EFFICIENT SERIES IMPEDANCE EXTRACTION

USING EFFECTIVE INTERNAL IMPEDANCE

 

Copyright

by

Beom-Taek Lee

1996

 

Approved By

Dissertation Committee:

_________________________________

(Dean P. Neikirk, Supervisor)

_________________________________

(Francis X. Bostick Jr,)

_________________________________

(Hao Ling)

_________________________________

(Mircea D. Driga)

_________________________________

(Lawrence T. Pileggi)

_________________________________

(Paul S. Ho)

 

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

August 1996

 

Supervisor: Dean P. Neikirk

Abstract

As semiconductor device speeds continue to increase it is becoming more critical to accurately model the parasitic effects of the interconnects (both on-chip and off-chip) between the digital circuit elements. The simulation of signal propagation along the interconnects using SPICE or some other circuit simulator is desirable. Therefore, electrical circuit representation of the interconnects (i.e., an equivalent electrical circuit representation of the physical interconnect) is required. A critical part of such extraction is the determination of the series impedance produced by finite conductivity wires and power/ground planes. For instance, resistance and inductance are frequency dependent due to the skin and proximity effects. Such frequency dependent effects can be determined using Maxwell's equation solvers, but using this approach as the first step in parameter extraction is computationally intensive, and frequently too slow. Faster and more efficient geometry-to-circuit extraction is necessary for regular lossy transmission lines and complex three dimensional structures.

This dissertation presents an efficient and accurate quasi-static methodology of evaluating the series impedance of interconnects based on the effective internal impedance. Three effective internal impedance models are developed, which pre-characterize the internal behavior of conductors and assigns a complex impedance to the surface of the conductor. Therefore, the effective internal impedance replaces the conductor interior by a surface, and considerably saves computation time. This approach must be coupled with an electromagnetic field solver. For example, the conformal mapping technique can be combined with the effective internal impedance; here this approach is applied to various planar transmission lines, and shown to be numerically efficient and reasonably accurate. The effective internal impedance is more effectively incorporated with the surface current integral equations, the efficiency and accuracy of this technique is examined in this dissertation through several two and three dimensional structures of inter-chip interconnects, e.g. multichip modules (MCM) and print circuit boards (PCB), and comparisons are made to the rigorous quasi-static techniques of the volume filament technique (VFM) and the partial element equivalent circuit method (PEEC). The methodology using the effective internal impedance is shown to be fast and accurate and to be integrable with various circuit simulators.

 

TABLE OF CONTENTS

 

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2. Effective Internal Impedance . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Standard Impedance Boundary Condition. . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Effective Internal Impedance of Rectangular Conductors . . . . . . . . . . . . . 11

2.2.1 Plane Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Modified Plane Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Transmission Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Comparisons of Three Effective Internal Impedance Models . . . . . . . . . . 20

2.4 Application of Effective Internal Impedance in Lossy Transmission Line

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Boundary Element Method with the Standard Impedance

Boundary Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 Boundary Element Method assuming Impedance Sheet carrying

Surface Current on the Conductor Surface . . . . . . . . . . . . . . . . . .30

2.4.3 Other External Field Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter 3. Conformal Mapping Technique . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Conformal Mapping and Partial Differential Equations . . . . . . . . . . . . . . 34

3.2 Conductor Loss Calculation using Conformal Mapping Technique . . . . . . 36

3.2.1 Conformal Mapping Technique using Effective Internal Impedance

and Transverse Resonance Method. . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2 Perturbation Method combined with Conformal Mapping

Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.3 High Frequency and Low Frequency Limit . . . . . . . . . . . . . . . . 41

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Symmetric Twin Rectangular Conductors . . . . . . . . . . . . . . . . . . 44

3.3.2 V-shaped Conductor-backed Coplanar Waveguide . . . . . . . . . . . . 48

3.4 Discussion and Further Consideration . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.1 Standard Impedance Boundary Condition and Effective Internal

Impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56

3.4.2 Integration and Parameter Evaluation in Schwarz-Christoffel

Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.3 Limits of Proposed Conformal Mapping Technique . . . . . . . . . . . .58

Chapter 4. Surface Ribbon Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 Integral Equation Method using Standard Impedance Boundary Conditions

and Volume Filament Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1.1 Integral Equation Method and Spectral Domain Method . . . . . . . . . 62

