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Time domain finite element analysis of microwave planar networks

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TIME DOMAIN FINITE ELEMENT ANALYSIS
OF MICROWAVE PLANAR NETWORKS
by
Jonathan O. Y. Lo, BA. Sc.
A thesis submitted in conformity w ith the requirem ents
for the Degree of Master of Applied Science,
Graduate D epartm ent of Electrical Engineering, in the
University of Toronto
© Copyright by Jonathan O. Y. Lo 1992
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TIME DOMAIN FINITE ELEMENT ANAL YSIS
OF MICROWAVE PLANAR NETWORKS
by
Jonathan O.Y. Lo 1992
A thesis submitted in conformity with the requirments for the
Degree of M aster of Applied Science, G raduate Department of
Electrical Engineering, in the University of Toronto
ABSTRACT
The point-matched time domain finite elem ent method is presented for
analyzing microwave planar circuits. The method employs the leap-frog scheme
to numerically simulate the wave propagation inside the circuit by solving a
boundary value problem a t each time step starring from £=0. The main feature
of this technique is th at the spatial domain is discretized with first-order finite
elements while the tim e domain is discretized w ith finite differences. A userfriendly CAD package called MICSim Version 1.0 is w ritten for the required finite
element modeling and analysis. Applications from microwave integrated circuits
are described and practical problems are solved.
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u
ACKNOWLEDGEMENTS
I would like to express my gratitude to my supervisor, Dr. Adalbert
Konrad, for his skilful guidance, continuing encouragement, extraordinary
patience, and generous assistance given to me during the course of this project.
I m ust also thank him for giving me the opportunities to present this work in
numerous conferences. Financial support from the departm ent is gratefully
appreciated.
I would also like to thank Dr. Lavers for introducing Dr. Konrad and the
opportunity to work in the lab as research assistant over past summers.
Special thanks to Dr. Dmitrevsky for his critical reading of the manuscript
and th e invaluable comments th a t he provided.
I am further indebted to Mr. Qiushi Chen and Mr. Sing Lee for their
numerous discussions, good advices, and unselfish assistance.
Finally, my thanks go to my family and friends for their understanding and
patience during the completion of this thesis.
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Ill
TABLE OF CONTENTS
CHAPTER 1
1.1
1.2
INTRODUCTION
Research Objectives
Thesis Summary
CHAPTER 2
MATHEMATICAL MODELING AND EXISTING
ANALYSIS TECHNIQUES
2.1 P lanar Circuit Model
2.2 Basic Equations
2.3 Existing Analysis Techniques
2.3.1
Modal Expansion Approach
2.3.2
Segmentation and Desegmentation Method
2.3.3
Contour-Integral Approach
2.3.4
Silvester’s Method
2.3.5
Time Domain Finite Difference Method
CHAPTER 3
3.1
3.2
3.3
3.3.1
3.3.2
3.4
3.4.1
3.4.2
3.5
3.6
METHOD OF ANALYSIS
Introduction
Governing Equations
Point-Matched Time Domain Finite Element Method
Finite Elements in Spatial Domain
Finite Differences in Time Domain
Boundary Conditions
Between the Coupling Ports
At the Coupling Ports
Stability and Convergence Criterion
Excitations
CHAPTER 4
COMPUTER IMPLEMENTATION OF THE METHOD
4.1 General
4.2 Computer Package Description
4.2.1
Grid Generation
4.2.2
Problem Definition
4.2.3
Analysis
4.2.4
Post-Processing
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1
1
3
4
4
6
7
8
8
9
10
10
12
12
12
14
14
18
19
19
20
23
25
27
27
28
28
29
31
32
iv
CHAPTER 5
5.1
5.2
5.3
5.3.1
5.3.2
5.4
5.4.1
5.4.2
5.4.3
5.4.4
5.4.5
5.5
Introduction
Fringing Field Considerations
Analysis of Stripline Circuits
Step in Width
Two-Port Rectangular Circuit
Analysis of Microstrip Discontinuities
Rectangular Stub Discontinuities
Microstrip Low-Pass F ilter
A nnular Ring
45° Bend Discontinuity
Microstrip Branch Line Coupler
Lim itation of the Proposed Method
CHAPTER 6
6.1
6.2
6.3
6.4
APPLICATION TO STRIPLINE CIRCUITS AND
MICROSTRIP DISCONTINUITIES
ANALYSIS OF LOSSY DISPERSIVE PLANAR
NETWORKS
Introduction
General Equations
Analysis of a lossy Meander Line
Time Response of a Dispersive Microstrip Line
CHAPTER 7
CONCLUSIONS
REFERENCES
APPENDIX
33
33
33
34
34
38
39
39
42
44
46
49
52
56
56
57
59
61
63
65
COMPUTER PROGRAM LISTINGS
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69
1
CHAPTER 1
INTRODUCTION
1.1
Research Objectives
In microwave integrated circuits, a discontinuity arises whenever there
is an abrupt variation from a straight and homogeneous line structure. This
nan be in the form of a bend, a step in width, a T-junction, or even a change
in the dielectric constant [2-4].
Though structurally simple, these discontinuities are the basic building
blocks of many stripline and microstrip circuits such as filters, branch line
couplers and even patch antennas.
Hence the accurate analysis and
characterization of these discontinuities rem ain an im portant and ongoing
issue in microwave integrated circuit design [1,2].
Conventional analyses use one-dimensional transm ission line models
an d lumped elements [2-4]. However, these approaches are based on quasi­
static approximations and therefore their results are accurate for relatively low
frequencies only.
A b etter approach would be to simulate the wave
propagation and scattering phenomena so th a t the results are also valid a t
higher frequencies. This idea is reflected in the recent development of tim e
domain finite difference methods [5-8]. The advantage of these methods over
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frequency domain methods such as the spectral-domain approach [9] and
moment methods [10,11] is th a t th e solution scheme is explicit and does not
require th e storage and inversion of a matrix. Hence, the method is feasible
for solving a large number of unknowns. In addition, in tim e domain analysis
both tim e domain and frequency domain information are readily available.
However, due to the nature of finite differences, the implementation of these
methods to circuits of arbitrary shape is difficult.
In 1987 Cangellaris et al. presented a point-matched tim e domain finite
elem ent method for scattering analysis [12]. The unique feature of this
technique is th at the spatial domain is discretized w ith finite elements but the
tim e domain is discretized w ith finite differences.
The objective of this research study is to investigate th e feasibility and
effectiveness of this point-matched time domain finite elem ent method as a
num erical tool to analyze microwave planar networks using the planar circuit
model as proposed by Okoshi and Miyoshi [16].
This method has the
advantage of the tim e domain analysis and yet it is more versatile than the
finite difference method due to the nature of the finite element formulation.
It is the purpose of this research to apply this technique to a variety of passive
networks, to allow the fast analysis, and to a lesser extent, the synthesis of
practical microwave integrated circuits.
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t
I
3
1.2
Thesis Summary
The results of the work reported in this thesis include the development
of a user-friendly CAD package called MICSim Version 1.0 for th e required
finite elem ent modeling and analysis using the proposed method.
Also
included is a post-processor which allows the user to display the spatial
waveforms a t various time steps using a colour-coding scheme. The package
is discussed in detail in chapter 4.
C hapter 2 describes the p l a n a r circuit model and briefly reviews some
of the available analysis techniques.
C hapter 3 presents the point-matched time domain finite element
method.
The governing equations, the numerical discretization, and the
boundary conditions are discussed. The necessary stability and convergence
criterion, as well as the excitations are also examined.
C hapter 5 contains and discusses the numerical results when the
method is applied to various configurations of stripline and m icrostrip circuits.
C hapter 6 extends the proposed technique to the analysis of lossy planar
networks by incorporating the conductor and dielectric losses of the circuit into
the planar circuit model.
C hapter 7 summarizes the thesis and presents some conclusions related
to this study.
The appendix provides the source listings of the com puter programs.
I
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4
CHAPTER 2
MATHEMATICAL MODELING
AND EXISTING ANALYSIS
TECHNIQUES
2.1
Planar Circuit Model
Electric circuits can be classified according to the size of the circuit
components compared to the wavelength. Everyone is familiar with zero­
dimensional circuits consisting of R, L, and C discrete circuit elements which
are small compared to the wavelength. Transm ission lines, on the other hand,
are one-dimensional circuits because the length of a transmission line is large
compared to the wavelength. A rectangular waveguide operating at microwave
frequencies is a good example of a three-dimensional circuit since all three
dimensions are large compared to the wavelength.
Planar circuits can be classified as two-dimensional circuits.
This
general concept was proposed by Okoshi and Miyoshi for electrical components
th at have two dimensions comparable to the wavelength and a third dimension
th at is much sm aller th an the wavelength [16].
From this view point, three types of p lan ar circuits are possible. They
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5
are the:
(i)
open type,
(ii)
triplate type, and
(iii)
cavity type,
as shown in Fig. 2.1 below.
Conductor
Ground P lane
Conductor
Ground Plane
G round Plane
(b]
(c)
Fig. 2.1 Three types of planar circuits.
(a) Open type, (b) Triplate type, (c) Cavity type.
The triplate (stripline) type planar circuit consists of a conductor strip
sandwiched between two ground conductor plates with a spacing of d from
each of them. The open (microstrip) type, in contrast, consists of a conductor
strip placed on top of a ground conductor plate with a spacing of d. The cavity
type planar circuit, on the other hand, is essentially a "thin" waveguide with
its length and width comparable to th e operating wavelength. As will be
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described in the next section, the analysis of planar circuits requires the
solution of the two-dimensional Helmholtz’s wave equation with appropriate
boundary conditions. Specifically, the homogeneous Neumann condition (zero
derivative) is used for the triplate and the open type, and the homogeneous
Dirichlet condition (zero amplitude) is used for the cavity type. The thesis will
be focused on the triplate and open type circuits, which presently play a major
role in microwave integrated circuits.
2.2
Basic Equation
One of the distinct advantages of the planar circuit model is th a t there
is no geometric restriction on the circuit. For the triplate and the open type,
the conductor strip can be arbitrarily shaped. An example is shown in Fig. 2.2
below.
Port 1
J Port 2
\ __/*
Fig. 2.2 An example of a planar circuit.
The space between the ground planets) and the conductor is filled with
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7
a dielectric m aterial of perm ittivity e and perm eability fi. Since the thickness
of this dielectric is small compared to the operating wavelength, it follows
directly from the Maxwell’s equations that the behaviour of the planar circuit
may be described in term s of a voltage wave function V(x,y,t), which must
satisfy the scalar Helmholtz equation [14-25]. In time-dependent form, it is
giver by
3*V
V • V V - fie
= 0
(1.1)
dt2
A planar circuit may have N ports a t which connections can be made.
The ports are assumed to be "narrow" enough so th a t the voltages are
relatively constant along th eir widths. The boundary conditions between the
ports (O are the homogeneous Neumann type since current flow m ust be
tangential. Dirichlet boundary conditions may specify a given source voltage
at the ports. Absorbing boundary conditions are used for ports with matched
loads because they allow th e incident wave to propagate through the ports
without reflections. A detailed description of these boundary conditions is
given in chapter 3.
2.3
Existing Analysis Techniques
The planar circuit model has been proved to be effective and practical
because it allows the analysis of microwave circuits of arbitrary shape within
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8
a relatively short computation time. It has resulted in a large number of
publications [14-25].
Some of these available methods are outlined and
discussed in the following sections.
2.3.1 Modal Expansion Approach
In 1972 Bianco and Ridella [19] presented a modal expansion approach
for the analysis of planar circuits hoseJ on the simplified version of the partialfraction representation of the general iV-port admittance m atrices. Basically,
it involves the modal expansion of the voltage field inside the circuit as a
infinite series of orthogonal functions. The impedance and the admittance
m atrices ave then w ritten in term s of these series. The shortcoming of this
approach is th at the analysis is restricted only to drcuits of rectangular shape.
