PREDICTIVE STATISTICAL ANALYSIS OF EMBEDDED MEANDER...

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PREDICTIVE STATISTICAL ANALYSIS OF

EMBEDDED MEANDER RESISTORS VIA

MEASUREMENT OF CANONICAL BUILDING

BLOCKS

A Thesis Presented to

The Academic Faculty

by

Lawrence A. Carastro

In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy In Electrical and Computer Engineering

Georgia Institute of Technology April 2002

Copyright ©2002 by Lawrence A. Carastro

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PREDICTIVE STATISTICAL ANALYSIS OF

EMBEDDED MEANDER RESISTORS VIA

MEASUREMENT OF CANONICAL BUILDING

BLOCKS

Approved:

____________________________________

Martin A. Brooke ____________________________________ Steveen Kenney

____________________________________

John Dorsey

Date Approved__________________

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DEDICATION

To My Wife Sheri

ACKNOWLEDGEMENT

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SUMMARY

A further developed passive device modeling methodology that accurately

predicts the high frequency electrical behavior, and the statistical variation of embedded

passive devices on a generic substrate, is presented. This methodology accurately models

new passive devices by utilizing a small number of test structures to deembed the lumped

element circuit values from canonical building blocks.

In essence, a nonlinear optimizer is used to find the optimal building block

models, by fitting extensive high frequency measurements of the test structures behavior.

This process is repeated on numerous fabricated test structures producing a relative

variation in the lumped element circuit values of the building block models. A predicted

range of operation for the actual modeled device is then generated using a MonteCarlo

simulation tool, and the relative variations of the building block models.

Two sets of thirty-two test structures are fabricated by depositing gold on a 96 %

Alumina substrate, and s-parameter measurements taken. The Hspice Optimization tool

uses these measurements to deembed the lumped element circuit values from the building

block models; which are contained within the fabricated test structures. Relative

variations of the deembedded element values are calculated and added to the device

model circuit file, for simulation in the Hspice MonteCarlo tool. By means of

comparison, between the range predicted by the MonteCarlo simulations and the actual

measured data, the statistical variations of the component values were shown to provided

an accurate representation of the overall passive device performance.

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This method has been experimentally verified for a 9-segment meander resistor,

by demonstrating that the variation in the complete equivalent circuit models, based only

on the circuit building blocks, can be used to predict such variations in the actual

fabricated device. An accurate statistical modeling of the same 9-segment meander

resistor utilizing only one test structure to deembed building block equivalent circuits will

also be successfully proven in this research.

To automate this research, Unix scripts were developed which act as interface

between user and computer by; automatically, running the complete Hspice Optimization

simulations, calculating the relative variations of the deembedded element values,

generating the device model circuit file, running the Hspice MonteCarlo analysis,

extracting and converting all s-parameters to polar coordinates, and saving and localizing

all circuit and optimization files.

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CHAPTER 1

INTRODUCTION

Passive components are referred to as “glue components” because they “glue”

integrated circuits together to make a system [1]. Currently, the majority of passive

components functioning in electronic systems are discrete, and dominate the area of a

printed wiring board in a typical electronic product. For example, a cellular phone may

consist of only about 20 integral circuits compared to 300-400 passives. Clearly, discrete

passives substantially influence system cost, size, and reliability. To meet the

requirements for the next generation of electronic packaging (i.e. smaller, lighter, faster,

cheaper, and more reliable), alternatives to discrete passives must be realized.

Recent advances in technology have allowed the integration of more functionality

onto a single IC by increasing on-chip transistor counts. This integration has lead to an

increase in passive component board area, and is now a limiting factor for further

reductions in board size. These further reductions would result in much smaller, lighter,

and more reliable systems, and potentially impact every component using electronic

circuits. Therefore, the advantages of board reduction or miniaturization have forced the

development of new technologies to remove passives from the top of the printed circuit

board, and to embed them within a substrate or a package.

Integral Passives defined, as any functional passive element that is formed on a

layer within the interconnecting substrate, is that new developing technology. Integral or

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embedded passives have the potential for increased reliability, improved electrical

performance, size shrinkage, and cost reduction. This potential has lead to a constant

focus on higher levels of system integration and miniaturization. Processes such as thick

and thin film processing have enabled materials to be deposited at the mil and micron

level for linewidths and spacing within multichip modules. Techniques such as low

temperature cofired ceramic (LTCC) processing can use thick film printing and stacking

of passive devices in a multilayer low-cost substrate. LTCC processing can create up to

30 layers of passive component integration with the possibility of combining it with an

integrated circuit within a standard IC size package. For the accurate design and

fabrication of these compact high performance systems, accurate modeling of embedded

passive components is becoming very important.

However, designing circuits with embedded components is non-trivial due to

electromagnetic interactions that lead to parasitics, and ultimately non-ideal frequency

behavior. Passive devices generally have complex geometries, non-uniform current flow,

and correspondingly complex field patterns; therefore, suffer from parasitic effects that

influence the electrical behavior of the device at different frequencies. For this reason,

accurate models of embedded passive components and the examination of component

value statistical variation are crucial for designing and characterizing high performance

systems.

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1.1 Thesis Organization

In this dissertation, a statistical analysis procedure is detailed and applied to a 9-

segment meander resistor, using a novel passive device modeling methodology. The first

part of this thesis examines the development and procedures of the modeling method.

The second section details the improvements to the modeling methodology, and

demonstrates the application of the analysis procedure to the statistical variation of a 9-

segment meander resistor using two separate deembedding procedures. The first

procedure uses two test structures to model four building blocks, which are comprised of

twenty-two different deembedded element values. The second procedure uses only one

test structure to model four building blocks, which are also comprised of twenty-two

different deembedded element values.

The statistical modeling of the 9-segment meander resistor with interactions is

examined, and good results are shown for the device fabricated on a 96% Alumina

substrate. A brief chapter-by-chapter outline of the thesis is given below.

Chapter 2 discusses several methods developed for the characterization,

modeling, and design of microwave and millimeter-wave passive structures. An

overview of various lumped element modeling methods and how they apply to statistical

prediction is presented.

Chapter 3 discusses the basis for this research which is the work done by Dr. Ravi

Poddar, Dr. Nan Jokerst and Dr. Martin A. Brooke in 1997 entitled “Accurate, High

Speed Modeling of Integrated Passive Devices” [28]. A test case using this modeling

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methodology is examined in detail for a 9-segment meander resistor, which demonstrates

that the predictive capability is both accurate and very fast.

Chapter 4 presents a detailed account of the improvements made to the passive

device modeling methodology detailed in chapter 3. The same 9-segment meander

resistor test case will be examined in detail and compared to the original methodology.

Chapter 5 will detail the successful modeling of the same 9-segment meander

resistor; however, only using one test structure to model the four building blocks and the

subsequent 22 deembedded element values. The results will be compared to the original

methodology (detailed in Chapter 3), and the improved methodology (detailed in Chapter

4).

Chapter 6 will present the statistical analysis procedure developed under this

research program. A question concerning whether the deembedded element values are

correlated or independent of one another, and how this applies to the construction of the

modeled device circuit file is discussed. Several statistical analysis techniques used to

evaluate and render analysis results will also be explained in detail.

Chapter 7 will detail the application of the statistical analysis procedure to a 9-

segment meander resistor using the improved modeling methodology detailed in Chapter

4. A comparison between correlated and independent element value variations, which

directly influence the construction of the modeled device circuit file, used in the

MonteCarlo analysis will be shown.

Chapter 8 will detail the application of the statistical analysis procedure to a 9-

segment meander resistor using only one test structure to model the canonical building

blocks as described in Chapter 5. A comparison between the results of Chapter 7

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(correlated and independent) and the results using only one test structure (independent)

will also be shown.

Chapter 9 will act as a tutorial for the operation of the Unix scripts, which

automate the entire modeling and statistical analysis process.

Chapter 10 will demonstrate the application of the statistical analysis procedure

by comparing MonteCarlo predictions with actual measured devices. Conclusions will

also be extracted regarding this analysis procedure along with recommendations for

further research.

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CHAPTER 2

BACKGROUND

2.1 Introduction

As presented in Chapter 1, accurate modeling of embedded passive components

and the examination of component value statistical variation are crucial for the design,

and fabrication of high performance systems that utilize these devices. Highly

miniaturized passive components, fabricated on GaAs, and high speed silicon substrates,

are extensively used in the microwave/RF community for use in microwave circuits;

whereas, good frequency dependent models must be obtained for successful design. As a

result of this requirement, much of the work in the area of passive component modeling

originates from the microwave engineering community [4].

A number of numerical methods have been developed for the characterization,

modeling and design of microwave and millimeter-wave passive structures. These

methods can be divided into 3 basic areas. The first is the numerical full-wave method,

which applies Maxwell’s equations to the structure and computes the electric and

magnetic field patterns. This method is the most flexible and general, however

simulation times are very long for complex structures.

Second is the analytical equation based method, which is derived from first

principles or from curve fitting using simulation or measurement. Generally speaking,

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computer efficiency increases with the amount of analytical effort required, but at the

same time, versatility of the method is reduced [4]. This is due to the limited number of

simple shapes that analytical forms are developed for, under some simplifying

hypotheses. This method is accurate and extremely fast.

Third is the measurement-based method that can be applied to any arbitrary

structure, which takes into account processing effects, and is very accurate at any

frequency range.

On a whole, applying any of these three methods to non-lumped element

simulation and modeling methods is difficult and time-consuming due to the complexity

of the modeled device. However, when lumped element models are used, the electrical

behavior information of the device at different frequencies must be synthesized to

deembed the equivalent circuit element values. To accomplish this goal, the

deembedding process incorporates four different methods: 1) Physical Equations, 2) EM

simulations, 3) Algorithms, and 4) Measurements.

In this chapter, an overview of various lumped element modeling methods and

how they apply to statistical prediction will be presented. Details can be found in the

various references.

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2.2 Lumped Element Models

Passive device modeling using lumped element models yields circuits or sub-

circuits that are easily and quickly modeled using a Spice type circuit solver. The

generated circuit models simulate very quickly, usually on the order of minutes or

seconds; therefore, providing a major speedup over methods that do not utilize lumped

elements [4]. These highly flexible circuit level models can be incorporated into the

circuit design stage to investigate affects on circuit performance. Recently, computer-

aided design tools such as Hspice have become indispensable in IC design; whereas,

accurate circuit simulation is dependent on both the structural validity of the device

models and the accuracy of the values used as model parameters. Therefore, the

extraction of an optimum set of device model parameter values is crucial in

characterizing the relationship between the model and the measured behavior. Fig. 2.2-1

is a schematic of a generic lumped element equivalent circuit model for a spiral inductor.

