
wavenumber error for different directions of the plane wave is plotted in Fig. 2.6 for a node cell size of λ/20 (5mm at 3 GHz in free space). The TLM nodes CAN and PCN show zero error in axis direction and the maximum error in the direction of the space diagonal (ϕ = 45°, θ = 54.7°). The finite difference node FDN shows minimum error in space diagonal direction and maximum error in axis direction. This behavior is common for FDTD schemes. The magnitude of the error of the FDN is much larger than the error of the CAN and PCN.
The numerical accuracy becomes worse as the node becomes larger compared to the wavelength. The analytical solution for the worst case error of cubic CAN and PCN (that is, in direction of the space diagonal) is found as
∆+=∆ xkxknum 3
2cos43
41arccos (2.91)
For small node cells, this simplifies to
( )
∆−∆≈∆ 2
7211 xkxkxknum (2.92)
Note that the dispersion error for propagation in axis direction is always zero for cubic CAN and PCN. The analytical solutions for the dispersion error of cubic FDN is found as
∆−∆=∆qxkqxkxknum 2
tan2 (2.93)
for plane wave propagation in axis direction (q = 1), plane diagonal direction (q = 2), and space diagonal direction (q = 3), respectively. This simplifies for small node cells to
32
Fig. 2.6: Numerical dispersion error for CAN / PCN (top) and FDN (bottom) for cubic node geometry (size λ / 20) and loss-free material. For CAN / PCN (top), the error range is 0...0.14%. For FDN (bottom), the error range is 0.28%...0.83%
φφφφ
φφφφ
ΘΘ ΘΘ ΘΘ ΘΘ
33
( )
∆−∆≈∆ 2
1211 xkq
xkxknum (2.94)
Note that the worst case error for CAN and PCN is only about half the best error of FDN and about one sixth of the worst case error of FDN. Fig. 2.7 shows this behavior as a function of the node cell size.
Fig. 2.7: Numerical dispersion error for CAN / PCN and FDN for cubic node geometry of variable size up to λ / 2 (f = 3 GHz, air, λ0 = 100mm, solid lines: exact analytical formulas, dashed lines: approximate formulas for small node cells)
Non-cubic node cells (loss-free material) The accuracy of the discretization schemes decreases as the node cell
shape deviates from the cubic form. In the following, for ease of comparison, the volume of the node cell is kept constant. Note that the nodes of type CAN and PCN produce now different results. Fig. 2.8 shows the dispersion error for a weakly non-cubic node of aspect ratio ∆x : ∆y : ∆z = 1:1:2 and size ∆z = λ / 12.6. It can be seen that the PCN produces lower error levels than CAN and FDN do. Furthermore, CAN is dispersion-free in axis direction as long as the node cell cross-section perpendicular to this axis is square. As the node cell shape deviates even more from the cubic case, the error distribution shows stronger dependence on the direction of
34
the incident plane wave, and as a general rule, the worst case errors grow strongly. However, PCN continues to produce much smaller errors than CAN and FDN. To give an example, Fig. 2.9 shows the dispersion error for a node of aspect ratio ∆x : ∆y : ∆z = 1:2:4 having the same volume as in the previous cases (∆z = λ / 10).
Fig. 2.8: Numerical dispersion error for CAN, PCN (next page, top), and FDN (next page, bottom) for weakly non-cubic node (side ratio 1:1:2) and loss-free material. The error ranges are 0...0.79% for CAN, 0...0.19% for PCN, and 0.23%...2.13% for FDN
φφφφ
ΘΘ ΘΘ
35
Fig. 2.8, continued: Numerical dispersion error for PCN (top) and FDN (bottom)
φφφφ
ΘΘ ΘΘ ΘΘ ΘΘ
φφφφ
36
Fig. 2.9: Numerical dispersion error for CAN (top), PCN (bottom), and FDN (next page, top) for strongly non-cubic node (side ratio 1:2:4) and loss-free material. The error ranges are 0.23%...5.47% for CAN, 0...0.35% for PCN, and 0.16%...3.43% for FDN
φφφφ
φφφφ
ΘΘ ΘΘ ΘΘ ΘΘ
37
Fig. 2.9, continued: Numerical dispersion error for FDN
Lossy materials A plane wave propagating through a homogeneous, lossy medium
(permittivity εr, conductivity σ) will be affected both in phase and magnitude by errors introduced by the numerical scheme. These errors will superimpose the aforementioned errors due to node cell shape etc.
Consider the following specific case: a cubic node cell of size 5mm, filled with a material having εr = 2.2, at f = 3 GHz. As before, FDN shows the smallest error in space diagonal direction and the worst case error in axis direction. PCN and CAN are identical, having no error in axis direction and the worst case error in space diagonal direction. Fig. 2.10 shows the error ranges as a function of the conductivity of the material. As it was the case for the loss-free material, the magnitude error of FDN is about 2...6 times larger than the worst case error of PCN and CAN. Furthermore, the magnitude error is very small for common dielectric materials, but it grows approximately with a power law and becomes significant for larger conductivity (σ > ≈ 0.1 S/m is common for, e.g., many biological materials). The phase error is almost unaffected at small conductivity. However, it shows an interesting zero at higher conductivity. At this point, the plane wave propagates without phase error (but with non-zero
φφφφ
ΘΘ ΘΘ
38
magnitude error) through the mesh in any direction. The dispersion zero occurs around a conductivity given by
rεεωσ 0≈ (2.95)
This equation holds exactly in the case where 0→∆x .
Fig. 2.10: Numerical dispersion error for CAN / PCN and FDN for a cubic node of ∆x = 5mm, εr = 2.2, f = 3GHz. Solid lines: phase error. Dashed lines: magnitude error over a distance of λ = 67.4mm. Note that the minimum error for PCN / CAN is zero (in axis direction)
σσσσ
39
2.4 Incorporation of Boundary Conditions
A material boundary separating two regions will be placed at the boundary of an FDTLM node cell. If both regions are part of the computational domain and discretized by SCN, the boundary is called an internal boundary. If one material is not part of the computational domain, it is called an external boundary.
In a TLM mesh representing a homogeneous region, the reflected voltage on a given link-line of a given node equals the incident voltage on the corresponding link-line of the adjacent node and vice versa. Mathematically, the connection between link lines of all nodes of the computational domain can be described with the connection matrix [C]. This matrix is very sparse. The connection between link-lines of nodes of a homogeneous region is realized by setting the appropriate elements of the connection matrix to one (note that the so-called Hybrid Nodes [9], [20] can produce a reflection at the boundary between two nodes even in a homogeneous region). Internal and external boundaries are represented by reflection and transmission coefficients. These coefficients are elements of the connection matrix.
Internal boundary For the nodes of type PCN and CAN, the reflection and transmission
behavior of a plane wave with normal incidence on the material boundary is accurately described by the reflection and transmission coefficients of the transmission line junction formed by the respective link-lines.
Between node A of material (εA, µA) and node B of material (εB, µB), the reflection coefficients ΓA, ΓB and the transmission coefficients TA→B, TB→A are given by
A
A
B
B
A
A
B
B
A
εµ
εµ
εµ
εµ
+
−=Γ (2.96)
AB Γ−=Γ (2.97)
ABAT Γ+=→ 1 (2.98)
BABT Γ+=→ 1 (2.99)
40
Note that for homogeneous material, Γ = 0 and T = 1. External boundary Physically, the external boundary is represented by the reflection
coefficient ΓPlaneWave seen by a plane wave with normal incidence onto this boundary. This coefficient is determined by the material parameters. Numerically, the link-line touching the external boundary is terminated with an appropriate reflection coefficient ΓLinkLine. The mapping between link-line voltages and currents at one hand and the field components at the other hand must be considered in calculating ΓLinkLine. In general,
PlaneWaveLinkLine Γ≠Γ (2.100)
Consider an SCN representing a material of impedance Zm and touching an external medium of impedance Zext. Consider the +x-directed, y-polarized link line touching this external medium. Voltages and currents on this link line are related to Ey and Hz by
( )incxy
refxyy VV
yE ++ +
∆= 1 (2.101)
( )incxy
refxy
xyz VV
ZzH ++ −
∆= 11 (2.102)
At the boundary
zexty HZE = (2.103)ref
LinkLineinc VV Γ= (2.104)
and thus, for the general case ({i,j,k} ∈ {x,y,z})
ijext
ijext
ijLinkLine
ZjkZ
ZjkZ
∆∆+
∆∆−
=Γ , (2.105)
It follows for PCN (see equation (2.33))
mext
mextijLinkLine ZZ
ZZ+−=Γ , (2.106)
and for CAN (see equation (2.24))
41
mext
mext
ijLinkLine
ZjkZ
ZjkZ
∆∆+
∆∆−
=Γ , (2.107)
If the external medium is perfectly conducting, ΓLinkLine = –1 for every type of node. On a perfect magnetic wall, ΓLinkLine = +1 for every node. If an absorbing boundary is modeled as matched medium (also known as ZRT boundary which stands for Zero Reflection Termination),
0,, =Γ ijZRTLinkLine (2.108)
for PCN, but
kjkj
ijZRTLinkLine ∆+∆∆−∆=Γ ,, (2.109)
for CAN. If an imperfectly conducting boundary is modeled,
mS
mSijZSLinkLine ZZ
ZZ+−=Γ ,, (2.110)
for PCN, but
mS
mS
ijZSLinkLine
ZjkZ
ZjkZ
∆∆+
∆∆−
=Γ ,, (2.111)
for CAN. Here, the metal surface impedance ZS is given by [12]
σµµπ
δσrel
SfjjZ 0)1(1 +=+= (2.112)
Two examples will illustrate the accuracy of the ZRT absorbing boundary and the lossy conducting boundary modeled by a complex reflection coefficient, respectively.
Consider the first example, an air-filled rectangular waveguide of cross-section a × b with a > 2b, discretized with cubic nodes. In some transverse plane, the link-lines shall be terminated by a ZRT absorbing boundary, whereas another transverse plane is the port plane. Using the algorithm for the scattering parameters of a 3D structure (see chapter 3.4), the input reflection coefficient 11S=Γ can be calculated. For a perfectly absorbing
42
boundary, the result should be 0=Γ . In practice, however, Γ is found to vary with frequency as shown in Fig. 2.11 (left axis). The fundamental TE10 mode of a rectangular waveguide is known to be composed of two plane waves which travel in zig-zag. Thus, the problem under consideration here is equivalent to a plane wave incident onto the ZRT boundary under an angle Θ , and parallel-polarized in the direction of a coordinate axis. Hereby, Θ is the angle between the direction of propagation of the plane wave and the boundary normal. It follows from the characteristics of the TE10 mode that gλλ /cos 0=Θ , which is the ratio of free-space wavelength and guided wavelength (see Fig. 2.11, right axis).
The reflection coefficient of the ZRT boundary is found to depend on the angle of incidence of the plane wave (parallel-polarized to a coordinate axis) as
)()(
900
10
cos1cos1
cutoffatwaveguidefatincidencenormal
forfor ∞→
=Θ=Θ
=Θ+Θ−=Γ (2.113)
Fig. 2.11: Input reflection coefficient of a rectangular waveguide terminated with ZRT absorbing boundary (left) and the incidence angle of the equivalent plane wave (right) as a function of frequency
In the second example, consider an air-filled rectangular waveguide of cross-section a × b with a = 8∆x = 40mm, b = ∆y = 5mm. The conductivity σ of the metal being 15.62 × 106 S/m (brass). The propagation constant γ = α – jβ can be found from an SCN network in form of an one-node thick
43
slice (this algorithm will be described in chapter 3.3). Depending on the slice thickness ∆z, the numerical accuracy will vary. Reference values of the propagation constant are found using the power-loss method [12].
The error of β and α is averaged over both the frequency range 1.1 fc < f < 1.9 fc (cutoff frequency fc = 3.75 GHz) and the node shape range 0.5 < ∆z/∆x < 2. The average error in the phase constant β is found to be 0.80% for PCN and 1.28% for CAN, and the average error in the attenuation constant α is found to be 1.88% for PCN and 2.44% for CAN. Note that because of the rough mesh, the error magnitudes are rather large. In case of CAN, if the reflection coefficient was mistakenly set according to equ. (2.110), then the average attenuation constant error was 11.5%.
44
2.5 Conclusions
In this chapter, the basic elements of the discretization process in the frame of the FDTLM method were developed and evaluated, namely the symmetrical condensed node, SCN, and the implementation of material boundaries.
First, different discretization schemes (CAN, PCN, FDN) are derived from Maxwell’s equations in a concise, unified manner. Previously, only FDN has been derived from Maxwell’s equations directly, whereas the other two nodes were derived from current and voltage analogies only. In the process of averaging and finite differencing, differences between the node schemes clearly arise. Ambiguities in the averaging process are detected, which will give rise to spurious solutions of the FDTLM method.
Second, the dispersion analysis of the three node schemes reveals significant differences in the level of numerical dispersion of the nodes. The dispersion errors in axis as well as diagonal directions are derived analytically. The dispersion error of the FDN is between two and six times larger than the worst-case dispersion error of cubic CAN and PCN. Investigations of non-cubic and lossy cases clearly show the superiority of PCN against the CAN and FDN schemes. The investigation of lossy cases shows that the phase error depends mainly on mesh size and is independent on the conductivity for small losses, whereas the magnitude error depends with a power law on the conductivity of the material.
Third, the incorporation of boundary conditions is investigated. It is shown that external boundaries can be treated as reflection coefficients on the attached link-lines. Formulas are given which take non-cubic nodes into account, both for CAN and PCN. The influence of the node shape onto the numerical reflection coefficient has not been considered in several previous publications. Furthermore, it is shown that the use of the ZRT as an absorbing boundary is of limited accuracy if the incidence is far from being normal to the boundary. An analytical formula is given for the numerical reflection of a ZRT.
45
3 Algorithms for the Application of the FDTLM Method
Microwave circuits are usually described by scattering parameters, represented by the S-matrix. During the numerical analysis of the microwave circuit, the electromagnetic fields are calculated first, and the scattering parameters are extracted subsequently. As a matter of fact, the field solution contains much more information about the physics of the problem than the S-matrix does. Therefore, locally highly accurate fields need a very high numerical effort, whereas highly accurate scattering parameters can be extracted even from less accurate field solutions as obtained from a rather rough mesh. In the following, the focus is set on obtaining accurate scattering parameters rather than field solutions.