As a result, the method has not been used extensively in planar circuit
analysis.
2.3.2 S eg m en t ■"'on a n d D eseg m en tation M ethod
W hen the circuit is made up of simple shapes such as rectangles, circles,
or rings, the segmentation method [4,21,22] can be used. The basic idea is to
divide th e circuit configuration into simpler "segments” th a t have regular
shapes in which Green’s functions are available. The overall performance of
the composite network can be then calculated by ’linking" th e characterization
OS or Z m atrices) of each of the segments. The desegmentation method [4,23],
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9
in contrast, is the complementary process of the segmentation method. In this
procedure, various segments are added to the circuit to result in a simpler
configuration to obtain the overrul frequency dependent param eters. The main
disadvantage of these methods is th a t they cannot be extended to the analysis
of planar circuits of arbitrary shape. These methods can only be applied when
the circuits are relatively simple in nature.
2.3.3 C o n to u r-In teg ral A p p ro ach
This is the method th a t was proposed by Okoshi and Miyoshi [16] for the
analysis o f triplate type p l a n a r circuits. The method is based on Green’s
theorem in cylindrical coordinates. R ather than solving for the fields of the
entire circuit, the wave equation is first converted to an line integral equation
along the circuit periphery by using Weber's solution for cylindrical waves.
The integral is then discretized as a m atrix equation and the equivalent circuit
param eters are then derived from th e solution of this m atrix equation. The
drawback of this technique is the inherent fact th a t the field solution inside
the circuit is not known. From the circuit designer's point of view, it is
advantageous to know the field p attern inside the circuit so th a t th e overall
performance can be optimized by accordingly altering the shape of the circuit
Hence, for CAD applications, an approach such as the finite elem ent method
[14,15,24], the finite difference m ethod [5-8], or the moment method [10,11] is
usually preferred.
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10
2.3.4 Silvester’s Method
The idea behind Silvester’s method [24] is based on finding the short|
circuit eigensolutions to Helmholtz’s equation and the bounded solution to
Laplace’s equation over the same region through a change of variables. This
is because for num erical solutions, homogeneous boundary conditions and an
inhomogeneous differential equation are usually preferred. These two field
problems are then solved with the finite element method, and the solution to
the original problem is constructed from these two solutions using the super­
position principle. An advantage of this method is that the resulting network
(RfWP1
adm ittance m atrices are in partial-fraction form. Thus, it is only necessary to
solve the fields once. There is no need for repeated calculations a t different
frequencies. However, the method has not been used extensively in planar
circuit analysis despite of this unique feature. In a recent study by Konrad et
al. [25], it was concluded that the method has a serious shortcoming because
it treats the planar network model as a bounded domain. In the formulation,
the ports are treated as the boundary of the network but in fact they are
connected to other networks. Hence, the method is rarely used today despite
of its early contribution.
2.3.5 Time Domain Finite Difference Method
A version of the tim e domain finite difference method was presented by
Gwarek [7,8] for th e analysis of planar circuits. In this method, the circuit is
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11
divided into mesh points separated by the distance h.
The Helmholtz’s
equation is discretized and solved numerically very much sim ilar to the well
known Yee’s algorithm [26] for electromagnetic radiation and scattering
calculations.
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12
CHAPTER 3
METHOD OF ANALYSIS
3.1
Introduction
In this chapter, the point-matched time domain finite element method
[12,13] is presented for analyzing microwave planar circuits. The method
employs the leap-frog scheme to numerically sim ulate th e wave propagation
inside the circuit by solving a boundary value problem at each time step
starting from £=0. The main feature of this technique is th at the spatial
domain is discretized w ith first-order finite elements while the tim e domain is
discretized w ith finite differences. Hence, the method combines both the
flexibility of the finite elem ent method and the simplicity of the finite
difference method.
Next, the iiiplem entation of the necessary boundary
conditions are discussed. The condition for stability and convergence is also
examined.
3.2
Governing Equations
The proposed m ethod of analysis is based on the leap-frog scheme where
th e finite element method is applied in space and the finite difference method
is applied in tim e [12,13]. To apply this method, we m ust solve the following
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13
pair of partial differential equations [7,8.14.15,17]
dJ(x,y,t)
V V(x,y,t) - -fid --------dt
(3.1)
e dVix,y,t)
---------------d
dt
(3.2)
V . eJ0c,y,f)
where d is th e substrate thickness and J is the surface current density vector
of the plane conductor which has both x and y components. By eliminating
it can be shown th at these equations are equivalent to the time-dependent
Helmholtz equations as described in the chapter 2.
It is interesting to note th a t in essence (3.1) and (3.2) are the twodimensional generalization of the well known one-dimensional transm ission
line equations [27,28]
dl(x,t)
avcx,t)
- T
dx
dl(x,t)
dx
dt
_ n
dV(x,t)
dt
where V and I denote the voltage and current functions, respectively, and L
and C denote the inductance and capacitance per unit length of the
transm ission line, respectively. Note the direct correspondence between L and
ftd and between C and e/d . La chapter 6, it will be shown th a t it is possible
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14
to include both conductor and dielectric losses through this one-dimensional
transm ission line analogy.
3.3
Point-Matched Time Domain Finite Element Method
3.3.1 Finite Elements in Spatial Domain
The first step in the point-matched finite element method is to discretize
the region of interest ixy plane) into a finite number of regions called elements.
Each element has several points called interpolation nodes. As a result, the
continuous functions V and J can be approximated by
M
V(x,y,t) = £ §S.x,y)Vi(t)
i=l
(3.5)
N
J(x,y,t) = £ \fpc,y)Jf.t)
7=1
(3.6)
where M and N are the number of nodes of the V and J finite element grids,
respectively, and <j>j(x,y) and \f£x,y) are th e basis functions which interpolate
the field w ithin each elementusing the values at the interpolation nodes.
Substitution of (3.5) and (3.6) into (3.1) and (3.2) yields
N
dJp)
1 M
£ VM,y) ------- ---------2 V W a t a W )
7=1
dt
fid t= l
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(3.7)
15
M
dVjd)
i= l
dt
£ 0«Ocry )
d
=
e
N
E v *yfcyyjft)
(3-8)
>1
The only unknowns in the above equations are the nodal values of th e
voltage VJj.) and the current density Jft) because
and y/x,y) are known
functions of position. In other words, the unknowns are
Vt(t),
i
JJLt) = [JjJ.tUjJdX0}T,
=
> 1 ,2 ,...^V
(3. 9)
(3.10)
This method is called the point-matched finite element method because
the basis functions 0j(x,y) and y/x,y) are defined to be
<bt(x,y) = 1
= 0
a t x=xit y=y,
a t other nodes, and
(3.11)
%bc,y) = 1
= 0
a t x=xjt y=y,
a t other nodes,
(3.12)
so that (3.7) and (3.8) are enforced a t each point. As a result, (3.7) and (3.8'
can be reduced to
d J /t)
---------
1
=
-
dt
dt
v
> 1,2,...^
(3.13)
fid i=1
dV tf)
---------
M
E
■
=
d
-
—
N
E v * v/Xi>yiWft),
£ j= i
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(3.14)
16
where for simplicity the notations
V ^ te y )
Xjy. = V^O^y,), and
V • \fpc,y)
(3.15)
(3.16)
have been used.
Although it is possible to use only a single mesh for both V and J so that
M = N, it is preferable to distribute the voltage and current density nodes such
th a t they form a pair of complementary grids as shown in Fig. 3.1.
•-------
Fig. 3.1
grid fo r V
—
gr i d f or J
A pair of complementary grids for V and J.
Note th at the voltage grid is slightly larger than the current density
grid. This means th at the circuit will be modeled with some errors. However,
the advantage of doing so is th at each voltage elem ent contains an
interpolation node for the current density and each current density element
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17
also contains an interpolation node for the voltage. As for th e rest of the
voltage nodes th a t are not contained in the current density elem ents, they are
governed by the associated boundary conditions. This complementary grid
formulation greatly simplifies computation and programming effort. As a
result, only the interpolation functions associated with the surrounding nodes
will contribute to the summation in (3.13) and (3.14). Hence, (3.13) and (3.14)
can be further reduced to
dJp)
1
K
----------£ V Q foyJV fr), j=l,2,...JST
fid i - i
dt
dVp)
dt
=
•
d
K
— E V ‘ YiC**ytWO»
e l-i
(3.17)
(3.18)
where K is the num ber of nodes of an elem ent (here K=4). As fa r as th e choice
of elements is concerned, notice th a t the gradient and divergence operators
require the interpolation functions to have continuous first derivatives inside
the elements. In the implementation, the linear isoparametric quadrilateral
element [29,30] is used. However, it should be noted th at th is freedom of
choice of elem ents is another advantage of th e finite element formulation. By
choosing the basis function, one can use a variety of fields w ithin the element,
in contrast w ith the finite difference method where only a linear one is
permitted.
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18
3.3.2 Finite Differences in Time Domain
The second step in the leap-frog scheme is to approximate the time
derivatives in (3.17) and (3.18) by finite differences with V and J discretized
in time in equal intervals to . U s in g the s im p lif ie d notation
V* = V(t=nto)
J*-m =
V2)to)
(3.19)
(3.20)
n=0,l,2,...,
the time derivative in (3.17) and (3.18) can be expressed as:
dJj
|
Jjn+V2 -
dt
| nto
(3.21)
tot
dVt |
V r l - V"
(3.22)
dt
I (n+lJ2)to
to
By substituting (3.21) and (3.22) into (3.17) and (3.18), respectively, the
following explicit formulae for finding Vi and J i are obtained:
jn + v z
=
to
jn .x n
M
vrl
=
v?
dto
K
£ v
i=i
K
- ----- £ V •
e
(3.23)
(3.24)
1=1
Hence, by knowing the initial conditions a t t=0, (3.23) and (3.24) can be
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19
used to sim ulate the wave propagation by alternately updating Ji and Vt a t
n=l,2,... . As can be seen, the 'leap-frogging" stems from th e fact th a t V and
J values a t each point are "chasing" each other a t half time step intervals [13].
3.4
Boundary Conditions
At each tim e step, the appropriate boundary conditions m ust be enforced
along the circuit periphery. For analyzing microwave stripline and microstrip
circuits, two cases are possible:
3.4.1 Between the Coupling Ports
The boundary conditions along the circuit periphery where there are no
external connections are of the homogeneous Neumann type since current flow
m ust be tangential. In other words, the relation
BV
= 0
(3.25)
dn
holds, where n denotes the outward normal to the periphery. In the finite
element method, the elements are distorted to model the circuit boundaries
and (3.25) can be satisfied by interpolating the voltage fields along the inward
normal directions within elements along th e boundary. This is illustrated in
Fig. 3.2.
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20
Fig. 3.2 Example of a Neumann boundary condition approximation.
At each tim e step, instead of using (3.23) and (3.24), the interior voltage
values are used to update the open boundary nodes of the finite element model.
3.4.2 At the Coupling Ports
At th e coupling ports of the microstrip circuit, the open circuit relation
in (3.25) in no longer valid. At these boundaries, the waves must be allowed
to propagate out of the ports and a t the same tim e the incident source voltage
waves m ust also be allowed to propagate into the ports. To achieve these
conditions, the so-called "absorbing boundary conditions" m ust be used.
Absorbing boundary conditions are also called radiation boundary
conditioris because originally they have been used in electromagnetic radiation
problems to minimize computational space. When they are applied, the
solution region can be truncated while still causing m in im a l reflections of an
outgoing wave. As a result, the distance from th e object of interest to the
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21
outer boundary is minimized, thereby reducing the num ber of unknowns and
increasing the com putational efficiency.
There is a large am ount of research devoted to this subject over the past
years [12,31-33]. However, not all of them are suitable to this application.
