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Modelling of multilayered interconnects in microwave and high speed circuits

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MODELLING OF MULTILAYERED
INTERCONNECTS IN MICROWAVE AND
HIGH SPEED CIRCUITS
by
Firas F. Jatou
A thesis submitted in conformity with the requirements for
the Degiee of Master of Applied Science in the Department
of Electrical and Computer Engineering in the
University of Toronto
© Copyright by Firas F. Jatou, 1995
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Table Of Contents
AKNOWLEDGMENTS
L IST OF FIGURES..................................................................................................... iv
LIST OF TABLES .................................................................................................................................vii
1. IN TR O D U CTIO N .....................................................................................................I
1.1 Motivation
1.2 Objectives
1.3 Methodology
1.4 Overview
2. A P P R O A C H ....................................................................................................................... 7
2.1 Formulation o f the G reen’s Function
2.2 M ethod of Moments
2.3 Determination o f Network Parameters
2 .3 .1 Characteristic Impedance (Z0)
2.3.2 Propagation Constant
2.3.3 Impedance and Scattering Parameters
3. NUMERICAL IM PLEM ENTATION................................................................ 17
3 .1 Computer Program
3.1.1 Input File
3.1.2 Impedance M atrix Formulation
3.1.3 Network Parameters
3.2 Parallel Implementation
3.2.1 Domain Partitioning
3.2.2 Load Balancing
3.2.3 Performance Evaluation
3.3 Conclusions
ii
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4. VALIDATION
4.1 Results
4.1.1
4.1.2
4.1.3
4.1.4
31
o f Uni-Planar Discontinuities
Series Gap
Right Angie Bend with Non-Uniform Discretization
Four-port Cross
Coupled Line Filter
4.2 Results o f Multilayered Discontinuities
4.2.1 Stripline Gap Junction
4.2.2 Tw o-iayer Resonator Band-Pass Filter
4.3 Sum m ary o f Numerical Results
5. TIM E-DOM AIN ANALYSIS
42
5.1 The M ethod
5.2 The C om puter Program
5.3 Simulation of a Linear System
5.3.1 The Cross Junction
5.3.2 L-Type Coupling Structure
5.4 Conclusion
6. CONCLUSIONS
6. 1 Concluding Remarks
6.2 Improvements and Recommendations
Appendix A. INPUT GENERATION PROGRAM
Appendix B. TIM E-DOMAIN ANALYSIS PROGRAM
in
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60
LIST OF FIGURES
Figure 1-1
(a) Edge-coupled filter, (b) M eandering edge-coupled filter, (c)
Two-layer coupled filter using less area.
2
Figure 1-2
Three types of multilayered m icro strp layouts (a) Open (b) C overed
(c) Shielded.
4
Figure 2-1
A homogeneously filled cavity il.ustra.ing the effect o f an electric
current J s and its contribution to the electric field at point P.
8
Figure 2-2
Side view illustrating how multilayers t f dielectric slabs are reduced
to one source layer sandwiched by equivalent impedance walls.
9
Figure 2-3.
(a) Typical interconnect discretizition into Method of M om ent
elem ents, (b) Non-uniform piecewise sinusoidal basis function, (c)
Basis function shape used for ports.
10
Figure 2-4
Voltage excitation at the input port of a discretized interconnect
geometry.
13
Figure 3 -1
Flow chart illustrating the main com ponents of the computer code.
18
Figure 3-2
(a) Typical interconnect discretization, (b) Non-uniform piecew ise
sinusoidal basis function described in Table 3-2.
20
Figure 3-3
Description of the output matrix for a three-layer exam ple (a)
G eneral impedance matrix, (b) Impedance submatrix (c) Physical
interpretation of one of the im pedance terms.
21
Figure 3-4
The conducting strips extending from the ports towards the
discontinuity are extracted for the network parameter measurement.
In turn the nature o f the discontinuity or the location does not affect
the routine (a) Complete interconnects structure (b) Extracted current
distributions.
24
Figure 3-5
Com parison of three com putational techniques: (a) serial
implementation, (b) integral partitioning, and (c) frequency
partitioning.
25
Figure 3-6
Illustration of data movement between processors with and w ithout
affinity.
29
Figure 3-7.
Performance results o f the different techniques using a sim ple
geom etry with 1(H) cells.
30
iv
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Figure 4-1
(a) Top-view o f the series gap structure analyzed (£r = 9.7, w = h =
0.025". b = c = 0.25". The gap is 0.009" [ 10]). (b) M agnitude of S
for the series gap discontinuity.
33
Figure 4-2.
(a) Layout and discretization of the right angle bend analyzed with
£,^9.8, h = 0 .6 3 5 mm. a = 3.18 mm, vv-2.545 mm, (b) Magnitude of
the S j2 parameter.
34
Figure 4-3
(a) Top-view o f the 4-port cross circuit analyzed with £r = 9.8. h =
0.635 mm, a = 3.18 mm. b = I = 5.0 mm, u j = 1 mm, u'2 = 2 mm (b)
Magnitude of the S ln parameters.
36
Figure 4-4
(a) Sketch o f the two resonator coupled line filter analyzed [9|. (b)
Magnitude o f the S [2 parameter o f the initial filter with resonator
separations o f 0.025”, (c) Magnitude o f the S 12 parameter of the
optimized filter (er = 9.7, w = h = 0.025", / = 1.002", b = 0.25” , a =
0.4” , seperations are 0.005", 0.025", 0.005")
37
Figure 4-5
(a) Sketch of the two-layer stripline gap junction analyzed (b) The
magnitude o f the S ^ parameter (£r = 2.2, t = 0.254 mm, vv = 1.75
mm, a = b = 5.0 m m , / = 10.1 GHz).
39
Figure 4-6
(a) The suspended stripline bandpass filter analyzed [12], (h)
Magnitude o f the S [2 parameter compared with FDTD results [40]
and measurements.
40
Figure 4-7
The magnitude of the current for a microstrip transmission line as a
function of the distance from the sidewalls.
41
Figure 5-1
Obtaining an equivalent circuit from the transfer function. The first
LC pair can be evaluated based on one o f the transfer function’s
zeros, such as z l in this case.
44
Figure 5-2
Ttme-Domain analysis program flow chart.
45
Figure 5-3
(a) The m icrostrip cross structure used for the time-domain analysis
(b) The schematic layout of the cross using LIBRA.
47
Figure 5-4
Comparison o f the computed S-parameter data and the polynomial
transfer function obtained using this method, (a) S j2 and S 14, (b)
48
s 13Figure 5-5
Time-domain response of the cross structure for an input excitation
(a) Gaussian pulse (b) Step with 5xlOy Volts/sec risetime.
49
Figure 5-6
The transmitted and the reflected waves o f the cross junction based
on transmission line theory ( T = -0.5 since the lines have the same
characteristic impedance).
50
v
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Figure 5-7
Time-domain response to a 1 kilovolt human-metal electrostatic
discharge, fa) Response showing complete discharge, fb) The
high-frequency com ponent of the discharge is show n to occur only at
the initial rise.
51
Figure 5-8
The L-Type coupling structure.
52
Figure 5-9
The output response, S ^ . showing a gradual decline in transmission
as frequency is increased. The approximated transfer function is
shown to deviate at higher frequencies.
53
Figure 5-10
Time-domain voltage on main line showing the effect o f coupler
loading.
54
Figure A -l
The GUI drawing program used to create the input file.
59
vi
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LIST OF TABLES
Table 3-1
Input tile description
Id
Table 3-2
M om ent method element description
Id
Table 3-3
Program descriptions
23
Table 5-1
M atlab routines used in the com puter code
44
vii
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ABSTRACT
MODELLING OF MULTILAYERED INTERCONNECTS
IN MICROWAVE AND HIGH SPEED CIRCUITS
Firas F. Jatou
Master of Applied Science
1995
Department of Electrical and Com puter Engineering
University of Toronto
The increasing com plexity of the functions performed by microwave and VLSI circuits has
made it necessary to resort to three dimensional m ultilayer interconnects. In this thesis, a fullwave
integral equation technique is applied in the frequency domain to study multilayered structures. A
computational code is written describing general multilayered circuits in terms o f their electrical
characteristics, such as scattering and im pedance parameters. Computer sim ulations are
conducted to validate the code.
Various parallel processing techniques are successfully
implemented and com pared in terms of computational efficiency. A method is also presented that
allows for the determ ination of an equivalent analytical system representation o f multilayered
interconnects from frequency-domain data. The equivalent system can then be used for
time-domain analysis of the interconnects.
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AKNOWLEDGMENTS
I w ish to extend my sincere gratitude to a kind supervisor. Prof. T.E. van Deventer for her
invaluable guidance and support throughout the course of this work.
Thanks are due to my lawyer, Susan Nessan, for her constructive criticism of this document. I
would like to thank my parents and all my colleagues in the Electrom agnetics Group for their help
and encouragement. This work was also supported by the Bell Canada/NSERC Chair in
Electrom agnetics at the University o f Toronto.
i
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Chapter 1
INTRODUCTION
1.1 Motivation
The birth o f electrical circuitry in the early tw entieth century has proved to be a significant
source of technological advancement. Prior to the late 1960’s most known electrical components
were categorized into three principal classes: lum ped, distributed, and w aveguide. The mid 1970’s
marked the beginning o f a new modus operandi w ith the introduction o f the planar circuit. Thin
conductors printed on top of dielectric slabs becam e significantly popular. Today, such planar
circuit technology is widely used in all types o f applications ranging from the com m on logic
circuit to the high frequency microwave and millimeter-wave integrated circuit.