Practically, one proceeds as follows. The computational domain is limited by boundaries. Semi-infinite waveguides are connected to the computational domain, thus forming ports. The scattering parameters refer to the eigenmodes of the waveguides at the port planes. Therefore, the analysis of a microwave circuit can be divided into three steps: • calculation of the eigenmodes of the waveguides at the ports, • calculation of the field distribution inside the computational
domain, • expansion of the field distributions at the port planes in terms of
waveguide eigenmodes and derivation of scattering parameters. An FDTLM node is described by a scattering matrix of size 12×12. By
combining the scattering matrices of all nodes of a computational domain, a given structure can be analyzed in form of a matrix algebra problem. This eases notation, programming, as well as the actual computation of the structure.
In the following, matrix algorithms are presented for the treatment of problems appearing during the analysis of microwave circuit structures, such as the resonance frequency of a cavity resonator, the eigenmodes of a waveguide, and the scattering parameters of a 3D structure.
Previously published algorithms are critically reviewed. A focus is set on the computational efficiency of the algorithms.
46
3.1 Cavity Resonance Frequency
The scattering matrices of all nodes of the computational domain can be put together in a block-diagonal matrix [ ]S . Each row and column of [ ]S contains between 4 and 6 non-zero complex entries. If ( )incV is the vector of incident waves on the link-lines and ( )refV denotes the vector of reflected voltage waves on the link-lines, the following general relation holds
( ) [ ] ( )incref VSV ⋅= (3.1)
As it has been described in chapter 2.4, the connection between the link-lines of all nodes of the computational domain as well as the connection to boundaries can be described by the connection matrix [ ]C . The matrix [ ]C is very sparse. It contains the reflection coefficients to boundaries in the main diagonal, and reflection and transmission coefficients describing the connection between the link-lines of adjacent nodes. The connection equation reads
( ) [ ] ( )refinc VCV ⋅= (3.2)
Note that the inverse of [ ]C is easily found because only inversions of submatrices of size 2×2 have to be performed. Furthermore, [ ] [ ]CC =−1 for a homogeneous computational domain (without dielectric interfaces, i.e. all transmission coefficients are equal one).
For a resonator at resonance, equs. (3.1) and (3.2) are combined to give
[ ] [ ] [ ]{ } [ ] [ ]{ } 0DetDet 1 =−=⋅− −CSCSE (3.3)
where [ ]E denotes the unit matrix. The search of a singularity in the equ. (3.3) while varying the frequency is rather involved if [ ]S is large.
Several methods for detecting a singular matrix are described in chapter 3.5. For large problems, the most efficient method is based on an eigenvalue search. This method will be described now. Consider first a resonator problem with homogeneous dielectric material, regularly meshed with cubic nodes. In this particular case, all entries of the matrix [ ]S have the same frequency dependence, namely, ( )2/exp xkj m ∆− (see equs. (2.28)-(2.30), (2.32)). Then, equs. (3.1) and (3.2) can be written as
[ ][ ] ( ) ( )refrefcenter VVCS ⋅=⋅ λ (3.4)
where the resonance frequency is found from the eigenvalue λ as
47
∆=
εµπλ
xjfres
lnRe (3.5)
The resonance frequency is purely real (within numerical accuracy of the iterative eigenvalue search) for loss-free resonators, otherwise, it is complex.
Instead of searching the complete eigenvalue solution of equ. (3.4), it is much more convenient (for large problems) to find the smallest eigenvalue only. The problem can be re-formulated as follows. Supposing
fffres ∆+= 0 , the eigenvalue problem can be written as
[ ] ( )[ ] [ ]( ) ( ) ( )refref VVCfSE min0 λ=⋅⋅− (3.6)
and solved for the smallest eigenvalue λmin to find the resonance frequency as
( )
∆−+=
εµπλ
xjffres
min0
1lnRe (3.7)
The better the guess frequency 0f , the smaller the magnitude of λmin.
Since λmin is small, it can be approximated as (considering the Taylor expansions of sine and cosine for small arguments)
( ) ( ) εµπεµπλ εµπ fxjfxe fxj ∆∆−∆∆≈−= ∆∆ 222
min 21 (3.8)
Thus, in the neighborhood of the resonance frequency, ( )minIm λ is a linear function of frequency, and ( )minRe λ is a quadratic function. The trace of λmin in the complex plane for frequencies close to a resonance is a parabola and is identical for all resonances of a given structure.
For a practical problem, however, a graded mesh will be used, and the resonator structure may contain dielectric interfaces. Then, the frequency dependence of the matrix entries cannot be extracted anymore as it was the case in equ. (3.4). It turns out, however, that ( )minIm λ is an almost linear function of frequency in a rather wide neighborhood of the resonance frequency. As a result, solving equ. (3.6) for two guess frequencies 2,1f allows to find a much improved frequency by a simple linear interpolation of ( )2,1minIm λ (refer to the benchmark in chapter 4.6 for details).
A problem associated with determining a cavity resonance frequency arises from the known property of the TLM method being overdetermined
48
by a factor of two. For a given physical resonance mode, equ. (3.3) yields two different resonance frequencies. These two frequencies are usually slightly different from each other as a result of numerical dispersion. They can, however, also occur at the same frequency giving rise to a double root in equ. (3.3). In order to determine which singularity actually corresponds to a physically meaningful resonance mode, the field distributions have to be calculated. ( )refV can be found from solving equs. (3.1) and (3.2) at the resonance frequency (one element of the solution vector voluntarily set to one) or as the eigenvector of the eigenvalue equs. (3.4) or (3.6). Once ( )refV is known, the distribution of the electromagnetic fields follows using equ. (3.2) from equs. (2.37)-(2.48).
The check whether the fields are physically meaningful can be done locally, that is, for each node separately. For a simple test, the field components, which are known at the boundary of a SCN, must be transformed to the node center by applying the propagation constants of the respective link-lines. Then, each field component is related to two pairs of complex values in the node center. For example, Ex is defined on four link-lines: +yx, –yx, +zx, –zx. After the shift towards the center of the node, the link-lines +yx and –yx give one pair of complex values, and the link-lines +zx and –zx give another pair. For a physical solution, the phases of a given pair of values are identical, whereas for a non-physical solution, they differ by 180 degrees. Two points should be considered while applying this test: First, the test should of course only be applied to field components having a substantial magnitude (otherwise the phase becomes arbitrary). Second, the field may change sign inside the volume occupied by a SCN, that is, there will be a phase jump for a physical solution. For a rather rough mesh, more than 95% of the phase tests follow the above rule, therefore, a clear decision is possible. This ratio improves the finer the mesh is. It should be mentioned that the described test gives clear results even for solutions of low accuracy, that is, the test can be done before the resonance frequency has been iterated to high accuracy. Depending on the problem at hand, the test frequency can be more than 1% off the final resonance frequency. This allows to stop iterating non-physical solutions early.
A difficulty arises if the resonance frequencies of physical and non-physical solutions are (almost) identical. Then, none of the two eigenvectors gives a meaningful field distribution. However, there exists always a linear superposition of the two eigenvectors which does represent a physical solution. The weighting coefficient of the superposition is found by enforcing (in a few nodes) the center-transformed field phases to be
49
equal. The superposed field can be validated as a physical solution by the check described above.
As resonators can have physically degenerate resonances (two or more different resonance modes having the same resonance frequency), the numerical solution can give twice as many degenerate eigensolutions. Again, a suitable superposition of all these eigenvectors gives the correct number of orthogonal, physical eigensolutions. In general, physically degenerate resonances will have numerically distinct resonance frequencies if a graded mesh is used, because of the numerical anisotropy introduced by the graded mesh. The resonator benchmark in chapter 4.1 will show some details.
50
3.2 Waveguide Mode Cutoff Frequency
The calculation of the cutoff frequency of a mode in a waveguide is a resonance problem [17]. In this case, the waveguide is considered to be homogeneous in propagation direction. At cutoff, the wave resonates in the transverse plane. Thus, the computational domain can be a one node thick transverse slice. At cutoff, all modes (including hybrid modes) degenerate to be either of type TE (no electric field in waveguide direction) or TM (no magnetic field in waveguide direction). Without loss of generality, the waveguide direction shall now be the z direction, and the transverse plane shall be the xy-plane.
For the case of TE-cutoff, only the field components Ex, Ey, Hz may exist. These components will not be affected by a magnetic wall boundary condition in the xy-plane. Thus, if magnetic walls are used to terminate z-directed link-lines, the transverse slice forms a resonator and the resonance frequencies will correspond to cutoff frequencies of TE modes (including hybrid modes degenerating to TE at cutoff).
The field components Ez, Hx, Hy exist in the case of TM cutoff. These components will not be affected by a perfect electric conductor boundary condition in the xy-plane. The eigenfrequencies of the resonator slice are the cutoff frequencies of TM modes.
Once the boundary condition is fixed in the xy-planes, the eigensolutions of the slice resonator can be found as described in the previous paragraph. This algorithm is based on the search for a singularity of a square matrix of size 12×Nnode, where Nnode is the number of SCN in the computational domain.
An alternative way of calculation of the singularity was suggested in [17]. By using matrix algebra, internal connections as well as the connections to the boundaries are substituted, and the resulting matrix (of which either the zero of the determinant or the minimum of the smallest singular value has to be found by varying the frequency) will have a size of 2×Nnode only. The matrix algebra to be performed for each frequency, however, includes two inversions of matrices of size 2×Nnode.
51
3.3 Waveguide Eigenmode Problem
A problem which is often encountered in microwave circuit analysis is to find eigensolutions (propagation constant, field distribution) of waveguides at a given frequency. If the waveguide cross-section is homogeneous (i.e. without dielectric interfaces), then the cutoff frequency of a mode determines the propagation constant and thus the complete eigensolution at any frequency. In a general case, however, the problem has to be formulated at each specific frequency.
In the following, the waveguide is considered to be homogeneous in propagation direction (+z direction). The waveguide eigensolution is found from a one-node thick meshed slice of SCN in xy-plane.
First, internal connections between nodes and the connections to boundaries are eliminated using matrix algebra. The general scattering equation (3.1) can be written as
[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]
⋅
=
incz
incy
incx
zzzyzx
yzyyyx
xzxyxx
refz
refy
refx
VVV
SSSSSSSSS
VVV
(3.9)
and the connection equation (3.2) can be written as
( ) [ ] ( )refxx
incx VCV ⋅= (3.10)
( ) [ ] ( )refyy
incy VCV ⋅= (3.11)
where [ ]xC and [ ]yC describe connections between nodes and onto boundaries for all x-directed and y-directed link-lines, respectively. Substitution of all incident and reflected voltages in the transverse plane gives the so-called intrinsic scattering matrix [ ]ismS
( ) [ ] ( )inczism
refz VSV ⋅= (3.12)
with
[ ] [ ] [ ] [ ]{ } [ ] [ ]zzxzxxxzxism RRRCRS +⋅−⋅=−− 11 (3.13)
where
[ ] [ ] [ ] [ ] [ ]{ } [ ]yxyyyxyxxxx SSCSSR ⋅−⋅+=−− 11 (3.14)
[ ] [ ] [ ] [ ] [ ]{ } [ ]yzyyyxyxzxz SSCSSR ⋅−⋅+=−− 11 (3.15)
52
[ ] [ ] [ ] [ ] [ ]{ } [ ]yxyyyzyzxzx SSCSSR ⋅−⋅+=−− 11 (3.16)
[ ] [ ] [ ] [ ] [ ]{ } [ ]yzyyyzyzzzz SSCSSR ⋅−⋅+=−− 11 (3.17)
The connection matrices [ ]yxC , are singular if a boundary reflection coefficient is zero. This is the case for the ZRT boundary. However, setting the reflection coefficient to a small, non-zero value (e.g., 10–6) introduces a negligible error in the simulation and prevents the connection matrices from becoming singular.
The above calculation of [ ]ismS mainly requires two inversions of matrices of size 4×Nnode. The cost for the inversion of the connection matrices can be neglected as they are at most blockdiagonal with blocks of size 2×2. Note that the conventional algorithm [20] requires four matrix inversions instead of only two.
The above method of elimination of 'inner' link-lines of a domain can be used for any kind of spatial diacoptics. In that case, 'external' link-lines have to be re-normalized with respect to their respective admittance before being connected to other domains by
[ ] [ ] [ ] [ ]5.05.0 −⋅⋅= YSYS ismnormalized
ism (3.18)
where [ ]5.0Y denotes a diagonal matrix containing the square roots of the respective link-line admittances. For the determination of propagation constants (or eigensolutions) of a waveguide, however, this normalization is not necessary.
In the next step, the intrinsic scattering matrix is cut in four parts by simply rearranging rows and columns to form
[ ] [ ][ ] [ ]
⋅
=
inc
B
incA
BBBA
ABAAref
B
refA
VV
SSSS
VV
(3.19)
where the planes A and B are located at z = z0 and at z = z0 + ∆z, respectively (Fig. 3.1). The total voltages, vA and vB, and the total currents, iA and iB, are related to the voltage waves by the well-known relations
refBA
incBABA VVv ,,, += (3.20)
refBA
incBABA VVi ,,, −=± (3.21)
where the minus sign in equ. (3.23) refers to iB (see the current direction definition in Fig. 3.1).
53
single SCN
yx
z z0 z0 + ∆z
iA iB
vA vB
slice of SCN
Fig. 3.1: Definition of voltages and currents on a network of SCN in the transverse plane of a waveguide. Note that positive currents flow in positive axis direction
An eigenvalue problem can be formulated by assuming that, similar to the scattering matrix of an uniform transmission line, the following relations hold
( ) ( ) ( ) ( )incA
incA
refB VVzV ⋅=⋅∆−= λγexp (3.22)
( ) ( ) ( ) ( )refA
refA
incB VVzV ⋅=⋅∆−= λγexp (3.23)
By solving equ. (3.19) for, e.g., ( )incAV , one obtains
[ ] [ ] [ ] [ ] [ ]{ } [ ]{ } ( ) 01 =⋅⋅⋅−⋅⋅−−⋅ − incAAAABBBBA VSSESSE λλλ (3.24)
which can be solved for the propagation coefficient λ by searching the zero of the determinant. The size of the involved matrices is 2×Nnode. This algorithm is computationally very expensive, because a matrix inversion is necessary before the determinant can be evaluated. Furthermore, the propagation constant may be complex (in the case of a lossy waveguide) which makes the search for the singularity even more involved. Thus, another method is described below which is much more efficient [5], [21]. It derives first the ABCD-matrix of the structure and enforces the conditions stated by equs. (3.22), (3.23) thereon. The resulting standard eigenvalue problem can be solved for the propagation constants directly.