This equivalent circuit can also model a capacitor when the circled capacitor is added.

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Ls Rs

Cs

C

CsiRsi Csi Rsi

Cox Cox

Capacitance and Resistance of Silicon Substrate Modeled by Csi and Rsi

Capacitance and Resistance of Silicon Substrate Modeled by Csi and Rsi

Oxide CapacitanceBetween

Spiral and SiliconModeled by Cox

Oxide CapacitanceBetween

Spiral and SiliconModeled by Cox

Inductance of SpiralModeled by Ls

Inductance of SpiralModeled by Ls

Feed-Through Path ModeledBy Series Capacitance Cs

Feed-Through Path ModeledBy Series Capacitance Cs

Resistance of Metal TraceModeled by Rs

Resistance of Metal TraceModeled by Rs

Added for Capacitor ModelModeled by C

Added for Capacitor ModelModeled by C

Figure #2.2-1: Lumped Element Circuit

The extraction of lumped element circuit values fall into two basic classes, the

first obtains electrical behavior data (s-parameters) by calculation, using an EM

simulation tool, or by measurement from an actual fabricated device. A unique

equivalent circuit model is created, and the element values deembed using an

optimization routine such as Levenberg-Marquardt or polynomial equations in a black-

box configuration. The model is then simulated in a Spice type circuit solver. The

second is a direct calculation of the lumped element values using scalable process and

configuration dependent physical equations, or algorithms derived from first principles

and simplified using assumptions. The lumped element values are inserted into a unique

equivalent circuit model and simulated in Spice type circuit solver.

Individual modeling goals further divide the lumped element extraction process

into three groups. The first group is the scalable models whose lumped element values

are a function of configuration such as length, width, and metal trace separation. The

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second group is the subset models that partition a device into subsets and model the

individual subset, which are then assembled to create the device model. The third group

is the polynomial equation models that use transfer functions and predictive polynomial

equations to model the device.

2.2.1 Scaleable Lumped Element Models

Scaleable lumped element models can be extracted using four different methods.

The first method uses physical equations and optimization routines to deembed element

values using fixed and free variables, and empirical and semi-empirical methods.

In the paper by Lakshminarayanan, B., Gordon, H.C., and Weller, T.M., the

lumped element values are deembedded using a combination of circuit optimization (free

variables using measured s-parameter data) and process and configuration dependent

equations (fixed variables) [5]. The free variables improve the prediction of substrate

dependencies while the fixed variables ensure scalability for different capacitor values.

The inclusions of the two RLC branches model higher order resonances and are

optimized using the original element values as initial guesses. This model accurately

captures significant changes in frequency response by changing only the height of the

microstrip substrate; therefore, parameter values for intermediate capacitor sizes are

predicted with a high degree of accuracy. The uniform variation of parameters is a

consequence of model’s close ties to physical properties. Fig. 2.2.1-1 shows the cross

section of the ceramic substrate and the lumped element equivalent circuit used to model

the capacitor.

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Electrodes

Metalization

MicroStrip

Substrate

GroundPlane

t

εc

rε

Cross Section of CMC In Typical Micro strip MountCross Section of CMC In Typical Micro strip Mount

P1 P2ESL ESR

Cs

Cg Cg

C

L1 R1C1

L2 R2C2

Substrate-Dependent Model for Ceramic Multilayer Chip Capacitors

Substrate-Dependent Model for Ceramic Multilayer Chip Capacitors

Figure #2.2.1-1: CMC Substrate and Equivalent Circuit Model

In the paper by Ronkainen, H., Kattelus, H., Tarvainen, E., Ruhisaari, T., and

Andersson, M., [6] a systematic study of modeling, design, and fabrication of integrated

planar inductors on silicon with different inductor geometries and number of turns was

examined. This methodology developed a purely empirical and a semi-empirical lumped

element model of an inductor. Purely empirical models are basically useless in the

physical layout of inductors, and are also not capable of describing device and process

parameters changes; however, they do accurately reproduce measured high frequency

features. The accuracy is improved by increasing the number of sub-partitions using the

distributed model shown on the left in Fig. 2.2.1-2.

The more physical or semi-empirical model, models components of the equivalent

circuit using functions related to inductor layout and material parameters. These

functions or equations provide scalability, and the minimal number of optimized

parameters provides the speed [6]. The probe pads are modeled separately to determine

their parasitics, which enhances accuracy. This semi-empirical model was found to be

the optimum method in terms of speed and accuracy due to the scaleable design equations

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and minimal optimized parameters. Fig. 2.2.1-2 shows the equivalent circuit for the

distributed (empirical) and semi-empirical inductor model.

CP

RS LS

RIN ROUT

CIN COUT

Equivalent Circuit of Distributed Inductor Model

Equivalent Circuit of Distributed Inductor Model

Cint

R L

Csub1 Csub2

C1 C2

Rsub1 Rsub2

Rsub12

nin nout

Semi-Empirical Sub circuit Model for Planar InductorsSemi-Empirical Sub circuit Model for Planar Inductors

Figure #2.2.1-2: Equiv. Circuit for Distributed and Semi-Empirical Inductor Model

The second method uses a custom program to calculate the coefficients in the

physical equations using an EM solver.

In the paper by Kmayman, N. and Jain, N., [7] an EM solver is used to calculate

coefficients in the design equations. Basically, the designer inserts the size and the EM

solver solves the unknown coefficients, which yield the element values of the lumped

element circuits [7]. The method is completely automated using shell-scripts as interface

between user and computer. The models are separated into different families according

to structures, to improve model accuracy and speed by reducing unknowns in equations.

Newton-Gauss’s singular-value-decomposition algorithm is also used to improve

convergence and simulation time. Fig. 2.2.1-3 shows the design equations and the

equivalent circuit for a MIM Capacitor and a square spiral inductor.

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C L RsCp1 Cp2

Rp Rp1

ωC. tanδn.=

Rs r 2ψ. f.. 106.=

2 GHz < f < 35 GHz

25 µm< l < 400 µm30 µm< w < 100 µm

h = 200 µm εr=4

Capacitor

Substrate

C α12. l w( ). α2l. w.= pF

L α31w. α4w. α5l. α6= nH

r α1lw. α8w. α9l. α10= Ω/ Hz

Cp1 α112. l w( ). α12l. w. α13= fF

Cp1 Cp2=

Model & Scalable Design Equations for MIM Capacitor

Model & Scalable Design Equations for MIM Capacitor

L RsCp1 Cp2

C

2 GHz < f < 35 GHz

150 µm< l < 1 mm1.5 < turn < 5.5

h = 200 µmεr=4

Capacitor

Substrate s =10 µm

w =30 µm

L α11. ln l( ). α2= nH

r α3l. ln l( ). α4= Ω/ Hz

Cp1 α5w. l total. α6= pF

Cp2 α7w. l total. α8= pF

Rs r 2ψ. f.. 106.=

Model & Scalable Design Equations for Square Spiral Ind

Model & Scalable Design Equations for Square Spiral Ind

Figure #2.2.1-3: Equivalent Circuits and Design Equations for MIM Capacitor and

Square Spiral Inductor

The third method optimizes the Q-factor and Inductance using a geometric

programming method.

In the paper by del Mar Hershenson, M., Mohan, S.S., Boyd, S.P., and Lee, T.H.,

[8] a simple and efficient CAD tool for designing on-chip spiral inductors for use in a

variety of RF circuits is proposed. This methodology uses the physical dimensions of the

inductor as design parameters to compute globally optimal trade-off curves between

competing objectives such as Q-factor and total inductor area. The Design goals are

translated into geometric programs, which permits circuit optimization efficiently and

globally [8]. Fig. 2.2.1-4 shows the square spiral inductor geometry and the equivalent

circuit used to model the inductor.

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WS

d out

d in

Square Inductor Layout and Geometry

Square Inductor Layout and Geometry

Ls Rs

Cs

CpRp CpRp

1 2

Lumped Element Equivalent Circuit Model

Lumped Element Equivalent Circuit Model

Figure #2.2.1-4: Inductor Geometry and Equivalent Circuit

The fourth method is a neural network model that is trained using EM solver

generated s-parameter data.

In the paper by Wang, S., Wang, F., Devabhaktuni, V.K., and Zhang, Q.J., [9] a

sub neural network has been developed that establishes the mapping between original

model input space (physical/geometrical parameters) and approximate circuit model input

space. The space-mapped resistor model (SMRM) was simulated using Hspice; whereas,

the output prediction closely matched the EM simulated data. The model simulated in

minutes and was much faster than original EM simulation. This methodology proves that

hybrid neural models are computationally efficient and have an accuracy that is

comparable to EM simulation. Fig. 2.2.1-5 shows the neural model structure and the

equivalent circuit.

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IS 21RS 21IS 11

fL R W R

R 1 R 2 L 1 C 1 C 2

CircuitModel

RS11

Hybrid Neural Model Structure

Hybrid Neural Model Structure

L1R1

C1

C2

R2

C1

R2

Equivalent Circuit Model

Equivalent Circuit Model

Figure #2.2.1-5: Neural Model and Equivalent Circuit

2.2.2 Sub-Set Lumped Element Models

Sub-set lumped element models can be extracted using three different methods.

The first method divides the model into sub-units and models a single sub-unit.

The method described in the paper by Belk, N.R., Frei, M.R., Tsai, M., Becker,

A.J., and Tokuda, K.L., [10] decomposes the inductor into repeated sub-units and solves

for the field dependent three-dimensional charge and current distributions to determine

the electrical characteristics of the inductor from a suitable combination of sub-unit

characteristics [10]. Basically, the designer assembles the sub-units, determines the

coupling, and then accurately extracts the electrical characteristics. Skin effect increases

resistance at higher frequencies and starts to reduce Q-factor as shown in Fig. 2.2.2-1.

Fig. 2.2.2-1 also shows the inductor and sub-unit configuration, and measured vs.

modeled inductance and Q-factor.