Recent advances in material technology and fabrication have allowed designers to include the
integration o f digital functions with analog m icrow ave circuits. The benefits from this increase in
integration include system size and weight reduction, enhanced performance, and reduced system
cost [I]. However, the interconnects of integrated circuits limit the density of active devices and
cause propagation delay, power consumption, and noise problems [2]. T h e increasing complexity
o f the functions perform ed by microwave and V L SI circuits has made it necessary to resort to
1
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three dimensional m ultilayer interconnects. The coupling among various layers in multilayered
circuits occurs either by direct connection using via holes, o r by vertical electrom agnetic
coupling. Whereas vertical coupling is viewed as a parasitic effect in fast digital networks, it has
found a wide range of applications in high frequency circuits such as passive filters and
directional couplers. In recent years, good results have been obtained by fabricating passive filters
directly onto substrates, such as G aAs, along with other devices [31. In microwave filter design
however, high coupling is difficult to obtain from edge coupled planar microstrip elem ents (Fig. 1
(a)) due to the need for im practical spacing between conductor edges. In order to obtain good
performance, edge-coupled planar filters may end up being long and narrow, and the amount of
circuit real estate taken up is large compared to the rest of the circuit. Depending on the
(b)
Figure 1-1 (a) Edge-coupled fiUcr. (b) Meandering edge-coupled fil­
ter. (c) Two-layer coupled filter using less area.
2
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fabrication technique employed, an additional problem o f such circuits arises when considerable
shrinkage o f the interconnect structure occurs during production, which could raise the filter’s
unit cost appreciably [3]. Another technique employs the m eandering edge-coupled filter
geometry in a limited package size is, as shown in Fig. 1-lb. Although this modification does
reduce the length, higher modes m ight appear since the filter is housed in a more rectangular
cavity. W ith the use of multilayered architecture the aspect ratio (ratio o f length to width o f
substrate) can be kept small, as w ell as keeping housing narrow so undesired modes are
elim inated (Fig. 1-lc).
There are three basic classes o f m icrostrip circuits: open, covered, and shielded, as shown in
Figure 1-2. In order to prevent radiation losses and electrom agnetic interference with the outside
environm ent, most microwave integrated circuits (MIC’s) are placed w ithin a metallic packaging
structure. Such housing introduces an additional conducting body which can significantly affect
the behavior o f the circuit. Most o f the Computer Aided D esign (CAD) models used in the
available CAD tool, LIBRA, do not adequately account for such effects [4], For this reason,
accurate analysis and modelling o f multilayered circuits are im portant for the design o f any
com ponent or system.
1.2 Objectives
The general goal of this research is to develop and apply a computational code using a
full-wave analysis technique to describe general multilayered circuits in terms of their
electrom agnetic characteristics, such as scattering and im pedance parameters. These parameters
can then be used to obtain accurate m odels fo r basic multilayered building blocks used to describe
more com plex systems. The specific objectives of this research are
• A pply a fullwave method in an efficient computer program for analyzing general shielded
m ultilayered circuits.
" D evelop a method to describe multilayered geometries as generalized N-port networks.
3
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Figure 1-2. Three types of multilayered microstrip layouts
(a) Open (b) Covered (c) Shielded.
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• Establish a technique to obtain time-domain responses o f such multilayered circuits.
• Validate the method by obtaining numerical results for various well know n planar and
multilayered structures.
1.3 Methodology
During the past tw o decades, several numerical techniques have been developed to investigate
planar interconnects for MIC applications. The selection of one technique over another is
primarily driven by the desire to obtain accuracy and'or computational efficiency. Some of the
techniques applied can be categorized as 1 me-<-oirain methods such as: finite difference
time-domain method [!}, the time-domai i
nite-e ement method [3], and the transm ission line
method [5], while others fall under the frequency domain category such as: conformal-mapping
technique [I], hybrid quasistatic analysis [6], and fullwave analysis using integral equation
techniques [7,8]. O f the methods developed to date, fullwave techniques provide the most
accurate results, at the expense of high computing cost [1,5]. Nevertheless, the accuracy
demanded from the latest MIC technology [9] has made it necessary tc resort to computationally
intensive methods in order to obtain more accurate CAD tools.
In this thesis, a fullwave space-domain integral equation (SDIE) technique is applied to study
multilayered structures. This involves a rigorous electrom agnetic description o f planar
interconnect geom etries with die following assumptions:
• the thickness o f the interconnects are assumed to be negligible
• the ohmic loss o f the conductors as well as the dielectric losses are not taken into account
The fullwave approach has been previously applied successfully to the study o f uniplanar
microstrip [10] and coplanar waveguide [11] structures. For the sake of clarity, the theoretical
analysis presented in this thesis is applied to the particular problem o f planar m etallic intercon­
nects, as shown in Figure 1-1, although the generality o f the method allows fo r the analysis o f
microstrips, patches, as well as slots.
5
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1.4 Overview
In Chapter 2, the theoretical approach is described, where the dyadic Green's function for the
electric field in the structure is used in an integral equation formulation. The choice o f basis and
testing functions resulting from the application of the method o f moments is discussed along with
the formulation o f the resulting im pedance matrix. In addition, the determination o f the network
parameters from such impedance matrices is also provided. C hapter 3 outlines the numerical
implementation o f the theory discussed in Chapter 2. Validation of the developed code is given in
Chapter 4 for various uni-planar as well as multilayered geometries. In Chapter 5, an approach is
described which uses previously obtained frequency-domain results to compute time-domain
responses as well as an equivalent continuous transfer function. The conclusions o f this thesis and
recommendations for future work are presented in Chapter 6.
6
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Chapter 2
APPROACH
In this chapter, the theoretica approach is described, where the G reen’s function for the
electric field in the structure is used in an integral equation formulation [12]. The choice of basis
and testing functions resulting from the application o f the m om ents method is discussed along
with the formulation of the resulting impedance matrix. A fter performing the matrix inversion the
electric current distribution on the interconnects is obtained and is combined with conventional
transmission line theory to solve for the network parameters of the structure [13].
2.1 Formulation of the Green’s Function
For a general cavity, the governing integral equation for the electric field at any point P is
given by:
E (*, y, z) = JJG * (x, y , z| x \ / , z ') • 7, (x , y ', z') d y 'd z '
(2- 1)
S
where G*is the electric field dyadic Green s function and 7*denotes the unknown electric current
7
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flowing in th e m icrostrip conductors. Both y and z directions o f current are considered allowing
the analysis o f a wide range o f planar structures. The electric dyadic G reen 's function o f the
problem represents the electric field at point P caused by an infinitesimal electric current source at
point P '. T he total field E {r) is the sum o f all the fields caused by arbitrary planar currents
present in the volum e V (Fig. 2-1). The G reen’s function for tl
problem is obtained by satisfying
the boundary conditions o f the structure. Since the walls of the cavity are assum ed to be perfectly
conducting, the tangential electric field m ust vanish at y = 0, b, z = 0. I and at .r = 0. a. The
X
Figure 2-1. A homogeneously filled cavity illustrating the effect of an electric current
Ts and its contribution to the electric field at point P.
boundary problem pertinent to multilayered interconnect problems is solved here by deriving a
generalized G reen’s function that accounts for any number o f dielectric layers within a shielded
structure. D ue to the existence of various dielectric layers, boundary conditions o f continuity of
electric and m agnetic field tangential to the interfaces between them must be met [ 141. The
remaining boundary condition to be applied is the Dirichlet condition of vanishing tangential
8
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electric field on the surface of the metallic interconnects. T o obtain a general form ulation o f the
problem, the G reen’s function is expressed with reference to a point source w ithin one o f the
homogeneous layers, i.e., the source layer (Fig. 2-2). The layers above and below this source
layer are replaced by impedance boundaries obtained through the application o f transm ission line
theory [ I 6J.
Impedance
z Walls
/’
is®*
[
/
Source
Layer
Figure 2-2. Side view illustrating how multilayers of dielectric slabs arc reduced to one source layer
sandwiched by equivalent impedance walls.
2.2 Method of Moments
In order to obtain the surface current distribution
current isexpanded into
J s o f an arbitrary interconnect layout, the
a set o f orthogonal functions [15]. Each o f these expansion functions is
defined over a subsection of the layout, as shown in Fig. 2-3(a). One suitable expansion series is:
Nr
V
r) = 5 / y
P
f
^
8P ^
(2' 2 a)
Ni
Vr>=2/* v °
q
V y)
( 2 - 2
b
)
where Iy and l z are unknow n amplitude coefficients, and Np, N q are the total num ber o f segments
in the y and z directions respectively. The expansion fu n c tio n /is used to describe the current
9
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Z-dlrected bails function
Y-dlractad basis function
z
(a)
y
(c)
Figure 2-3. (a) Typical interconnect discretization into Method of Moment elements, (b)
Non-uniform piecewise sinusoidal basis function, (c) Basis function shape used for ports.
distribution in the longitudinal direction of current while g describes the current distribution in the
transverse direction (Fig 2-3 b). T he subsectional sinusoidal basis function, /, and the pulse
distribution function, g are as follows:
sin M y - y , - t )
. y ; . / < y < yi
sin M y[
/( * ) =
sin ks (yi + l - y )
• y i < y < y i+ i
sinAr.L
5 /i*l
0
, elsewhere
10
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(2-3 a)
where ly = y t -
and ks a scaling param eter which is a function o f the frequency of operation
and the dielectric permitivity o f the source layer and is defined based on [16]. To satisfy the edge
conditions at the ports, a modified version o f / i s used where current exists only in the region
Z j\
-
as show n in Fig. 2-3 (c).
Figure 2-3(a) shows that the orthogonal surface currents Jy and J z do not share a common
centre. The centres o f the two sets o f subsections need to be offset w ith respect to each other
because a z-directed current cannot induce current in a y-directed subsection located in the same
position. This modal layout has been show n to provide good results for its ability to handle the
current com ponents at conductor edges and com ers [17]. The integral equation (2-1) can now be
w ritten in the following form:
p
' L I ^ y ) - L{f g ) =
(:M )
P
w here L is the integral operator, and h is the vector function o f the electric field associated with
the current J s given in equation 2-2. A suitable inner product (a, b) is chosen for the problem in
term s o f an integral over the conducting surface for which the distribution o f equation 2-2 is
defined:
(a,b) = j a • b ds
(2-5)
S
are defined in the range o f the operator L, and
T hen a set o f weighting, or testing, functions
the inner product of equation 2-5 is used with each Wj to get
p
l a
p
if- g) ,wt) ■I (xy) = ig (£ ) .w,)
(2-6)
A G alerkin’s method is applied to this problem, the weighting functions are taken to be
identical to the basis functions. In this way, the problem has been reduced to a set o f linear
algebraic equations which can be w ritten in matrix notation as [ 10]:
11
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M oM
where Umom\ *s
(2-7)
t M oM
vector ° f unknown y and z current amplitudes, [Vexc] is the excitation vector
which is identically zero everywhere except at the position of the source, and [ZW(),vfl is the
moment method im pedance matrix, which has the form:
( 2 - 8)
The subm atrices represent the coupling between the y and ^-directed current elements. Since
only planar currents are considered in this thesis, there are no .t-components in the moment
method im pedance matrix in equation (2-8).