The procedure is as follows. First, the scattering matrix (equ. (3.19)) has to be transformed to an ABCD-matrix. This can be done either by calculating first the Z-matrix or by calculating first the T-matrix. The algorithm which uses the T-matrix needs less computational effort and turns out to be better conditioned. For completeness, the way via the
54
calculation of the Z-matrix is briefly described [21]. The most simple (and most expensive) way is
[ ] [ ][ ] [ ] [ ] [ ]{ } [ ] [ ]{ }ismism
BBBA
ABAA SESEZZZZ
+⋅−=
−1 (3.25)
Note that the two factors on the right-hand side of (3.25) commute. More efficiently, similar results are obtained from
[ ] [ ] [ ]{ } [ ] [ ]{ } 122
−−⋅+= ZEZEZ AA (3.26)
[ ] [ ] [ ]{ } [ ] [ ]1ZSZEZ ABAAAB ⋅⋅+= (3.27)
with
[ ] [ ] [ ]{ } 11
−−= AASEZ (3.28)
[ ] [ ] [ ] [ ] [ ]ABABAA SZSSZ ⋅⋅+= 12 (3.29)
The submatrices of the ABCD-matrix are then calculated according to (see Fig. 3.1)
[ ] [ ] [ ] 1−⋅= BAAA ZZA (3.30)
[ ] [ ] [ ] [ ] [ ]BBBAAAAB ZZZZB ⋅⋅−= −1 (3.31)
[ ] [ ] 1−= BAZC (3.32)
[ ] [ ] [ ]BBBA ZZD ⋅−= −1 (3.33)
In addition, [ ] [ ]BBAA ZZ = and [ ] [ ]BAAB ZZ = . The algorithm requires either one inversion of a matrix of size 4×Nnode plus one inversion of a matrix of size 2×Nnode, or three inversions of matrices of size 2×Nnode. Note that in [10], [20] five inversions of matrices of size 2×Nnode are needed instead of three. However, a better method to calculate the ABCD-matrix is as follows [33]. The transfer matrix [ ]T can be easily found from the scattering matrix as
[ ] [ ] [ ] [ ] [ ]BBBAAAABAA SSSST ⋅⋅−= −1 (3.34)
[ ] [ ] [ ] 1−⋅= BAAAAB SST (3.35)
[ ] [ ] [ ]BBBABA SST ⋅−= −1 (3.36)
[ ] [ ] 1−= BABB ST (3.37)
55
by performing only one inversion of a matrix of size 2×Nnode. The ABCD-matrix is then related to the T-matrix by
[ ] [ ] [ ] [ ] [ ]{ }BBBAABAA TTTTA +++=21 (3.38)
[ ] [ ] [ ] [ ] [ ]{ }BBBAABAA TTTTB +−+−=21 (3.39)
[ ] [ ] [ ] [ ] [ ]{ }BBBAABAA TTTTC ++−−=21 (3.40)
[ ] [ ] [ ] [ ] [ ]{ }BBBAABAA TTTTD +−−=21 (3.41)
The derivation of the Z-matrix involves more matrix inversions and is computationally less stable (in the case where many or all nodes are cubic) than the T-matrix algorithm.
It has been shown [5] for a loss-free system of coupled transmission lines that [ ] [ ] [ ] [ ]DBBA ⋅=⋅ (3.42)
[ ] [ ] [ ] [ ]CDAC ⋅=⋅ (3.43)
[ ] [ ] [ ] [ ] [ ]EAACB −⋅=⋅ (3.44)
Experience shows, however, that these relations also hold if losses are included. The eigenvalue problem is now found from the assumption that each current and voltage at a certain position in the waveguide cross section undergoes the same phase shift (and attenuation) while traveling from plane A to plane B. That is
( )( )
[ ] [ ][ ] [ ]
( )( )
( )( ) ( ) ( )
( )
⋅∆−=
⋅
=
A
A
B
B
B
B
A
A
iv
ziv
iv
DCBA
iv
γexp, (3.45)
The inverse of the ABCD-matrix greatly simplifies using equs. (3.42)-(3.44). Finally, the following eigenvalue equations are obtained [ ] ( ) ( ) ( )AA vzvA ⋅∆=⋅ γcosh (3.46)
[ ] ( ) ( ) ( )AA iziD ⋅∆=⋅ γcosh (3.47)
The size of each of the matrices [ ]A and [ ]D is 2×Nnode.
Since cosh() is an even function for a complex argument, two solutions for the propagation constant γ exist, having the same magnitude but opposite sign. These two solutions correspond to one wave, that is, a wave
56
traveling in a given direction with a phase constant β and attenuated by the attenuation constant α can also be seen to travel in the opposite direction with –β, amplified by –α. This ambiguity is readily overcome if the attenuation constant α is forced to be non-negative. Another ambiguity follows from the fact that the problem at hand is symmetric regarding the direction of the wave propagation. As a consequence, there are two (attenuated) waves attributed to each eigenvalue. The propagation constants
2,1γ of these two waves are complex conjugate, ∗= 21 γγ . In order to obtain correct solutions for 3D scattering problems (see chapter 3.4), all modes should travel in the same direction, that is, for example, all phase constants β should be chosen positive (because of conj(cosh(w))=cosh(conj(w)), the conjugate of the eigenvalue can be taken instead).
Once the voltage eigenvector is known from equ. (3.46), then the current eigenvector is found directly from equs. (3.45), (3.47) as [ ] ( ) ( ) ( )AA izvC ⋅∆=⋅ γsinh (3.48)
Finally, by using equs. (3.20), (3.21), the incident and reflected voltages can be calculated
( ) ( ) ( ){ }AAinc
A ivV +=21 (3.49)
( ) ( ) ( ){ }AAref
A ivV −=21 (3.50)
Note that voltage and current vectors have to be calculated from the same eigenvector, say from equ. (3.46) using equ. (3.48), because an eigenvector is defined except for a scaling factor.
If the incident and reflected voltages of a given eigenmode are known on all link-lines in the waveguide cross-section, then the transverse field components are easily calculated from equs. (2.37)-(2.48). From the transverse field components, the longitudinal component of the Poynting vector and thus the net power flow can be derived from
( ) ( ) ( ) ( ){ }∫∫ ∗∗ ⋅−⋅=
sectioncross
xyyxz dAHEHEP (3.51)
The power is real for a propagating mode and imaginary for an evanescent mode. Note that every physical solution is accompanied with a non-physical solution, for example, a mono-mode transmission line shows two modes with real power, one of them being non-physical (see chapter
57
4.3 for details). The eigenvectors ( )incAV and ( )ref
AV can now be normalized such that the modal power equals 1W (or ±1jW for an evanescent mode). The power normalization is a prerequisite of the scattering parameter calculation described in the following chapter.
58
3.4 Scattering Parameters of 3D Structures
In the following, two algorithms will be derived which allow for the calculation of multi-port, multi-mode scattering parameters of a three-dimensional junction of waveguides. The first algorithm is a generalization of a known scattering parameter extraction procedure for sliceable structures [21]. This algorithm involves matrix inversions. Therefore, it is numerically costly. The second algorithm presented thereafter is based on the solution of a linear equation system, which is numerically much cheaper than an inversion.
The junction volume, denoted as the computational domain, is limited by boundaries and meshed with SCN. Semi-infinite waveguides are attached perpendicularly onto the boundary of the computational domain. A 2D analysis of the waveguides' cross-sections as described in the previous chapter yields the field distribution of their eigenmodes. The fields inside the computational domain are calculated several times for different excitations at the ports. Every field calculation in the 3D computational domain gives a field distribution at the port boundaries, which will be expanded in eigenmodes of the attached waveguides. The expansion coefficients allow for the calculation of the scattering parameters.
The governing equations of the FDTLM method (equs. (3.1), (3.2)) can be written as
( )( )
[ ] [ ][ ] [ ]
( )( )
⋅
=
inc
P
incI
PPPI
IPIIref
P
refI
VV
SSSS
VV
(3.52)
( ) [ ] ( )refII
incI VCV ⋅= (3.53)
where the subscript I refers to link-lines of inner connections as well as of connections to the boundaries, whereas the subscript P refers to link-lines to the port boundaries. In a first step, the inner connections are eliminated, resulting in
( ) [ ] ( )incP3D
refP VSV ⋅= (3.54)
[ ] [ ] [ ] [ ] [ ] [ ] [ ]{ } [ ]IPIIIIPIPP3D SCSECSSS ⋅⋅−⋅⋅+= −1 (3.55)
Note that this formulation includes the inversion of a large, sparse, square matrix, [ ] [ ] [ ]{ }III CSE ⋅− , which is computationally expensive. Instead, the computational domain can be divided in small sub-regions, the internal connections of each sub-region are eliminated separately, and the sub-regions are connected together thereafter. This so-called (spatial)
59
diacoptics or decomposition technique circumvents the solution of one large linear equation system at the cost of solving several medium-sized equation systems. Using diacoptics, the complexity of the algorithms increases significantly in the case of a general 3D structure. However, for the case of a structure which can be cut entirely in transverse slices, the concept using scattering transfer matrices will give a rather simple algorithm [21].
A point worth to consider regarding spatial diacoptics is the computational effort. Note that the matrix to be inverted in equ. (3.55) is large, but very sparse. The sparsity allows for efficient direct as well as iterative solution methods. Furthermore, the matrix inversion can be replaced by the solution of one or a few linear equation systems of similar size, which will be shown later. Any diacoptics algorithm, however, will loose sparsity quickly and has then to deal with dense matrices of smaller size. Thus, the advantages using diacoptics techniques are in general not obvious. Some further remarks on computational efficiency are given in chapter 3.5.
Consider now a three-dimensional multi-port problem. In order to obtain the scattering matrix, all modes at all ports are matched to the respective eigenmodes of the connected waveguides, and one port after another is excited with one eigenmode. From that, the scattering parameters of the structure will be found.
Equ. (3.54) can be written as
( )( )
[ ] [ ][ ] [ ]
( )( )
⋅
=
−−
−−inc
othP
incexcP
oth3D,othexc3D,oth
oth3D,excexc3D,excref
othP
refexcP
VV
SSSS
VV
,
,
,
, (3.56)
where the subscript exc denotes link-lines of the excited port, and the subscript oth refers to the link-lines of all other (non-excited) ports. Incident and reflected voltages of a matched (non-excited) port are related by the matched port matrix [ ]PM
( ) [ ] ( )refothPP
incothP VMV ,, ⋅= (3.57)
This matrix will be derived below. Equ. (3.56) can now be reduced to give a one-port problem
( ) [ ] ( )incexcPred
refexcP VSV ,, ⋅= (3.58)
where
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waveguide
incVP
refVP
incbV
refbV
3D computational domain
port planea b
y
x z
slice port inc
aV
refaV
Fig. 3.2: Definition of incident and reflected waves at the port boundary of a three-dimensional structure
[ ] [ ][ ] [ ] [ ] [ ] [ ]{ } [ ]exc3D,othPoth3D,othPoth3D,exc
exc3D,excred
SMSEMS
SS
−−
−−
−
⋅⋅−⋅⋅
+=1 (3.59)
The expansion of a field distribution at a port in terms of eigenmodes of the connected waveguide is done as follows. Referring to Fig. 3.2, note that a mode incident to the 3D structure (denoted as power wave a) is represented on a FDTLM mesh as a superposition of incident voltages ( )inc
aV and reflected voltages ( )refaV . The same holds for an eigenmode
leaving the 3D structure (denoted as power wave b, being a superposition of ( )inc
bV and ( )refbV ). Furthermore, since a and b point in opposite directions,
the following relations hold
( ) ( ) ( )refb
inca
incP VVV += (3.60)
( ) ( ) ( )incb
refa
refP VVV += (3.61)
Any field in the transverse plane of a waveguide can be represented by a linear combination of eigenmodes, thus
( ) ( )∑ ⋅=i
iirefincrefinc
a avV ,, (3.62)
( ) ( )∑ ⋅=j
jjrefincrefinc
b bvV ,, (3.63)
where ( )iincv is the ith eigenvector of incident voltages as obtained from
the 2D eigenvalue analysis of the waveguide (see equs. (3.49), (3.50)), and ( )i
refv is the ith eigenvector of reflected voltages.
61
There is no power wave incident onto a matched, non-excited port (a=0 in Fig. 3.2), therefore, all coefficients ia are zero. Equs. (3.60), (3.61) read
( ) ( ) [ ] ( )othrefoth
jothjj
refoth
incothP bvbvV ⋅=⋅=∑ ,, (3.64)
( ) ( ) [ ] ( )othincoth
jothjj
incoth
refothP bvbvV ⋅=⋅=∑ ,, (3.65)
where the square matrices [ ]incothv , [ ]ref
othv contain all normalized eigenvectors as columns (in the same order), and the vector ( )othb contains all weighting coefficients. Referring to equ. (3.57),
[ ] [ ] [ ]{ }1diag
−⋅= inc
othrefothP vvM (3.66)
is a block-diagonal matrix containing the eigenmode expansions of all matched (non-excited) ports.
If all eigenvectors are taken into account, then [ ]incothv is square and if its
inverse exists, then equ. (3.66) applies. On the other hand, if only a few eigenvectors are considered (e.g., those of all propagating and a few evanescent modes), then [ ]inc
othv is of size m × n, m > n, and the inverse cannot be calculated directly. Fortunately, the singular value decomposition allows to calculate [ ]PM as follows (superscript H denotes hermitian)
[ ] [ ] [ ] [ ] [ ]( ){ }Hnmcolumnsnnnrowsnnnnm
refP UVvM
×−××× ⋅Σ⋅⋅= 1diag (3.67)
where [ ]PM is of size m × m and [ ]V , [ ]Σ , and [ ]U are given by the singular-value decomposition of the (non-square) [ ]incv as [42]
[ ] [ ] [ ] [ ]Hinc VUv ⋅Σ⋅= (3.68)
Here, [ ]U is an unitary matrix of size m × m, [ ]V is a unitary matrix of size n × n, and [ ]Σ is a diagonal matrix of size m × n which contains the so-called singular values (unitarity means [ ] [ ]HUU =−1 , [ ] [ ]HVV =−1 ). Thus,
[ ] [ ] [ ] [ ] 111 −−−⋅Σ⋅= UVvinc is found to be of size n × m.