The "perfect" absorbing condition based on Huygen’s principle is expensive in
computer storage and tim e consuming because it involves the integration of
dyadic Green’s functions over the boundary [31]. Bayliss and Turkel’s method,
on the other hand, requires the use of an origir .r center in the computational
domain [32]. W hat is needed is a simple algorithm th a t involves only local
operations in both time and space so th a t it is consistent with the pointm atched finite element method. In the author’s implementation, the absorbing
boundary conditions by Taflove and Brodwin [33] are used.
The idea behind th e Taflove and Brodwin’s method is based on relating
th e field components a t points on the boundary to the points just inside th e
boundary a t the previous tim e step. For an uniform square grid w ith a mesh
size of h , this can be easily accomplished by choosing the tim e step relation to
be
uto = h,
(3.26)
where u is the velocity of propagation, M is the size of th e time step, and h is
th e element size. To account for all possible angles of incidence of the outgoing
wave a t the boundary, th e average of the nearest three interior points at one
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22
previous time step is applied to each point on the boundary. This is illustrated
in Fig. 3.3 below.
40
45
10
5
•
44
39
9
4
43
38
8
3
42
37
7
2
41
36
6
1
Fig. 3.3 Absorbing boundary conditions for V.
Thus, if points 1 to 5 are taken to be a port with a matched load, the
absorbing boundary conditions for points 2, 3, and 4 would be as follows:
V2n =
(V6"*1 + V7n*1 + V8b1)/3,
(3.27a)
V3n =
(V7n’1 + Vy1'1 + V9b1)/3,
(3.27b)
V4n =
iVan'1 + VJ1'1 + V10Rl)/3.
(3.27c)
The absorbing boundary conditions for the comer points 1 and 5 are
V,*
=
(Vg"1 + 2V7nl)/3,
(3.27d)
Vsm =
(V10nl +2V9» 1)/3.
(3.27e)
For a pert with source voltage (points 41 through 45 in Fig. 3.3), these
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23
absorbing boundary conditions m ust be modified because the reflected wave
created in the circuit m ust be allowed to propagate back through the port and
a t the same time the source voltage m ust be allowed to propagate into the
port. Otherwise, multiple reflections would be produced. The correct boundary
conditions for such a port are as follows:
1 + 2V37'll)/3 l}At) + f(nAt),
(3.28a)
V<* =
W
V<2" =
(V36n l + V3 / 1 + VMn l)/3- f ( [ n - l}Af) + f(nAt),
(3.28b)
V J1 =
t t V 1 + V ^ 1 + V3aril)/3-f({n-l}At)+f(nAt),
(3.28c)
V<S =
(V *-1 + VW*1 + V « rlV3- f a n - l W ) +f(nM),
(3.28d)
= ( V ^ 1 + 2V39n l)/3 - /«n-l}A f) + f(n A t\
(3J29e)
where f (t) is the continuous source voltage function.
3.5
Stability and Convergence Criterion
The two partial differential equations in (3.1) and (3.2) constitute a
hyperbolic system. For the case of the uniform square grid w ith elem ent size
of h, it has been proven [34,35] th a t the solution to the leap-frog scheme is
stable if th e relationship
uAt £ h
(3.30)
is satisfied everywhere inside the grid. In the above inequality, u denotes the
velocity of propagation and At is th e tim e step size. This also im plies th a t for
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24
accurate solutions, both the time increment and the spatial division have to be
sm all
For the case of a grid with distorted elements, the criterion is yet to be
proven. However, Cangellaris et al. pointed out th a t the stability criterion in
(3.30) can still be used provided th a t "each of its irregular elements is large
enough to contain a regular element of the grid" (12].
An added complication to this issue is the presence of absorbing
boundary conditions at the artificial boundary. A theoretical solution to this
stability problem is given by Gustafsson, Kreiss, and Sundstrom [36]. The
main points of th eir study are summarized as follows [12]:
An initial boundary value problem model is stable if and only if
(1)
the stability condition of (3.30) is satisfied everywhere inside the grid;
(2)
the model adm its no wave solutions th a t grow in magnitude, and
(3)
the model adm its no wave solutions th a t travel inward from the
artificial boundary to the interior of th e computation domain.
The first two conditions can be met by choosing the appropriate spatial
and time subdivisions as well as the voltage source functions at the source
excited ports. The third condition is really a statem ent of the importance of
the effectiveness of the absorbing boundary a t the artificial boundary. In other
words, the absorbing boundary conditions m ust be "absorbing” enough,
otherwise the numerical solution will be unstable.
For th e convergence criterion, Lax and Richtmyer [37] have shown th a t
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25
stability is the necessary and sufficient condition for convergence provided th a t
the equations are linear and consistent with a properly posed initial value
problem. Because the two partial differential equations in (3.1) and (3.2)
constitute a properly posed initial value problem and the leap-frog scheme is
a consistent difference approximation to this problem, the stability criterion in
(3.30) is the sufficient condition for convergence.
3.6
Excitations
The frequency response of the circuit can be obtained simply by applying
a sine wave at the excitation port until steady-state is observed a t the output
port. Another approach would be to used a G aussian pulse to exdte the
circuit, and from the transient response a t the output port to extract the
frequency response using the Fourier transform . This approach has the
advantage th at the results for a wide range of frequencies can be obtained
using one computation as demonstrated in [5,6]. However, because of the
dispersive nature of the finite difference approximation [34], a Gaussian pulse
of the form
g it) = exp U t - Q W
(3.31)
might change its form as it propagates if th e duration is too "short".
CangeQaris et. aL [12] have shown th a t the following conditions m ust be
satisfied
uT > 4h
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(3.32)
26
in order for the dispersive effect to be negligible.
For the sine wave, the spatial subdivision h and th e time increment /Si
should be chosen in such a way th at they are compatible w ith the wavelength
and th e period, respectively.
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27
CHAPTER 4
COMPUTER IMPLEMENTATION
OF THE METHOD
4.1
General
The analysis of planar circuits using the point-matched finite element
method follows these basic steps:
(i)
GRID GENERATION
The circuit has to be subdivided into linear quadrilateral elements.
Hence, th e nodal coordinates, elem ent composition, and th e relative
perm ittivity of each element has to be generated.
(ii)
PROBLEM DEFINITION
The appropriate boundary conditions and other m aterial properties of
the circuit m ust be assigned. The desired time increment also has to be
specified.
(iii)
ANALYSIS
The sim ulation is performed and th e assigned boundary conditions are
enforced according to the data specified in the above two steps.
(iv)
POST-PROCESSING
A fter th e problem is solved, the fields a t various tim e steps are
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28
displayed so as to allow the user to visualize the scattering phenomena and to
extract the desired information.
4.2
Computer Package Description
A CAD package called MICSim Version 1.0 consisting of four different
programs (GRID, PROB, SOLV, and POST) is w ritten for these purposes. The
package is user-friendly. Interactive commands are used in the programs.
Features like help menu, file directory, d ata retrieval, automatic data saving,
and two-dimensional field plotting using a digital colour-coding scheme are
available. The four programs are briefly described below.
4.2.1 Grid Generation
A simple b u t effective grid generator called GRID is w ritten for this
purpose. The grid generation is based on th e idea of "breaking" up the circuit
into various segments th a t are either quadrilaterals or circular arcs. The
circuit is then constructed by "joining" these segments with the JOIN
command. The only input required by th e u ser to generate these segments are
the coordinates of the four comers in a counter-clockwise fashion as well as the
number of divisions along the four edges. The segments are identified w ith a
6-character (or less) name, and the user is able to switch between various grid
models by using th e MODE command. As a result, a large complicated planar
network can be constructed in this fashion.
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29
The program uses a book-keeping file called MODELS .DIR to keep track
of the models. To build a segm ent, the MAKE command is first used to specify
the model name, and then th e QUAD or ARC command can be used to
construct the desired grid. Once the grid is constructed, th e voltage elements,
current density elements, as well as voltage boundary nodes can be displayed
w ith the DISP command Also available are the COPY command, which is
used for model duplication, the TRAN command, which is used for coordinate
translation, and the ROTA command, which is used for rotation around the
origin. A brief description of all of the available commands can be found by
using the HELP command
A grid model is stored as four separate d ata files. They are
(i)
modelname.NOD
-
Contains the num ber of nodes and
the coordinates for both grids.
(ii)
modelname.YAM.
-
Contains the element lists and the
complementary interpolation nodes for
each of the elements for both grids.
(iii)
modelname.BND
-
Contains the list of boundary nodes and
elements.
(iv)
modelname ~MTL
-
Contains the relative perm ittivity of each
current density elem ent.
4.2.2 Problem Definition
The problem definer code PROB is w ritten in a sim ilar fashion. Its
function is to allow the u se r to assign the desired boundary conditions for the
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30
grid models. To define a problem, a problem file is first created with the
MAKE command by specifying the nam e of the problem file as well as the
name of the grid model. The user can then specify the source excitations, port
term inations, and m aterial properties of the circuit. These information are
stored as two data files as follows:
(i)
filen a m e^B1
-
Contains the number of ports, the
corresponding node, and the port sources
and terminations.
(i)
filename.VB2
-
Contains the m aterial properties as
defined by the user.
A problem file can be saved w ith the SAVE command and retrieved with
the RETR command. Commands such as COPY, DELE, and DIRE are also
available.
The data file called FILES.DIR is used for booking-keeping
purposes. Before a problem can be solved, it m ust be selected by the user with
the SELE command so as to create the following data files:
(i)
(ii)
NEUMAN1.DAT
DIRICH 1J)AT
(iii) SOLV.DAT
-
Contains the nodes for open-circuited
boundary.
-
Contains the nodes for short-circuited
boundary.
-
Contains the name of the problem file to
be solved and the time increm ent as
specified by the user. I t also contains
inform ation about how the field solutions
are saved.
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31
4 J 2 J A n a ly s is
The program SoLV can be considered as the "core” of the package. Its
role is to sim ulate the wave propagation inside the circuit and to enforce the
assigned boundary conditions a t each tim e step. The field solutions are saved
according to th e param eters as stored in SOLV.DAT. This is a pure "number
crunching” program and is the only program in the package th at, does not
require any in p u t from the user. Before th e time iterations, the following data
files are computed by the program:
(i)
INTERV.DAT
-
Contains th e complementary interpolation
nodes in local coordinates for each of
the voltage elements.
(ii)
INTERJ.DAT
-
Contains the complementary interpolation
nodes in local coordinates for each of
the current density elements.
(iii) NEUMAN2.DAT
-
Contains the interpolation nodes for
Neumann boundary condition approximations.
(iv)
-
Contains the interpolation nodes for
absorbing boundary condition approximations.
ABSORB3.DAT
#
These d a ta files are read back by th e program during each time step.
The advantage of doing so is th at it makes debugging very simple. The output
files are
(i)
PORTVTN.OUT
-
(ii)
PORTVOUT.OUT -
Contains the input voltage waveforms at
the ports.
Contains th e output voltage waveforms a t
the ports.
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32
(in)
PORTVTOT.OUT -
Contains the total voltage waveforms a t
the ports.
(iv)
VXY.OUT
Contains the spatial voltage solution
V(x,y,t) a t various time steps.
(v)
JXXY.OUT
Contains the spatial current density
solution J z(x,y,t) at various time steps.
(vi)
JYXY.OUT
Contains the spatial current density
solution Jy(x,y,t) a t various time steps.
(vii)
POST.DAT
Contains the d ata needed for graphical
display.
4.2.4 Post-Processing
After the problem is solved, the program called POST can be used to
display the field solutions a t various tim e steps. The two-dimensional fields
are plotted according to a digital colour-coding scheme so as to provide the user
a visualization of th e wave scattering phenomena inside the circuit. Both the
voltage and current density solutions can be displayed with the DISP
command. The user also has the option of selecting the level of display
resolution with the OPTI command.