2.3 Determination of Network Parameters
The current distribution is obtained from the adm ittance matrix by exciting the structure at the
port locations. T he voltage excitation used, often called the delta gap model, is equivalent to
creating an infinitesimal gap at the port cell and placing a voltage generator at this location (Fig.
2-4). A ll other elem ents o f the voltage vector are set to zero to maintain a short-circuit at the
interfaces between the elements. In other words, the excitation vector may be written as:
(2-9)
where 8 . is the discrete delta distribution vector with the /th element being equal to I and all
others are 0.
Although such a m odel for the excitation mechanism is an idealization, it does provide
accurate results [18]. It w as shown that the use o f a voltage gap generator instead o f a coaxial line
feed excitation does not affect the phase nor the m agnitude of the reflection coefficient [ 10).
Once the current distributions are obtained, the circuit can be described through several
different netw ork param eters such as:
12
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Figure 2-4. Voltage excitation at the input port of a discretized interconnect geometry.
• Input impedance
• Reflection coefficient
• Scattering matrix
• Z, ABCD, Y, T, H matrices
These network param eters depend greatly on tw o characteristics that describe the current
distribution along a uniform line; namely the characteristic impedance and the propagation
constant.
2.3.1 Characteristic Impedance (Zo)
There exist various definitions of ZQ for m icrostrip line problems [10] and sim ilarly for
multiconductor problem s [20], The method used in this research does not depend on a certain Zq
definition. As a result, the characteristic impedance o f the various ports is not considered; all
values presented are normalized parameters. T he normalized impedance at a certain port
corresponds to the characteristics of the line at the port, and not some external ZQ. It has been
shown that the norm alized 5-parameters, for exam ple, can be used to obtain 5-param eters relative
to any arbitrary characteristic impedances using a linear transformation [21]. T he method allows
for such a transform ation provided that the true characteristic impedances of the lines are known.
This is achieved in this work by a post-processing Matlab program which converts scattering
parameters to generalized scattering parameters. G eneralized scattering param eters provide the
electrical scattering param eters o f a multiport system having different characteristics impedances
13
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at different ports [21].
2.3.2 Propagation Constant
T h e expression for the phasor current at any coordinate z along the line is:
(2-10)
The values o f the phasor coefficients // and /? are determ ined by four factors: the voltage
excitation at z= 0, the attenuation o f the line, the line length, and the load impedance. The com plex
propagation constant y may be w ritten as
( 2 - 11)
y = a + ;P
where a is the attenuation factor w hich, for shielded geom etries, represents the ohmic losses of
the interconnects and the dielectric losses o f the layers, causing the wave to diminish steadily. The
phase constant P is a function o f the whole geometry. Using an optim ization technique based on
the Levenberg-M arquardt method [22], the expression in equation (2-10) is curve-fitted to the
m om ent method solution. From this expression, the reflection coefficient is easily computed,
providing the normalized input im pedance [10]. For a given current distribution on a microstrip
line i, the normalized input im pedance would be:
2.3.3 Impedance and Scattering Parameters
A s previously outlined, follow ing the method o f m om ents solution, a matrix equation is
obtained as:
(2-13)
[ V l " [ZAfoAf] [7AfoAf)
w here Z m 0m is the moment m ethod matrix, [Vejtc] and
are l^ e excitation vectors and the
14
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current distribution respectively. The size of the matrix Z m 0m depends on the num ber of cells
used in the discretization o f the interconnect layout. F or example, the layout in Fig. 2-3, would
require 16 z-directed basis functions and lO y-directed basis functions to adequately describe the
layout with the grid discretization chosen. T he discretization procedure is automated for
uniformly discretized layouts by using the com puter program ‘M AKEINPUT’ described in
Appendix A. From equation (2-13), the network im pedance parameters are obtained as follows:
(2-14)
where Z imp is the im pedance matrix o f size NxN, with N representing the num ber o f ports of the
network. In order to extract Zjmp from equation (2-13), the network has to be excited with N
independent voltage excitations, as described in [23], where the nth excitation is given by:
T
(2-15)
where V, is the ith elem ent o f the excitation vector v exc, and
n
For each excitation, N associated input im pedances are generated. As a result, N
2
input
impedance terms arc created. The transformation from the input impedance vector to generate the
impedance matrix elem ents is as follows:
(2-16)
where [A] is the transform ation matrix outlined in [23]. O nce the normalized im pedance matrix is
found the scattering matrix [S] can be evaluated by:
(2-17)
15
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For the particular case o f two port networks, the choice of excitations given in equation 2-15 will
not provide independent sets as shown:
vexc(1) = f I -U
(2-18 a)
Vexc(2)= [-l 1}
(2 -18 b)
In this particular situation, even and odd mode excitations are used represented by [ I 1] and [1 -1),
for the even and odd modes respectively. The input im pedance, Z in, at each port is found for both
the even and the odd excitations. Together they are used to find the two-port impedance matrix:
, yodd
ye v e n
"11
/n
~
yeven
"
z 12
=
in
(2-19 a)
2
_
yodd
"
in
^2
( 2 '
, 9
b
)
T he scattering parameters are found from equation (2-17). Other network parameters are also
obtained through the use of linear transformations o f the im pedance matrix [24].
16
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Chapter 3
NUMERICAL IMPLEMENTATION
In this chapter the com putational implementation o f the method discussed in chapter 2 is
presented. A n explanation is provided o f how the geometry is described by means of the input file
to the program and how the algorithm efficiently produces the im pedance matrix. The electrical
com putation of the characteristics is explained in detail. Also, various parallel processing
im plem entations [251
developed and compared in terms o f computation and storage
efficiency. Convergence of the method is also addressed briefly.
3.1 Computer Program
The program is a user-oriented code based upon the m ethod of m om ents, for treating perfectly
conducting thin dielectric planar interconnects. The general flow o f the com puter program written
in FORTRAN can be subdivided into the steps shown in Fig. 3-1. The program employs a
non-uniform discretization o f the interconnects in order to accurately represent the la y o u t The
m atrix is used to compute the electrical quantities of interest such as current distribution, input
im pedance, or scattering parameters.
3.1.1 Input File
The param eters o f the input file that describe the multilayered geom etry to be analyzed are
n
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READ CAVITY MODEL
&DEFINE STRIP GEOMETRY |
/ COMPUTE IMPEDANCE^
SUBMATRICES FOR
I
EACH LAYER
/
_______________ I _______________
PLACE SUBMATRICES IN
GENERAL IMPEDANCE MATRIX
r_____
INVERT MATRIX I
1-----------
I
1_____________ ^
^
( COMPUTE CURRENT DISTRIBUTION
V
FOR GIVEN EXCITATIONS
( CALCULATE NETWORK
V.
PARAMETERS
Figure 3-1. Flow chart illustrating the main components of the computer code
listed in Table 3-1. Additional parameters in the input file include the frequency o f operation as
well as the level o f accuracy of the impedance matrix elements which is governed in part by the
summation limits m stop and nstop, which are further explained in section 3.1.2. The metallic
interconnects are subdivided in rectangular cells (see Fig. 3-2) which are described by the
parameters in Table 3-2. To input the layout definition in a user friendly manner, an interface
program w as written to convert bitmap images o f interconnect layouts into the format o f Table 3-2
(see also A ppendix A). Since bitmap images do not provide length o r width o f the mapped biLs,
this program can only be used to create uniformly discretized elemenLs. One may use this tool to
create a ‘skeleton’ layout and manually add a few non-uniform elements at edges for optimizing
the response o f the interconnects.
18
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Table 3-1: Input file description
Physical
variable
Description
a
Height
b
Width
I
Length
Dielectric
nl
N um ber of layers
layers
si
Source layer number
C avity
dimensions
Relative permittivity of layer n
Metallic
Interconnects
Operational
Frequency
Accuracy
It
Loss tangent o f layer n
nU
Num ber of interconnect layers
ne
N um ber of elements
######
Elem ent description (Table 3-2)
start
Starting frequency
nf
N um ber o f frequency points to analyze
increment
Frequency increment
mstop, nstop
Summation limits
Table 3-2:
Moment method elem ent description
Parameter
Description
(y,z)
Coordinates of the test function
front
The length o f the cell directly in front o f the above point
back
T he length o f the cell directly behind the above point
width
The width o f both cells
port
‘ 1’ if the cell is a port. ‘O’ if it is not.
19
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(b)
Figure 3-2. (a) Typical interconnect discretization, (b) Non-uniform piecewise sinusoidal
basis function described in Table 3-2
3.1.2 Impedance Matrix Formulation
T he impedance matrix is com puted based on the geom etry o f the structure. This entails the
com putation o f the impedance subm atrices corresponding to each layer at the operating frequency
(Fig. 3-3). A s illustrated in Fig. 3-3, the impedance submatrix Z*/y (/, j = y, /.; k, I = 1,2
w here N is the number o f cells) represents blocks o f the impedance matrix given by:
m sto p n s to p
tfil -
^
m
^
(m , n ) f - k ( m ) T y k ( n )
^i(n)
n
20
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N,
LAYER
(1.1)
LAYER
<U>
LAYER
(1.3)
LAYER
(2.1)
LAYER
(2.2)
LAYER
(2.3)
z “12
LAYER
(3.1)
LAYER
(3.2)
LAYER
(3.3)
S,
_ _ 12 VV
7?