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123

Optimized Three-Turn InductorOptimized Three-Turn Inductor Fit of Algorithms to Measured DataFit of Algorithms to Measured Data

Figure #2.2.2-1: Optimized Three-Turn Inductor

The second method divides the model into unconnected concentric rings and

optimizes the Q-factor and Inductance as a function of area using a design space method.

In the paper by Rejaei, B., Tauritz, J.L., Snoeij, P., and Kayali, S., [11] this

methodology replaces an N-turn spiral by a series connection of N circular rings, and

assumes the rings are disconnected by modeling them as a 2N-port system with N Inputs

and N Outputs [11]. Assumptions such as one-dimensional/planar approximations and

circular symmetry simplify the field equations yielding an equivalent lumped element

model whose components are expressed in terms of electric and magnetic Green’s

functions. The lumped element model is too complicated for practical purposes;

therefore, by restricting m to a small integer only a limited number of resonant sub-

circuits are used. The measured vs. simulated results for the inductance (above) and loss

(below) illustrates that the models tend to overestimate the inductance, this is possibly

due to the 2D nature of calculations. The DC current was measured as a function of the

input voltage yielding resistor values in good agreement with the predicted ones. Fig.

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2.2.2-2 shows the spiral inductor and concentric rings configuration and some measured

vs. modeled data for the inductance and impedance.

R R

SS

Spiral Inductor and Concentric Ring Model

Spiral Inductor and Concentric Ring Model

Measured (solid line) vs. Simulated (dashed line)

Inductor Results

Measured (solid line) vs. Simulated (dashed line)

Inductor Results

Figure #2.2.2-2: Spiral Inductor Model and Measured Data

The third method divides the device into segments; whereby, the modeled device

is a combination of individually modeled segments. The lumped element model also

includes inductive and magnetic coupling, by adding magnetic coupling to the substrate

and including inductive coupling between metal traces and other devices.

In the paper by Pieters, P., and Beyne, E., [12] a method for the realization of an

optimum quality factor for spiral inductors given an inductance value and frequency

range is developed. This method approximates spiraling coils as concentric rings and

solves the algorithms for each separate ring. The equivalent circuit element values are

the function of algorithms and relate to physical effects of the component. An

optimization routine finds the highest possible Q-factor as function of inductor area and

frequency. This methodology creates an inductor design space that accurately predicts

inductor behavior for any possible geometry. The Design space for the evaluated

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geometrical parameter range is created by plotting the quality factor in a plane of

component values vs. frequency [12]. This research observed that for small inductances

values the highest quality factor was achieved at high frequencies, and the other way

around for larger inductances. Fig. 2.2.2-3 shows the equivalent circuit model and some

measured vs. modeled data for S11 and S22 magnitude, and also the Q-factor.

L R

C3

C1 C2Q

Imag ZinRe Zin

=

Lumped Element Model for Multi-turn Spiral Inductor

Lumped Element Model for Multi-turn Spiral Inductor

Comparison of Measurements and Calculated ModelsComparison of Measurements and Calculated Models

Figure #2.2.2-3: Lumped Element Equiv. Circuit & Measured vs. Modeled Data

In the paper by Arcioni, P., Castello, R., Perregrini, L., Sacchi, E., and Svelto, F.,

[13] a classical and an improved equivalent circuit model for inductors has been

developed. The classical equivalent circuit was unable to distinguish between losses in

metal strips and losses in substrate. The improved equivalent circuit introduces a

substrate resistance into the lumped element model, which is magnetically coupled with

the inductor and accurately describes the loss mechanism in both the metal and in the

low-resistivity substrate [13]. The separate characterization of the metal and substrate

losses allows the designer to separately evaluate their contribution to the inductor’s Q-

factor. The separate s-parameter measurements of the entire inductor and the pad isolate

the individual properties. There is a noticeably better fit for the improved equivalent

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circuit as shown in Fig. 2.2.2-4. The extracted parameters were compared to the expected

ones estimated from geometrical and technological data calculated using the Grover

method [14] yielding errors of 30% for capacitance values, (15%) for inductance values,

and (8%) for resistance values. The s-parameter measurements of the actual device were

compared to modeling results yielded good agreement. These measurements take into

account the physical phenomena underlying the coupling between metal and substrate.

The new proposed model is more accurate and highlights losses contributing at a given

frequency. Fig. 2.2.2-4 shows the classical and improved equivalent lumped element

circuit and some measured vs. modeled data for S21 magnitude and the Q-factor.

LYL RS

1:1

RSUB

CF

Improved Equivalent Circuit

Improved Equivalent Circuit

L

YL

RS

CF

Classical Equivalent Circuit

Classical Equivalent Circuit

Measured vs. Classical and Improved Model

Measured vs. Classical and Improved Model

Measured Device vs. Modeled Q-Factor

Measured Device vs. Modeled Q-Factor

Figure #2.2.2-4: Classical and Improved Equivalent Circuit Model and Measured

vs. Modeled Data

In the paper by Koutsoyannopoulos, Y.K., and Papananos, Y., [15] a generic and

process independent model for simulating the performance of arbitrarily shaped and

multi-layer inductors on silicon substrates was developed. This method breaks-up the

modeled device into segments and adds a mutual inductance parameter to the lumped

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element model. The figures on the right in Fig. 2.2.2-5 show all possible relative position

cases between two segments, this information is used to calculate the mutual inductance

between the traces. The Spiral Inductor Simulation Program (SISP) is generic in terms of

geometry and technology and performs a fast segment-by-segment extraction of the

equivalent SPICE subcircuit within seconds [15]. The performance prediction was

successful and accurate for the inductance and quality-factor and critical resonance

frequency. The relative percent errors were below 2% for (L) and (Q), below 5% for

resonance frequency. Fig. 2.2.2-5 shows the segment configuration, equivalent circuit,

magnetically coupled configurations, and measured vs. modeled data for the inductance

and Q-factor of a square spiral and octagonal inductor.

GsCs GsCsRsub

Nth Segment

SubstrateModeling

L R

Coupled to imageCurrent

Cp CpInsulator

Capacitance

K

R

Coupled to AllSegments Cf2

To AdjacentSegment Nodes

Zin Segment

Cf1

Equivalent Two-Port for One Segment of Spiral Inductor

Equivalent Two-Port for One Segment of Spiral Inductor

ρ

l

m

(a) (b)

ρ

m

ρ q

l

(c)

φ

l y

m(d)

ρ φ

l B

DC

v

A

m

µ

(e)

da

vm

D

BAφ

C

(f)

ρφ

l

D

C

Am

µ1

2m

Magnetically Coupled Segments in Special

Configurations

Magnetically Coupled Segments in Special

Configurations

Well

W S

d

thsio2

hsi

Parallel-Coupled Spiral SegmentsOver Si Substrate

Parallel-Coupled Spiral SegmentsOver Si Substrate

Comparison between Measured and Modeled L and QComparison between Measured and Modeled L and Q

Figure #2.2.2-5: Segments, Equivalent Circuit, Magnetically Coupled Segment

Configurations, and Measured vs. Modeled Data

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2.2.3 Polynomial Equation Models

Polynomial equation models can be extracted using two different methods. The

first method uses a high permittivity circular insert to model substrate effects and

calculate capacitance.

In the paper by Delaney, K., Barrett, J., Barton, J., and Doyle, R., [17A] a high

permittivity circular insert is used to enhance component performance and to help model

substrate effects. A proportional relationship between capacitance and number of inserts

was verified [16]. Three methods were used to complete the full electrical modeling

analysis; analytical modeling, boundary element modeling, finite element modeling. The

finite element method provided the best results however was very slow. A computer

program was written to apply the analytical model, which became the fastest performance

predictor of the three methods investigated. The analytical method calculates the

capacitance contribution of the insert and the LTCC material, and the MICAP program

calculates the fringing effects at interfaces between the two materials; whereas, defects

due to non-optimized process conditions were predicted. Fig. 2.2.3-1 shows the lumped

element equivalent circuits for a buried capacitor, resistor, and the equations used to

determine the element values.

22

Z R1R2

1 ω2R2

2. C2.1i ωL.

ωR22. C.

1 ω2R2

2. C2.=

C α β log ω( )2. λ log ω( )

3.=

D A Blog ω( )4. CD. log ω( )

12.=

CT KN. k. Nper.

E0ψ.

4 d.D2 λ D

tan θ( ) λ D( ).=

CPLATEε DE0

.

tLW.( ) Nψ. γ tan θ( ). γ tan θ D( ).

t tan θ( ). D2

4.=

Rc

Cg

Rg

Ci

Ri

RL

Ci

Ri

R

Resistor

Glass Matrix

InterfacialBarrier

Conductive Particle

[A]

[B]

Resistor

InterconnectingConductors

Equivalent Circuits for Buried Resistor

Equivalent Circuits for Buried Resistor

RcLc

Cp

Rp

CapacitorInterconnecting

Conductors

Equivalent Circuits for Buried Capacitor

Equivalent Circuits for Buried Capacitor

Lumped Element EquationsLumped Element Equations

Figure #2.2.3-1: Lumped Element Equivalent Circuit for Buried Capacitor and

Resistor, and Lumped Element Equations

The second method uses an interpolation technique and the ratio of two

polynomials to deembed lumped element values.

In the paper by Kwang Lim Choi, and Swaminathan, M., [17] a broadband

transfer function of any structure as a ratio of two polynomials is generated using a

minimum number of either measured or simulated data [17]. All lumped element

equivalent circuits or transfer functions generated may be used in Spice and simulated in

both frequency and time domain. Fig. 2.2.3-2 shows the geometrical representation,

equivalent circuits, material and physical parameters, and polynomial equation of a three

turn spiral inductor.

23

Zin s( )a 0 a 1 s. a 2 s2. a3 s3.

b 0 b 1 s. b 2 s2. b 3 s3.=

1114

106

98

8

Metal width = 8

Metal Dimensions of Inductor in Mils

Metal Dimensions of Inductor in Mils

= 5.6

tan δ= 0.001 α= 3e7 S/m

10

10

= 5.6

tan δ= 0.001

Metal

Ground

0.3

0.270 pF

Port 1

0.793 Ω

0.431 Ω

0.399 pF

6810 Ω

419940 Ω

3.543 nH

9.532 nH

εr

εr

Equivalent Circuits and Material and Physical Parameters of Inductor in MilsEquivalent Circuits and Material and

Physical Parameters of Inductor in Mils

Polynomial EquationPolynomial Equation

Figure #2.2.3-2: Geometrical Representation, Equivalent Circuit, and Polynomial

Equation Model for Three-Turn Spiral Inductor

2.3 Statistical Variation Prediction

Process parameters in contrast to electrical parameters are the optimum basis for

statistical circuit modeling of passive devices [21]. Therefore, to statistically model a

device material properties such as oxide and metal thickness variations and physical

properties such as dielectric constant and resistivity variations are required. In

methodologies that utilize lumped elements models, process variations equate to element

value variations.