4.1.2 Volume Filament Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Surface Ribbon Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.1 Discretization and Weighted-averaged Effective Internal

Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

4.2.2 Internal Inductance and External Inductance . . . . . . . . . . . . . . . . 71

4.2.3 High and Low Frequency Behavior of Surface Ribbon Method . . . 74

4.2.4 Surface Ribbon Method vs. Integral Equation Method using

Standard Impedance Boundary Condition . . . . . . . . . . . . . . . . . . 74

4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3.1 Single and Twin Circular Conductors . . . . . . . . . . . . . . . . . . . . .75

4.3.2 Single and Twin Rectangular Conductors . . . . . . . . . . . . . . . . . . 76

4.3.3 A Microstrip Line : Comparison of Surface Ribbon Method and

Integral Equation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.4 Asymmetric Coupled Striplines. . . . . . . . . . . . . . . . . . . . . . . . . .84

4.3.5 Crosstalk on Lossy Transmission Lines . . . . . . . . . . . . . . . . . . . 86

4.4 Discussion and Further Considerations . . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter 5. Three Dimensional Surface Ribbon Method . . . . . . . . . . . . 90

5.1 Partial Element Equivalent Circuit Technique. . . . . . . . . . . . . . . . . . . . . 91

5.2 Three Dimensional Surface Ribbon Method . . . . . . . . . . . . . . . . . . . . . 93

5.2.1 Discretization and Effective Internal Impedance . . . . . . . . . . . . . . 93

5.2.2 Current Continuity Condition and Mesh Analysis . . . . . . . . . . . . . 96

5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3.1 Coupled Right-angled Bends . . . . . . . . . . . . . . . . . . . . . . . . . . .98

5.3.2 Coupled Meander Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.3 A Microstrip Line over a Meshed Ground Plane . . . . . . . . . . . . . 105

5.4 Discussion and Further Considerations . . . . . . . . . . . . . . . . . . . . . . . .110

Chapter 6. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111

Appendix Analysis of Coupled Lossy Transmission Lines . . . . . . . . .114

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125

 

LIST OF TABLES

2.1: Comparison of run time on an IBM RISC 6000 for boundary element method with the standard impedance boundary condition (SIBC) and without the standard impedance boundary condition (SIBC). BEM with SIBC is at least 100 times faster than BEM in assembling a matrix, and at least 5 times in solving matrix with gaussian elimination algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

 

3.1: Comparison of run times on an IBM RISC 6000 for various conductor loss calculations. SRM [21, 22] and VFM [24] use gaussian elimination as a matrix solver, *: 24-point Gaussian quadrature with 10 segments between two singular points in conformal maps, **: 100 point integration in the mapped domain. . . . . . . . . .56

 

4.1: Comparison of run times on an IBM RISC 6000 for volume filament method (VFM), surface ribbon method (SRM), and integral equation method. SRM and VFM use gaussian elimination as a matrix solver, a uses fine segments, [beta] uses minimal segments, [gamma] assumes two plates for the signal line, and * is negligible time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82

 

4.2: Comparison of run times on an IBM RISC 6000 for volume filament method (VFM) and surface ribbon method (SRM). SRM and VFM use gaussian elimination as a matrix solver, * uses minimal segments, and ** is negligible time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

 

5.1: Comparison of the number of unknowns used and run times on an IBM RISC 6000 for partial element equivalent circuits method (PEEC), three-dimensional surface ribbon method (3DSRM), and 2-D approximation using the volume filament method (VFM). Gaussian elimination is used as a matrix solver, and * is in case of 5 periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

 

5.2: Comparison of the number of unknowns used and run times on an IBM RISC 6000 for three segmentation schemes of partial element equivalent circuits method (PEEC) and three-dimensional surface ribbon method (3DSRM). Gaussian elimination algorithm is used as a matrix solver, and * is in case of 9 periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108

LIST OF FIGURES

 

1.1: Critical frequency vs. resistance of interconnects with varying in-ductance of 1, 3, and 10 nH/cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

2.1: Plane wave model for the effective internal impedance. Transverse Magnetic (TM) wave is incident onto all surfaces of the conductor. Effective internal impedance is calculated by solving diffusion equation locally. . . . . . . . . . . . 15

2.2: Wave reflection and transmission from a lossy conductor in air . . . . . . . . . . . 15