The excited port is treated as follows. If one mode is incident onto the 3D structure, and all leaving modes are matched, equs. (3.60), (3.61) combine to
( ) ( ) [ ] ( )excrefexcn
incexcn
incexcP bvvaV ⋅+⋅=, (3.69)
62
( ) ( ) [ ] ( )excincexcn
refexcn
refexcP bvvaV ⋅+⋅=, (3.70)
Suppose the power of the exciting mode is normalized, then an=1. Substitution into equ. (3.58) gives a linear equation system for ( )excb
[ ] [ ] [ ]{ } ( ) ( ) [ ] ( )nincexcredn
refexcexc
incexc
refexcred vSvbvvS ⋅−=⋅−⋅ (3.71)
Note that if another mode of the same port is used as excitation, only the right side of equ. (3.71) changes, but the decomposition of the matrix remains the same. The coefficients ( )excb are the elements of the S-matrix describing the scattering from the excitation mode into all modes of the excitation port (in case of a mono-modal port: Sii). The scattering from the excitation mode into all modes of all other (non-excited) ports (in case of mono-modal ports: Sij, i≠j) is found from
( ) [ ] ( ) [ ] [ ] [ ]{ } [ ] ( )incexcPexc3D,othPoth3D,othoth
incoth
refothP VSMSEbvV ,
1, ⋅⋅⋅−=⋅= −
−− (3.72)
where ( )incexcPV , is given by equ. (3.69) after the system (3.71) has been
solved. The linear equation system (3.72) can then be solved for ( )othb . Note that the matrix inversion in equ. (3.72) has already been done while obtaining [ ]redS (equ. (3.59)).
The above algorithm is similar to (but more general than) the procedure developed to solve for the scattering parameters of sliceable, cascaded waveguide structures [21].
The above algorithm requires the inversion of a large, though sparse matrix in order to obtain [ ]DS3 (see equ. (3.55)). This inversion is the main contribution to the numerical cost of the algorithm. Instead of a matrix inversion, the following variation of the algorithm needs only one solution (or a few) of a linear equation system of similar size. In fact, this linear equation system has to be solved for each excited port (i.e., once for one-ports and reciprocal two-ports, and twice for non-reciprocal two-ports). Note that the numerical cost of a matrix inversion is much higher than that of a linear equation system solution, therefore, the acceleration of the algorithm is significant.
By applying a distinction between voltages on internal link-lines, ( )IV , on link-lines connected to the excited port, ( )excPV , , and those connected to non-excited ports, ( )othPV , , equ. (3.52) can be written as
63
( )( )( )
[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]
( )( )( )
⋅
=
incexcP
incothP
incI
PexcPexcPothPexcIPexc
PexcPothPothPothIPoth
PexcIPothIII
refexcP
refothP
refI
VVV
SSSSSSSSS
VVV
,
,
,,,
,,,
,,
,
, (3.73)
Using equs. (3.53) and (3.57), one obtains
[ ] [ ] [ ]{ } [ ] [ ]{ }[ ] [ ]{ } [ ] [ ] [ ]{ }
( )( )
[ ][ ] ( )inc
PexcPexcPoth
PexcI
refPoth
refI
PothPothPothIPoth
PothPothIII
VSS
VV
MSECSMSCSE
⋅
=
⋅
⋅−⋅−
⋅−⋅−
,
,
,,
,
(3.74)
In addition, by equs. (3.66), (3.69), and (3.70)
( ) [ ] ( ) ( ) [ ] ( )( )nrefexcPexcn
incexcn
refPexcPexc
incPexc vMvaVMV ⋅−⋅+⋅= (3.75)
Combining equs. (3.73), (3.74), (3.75) gives
[ ] [ ][ ] [ ]
( )( )
[ ][ ] [ ] ( ) [ ] ( )( )n
refexcPexcn
incexcPexc
PexcPoth
PexcIn
refPoth
refI
PothPothIPoth
PothIII
vMvQSS
a
VV
AAAA
⋅−⋅⋅
⋅
=
⋅
−1
,
,
,,
,
(3.76)
where
[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ]PothPexcPexcPothPothPothPothPothPoth
IPexcPexcPothIPothIPoth
PothPexcPexcIPothPothIPothI
IPexcPexcIIIII
QSMSEAQSCSAQSMSAQSCSEA
,,,,
,,,,
,,,,
,,
⋅−⋅−=⋅−⋅−=⋅−⋅−=⋅−⋅−=
[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]PexcPexcPexcPexc
PothPothPexcPexcPexcPothPexc
IPexcPexcPexcIPexc
SMEQMSMQQCSMQQ
,
,1
,
,1
,
⋅−=⋅⋅⋅=⋅⋅⋅=
−
−
For single port problems (higher order modes are allowed), equ. (3.76) simplifies to
[ ] ( ) [ ] [ ] ( ) [ ] ( )( )nrefexcPexcn
incexcPexcPexcIn
refIII vMvQSaVA ⋅−⋅⋅⋅=⋅ −1
, (3.77)
After the linear equation system (equ. (3.76)) has been solved, transmission scattering parameters are found from equ. (3.65), which represents a linear least square problem of small size
64
( ) [ ] ( ) ( )
⋅=⋅= n_,2_
n_,1_
, , ModePexcModePoth
ModePexcModePoth
nothothincoth
refothP S
SabbvV (3.78)
where 1=na if the power of the exciting mode has been normalized. The reflection scattering parameters at the excited port are obtained using equs. (3.69), (3.70) which form the linear least square problem
( )( )
( ) [ ]( ) [ ] ( )
( )
⋅=
⋅
=
n_,2_
n_,1_
,
ModePexcModePexc
ModePexcModePexc
nexc
exc
nincexcn
refexc
refexcn
incexc
refPexc
incPexc
SS
ab
ba
vvvv
VV
(3.79)
If the power of the exciting mode has been normalized, then 1=na , and the above equation system simplifies somewhat.
The main contribution to the numerical cost of the above algorithms
stems from either the inversion of a large, sparse matrix necessary to obtain [ ]3DS (see equ. (3.55)), or the solution of a large, sparse linear equation system (see equ. (3.76)). This step needs to be performed once for a given frequency (matrix inversion), or once for each excited port at a given frequency (solving equation system). Note that a matrix inversion is numerically much more involved than the solution of a large, sparse, linear equation system. An additional point which can make the second algorithm more efficient is the following. Suppose that the field analysis is required at several nearby frequencies (or if a material parameter changes gradually [16]). Then, the calculation results at one frequency may serve well as starting vector for an iterative linear equation system solver at the next, neighboring frequency. The algorithm which is based on the solution of a linear equation system can also be subject of other numerical procedures which speed up the solution. This will be discussed in the next chapter.
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3.5 Implementation Aspects
In the previous chapters, methods and algorithms have been presented which allow to cast the RF electromagnetic field problem in the form of simple algebraic formulas. In order to solve these algebraic formulas, the requirements in terms of computer memory and time are significant and grow with the complexity of the problem under consideration. Thus, relevant real world problems can only be treated if the overall solution process is optimized. Three important points have to be considered:
First, the numerical method itself should be numerically efficient and appropriate for the problem (Consider for example the differences between a full-field solution and a physical optics based approach: both methods have advantages for some categories of problems, and have drawbacks for others.) Once the solution method for a given problem has been fixed, the relevant parameters of the method must be chosen carefully (for example, in the FDTLM method, the chosen mesh will affect the solution process fundamentally).
Second, efficient programming (programming languages, libraries, and compilers) has to be used on powerful (fast and with large memory) computers. For example, the question of using an iterative or a direct solver is not simple to answer from the mathematical point of view, but it becomes even more complex with regard to the actual implementation of the algorithm in the numerical library, and, last but not least, the computer architecture (e.g., memory resources and access time, processor speed) should be considered.
Third, use efficient algorithms from modern numerical algebra to reduce time and memory requirements. This seems to be obvious but difficulties arise due to the fact that these algorithms are often very specific, potentially unstable, and not yet extensively tested.
It can be seen that a successful implementation and application of any method needs the cooperation of mainly three fields: electromagnetics (first of the above mentioned points), computer science (second point), and mathematics (third point). The presented thesis is written from an application point of view, and therefore, the electromagnetics part predominates. In this chapter, however, some easy-to-implement alternatives from the numerical algebra side are described and partly evaluated. The effect of some of these algorithms will become clear in later chapters, when example structures will be analyzed. It will become clear
66
that an inappropriate numerical solution method can make the analysis of a given problem very costly if not simply impossible.
Sparse Matrix Techniques Discretizing differential equations with a local operator, which is the
case for all finite-element and finite-difference methods (FEM, FDTD, TLM, ...), will result in a huge linear equation system. Each unknown, however, depends on very few other unknowns only. Thus, the resulting system matrix will be very sparse (that is, it has few non-zero entries). One can and should profit from matrix sparsity in two ways: First, as only the non-zero elements have actually to be stored (e.g., in a list), memory is saved. Second, matrix-vector multiplication is much faster with a sparse matrix than with a full matrix. Indeed, for a square matrix, the cost of it is proportional to N if the matrix is sparse, and proportional to N 2 for a full matrix (where the cost is defined as the number of multiplications, and N is the number of elements of the vector). Since many algorithms in the field of numerical algebra are based on matrix-vector multiplications, most programs will be more powerful when using sparse matrix techniques.
Linear Equation Systems and Matrix Inversion Once a problem has been cast in the form of a system of linear equations,
the question arises how to solve it efficiently. Consider the form [ ] ( ) ( )bxA =⋅ , where the square matrix [ ]A and the vector ( )b of length N are given, and the vector ( )x contains the unknowns.
The solution should avoid the inversion of [ ]A , although formally ( ) [ ] ( )bAx ⋅= −1 , because the inversion is much more costly than other solution procedures, and because [ ] 1−A is usually dense or full even if [ ]A is very sparse (there is ongoing research in the field of numerical algebra to find out under what circumstances the inverse of a dense matrix becomes sparse). As a result, the 3D scattering parameter algorithm in the version which involves the inversion of the system matrix (see equ. (3.55)) is limited to rather simple structures, whereas the version which is based on the solution of a linear equation system (see equ. (3.76)) can handle more complex problems. Similarly, any domain-decomposition algorithm (e.g., [21]) necessarily involves many matrix inversions and deals then with full matrices, it is therefore limited to special cases: each sub-domain shall be simple (since a matrix inversion of the related equation sub-set is necessary), and preferably some sub-domains shall be similar so that the
67
related matrix-inverse can be re-used (e.g., in the case of semi-periodic structures and for many filter structures).
There exist several algorithms to solve [ ] ( ) ( )bxA =⋅ efficiently for ( )x . In the case of equ. (3.76), [ ]A is found to be square, sparse and non-singular, but complex and non-hermitian. An appropriate direct (non-iterative) method is the LU-decomposition technique. In fact, this is the Gaussian elimination procedure with pivoting. It generates a lower triangular matrix [ ]L , an upper triangular matrix [ ]U , and a permutation matrix [ ]P such that
[ ] [ ] [ ] [ ]APUL ⋅=⋅ (3.80)
The unknown vector ( )x can then be found quickly by back-substitution (see Fig. 3.3, top).
A significant improvement of the LU-decomposition of a sparse matrix can be achieved by an appropriate ordering of the matrix rows and columns prior to the decomposition. This is an area of current research since, for non-symmetric systems, few orderings are established [13]. Nevertheless, the so-called column-based minimum degree ordering was found to give impressive speed improvements compared to standard LU (see Fig. 3.3, bottom, for the MATLAB implementation). The preordering takes only a few seconds, so it comes practically for free. This implementation is equivalent to MATLAB’s ‘Backslash’ division, but it keeps the matrices [ ]L and [ ]U accessible as preconditioners for iterative solutions, as described below.
>> % solving Ax=b by LU-decomposition>> [L,U,P] = lu(A) ;>> y = L \ (P*b) ;>> x = U \ y ;>> clear L U P y ;
>> % solving Ax=b by LU-decomposition with>> % column minimum degree ordering>> n = length(b) ;>> R = sparse(colmmd(A),[1:1:n],ones(n,1)) ;>> [L,U,P] = lu(A*R) ;>> y = L \ (P*b) ;>> xp = U \ y ;>> x = R * xp ;>> clear L U P R n xp y ;
Fig. 3.3: Solving a linear equation system Ax=b by LU-decomposition without (top) and with pre-ordering (bottom) in MATLAB
68
A point to consider while dealing with the LU-decomposition of sparse matrices is the so-called fill-in. The numbers of non-zero elements (nnz) in the triangular matrices [ ]L and [ ]U , nnz(L) and nnz(U), are usually much larger than nnz(A). Therefore, for large problems, the LU-decomposition and thus the direct solution of the equation system is impossible due to memory constraints. A remedy could be the iterative solution for the unknown vector ( )x , starting from some ( )0x . Unfortunately, the iterative algorithm might not converge. Then, one could start with the so-called incomplete LU-decomposition, which solves the system in some approximate way, and continue with an iterative solver until the exact solution is found with some specified accuracy.
The incomplete LU-decomposition sacrifices some of the elements of the triangular matrices [ ]L and [ ]U , therefore lowering the count of non-zero matrix entries and weakening the memory requirements. The difficulty lies in the decision which matrix entry is unimportant so that it can be set to zero. For example, the MATLAB function luinc requires a scalar threshold parameter, below which all matrix elements are neglected. The goal of an incomplete factorization is to use the triangular matrices [ ]L and [ ]U as preconditioners for iterative solvers in order to improve the convergence behavior of the latter. For non-symmetric matrices, however, ”... the performance of ILU [incomplete LU] preconditioners is very unpredictable ... the preconditioned system can perform much worse than the original matrix with respect to convergence of the iterative method [13].”