For element values deembedded using Measurement the process would be to take

measurements of multiple devices, find a uniform parameter distribution in the lumped

element values, and insert percent device deviations into the MonteCarlo simulation tool.

The MonteCarlo simulation tool generates, a range of device performance.

24

For element values deembedded using Physical Equations metal deposition

variations would need to be obtained and inserted into the modeling equations. A large

number of lumped element models would need to be created and the individual element

values examined to calculate a uniform parameter distribution. The process follows the

method described above to obtain overall model performance.

For element values deembedded using EM Simulation and Algorithms process

variations would be entered into the calculations yielding a large number of lumped

element models. As in the categories above a uniform parameter distribution is generated

and MonteCarlo simulations run to accurately predict a device range of operation.

Ideally, the designer would like to demonstrate that the variation in the equivalent

circuit models, based only on the data used to extract them, can be used to predict

variations in actual fabricated devices; therefore, providing an overall accurate

representation of the device performance.

2.4 Discussion

A unique variety of methods for modeling passive devices using lumped element

models have been presented. All methods preformed their functions in diverse ways with

varying computational requirements and goals. While some methods focused on sub-

dividing the device and modeling just that sub-division, others moved towards scalability

and optimization of design space. Still, others focused on enhancement techniques

applied to existing lumped element models and substrates by adding inductive coupling

and substrate inserts. Finally, there was the neural network approach and the broadband

25

transfer function technique. Each methodology added its own uniqueness to the process

of deembedding lumped element values for insertion into a circuit solver.

To arrive at an optimum modeling methodology means defining the trade offs

between speed and accuracy for the different methods, as shown in Fig. 2.4-1.

Method Speed Accuracy NotesPhysical Equations Excellent Moderate Minimal high frequency information.

Physical Equation and Measurement

Very Good Very Good Optimizing only substrate effects.

EM Simulations Slow Excellent Slow due to 3-D field equations, however very accurate.

Algorithms Excellent Good Approximations and assumptions, loss in accuracy.

Measurement Moderate Excellent Contains actual information from substrate (non-ideal material properties) and device.

Figure #2.4-1: Speed/Accuracy Summary for Modeling Methodologies

A purely Physical Equation based method is fast, however does not model high

frequency behavior well. An Algorithm based method is fast and models high frequency

behavior well due to its close ties to numerical methods; however, has no way of

modeling non-ideal material properties. An EM solver based method equates to high

accuracy however speed is definitely lost in 3-D field equation calculations. A

Measurement based method models high frequency behavior well and contains nonideal

material properties, which would be impossible to model mathematically; however,

convergence problems can be open ended in terms of time and the device must be

fabricated before it can be modeled.

It appears that, to develop the consummate passive device modeling methodology,

a modeling method that employs the accuracy of “measurement” coupled with the speed

of “physical equations” must be used.

26

2.5 Summary

The methods developed for the characterization, modeling, and design of

microwave and millimeter-wave passive structures, and an overview of various lumped

element modeling methods and how they apply to statistical prediction was presented in

this chapter. The next chapter will discuss in detail a novel modeling methodology used

as the basis for this research. A test case will be provided and will demonstrate the

accuracy and speed of this modeling methodology.

27

CHAPTER 3

PASSIVE DEVICE MODELING METHODOLOGY

3.1 Introduction

The point of departure for this research is the work done by Dr. Ravi Poddar, Dr.

Nan Jokerst and Dr. Martin A. Brooke in March of 1997 entitled “Accurate, High Speed

Modeling of Integrated Passive Devices” [22]. The fundamental idea behind the

modeling procedure (proposed in [22]) is that most designed passive structures are

comprised of several key geometrical building blocks, that is, they can be constructed

from several building block cells representing individual parts of the structure. These

building blocks can be defined in a number of ways, but careful selection can result in

relatively few building blocks needed. The theory being, if accurate models for each of

the building blocks along with interaction information can be obtained, then any structure

comprised of those building blocks can be modeled accurately using the individual block

models [23].

The building block equivalent circuits are derived from fabricated test structures

and measurements using optimization and extraction routines. Passive RLC models for

each embedded building block are extracted and take into account effects of process

nonidealities and unknown material properties. The objective here is to predict the

28

electrical behavior of arbitrary geometry passive devices in a standard circuit simulator,

thus enabling a major increase in modeling speed over methods that do not utilize lumped

elements.

To be a practical method it is necessary that the number of test structures needed,

to accurately model all of the building blocks required, be as few as possible. In the test

case detailed below, the computed fundamental building block models were used to

predict the behavior of a 9-segment meander resistor, and will show that the number of

test structures can be much fewer than the number of parameters and building blocks

used. Specifically, four building blocks will be modeled with 18 parameters and

extracted from only two test structures. The test case will demonstrate that the predictive

capability of the resulting models for new devices, fabricated from the building blocks, is

both accurate and very fast.

3.2 Design and Modeling Flowchart

The modeling methodology can be described in a flow diagram, as shown in Fig.

3.2-1. The first step in the modeling process is to identify what devices to model in a

process, and identify building blocks and design rules. The next step is to design and

fabricate test structures to characterize. Next, the test structures are characterized for

building block circuit extraction. The measured data is then used to set up optimizations

and determine initial guesses to extract equivalent circuits of the test structures and

building blocks. Once successful optimizations have been achieved, then the building

blocks with associated models and design rules are combined in a library.

29

Accurate ResultsAccurate Results

Simulate in SPICESimulate in SPICE--type type simulatorsimulator

DRCDRC

Design Desired Passive Design Desired Passive DeviceDevice

Canonical Block/Design Canonical Block/Design Rule LibraryRule Library

Generate Design Rules Generate Design Rules Associated with Each Associated with Each

BlockBlock

Model CanonicalModel CanonicalBlocksBlocks

PassPassPass

FailFailFail

Design &Design & FabFabTest StructuresTest Structures

Figure #3.2-1: Design and Modeling Flowchart

3.3 Detailed Resistor Modeling Procedure

3.3.1 Devices Modeled

Most resistors are designed in straight lines or serpentine structures; however, the

layout of choice for high frequency applications is usually straight lines. It is attractive to

explore serpentine or meander resistors for high frequency applications for several

reasons. First, meander resistors are more efficient in substrate area for the same

resistance value, and, if modeled correctly, could have larger application in the high

frequency arena. Secondly, the meander structure presents a difficult modeling problem

due to higher levels of parasitics, such as coupling effects between the segments of the

structure, potentially affecting the overall system response [24].

30

3.3.2 Identify Building Blocks

In the test case below, meander resistors fabricated with 25 µm linewidths and

spacings were considered. The meander (serpentine) geometry dictated three

fundamental building blocks that required characterization: 1.) A square building block

with connections on opposite sides; 2.) A U-shaped section connecting two parallel

segments of the resistor together; and 3.) A coupled block segment to characterize line-

to-line coupling behavior on a per square basis. Due to testing requirements, a probe pad

was added as a fourth building block. Coupling was only considered with respect to

nearest neighbors. A current flow visualization tool could determine the building block

sizes [25].

Current visualization software was used to predict current flow through a

representative serpentine resistor, shown in Fig. 3.3.2-1. The cutoff points for each of the

primitives were at the areas where the current contours stopped changing rapidly,

indicating constant current flow between the boundaries. Using this approach, the pad

primitive was taken to be the large pad square plus one adjacent line square, the material

square and coupled material square were taken as one unit of material square each, and

the U shaped primitive was represented by 3 squares on each of the horizontal and

vertical axes of the U shape.

31

Figure #3.3.2-1: Current Distribution Plot of Serpentine Resistor Structure

3.3.3 Equivalent Circuits

Each building block is modeled as a Spice compatible RLC circuit. For simple

uncoupled building blocks, such as a piece of straight line, simple RLC models based on

the partial element equivalent circuit (PEEC) are used for equivalent circuits [26]. In the

case of blocks where coupling is examined, e.g. coupled material squares (corresponding

to coupled adjacent lines or interacting material squares), coupled PEEC models joined

by coupling capacitances and mutual inductances are used, as shown in Fig. 3.3.3-1.

32

Building BlocksBuilding Blocks1 3

2 4

1 2

R=0 .09 Oh mL= 1E -11 HC = 1.1e-1 5 FCM =0.4 e- 15 FCC = 1.4e-1 5 FLM = 0 .20

LC

R L R

C C CC

LC

R L R

C C CC

LM

LMCM

1

2

3

4R=0 .08 OhmL= 1 E- 11 HC = 2.7 e-15 FCC = 1. 2e -15 F

L

C

R L R

CC C C

1 2

PEECsPEECs

Figure #3.3.3-1: Uncoupled and Coupled PEEC Circuits With Associated Building

Blocks

3.3.4 Test Structures

Once the building blocks have been defined, the next step is to characterize and

develop models. This is achieved through the use of carefully designed test structures.

The test structure set is designed to include all of the predefined building blocks. The

equivalent circuit of the test structure is made up of the equivalent circuits of each

building block. Once designed and fabricated in the process of interest, high frequency

measurements are taken and used to characterize each of the building blocks [23].

By virtue of device measurements, no assumptions are made regarding material

properties, layered dielectrics, or imperfect substrates, because all of these effects are

represented in the measured data. This makes the modeling procedure entirely process

independent.

33

To model the four stated building blocks, two test structures were fabricated, as

shown in Fig. 3.3.4-1. The first test structure is simply a line with probe pads on the two

ends. It is used to characterize basic uncoupled material parameters, including self-

resistance, inductance, and capacitance.

The second test structure is a 3-segment meander resistor. This structure allows

passive characterization of the U-shaped corner segments as well as line-to-line mutual

inductance and coupling capacitance.