2.3: Transmission line model for the effective internal impedance. A rectangular conductor is segmented into flat rectangles and four squares. The effective internal impedance is calculated from transverse resonance method and telegraphist's equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

2.4: Each right-angled triangle is segmented into several isosceles triangles to take care of current crowding towards the corner. Effective internal impedance is calculated from transverse resonance method and telegraphist's equations of non-uniform transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5: Comparison between three effective internal impedance models and the surface impedance for a rectangular conductor ( 20 um wide, and 4 um thick ). A(solid line): transmission line model; B(dashed line): modified plane wave model; C(dotted line): plane wave model; D(8): the surface impedance calculated by the volume filament method [24]. . . . . . . . . . . . . . .22

2.6: Comparison of DC internal inductance and total DC inductance with different effective internal impedance models in conjunction with the surface ribbon method [21, 22] for w/h ratio of 1 to 100. These are compared to each other and the results of the volume filament method [24]. A(solid line): volume filament method; B(dotted line): transmission line model; C(dashed line): plane wave model; D(dot-and-dashed line): modified plane wave model . . . . . . . . . . . . . 24

 

2.7: Comparison of resistance and inductance between boundary element method without the standard impedance boundary condition (full BEM) and boundary element method with the standard impedance boundary condition (surface BEM). A(8): BEM without SIBC; B(solid line): BEM with the surface impedance of isolated conductor as SIBC; C(dashed line): BEM with the surface impedance of twin coupled conductors as SIBC, which is calculated by the boundary element method. The conductor surface is divided into 90 uniform elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8: Comparison of resistance and inductance between boundary element method without the standard impedance boundary condition (full BEM) and boundary element method with the standard impedance boundary condition (surface BEM). A(8): BEM without SIBC; B(solid line): BEM with transmission line model as SIBC; C(dashed line): BEM with modified plane wave model as SIBC; D(dotted line): BEM with the surface impedance as SIBC, which is calculated by the volume filament method [24]. . . . . . . . . . . . 29

3.1: Conformal mapping process from the original plane to the intermediate plane, and finally to the mapped plane . . . . . . . . . . . . . 37

3.2: Resistance and inductance comparison between the conformal mapping technique with different effective internal impedance models and the surface impedance and the volume filament method for twin rectangular conductors. A(solid line): transmission line model; B(dashed line): modified plane wave model; C(dotted line): the surface impedance; D(8): volume filament method [24]; E(thick solid line): inductance calculation using perturbation method (3.19). 100 integration points used in the mapped domain, 160 unknowns used in VFM. . . . . . . . . . 46

3.3: Comparison between two effective internal impedance models and the surface impedance for the parallel plate conductors at d = 5t, d = t/2, and d = t/6 ( 20 um wide, 4 um thick, and 4 um gap ). A(solid line): transmission line model; B(dashed line): modified plane wave model; C(dotted line): the surface impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4: V-shaped conductor-backed coplanar waveguide (VGCPW). The angle corresponds to an anisotropically etched groove in (100) Si. . . . . . . . . . . . . . . . .49

3.5: CPW gap (b-a) as a function of the V-groove distance d for a constant impedance(50 ohm) design. Distance d = 16 um for a 50 [Omega] V-groove microstrip line ( i.e., no CPW ground planes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6: Comparison of various modeling techniques for the conductor loss of a 50 [Omega] CPW and a 50 [Omega] VGCPW. A(solid line): conformal map assuming conductor thickness is zero; B(dotted line): conformal map of finite thickness conductors; C(dashed line): surface ribbon method [21, 22]; D(8): volume filament method [24]; E(thick solid line): perturbation method (3.13). . . . . . . . . . . . . . . . . . 51

3.7: A 50 [Omega] microstrip line with V-shaped ground plane, designed using the boundary element method (BEM) [43, 44] and a 50 [Omega] microstrip line, designed using reference [26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.8: Comparison of various modeling techniques for resistance and inductance of a 50 [Omega] microstrip line. A(solid line): conformal map assuming conductor thickness is zero; B(dotted line): conformal map of finite thick conductors using the effective internal impedance of the rectangular conductor; C(dashed line): conformal map of finite thick conductors using the effective internal impedance of the flat conductor; D(dot-and-dashed line): surface ribbon method [21, 22]; E(8): volume filament method [24]; F(thick solid line): perturbation method (3.14). . . . . . . 53