In contrast to direct methods, iterative methods do not modify the matrix [ ]A . Therefore, the excess computer memory required during the solution process is small (a few vectors of size N) and often predictable. There exist various algorithms for the iterative solution of large, sparse linear equation systems [40]. Nowadays, the so-called Krylov subspace methods (A.N. KRYLOV [1863–1945], a russian maritime engineer, published the method in 1931 and used it to solve eigenvalue problems of order 6) are recognized to be most efficient in terms of convergence, stability, and memory requirement. The Stabilized Bi-Conjugate Gradient method (bicgstab) and the Generalized Minimum Residual method (gmres) are suited for non-symmetric, non-hermitian problems. The number of iterations required to achieve a specific accuracy can be greatly reduced by using preconditioning matrices and/or approximate solutions. The latter can be found from the solution at a neighboring frequency, provided the frequency steps are small. A very good choice of preconditioners are the LU decomposition matrices. This is not surprising because the LU matrices
69
represent the actual solution of the problem. However, the LU matrices found at one frequency are good preconditioners in a rather wide frequency interval.
Eigenvalue Problems Eigenvalue problems arise in the course of a problem solution using the
FDTLM method in two cases: finding the propagation constant of a waveguide (chapter 3.3), and finding the resonator resonance frequency (chapter 3.1). The latter problem will be described below as part of matrix singularity tests. The first problem is 2D, therefore small with respect to the number of unknowns, and the involved matrix is dense. It can be solved using standard algorithms which will give the full eigenvalue spectrum. In general, however, only a few eigensolutions, namely those of the propagation mode(s) and some evanescent modes, are required. It turns out that the corresponding eigenvalues are among the smallest in magnitude of all eigenvalues of the matrix (the lower non-physical modes, see chapter 4.3, also correspond to eigenvalues of small magnitude). A few smallest eigenvalues can be found much faster using an iterative method (see [2] and the MATLAB command eigs) than the complete eigenvalue spectrum.
Test for Matrix Singularity The algorithm for the calculation of cavity resonance frequencies
(chapters 3.1 and 3.2) requires the system matrix to be singular at the resonance frequency in order to have a non-zero field distribution. Because the system matrix depends on the frequency in a non-linear way, the solution method of choice is to vary the frequency until the system matrix becomes singular. Therefore, two points are important for the choice of the mathematical procedure: First, the check for singularity has to be quick because it is repeated again and again until the resonance is found. Second, the parameter describing the ‘singularity of the matrix’ should allow for the detection and iteration of a resonance with as few frequencies as possible tested for singularity.
Different methods which allow to find a singular matrix can be characterized in view of this two requirements as follows.
The determinant of the system matrix will be zero at the resonance frequency. The calculation of the determinant is quite fast since the system matrix is very sparse. The determinant does not has poles (as it is the case for matrices which arise from other numerical methods such as, e.g., the
70
method of lines) because the entries of the system matrix are always finite and do not differ too much in magnitude. There are, however, drawbacks of this method. Since the determinant of a large matrix is a sum of many terms, its zero may be inaccurate due to the accumulation of round-off errors. Since every element of this sum is actually a product of many factors, underflow or overflow may occur. The latter means that for a large matrix, the determinant may become numerically zero in a rather wide interval around the actual resonance frequency.
The smallest singular value of the system matrix will be zero at the resonance frequency. This method finds also the multitude of the singularity (see Fig. 4.1). By tracking several of the smallest singular values over frequency, resonances can be detected using large frequency steps and can be iterated efficiently. The drawback of the method is the numerical cost of calculating the singular-value decomposition. This can be done by the so-called QR algorithm [42], which transforms the system matrix, thereby loosing sparsity (accordingly, MATLAB’s svd command only works with full matrices). This means that this method is limited to small problems due to memory and time constrains. Furthermore, the method provides all singular values of the matrix, and the smallest with the worst accuracy, which is not optimal for the singularity test. Iterative methods for sparse matrices calculating only one singular value (or a few) are available (e.g., MATLAB’s svds command), but they trace the problem back to an eigenvalue problem, which can be used directly to find a singularity (see below). Finally, it should be said that the problem is subject of current research in the numerical algebra domain and more efficient algorithms can be expected in the future [2].
The condition number of a matrix is a measure for singular behavior. The solution of a linear equation system involving a given matrix is the more inaccurate the larger the condition number of this matrix is. Thus, the condition number of a singular matrix is infinite. A popular way is to calculate the condition number as the ratio of the largest singular value to the smallest singular value (e.g., MATLAB’s cond command uses this definition). As a result, the calculation of the condition number is too expensive for large problems. Instead, however, an estimate of the condition number can be found cheaper.
A condition number estimate can be found from a LU decomposition of the matrix. The underlying iterative algorithm (condest) is based on the LU decomposition of the matrix in the inner loop. It needs a significant amount of memory. As described above, pre-ordering (colmmd) should be used in order to increase speed and to decrease memory requirements. The
71
condition number estimate as a function of frequency is a non-continuous function. The steps of this function prevent efficient interpolations and result in a slow iterative procedure because the determination of the frequency where the singularity occurs needs to calculate the condition number estimate at many points.
A standard eigenvalue problem as stated in equ. (3.6) can be solved for the smallest eigenvalue (zero at singularity) by iterative methods [2]. As an iterative method, it requires less memory than a LU decomposition and is particularly suited for large problems. Advantageously, the smaller the magnitude of the eigenvalue to be calculated, the faster the algorithm. Nevertheless, these algorithms take a significant amount of time because several linear equation systems have actually to be solved during the iteration. Note that the implementation in MATLAB’s eigs command uses the so-called Arnoldi method [2], which is iterative but very inefficient in terms of memory because it calls several times a LU decomposition.
The above methods all have advantages and drawbacks. For small problems, repeated calculation of the determinant or the calculation of the smallest singular value in combination with a smart zero search strategy can be applied. For medium-sized problems, the determinant calculation turns out to be as fast as the calculations of the smallest eigenvalue or the condition estimate. For large problems, where the determinant calculation fails, the condition estimate is usually faster than the eigenvalue iteration. The eigenvalues, on the other hand, allow for an efficient zero search strategy as it has been described in chapter 3.1. For very large problems, an iterative eigenvalue algorithm is the method of choice [2], but its implementation should avoid matrix decompositions.
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3.6 Conclusions
In this chapter, algorithms were developed which allow to solve typical classes of problems arising during the analysis of microwave circuits.
First, the calculation of cavity resonance frequencies and waveguide cutoff frequencies is described. It is shown that large problems are best treated in the form of a standard eigenvalue problem. Spurious solutions appear as non-physical resonance frequencies nearby the physical solution. A method is described which allows to identify non-physical solutions in a mathematically simple and stable way.
Second, the well-known procedure for the calculation of waveguide eigenmodes is derived. This method is optimized for smallest computational cost. Ambiguities of the solutions are described.
Third, an algorithm for the calculation of three-dimensional multi-port, multi-mode structures is described. The three steps of the procedure, namely, the eigenmode excitation, the field calculation, and the extraction of the scattering parameters, are derived in a comprehensive way. One version of this algorithm is based on an inversion of the large system matrix and is found to be a generalization of the previously known S-parameter algorithm for sliceable structures. A second version of the algorithm is developed and needs the solution of a linear equation system based on the large system matrix, what comes at a much smaller computational cost than a matrix inversion.
Finally, aspects of the implementation of the algorithms are discussed. It is pointed out that the choices of both the FDTLM algorithm and the numerical solution procedure affect the overall cost needed to solve a problem. It is shown that a resonator problem is best treated as a standard eigenvalue problem in a form where the smallest eigenvalue is actually sought. This method, however, is still iterative and rather involved so that the problem size, which can be solved on today’s workstations, is limited to a few thousands FDTLM nodes. This number may grow in the near future since sparse eigenvalue solver are a topic of intense research. On the other hand, the scattering parameter algorithm is based on the solution of a linear equation system. Therefore, it can handle problems involving 10...20'000 nodes on a modern workstation if an efficient (iterative) matrix solver is used.
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4 Application of the FDTLM Method to Passive Microwave Circuit Elements
In the following, a series of benchmark problems is solved using the algorithms which have been developed in the previous chapter.
At the beginning (chapters 4.1-4.4), simple structures are investigated in order to establish characteristic properties of the FDTLM method, such as convergence order and the appearance of spurious solutions. Some of the example structures are accessible to analytical FDTLM solutions, giving further insight into the nature of the method.
Later on (chapter 4.5), the scattering parameter algorithm is validated on a simple waveguide bend structure.
Finally (chapters 4.6-4.7), realistic problems are investigated. The solutions are compared to FEM results, and the computational effort is discussed.
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4.1 Cavity Resonance Frequencies and the Non-physical Solutions
A simple, loss-free cavity resonator is analyzed in order to evaluate the accuracy of FDTLM solutions using different meshes. Physically degenerated resonance modes are shown to be numerically distinct due to the influence of numerical dispersion. Non-physical solutions appear but can be identified using a simple local field test.
The cavity of interest is an air-filled cube of size (40mm)3. This resonator has a threefold degenerated fundamental resonance at 5299.6 MHz, and the next (single) higher resonance at 6490.6 MHz. The FDTLM method using a mesh of cubic nodes gives the results shown in Tab. 4.1. For example, a mesh of 4×4×4 nodes results in an error of the fundamental resonance frequency of 1.3%.
Mesh Resonance frequency Error
2×2×2 (cube size 20mm) 4997 MHz –5.7%
3×3×3 (cube size 16.67mm) 5173 MHz –2.4%
4×4×4 (cube size 10mm) 5230 MHz –1.3%
5×5×5 (cube size 8mm) 5255 MHz –0.8%
6×6×6 (cube size 6.667mm) 5269 MHz –0.6%
8×8×8 (cube size 5mm) 5282.5 MHz –0.32%
10×10×10 (cube size 4mm) 5288.5 MHz –0.21%
12×12×12 (cube size 3.333mm) 5292.0 MHz –0.14%
16×16×16 (cube size 2.5mm) 5295.37 MHz –0.08%
Tab. 4.1: Accuracy of cubic cavity resonator resonance frequencies using regular (non-graded) mesh of various size
Since the physical singularity is threefold and the FDTLM method adds a non-physical solution to each physical solution, the numerical singularity is six-fold. If a graded mesh is used, however, the singularities will appear at slightly different frequencies because of the numerical anisotropy introduced by the graded mesh. This is shown in the following example.
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A cubic cavity resonator of size (40mm)3 is meshed with 4×4×4 nodes. The node dimensions are • in x-direction: 8.5mm, 9.5mm, 10.5mm, 11.5mm, • in y-direction: 7.75mm, 9.25mm, 10.75mm, 12.25mm, • in z-direction: 7mm, 9mm, 11mm, 13mm.
A frequency sweep of the smallest singular values (see Fig. 4.1) shows six distinct singularities around 5300 MHz. The actual values of the resonances are given in Tab. 4.2. By checking the eigensolutions of physical relevance (the method was described in chapter 3.1), three solutions are found to be physical. For the solution to be characterized as physical, the phases of the center-transformed field components of all nodes should not jump (by 180 degrees) across the node center. For the simple cavity under consideration, the physical solution is also characterized by having only one non-zero electric field component, whereas the non-physical solutions have all field components non-zero. This additional check confirms the proposed method.
Fig. 4.1: Magnitudes of the 6 smallest singular values around the fundamental resonance frequency of a cubic resonator
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Resonance frequency Error in frequency Physical solution or not ?
4978.5 MHz –6.1 % non-physical 5049.5 MHz –4.7 % non-physical 5127.7 MHz –3.2 % non-physical 5222.74 MHz –1.4 % physical, Ex=Ey=0 5224.02 MHz –1.4 % physical, Ex=Ez=0 5224.93 MHz –1.4 % physical, Ey=Ez=0
Tab. 4.2: Fundamental resonance frequencies of a cubic resonator obtained by a graded mesh and the distinction between physical and non-physical solutions
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4.2 Rectangular Waveguide Cutoff Frequencies and the Influence of Mesh Grading
The cutoff frequencies of rectangular waveguides are calculated using meshes of different densities. It is shown that the accuracy of the solution depends also on the shape of the nodes and the type of node chosen. For the particular case of the TEn0 modes of a dielectric slab loaded waveguide, an analytical solution can be derived.
Consider an air-filled rectangular waveguide (cross-section dimensions ba 2= ). The cutoff frequencies of some modes are calculated using
different meshes of cubic nodes (Tab. 4.3). It can be seen that those modes are exact which resonate in the direction of a coordinate axis only (i.e. TEn0, TE0n). All other modes are the more accurate the finer the discretization is.
mode 2×1 nodes 4×2 nodes 8×4 nodes 16×8 nodes
TE10 exact exact exact exact
TE20, TE01 exact exact exact exact
TE11, TM11 –10.6 % –2.2 % –0.52 % –0.13 %
TE21, TM21 — –5.7 % –1.32 % –0.32 %
TE30 exact exact exact exact
Tab. 4.3: Accuracy of rectangular waveguide mode cutoff frequencies obtained using different mesh densities (regular mesh of cubic nodes).
The accuracy with which a FDTLM mesh models the (non-evanescent) electromagnetic fields is seen to increase with the square of the mesh density. That is, by dividing the spatial discretization step by a factor of two, the error decreases by a factor of four. This behavior was shown in Tab. 4.1 for a cavity resonator as well as in Tab. 4.3 for waveguide cutoff frequencies. The square law corresponds to a linear dependence in a double-logarithmic plot as shown in Fig. 4.2.
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Fig. 4.2: Quadratic increase (linear in log-log) of the accuracy versus mesh density shown for cubic resonator TE110 resonance (see Tab. 4.1) and rectangular waveguide TE11 and TE21 cutoffs (see Tab. 4.3)
Note that the matrix singularities of the corresponding cutoff frequencies are double singularities, indicating the coincidence of a physical and a non-physical solution. In case that non-cubic nodes are used in the mesh, the frequencies of the two singularities will slightly differ from each other. It can be decided locally from the field distribution which solution is the physical one, as it has been described in chapters 3.1 and 4.1.