Test Structure 1Test Structure 1

Coupled Square PrimitiveCoupled Square Primitive

U-Shaped BendPrimitive

U-Shaped BendPrimitive

Pad PrimitivePad PrimitiveMaterial Square PrimitiveMaterial Square Primitive


Figure #3.3.4-1: Test Structures and Primitives for Meander Resistor Modeling

It is interesting to note that a structure with first order coupling is actually a 4-port

structure, whereas the test structures are only 2-port devices. Thus only standard,

repeatable 2-port measurements are necessary. A 4-port device is more difficult to

measure in practice than a 2-port, since many different excitation and loading iterations

are required. The method of simply measuring two port structures and extracting all

required multiport information is a significant advantage over attempting to measure

coupling between physically disconnected devices [23].

34

3.3.5 Processing and Measurement

The test structure design was fabricated by depositing gold on a 96 % alumina

substrate, which had a surface roughness of approximately +/- 1.5 µm. A

photomicrograph of the fabricated test structures is shown in Fig. 3.3.5-1.

The test structures were measured using network analysis techniques, a DC curve

tracer, and a high precision multimeter. For the high frequency measurements, a HP

8510C network analyzer was used with a Cascade Microtech probe station and ground-

signal-ground configuration probes. Calibration was accomplished using a calibration

substrate and utilization of the line-reflect-match (LRM) calibration method. Data was

gathered for each of the test structures at over 200 frequency points between 500MHz

and 20GHz, and were stored with the aid of computer data acquisition software and

equipment. DC I-V measurements of the test structures were also made in order to

ascertain component resistances. At DC, parasitic capacitance and inductance have no

effect on the response and the measured resistance value, and, once properly apportioned,

can be used directly in the models of the building blocks.

35

Figure #3.3.5-1: Photograph of Fabricated Structures for Meander Resistor

Modeling

3.3.6 Modeling and Parameter Extraction

Circuit models must be generated for each of the defined building blocks. The

circuit topologies and parameters for the uncoupled and coupled building blocks are

shown in Fig. 3.3.6-1. These circuits represent models for the building blocks only; the

test structures and resistor circuits are comprised of many building block circuits

connected in accordance with the structure geometry.

36

Test Structure 1

Deembed MaterialSquare

Pad

R =0 . 08 O h mL = 1 E- 11 HC = 2.7 e- 1 5 FC C = 1.2e - 15 F

L

C

R L R

C C C C

1 2

R =0 .0 8 O hmL = 1 E- 11 HC = 2.7 e- 15 FC C = 1. 2e - 15 F

L

C

R L R

C C C C

1 2

Test Structure 2

Deembed

1Block

Coupled Pair

PEECsPEECs

R =0 .09 O h mL = 1 E -1 1 HC = 1 .1 e -1 5 FC M =0 .4 e - 1 5 FC C = 1 .4 e-1 5 FL M = 0 .2 0

LC

R L R

C C CC

LC

R L R

C C C C

L M

L MCM

1

2

3

4

R =0 . 08 O h mL = 1 E- 11 HC = 2.7 e- 1 5 FC C = 1.2e - 15 F

L

C

R L R

C C C C

1 2

Building BlocksBuilding Blocks

Figure #3.3.6-1: Building Blocks, Equivalent Circuits and Parameter Values for

Meander Resistor Modeling

The extraction of the circuit model parameters was achieved in several steps. Due

to the highly nonlinear nature of the generated system equations with respect to circuit

parameter values, a procedure of hierarchical optimization with respect to measured s-

parameter and DC resistance data was chosen [27]. All optimizations and simulations

were done using the Hspice circuit simulator on Sun Sparcstation 20 series workstations.

The starting point or initial guess of the circuit parameters was crucial for correct

optimization results. Therefore, an initial optimization was done assuming that each test

structure was comprised of one building block utilized repetitively across the length of

the structure on a per square basis. This method was very effective for obtaining a good

starting point for the optimization of the test structure circuits. Optimizations were

37

performed for frequencies up to 10 GHz, and results for the meander resistor test

structure (shown in Fig. 3.3.4-1) are shown in Fig. 3.3.6-2.

0

0.05

0.1

0.15

0.2

0.25

1.00E+08 1.00E+09 1.00E+10 1.00E+11

Frequency (Hertz)

S11

S11(R) MeasuredS11(R) ModeledS11(I) MeasuredS11(I) Modeled

Real

Imaginary

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.00E+08 1.00E+09 1.00E+10 1.00E+11

Frequency (Hertz)

S21

S21(R) Measured S21(R) ModeledS21(I) MeasuredS21(I) Modeled

Real

Imaginary

Figure #3.3.6-2: Measured vs. Modeled Results for the Meander Resistor Test

Structure: (a) S21 Real and Imaginary Response; (b) S11 Real and Imaginary

Response

3.3.7 9-Segment Meander Resistor

The computed fundamental building block models were used to predict the

behavior of a 9-segment meander resistor. An equivalent circuit of the resistor was

constructed by replacing each building block in the structure with its equivalent extracted

circuit, as shown in Fig. 3.3.7-1. Since only first level coupling was taken into account,

38

each material square in each segment of the resistor was coupled to its nearest neighbor

by a pair of mutual inductances and a coupling capacitance. As inferred from the circuit

description, the resulting circuit using 25µm linewidths and 300µm line lengths per

segment 9-segment resistor was a complex, highly interconnected system, consisting of

approximately 700 nodes. The longest path length of the resistor was approximately 0.35

wavelengths long at 10GHz.

Coupled LineBuilding Block

U Building Block

Uncoupled LineBuilding Block

Figure #3.3.7-1: Meander Resistor and Associated Building Blocks

3.4 Results

In spite of the large circuit size, the AC small signal circuit simulation was

completed in less than 2 minutes. The predicted s-parameters, shown in Fig. 3.4-1, were

compared to measured values for the same structure. Both real and imaginary parts of S11

and S21 were accurately predicted up to 5GHz. In comparison, the same structure was

39

designed and simulated in a method of moments solver with a 3GHz meshing frequency.

The structure required 72 minutes to complete, while consuming approximately 50MB of

system memory while utilizing 2 processors in a multiprocessing Sun workstation. Thus,

for this example, a speedup factor of approximately 35 was demonstrated.

-0 .1

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

1 .0 0 E +0 8 1 .0 0 E +0 9 1 .0 0 E +1 0 1 .0 0 E +1 1

F requency (H ertz )

S11

S 1 1 (R ) M e a s ure dS 1 1 (R ) P re dic te d (This P a pe r)S 1 1 ( I) M e a s ure dS 1 1 ( I) P re dic te d (This P a pe r)

-0 .8

-0 .6

-0 .4

-0 .2

0

0 .2

0 .4

0 .6

0 .8

1

1 .0 0 E +0 8 1 .0 0 E +0 9 1 .0 0 E +1 0 1 .0 0 E +1 1

F requency (H ertz)

S21

S 2 1 (R ) M e a s ure dS 2 1 (R ) P re dic te d (This P a pe r)S 2 1 ( I) M e a s ure dS 2 1 ( I) P re dic te d (This P a pe r)

R ea l

Im ag ina ry

R ea l

Im ag ina ry

Figure #3.4-1: Measured vs. Predicted Results for 9-Segment Meander Resistor: (a)

S21 Real and Imaginary Response. (b) S11 Real and Imaginary Response

40

3.5 Summary

This methodology shows that with enough frequency points and sufficient

sensitivity, accurate equivalent circuit extraction of individual building blocks from multi

building block test structures is possible. The fact that a non-singular system of

equations can be formulated for a linear network, with each equation generated by a

different frequency point, leads to the idea that simply by sampling the system over

frequency, we can obtain all the information necessary to deembed the various circuit

parameters which comprise that system [28]. Therefore, many more than one unknown

sets of equivalent circuit parameters can be de-embedded from the measurements of one

test structure. This theory will be proven for the same 9-segment meander resistor and

detailed in Chapter 5.

41

CHAPTER 4

PASSIVE DEVICE MODELING METHODOLOGY

IMPROVEMENTS

4.1 Introduction

Ideally, the designer would like to demonstrate that the variation in the equivalent

circuit models, based only on the data used to extract them, can be used to predict

variations in the actual fabricated device. Therefore, the more accurate the measured vs.

modeled results of the test structures, the more accurate the modeled device, and

consequently the tighter and more realistic the predicted range of operation.

The passive device modeling methodology described in Chapter 3 has been very

successful in modeling both 2D and 3D passive device structures. However, the

problems with this method lie in the occasional long convergence time experienced when

using the optimization routine. An initial guess and a range of possible element values is

required by the optimizer; whereas, inaccurate initial guesses coupled with non-inclusive

element value ranges and an excessive number of optimized elements, lead to long

convergence times or no convergence at all.

One solution to this problem is to decrease the number of optimized element

values by using physical equations (Table 4.1-1) to calculate some of the lumped element

42

circuit values. In the paper by Ronkainen, H., Kattelus, H., Tarvainen, E., Ruhisaari, T.,

and Andersson, M., entitled “IC compatible planar inductors on silicon,” a more physical

semiempirical model was developed. Several components in the lumped element

equivalent circuit, shown in Fig. 4.1-1, were modeled by functions related to inductor

layout and material parameters, and the remaining were deembedded through circuit

optimization [6].

Cint

R L

Csub1 Csub2

C1 C2

Rsub1Rsub2

Rsub12

nin nout

Semi-Empirical Sub circuit Model for Planar InductorsSemi-Empirical Sub circuit Model for Planar Inductors

Figure #4.1-1: Semi-Empirical Lumped Element Circuit Model

This equivalent circuit and the method used to arrive at the circuit values was

applied to the modeling method described in Chapter 3. The equations in Table 4.1-1

were also singularly applied to the existing elements of the PEECs, from Chapter 3, in an

attempt to increase measured vs. modeled test structure accuracy.

43

Equation Number

L 1.5 µ 0. N2. D. e

3.7 N 1( ). W S( ).

D. DW

0.1.=

µ 0 is vacuum permeability, N number of turns, D length of side of rectangular inductor, and S and W are spacing and width of inductor traces, respectively

(1)

R ρlA

. 1 k f.( )=

ρ is resistivity of inductor metallisation, l length of inductor, A cross-section of wire, and k is empirical parameter

(2)

C 1 C 2 ε oxA ind A pads

t ox.=

ε r is permitivity of oxide, Aind ( Apads ) total area of inductor (pads), and tox oxide thickness

(3)

Table #4.1-1: Semiempirical Expressions for Inductor Model

The outcome from both attempts yielded less than accurate results. It was

apparent that the circuit topology in Fig. 4.1-1 could not hold the circuit behavior

information necessary to model the test structures, into the GHz. After this attempt

failed, the focus on improving the building block equivalent circuit model was contained

within the existing PEECs’.