3.9: Comparison of conductor loss for several 50 [Omega] designs. A(8): b - a = 15 um d = 100 um (normal 50 [Omega] CPW); B(dotted line): b - a = 21 um d = 31 um (50 [Omega] VGCPW); C(solid line): b - a = 40 um d = 20 um (50 [Omega] VGCPW); D(O): 50 [Omega] microstrip line; E(dashed line): 50 [Omega] microstrip line with V-shaped ground plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1: Modeling a lossy thick rectangular conductor in the integral equation method (IEM). (a) configuration of a thick rectangular conductor, (b) equivalent one thin plate with the surface impedance of a flat conductor, (c) equivalent two thin plates with the transfer impedance boundary condition. . . . . . . . . . . . . . . . . . . . . 62

4.2: Discretization of the conductor inside for the use of the volume filament method (VFM). The conductor is segmented into small rectangular filaments, and the current on each filament is approximated with appropriate basis functions. . . . 65

4.3: Discretization of the conductor surface for the use of the surface ribbon method (SRM). The conductor surface is segmented into small ribbons, the effective internal impedance (EII) is assigned at each ribbon, and the current on each ribbon is approximated with appropriate basis functions. . . . . . . . . . . . . . . . . . . . .68

4.4: Non-uniform discretization and the weighted-averaged effective internal impedance. The conductor surface is sequentially divided into non-equal width segments with a certain width ratio and the weighted-averaged effective internal impedance (EII) is assigned at each ribbon. A(solid line): position-dependent effective internal impedance; B(dotted line): weighted-averaged effective internal impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5: Comparison of the actual geometry and the equivalent model for the surface ribbon method (SRM). (a) Actual geometry with the descriptions of fields and material properties, (b) At low frequency equivalent model for the surface ribbon method (SRM) with the descriptions of fields and material properties, (c) At high frequency equivalent model for the surface ribbon method (SRM) with the descriptions of fields and material properties. . . . . . . . . . . . . . . . . . . . . . . . . 73

4.6: Comparison of resistance and inductance between surface ribbon method (SRM), and boundary element method (BEM). A(8): BEM; B(solid line): SRM. . . . . 78

4.7: Comparison of resistance and inductance between volume filament method (VFM), surface ribbon method (SRM), and boundary element method (BEM). A(8): BEM; B(dotted line): SRM using transmission line model; C(dashed line): SRM using modified plane wave model; D(solid line): VFM. . . . . . . . . . . . .79

4.8: Minimal segmentation of a microstrip line. Signal line is segmented into 4 ribbons and the ground plane is divided into 5 ribbons, where the width of each ribbon is dependent only on the dimensions of the structure such as signal line width, thickness, and height above the ground. . . . . . . . . . . . . . . . . . . . . . . . . . .81

4.9: Comparison of resistance and inductance of a microstrip line between volume filament method (VFM), surface ribbon method (SRM), and integral equation method (IEM) using the surface impedance boundary condition (SIBC). A(8): VFM; B(solid line): SRM; C(dotted line): SRM with minimal segments; D(dashed line): IEM; E(dot-and-dashed line): SRM assuming two plates for the signal line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.10: Comparison of resistance and inductance calculated using different segmentations in SRM. Resistance and inductance are normalized by the results of VFM. A(solid line): fine segments; B(dot-and-dashed line): one segment for the ground plane; C(dashed line): three segments for the ground plane; D(dotted line): five segments for the ground plane. Minimum segmentations use four segments for the signal line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.11: 50 [Omega] coupled asymmetric striplines. The signal lines are 10 um wide, 4 um thick, 10 um gap over a bottom ground plane and between signal lines, conductivity of copper, and 38 um thick dielectric. . . . . . . . . . . . . . . . . . . . 84

4.12: Comparison of resistance and inductance of coupled asymmetric striplines between volume filament method (VFM), surface ribbon method (SRM) with the same segmentation scheme as VFM, and surface ribbon method (SRM) with minimal segments. A(8): VFM; B(solid line): SRM; C(dotted line): SRM with minimal segments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.13: Example circuit configuration of coupled lossy transmission lines. . . . . . . . . 87