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33
CHAPTER 5
APPLICATION TO STRIPLINE
CIRCUITS AND MICROSTRIP
DISCONTINUITIES
5.1
Introduction
As previously stated, one of the advantages, of u s i n g the planar circuit
approach for analyzing microwave integrated circuits is th a t the model is
applicable to a wide variety of circuits w ith no geometrical restriction. In this
chapter, numerical results are presented when the point-matched time domain
finite element method is applied to various configurations of stripline and
microstrip circuits. Comparison is made w ith other analysis techniques when
possible. The limitations of th e method are also discussed.
5.2
Fringing Field Considerations
In the planar circuit model, it is assumed th a t the current flow is
tangential along the circuit periphery. In other words, the circuit periphery
constitutes a perfect "magnetic wall". In reality, however, a f r i n g in g field is
always present. For stripline circuits, vhis f r in g in g effect can be accounted for
by shifting th e magnetic wall by a certain distance 5 from the physical
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34
boundary according to the formula [4,16]
8 = 2d(ln2)/jr,
(5 . 1)
in order to sim ulate the static f r in g in g capacitance.
For microstrip circuits, a more complicated model is needed. This is
because th e propagating Held is not purely TEM but rather quasi-TEM due to
the open structure.
In such case, the perm ittivities and widths of the
microstrip lines can be adjusted according to the Wheeler, Schneider and
Hammerstad formulae [38]. The author finds th at this model is adequate for
most applications. However, a more accurate representation based on the
planar waveguide model of the microstrip line [39] can be used in the case
where the frequency dependence of the effective widths and effective
perm ittivities m ust be accounted for.
5.3
Analysis of Stripline Circuits
5.3.1 Step in Width
The method is first applied to a stripline step junction \.ith dimensions
as shown in Fig. 5.1.
Port 2
W,
Wt=4 d2 mm, W2=6.80 mm, L x-L 2=17.375 mm,
Ep—2.5, d=3.5 mm.
F'g. 5.1 A ■’tripline step junction.
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35
0.8
?w
&
n
o
0.4
e
£
03
-03
-0.4
0.05
0.1
0.15
03
035
0.3
0.35
0.4
0.45
0.5
Fig. 5.2(a) Time response of port 1 voltage
of th e step junction in Fig. 5.1.
0.8
>
s&
r*
c
£
^
0.4
0.2
-
0.2
-0.4
0.05
0.1
0.15
0.2
0.25
03
0.35
0.4
0.45
05
Tune(ns)
Fig. 5.2(b) Time response of port 2 voltage
of th e step junction in Fig. 5.1.
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<
6
3
0.
S
e
£
0.05
0.1
0.15
0.2
0.25
0.3
0J5
0.4
0.45
0.5
Fig. 5.2(c) Time response of port 1 input current
of the step junction in Fig. 5.1.
<
E
e
g
a
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0J
Tiine(ns)
Fig. 5.2(d) lim e response of port 2 output current
of th e step junction in rig. 5.1.
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37
Fig. 5.2 (a) and (b) show the time response of the port voltages under the
excitation of a Gaussian pulse of T = 20 ps and ta = 0.06 ns. The fringing fields
are included in the analysis by extending Wx and W2 to ^ ’=5.56 mm and
W2’=8.34mm according to (5.1). The stripline is modeled w ith a finite element
grid of mesh size h = 0.695 mm, and th e simulation is performed with a time
increment of A/=3.663 ps. The results show excellent agreem ent w ith the
expected reflection coefficient of -0.2 and transm ission coefficient of 0.8 from
transmission line theory [27,28].
Fig. 5.2 (c) and (d) show the tim e response of port 1 input current and
port 2 output current, respectively. Again, very good agreem ent has been
found w ith the reflection coefficient of 0.2 and the transm ission coefficient of
1.2 from transm ission line theory [27,28].
In reality, the stripline will have losses and the phase velocity will then
vary w ith frequency. As a result, the exciting pulse will lose its shape during
propagation.
However, since the circuit is assumed to be lossless, such
dispersive effects are absent. Losses can be accounted for by introducing
m aterial param eters such as the conductivities of the dielectric and the
conductor into the planar circuit model so th a t these frequency dependent
characteristics are incorporated in an approximate m a n n e r . This formulation
is discussed in detail in chapter 6.
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38
5.3.2 Two-Port Rectangular Circuit
The next example considered is a simple two-port rectangular stripline
circuit. The dimensions of the circuit are shown in Fig. 5.3 below.
1
w
T
a=54.8mm, b=46.5 mm, w=5.2 m m , ^=2.5, 2d=6.94
mm
Fig. 5.3 A rectangular stripline circuit.
40
S21 dB
&
C tr r e n t A iily d *
- B - G w irek (FDTD)
E x p tr m tit
-20
-4 0
-6 0
1
1.5
2
2.5
Frequency (GHz]
3
4
Fig. 5.4 Transmission magnitude of a rectangular stripline circuit.
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39
For planar circuit modeling, the dimensions are adjusted to a’=57.9 mm,
b’=49.6 mm, and w’=8.27 mm. In the analysis, a grid size of 2.757 mm and a
time increment of 14.5 ps are used. Fig. 5.4 shows the computed transm ission
magnitude in the frequency range from 1 to 4 GHz by exciting the circuit w ith
sine waves of various frequencies.
Also shown in the figure are the
experimental values and numerical results obtained by Gwarek using; j. tim e
domain finite difference approach with th e same mesh size [8]. On average,
the results are within a 5% range of each other.
5.4
Analysis of Microstrip Discontinuities
5.4.1 Rectangular Stab Discontinuities
Open circuit stubs are often used as impedance matching networks in
high frequency microwave integrated circuits. The need for such matching
networks arises because in microwave am plifier design, both the input port
and the output port m ust be properly term inated in order to deliver m a x im u m
power to the load. For broad-band applications, low impedance stubs are
required. When realized as single stubs, the w idths are often large enough to
cause possible excitation of unwanted higher modes.
To overcome th is
lim itation, the stubs are sometimes realized as double stubs connected in
parallel on both sides of th e main line. The impedance of each of the two stubs
is equal to twice the impedance of the original stub [4]. Both structures are
studied in this section.
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40
F.'g. 5.5(a) and (b) show the dimensions of a single and double open stub
discontinuity, respectively.
0.51 mm
1
1.75 mm
1
*
T
0.23 mm
Fig. 5.5(a) A single stub discontinuity.
0.51 mm
7. B5 rrrn
i
T
0 . 2 3 mm
Fig. 5.5(b) A double stub discontinuity.
Both circuits have a dielectric constant of 9.9 w ith a substrate thickness
of 0.254 mm. In the finite elem ent models, the single stub is modeled with
square elements of size /i =0.046 mm, and the double stub structure is modeled
with A=0.02875 mm. Figures 5.6(a) and (b) show the transm ission magnitude
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as a function of frequency in comparison with the calculated and measured
results from Giannini et. al. [40]. As can be seen in the figures, the pointmatched finite element method is able to predict the field behaviour within a
5% error range at lower frequencies on average. The large discrepancies at
high frequencies seem to suggest th e need for a grid w ith finer spatial
subdivisions.
S21 (00)
*
C m it t a l f u
-a- sai
B tn n n t
-10
-20
-30
6
10
10
14
Frtqotncy (GHz)
22
20
Fig. 5.6(a) Insertion loss of the single stub.
-2 0
-30
Fig. 5.6(b) Insertion loss of the double stub.
i
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42
5.4.2 Microstrip Low-Pass Filter
The next example considered is a m icrostrip low-pass filter which has
been measured and analyzed by Lee [41] using a three-dimensional tangential
vector finite element method. The circuit is built on a substrate w ith ^= 2.2
and has a thickness of 0.794 mm. Its dimensions are shown in Fig. 5.7 below.
2.34 rnn
Port 1
20.32 n n
1
Port 2 2.413 n n
3.83 n n
Fig. 5.7 A planar microstrip low-pass filter.
The time step used is A£=1.256 ps. For simplicity, only square elements
w ith /i=0.254 mm are used in the sim ulation. As a result, the locations and
the widths of the ports are modeled with some errors but the dimensions of the
rectangular patch are modeled exactly.
Fig. 5.8(a) show the calculated return and insertion losses in comparison
w ith the m easured and analytical values from Lee [41] in the range from 0 to
20 GHz. The results are well within a 5% range of each other. In particular,
the point matched finite element method is able to predict the transm ission-
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43
zero at 7 GHz. The discrepancies at high frequencies are most probably due
to radiation, which is neglected in the planar circuit model.
Sli (dB)
-10
-2 0
-30
-40
#
-30
-60
C irrM l A K |» *
Expanaaat
0
3
20
13
10
Frtqitncy (GHz]
Fig. 5.8(a) R eturn loss of the low-pass filter in Fig. 5.7.
S21 (dB)
*
*
-1 0
-2 0
-30
-40
V
-50
-60
C ir iM t M a p t a
-B - U »
0
5
to
13
20
Frequency (GHz)
Fig. 5.8(b) Insertion loss of the low-pass filter in fig. 5.7.
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44
5.4.3 Annular Ring
One of the disadvantages of the finite difference method is th at a stairstepped. approximation m ust be used to model a curved boundary. It has been
shown th a t the errors due to th is approximation may contaminate the held
calculation significantly if the grid is not fine enough [42]. The finite element
method, however, overcomes th e lim itation by allowing th e elements to be
distorted to model the boundary. This point is illustrated in the following
example.
Fig. 5.9(a) shows the dimensions of a microstrip annular ring. This
structure has the interesting characteristic th at th e filtering properties can be
controlled by varying the ring w idth and the locations of the two coupling ports
as dem onstrated by DTnzeo et al. using a modal expansion approach adapted
for two-port circular ring geometry [43]. For finite elem ent calculations, the
widths have been modified according to the Wheeler, Schneider, and
H am m erstad formulae [38]. Note th at by taking advantage of the symmetry,
only h a lf of the problem needs to be solved. Fig. 5.9(b) shows the grid that is
used for the calculations.
rl = 4.0 mm
r2 : 4.6 mm
W : 0.6 mm
W
Fig. 5.9 (a) A 2-port microstrip annular ring.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 5.9(b) A finite elem ent grid for the problem
as shown in Fig. 5.9(a).
Fig. 5.10(a) and (b) show the calculated frequency responses along with
the experimental and computed d ata from Dlnzeo et al. [43]. By comparing
the results, one finds th a t there is a slight shift in dip frequencies of about 6%
in the return loss of the finite element calculation. This is probably due to the
fact that in the modal expansion analysis of Dlnzeo, effective param eters are
employed a t each frequency, whereas in the current analysis, the effective
param eters are computed using the Wheeler, Schneider and Hammerstad
formulae a t only one frequency.
5an m i
-s
-10
2
1
4
S
S
7
t
•
C w ifft I M l,M i
a
okm
8
10
tt
12
R a m c r WltaJ
Fig. 5.10(a) Insertion loss of th e a n n u l a r ring problem.
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46
10
-5 '
-1 0
-1 5
-20
2
3
4
5
6 7
B
Preqieicy (GHz)
9
10
11
12
Fig. 5.10(b) Return loss of the annular ring problem.
5.4.4 45° Bend Discontinuity
The next example considered is a 45° microstrip bend with dimensions
as shown in Fig. 5.11(a). The bend has been analyzed by Lee et al. [44] with
a non-orthogonal FDTD algorithm and verified with simulation results from
the commercial CAD package TOUCHSTONE. A possible finite elem ent grid
model is shown in Fig. 5.11(b).
2.4 mm
45
Fig. 5.11(a) A 45° microstrip bend discontinuity.