After thorough investigation, improvements to the overall existing methodology

detailed in Chapter 3 were observed and implemented. These improvements resulted in a

substantial increase in measured vs. modeled accuracy for the test structures as well as

the modeled device. This chapter will follow the format of chapter 3 in detailing the

improvements made to the original methodology. The 9-segment meander resistor will

44

be modeled again using both versions (original & improved) of the methodology, and

will display the modeled vs. an actual measured device results.


The improvements to the novel passive device modeling methodology are

described using the same flow chart with the enhancements localized in the “Model

Canonical Blocks” section, shown in Fig. 4.2-1. The first step, as before, in the modeling

process is to identify what devices to model in a process, and identify building blocks and

design rules. The next step is to design and fabricate test structures to characterize.

Next, the test structures are characterized for building block circuit extraction; this is one

of the enhanced areas. The measured data is then used to determine initial guesses and to

set up optimizations to extract equivalent circuits of the test structures and building

blocks; this is also enhanced. Once successful optimizations have been achieved, then

the building blocks with associated models and design rules are combined in a library.

45

DRCDRC


Simulate in SPICESimulate in SPICE--type simulatortype simulator

Design Desired Design Desired Passive DevicePassive Device


Generate Design Rules Generate Design Rules Associated with Each BlockAssociated with Each Block


PassPassPass

FailFailFail

Design &Design &FabFabTest StructuresTest Structures

ImprovementsImprovements




As stated previously, the serpentine (meander) geometry of the resistor dictated

four fundamental building blocks that required characterization: 1.) A square building

block with connections on opposite sides; 2.) A U-shaped section connecting two

parallel segments of the resistor; and 3.) A coupled block segment to characterize line-

to-line coupling behavior on a per square basis; and 4.) A probe pad to connect to the test

equipment. Nearest neighbor coupling was also included between the material squares.

The current visualization software used by Dr. Poddar to predict current flow

through a representative serpentine resistor is shown in Fig. 4.3.1-1. The cutoff points for

46

each of the primitives are at the areas where the current contours stop changing rapidly,

indicating constant current flow between the boundaries. Using this approach, the pad

primitive was taken to be the large pad square plus one adjacent line square, the material

square and coupled material square were taken as one unit of material square each, and

the U-shaped primitive was represented by 3 squares on each of the horizontal and

vertical axes of the U shape. However after close examination, it was observed that the

current contours are continually changing throughout the corner primitive. Therefore

instead of modeling them as one block, each piece needs to be individually deembedded

to accurately reproduce the complex electrical behavior as shown in Fig. 4.3.1-1.

Close-UpClose-Up

Test Structure 2Test Structure 2U-Shaped BendBuilding BlockU-Shaped BendBuilding Block

U-Shaped BendBuilding BlockU-Shaped BendBuilding Block

Figure #4.3.1-1: Current Distribution Plot of Meander Resistor Structure

47


As stated previously, each building block is modeled as a Spice compatible RLC

circuit. For simple uncoupled building blocks, such as a piece of straight line, simple

RLC models based on the partial element equivalent circuit (PEEC) are used for

equivalent circuits [26]. In the case of blocks where coupling is examined, e.g. coupled

material squares, coupled PEEC models joined by coupling capacitances and mutual

inductances are used. The corner primitive was reevaluated and modeled as three

independent PEECs’; 1.) One for the input/output; 2.) One for the two corners; and 3.)

One between the corners. The test structures, the improved building block equivalent

circuits, and the respective PEECs’ are all shown in Fig. 4.3.2-1.



U-Shaped BendPrimitives


Pad PrimitivePad Primitive Material Square PrimitiveMaterial Square Primitive


Building Blocks / PEECsBuilding Blocks / PEECs

1 3

2 4

R=0 .09 OhmL= 1E- 11 HC = 1.1e-1 5 FCM=0.4e-15 FCC = 1.4e-1 5 FLM = 0.20

LC

R L R

C C CC

LC

R L R

C C C C

L M

LMC M

1

2

3

4

1 2

R =0 .08 O hmL= 1 E- 11 HC = 2.7 e- 15 FCC = 1. 2e- 15 F

L

C

R L R

C C C C

1 2

1 2

R =0.08 OhmL= 1 E- 11 HC = 2.7 e- 15 FCC = 1.2e- 15 F

L

C

R L R

C C C C

1 2

.

R= 0.4 O hmL= 3 .7 E- 11 HC = 5.3e-1 5 FCC=2 7e- 15 F

L

C

R L R

CC CC

1 2

CC CC

2L

C

R L R1

1

2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

Figure #4.3.2-1: Uncoupled and Coupled PEEC Circuits with Associated Building

Blocks

48


The building blocks are defined as in Chapter 3, and include the improvement to

the U-shaped bend primitive. The developed models are characterized using the same

carefully designed test structures, which include all of the predefined building blocks.

The structures are designed, fabricated, and measured to characterize each of the building

blocks.

By virtue of device measurements, no assumptions are made regarding material

properties, layered dielectrics, or imperfect substrates, because all of these effects are

represented in the measured data. This makes the modeling procedure entirely process

independent.

To model the four stated building blocks, the same two test structures were

fabricated, as shown in Fig. 4.3.2-1. The first test structure is a line with probe pads on

the two ends, and is used to characterize the basic uncoupled material parameters,

including self-resistance, inductance, and capacitance. The second test structure is a 3-

segment meander resistor. This structure includes the enhanced characterization of the

entire U-shaped corner segments as well as line-to-line mutual inductance and coupling

capacitance.


The test structure design was fabricated by depositing gold on a 96% Alumina

substrate. A photomicrograph of the fabricated test structures is shown in Fig. 4.3.4-1.

The test structures were measured using network analysis techniques. For the

high frequency measurements, a HP 8510C network analyzer was used with a Cascade

49

Microtech probe station and ground-signal-ground configuration probes. Calibration was

accomplished using a calibration substrate and utilization of the line-reflect-match (LRM)

calibration method. Because measured vs. modeled accuracy is directly related to the

number of evaluated frequency points, data was gathered for each of the test structures at

over 400 frequency points between 45MHz and 20GHz, and were stored with the aid of

computer data acquisition software and equipment.

Test Structure 2Test Structure 2Test Structure 1Test Structure 1

9-Segment Meander Resistor9-Segment Meander Resistor

Figure #4.3.4-1: Photograph of Fabricated Structures for Meander Resistor

Modeling



circuit topologies and parameters for the uncoupled and coupled building blocks, pad

building block, and improved multi-block U-shaped bend are shown in Fig. 4.3.5-1.

These circuits represent models for the building blocks only; the test structures and

50

resistor circuits are comprised of many building block circuits connected in accordance

with the structure geometry.

Test Structure 1


Pad

R = 0 . 08 O h mL = 1 E - 1 1 HC = 2 .7 e- 1 5 FC C = 1 . 2 e - 15 F

L

C

R L R

C C C C

1 2

R =0 .0 8 O h mL = 1 E - 1 1 HC = 2.7 e - 1 5 FC C = 1. 2 e - 15 F

L

C

R L R

C C C C

1 2

Test Structure 2

Deembed

Multi-Block

Coupled Pair

PEECsPEECs

R = 0 . 0 9 O h mL = 1 E -1 1 HC = 1 .1 e -1 5 FC M = 0 .4 e - 1 5 FC C = 1 .4 e-1 5 FL M = 0 .2 0

LC

R L R

C C CC

LC

R L R

C C C C

L M

L MC M

1

2

3

4

Building BlocksBuilding Blocks

.

R = 0 . 4 O h mL = 3 .7 E- 1 1 HC = 5 .3 e -1 5 FC C = 2 7 e - 1 5 F

L

C

R L R

CC CC

1 2

CC CC

2L

C

R L R1

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

Figure #4.3.5-1: Building Blocks, Equivalent Circuits and Parameter Values for


In an attempt to increase the measured vs. modeled test structure accuracy, several

different aspects of the circuit model extraction process were performed, and are shown

in Fig. 4.3.5-2. Because the number of element values extracted and the speed of

convergence are inversely proportional, using the smallest number of optimized elements

51

is desired. Several different circuit configurations and number of deembedded element

values were applied to test structures one and two, and analyzed using the Hspice

Optimization Tool in an attempt to evaluate speed vs. accuracy trade offs.

Test Structure 1

Test Structure 2

Deembed

Deembed

Different CircuitConfigurations

Different CircuitConfigurations

1. Semi-Empirical

2. No Substrate Resistance

3. Substrate Resistance Included

1. Semi-Empirical

2. No Substrate Resistance/1Block Corner

3. Substrate Resistance Included/1Block Corner

4. No Substrate Resistance/MultiBlock Corners

5. Substrate Resistance Included/MultiBlkCorners

Building Blocks1Block

Multi Block

Coupled Pair

MaterialSquare

Pad

Circuit Model Topologies

Figure #4.3.5-2: Test Structures and Primitives for Meander Resistor Modeling

To simplify accuracy evaluation, percent error between measured vs. modeled

was calculated at each frequency point and is shown in the plots below. After looking

deeper into the Hspice Optimization Tool [31], it was noticed that iterative optimization

runs increase measured vs. modeled accuracy. Therefore, Unix scripts were developed to

automatically run 18 iterative optimizations per measured device, while at the same time

increasing the accuracy settings and decreasing the range of possible element values

within the optimization circuit file. The building block characterization process executed

52

1152 individual optimizations to arrive at 32 unique equivalent circuit models of test

structure one, and 32 unique equivalent circuit models of test structure two.

The entire Optimization/MonteCarlo analysis process together with data

calculations, conversions, storage, and circuit file creation was automated using Unix

shell scripts. The detailed automated process will be showcased in Chapter 9.

4.3.6 Test Structure 1 Results

After inspecting the PEECs’ of test structure one and two used in Chapter 3, it

was found that the substrate resistance was held constant at 10Mega ohms while the

remaining building block element values were deembedded using the Hspice

Optimization Tool. Knowing the importance of substrate dependencies, this appeared to

be an area in need of improvement. In the paper by Arcioni, P., Castello, R., Perregrini,

L., Sacchi, E., and Svelto, F., [13] an improved equivalent circuit for silicon-integrated

inductors was presented. This model introduces a substrate resistance, which is

magnetically coupled with the inductor and accurately describes the loss mechanism in

both the metal and in the low-resistivity substrate. The separate characterization of metal

and substrate losses permits the designer to separately evaluate their contribution to the

inductor’s Q-factor. Therefore, the optimization of the PEEC’s with and without

deembedded substrate resistance was performed on test structure one and two for both

sets or 32 fabricated devices.