12
7P
12
(a)
Figure 3-3. Description of the output matrix for a three-layer example (a) General impedance
matrix, (b) Impedance submatrix (c) Physical interpretation of one of the impedance terms.
where T and B represent the testing and basis function integrals over the discretized cell elem ents
k and /, respectively. Each matrix elem ent requires the com putation o f the two basis and testing
function integrals to account for the coupling between any two elem ents. A Galerkin procedure is
used in this algorithm where the testing and the basis functions are the same with:
y^i
-v/
y y f) =
|
COSky y ' f ( y ' ) d y +
J
COSky y ' f ( y ' ) d y '
(3-2 a)
■v r - 1
T fy )
= J s in kz z ’g ( z ) d z '
(3-2 b)
<7-1
w here the basis fu n ctio n s,/an d g, shown in equation (3-2) are defined in chapter 2 and kv k y are
the w avenum bers in the z and y directions respectively, which are dependent on the cavity size
[16] as follows:
21
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Similarly, the basis functions in the case for a --directed current source:
(3-4 a)
(3-4 b)
Upon exam ining the above equations, it can be noticed that the integrals are only dependent on
one variable, either the z or the y coordinate. This means that interconnect elements having a
common coordinate (aligned in the z or y direction) as well as having the sam e length (integration
limits) w ould produce the same integral. In order to reduce computation time, an initial search is
conducted in the program to find the cells containing similar integrals to eliminate any
redundancy in computation. In many cases, this initial step reduces the computation time of the
integrals by approxim ately 50-75%. The output file from the program contains the impedance
matrix as w ell as the z and y coordinates o f each cell which are necessary to compute the network
parameters. This is further explained in the next section.
3.1.3 Network Parameters
In chapter 2, it was shown how the im pedance matrix can be used to obtain the current
distribution on the conducting strips within the shielded geometry. From the knowledge of the
current distribution, the various network param eters are obtained using the steps shown in section
2.3. The m ethod is implemented using a function, Networkp, written for the MATLAB
22
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environment, which requires the input parameters shown in Table 3-3 . The portion o f the current
Table 3-3: Program Descriptions
Program
Description
1
A utom atic (using KAP)
Automatic parallelization software
2
Sum m ation partitioning
Parallelizing the loops over m and n
3
Individual elements
Integral evaluation and m atrix tiling
4
Individual elements with locality
Same as above with layout consideration
5
Frequency partitioning
One frequency per processor at a tim e
6
Frequency partitioning with inte­
gral sharing
The M oM integrals are evaluated once and
then shared by all others
Program
No.
distribution extending from the ports, whic is represented by the variable strip, is extracted from
the admittance m atrix, y. This current constitutes only a portion of the total currents obtained. This
allows the method to be versatile in obtaining the network parameters for arbitrary interconnect
structures without relying on the shape, location of the discontinuity, or the location o f the ports
(Fig. 3-4). In other words, the network parameters can be computed from the current distribution
away from the discontinuity. As a result, only the current along the strips extending from the ports
is needed.
3.2 Parallel Implementation
Recently, parallel processing techniques have been successfully applied to a w ide range of
electromagnetic problem s, ranging from the study o f transients in pov/er system s [19] to their
implementation in w idely available codes such as the Numerical Electrom agnetics C ode (NEC)
[26]. This section discusses the effectiveness o f several parallelization techniques as applied to
multilayered structure problems, and compares their performance from a hardw are point o f view.
The formulation o f the problem is presented along w ith its numerical im plem entation. Various
techniques are successfully implemented and com pared in terms o f efficiency. The m ost efficient
23
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current distribution
Interconnect
Discontinuities
port
mici Dstrip
extraction
port n
(a)
(b)
Figure 3-4 The conducting strips extending from the ports towards the discontinuity are extracted for
the network parameter measurement. In turn the nature of the discontinuity or the location docs not
affect the routine: (a) Generic interconnects structure (b) Extracted current distributions at the ports.
techniques for shared-merr.ory m ultiprocessors are discussed in detail.
T h e parallel processing techniques applied in this research were developed to run on the
Kendall Square Research (KSR1) shared memory machine available at the University of Toronto.
The KSR1 is a highly parallel com puter that can be configured with up to 1,088 processors [271.
These processors are superscalar RISC devices with separate integer and floating point processing
units. The processors have the ability to perform separate add and multiply instructions in each
clock cycle, giving the processors a maximum rating o f 40 M FLOPS. The available m achine is
equipped with 32 processors, providing a maximum rating o f 1.28 GFLOPS. The total m em ory of
the system is 1 Gbyte.
F o r com plex structures containing a large number of cells, the parallel implementation
overhead becomes very small relative to the computation time o f the large impedance matrix [ 19,
26]. To allow for a true com parison o f the overhead tim e caused by the various parallel
im plem entations, all perform ance tests over domain partitioning and load balancing were run on a
24
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simple structure containing 100 cells in total.
3.2.1 Domain Partitioning
As one can deduce from the serial implementation o f the im pedance formulation outlined in
Figure 3-5(a), partitioning of the w orkload can be done effectively over three sets of variables: the
sum m ation parameters m and n, the individual discretized cell elem ents that make up the
interconnects, and the operating frequency. To take advantage o f these different partitioning
schem es, several FORTRAN program s were written and are listed in Table 3-3.
Input Data
Frequency loop
[
Input Data
Input Data
Frequency loop
MoM integrals
Summation loop 1
MoM integrals
Summation loop 1
MoM integrals
-J l
Summation loop 2
i i •
45
i i Frequency loop
I
I
I
I
I
I
Summation loop 1
1
H Summation loop 2
i l T
I
1 Il
Summation loop 2
MoM integrals
MoM integrals
\ \ : .T 7
(a)
\ ' (_L
\
\ I
*
\ V f
1 M I
\
v.
A
s. s
/
(c)
(b)
Figure 3*5. Comparison of three computational techniques: (a) serial implementation, (b) integral
partitioning, and (c) frequency partitioning.
First, a program was created by the KSR automatic parallelization pre-processor, called K AP
[28], fo r comparison purposes. T he second program is parallelized with respect to the m ode
sum m ations m and n. This version attem pts to provide good overall efficiency by achieving better
25
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load balancing, and w ill be discussed in further detail in the next section. In the third program, the
parallelization is done with respect to the discretized geometry (i.e. over the individual cell
elements) and is illustrated in Figure 3-5(b). The m ain drawback of this method is that processes
are created through every summation iteration which effectively reduces the efficiency by
increasing the overhead cost of process creation. Program 4 is an enhanced version of program 3.
Here memory locality is taken into consideration by minimizing data movement. Alternatively,
when parallelization is performed over frequency, no data dependence is encountered in the
matrix formulation. This ensures that invalidation latency [29] does not occur. Such frequency
partitioning is im plem ented in programs 5 and 6. T his latter program first com putes the moment
integrals before initiating the frequency dependent calculations. Taking advantage of the shared
memory architecture, the integrals are placed in global memory so that all processes can access
them independently, as illustrated in Figure 3-5(c).
3.2.2 Load Balancing
The overall efficiency o f all the techniques described above is highly governed by the balance
of the workload allocated to each processor. In program ., the summation limits m stop and nstop,
not only represent the extent of the partitioning dom ain but are also large relative to the number of
available processors. In this end, the w ork can be distributed evenly across the different
processors, w hich ensures an adequate load balance.
To achieve a favorable load balance in program s 3 and 4, the memory organization must be
taken into consideration. From the viewpoint of a single processor on a KSR/Serics computer, the
memory is organized in a hierarchical fashion w here data that resides in the low est level of the
hierarchy, although limited, can be accessed faster by the processor. For m ost o f the structures
analyzed, the data representing the number o f cells can be maintained in the local cache, giving
the program a good locality o f reference. If the complexity of the geom etry requires a larger
number of discretization cells, some data will reside in the next level o f m em ory which may cause
higher subcache misses. The latency generated would then occur evenly across all processors
26
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while still maintaining an optimum load balance. Furtherm ore, because the com putation required
for each impedance elem ent is independent of the geographic location of the cells, load balancing
is ensured as long as the elements are partitioned evenly across all processors.
In most microwave applications, characterization o f the M IC ’s is only required at a num ber of
frequencies com parable to the number of available processors. For such problems, load balancing
in the frequency partitioning case (programs 5 and 6 ) m ight not be optimum. Nevertheless, this
imbalance is reduced as larger number o f frequency points are analyzed, for exam ple, in the case
o f transient analysis.
3.2.3 Performance Evaluation
In all cases considered, parallel implementation provides better speedup for m ore complex
structures. This is m ainly due to the fact that having m ore cells means that a larger portion of
program time is spent on true computation. O verhead tim e owing to process creation becomes
negligible relative to the impedance matrix computation.
The parallelization techniques of programs 1 and 2 are not effective in providing any speedup
to the overall com putation. In the case of program 1, performance monitoring program s [27]
available on the K SR m ultiprocessor show that m ost o f the processors remain idle 87% of the
time. The reason fo r the poor performance o f this program is that the autom atic parallelism
preprocessor (KAP) only transforms loops with com pletely independent data, w hich occur solely
at the input and output stages o f the program. On the other hand, the poor perform ance o f the
second program, is caused by invalidation latency as w ell as memory contention, as various
processors attempt to update the same impedance elem ents for different values o f m and n.
Loop parallelization o f integral computation is achieved by programs 3 and 4 through the use
o f tiling, the process w hereby the execution o f a single do loop is transformed into parallel
execution o f multiple tiles, o r segments, of that oop [28J. M oderate speedup is realized with this
method because the integral computation is only a portion o f the total com putation. Therefore,
27
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much o f the original program is maintained in its serial form (Fig. 3-5). Another reason for the
m odest performance of program 3 was observed through the use o f available performance
m onitors. Computation o f some o f the impedance elements w as performed on a different
processor during each summation iteration. This problem was solved in program 4 by the creation
of affinity regions [31], These affinity regions force the same part o f the impedance matrix to be
executed on the same processor for every loop iteration, thus im proving cache performance by
m inim izing data movement. For the purpose o f illustration. Figure 3-6 shows the data movement
between processors throughout the mode summation process. A s an example, the data being
worked on by processor 1 during the first iteration will be subsequently updated by processor 9,
unless data affinity is used. In addition, a KSR compiler directive using a wavefront strategy [311
was used in program 4 in order to avoid deadlock when all processors try to access data belonging
to one particular cell at the sam e time. Results in Figure 3-7 show that program 4 outperforms
program 3 in term s of speed and scalability.