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47
Fig. 5.11(b) A finite elem ent grid for the 45° bencL
The calculations using this grid, however, are found to be unstable. A
close examination n? the solutions reveals th a t non-physical reflections are
created a t the junction. To correct this situation, the grid is modified to the
one as shown in Fig. 5.11(c), where the abrupt edge is "smoothed out" by a
section of a circular arc with an inner radius of 2.4 mm and an outer radius
of 4.75 mm. The resulting calculated retu rn loss along w ith the results from
Lee et at. [44] Is shown in Fig. 5.12. As expected, the finite elem ent analysis
gives the right order of magnitude with the modified grid b ut fails to give the
overall behaviour of the bend. The lim itation imposed by th is instability
problem is examined in detail in section 5.5.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 5.11(c) A modified finite elem ent grid for the 45° bend.
-1 5
-20
-2 5
-3 0
#
-3 5
C irra n t A u i y s s
□ Im t! aL
-4 0
TOUCHSTONE
-4 5
-5 0
-5 5 -6 0
0
2
F requency (GHz)
Fig. 5.12 Return loss of th e 45° bend using the
modified grid shown in Fig. 5.11(c).
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49
5.4.5 Microstrip Branch Line Coupler
The branch line coupler is another commonly used device in microwave
integrated circuits. An example of a branch line coupler is shown in Fig. 5.13
below.
9.75 mm
Port 1
Port 4
2.413 mm
9.75 mm
3.96 mm
Port 2
Port 3
Fig. 5.13 A example of a branch line coupler.
The function of th e branch line coupler is to provide equal power
distribution at th e output ports (3 and 4) when excited a t either one of the
input ports (1 or 2). This occurs when the center-to-center distance (9.75 mm)
between the four main lines is a quarter of the operating wavelength. In
addition, the branch line coupler provides isolation a t the adjacent port a t the
same frequency. To model this circuit, a finite elem ent mesh of A=0.295 mm
is used. As before, the dimensions are adjusted to th e W heeler, Schneider, and
Hammerstad formulae. The computed param eters are shown in Fig. 5.14(a)
through (c) in comparison w ith experimental results and calculated results
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50
from Sheen et al. obtained using a three-dimensional finite difference time
domain method (FDTD).
S11 dB
-10
-20
-30
X
C u r r e it Am lysis
- B - 30 POTD
E x p s r m s it
-50
♦
0
2
3
5
6
Freqiency (GHz)
7
8
9
10
Fig. 5.14(a) Return loss | S n | of a branch line coupler.
S21 dB
-10
,* *
-20
-30
it
c u r r a s t Am lysis
- B - 30 F0TD
E x p s r m s it
-5 0
0
1
2
3
S
6
Freqiency (GHz)
7
a
9
10
Fig. 5.14(b) Isolation | S 2l | of a branch line coupler.
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51
531 dB
-10
-20
-30
*
C u rre it Am lysis
- B - 3D rDTD
E x p s rm s it
-5 0
0
1
2
3
4
5
6
Freqiency (GHz)
7
B
9
10
Fig. 5.14(c) Coupling | S 31 | of a branch line coupler.
S41 dB
-10
-20
30
curreit Antlysis
-B- 30 FDTD
-4 0
E x p s r m s it
-50
0
1
2
3
4
S 6
Freqiency (GHz)
7
a
9
10
Fig. 5. 14(d) Coupling \S 4l\ of a branch line coupler.
I
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52
Overall, the results show a less than 5% error except for the return loss
calculation around the dip frequency of 6.5 GHz. This is likely due to the
exclusion of electromagnetic coupling between the four lines in the analysis.
5.5
Limitation of the Proposed Method
In this chapter, various stripline and microstrip circuits are analyzed
with the point-matched time domain finite element method. The method gives
reasonably accurate results when the finite element grids are constructed with
rectangular elements and/or with elements th a t are distorted in a conformal
way, such as the ones shown in Figures 5.9(b) or 5.11(c). However, when this
is not the case, such as the one shown in Fig. 5.11(b), the results will be
unstable even though it appears to satisfy the criterion in (3.30).
To investigate the cause of this problem further, the time response of a
single m icrostrip line is analyzed using three difiexent grids as shown in
Figures 15.5(a) to (c).
»
Fig. 5.15(a) A microstrip line modeled with square elements.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 5.15(b) A microstrip line modeled w ith
"gradually" distorted elements.
Fig. 5.15(c) A microstrip line modeled w ith
distorted elements along the width.
The calculated tim e responses under the excitation o f a Gaussian pulse
of 7=20 ps and £o=0.06 ns a t one end of th e line are shown in Figures 5.16(a)
through (c). The solid line indicates the voltage waveform observed at the
input port and the dashed line indicates th e voltage waveform a t the output
port. As can be seen, th e microstrip line modeled with square elements
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
produces the expected output. The two microstrip lines modeled with distorted
elem ents, on the other hand, show additional reflection as well as attenuation.
These results confirm the earlier observation th at the point-matched finite
elem ent method is only suitable for grids th a t are constructed with conformally
distorted elements when using the linear quadrilateral elements.
0.6
>
00
-3
o
0.4
>
e
£
02
0.05
0.1
0.15
025
033
Tune(ns)
Fig. 5.16(a) Time response of a microstrip line modeled
with the grid shown in Fig. 5.15(a).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
0.8
0.6
>
&
m
o
0.4
>
e
£
-0.21
0
0.05
0.1
0.15
--------- '
■
0.2
0.25
'--------
03
035
Time(ns)
Fig. 5.16(b) Time response of a microstrip line modeled
with the grid shown in Fig. 5.15(b).
0.8
0.6
>
Ofl
C
n
o
0.4
>
e
o
Q.
03
0.05
0.1
0.15
035
03
035
Fig. 5.17(c) Time response of a microstrip line modeled
with the grid shown in Fig. 5.15(c).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
CHAPTER 6
ANALYSIS OF LOSSY
DISPERSIVE PLANAR
NETWORKS
6.1
Introduction
For theoretical and computational convenience, the analysis of
microwave integrated circuits using the plan ar circuit usually assumes the
circuits are lossless [14-25]. This assum ption, however, cannot be justified
when the planar dimensions of the network are much larger th an the
operating wavelengths. In such case the wave attenuation and dispersion due
to m aterial losses m ust be accounted for in th e analysis.
The objective of this chapter is to deal with the analysis of planar
dispersive networks by incorporating conductor and dielectric losses into the
planar circuit model. The general equations are presented and discussed. The
numerical results of a periodic meander line and a single dispersive m icrostrip
line are also dem onstrated.
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57
6.2
General Equations
In chapter 3, it was shown th at the planar circuit is governed by the
following pair of equations [7,8,14,15,17]
(6.1)
at
d
(6.2)
dt
where d is the substrate thickness and J is the surface current density vector
along the conductor strip. In addition, it was noted th a t these two equations
in essence are the two-dimensional generalization of the well-known one­
dimensional transm ission line equation [27,28].
at
(6.3)
at
c dVfct)
& "
(6 4)
at
In the above pair of equations, V and I denote the voltage and current
functions, respectively, and L and C denote the inductance and capacitance per
u n it length of the transm ission line, respectively.
The general lossy
transm ission line equations are given as [27,28]
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58
U fa # =- S i(x ^ ) - L ^ ^ -
(6.5)
— =-GV(xj) -C- ^ ^
dr
dr
(6.6)
dr
dt
where R is the resistance per u n it length due to conductor loss, and G is the
conductance per unit length due to dielectric loss. Hence, through the same
argum ent, the general lossy equations of a planar circuit would be
W (W )=- tf /( W )
VJ(W )=-Gxn w ) - C, —
^
dt
dt
(6-7)
(6.8)
where R s and G, are the resistance and conductance of an u n it square of the
circuit, respectively. For stripline and microstrip type planar circuits, they can
be approximated by
*,=—
(6.9)
° c 'c
(6.10)
where ac and tc are the conductivity and thickness of the conductor strip, and
<5d is th e conductivity of th e dielectric. L, and C, are th e inductance and
conductance of an unit square of th e circuit and they are given by ftd and e /d ,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
respectively.
By following the same procedure as described in chapter 3, (6.7) and
(6.8) are discretized as follows:
where the interpolations
(6.13)
2
(6.14)
have been used.
&3
Analysis of a Lossy Meander Line
Fig. 6.1 shows the geometry of a periodic meander line. M eander lines
are often used in microwave integrated circuits as delay or slow-wave lines.
To illustrate th e effects of m aterial losses, Fig. 6.2 presents the t r a n s m i ssio n
magnitude as a function of period for such a structure w ith a substrate
thickness of 0.5 mm and ^=2.2 a t 5 GHz. For comparison, th ree cases have
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60
been computed : 1) w ithout losses, 2) with dielectric loss only by considering
the conductivity of the dielectric to be 0.005 S/m, and 3) with both dielectric
and conductor losses by considering the same substrate and the thickness
conductivity of the conductor to be 1 pm and lxlO 7 S/m, respectively.
W=2.4 mm, S=7.2 mm, L=18.4 mm
Fig. 6.1 An example of a m eander line.
T ra n sm issio n Magnitude
03Lossless
0 .0 3 -
Oielactrtc Loss Only
Botn Losses
0.750.7
t
2
3
Number o f Periods
Fig. 6.2 Transmission m agnitude as a function
of period at 5 GHz.
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61
6.4
Tune Response of a Dispersive Microstrip Line
In addition to wave attenuation, the presence of m aterial losses might
also introduces w hat is known as dispersion [27,45] on a microstrip line. In
this case, a signal containing many frequency components will "change" its
shape during propagation because the phase velocity varies with frequency.
This effect is demonstrated in the following example.
A microstrip line w ith a length of 250 mm and a width of 5 mm is
considered here. The substrate has a relatively perm ittivity of 2.5 and a
thickness of i mm. The loss param eters of the line are as follows:
Cc = 1X106 S/m,
te = 0.2 pm, and
ad = 0.0001 S/m.
Fig. 6.3(a) shows the input wavefoim, which is the first half-cycle of a
2 GHz sine wave. The simulation is performed with a spatial subdivision of
2.5 mm and a tim e increment of 13.17 ps. Fig. 6.3(b) shows the resulting
waveform observed a t the end of the line. Note th at in addition to attenuation,
the output wavefcim is also distorted due to the presence of multi-frequency
components.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
1.2
0.8
>
w
0.6
u
on
<
9
O
>
0.4
3
Q.
C
-
0.2
0
0.4
0.1
0.6
0.7
0J
0.9
T im e(nj)
Fig. 6.3(a) The first half-cycle of a 2 GHz sine wave
as th e input of a dispersive microstrip line.
0.4
0.35
Output Voltage (V)
03
0.15
0.1
aos
•0.03
-
0.1
0.6
0.7
0.8
0.9
1.2
U
1.4
Tune (ns)
Fig. 6.3(b) Time response of a dispersive microstrip line
under the excitation shown in Fig. 6.3(a).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
CHAPTER 7
CONCLUSIONS
The point-matched tim e domain finite element method has been
presented and applied to various configurations of microwave planar circuits.
This formulation is more flexible than the finite difference method because it
allows the use of distortahle elem ents to model the circuit boundary and avoids
the reed of stair-stepped approximation. La addition, the solution scheme is
explicit and does not require the storage and inversion of a m atrix. Hence, the
method is feasible for solving a large number of unknowns. However, as
dem onstrated in one of the num erical examples, more work has to be done in
investigating the stability o f the method in relation to the necessary grid
generation for the case of non-uniform grid.
It m ust be stressed th a t the planar approach to microwave integrated
circuitry is only an approxim ate technique and therefore cannot be expected
to provide extremely accurate results for all problems. La such cases, a threedimensional full-wave analysis should be employed. However, because of its
*■
dimensional modeling nature, it does offer more accurate results and
overcomes the lim itation imposed by the conventional one-dimensional
approach.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Possible further work includes:
(i)
The use of higher order elements so as to reduce the number of
elements required for finite element modeling.
(ii)
T he inclusion of electromagnetic coupling and radiation loss so
as to allow the sim ulation of the entire planar circuit board.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
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Trans. Microwave Theory Tech., vol. MTT-40, no. 2, pp. 378-388, Feb.