In the following, it is shown that the accuracy of the solution degrades if the node shape deviates from being cubic. Consider the cutoff resonance of the TE10 mode, which can be seen equivalent to the propagation of a plane wave in transverse axis direction. The cutoff frequency should be independent from the shape of the nodes, in practice, however, errors occur. For example, a mesh of four similar nodes in the cross section of the waveguide is shown in Fig. 4.3. The errors caused by a variation of the node dimensions ∆y and ∆z are shown in Fig. 4.4 for the PCN. The altitude lines of such an error plot are shown in Fig. 4.5 for different nodes. It compares also the so-called hybrid node schemes [20], [9]. It can be seen that CAN is by far the least accurate node, and the PCN performs at least as good as the hybrid node schemes.
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∆x
X
Y
Z
∆y
∆z
Fig. 4.3: Waveguide cross section discretized with four identical FDTLM nodes
Fig. 4.4: Accuracy of TE10 cutoff frequency while using non-cubic nodes (PCN)
The CAN, despite of being the least accurate node in general, shows no dispersion as long as ∆y = ∆z (see Fig. 4.5). Further evidence can be found analytically. Consider a dielectric slab loaded waveguide (Fig. 4.6). Each region is discretized by one node (CAN) only. The analytical solution of the resonance condition (zero of the determinant in equ. (3.3)) simplifies for ∆y = ∆z to
( ) ( )2010 tantan xkxk rr ∆−=∆ εε (4.1)
which is the exact analytical solution for the cutoff frequencies of the TEn0 (n=1,2,...) modes [12].
∆y / ∆x
∆z / ∆x
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Fig. 4.5: Accuracy of various types of FDTLM nodes of non-cubic geometry. The error magnitude lines correspond to an error of the cutoff frequency of 0.1% (solid), 0.5% (dashed), 2.5% (dash-dotted), respectively
X
Y
Z
εεεεrel
X
Y
Z
εεεεr
∆x1 ∆x2
Fig. 4.6: Dielectric slab loaded rectangular waveguide cross section discretized with two FDTLM nodes (CAN)
On the other hand, setting ∆x1 = ∆x2 = ∆x, εr = 1, but ∆y ≠ ∆z, yields
+⋅
+−⋅
+=
∆∆+
∆∆∆−
∆∆∆−
∆∆∆−
yz
zyxkj
zyxkj
yzxkj
eee000
1110 (4.2)
which simplifies to the exact solution (λc = 4∆x / n, n=1,2,...) only in case that ∆y = ∆z.
∆z / ∆x ∆z / ∆x
∆y /
∆x
∆y /
∆x
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4.3 Non-physical Solutions in the Waveguide Eigenmode Algorithm
This chapter discusses the appearance of non-physical solutions in the 2D eigenmode algorithm (see chapter 3.3). Because both physical and non-physical modes can be propagating, thus carry real power, it is important to know how to recognize a non-physical eigensolution.
In the following, eigensolutions of a rectangular waveguide are analyzed mainly analytically. Consider a rectangular waveguide cross-section, discretized with four identical cubic SCN. By applying a magnetic wall symmetry, the computational domain contains N = 2 nodes only (Fig. 4.7). As pointed out in chapter 4.2, the FDTLM method obtains the cutoff frequencies of the TEn0 (n=1,3,5,...) modes exactly. At other frequencies but cutoff, however, numerical dispersion will result in an error in the propagation constants. The 2D eigenvalue problem (see equ. (3.46)) results in 2N = 4 eigenvalues and eigenvectors. An analytical solution gives the eigenvalues
( ) ( )( )0cos222223arcosh1 kj III ∆+++∆
=+= βαγ (4.3)
( ) ( )( )0cos222223arcosh1 kj IIIIII ∆−−−∆
=+= βαγ (4.4)
( ) ( )( )0cos222223arcosh1 kj IIIIIIIII ∆−++−∆
=+= βαγ (4.5)
( ) ( )( )0cos222223arcosh1 kj IVIVIV ∆+−−−∆
=+= βαγ (4.6)
where ∆ denotes the node dimension. The eigenvectors are of the form ( ) ( ) ( ) ( ) ( ){ }4321 ,,, VVVVv = (see Fig. 4.7) and given by
( ) ( ) ( ) ( ) ( ){ }1,0,21,0 −−=Iv (4.7)
( ) ( ) ( ) ( ) ( ){ }1,0,12,0 −=IIv (4.8)
( ) ( ) ( ) ( ) ( ){ }0,1,0,21−=IIIv (4.9)
( ) ( ) ( ) ( ) ( ){ }0,1,0,21+=IVv (4.10)
The total link-line voltages V1...4 are proportional to the transverse electric field components. Therefore, ( )Iv and ( )IIv describe the exact field magnitudes of the TE30 and TE10 modes, respectively. On the other hand, ( )IIIv and ( )IVv describe non-physical solutions.
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magnetic wallsymmetry plane
V1 V3
V4V2
xy
z
node 1 node 2
Fig. 4.7: Rectangular waveguide cross section discretized with two FDTLM nodes (symmetry plane applied)
Fig. 4.8 shows the real eigenvalues, ( )∆γcosh , as a function of normalized frequency according to equs. (4.3)-(4.6).
Fig. 4.8: Matrix eigenvalues as a function of normalized frequency
Recalling the properties of the area cosine hyperbolicus function, namely
−≤−≥≥
≥
≥+≤≤+
≥+=
111
1
0,0,0
0,0)(arcosh
zz
z
forajaforbjbforaja
zπ
π (4.11)
k0 ∆ / π
cosh
(k0∆
)
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and considering α = a / ∆ and β = b / ∆, it is found that an evanescent mode solution corresponds to a matrix eigenvalue ( ) 1cosh >∆γ , cutoff occurs at ( ) 1cosh =∆γ , a propagating solution has ( ) 1cosh1 −>∆>+ γ , and solutions with ( ) 1cosh −≤∆γ are non-physical. Thus, propagating solutions may be either physical or non-physical.
Consider Fig. 4.9 which shows the attenuation constant α and the phase constant β as found from the simulation of the waveguide structure. Due to the properties of the arcosh(...) function, a maximum frequency is given as fmax = c / 2∆. At this point, where 1/0 =∆ πk , all propagating modes have the same phase constant of β = π / ∆ and the node dimension equals half a wavelength. At lower frequencies, the fundamental mode TE10 (index II in Fig. 4.9) has to be distinguished from its non-physical, though propagating, counterpart (index III). A practical upper frequency limit can be established in that at reasonably low frequencies (approx. f < c / 4∆), the non-physical mode is faster than the speed of light and can be crossed out (at higher frequencies, it can still be detected from its field distribution as it was described in chapter 4.1). For inhomogeneous waveguides involving dielectrics, this frequency limit lowers.
Fig. 4.9: Phase constant (top) and attenuation constant (bottom) of the four eigensolutions as a function of normalized frequency. The dashed line (top) describes the wavenumber of a plane wave. The block arrow depicts a practical upper frequency limit, where the first non-physical mode (III) becomes slower than the speed of light
k0 ∆ / π
α ∆
β ∆
π
π
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The propagation modes, either physical or non-physical, carry real power. Experience shows, however, that the scattering between them is zero (within numerical limits), that is, a 3D structure does not couple between physical and non-physical fields.
Although the described situation may be slightly different for practically relevant transmission lines, experience shows that the two smallest matrix eigenvalues ( )∆γcosh correspond to the fundamental propagating mode and its non-physical counterpart (index II and index III in Fig. 4.8). Consequently, instead of solving the eigenvalue problem (equ. (3.46)) for the complete eigenvalue spectrum, which is numerically costly, an iterative eigenvalue solver can search a few (at least two for a single-mode solution) of the smallest eigenvalues only.
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4.4 Scattering Parameters of a Waveguide Filter and Accuracy Analysis
A double-post waveguide resonator filter will be used to evaluate the accuracy convergence of the FDTLM analysis of a 3D structure. In addition, the influence of correct matching of evanescent modes at the ports will be discussed.
Two inductive metallic posts in a rectangular waveguide form a resonator (Fig. 4.10). The structure shows band-pass behavior at resonance. A magnetic wall symmetry plane reduces the computational domain. By assuming perfectly conducting materials only, the fields can be expanded in terms of TEn0 modes and the solution becomes independent of the height of the structure.
Fig. 4.10: Drawing of the single resonator waveguide filter (waveguide width 2x11.43mm, overall length 33.43mm, cavity length 22mm, post 5.715mm x 2.8575mm)
The filter characteristics were calculated using the FDTLM method employing meshes of different densities. A comparison with the FEM (HFSS) as well as with a high-resolution mode-matching approach [35] will be given. The mode-matching method is particularly well suited for this kind of problems, and therefore it is considered as the reference. The accuracy obtained using the different approaches is given in Tab. 4.4. Fig. 4.11 shows the respective meshes used for the FDTLM method and FEM solutions.
It was shown in the preceding chapter (see Fig. 4.2) that the convergence of FDTLM is of second order, that is, the accuracy improves by a factor of four if the mesh grid size is cut by two. This is a well-known behavior of any finite-difference method using linear central difference operators to solve Maxwell’s equations. This relation, however, holds only if a plane wave propagation in homogeneous medium, discretized by a non-graded mesh, is considered. In particular, field singularities (e.g., at material edges and wedges) decrease the convergence order. Suppose that a halved mesh
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grid size increases the accuracy by 2p, then 2 ≥ p ≥ 1, whereas p = 2 holds for the simplest plane wave case, and p ≈ 1 holds for the neighborhood of a metallic fin (the latter case has been discussed in [41]). The waveguide filter of Fig. 4.10 exhibits both singularities (the edges of the posts) and a plane wave section (the waveguide part), and p ≈ 1.45 is found from the FDTLM solutions tabulated in Tab. 4.4.
Although a comparison of the accuracy between FDTLM and FEM solutions depends on the actual mesh shape and is therefore somewhat arbitrary, it can be seen from Tab. 4.4 that FEM reaches a higher accuracy (for the given problem) for a given number of unknowns. The difference is small and might be due to the fact that FEM allows more easily for locally refined meshes.
Mode-matching method 50 modes fres = 9136 MHz
FDTLM, rough mesh 344 unknowns (30 nodes) – 1.50 %
FDTLM, fine mesh 1'408 unknowns (120 nodes) – 0.54 %
FDTLM, high resol. 4928 unknowns (416 nodes) – 0.20 %
FEM, rough mesh 189 unknowns + 0.91 %
FEM, fine mesh 1'452 unknowns + 0.32 %
FEM, high resolution 52'500 unknowns + 0.07 %
Tab. 4.4: Resonance frequency of the single-resonator filter obtained with different numerical methods. The respective 'rough' and 'fine' meshes are depicted in Fig. 4.11
The complete scattering parameter plots as obtained with the FDTLM method on rough and fine meshes, respectively, are shown in Fig. 4.12.
The scattering parameter algorithm of chapter 3.4 can take up to 2×Nport_node eigenmodes into account. These include the actual propagating mode(s), non-physical propagating mode(s), evanescent modes, and many numerical artifacts. Experience shows that no power is scattered from a physical, propagating mode excitation into a non-physical propagating mode. Evanescent modes, however, can physically store reactive energy and have to be considered. This can be done either by actually connecting some length of waveguide between junction and port plane in order to let
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the evanescent modes decay, or, if the port plane is located nearby the junction, by matching the evanescent modes properly.
This is illustrated in Fig. 4.13. Four different meshes similar to the fine mesh of Fig. 4.11 (top right), but with, respectively, one, two, three, and four layers of nodes between ports and junction are analyzed. If the complete matching matrix [ ]PM is used, then all four simulations yield identical scattering parameter magnitudes and perfect power balance ( 12
212
11 =+ SS ). If only the fundamental mode is considered at the ports, both accuracy and power balance are inaccurate but improve if the connected waveguide becomes longer. This means that indeed the port plane can be located where evanescent modes still exist, as long as these evanescent modes are taken into account in the port field expansion.
E wall
M wall
portport
E wall
M wall
portport
Fig. 4.11: Drawings of different meshes used for the filter analysis. Top: FDTLM meshes. Bottom: FEM meshes. Left: rough meshes. Right: fine meshes.
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Fig. 4.12: Scattering parameters (top: magnitudes, bottom: phases) of the filter obtained using the FDTLM method with a rough mesh of 30 nodes (solid lines), the FDTLM method with a fine mesh of 120 nodes (dash-dotted lines), and the mode-matching method with 50 modes (reference solution; dashed lines), respectively. The FDTLM meshes are sketched in Fig. 4.11
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Fig. 4.13: Accuracy of port matching with the fundamental mode only. Different numbers of node layers (1...4) connected between post and port. Top: The longer the waveguide section, the more |S21| converges to the solution which uses the complete mode spectrum. Bottom: The power balance converges to the correct value of one
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4.5 Higher Order Mode Scattering Parameter Extraction of a Waveguide Bend
The rectangular waveguide H-plane bend shown in Fig. 4.14 (left) is analyzed in order to evaluate the extraction of higher order mode scattering parameters. In addition, this example shows port planes which are perpendicular, and not parallel, to each other. A convergence analysis of the solution’s accuracy reveals frequency-dependent convergence orders in the range between 1.5 and 2.
uncompensated bend matched bend
Fig. 4.14: Rectangular waveguide H-plane bend: standard (left), matched (right)
The rectangular waveguide has an aspect ratio of the cross section of 4:1. This implies mono-modal behavior for frequencies fc < f < 2fc, where fc is the cutoff frequency of the fundamental TE10 mode. Between 2fc and 3fc, both modes TE10 and TE20 propagate, and for 3fc < f < 4fc, three modes (TE10, TE20, TE30) are above cutoff.
A n-port device having m modes at each port is described by a generalized scattering matrix (GSM) of size nm × nm. In the case of a waveguide bend with three modes at each port, the GSM is of size 6 × 6. The modal scattering parameters versus frequency for fundamental mode excitation at port one (these six parameters give the first column of the GSM) are shown in Fig. 4.15, Fig. 4.16, and Fig. 4.17. The FDTLM results are obtained from a mesh of cubic nodes of size of 1/16 of the waveguide width. They coincide well with the reference results obtained by high-resolution FEM (HFSS). Additional reference data were taken from a mode-matching approach [4]. Because the structure is considered loss-free, the power balance can be checked by summing up the squares of the magnitudes of one column of the GSM. This test reveals an accuracy in the order of numerical accuracy (< 10–11).