The percent error between the actual measured device and the modeled device

was calculated for test structure one, and is shown in Fig. 4.3.6-1 and Fig. 4.3.6-2. The

reflection coefficient (S11), which models the terminating impedance at the input of the

53

device, shows a 0.5% difference between no substrate resistance (no deembedded) and a

deembedded substrate resistance. There seems to be an improvement in both the high

and low frequency for the models that deembedded the substrate resistance.

Test Structure 1 No Substrate Resistance S11 Real Measured vs . Modeled

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

PercentError

Test Structure 1 Substrate ResistanceS11 Real Measured vs. Modeled

-0.5

0

0.5

1

1.5

22.5

3

3.5

4

4.5

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

Percent Error

No Substrate Resistance Deembedded~3 percent Error

Between Measured vs. Modeled

No Substrate Resistance Deembedded~3 percent Error


Substrate Resistance Deembedded~2.5 percent Error




Figure #4.3.6-1: S11 Measured vs. Modeled Percent Difference for 32 Fabricated

Structures and 32 Modeled Structures

The transmission coefficient (S21) that models the actual impedance of the

overall device shows a 0.25% difference between no substrate resistance (no

deembedded) and a deembedded substrate resistance. An improvement can clearly be

seen at low frequency for the models that did not deembed the substrate resistance;

whereas, a high frequency improvement is shown for the models that deembedded the

substrate resistance. The dividing line that separates the low and high frequency points

seems to be centered at approximately 3.5GHz. At this point substrate parasitics that are

a function of physical properties such as dielectric constant and resistivity, and skin effect

which are a function of material properties, begin to influence the circuit behavior.

54

Test Structure 1 Substrate ResistanceS21 Real Measured vs. Modeled

-0.5

0

0.5

1

1.5

2

2.5

3

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

Percent Error

Test Structure 1 No Substrate Resistancei S21 Real Measured vs. Modeled

-0.5

0

0.5

1

1.5

2

2.5

3

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

PercentError

No Substrate Resistance Deembedded~0.5 percent Error


No Substrate Resistance Deembedded~0.5 percent Error








Due to the focus on low frequency behavior (i.e. less that 5GHz), the evaluation

of test structure one clearly points to “not deembedding” the substrate resistance.


As stated previously, close examination of the current visualization plot found

that the current contours were continually changing throughout the corner primitive.

Therefore instead of modeling them as one block, each piece was individually

deembedded to accurately reproduce the complex electrical behavior of the corners.

Optimizations were then performed on test structure two; 1.) With and without the

substrate resistance; and 2.) Deembedding a single block corner vs. deembedding a multi-

block corner, for all 32 fabricated devices.

55


was calculated for test structure two, and is shown in Fig. 4.3.7-1 and Fig. 4.3.7-2. The

reflection coefficient (S11) for “no substrate resistance (no deembedded)/multiBlk

corner” shows a 1.5% improvement over “no substrate resistance (no deembedded)/1Blk

corner,” a 3.5% improvement over “substrate resistance (deembedded)/multiBlk corner,”

and a 7.5% improvement over “substrate resistance (deembedded)/1Blk corner.” The

building block model that “does not deembedded a substrate resistor” and that

“characterizes the additional corner primitives” is the optimum circuit model for S11 of

test structure two.

Test Structure 2 Substrate Resistance MultiBlk Corner S11 Real Measured vs. Modeled

-20

24

6

8

10

12

14

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

Percent Error

Test Structure 2 Substrate Resistance 1Blk Corner S11 Real Measured vs. Modeled

-505

10152025303540

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

Percent Error

Sub Resistance Deembedded MultiBlk Corner~6 percent Error


Sub Resistance Deembedded MultiBlk Corner~6 percent Error


Sub Resistance Deembedded 1Blk Corner~10+ percent Error


Sub Resistance Deembedded 1Blk Corner~10+ percent Error


Test Structure 2 No Substrate Resistance 1Blk Corner S11 Real Measured vs. Modeled

-2

0

2

4

6

8

10

12

14

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

Percent Error

No Sub Resistance Deembedded MultiBlk Corner~2.5 percent Error


No Sub Resistance Deembedded MultiBlk Corner~2.5 percent Error


No Sub Resistance Deembedded 1Blk Corner~4+ percent Error


No Sub Resistance Deembedded 1Blk Corner~4+ percent Error


Test Structure 2 No_Multi S11 Real Measured vs. Modeled

-202468

101214

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

Percent Error



56

The transmission coefficient (S21), which models the actual impedance of the

overall device, shows an overall percent error of 1.0%. As seen in the S21 plots, the

building block model that “does not deembedded substrate resistance” and that

“characterizes the additional corner primitives” is the optimum circuit model for S21 of

test structure two.

Test Structure 2 No Substrate Resistance 1Blk Corner S21 Real Measured vs. Modeled

-2

0

2

4

6

8

10

12

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

Percent Error

Test Structure 2 Substrate Resistance MultiBlk Corner S21 Real Measured vs. Modeled

-2

0

2

4

6

8

10

12

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

PercentError

Test Structure 2 Substrate Resistance 1Blk Corner S21 Real Measured vs. Modeled

-5

0

5

10

15

20

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

Percent Error

No Sub Resistance Deembedded / MultiBlk Corner~1.0 percent Error


No Sub Resistance Deembedded / MultiBlk Corner~1.0 percent Error


Sub Resistance Deembedded / MultiBlk Corner~3+ percent Error


Sub Resistance Deembedded / MultiBlk Corner~3+ percent Error


No Sub Resistance Deembedded / 1Blk Corner~1.5+ percent Error


No Sub Resistance Deembedded / 1Blk Corner~1.5+ percent Error


Sub Resistance Deembedded / 1Blk Corner~10+ percent Error


Sub Resistance Deembedded / 1Blk Corner~10+ percent Error


Test Structure 2 No_Multi S21 Real Measured vs. Modeled

-202468

1012

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

PercentError



57


The original/improved fundamental building block models were used to predict

the behavior of a 9-segment meander resistor using the original (Chapter 3) and improved

modeling methodology. An equivalent circuit of the resistor was constructed by

replacing each building block in the structure with its equivalent extracted circuit, as

shown in Fig. 4.3.8-1. Since only first level coupling was taken into account, each

material square in each segment of the resistor was coupled to its nearest neighbor by a

pair of mutual inductances and a coupling capacitance. As inferred from the circuit

description, the resulting circuit using 30µm linewidths and 510µm line lengths per

segment 9-segment resistor was a complex, highly interconnected system, consisting of

approximately 700 nodes. The longest path length of the resistor was approximately

0.175 wavelengths long at 20GHz.

MaterialSquare

R = 0 .0 8 O hmL = 1 E- 1 1 HC = 2.7 e-15 FC C = 1. 2e -15 F

L

C

R L R

C C C C

1 2

Pad

R = 0 .08 O h mL = 1 E- 11 HC = 2.7 e- 1 5 FC C = 1.2e -1 5 F

L

C

R L R

C C C C

1 2

Coupled Pair

R = 0 . 09 O h mL = 1 E -11 HC = 1 .1 e-1 5 FC M = 0.4 e- 1 5 FC C = 1 .4 e -1 5 FL M = 0 .2 0

LC

R L R

C C CC

LC

R L R

C C C C

LM

L MCM

1

2

3

4

.

R= 0 .4 O h mL= 3 .7 E- 1 1 HC = 5.3 e -1 5 FCC = 2 7 e -1 5 F

L

C

R L R

CC CC

1 2

CC CC

2L

C

R L R1

Multi-BlockL

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

Pad

Pad

Figure #4.3.8-1: 9-Segment Meander Resistor and Associated Building Blocks

58

4.4 Results

The AC small signal circuit simulation as before was completed in less than 2

minutes. Comparisons in the form of percent differences from the actual measured

device are shown in Fig. 4.4-1. The “improved” modeling methodology clearly

outperforms the original, yielding an improvement of approximately 1.0% (which is

actually a 50% improvement over the original 2% error) at low frequencies and up to

14.0% at frequencies greater than 3GHz. This is due to the enhanced modeling of the

corner primitives, which more accurately follows the complex high frequency electrical

behavior as the current moves through the corners.

Actual Measured 9 Segment Meander Resistor vs. Original and Improved Characterization MethodActual Measured 9 Segment Meander Resistor vs. Original and Improved Characterization Method

9 Segment Modeled ResistorOriginal Characterization Method vs. Improved Method

-2.00E-01

-1.00E-01

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

8.00E-01

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11/S21Real

S11 Measured DeviceS11 Original MethodS11 Improved MethodS21 Measured DeviceS21 Original MethodS21 Improved Method

S11R Percent Difference From Measured Device Original Method vs. Improved Method


9 Segment Modeled ResistorCalculated Measured vs. Modeled Percent Difference

Original Characterization Method vs. Improved Method

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21 RealPercent

Error

Improved MethodOriginal Method



-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11 RealPercent

Error




Figure #4.4-1: 9 Segment Meander Resistor Measured vs. Original and Improved

Characterization Methods

59

4.5 Summary

Improvements to the overall existing methodology, detailed in Chapter 3, have

been presented, yielding a substantial increase in measured vs. modeled accuracy and

tool/user interface. The 9-segment meander resistor was again used to showcase the

improvements made to the existing methodology. The percent improvement results

shown in Fig. 4.5-1 clearly demonstrate a 1.0% improvement at frequencies below 3GHz

and a much greater percent improvement for higher frequencies.

Original Method vs. Improved Method~1 Percent Difference Up To 3GHz

Original Method vs. Improved Method~1 Percent Difference Up To 3GHz

9 Segment Modeled ResistorCalculated Percent Improvement


-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11/S21 RealPercent

Improvement

S11 Percent ImprovementS21 Percent Improvement

Figure #4.5-1: 9 Segment Meander Resistor Original Characterization Method vs.