1992.
[41]
J J 1. Lee, "Analysis of passive microwave devices by using a threedimensional tangential vector finite elements,”Int. J. Numer. Modelling:
Electronic networks, devices and fields, vol. 3, pp. 235-246, 1990.
[42]
A.C. Cangellaris and D.B. W right, "Analysis of th e numerical errov
caused by the stair-stepped approximation of a conducting boundary in
FDTD simulations of electromagnetic phenomena," IEEE Tranz Antenna
Propagat., vol. AP-39, no. 10, pp. 1518-1525, Oct. 1991.
[43]
GJDTnzeo, F. Giannini, and R. Sorrentino, "Microwave planar networks:
The annular structure," Electron. Lett., vol. 14, pp. 526-528, Aug. 1978.
[44]
J.F . Lee, R. Palandech, and R. M ittra, "Modeling three-dimensional
discontinuities in waveguides using nonorthogonal FDTD algorithm,"
IEEE Trans. Microwave Theory Tech., vol. MTT-40, no. 2, pp. 346-352,
Feb. 1992.
[45]
R.E. Collin, Field Theory o f Guided Waves. McGraw-Hill, 1960.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A PPEN D IX
COM PUTER PROGRAM LISTIN G S
The source codes (GRID, PROB, SOLV, POST) for the CAD package
MlCsim 1.0 are provided. Also shown is th e program STUB3, which is capable
of analyzing a simple two-port rectangular circuit. All programs axe w ritten
in FORTRAN.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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grid.for
Mon Apr
300
DO 3 0 0 1 - 1 , 1 0 0
I F (MODNAM I I ) , r . g . m o d e l |
w here - 1
GOTO 3 0 0
ENDir
CONTINUE
300
ir
6 12:31:14 1992
IN TEG ER' S
RE A L 'S
IN TEG ER' S
CHARACTER'S
CHARACTER'60
LOGICAL
CHARACTER' 6
I N T E G E R '3
THIN
C
lM h«fB .L O .O )
WHITE ( ' , • ) '
W RI T E( • , * 1
ELSE
THEN
ERROR : M o d e l
C
C
C
Not
I our,a
t
!'
DELETE ACTUAL F I L E S HERE !
O P E N (U N IT - 3 0 ,
C LO SE(U NIT-30,
OPENIUNIT-30,
C LO SE(U N IT-30,
O PE NIU N IT-40,
CLO SE(U N IT-40,
OPEN(UN IT-S0,
CLOSE ( U N I T - S 0 ,
C
C
C
FILE-M ODNAM(where)/ / ' . NOO',
S T A TU S-'D ELE TE')
FILE-MODNAM(where)/ / ' .E L M ',
S T A TU S-'D ELE TE')
F I L E - M O D N A M ( w h e r e ) / / ' . H N D ',
S T A T U S - 'D E L E T E ’ I
FILE-M ODNAM(where)/ / ' . H T L ',
S T A TU S-'D ELE TE')
IF
(w hich.E Q .O )
WRITE I ' , ' ) '
W R I T E ! ' , •)
ENDIF
S T A T U S -'O L D '|
S TA TU S- ’ O I . D ' )
CALL m y s a v e
10
30
w here - 0
DO 3 0 1 - 1 , 1 00
I F (MODNAM( 1 ) , E 0 . J m o d e l )
w here - 1
GOTO 30
ENDIF
CONTINUE
30
IF
w h ich - 0
RETURN
END
SUBROUTINE
I N T E G E R 'S
CHARACTER* 6
CHARACTER*SO
LOGICAL
IN T E G E R 'S
COMMON
(w h ere.E 0 .0 )
W R I T E ! ',') '
W R IT E !',')
GOTO 9 9 9 9 9
ENDIF
d ire ct
w h i c h , NUI9400
MOONAMIOtlOO)
MODLAB( 0 : 1 0 0 )
SAVDIR, SAVMOD
1
/M O D E L /
w h ic h ,
NUMMOD, MOONAM,
MODLAB,
S A V D IR ,
WRITE( • , • )
WRITE( ' , ' )
W R IT E (',‘ )
SAVMOD
t
W R IT E !',')
4
'
* MODELS'
(w h ich )
WRITE( ' , ' ( A M ' ) ' E n t e r J o i n i n g M odel
R E A D I',1 0 )
) model
FORMAT ( A t )
W R I T E ! ', •)
I F O m o u e l . EO. *
’ ) GOTO 99999
CALL UPCAS E( ) m o d e 1)
END IF
9999
!’
S T A T U S - ' O L D 'I
1
M OD LA B( w her e) - '
IF (w h ere.E O .w h ich )
SAVDIR - .T R U E .
THEN
• • • ERROR : No A c t i v e M o d e l
ST A T U S-'O U )' I
REMOVE MODEL FROM DIRECTORY HERE !
NUMMOD - NUMMOO M O O N A M (w here) - '
NPV, NEV, E L M N T V ( 3 8 0 Q , J |
XV( 4 0 0 0 ) , YV ( 4 0 0 0 1
w h i c h , NUM40D
MODNAM( 0 ; 1 0 0 )
MODLAB( 0 : 1 0 0 )
SAVDIR, SAVMOD
Imodel
w h e r e , 1 , NEUGE, NN O D E(S ), E D G E ( S , ] DO) , NPTS,
FROM, TO, ] , k ,
1
R IA L 'S
u i , y l , x S , y S , e r r o r * , e r r o r y , * o , yo
REAL M
e r (SI
COMMON /V G R I D / NPV, XV, YV, NEV, ELMNTV
COMMON /MODEL/ w h i c h , NUMMOD, MODNAM, MODLAB, SAVDIR, SAVMOD
W R IT E !',•)
W R IT E (',' )
DIRECTORY OF A V A IL A B LE ',
Name : '
THEN
THEN
* • * ERROR : M o d o l N o t F o u n d
'
JO IN I N G EDGE(S)
1'
S PE C I F IC A T I O N S '
' E d q e ( a ) m u s t b e STRAIGHT a n d d o l l n e d
'd i r e c t I o n a i r ng '
' CURRENT MODF , . '
i n ccw ' ,
W RITE)*, • )
40
100
300
DO 3 0 0 1 - 1 , 1 0 0
I F (M O O N A M ! I ) .N E . '
' ) THEN
W R IT E !*,100)
MOONAM ( I ) , HOOLAR(I)
FORMAT!' ' , '
' , A 6 ,' :
',A tO )
ENDIF
CONTINUE
WRITE( ' , ' )
C
RETURN
END
C
c
-------
c
SUBROUTINE
loll)
4S
50
W R I T E ( * , ' (AM ' I ' E n t e r N u m b e r o f J o i n i n g E d g e ( s ) ( H a x . S ) : '
R E A D !',')
NEDGE
W R I T E ! ', •)
I F (NEDGE.EQ.O)
COTO 9 9 9 9 9
I F ( ( N E D G E . L T . l ) . A N D . ( N E D G E . G T . 5 ) ) THEN
W R I T E ) ' , ' I ' • " ERROR : I n v a l i d N u m b e r o f J o i n i n g E d g e ( s )
W R IT E !',‘ I
COTO 40
ENDIF
DO 9 0 0 1 - 1 , NEDGE
W R IT E (*,tO ) I
FORMAT I ' ’ , ' EDGE • ' , 1 3)
W R I T E ! ' , ' ( A M ' l * From ?
R E A D !',* )
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6 12:31:06 1992
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CONTINUE
IF (w h ere.E Q .O ) then
W H I T E ! * , * ) * *** ERROR :
W R I T E ! * , •)
GOTO 3 0
EESE
PORT( 1 , k) - w h e r e
ENDIF
CONTINUE
CONTINUE
390
400
DO 400 1 - 1 , NPORT
TERM(l) - 0
FUNC (1 ) - 0
DO 390 ) - l , 9
D A T A d ,)) - 0.0
CONTINUE
CONTINUE
SAVE PROBLEM F I L E HERE
I n v a l i d No. o l
IF
Nodes
OPEN ( U N I T - 1 1 , F I L E - F I L N A M ( w h e r e ) / / ’ , P B l ' , STATUS-'UNKNOWN* I
W R I T E d l, 100)
NPORT
FORMAT( 1 3 )
DO 1 2 0 1 - 1 , NPORT
W R I T E I l l ,110)
1, NNODE( 1 ) , P E P S ( 1 | , TERM( 1 ) , FUNC( I |
W R IT E d l,1 1 1 )
(PORT ( 1 . 3 1 , J * l , NNODE ( 1 ) )
W R IT E d l, 112)
(DATA ( 1 , 1 ) , 1 * 1 , 9 )
FORMAT (21 6 , El S . 7 , 21 6)
FORMAT( 4 0 1 6 )
FORMAT(9E15 . 7 )
CONTINUE
CLOSE ( U N I T - 1 1 , S T A T U S - ' K E E P ' )
100
c
110
11 1
112
120
OPEN ( U N I T * ! 2 , F I L E - F I L N A M ( w h e r e ) / / ' . P B 2 ' ,
WRITE( 1 2 , * )
a
W RITE(12,*>
m ur
W R IT E 0 2 ,* )
slgm ad
W RITE(I2, • )
slg m ac
W R I T E ( 1 2 , *)
tc
CLOSE ( U N I T - 1 2 , S T A T U S - ' K E E P ' )
c
c
SA V FI L -
.T R U E .
C
99999
RETURN
END
C
C
SAVFIL ENDIF
-------------------- ------------------------------------------------------------------------------------------ ----------------------
c
SUBROUTINE
INTEGER* 2
INTEGER*2
t
t
m y s a v - (w here)
w hore
NPORT, NNODE( 1 0 ) , P O R 7 ( 1 0 , 4 0 ) , TER M(1 0 ) ,
func(io )
REAL M
P E P S ( I O ) , C.'T.\ ( 1 0 , 9)
INT EG ER* ?
NPV
REAL’ S
XV ( 4 0 0 0 ) , Y V ( 4 0 0 0 )
REAL’ S
d , m ur, s l g n a d , s lg m a c , t o
INTEGER*2
w h i c h , NUHFIL
CHARACTER*6
F I L N A M ( 0 ; 1 0 0 ) , MODNAM( 0 : 1 0 0 I
CHARACTER*6 0
FIL L A B (0:100)
LOGICAL
SAVDIR, SA V FI L
IN TEG ERS
1, j , c o u n t
COMMON / P O R T S /
NPORT, NNODE, PORT, P E P S , TERM, FUNC, DATA
COMMON /MO DEL/ NPV, XV ,YV
COMMON /M A TRL / d , m u r , s l g m a d , s l g m a c , t c
COM ION/ F I L E S /
w h ich ,
NUH FIL , FILNAM, MODNAM, F I L L A B , SAVDIR,
SA V FI L
C
C
C
30
.FALSE,
SUBROUTINE s e l e
CHARACTER*60
CHARACTER*6
IN TE G ER '2
REAL*4
INTECER*2
REAL*8
CHARACTER* 1
INTEGER*2
REAL* 4
COMMON / P O R T S /
COMMON /MODEL/
(U le .E Q ,'
W R IT E!*,*)
W R I T E ( * , *)
GOTO 9 9 9 9
ENDIF
c t ( t I t l e , f 1le ,m o d e l)
t ! • '.a
( l i e , model
NPORT, NNODE ( 1 0 ) , PORT ( 1 0 , 4 0 ) , TERM ( 1 0 ) ,
FUNC( 1 0 )
P E P S ( I O ) , D A T A ( 1 0 , 9)
NPV
XV( 4 0 0 0 ) , YV ( 4 0 0 0 )
BOUND( 4 0 0 0 )
NST EP, t y p e , N, 1 , 1, I P , NTYPE, ELF.M1, ELEM2,
ELEM3, NNEU, NDIR
ste p
NPORT, NNODE, PORT, P E P S , TERM, FUNC, DATA
NPV, XV, YV
IF
IF
20
STATUS-'UNKNOWN'I
RETURN
END
SAVE PROBLEM F I L E DIRECTORY
10
( ( S A V F I L ) . A N D , ( w h e r n . N E . 0 ) ) THEN
(SAVDIR) THEN
OPEN ( U N I T - I O , F I L E - ' F I L E S . D I R ' , STATUS-'UNKNOWN')
WRITE( 1 0 , 1 0 )
NUHFIL
FORMAT ( 1 3)
count - 1
DO 3 0 1 - 1 , 1 0 0
I F (F IL N A M (i).N E .'