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Fig. 4.15: Magnitudes of the scattering parameters of a rectangular waveguide H-plane bend. Excited is the fundamental mode at port 1. The figure shows reflection (r) into the first mode at port 1 and transmission (t) into the first mode at port 2. Solid lines: FDTLM, dashed lines: FEM (HFSS), markers: mode-matching approach [4]
Fig. 4.16: Magnitudes of the scattering parameters of a rectangular waveguide H-plane bend. Excited is the fundamental mode at port 1. The figure shows reflection (r) into the second mode at port 1 and transmission (t) into the second mode at port 2. Solid lines: FDTLM, dashed lines: FEM (HFSS)
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Fig. 4.17: Magnitudes of the scattering parameters of a rectangular waveguide H-plane bend. Excited is the fundamental mode at port 1. The figure shows reflection (r) into the third mode at port 1 and transmission (t) into the third mode at port 2. Solid lines: FDTLM, dashed lines: FEM (HFSS)
The transmission behavior of a waveguide bend can be improved over a narrow frequency range by simply introducing a protruding corner. A corner of size of 3a/8 (waveguide width a) is introduced as shown in Fig. 4.14 (right). The scattering parameter magnitude |S11| over the mono-mode frequency band, obtained with FDTLM (cubic nodes of size of a/16) and high-resolution FEM (HFSS), respectively, is shown in Fig. 4.18. The solutions are found to agree well.
The accuracy of the FDTLM solution increases with a finer mesh. The convergence order of the solutions was checked for different mesh resolutions by comparing the (complex) scattering parameter S11 with the FEM solution. Fig. 4.19 shows the convergence of the accuracy of the FDTLM solutions for the uncompensated waveguide bend and the matched bend, respectively, with respect to the FEM reference solution. The uncompensated bend was analyzed with the FDTLM method using five different meshes having, respectively, 4, 8, 12, 16, and 20 nodes over the waveguide width a. A convergence order can be calculated from each pair of solutions. The convergence order is found to vary somewhat for the different solution pairs, but a general frequency dependence can be observed. The matched bend was analyzed using two meshes only,
93
therefore, only one convergence order is obtained. This convergence order, too, shows a frequency-dependent behavior.
Fig. 4.18: Magnitude of reflection coefficient of a matched H-plane waveguide bend. Waveguide width a = 7.112mm. The –20dB bandwidth is about 10.2 GHz centered in the waveguide (Ka) band. Solid line: FDTLM (node size a/16), dashed line: FEM (HFSS)
In general, the convergence order of the matched bend is lower than that of the uncompensated bend. Furthermore, for a given FDTLM mesh resolution, the solution of the matched bend is less accurate than the solution of the uncompensated bend: For example, the error of the transmission zero frequency of the uncompensated bend is –0.40%, whereas the error of the reflection zero frequency of the matched bend is +1.60%, both for node size a/16. This behavior can be attributed to the additional field singularity introduced by the protruding corner, which reduces both the overall convergence rate and the accuracy of the solution.
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Fig. 4.19: Convergence order of the complex input reflection coefficient of the uncompensated waveguide bend and the matched bend, obtained by FDTLM with meshes of different resolution. The uncompensated bend solution corresponds to meshes of cubic cells with, respectively, 4, 8, 12, 16, and 20 nodes over the waveguide width a. The matched bend solution is based on meshes with resolutions of a/8 and a/16
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4.6 Resonance Frequencies of High-Permittivity Dielectric Resonators
The analysis of high-permittivity dielectric resonators is a challenge for many field solvers: the high dielectric contrast requires an accurate modeling of the material boundaries, field singularities may arise at dielectric wedges much like at metallic ones, and the free-space wavelength and thus the local variation of field magnitudes differs largely for different regions of the computational domain.
In the following, a brick-shaped, high-permittivity dielectric resonator in a perfectly conducting, air-filled cavity is analyzed with the FDTLM method and the results are compared to FEM (Ansoft HFSS 8.0). Fig. 4.20 shows the resonator (εrel = 80) of size 30 × 25 × 5mm3, centered in a cavity of size 50 × 50 × 15mm3. The dielectric material is representative for temperature-compensated ceramics based on Rutile (TiO2).
x y
z
Fig. 4.20: Dielectric brick resonator centered in an air-filled, metallic cavity. Symmetry planes cut the computational domain by eight. The dotted lines show a rough initial mesh with 18 FDTLM node cells.
The fundamental resonating mode of the dielectric resonator resembles the TM101 mode which would exist in a magnetic-wall rectangular waveguide cavity. Accordingly, the second, third, and fourth resonances of the brick resonator are similar to the TM201, TM102, and TM202 modes, respectively. The computational domain can be cut by 8 using three appropriate symmetry planes. Using high-resolution FEM, the resonance frequencies of the fundamental four modes are found to be 1930.6 MHz, 2203.3 MHz, 2327.5 MHz, and 2629.7 MHz, respectively.
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Accuracy versus Mesh Density for the FDTLM Solution A very rough mesh using 18 nodes only is depicted by the dotted lines in
Fig. 4.20. This mesh is governed by the geometrical dimensions of the structure and the fundamental rule to keep nodes as cubic as possible. The accuracy of a FDTLM analysis using this mesh is expected to be rather poor since the node cells are large and still strongly non-cubic.
In the following, this mesh is refined successively by dividing each node cell in, respectively, 8 ( = 23), 27 ( = 33), and 64 ( = 43) cells of equal shape. The accuracy of the FDTLM solution is compared to the 'true' resonance frequency of 1930.6 MHz. The results are shown in Tab. 4.5. The FDTLM solution converges with a convergence order of 1.87...1.94. The lower convergence order is found for rough meshes and higher for fine meshes. Note that this mesh refinement process is geometrical and non-adaptive and, therefore, sub-optimal (from a knowledge of the field solution one would favor a finer mesh in the dielectric, in particular near the dielectric boundaries, and a rough mesh in the air).
18 nodes 144 nodes 486 nodes 1152 nodes
–6.61 % –1.81 % –0.83 % –0.48 %
Tab. 4.5: Error in the fundamental mode resonance frequency as obtained with FDTLM using different meshes. A convergence order of approximately 1.90 is found
A comparison of accuracy and convergence between FDTLM and FEM can be made based on the number of unknowns in the eigenvalue problem. Hereby, one FDTLM node gives 12 unknowns and one tetrahedron in the FEM contributes about 6 unknowns (a tetrahedra of 0th order has 6 unknowns, whereas a 1st order tetrahedron has 20 unknowns [19] but the latter is rarely used in HFSS). Fig. 4.21 shows the error of the resonance frequency versus the number of unknowns of the eigenvalue problem. Note that the FEM solver makes an iterative, adaptive mesh refinement based on the field solution of the previous run. The number of tetrahedra added in each run was set to 20% and 100%, respectively, in Fig. 4.21. The starting mesh can be chosen to be based on geometry only (point A in Fig. 4.21). Alternatively, the mesh can be seeded denser in regions of high permittivity (point B in Fig. 4.21). The adaptiveness of the FEM solver together with the ability of the tetrahedra-based mesh to refine locally makes the FEM solution much better for fine meshes. However, FEM meshes which are not
97
yet adaptively refined reach about the same accuracy as FDTLM. It can also be seen from Fig. 4.21 that refinement in small steps (which involves many field solution runs) results in a better adapted mesh if compared to a case where a large number of tetrahedra is added at each refinement step, as one would expect.
Fig. 4.21: Fundamental mode resonance frequency error of the dielectric brick resonator for various meshes. The FEM solution uses adaptively refined meshes. The figure shows the respective use of large refinement steps (square marks) and small refinement steps (triangle marks). The FEM starting mesh was made according to the geometry (point A, 62 tetrahedra), or, alternatively, by seeding a finer mesh in regions of high permittivity (point B, 180 tetrahedra)
Application of the Eigenvalue Algorithm The search for a matrix singularity in equ. (3.3) and thus the calculation
of a resonance frequency is rather involved. The best procedure starts with a frequency sweep on a rough mesh, which detects all singularities with low accuracy. Then, the test of spurious solutions (see chapter 3.1) allows to cross out non-physical resonances. Finally, using a fine mesh, the accurate resonance frequency is found in the vicinity of the previously detected, physical, low-accuracy singularity.
The initial search on the rough mesh can be done by various numerical methods as they were described in chapter 3.5. For example, Fig. 4.22 shows the frequency dependence of the magnitude of the system matrix'
98
determinant (see equ. (3.3)) and the condition number estimate of the system matrix for the 144 node mesh. Singularities are detected near 1460 MHz and 1900 MHz. The test reveals that the former resonance is non-physical, whereas the latter resonance is physical.
Fig. 4.22: Resonance frequency search for the dielectric brick resonator (symmetry planes applied) for rough mesh (144 nodes). Both determinant and condition number indicate singular matrices for f ≈ 1460 MHz and f ≈ 1900 MHz
In a second step, the accurate resonance frequency will be found from the fine mesh (1152 nodes) in the neighborhood of 1900 MHz. Experience shows that a rough mesh underestimates the 'true' resonance frequency. The eigenvalue algorithm (chapter 3.1) allows to find an eigenvalue λmin=0 efficiently by linearly interpolating Im(λmin(f)) over frequency. Although this is a linear function only for a cubic mesh and a homogeneous computational domain, it is still an 'almost linear' function in the neighborhood of a resonance for a resonator with high dielectric contrast and non-cubic node cells. Fig. 4.23 shows λmin(f) in the frequency band of interest. The resonance frequency (Im(λmin(fres))=0) is 1921.472 MHz. Two possible iteration paths are presented in Tab. 4.6. After having solved the eigenvalue problem for only three different frequencies, the linear interpolation process gives the accurate solution within reasonable limits. Air-filled cavities or resonators employing low-permittivity dielectrics will show a much wider frequency band in which Im(λmin(f)) is a linear function.
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Fig. 4.23: Real and imaginary part of the smallest eigenvalue in the vicinity of the resonance frequency
first guess frequency f1 = 1900 MHz –1.12 %
second guess frequency f2 = 1930 MHz +0.44 %
interpolating f1 & f2 f3 = 1916.126 MHz –0.28 %
interpolating f2 & f3 fres = 1921.657 MHz +0.0096 %
first guess frequency f1 = 1910 MHz –0.60 %
second guess frequency f2 = 1920 MHz –0.08 %
interpolating f1 & f2 f3 = 1923.145 MHz +0.09 %
interpolating f2 & f3 fres = 1921.476 MHz +0.0002 %
Tab. 4.6: Two examples for iterating Im(λmin(f)) = 0 using linear interpolation. In both cases, the eigenvalue problem is solved at 3 frequencies. The third column shows the error with respect to the zero at 1921.472 MHz
Re(
λλ λλ min)
Im( λλ λλ
min)
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4.7 Microstrip Patch Antenna on Corrugated Substrate
A microstrip patch antenna made on a synthesized substrate is analyzed in order to show the ability of the proposed FDTLM algorithms to solve challenging problems. The antenna substrate shows periodic corrugations, which results in an artificially enhanced permittivity and a reduced patch size [18]. The structure has been analyzed using FEM software (HFSS) and was built and measured for means of comparison. The analysis of the structure is cumbersome due to the very large number of geometric details.
Principle of Operation Periodic corrugations on the metallized backside of a micrcostrip
dielectric substrate enhance permittivity, possibly even beyond the bulk material permittivity. This is achieved by corrugations whose depths and period are much smaller than the wavelength. By periodically altering the height of the substrate along a microstrip line, the effective capacitive and inductive loadings of the line are increased. This effect decreases the wave velocity and may change the effective line impedance. It should be noted that the currents (and thus the line, too) have to be directed perpendicular to the direction of the corrugations in order to obtain the maximum slowing effect (see Fig. 4.24).
strip
corrugated substrate
back–metallization
Fig. 4.24: 3D-view of a microstrip line on corrugated substrate
Applying this principle, rectangular, circular, and ring patch antennas of reduced size have been reported in [18]. A rectangular patch antenna will be used in the following as a benchmark for the FDTLM method.
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Patch Antenna Design The corrugated substrate used is polystyrene (εrel = 2.53, tan δ = 0.0005).
The period of the corrugations is 1mm, a section of thin dielectric (0.2mm thickness) of 0.5mm length is followed by a section of thick dielectric (1.2mm thickness) of 0.5mm length. The substrate size is 20mm × 25mm, containing 25 periods of corrugations. A square metallic patch of 16mm side length is centered on the substrate. The antenna is inset fed with a microstrip line quarterwave transformer. Fig. 4.25 shows the remaining dimensions.
magnetic wallsymmetry plane 4.5 9.9 7.85
corrugated substrate25mm x 20mm
square metal patch16mm x 16mm
2.0
0.35 1.
0
1.8
Fig. 4.25: Drawing of the inset fed square patch antenna on corrugated substrate (all dimensions in mm)
FEM Simulation The simulation of the antenna using the commercial FEM code HFSS
results in a resonance frequency of 3491 MHz (|S11| = –40dB) and an associated directivity of 4.6 dBi (gain of 3.7 dBi at σ = 30 × 106 S/m). The antenna shows an impedance bandwidth (|S11| < –10dB) of 1.37%. In the simulation, about 305'000 unknowns were used.
Measurements The antenna was built by milling and cutting a polystyrene (STYCAST)
block, which was then metallized using conductive silver paint. The antenna shape has been cut mechanically into the metallic paint. This method is simple and flexible, but it is of poor accuracy and has high conductive loss.
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Measurements of the antenna showed an input match of –7dB at 3620 MHz. The measured antenna gain was –1.3 dBi. The poor manufacturing accuracy results in a bad input match, and the low gain is due to the high conductor losses. Nevertheless, the effect of size reduction due to the corrugated substrate is clearly shown (a patch of similar physical size on the same substrate, but without corrugations, would resonate at 5.9 GHz).
FDTLM Simulation The meshes used for two different FDTLM simulations are depicted in
Fig. 4.26. For rough discretization, the cross-section of a substrate groove is
discretized with only 1×2 nodes. The complete mesh contains 4166 nodes. The microstrip port boundary (5mm × 5mm) contains 58 nodes. The simulation gave a resonance peak of –15.5 dB at 3215 MHz.