4.14: Frequency spectra of input pulse, rise time = 0.3 ns, producing an effective bandwidth of 2.26 GHz. . . . . . . . . . . . . . . . . . . . . . . . 87

4.15: Output voltage at the end of the active line and far end crosstalk of the quiet line. (a) Output voltage of the active line with 15 cm length and matched termination, (b) Far end crosstalk of the quiet line with 15 cm length and matched termination, (c) Far end crosstalk vs. line length with unmatched termination, (d) Far end crosstalk vs. line length with matched termination. A(dotted line): pure assumption; B(dashed line): assumption with DC resistance; C(solid line): lines considering the skin and proximity effects. . . . . . . . . . . . . . . . . 88

5.1: Discretization of the conductor inside for the use of partial element equivalent circuits (PEEC) method in an example of a right-angled bend. (a) The conductor is divided into small hexahedrons, and (b) an element is defined between adjoining nodes placed at the center of the hexahedrons; arrow indicates direction of current flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94

5.2: Discretization of the conductor surface for the use of the three-dimensional surface ribbon method (3DSRM) in an example of a right-angled bend. (a) The conductor surface is divided into small rectangular patches, and (b) a ribbon is defined between adjoining nodes placed at the center of the rectangles; arrow indicates direction of current flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3: Sub-divisions of the conductor interior for defining the effective internal impedance (EII) along the direction of current flows in an example of a right-angled bend. (a) y-axis, (b) z-axis, (c) x-axis; arrows indicate direction of current flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4: Circuital representation of the three-dimensional surface ribbon method (3DSRM) in case of a right-angled bend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5: Coupled right-angled bends. The lines are 10 um wide, 10 um thick, 10 um spacing, and conductivity of copper. Discontinuity section is defined on the bends from the corner of outer lines with l= 35 um. . . . . . . . . . . . . . . 99

5.6: Comparison of resistance and inductance of coupled right-angled bends between partial element equivalent circuits method (PEEC), three-dimensional surface ribbon method (3DSRM), and two-dimensional approximation using the volume filament method (VFM). (a) Discontinuity resistance and inductance with l = 100 um, (b) Total resistance normalized by resistance, (c) Total inductance normalized by inductance of uniform lines having the same DC resistance. A(8): PEEC; B(solid line): 3DSRM; C(dotted line): 2-D approximation using VFM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100

5.7: Coupled meander lines. The lines are 10 um wide, 10 um thick, 10 um spacing, 40 um period, and conductivity of copper. Two lines are mirror symmetric against . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103

5.8: Comparison of resistance and inductance of coupled meander lines between partial element equivalent circuits method (PEEC), three-dimensional surface ribbon method (3DSRM), and two-dimensional approximation using the volume filament method (VFM). (a) Per unit period resistance and inductance calculated with 10 periods, (b) Per unit period inductance calculated vs. the number of periods. A(8): PEEC; B(solid line): 3DSRM; C(dotted line): 2-D approximation using VFM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.9: A microstrip line over a meshed ground plane where the aperture is placed at an angle of 45 degrees with respect to the signal line. The signal line is 12 um wide, 2.5 um thick, 12 um over the ground, the mesh pitch is 100 um, the aperture is 50 um square, and the conductivity is that of copper. . . . . . . . . . . . . . . . . . 106

5.10: Comparison of resistance and inductance of a microstrip line over a meshed ground plane between partial element equivalent circuits method (PEEC), three-dimensional surface ribbon method (3DSRM), and a microstrip line over a solid ground. A(dashed line): PEEC with straight line approximation; B(dotted line): PEEC with one layer segment of the ground; C(8): PEEC with two layer segments of the ground; D(solid line): 3DSRM; E(dot-and-dashed line): a microstrip line over a solid ground. Meshed ground is considered by 5 apertures perpendicular to the signal line and 9 apertures along the signal line. . . . . . . . . . . . 107

5.11: Comparison of resistance and inductance of a microstrip line over a meshed ground plane with varying mesh pitch. (a) Resistance vs. mesh pitch, (b) High frequency inductance vs. mesh pitch. Aperture rati100 [%]. A(O and solid line): 3DSRM at 50% aperture; B(O and dotted line): PEEC with straight line approximation at 50% aperture; C(8 and solid line): 3DSRM at 25% aperture; D(8 and dotted line): PEEC with straight line approximation at 25% aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108