O f the techniques employed, frequency partitioning implemented in programs 5 and 6 provides
the best overall speedup. This is obviously due to the fact that the analysis is performed in the
frequency-dom ain, allowing the computation to be geared tow ards computing one frequency at a
tim e; thus providing a natural decoupling in terms of frequency. This method provided near-linear
response even for small structures as shown in Fig. 3-7. The figure also shows that the rate o f
speedup seems to decrease as the num ber o f processors is increased. This tendency is due to the
poor load balance caused by the increasing number of processors relative to the num ber of
frequency points used (which w as 20 points in this case). In program 6, the same frequency
partitioning technique is applied w ith the modification described in section 3.2.1. This provides
better than linear performance as shown in Fig. 3-7. Som e performance degradation is
encountered w hen larger numbers o f processors arc used due to the same reason ascribed above
for program 5.
In term s o f portability of the code, all parallel versions o f the method were implemented
through the u se of KSR-specific com piler directives. Therefore the programs cannot be directly
28
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i
S um m ation lo o p
c o u n te r = 1
Impedance
Matrix
|
j
T
With
y
data affinity /
\
Without
\
\
a'
\
S um m ation loop
c o u n te r ^ 2
Processors
1
2
3
4
5
6
7
8
S ...
Figure 3-6. Illustration of data irovement between processors with and without affinity.
compiled on other multiprocessor machines. T he techniques used in these program s however, are
quite portable and can be adapted to other shared-m em ory architectures. Timing resu io may difler
slightly depending on the network topology and the memory organization used.
3.3 Conclusions
The feasibility and the efficiency o f parallel processing techniques for the fullwave
frequency-domain problem have been demonstrated. Comparison of the various techniques shows
that static frequency partitioning o f the work provided the most noticeable speed im provem ent in
all tested cases. Tests also illustrated that the larger the interconnect structure being analyzed, the
greater the need for parallel computing. Speedup was found to approach linearity for more
complex structures.
29
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— ■- Linear
14
q
Integral partitioning
X
With affinity
X
Frequency partitioning
+
With integral sharing
Speedup
2
4
6
10
8
Number o f processors
12
14
16
Figure 3-7. Performance results ol the different techniques using a simple geometry w,'hl(X) cells.
Discretization of the interconnect layout was made simple w ith the use of the interface
program (bm2in). However, care m ust be taken in choosing the level o f discretization.
Convergence studies o f [18, 30] were used as a guideline in discretizing the structures, which
range from 12 to 100 cells per wavelength. In general, the length o f the individual cells did not
exceed X J2 0 , where Xq is the propagation wavelength at the centre frequency of the operating
range. F or example, the wavelength associated with 15 C H z is used for frequency analysis in the
range o f 1 - 30 GHz. The summation limits were kept constant at 800 for all structures analyzed.
T hese figures of m erit produced adequate results.
30
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Chapter 4
VALIDATION
In this chapter, numerical results obtained using the present code are compared with various
published data. The structures analyzed are chosen to highlight the m ethod’s abilities in handling
various configurations o f interconnects that may be encountered. The comparisons are m ade to
available CAD models where published measured data is not available. The CAD models used are
based mostly on closed form solutions and look-up tables obtained from experimental data [4],
The
geom etries
analyzed
fall
under
two sub-categories:
uni-planar and
multilayered
discontinuities.
4.1 Results of Uni-Planar Discontinuities
U ni-planar discontuinuities are considered here because they constitute a large portion of
modern-day M IC’s. In addition, the uni-planar geometries w hich form a sub-class of shielded
multilayered structures are studied to allow further com parison with other full-wave techniques
[ 10].
4.1.1 Series Gap
31
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Figure 4-1 show s a top view o f the series gap discontinuity analyzed in this work. This
represents one o f the simplest planar discontinuities. Results of the transmission coefficient S 12
compare more closely with measurements [ 10] than with the commercial CAD program LIBRA
[4], T his discrepancy is m ost likely due to the fact that the available CAD model only takes the
effect o f the side walls into consideration aw ay from the discontinuity. The gap analyzed was
discretized using 200 uniform overlapping elem ents o f length 1.8 mil each. This allows for the
analysis o f several gap widths by removing elements from the impedance matrix which
corresponds to increasing the gap width by 0.9 mil increments. This first step in the validation of
the method dem onstrates the ability to analyze two-port planar structures with uniformly
discretized interconnects.
4.1.2 Right Angle Bend with Non-Uniform Discretization
T he structure depicted in Fig. 4-2 is analyzed to extend the verification to non-uniformly
discretized structures. In addition, the right angle bend provides a further numerical test o f the
developed code: the transverse direction o f current is tested by having the second port
perpendicular w ith the first port. The structure is shown in Fig. 4-2 along with the comparison of
S j2 to that obtained from LIBRA, which agree reasonably well with the accepted model. The
ripples shown in the response obtained from LIBRA are believed to be due to a mismatch
between the port reference impedance of 50 Q and the true impedance o f the line, which is a
function of the line width, the substrate height, and the proximity o f the shielded structure to the
microstrip line. W ith the availability of non-uniform discretization, the interconnect subsections
can be further refined to monitor the current density as it flows through the bend. The cell width
used in this case was 2.54 mm and the length was 0.254 mm. With the application o f fine
discretization at the bend location, an approxim ation of a mitered bend, which is common in
today’s circuits, is achievable.
4.1.3 Four-port Cross
A s discussed in Section 3.1, structures w ith more than two ports are handled differently using
32
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port
port
2
(a)
-
10-
-1 5 -
IsJ' 20
(dB)
-2 5 -
- LIBRA
x Measurements [I2 |
o Present method
20
Frequency (GHz)
(b)
Figure 4-1. (a) Top-vicw of the series gap structure analyzed. (er =9.7, W=h = 0.025”, b =
0.25", g= 0.009” [10)]. (b) Magnitude of S12 for the series gap discontinuity.
I■
33
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1 | y-etemen!
r~
*~'* z-etem
> ent
f.-:i
1
port 2
(a)
-2
-12
-14
S 12 (LIBRA)
S 12 (present method)
S „ (LIBRA)
Sn (present method)
-16
-18
-2 0
20
35
40
30
Frequency (GHz)
45
50
(b)
Figure 4*2. (a) Layout and discretization of the right angle bend analyzed with e^=9.8,
h=0.635 mm, a=3.18 mm, vv=2.545 mm, (b) Magnitude of the S12 parameter.
34
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60
the method outlined in 3.1.3. To test the N-port parameters extraction method, a four-port cross
junction was analyzed (Fig 4-3). A s shown, the results obtained from this work agree w ith those
o f the available CAD package LIBRA. The small deviation is most likely due to the inability of
the available CAD tool to truly m odel the interaction or coupling among the four branches o f the
cross. In LIBRA, the complete circuit is modelled by cascading the electrical representation o f
each sub-component, i.e. the microstrip lines, and the cross junction.
4.1.4 Coupled Line Filter
T he final planar circuit presented here is a two resonator coupled line filter described by Fig.
4-4. The response of this edge-coupled filter is compared with LIBRA and with measurements for
validation [10]. First a simple approximation of the filter w as simulated where the resonator
inter-spacing was kept constant. A comparison of the non-optimal filter is shown is Fig. 4-4(b) to
the analysis made by LIBRA. Fig. 4-4(c) shows the comparison o f the optimized filter prediction
to measured results o f the two resonator filter. Close inspection o f the computed response shows
that there exists a sudden jum p in the S parameters which occurs at approximately 18 GHz. This
may be due to the fact that the discretization used was not fine enough relative to the guided
wavelength at the high frequency region. The average subsection length used was one tenth that
o f the wavelength at the centre frequency o f 10 GHz. A possible improvement of the results may
be achieved by varying the discretization according to the w avelength o f operation. Nevertheless,
the approximation achieved here already provides good results. To verify that the code is correctly
handling the existence o f additional dielectric substrates, the substrate for this structure was
divided into two equally thick substrates having the same relative permittivity and the sam e total
substrate height. The code was executed for several frequency points. The S-paramcters produced
w ere exactly the same as those o f the single substrate case.
4.2 Results of Multilayered Discontinuities
4.2.1 Stripline Gap Junction
A n extension of the planar series gap discussed in section 4.1.1 is the two-layer overlapping
35
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port 2
port 4
(a)
0.7
—
0.65
UBRA
x • o Present method
0.6
|S| 0.55
0.45
0.4
0.35
0.3
Frequency (GHz)
(b)
Figure 4-3. (a) Top-view of the 4-port cross circuit analyzed with e^=9.8, /t=30 mil, «=75
mil, b=t=200 mil, wt=25 mil, W2=50 mil (b) Magnitude of the S |Bparameters.
36
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(a)
•10
•20
-60
•70
LIBRA
Present method
-60
•00
6
16
10
Frequency (GHz)
(b)
-to
•20
x Measurement
- Present method
-«0
-7 0
10
12
Frequency (GHz)
(C )
Figure 4-4. (a) Sketch of the two resonator coupled line filter analyzed [9] (b) Magnitude
of the S 1 2 parameter of the initial filter with j j = s2 = s$ = 0.025” (c) Magnitude of the SI2
parameter of the optimized filter.(er =9.7, w = h = 0.025” , / = 1.002”, b = 0.25”, a = 0.4”,
J, = j 3 = 0.005”, J2 = 0.025”)
37
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transm ission line ends depicted in Fig. 4-6. This simple structure is very important in the design
o f m ultilayered microwave circuit because it offers increased coupling relative to end o r side
coupled planar lines, and may be used as a basic building block for more complicated circuits
[33]. Figure 4-6(b) shows the transm ission scattering param eter S 12 as a function of overlapping
distances d at an operating frequency o f 10 GHz. The results are compared with a method based
on a combination o f spectral dom ain immittance matrix approach and standard CAD m ethods
[33]. Although some slight deviation is shown in the transmission region the overall response
resem bles very much that obtained by the above mentioned method.
4.2.2 Two-layer Resonator Band-Pass Filter
A s an application o f this m ethod for the analysis o f m ultilayered circuits, the suspended
stripline filter shown in Fig. 4-7 is simulated and compared to m easured data obtained from [351.
W hile, in the past, such circuits could only be modelled by cascading uni-planar microstrip
param eters with overlap param eters described in section 4.2.1, the present method allow s for
consideration of the com plete circuit as a single unit w ithin a multilayered geometry, i.e.
according fo r the total electrom agnetic interact’on occurring within the filter. The results shown in
F ig 4-7(b) agree well, except for the slight frequency shift at roll-off beyond 10.5 GHz. T he shift
could be due to the fact that the fabricated circuit was subjected to slight adjustments for tuning
purposes w hich are not taken into account In this situation. In addition, it has been shown that the
grid size chosen to discretize the filter could slightly shift the frequency response [ 18].