' ) THEN
WRITE( 1 0 , 2 0 )
c o u n t , F I L N A M ( l ) , MODNAM(I) , F l L L A B ( l )
FORMAT( 1 3 , 2 A 6 , A 60 )
count - count t 1
ENDIF
CONTINUE
CLOSE ( U N I T - 1 0 , S T A T U S - ' K E E P ' )
SAVDIR - . F A L S E .
ENDIF
SO
'
•**
' ) THEN
ERROR ; N o A c t i v e
P roblem F l l o
OPEN ( U N I T - 2 0 , F I L E - m o d e l / / ' . BND’ , S T A T U S - ' O L D ' )
DO SO 1 - 1 , NPV
READ( 2 0 , • )
I P , NTYPE, ELEM1, ELEM2, ELEM3
I F ( N TY PE .E Q .O ) THEN
BOUND( 1 ) - T
ELSE
BOUND( 1 ) - ' N '
ENDIF
CONTINUE
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Hon Apr
prob.for
100
200
310
320
330
340
350
1000
C
6 12:31:06 1992
WRI TE f , 2 0 0 )
DATADATA(l,2)
W R I T E ! * , *)
E L S E I F ( F U N C ( l ) . E 0 . 3 ) THEN
Source : '//CHAR(228)//
WRITE!*,*) '
t A l*sln(2*pi*fl*t),
f o r 0 < t < T.
WR I T E ( * , 3 1 0 )
DATA ( 1 , 1 ) , DATA ( 1 , 2 )
DATA( 1 , 3 ) , DP“ A ( I , 4)
WRITE!*,320)
WRITE!*, 330)
DATA ( 1 , 3 ) , DATA ( 1 , 6 )
WRIT E!*,340)
DATA( 1 , 7 ) , DATA 1 1 , 8 )
DATA ( 1 , 9 )
WR I T E ( * , 3 3 0 )
E L S E I F ( F U N C ( l ) . E Q . 0 ) THEN
Source : N one.'
WRI TE! * , * ) '
WRI TE! * , * )
ENDI F
FORMAT!' ' , 1 5 X , ' A - ' , E 1 2 . 4 , '
T - ' , E12. 4 , '
to
, El 2 . 4 1
FORMAT! ' ' , 1 5 X , ' A - ' , E 1 2 . 4 , '
f - ' ,E12.4)
FORMAT!' ' , 1 5 X , ' A 1 - ' , E 1 2 . 4 , '
f l - ' , E 1 2 . 4)
FORMAT!' ' , 1 5 X , ’ A2 * ' , E 1 2 . 4 , '
12 - ' , E 1 2 , 41
13 » ' , E 1 2 . 4 I
FORMAT! ' ' , 1 5 X , ' A 3 - ' , E 1 2 , 4 , '
FORMAT( ' ' , 1 5 X , ' A4 - ' , E 1 2 . 4 , '
14 - ' , E 1 2 . 4 )
FORMAT! ' ' , 1 5 X , ' T
, E 1 2 .4)
E L S E I F ( T E R M ( l ) . E Q . 2 ) THEN
WRITE!*,* ) ’
Termination s O p en -C lrcu lt.'
WRITE!*,*)
E L S E I F ( TERM( 1 ) • EQ 3) THEN
S hort-C lrcult,'
WRITE!*,*) *
Termination
W R I T E ! * , *)
ENDI F
I F ( M 0 D ( 1 , 2 ) . E Q . 0 ) THEN
READ! * , • )
ENDI F
CONTI NJ E
WRITE!*,*)
WR ITE !*,•)
DO 5 0 0 1 - 1 , NPORT
WRITE(*, 20)
FORMAT( ' ' , * PORT*, 1 3 )
W R I T E ( * , *1
WRI T E! * , * ) 1 ( 1 | M a t c h b d - L o a d
[2) O p e n - C i r c u l i
( 31 S h o r ' - • ,
'C ircuit'
W RI T E ! *, •(
W R I T E ! * , ' ( AN) ' ) ' E n t e r S e l e c t i o n : '
READ!*, *)
TERM | 1 )
W RI T E ! *, •)
I F (TERM( 1) . EQ. 1) THEN
W R I T E ! * , * ) ’ SOURCE : H ] G a u s s i a n P u l s o
|2] s in e
t
'
13 ) Sum o f 4 s i n e w a v e s
[0 | None'
WRITE!*, *)
W R I T E ! * , ’ ( AN) ' ) • E n t e r S e l e c t i o n : *
READ!*,*)
FUNC( 1)
WRITE ( * , * 1
I F ( FUNC( 1 ) . E Q . 1) THEN
WRI TE! *, *)
’ G ( L ) - A * e x p ( - ( t - t o ) * 2 / T “ 2) *
W R I T E ! * , •)
W R I T E ! * , ' ( A N ) ' ) ' E n t e r A (V) : '
READ( *,*|
DATA( 1 , 1 )
WRI T E( • , ' ( A \ ) ' ) ' E n t e r T ( s e c ) ; '
READ!*,*)
DATA( 1 , 2 )
WRITE!*,' ( A \ ) ') ' E n t e r t o (sec) : '
READ!*,*)
DATA( 1 , 3 )
W R I T E ! * , *)
E L S E I F ( FUNC( 1 ) . E Q . 2 ) THEN
WRITE!*,*)
' G(l) - A * s l n ( 2 * p l* t* t ) '
W R I T E ! * , •)
WRI TEC* , ' ( A \ » ' ) ' E n t e r A (V) : '
READ!*,*)
DATA( 1 , 1 )
WR I T E ( * , ' ( A N ) ' ) ' E n t e r I ( Hr ) : '
READ!*,*)
DATA ( 1 , 2 )
W R I T E ! * , •)
E L S E I F ( FUNC( 1 ) . E Q . 3) THEN
WRI TE! *, *)
' G ( l ) - ' / / CHAR ( 2 2 8 ) / / ' A l * a l n ( 2 * p l * M * t ) ‘
' ,1 -1 ,.4 , for
0 < t < T'
W R I T E ! * , •)
W R I T E ! * , ' ( A N ) ' ) ' E n t e r A1 (V)
READ!*,*)
DATA ( 1 , 1 )
W R I T E ! * , ' ( A N ) ’ ) ' E n t e r f l (Hz)
READ!*,*)
DATA ( 1 , 2 )
W R I T E ! * , ’ ( A N ) ' ) ’ E n t e r A2 (V)
READ!*,*)
DATA( 1 , 3 )
W R I T E ! * , ' ( A N ) ' ) ' E n t e r f 2 (Hz)
READ!*,*)
DATA( 1 , 4 )
W R I T E ! * , ' ( A N ) ' ) ' E n t e r A3 (V)
DATA( 1 , 5 )
READ( • , * )
E n t e r 13 (Hz)
WRITE!*,' (AN)')
DATA ( 1 , 6 )
READ!*,*)
10
20
CI RCUI T MATERIAL'
'
C
2000
C
9999
'd
-', d
WR I T E ( * , 2 0 0 0 )
CHAR ( 2 3 0 ) / / ' r
WR I T E ( * , 2 0 0 0 )
CHAR( 2 2 9 ) / / ' d
WRITE(*,2000)
CHAR ( 2 2 9 ) / / ' c
WRITE ( * , 2 0 0 0 )
'tc
•',tc
WRITE!*,2000)
FORMAT! ' ' , 6X, AS, E 2 4 . 1 6 )
WRITE!*,*)
mur
slgmad
slgmac
RETURN
END
SUBROUTINE
I NTEGER*2
termln
which
LOGICAL
SAVFIL
(which,SAVFIL)
INTEGERS
NPORT, NNO D E ( I O ) , PORT ( 1 0 , 4 0 ) , T E R M ( I O ) ,
REAL*4
PEP S ( 1 0 ) , D A T A ( 1 0 , 9 )
COMMON / P O R T S /
NPORT, NNODE, PORT, P EP S , TERM, FUNC,
IF
(Which.EQ.0)
WRITE( *,*) '
W R I T B C , *)
GOTO 9 9 9 9
ENDI F
WRI TE I * , * )
WR I T E ! * , *)
'
THEN
• • * ERROR s No A c t i v e
PORT ( S )
Problem F l l o
PUNC ( 1 0 )
DATA
I'
TERMI NATI ON'
500
1
W R I T E ( * , ' ( A N ) ' ) ' E n t e r A4 (V)
READ( * , * )
DATA( 1 , 3 )
W R I T E ! * , ' ( A N ) ' ) ' E n t e r f 4 (Hz)
DATA( 1 , 8 )
READ(*,*)
WRITE!*,’ (AN)') ' E n t e r T (sec)
DATA ( I , 9)
READ( * , * )
W R I T E ! * , *)
ENDI F
ELSE
FUNC(I| - 0
ENDIF
CONTINUE
I*
1992
6 12:31:06
Hon Apr
prob.for
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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post.for
Mon Apr
6 12:31:36 1992
CALI, h e l p
GOTO 1 0 0 0
E L S E I F ( c o m m n d . E Q . ' O P T l ' ) THEN
CALL o p t 1
GOTO 1 0 0 0
ELSEIF ( c om mn d. EQ .'
' ) THEN
GOTO 1 0 0 0
ELSE
W R I T E ! * , •)
Command n o t r e c o g n i z e d ;
WR I T E ! * , • )
GOTO 1 0 0 0
ENDIF
please
re-enter'
help
SUBROUTINE
WRITE!*,*) »
WRITE ( * , * )
WRITE(*, • ) '
DI SP
'
'
•
EXI T
HELP
OPTI
L I S T OF AVAI LABLE COHMANDS*
dummy - s e t v i d e o m o d e ( SVRES16COLOR)
I F ( d u m m y . E Q . 0 ) THEN
dummy - s e t v l d e o m o d e (SDEFAULTMODE)
W R I T E ! * , * ) ' * • • ERROR f VGA g r a p h i c s
WR I T E ! * , *1
GOTO 9 9 9 9 9
ENDI F
density so lu tio n s.'
Eklt p o stp ro ce sso r.'
D is p la y t h i s menu.'
Set d is p la y o p tio n s.
card
required
C
C
C
•
C
c
C
ft
V
WRITE I * , *) ' Wh i c h s o l u t i o n
V
W R I T E ! * , *1
WRITE!*,*)
'
V: V o l t a g e '
WRITE(*,*I
*
JX;X c o m p o n e n t
of s u r f a c u c u i r o n t d e n s i t y '
WRITE(*,*I
' JY
: Y component of s u r f a c e c u r r e n t d e n sity*
W RIT E)* ,•)
'
NONE:
E x i t DI SP'
WRIT E!*,•)
10
WRITE!*,’ (A M ') *
Entor choice : ‘
READ ( * , 2 0 |
which
20
FORMAT(A4)
CALL UPCASE ( w h i c h )
I F ( w h i c h . E Q . ' N O I ( E ' ) THEN
WRI TE! *, *)
GOTO 9 9 9 9 9
ENDIF
IF ((w h ich .N E .'V
' ) .AND.(which.NE.'JX
'I.AND.