For fine discretization, the cross-section of a substrate groove is discretized with only 2×3 nodes. The complete mesh contains 12055 nodes. The microstrip port boundary (5mm × 5mm) contains 94 nodes. The simulation gave a resonance peak of –17.9 dB at 3373 MHz
The simulations used magnetic-wall symmetry, ZRT open boundaries, and loss-free materials.
Results and discussion The input reflection coefficient is shown in Fig. 4.27 (magnitude) and
Fig. 4.28 (Smith chart plot). The difference between the resonance frequencies obtained using the rough FDTLM simulation and FEM is about 8%. The difference between the resonance frequencies obtained using the fine FDTLM simulation and FEM is about 3.4%. The Smith chart of the input reflection coefficient shows similar behavior of all three simulations. The input match of this kind of patch antenna is very sensitive to an accurate modeling of the quarter-wave transformer in the feedline as well as the correct representation of the fields at the feedpoint inside the patch. Considering these difficulties together with the overall complexity of the problem, the results of both FDTLM and FEM are encouraging.
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port: 58 nodes
rough discretization: 4166 nodes
port: 94 nodes
fine discretization: 12055 nodes
Fig. 4.26: Discretization of the patch antenna on corrugated substrate, as used for the FDTLM simulation (top: rough mesh, bottom: fine mesh)
The comparison of numerical cost is based on single-frequency calculations, performed on a SUN ULTRASPARC II at 336 MHz. An iterative solver is used for the equation system of the FDTLM method (bicgstab without preconditioning). The cpu-time depends somewhat on the required accuracy (a relative residual limit of <10–5 was used and found to be enough even at the resonance peaks). Preconditioning would accelerate the solution process. LU preconditioners were tried for the rough mesh case. The LU preconditioners were calculated at one centered frequency and were found to be very efficient over a large (several percent) frequency band, resulting in a speed-up factor of 10...50. The LU decomposition for the fine mesh case was found to take too much memory. The iterative solver accepts a solution start vector. Taking the solution vector of a neighboring frequency as a start vector was found to have no effect unless the frequency step was as small as 1 MHz or so.
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Fig. 4.27: Simulation of the antenna input reflection coefficient (magnitude) using FDTLM and FEM, shown with measurements
Fig. 4.28: Smith chart plot of the antenna input reflection coefficient as obtained by FDTLM and FEM simulations
FEM (Hfss) FDTLM (fine) FDTLM (rough)
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The FEM simulation uses the largest number of unknowns (see Tab. 4.7) and benefits from an adaptive mesh refinement process (10 refinements in this case) prior to the final simulation. Therefore, the FEM simulation can be considered the most accurate. Although FDTLM uses a smaller number of unknowns and is limited by a rectilinear mesh without adaptive refinement, the results are still acceptable.
simulation resonance frequency
number of unknowns
memory cpu time per frequency
point
measurement 3620 MHz — — —
FDTLM, rough mesh
3215 MHz 50’000 330 MB 1500 sec
FDTLM, fine mesh
3374 MHz 145’000 1390 MB 7500 sec
FEM (HFSS) 3491 MHz 305’000 1450 MB 5200 sec
Tab. 4.7: Computer resources needed for the simulation of the patch antenna (processor Sun UltraSparc II at 336 MHz)
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5 General Conclusions and Outlook
Overall conclusions The objective of this thesis was to establish the FDTLM method as a
reliable tool for the analysis of real-world microwave circuit structures. To this end, various discretization schemes (nodes), namely the
Characteristic Admittance Node (CAN, [21], [23]), the Propagation Constant Node (PCN, [3], [24]), and the Finite Difference Node (FDN, [22]), were all derived from Maxwell's equations in a consistent way. Then, the numerical accuracy (dispersion) of these nodes was investigated for plane wave propagation in a periodic mesh, considering the effects of node cell size, non-cubic node cells, and lossy media. In some cases, analytical expressions for the dispersion error were derived. As a result, it can be said that the PCN is of superior accuracy compared to the CAN and the FDN. The dispersion level of the PCN is significantly lower than, e.g., that of FDFD and common FEM [37]. This can be attributed to the concept of link-lines incorporated in the PCN, which can make plane wave propagation dispersion-free (under some circumstances), regardless of the node cell size and even for lossy media. The number of unknowns attributed to a FDTLM node, however, is larger than the number of unknowns in a unit cell of FDFD or of most FEM schemes.
The incorporation of boundary conditions was briefly investigated. Dielectric and lossy metallic boundaries are easily incorporated into the FDTLM algorithm. The simple ZRT (Zero-Reflection Termination) open boundary [36], however, was found to perform well only for almost-normal plane-wave incidence. An analytical expression was found relating the numerical reflection coefficient of the ZRT boundary and the angle of incidence of a plane wave.
The second part of this thesis was devoted to the development of matrix algebra algorithms for the solution of typical problems arising during the analysis of microwave circuit structures. Specifically, algorithms are presented for the calculation of the resonance frequency of resonators, waveguide cutoff frequencies, eigensolutions of waveguides and transmission lines, and multi-port multi-mode scattering parameters of 3D structures. Particular attention was given to the numerical efficiency of these algorithms. For example, the numerical cost of the previously known
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waveguide eigensolution procedure [21] was reduced, and the procedures to calculate resonance frequencies and scattering parameters were made applicable to large, real-world problems by casting them, respectively, in the form of a standard eigenvalue problem and a linear equation system. The previously known algorithm for the calculation of the scattering parameters of a sliceable, aligned structure [21] was found to be a special case of the developed scattering parameter algorithm. The second part of the thesis concludes with some remarks about implementation aspects of the method, which are not new from a numerical algebra point of view, but those dealing with computational electromagnetics should nevertheless be aware of these aspects.
In the last part of this thesis the FDTLM method was applied to a number of benchmark problems. Some simple structures were used to analyze, respectively, the accuracy, the convergence order of the accuracy, and the influence of spurious solutions. Finally, a high-permittivity dielectric resonator and a microstrip patch antenna on corrugated substrate were investigated and the results were compared to high-resolution FEM results. The brick-shaped high-permittivity dielectric resonator is a structure to which the FDTLM method is particularly well suited, and the results are quite accurate for a reasonably rough mesh (e.g., 1000 nodes). The patch antenna example uses up to more than 12'000 nodes (approximately 150'000 unknowns) and shows the ability of the FDTLM method to analyze a complex dielectric structure which includes sensitive planar transmission line details (a quarter-wave microstrip transformer, which is a both phase and impedance sensitive structure, is accurately modeled).
Original contributions The main original contributions can be summarized as follows:
• Various FDTLM nodes are derived directly from Maxwell's equations, which has not been done before for the CAN and the PCN. A concise comparison of the numerical dispersion of the nodes includes lossy material and gives analytical expressions for several error bounds.
• The limitations of the ZRT open boundary for non-normal plane wave incidence are investigated and the numerical reflections of this matched boundary are partly cast into an analytical expression.
• Algorithms based on matrix algebra formulations are developed for various classes of microwave structures. The algorithm for resonance frequencies is formulated as a standard eigenvalue algorithm, which
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allows for the efficient solution of large problems. As a special class of resonance problems, waveguide cutoff frequencies can be calculated. The numerical cost of the previously known algorithm for the calculation of eigensolutions of waveguides was reduced in that less matrix inversions than before are needed and in that the eigenvalue problem will be cheaper since it has to be solved for a few smallest eigenvalues only. A multi-port multi-mode scattering parameter algorithm for 3D structures is developed, which is based on a single (or at most a few) solution of a linear equation system. As a result, very large problems can be analyzed with iterative equation system solvers. The extraction of scattering parameters from the solution vector of the linear equation system is described in a comprehensive way.
• Using a number of benchmark problems, the convergence of the accuracy of FDTLM solutions was found to be between first and second order, depending on whether the computational domain includes evanescent field regions due to field singularities or not. For example, air-filled, brick-shaped cavities are found to lead to second order accuracy exactly.
• Non-physical solutions are found both for resonance problems and waveguide eigensolutions. Using analytically calculated waveguide eigensolutions, ways are proposed to detect and separate physical and non-physical eigenmodes.
The algorithms presented in this thesis allow for an efficient analysis of microwave circuit elements. The benchmarks provide know-how regarding the specific applications of these algorithms. In conclusion, the FDTLM method can now be used with confidence to solve real-world problems.
Suggestions for further research Suggestions for further research are the following: • The modeling of evanescent fields at singularities such as edges and
wedges has first order accuracy convergence only. Thus, fine meshes have to be used in these areas in order to get high accuracy, which increases the number of unknowns of the overall problem drastically. It is therefore desirable to have special nodes which take the singularity of the fields at 90 degrees wedges and knife edges into account. The concepts proposed to this problem in the time-domain
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TLM method [44] are found to be not directly applicable to the frequency-domain TLM method.
• The calculation of a resonator resonance frequency is based on the search of the smallest eigenvalue of the related standard eigenvalue problem. Powerful algorithms for this class of problem are a matter of current research in the numerical algebra community and some of these algorithms are currently made available in libraries [2]. The iterative eigenvalue solver used in the run of this thesis (MATLAB 5.3 command eigs) is based on a LU factorization which is not suitable for large problems.
• Scattering parameters of a microwave circuit are often needed with high accuracy, but over a narrow frequency range only (consider, e.g., a multi-pole bandpass filter). Then, two features of the algorithm are important: First, a linear equation system has to be solved again and again, and the elements of the system matrix do not vary much from one frequency to the next. Second, only very few elements of the solution vector correspond to port values and are actually of interest for the extraction of scattering parameters. Assuming 'nice' behavior of the system matrix, it follows from the first point that the space which is spanned by the eigenvectors of the system matrix will not change much over the frequency band of interest. Consequently, a procedure which generates a 'good' approximation of the eigenspace of the system matrix at some frequency will be useful for the approximation of the behavior of the system over the entire frequency range. The second point further weakens the requirements for this approximate subspace since there are actually rather few dimensions of the eigenspace which are of interest and, therefore, have to be modeled accurately by the subspace. Practically, an appropriate subspace has to be generated at some expansion frequency. This comes at a numerical cost which is higher than that of a single frequency solution. Afterwards, the subspace is a 'good' approximation of the system in a frequency band in the vicinity of the expansion frequency. Since the subspace is of much lower dimension than the original system matrix, the solutions at all other frequency points in the band of interest can be calculated very quickly. Numerical algorithms for the generation of subspaces are known for quite some time [30] and are used for time-domain model generation since the 1990's [15], [7]. They were introduced for frequency-domain electromagnetics in the FEM software HFSS [43]. The underlying numerical methods are a topic of current research in the field of
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numerical algebra and unfortunately, they are still susceptible to convergence instabilities and numerical breakdowns. However, the potential speed-up of the solution process is so astonishing that only an implementation of these so-called model-order reduction techniques will keep any electromagnetics software, and so FDTLM too, competitive.
• One principal advantage of the FDTLM method compared to other finite difference methods is the link-line concept, which greatly reduces the dispersion. On the other hand, the FEM method benefits mainly from its tetrahedron-based mesh, which allows for geometrical flexibility, easy local refinements, and dispersion cancellation in unstructured grids [37]. The ultimate solution method would combine the advantages of both approaches, that is, this method should be based on tetrahedron-shaped FDTLM nodes, with 8 link lines and scattering in the center of the tetrahedron. To the knowledge of the author, no successful attempts in this direction have been published yet.
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[40] Y. Saad, Iterative methods for sparse linear systems, PWS Publ., Boston, 1996.
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Publications of the Author Related to this Thesis
[I] R. Vahldieck, D. Pasalic, and J. Hesselbarth, "S-parameter calculation of metal and dielectric discontinuities in guided wave structures using the FDTLM method," Proc. Second Int. Workshop on Transmission Line Matrix (TLM) Modeling, October 1997, Munich, Germany, pp. 217-227.
[II] J. Hesselbarth, R. Vahldieck, ”Mesh grading and cutoff frequencies in the frequency-domain TLM method,” Proc. IEEE MTT Symp., 1998, pp. 1551–1554.
[III] D. Pasalic, R. Vahldieck, J. Hesselbarth, ”The frequency-domain TLM method with absorbing boundary conditions,” Proc. IEEE MTT Symp., 1999, pp. 1669–1672.
[IV] D. Pasalic, R. Vahldieck, J. Hesselbarth, and J. Bornemann, "Zero-reflection absorbing boundary conditions in the frequency-domain TLM method and their application to planar circuit analysis," Proc. Third Int. Workshop on Transmission Line Matrix (TLM) Modeling Theory and Applications, October 1999, Nice, France, pp. 85-92.
[V] J. Hesselbarth, R. Vahldieck, ”Microstrip patch antennas on corrugated substrates,” Proc. 30th European Microwave Conf., 2000, pp. 282–285.
[VI] J. Hesselbarth, R. Vahldieck, ”Accuracy of the frequency-domain TLM method and its application to microwave circuits,” Int. J. Numerical Modelling: Electron. Networks, Devices and Fields, revised.
[VII] J. Hesselbarth, R. Vahldieck, ”Multi-port, multi-mode scattering parameter extraction with a numerically efficient frequency-domain TLM algorithm,” Proc. ACES 18th Ann. Rev. Progress Appl. Comp. Electromagnetics, 2002.
[VIII] J. Hesselbarth, R. Vahldieck, ”Resonance frequencies calculated efficiently with the frequency-domain TLM method,” IEEE Microw. Wireless Comp. Lett., submitted.
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Curriculum Vitae
Born on 13th April 1970 in Dresden, Germany
1976 – 1988 Primary- and Secondary school in Dresden
1988 Abitur from ”Martin-Andersen-Nexö”-Erweiterte Oberschule in Dresden
1988 – 1989 Military service
1989 – 1995 Studies in Telecommunications at the TU Dresden
1995 Graduation (Diplom-Ingenieur (TU) Elektrotechnik)
1995 – 1996 Research assistant with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, B.C., Canada
1996 – 1997 Design engineer with Daimler-Benz Aerospace Sensor Systems (now: EADS), Ulm, Germany
1997 – 2001 Assistant and Ph.D. student at the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland
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