A nother set of tests was conducted with this type o f geom etry to study the effects o f the
packaging, and in particular the side walls, on the performance o f shielded circuits. This is
especially important for suspended substrate structures where the electromagnetic interaction with
side walls can affect the circuit’s performance as much as the cover shields above and below.
F igure 4-8 shows the am plitude o f the current on a uniform stripline inside the shielded geom etry
used for the filter. The current is created by a voltage excitation at one end of the line. Packaging
analysis results of this type are useful for the designer in determ ining the placement o f side walls
38
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0
X
Menzcl
Present Method
0
0
0
0
0
0
0
0
Overlap d (mm)
(b)
Figure 4-5. (a) Sketch of the two-layer stripline gap junction analyzed (b) The magni­
tude of the Sp parameter (er = 2.2, t = 0.254 mm, w = 1.75 mm, a = b = 5.0 m m ,/= 10.1
GHz).
39
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— Measured
FDTD
x Present Method
-5
-10
-1 5
IX
-3 5
-4 0
-4 5
-5 0
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
Frequency (GHz)
(b)
Figure 4-6. (a) The suspended stripline bandpass filter analyzed [12]. (b)
Magnitude of the Si2 parameter compared with FDTD results [42] and mea­
surements.
40
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without affecting circuit performance. Several CA D packages which handle suspended substrates
do not take the side walls into consideration. They are simply added later for shielding and
packaging purposes. D ata in Fig. 4-8 shows that placing the side walls a minimum o f 1.5 cm away
from the stripline would maintain the circuit p*'rform ance estim ated by available CAD packages.
0.02
0.018 -
m
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0 .0 02 0.5
1.0
i.5
2.0
2.5
3.0
Distance to the walls (cm)
Figure 4-7. The magnitude of the current for a microstrip transmission line as a function
of the distance from the sidewalls.
4.3 Summary of Numerical Results
In this chapter, several structures were analyzed to provide such parameters as effective
dielectric constant, S-param eters, guided wavelengths, and current distributions. Com parison was
made to other fullwave analysis results and available C A D packages. The deviations encountered
suggest that further study is required to establish a discretization scheme that w ould ensure
optimum results consistently. Comparison to geom etries with more than two layers o f
interconnects is required to fully validate the com puter program. The lack o f published
measurements o f these circuits hinders such analysis. However, the method is show n to be an
effective tool in the analysis o f multilayered circuits.
41
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Chapter 5
TIME-DOMAIN ANALYSIS
Present day h:gn speed digital circuit design relies heavily on time-domain analysis o f the
signals along the circuit interconnects to study such effects as propagation delay and cross talk.
D iscontinuities along the interconnect paths can introduce signal delay as well as reflections that
not only affect the response along the path but can also affect other areas of the system. The latter
consequence is one form of electrom agnetic interference (EM I). In the previous chapters, the
application and the validation o f a fullwave method was presented which allows the user to obtain
the desired data at discrete frequency points. In the present chapter, a method is presented that
allow s for the determination o f an analytic representation o f the interconnects in the form of a
continuous frequency domain transfer function. The state-space [36] representation of the transfer
function is then used to perform the required time-domain analysis.
5.1 The Method
Tim e-dom ain analysis based on frequency-domain characteristics is usually performed [37] by
transform ing the input signal V(t) into the discrete frequency-domain signal V/n(CO) via the Fourier
transform :
42
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vin (co)
=
j V ( t ) e jm d t
(5-1)
—oa
Vin((H) along with the discrete transfer function G (CO), obtained from the frequency-domain
analysis, are then used to produce the output Vouf(CO) as follows:
Vou i« °) = ^ ( ( 0 ) 0 ( 0 ) )
(5-2)
The output voltage is then usually transformed back to the time-domain by using the inverse
Fourier transform. In this thesis, a different method is used where the system frequency response
data, C (CO), is fitted to a continuous line: r transfer function,G(s), using
a least squares
optimization method [22]. Once the transfer function is obtained, a built-in MATLAB
routine,
LSIM, is used to provide the time response of the polynomial function to any arbitrary input.
LSIM converts the num erator and denom inator o f the transfer function to a state-space
representation [22] o f the system
A B U( t )
C D Ly (0.
G(s)
(5-3)
where x represents the states o f the system, u(t) the input, y(t) the output.
The state-space system is then simulated for a discrete input u to provide the desired output y.
This method provides several advantages. First, the transfer function obtained from this procedure
can be easily represented by lumped elements [32]. Hence an equivalent circuit representation is
produced in addition to the time-domain output (Fig. 5-1). Second, since the analog transfer
funrtion obtained is defined through the entire region o f interest, problem s o f matching the
Fourier transformed input signal points with the original discrete transfer points are avoided.
Third, once the analog transfer function is obtained, any arbitrary input signal can be subjected to
the system without resorting to Fourier transformation for every input. Finally, t h : transfer
43
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function can be correlated to the physical dimensions o f the analyzed structure to find an
equivalent model representation. This can be done by altering the physical dimensions and
m onitoring the change in transfer function parameters.
G(c)_ A(s?-#Z1XS2- z 2 ) ( s 2- z 3 )( s 2- z 4)
/ s 2- p1)(s2- p2)(s?-p3)
Output
Input
Figure 5-1. Obtaining an equivalent circuit from the transfer function. 'ITie lirst LC pair
can be evaluated based on one of the transfer function’s zeros, such as zl in this case.
5.2 The Computer Program
T he computer code, written for the MATLAB environm ent, acts mainly as a front- ‘nd to
various built-in functions within the commercial MATLAB p ickage. The functions used am
show n in Table 5-1.
Table 5-1: Matlab routines used in the com puter code.
-............
Function
Description
INVFREQS
Analog filter leastsquare fit to frequency response data
LSIM
Simulation of continuous-time linear systems to arbitrary inputs
ROOTS
Computes the roots o f the polynomial
A s shown in the flowchart of Fig. 5-2, the program uses an iterative technique to find the orders
for the num erator and the denom inator for the transfer function to best fit the computed data. The
44
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variance in the transfer function elem ents is used as a measure o f how w ell the fitted functions
approximates the actual data. Since the function is created based on discrete data provided, it is
only valid for the frequency range o f the input. In addition, a larger num ber of frequency points
used in the input would produce a m ore accurate interpolation o f the true transfer function. The
input requirem ents for the program are further explained in Appendix B.
Input frequency
data & initial
guess for num. den
INVFREQS
(MATLAB)
Change
num. den
NO
Good ^
approximation?
YES
RCOTS
(MATLAB)
signal
LSIM
(MATLAB)
Poles &zcros
output
Time-domain
output
Figure 5-2. Time-Domain analysis program flow chart
45
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5 3 Simulation of a Linear System
5.3.1 The Cross Junction
In order to validate the method, a simple geometry was chosen for the analysis. The planar
microstrip cross w as designed to be symmetric in order to test the method for reciprocity (Fig.
5-3). For exam ple, die scattering parameter describing the transmission from port 1 to port 2 must
be identical to that o f port 1 to port 4. Similarly, the following conditions m ust hold:
S 11= S 22= S 33= S 44
Sl2=Sl4=S23=S34
(5-4)
S 13=S 24
The linear system m ust satisfy reciprocity. Therefore the reciprocal o f the above conditions
must also be equal to each other. The cross was analyzed and validated for 10 frequency points
from 100 M Hz to 12 GHz with a commercially available package which handles planar
microstrip cross junctions. Additional frequency points were obtained using this package in order
to save time as well as to show that the tim e-dom ain analysis method can be used with any
frequency-domain results (Fig. 5-4).
The program was executed for three scattering param eters, namely S 12, S 13, and S 14. Third
order polynom ials fit the frequency-domain data very well as shown in Fig.5-4. F or example, the
polynomial fitting the S 14 data is given by:
„ , ,
S 14(s)
14
- 0.3985s3 + 75.97s2 - 8796.3s + 420230
= ------- =---------------5---------------------------------s 3 + 158.07s2 + 19792s + 845120
(5' 5)
The associated roots o f the numerator and the denom inator of the polynomial are then obtained by
using the ROOTS function to produce:
,
S t j (s)
14
0 2 - 132320) 0 - 7 9 . 7 )
^
(s 2 - 13873) ( s + 60.92)
(5 6)
The transfer function above can then be used to find an equivalent LC representation. The LC
network representation lies beyond the scope of this thesis but is a relatively straightforward
46
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(a)
USCB-MSUBI
U C G V C < t« U C 0 /T
port
u lin
pr
pflf* •!
tL i
p o t «»3
W -50
L-100
ur;uB-M5uBi
- msubi
UCOVCR-UCOVt
wsub
U CO vCP-M CO V
MWALl-*#Al. «
DATA
U SU B
vsuei
C H -tO
I USUB-USUB1
| U C G V C R ttU C O vt
I UWAll-WWAl^
-2 0 0
BMQ-1
» C H-< ?
SOr t •4
(b)
Figure 5-3. (a) The microstrip cross structure used for the time-domain analysis (b)
The schematic layout of the cross using LIBRA.
extension o f the above transfer functions [32]. Note that there are no resistors in the representation
since the analysis w as performed assuming perfect conductors, and lossless dielectrics.
Fig. 5-5 shows the response o f a gaussian pulse injected at port 1 using the above m ethod. This
type of pulse is used often in time-domain analysis tools, such as finite difference time-domain
(FDTD) methods because it provides simple transform ation to the frequency-dom ain. As
discussed in section 5.1, this method offers the advantage o f allowing the sim ulation o f any
arbitrary signal with very little additional work. Fig 5-5(b) shows the response o f the sam e cross
structure w hen subjected to a 200 psec rise-time step. The response at the three ports approached
47
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6 .1
-
-5 0
-1 0 0
- 6 .3
- 6 .4
- 6 .5
-
-2 5 0
6.6
- 6 .7
-3 0 0
,s
10 '
10
o S-param eter data
- Transfer function
10
10,o
Frequency (Hz)
10
Frequency (Hz)
10
(a)
- 3 .5
-5 0
-1 0 0
at
CL
-5
-2 5 0
-3 0 0
- 5 .5
-3 5 0
—
6
.9
10,10
10.9
Frequency (Hz)
1010
10 '
Frequency (Hz)
(b)
Figure 5-4. Comparison of the computed S-paramcter data and the polynomial transfer function
obtained using ibis method, (a) S12and Si4. (b) S|3.