I
(which.NE.'JY
'II
GOTO 10
SUBROUTINE
UPCASE ( t o k e n )
CHARACTER**
token
I NTEGER*2
1
CHARACTER*1
letter
DO 1 0 0 1 - 1 , *
lo tte r - token(l:l)
I f i ( i CHAR ( l e t t e r ) . G E . 9 7 ) . AND. ( I C I I A R ( l e t t e r ) . I . E . 1 2 2 ) )
t o k e n ( 1 : 1 ) - CHAR ( I CHAR (1 e t t e r ) - 3 2 )
CONTINUE
RETURN
END
WRITE!*, • )
w h it e (*,*)
WRITE ( * , * )
WRITE( * , * )
RETURN
END
CJXSCALE,
C
C
END
100
REAL' S
SOLU( 4 0 0 0 ) , X I 4 0 0 0 ) , Y(4000)
I NTEGER*2
NE, ELMNT( 3 8 0 0 , 5 |
CHARACTER*9
CSTF.P, c t I me
RECORD / r c c o o r d / c i r p o s
COMMON / V G RI D/ NPV, h. . , TV, NEV, ELMNTV
COMMON / J G R . 0 / N P J , X J , T J , N E J , ELMNTJ
COMMON / RANGE/ x m l n , x ma x , y m l n , y ma x ,
t
c x m l n , c x m a x , c y m l n , c y ma x
COMMON / S U B S T / c d , c m u r , c s l g m a d , c s l g m a c , e t c
COMMON / S OLVE/ f i l e , m o d e l
COMMON / S C A L E S /
VSCALE, JXSCALE, JYSCALE, CVSCAI.E,
(
CJYSCALE
COMMON / S E T T I N G S /
le v el, s e t , g rid , ppause
SUBROUTINE
INCLUDE
d l s p (N ,NSTEP,clrtyp)
• FGRAPH.FD*
I NTEGER*2
I NTEGER*2
REAL*8
REAL' T
CHARACTER*9
C HARACTERS
CHARACTER*6
REAL* 8
C HARACTE RS
I NTEGER*2
CHARACTER*4
I NTEGER*2
I NTEGER*2
INTBGER‘ 4
CHARACTER*9
REAL* 8
N, NSTEP, c l r t y p
NPV, NEV, N P J , N E J , ELMNTV( 3 8 0 0 , 5 ) , E l M N T J ( 3 0 0 0 , V
XV( 4 0 0 0 ) , YV( 4 0 0 0 ) , X J ( 4 0 0 0 ) , YJ ( 4 0 0 0 )
x m l n , xma x, y m l n , ymax
c x m l n , cxmax, c y m l n , cymax
c d , cmur, c s l g m a d , c s l g m a c , e t c
f i l e , model
VSCALE( 1 0 ) , J X S C A L E ( 1 0 ) , J YS CA L E ( I O )
CVSCALE( 1 0 ) , C J XS C A L E ( 1 0 ) , CJ YS CALE( I O)
l e v e l (3), s e t , g r i d , ppause
which
d ummy , 1 , ]
l h r , tm tn, t s e c , 1 1 0 0 th , p h r , pmln, p s e c , plOOth
k
S CALE( 1 0)
x l , x2 , y l , y2
CALL s e t t e x t p o s l t l o n ( 2 , 6 2 , c u r p o s )
CALL o u t t e x t ( ' M I C S l m l . 0
1992')
CALL s e t t e x t p o s l t l o n ( 3 , 6 2 , c u r p o s )
CALL o u t t e x t ( ’ POST V e r s i o n 1 . 0 ’ )
CALL s e t t e x t p o s l t l o n ( 5 , 6 2 , c u r p o s )
CALL o u t t e x t ( ' PROB : ' / / f i l e )
CALL s e t t e x t p o s l t l o n ( 6 , 6 2 , c u r p o s )
CALL o u t t e x t ( ’ GRID i ' / / m o d e l )
CALL s e t t e x t p o s l t l o n ( 6 , 6 2 , c u r p o s )
IF ( c l r t y p . E Q . l )
CALL o u t t e x t ( ' MI CROS TRI P T Y P E ' )
IF ( c l r ty p .E Q .2 )
CALL o u t t e x t ( ' S T R I P L I N E T Y P E ' )
CALL s e t t e x t p o s l t l o n ( 1 0 , 6 2 , c u r p o s )
CALL o u t t e x t ( * d
• *//cd)
CALL s e t t e x t p o s l t l o n ( 1 1 , 6 2 , c u r p o s )
CALL o u t t e x t ( C K A R i - - a ; ' ' r
- '//cmur)
CALL s e t t e x t p o s l ’ i o n ( 1 . , u 2, c u r p o s )
CALL o u t t e x t ( C H A R I 2 2 9 ) / / ' d
- '//cslgm ad)
CALL s e t t e x t p o s l t l o n ( 1 3 , 6 2 , c u r p o s )
CALL o u t t e x t (CHAR( 2 2 9 ) / / ' c
- '//cslgm ac)
CALL s e t t e x t p o s l t l o n ( 1 4 , 6 2 , c u r p o s )
CALL o u t t e x t ( ’ t c
- '//etc)
I'
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*?ost.for
Hon Apr
6 12:31:36 1992
CALL s e t t e x t p o s l t l o n ( 1 6 , 6 2 , c u r p o s )
CALL o u t t e x t ( ' S O L U T I O N : ' / / w h i c h )
C
C
C
3
C
READ I N SO L UT I O NS
OUTPUT SCALES
DO 3 0 0 1 - 1 , N S T E P / N
IF (w hich.E O .'V
' ) THEN
READ|300, 200)
CSTEP, c t l m o ,
DO 1 5 0 3 - 1 , NPV
!
IF
30
36
40
(Which.EO.'V
' ) THEN
00 30 1*1,10
SCALE ( 1 ) - CVSCALE (1)
CONTINUE
OPEN ( U N I T - 3 0 0 , K I L E - ' V X Y . O U T ' , S T A T U S - ' OLD' 1
ELSEIF ( w h i c h . E Q . ' J X
' ) THEN
DO 3 6 1 - 1 , 1 0
S C AL E ( 1 ) * CJ XS CA L E ( 1)
CONTINUE
OPEN ( U N I T - 3 0 0 , F I L E - ' J X X Y . O U T ' , S T A T U S - ' O L D ' )
ELSE
DO 4 0 1 - 1 , 1 0
SCALE ( 1 ) - CJYSCALE ( 1 )
CONTINUE
OPEN ( U N I T - 3 0 0 , F I L E - ' J Y X Y . O U T * , S T A T U S - ' O L D ' )
ENDI F
162
164
160
DO 6 0 1 - 1 , 1 0
CALL s e t t e x t p o s l t l o n ( 1 9 * 1 , 6 8 , c u r p o s )
CALL o u t t e x t ( S C A L E ( 1 1 - 1 ) )
CONTINUE
.
mi
•
-
rem appalette
rem appalette
rem appalotte
rem appalette
rem appalette
rem appalette
rem appalette
rem appalette
rem appalette
rem appalette
16 2
164
• 3 6 1 3 1 C)
I3F260B)
I3F3A18)
•303C15)
1003611)
•003C2F)
I133C3C)
I1C323C)
(9. I00182C)
( 1 0 , I 2A2A2A)
(1.
(2,
<3,
(4.
(5,
(6,
(1,
(8,
170
DO 1 0 0 1 * 1 , 9
dummy • s e t c o l o i ( 1 0 - 1 )
dummy « r e c t a n g l e ( S G F I L L I N T E R I O R , 4 9 0 , 3 1 0 * ( 1 - 1 ) * 1 6 ,
(
626,310*1*16)
1 0 0 CONTINUE
dummy - s e t c o l o r ( 1 6 )
DO n o 1 - 1 , 9
dummy • r e c t a n g l e ( SCBORDER, 4 9 0 , 3 1 0 . ( 1 - 1 ) • 1 6 , 6 2 6 , 310*1 ' 1 6 )
1 1 0 CONTINUE
172
174
200
dummy - r e c t a n g l e
(SCBORDER,10, 1 0 , 4 / 0 , 470)
CALL s e t v l e w p o r t ( 1 1 , 1 1 , 4 6 9 , 4 6 9 )
I F ( x m a x - x m l n . C T . y m a x - y m l n ) THEN
x l - xmln - 0 , 0 6 * (xmax-xmtn)
x2 - xmax • 0 . 0 6 * ( x ma x - xm l n )
y l - y m l n - 0 . 0 6 * (x ma x- xmtn )
y2 - yml n * 1 . 0 6 * (xmax-xmtn)
ELSE
x l * xmln - 0 . 0 6 * (ymax-ymln)
x2 • x ml n * 1 . 0 6 * ( yma x- yml n)
y l - y m l n - 0 . 0 6 * (y ma x- ymln )
y 2 - ymax * 0 . 0 6 * ( y m a x - y m l n )
ENDI F
dummy - s e t w l n d o w ( . T R U E . , x l , y l ,
• 260
ELMNTV(J,k)
DO 1 6 0 3 - 1 , N P J
X ( 31 - X J ( 3 )
Y ( 31 “ Y J ( j )
CONTI NUE
NL - N E J
DO 1 6 4 3 - 1 , N E J
DO 1 6 2 k - 1 , 5
ELMNT ( j , k ) - ELMNTJ ( 3 , k)
CONTI NUE
C ONTI NUE
CALL d l s p s o l
( C S T E P , c t l m e , SOl . U, NE , ELMNT, X , Y , J XS CALE )
IF ( g r l d .E O .l )
CALL d l i p _ g r l d ( NE , E L MNT , X, Yl
E LS E
READ ( 3 0 0 , 2 0 0 )
CSTEP, c t l m e ,
( S O L U ( ) | , j - l , NPJ)
DO 1 7 0 3 - 1 , N P J
x ( 3)
-
Y(3)
- YJ (3>
XJ(3)
CONTI NUE
NE “ N E J
DO 1 7 4 3 - 1 , N E J
DO 1 7 / k - 1 , 5
ELMNT ( 3* k ) - ELMNTJ ( 3 , k i
CONTI NUE
C O NTI NUE
CALL d l s p s o l
( C S T E P , c t I m e , SOl . U, NE , ELMNT, X, Y, J YSCA1. E)
IF ( g r i d . C O . I I
CALL d t s p g r l d
( NE, ELMNT, X, V)
ENDIF
FORMAT ( 2 A 9 , 4 0 0 0 E 9 . 3 |
( p p a u s e . E Q . 1 1 ThEN
R E A D C , •)
ELS E
CALL GETT1 M (1 h r , I m l n , 1 s e c , 1 1 O O t h i
CALL C E T T I M ( p h r , p m l n . p s e c , p i O O t h )
DO 2 7 0 k - 1 , 3 0 0 0 0 0
270
CONTI NUE
I F ( p s e c - 1 s e c . I . T, 3)
•
300
GOTO 2 5 0
ENDIF
CONTI NUE
C
CLOSE
(UNIT-300,
STATUS-'KEEP'I
c
dummy * s e t v l d e o m o d e
WRI TE ( * , *1
x2,
y2|
c
99999
( SOl . U ( J) , J 1 , N P V |
C O NTI NUE
CONTI NUE
CALL d l s p s o l
( CSTF. P, c t l m e , SOl . U, NE , ELMNT, X , Y , V S C A L E )
I F ( g r i d , E Q . 1)
CALL d l s p g r i d ( NE , ELMNT , X, Y)
ELSEIF ( w h i c h . E Q . ' J X
' ) THEN
READ(300, 200)
CSTEP, c t l m e ,
(SOl . U( 1 ) , J 1 , NPJ)
IF
C
C
C
X(J) - XV(j)
Y | ) l “ YV()|
CONTI NUE
NE - NEV
DO 1 5 4 J - l , NEV
DO 1 5 2 k - 1 , 5
E L MN T ( j , k ) -
160
C
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
I
C
RETURN
( SDEFAULTMODE)
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