48
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P o rt 1 (input)
P o rt2 (output)
0.8
P o rt3 (output)
0.6
0.2
0.2
-
0.05
0.1
0.1 5
0.2
0.25
0.3
Tim e (nsec)
0.35
0.4
0.45
0.5
(a)
P o rt 1 (input)
Port2 (output)
0.8
Port3 (output)
0.6
o>
0.4
0.2
0
0.1
0.2
0 .3
0.4
0.5
0.6
0 .7
0.8
0 .9
1
Time (nsec)
(b)
Figure 5-5. Time-domain response of the cross structure for an input excitation (a) Gaussian
pulse (b) Step with 5xl09 Voits/sec risetime.
49
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half o f the incident signal which agrees w ith transmission line theory approxim ations.
Incident
_n_
i _ r
Reflected
Transmitted
Figure 5-6. The transmitted and the reflected waves of the cross junction based on transmission line
theory, r = -0.5 since the lines have the same characteristics impedance.
Fig 5-7 show s the response to a typical electrostatic discharge o f a human body charged to I
kilovolt [34]. This figure demonstrates the versatility o f the m ethod in handling not only
operational signals but also interference signals on integrated circuits. Fig. 5-7(a) shows the
difference betw een the response at pert 2 and port 3 to be substantial, regardless of the fact that
the two ports should be at the same potential since they are connected w ith perfect conductors that
are equidistant from the source. Figure 5 -7(b) shows the high frequency com ponents of the same
electrostatic discharge of Fig. 5-7(a). All frequency components below 90() MHz have been
removed to explain the existence o f a potential difference only at the beginning o f the discharge.
The S-param eters, as shown in Fig. 5-4, start to deviate from the quasistatic value of 0.5 at
approxim ately 900 MHz. Therefore, only the high frequency com ponent o f Fig 5-7(b) creates the
difference show n in Fig. 5-7(a). The difference is shown to occur slightly after the 1 nsec
occurrence o f the input discharge which is due to the delay in travelling along the microstrip lines.
This small difference can be quite critical in high speed digital circuits if, for exam ple, the input to
a high speed active device is based on the potential between the two ports. Such analysis can serve
50
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- -Port 1 (input)
- Port 2 (output)
— -Port 3 (output)
(Port 2 - port 3)
3.5
2.5
<
It
3
0.5
I
- 0 .5
10
Time (nsec)
(a)
0.25
0.2
0.15
0.1
0.05
1.5
2 .5
3 .5
4.5
Time (nsec)
(b)
Figure 5*7. Time-domain response to a 1 kilovolt human-metal electrostatic discharge, (a)
Response showing complete discharge, (b) The high-frequency component of the discharge
is shown to occur only at the initial rise.
51
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as a powerful tool in the design of low noise high speed digital circuits.
5.3.2 L-Type Coupling Structure
Further validation of this method w as achieved by analyzing the circuit shown in Fig. 5-8. The
layout is shown to consist o f a m icrostrip transmission line on a 0.006” thick dielectric substrate
w ith relative permittivity o f 9.8, loaded by an inverted ‘L’ shaped coupler. The separation, w, used
in the analysis was set equal to the w idth of the microstrip lines o f 0.006” , and the length, /, o f the
coupled section is 0.79”. The analysis may be done in practice to study the effects o f loading,
caused by adding couplers on the data transmission line extending from port 1 to port 2. Such
analysis can be useful in determ ining the maximum num ber o f allow able couplers that can be
placed on the line without the generation o f unacceptable deterioration to the signal at port2.
(b)
Figure 5-8. The L-Typc coupling structure (a) Top view (b) 3-dimentional view /r1=0.01”,
A2 = 0.006”.
52
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A s with the cross junction discussed in section 5.3.1, this structure w as also analyzed at a few
frequency points with the present full-wave method of chapter 2, and additional points were
obtained using LIBRA (Fig. 5-9). As shown, the response does not exhibit the smoothness that the
cross junction has, as shown in Fig. 5-3. This is due to the m ism atch in the reference port
im pedance used of 50 ohm and the characteristic impedance o f the 0.006” microstrip line. This
width is chosen to demonstrate that for more complex responses, a higher order filter
approxim ation is required to model the response adequately. Fig. 5-9 shows the 16th order
transfer function used to model the response which required 40 iterations to obtain.
. S-param eter data
- Transfer function
0.9
0.8
0.7
0.6
0.5
v
0.4
0.3
0.2
Frequency (GHz)
Figure 5-9. The output response, S12, showing a gradual decline in transmission as fre­
quency is increased. The approximated transfer function is shown to deviate at higher fre­
quencies.
Figure 5-10 shows the response o f a ‘square’ pulse having sm ooth rise and fall times o f 350
psec injected at port 1. This pulse input is chosen to demonstrate the effects loading may have on
a digital signal as it propagates on a bus line for example. The advantage o f having an analytic
transfer function o f the single coupler geom etry is that the output (the ‘O ne coupler’ curve o f Fig.
53
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5-9) can be re-injected into the mathematical system to obtain the output o f a tw o-coupler circuit.
This recursive process can be carried out as m any times as required to study the effects o f having
many couplers as show n in the 20 coupler exam ple in Fig. 5-9. Although the recursion technique
ignores the electrom agnetic interaction among the various couplers, it nevertheless serves as a
good low frequency model if the couplers are kept far apart.
One coupler
—
Five couplers
Twenty Couplers
0.8
Voltage
(v)
0.4
0.2
20
40
60
80
100
120
Time (nsec)
140
160
180
200
Figure 5-10. Time-domain voltage on main line showing the effect of coupler loading.
5.4 Conclusion
In this chapter, a method for obtaining tim e-dom ain signatures from frequency response data
was presented. T h e approach was demonstrated for a generic microstrip cross junction, where the
transmitted signals were acquired for various tim e-dom ain input signals. The linear simulation of
a 500 point tim e-dom ain input requires less than 4 seconds to compute on a SPARC 2 workstation
using MATLAB, show ing its numerical efficiency.
54
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The only disadvantage o f this method is its inability to handle certain responses such as the
reflection terms o r S n for the cross structure. This is due to the fact that the reflection coefficient
is negative which is an im proper transfer function. In other words, the program fails to find an
analog filter that can produce a negative pulse. Further improvements is required to handle these
special cases.
55
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Chapter 6
CONCLUSIONS
6.1 Concluding Remarks
In this research, results for the electromagnetic behavior of shielded multilayered geom etries
w ere obtained based on a general com puter code developed using a full-wave numerical
technique. The code was further enhanced in both versatility and efficiency, by im plementing
non-uniform discretization o f interconnect geometries allowing the analysis of more com plicated
circuits without resorting to the unnecessary increase of discretization density. The program w as
also im proved by providing the ability to automatically recognize and eliminate com m on
com putations o f basis functions through the use of links. Parallel processing techniques w ere
developed and compared for efficiency and storage. It was found that a frequency-based
partitioning o f the workload is superior to spatial partitioning o f the physical interconnect layout.
Also, basis function integral sharing was found to further enhance the parallel code.
To validate the code several geom etries were analyzed, chosen to test the various features o f
the method. These included tw o-ports, N-ports, multilayered conducting strips, and non-uniform
discretization. A time-domain analysis was also developed that could potentially be used to treat
such problem s as high frequency interference which may occur in integrated circuits. A sim ple
planar geometry, the cross junction, was analyzed to dem onstrate the effectiveness o f the method.
56
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6.2 Improvements and Recommendations
Although results presented in this thesis dem onstrate the accuracy o f this m ethod, certain
issues ought to be addressed. For example, the structure o f the FORTRAN program m ing language
allows for static m em ory allocation. This means that the number o f interconnect layers must be
built into the program and not a user-specified option. An obvious improvement would be the
conversion to “C ” programming language which would allow dynamic memory allocation as well
as easier linked list handling of basis function integrals. Furthermore, to analyze realistic
structures, the incorporation of conduction loss o f interconnects as well as dielectric loss should
be im plemented, thereby adding greatly to the code’s generality. Dielectric losses can be
incorporated into any structure since the dielectric primitivities of all layers are treated as
complex numbers by the computer code. However, validation of the code’s ability to handle such
lossy structures by comparison with published results has to be performed before applying the
code to new lossy structures. It is also recommended to perform further convergence analysis due
to introduction o f non-uniform discretization. This is to test the validity o f the assumption that
discretization lim its o f uniform analysis holds for non-uniform analysis. A nother highly
recommended study is that of the incorporation of lumped elements as well as vias into the
impedance matrix, in order to analyze a m uch wider range of circuits. This is seen as a natural
step towards the developm ent of a general purpose full-wave electromagnetic analysis tool for
multilayered circuits.
*"7
*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX A
INPUT GENERATION PROGRAM
T he generation of input files describing complex structures can be quite cumbersome. Such
structures may have to be discretized into hundreds of cells to ensure adequate analysis. As
m entioned in Chapter 3, each of these cells is described by six variables:
• coordinates of the test function
• length o f the cell directly in front o f the above point
• length o f the cell directly behind the above point
• width o f both ceils
• type o f cell
To avoid manual entry of cell locations and the possibility o f human input error, a com puter
program was written in the “C” language to automatically generate the input from a com puter
draw ing. The program is executed as follows:
M akelnput <deltaz> <deltay> <inputfi,ename>
T he program initiates a bitmap editor which offers a Graphical U ser Interface (GUI) to draw
the layout (Fig. A -l). The structure is first drawn in black and white, or any set o f two colors. The
object may also be scanned from a drawn or photographed image. Once the drawing is completed
and the file is saved, the program executes another system program which converts the bitmap file
to an ASCn file. This simply converts the Hexadecimal image bit to T ’ if the bit in on (black) and
to ‘O’ if the bit is off (white). T he execution of this step can be done manually as follows:
bmtoa -char 01 <filename.bmp> <newfilename>
58
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H^bttMpJSSBSBHHBBSSHSSi^BSIil^BSSSSSSSS
(tof ile } (ElMiO ri l l —: Ohm) l « » w t : <boh> Si m : 30*3(1
Clear
(
Set
(
(
Invert
I
ttw u k
(
c«Tr
(
Point
)
c
C am
)
(
B«ct<nql«
t* r k
)
r~ ii^"~ )
)
fFUlri Kactanqlg)
(_
Clrcl*
)
C Filled Circle")
C Flood Fill )
(
Set Mot Spot
)
C Clear Hot ^ o Q
c
t>^°
~t
Figure A-l. The GUI drawing program used to create the input file.
The program then analyzes the new ly formed matrix and creates all the y and z directed cells
necessary to describe the layout. The input parameters, deltaz and d eltay are used for all the cells
as the physical length, in meters, of the front and back o f each cell. The output is saved in the file
<newfilename> which can then be appended to the original input file discussed in Table 3-1,
which contains the complete geometry description. The procedure is repeated for each layer in the
structure.
Since the GUI bitmap editor handles constant bit sizes, the program can only create uniformly
discretized grids. However, the program can be used to create a geom etry resembling that bemg
analyzed as much as uniform discretization allows. The user m ay then manually edit the edge
cells to m ake the input file conform exactly to the desired layout.
59
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APPENDIX B
TIME-DOMAIN ANALYSIS PROGRAM
This analysis is made possible with the use o f the program T D A N A L Y S IS . It is invoked in the
Matlab environm ent as follows:
[z,p,y]= TD AN ALYSIS {H, w, na,nb, input, t)
where
H
w
na, njj
An array o f complex frequency response data obtained with the
frequency-domain method
An array o f corresponding frequencies of H
A "uess of the maximum order of the numerator and denominator of the
transfer function which is:
A „ s + A„ _ , s + . . . + A , s + A
T( s) =
input
t
z
p
The time-domain input signal to the system
The discrete time array associated with the input signal
An array o f the zeros o f the system
An array o f the poles o f the system
such that:
( s - z „ ) ( s - z „ _ l ) . . . ( s - z 0)
T( s ) =
where
n, m
y
are the final numerator and denominator orders
is the discrete time-domain output of the system w hen subjected with the
signal input at times specified by t
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REFERENCES
1.
T. Itoh, Num erical Techniques fo r Microwave a n d Millimeter-Wave Passive Structures,
New York: John W iley & Sons, 1989.
2.
V. N, Rayapati, B. Kaminska, “Performance analysis o f multilayer interconnections for
megabit static randor- access memory chip,” IE E E Trans, on Components, Hybrids, and
M anufacturing Technology, vol. 16, pp. 4 6 9-4/4, August 1993.
3.
Jesse Sheinwald. “MMIC compatible bandpass filter design: A survey o f applicable
techniques,” M icrowave Journal, pp. 26-41, M arch 1994.
4.
EEsof Inc., Circuit Element Catalog, vol. 1, W estlake Village, CA 1993.
5.
K. C. G upta, M.D. Abouzahra. Analysis and Design o f I 'lanar M icrowave Components,
IEEE Press, New York, 1994.
6.
C. N. C hang, J. F. Cheng, “Hybrid quasistatic analysis o f multilayer m icrostrip lines,” IEE
Proceedings-H, vol. 140, pp. 79-83, April 1993.
7.
C. N. C hang, J. F, Cheng, “Fullwave analysis of multilayer m icrostrip lines,” IEE
Proc.-M icrowave Antennas and Propagation, vol. 141, pp. 185-188, June 1994.
8.
M. Yang, R .’ 1. Johnston, “A generalized spectral domain dyadic green’s function and its
application to microstrip transmission lines in m ultilayer structures,” IE E E Transactions
on M icrowave Theory Tech., vol. 36, pp. 813-816, Jan. 1993.
9.
D.I. W u, D.C. C hang, B.L. Brim, International Journal o f M M ICAE, vol. 1, pp. 48-58,
1991.
10.
L.P. Dunleavy, P.B. Katehi, “A generalized m ethod for analyzing shielded thin microstrip
discontinuities,” IE E E Transactions on M icrowave Theory Tech., vol. 36, pp. 1758-1766,
Dec. 1988.
11.
N.I. Dib, P.B. Katehi, G.E. Ponchak, and R.N. Simons, “Theoretical and experimental
characterization o f coplanar waveguide discontinuities for filter applications,” IEEE
61
with permission of the copyright owner. Further reproduction prohibited without permission.
Transactions on M icrowave Theory Tech., vol. 39, pp. 873-882, June 1991.
12.
Personal communications with T.E. van Deventer, University o f Toronto, 1993.
13.
P.B. Katehi, ‘A generalized method for the evaluation of mutual coupling in microstrip
arrays,” IEEE Transactions on Antennas and Propagation, vol. AP-35, pp. 125-133, Feb.
1987.
14.
T .E . van Deventer, P.B. Katehi, A.C. Cangellaris “ An integral equation method for the
evaluation o f conductor and dielectric losses in high frequency interconnects,” IEEE
Transactions on M icrowave Theory Tech., vol. 37, pp. 275-280, Dec. 1989.
15.
R .F. Harrington, Field computation by moment methods, IEEE Press, New Jersey, 1993.
16.
T.E. van Deventer, “Characterization o f two-dimensional high frequency microstrip and
dielectric interconnects,” Ph.D. dissertation. Dept, of Electrical Engineering and
Com puter Science, T he University of Michigan, 1992.
17.
E.H . Newman, “A user’s manual for the electromagnetic surface patch code: ESP Version
IV ,” Technical Report No. 716199-11, The Ohio State University, 1988.
18.
J. Sercu, N. Fache, F. Libbercht, D. DeZutter, “Study o f gridding and cell-cell interactions
in the method of m om ents analysis o f arbitrary shaped pi.mar circuits,” IE E E M TT-S
Digest, pp. 753-756, Atlanta, Georgia, 1993.
19.
D. M. Falcao, E. Kaszjurew icz, H.L.S. Almeida, “Application o f parallel processing
techniques to the sim ulation of power system electromagnetic transients,” IE E E
Transactions on Pow er Systems, vol. 1, pp. 8, 1993.
20.
K. G. Balmain, S.G. Zaky, T.E. van Deventer, “ Research consortium in electrom agnetics
and electrostatic discharge,” Report no. 4 fo r Bell Canada, pp. 8-10, 1994.
21.
D. M. Pozar, M icrowave Engineering, Reading, M A: Addison-W esley, 1990.
22.
M ATLAB Reference Guide, Control Toolbox, T he M athW orks, Inc., 1992.
23.
P.B. Katehi, “Mutual coupling between m icrostrip dipoles multielement arrays,” IE E E
Transactions on Antennas and Propagation, vol. 37, pp. 275-279, M arch 1989.
24.
D . A. Frickey, “C onversions between S, Z, Y, H, ABCD, and T parameters which are
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
valid for complex source and load impedances,” IEEE Transactions on Microwave Theory
and Tech., vol. 42, pp. 205-211, Feb. 1994.
25.
F. Jatou, T.E. van Deventer, “Modelling o f High-Frequency M ultilayered Interconnects
Using Parallel Processing Techniques”, International Journal o f Numerical M odelling:
Electronic Networks, D evices and Fields, vol. 8, pp. 233-242.
26.
A.D. Millis, D.J. Brammer, “Implementation of the num erical electromagnetics code
NEC3 on a parallel com puter,” IEE International C onference on Computation in
Electromagnetics, pp. 47-49,1991.
27.
Kendall Square Research Corporation, KSR/Series Performance Analysis, Waltham, MA,
1993.
28.
Kendall Square Research Corporation, KSR Parallel Programming Manual, W altham ,
MA, 1991.
29.
T. M owry, A. Gupta, ‘T olerating latency through software-controlled prefetching in
shared-memory multiprocessors,” Journal o f Parallel and D istributed Computing, vol.
12, pp. 87-106, Dec. 1991.
30.
T.G. Livemois, “Numerical and experimental analysis o f metal-insulator-semiconductor
microstrip structures,” Ph.D. Dissertation, University o f M ichigan, 1991.
31.
Kendall Square Research Corporation, KSR Fortran Program m ing, Waltham, MA, 1993.
32.
A. S. Sedra, Theory and D esign o f Analog Filters, U niversity o f Toronto, 1989.
33.
W. Schwab, W . Menzel, “On the design of planar m icrowave com ponents using
multilayer structures,” IE E E Transactions on M icrowave Theory Tech., vol. 40, pp. 67-71,
Jan. 1992.
34.
Personal communications with Rakesh Saini, University o f Toronto, 1994.
35.
W.
Schwab, W. M enzel, “Com pact bandpass
filters w ith
improved stop-band
characterisdcs using planar m ultilayer structures,” IE E E M TT-S Digest, DD-6, vol. 3, pp.
1207-1209, Albu^ueque, New Mexico, 1992.
36.
Chi-Tsong Chen, Linear System Theory and Design, N ew Y ork, Toronto, Holt, Rinehart
and Winston, Inc., 1984.
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37.
K. G. B alm ain, S.G. Zaky, T.E. van Deventer, “Research consortium in electromagnetics
and electrostatic discharge,” Report no. 3 fo r Bell Canada, pp. 14-16, 1994.
38.
G. W . Slade, K. J. Webb, “Computation o f characteristic im pedance for multiple
m icrostrip transmission lines using a vector finite element method,” IE E E Transactions on
M icrow ave Theory and Tech., vol. 40, pp. 34-40, June 1992.
39.
K. G. B alm ain, S.G. Zaky, T.E. van Deventer, “ Research consortium in electromagnetics
and electrostatic discharge,” Report no. 5 fo r Bell Canada, pp. 19-23, 1994.
40.
W . L. Stutzman, G. A. Thiele, A ntenna Theory and Design, New York, John Wiley &
Sons Inc., 1981.
41.
R. C. D aigle, G. W . Bull, D. J. Doyle, “M ultilayer microwave boards: manufacturing and
d e s i g n M icrowave Journal, April 1993.
42.
Personal communications with Xidong Wu, University o f Toronto, 1995.
64
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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