Improved Characterization Methods

60

CHAPTER 5

DEEMBEDDING BUILDING BLOCK EQUAVILANT CIRCUIT

ELEMENT VALUES USING ONLY ONE TEST STRUCTURE

5.1 Introduction

Another modification to the existing methodology, detailed in Chapter 3, was the

removal of one of the test structures used to deembed two of the four building blocks.

This chapter will detail the deembedding of all 22 element values from four building

blocks using only one test structure, as shown in Fig. 5.1-1. These improvements

resulted in a substantial increase in measured vs. modeled accuracy for the test structures

as well as the modeled device. This chapter will follow the format of chapter 4 in

detailing the departure from a two-test structure deembedding procedure to a one-test

structure deembedding procedure. The 9-segment meander resistor will be modeled

again using this new version of the methodology. A comparison between the original,

improved and the improved using 1 test structure will be displayed via measured vs.

modeled device results.

61

Test Structure 2Test Structure 2Coupled Square PrimitiveCoupled Square Primitive U-Shaped Bend

PrimitivesU-Shaped Bend

Primitives

Pad PrimitivePad Primitive

Material Square PrimitiveMaterial Square Primitive

Figure #5.1-1: Test Structure and Building Blocks


The improvements to the novel passive device modeling methodology are

described using the same flow chart with the enhancements localized in the “Model

Canonical Blocks” section, shown in Fig. 5.2-1. The first step, as before, in the modeling

process is to identify what devices to model in a process, and identify building blocks and

design rules. The next step is to design and fabricate one test structure to characterize.

Next, the test structure is characterized for building block circuit extraction; this is the

enhanced area. The measured data is then used to determine initial guesses and to set up

optimizations to extract equivalent circuits of the test structure and building blocks. Once

62

successful optimizations have been achieved, then the building blocks with associated

models and design rules are combined in a library.

DRCDRC


Simulate in SPICESimulate in SPICE--type simulatortype simulator

Design Desired Design Desired Passive DevicePassive Device


Generate Design Rules Generate Design Rules Associated with Each BlockAssociated with Each Block


PassPassPass

FailFailFail


ImprovementsImprovements




As stated previously, the serpentine (meander) geometry of the resistor dictated

four fundamental building blocks that required characterization: 1.) A square building

block with connections on opposite sides; 2.) A U-shaped section connecting two

parallel segments of the resistor; 3.) A coupled block segment to characterize line-to-line

coupling behavior on a per square basis; and 4.) A probe pad to connect to the test

equipment. Nearest neighbor coupling was also included between the material squares.

63

Using the current visualization approach, the pad primitive was taken to be the

large pad square plus one adjacent line square, the material square and coupled material

square were taken as one unit of material square each, and the U-shaped primitive was

represented by 3 individually deembedded squares on each end, as shown in Fig. 5.3.2-1.


Each building block is modeled as a Spice compatible RLC circuit. For simple

uncoupled building blocks, such as a piece of straight line, simple RLC models based on

the partial element equivalent circuit (PEEC) are used for equivalent circuits [26]. In the

case of blocks where coupling is examined, e.g. coupled material squares, coupled PEEC

models joined by coupling capacitances and mutual inductances are used. The corner

primitive are modeled as detailed in Chapter 4, with three independent PEECs’; 1.) One

for the input/output; 2.) One for the two corners; and 3.) One between the corners. The

test structures, the improved building block equivalent circuits, and the respective

PEECs’ are all shown in Fig. 5.3.2-1.

64


1 3

2 4

R = 0 .09 Oh mL = 1 E -1 1 HC = 1 .1 e -1 5 FC M = 0 .4 e -1 5 FCC = 1 .4 e-1 5 FL M = 0 .2 0

LC

R L R

C C CC

LC

R L R

C C C C

L M

L MCM

1

2

3

4

1 2

R =0 .08 Oh mL = 1 E- 11 HC = 2.7 e- 1 5 FC C = 1.2e -15 F

L

C

R L R

C C C C

1 2

1 2

R =0 . 08 O h mL = 1 E- 11 HC = 2.7 e- 1 5 FC C = 1.2e - 15 F

L

C

R L R

C C C C

1 2

.

R= 0 .4 O h mL= 3 .7 E- 1 1 HC = 5.3 e -1 5 FCC = 2 7e - 1 5 F

L

C

R L R

CC CC

1 2

CC CC

2L

C

R L R1

1

2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2



Primitives



Figure #5.3.2-1: Uncoupled and Coupled PEEC Circuits with Associated Building

Blocks


The building blocks are defined as in Chapter 4, with characterization and model

development next. The process of characterization and model development is

accomplished through the use of one carefully designed test structures. In this chapter

only one test structure comprised of all the predefined building blocks will be used. The

equivalent circuit of the test structure is assembled of “building block equivalent circuits”

of which it is comprised. The structure is designed, fabricated, and measured to

characterize each of the building blocks.

65

An important issue when designing the test structure is that it must be sensitive to

all the building blocks. Wide band sampling with sufficient frequency resolution is

crucial in order to capture all the major reactive effects over a band of interest [4]. In

most cases, parasitics are small enough that they only manifest themselves at higher

frequencies and in order to deembed them they must be observed. Therefore, the Hspice

optimizations were run using 400 frequency points at a range of 45MHz to 20GHz to

insure that the output parameters were sensitive to all of the important circuit parameters

at some points over the entire frequency range.

As stated previously, by virtue of device measurements, no assumptions are made

regarding material properties, layered dielectrics, or imperfect substrates, because all of

these effects are represented in the measured data. This makes the modeling procedure

entirely process independent.

To model the four stated building blocks, only “test structure two” will be used, as

shown in Fig. 5.3.2-1. Test structure two is a 3-segment meander resistor, and will be

used to characterize the basic uncoupled material parameters (including self-resistance,

inductance, and capacitance) the entire U-shaped corner segments, as well as line-to-line

mutual inductance and coupling capacitance. High frequency measurements of the test

structures were taken and used to characterize each of the four building blocks.


The test structure design was fabricated by depositing gold on a 96% Alumina

substrate. A photomicrograph of the fabricated test structure is shown in Fig. 5.3.4-1.

66

The test structure was measured using network analysis techniques. For the high

frequency measurements, a HP 8510C network analyzer was used with a Cascade


accomplished using a calibration substrate and utilization of the line-reflect-match (LRM)

calibration method. Data was gathered for test structure two at over 400 frequency points

between 45MHz and 20GHz, and was stored with the aid of computer data acquisition

software and equipment.

9-Segment Meander Resistor9-Segment Meander Resistor


Figure #5.3.4-1: Photograph of Fabricated Structure for 9-Segment Meander

Resistor Modeling



circuit topologies and parameters for the uncoupled and coupled building blocks, pad

building block, and improved multi-block U-shaped bend are shown in Fig. 5.3.5-1.

These circuits represent models for the building blocks only; test structure two and the 9-

67

segment meander resistor circuits are comprised of many building block circuits

connected in accordance with the structure geometry.

Deembed

Test Structure 2

Test Structure 2

MaterialSquare

R = 0 . 0 8 O h mL = 1 E - 1 1 HC = 2 .7 e - 1 5 FC C = 1 . 2 e - 1 5 F

L

C

R L R

C C C C

1 2

Pad

R = 0 .0 8 O h mL = 1 E - 1 1 HC = 2.7 e - 1 5 FC C = 1. 2 e - 15 F

L

C

R L R

C C C C

1 2

Multi-Block

Coupled Pair

R = 0 . 0 9 O h mL = 1 E -1 1 HC = 1 .1 e - 1 5 FC M = 0 .4 e - 1 5 FC C = 1 .4 e -1 5 FL M = 0 .2 0

LC

R L R

C C CC

LC

R L R

C C C C

L M

L MC M

1

2

3

4

.

R = 0 . 4 O h mL = 3 .7 E- 1 1 HC = 5 .3 e -1 5 FC C = 2 7 e - 1 5 F

L

C

R L R

CC CC

1 2

CC CC

2L

C

R L R1

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

PEECsPEECsBuildingBlocks

BuildingBlocks

4 Building Block Primitives22 Deembedded Element Values

Figure #5.3.5-1: Building blocks, Equivalent Circuits and Parameter Values for


Using only one test structure for the deembedding process will simplify the

fabrication and extraction procedure, and will increase the measured vs. modeled test

structure accuracy. A comparison was made between the methodology used in Chapter 4

(Improved) and the one used in this Chapter (Improved 1TS). Both analyses used the

same building block circuit configurations as shown in Fig. 5.3.5-2, and were analyzed

using the Hspice Optimization Tool.

68

Test Structure 1 & Test Structure 2Used (Improved)

Test Structure 1 & Test Structure 2Used (Improved)

No Substrate ResistanceDeembedded

Building Blocks Circuit Model Topology


PadMulti Block

Coupled Pair

No Substrate ResistanceDeembedded


PadMulti Block

Coupled Pair

Only Test Structure 2Used (Improved 1TS)Only Test Structure 2Used (Improved 1TS)

Figure #5.3.5-2: Analysis Comparison Using Two Test Structures and One Test

Structure

To simplify accuracy evaluation, percent error between measured vs. modeled

was calculated at each frequency point and is shown in the plots below. Iterative Hspice

optimization runs increased measured vs. modeled accuracy. Unix scripts automatically

ran 18 iterative optimizations per measured device, while at the same time increasing the

accuracy settings and decreasing the range of possible element values within the

optimization circuit file. The building block characterization process executed 576

individual optimizations to arrive at 32 unique equivalent circuit models of test structure

two. The entire Optimization/MonteCarlo analysis process together with data

calculations, conversions, storage, and circuit file creation was automated using Unix

shell scripts. The detailed automated process will be showcased in Chapter 9.

69


An optimization analysis was performed on test structure two; 1.) Without

deembedding the substrate resistance; and 2.) Deembedding a multi-block corner, for all

32 fabricated devices.


was calculated for test structure two using the methodology detailed in this chapter, and

compared with the results from Chapter 4 as shown in Fig. 5.3.6-1. The reflection

coefficient (S11), using only one test structure to deembed, shows a 0.5% improvement

over the two-test structure method. The transmission coefficient (S21) also shows an

overall percent error improvement of 0.5%.

70

Two Test Structures Used~3 percent Error

Between S11 Real Measured vs. Modeled



One Test Structure Used~2.5 percent Error




Test Structure 2Two Test Structures Used to Deembed Element Values

S11 Real Measured vs. Modeled

-1

1

3

5

7

9

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11 Real% Error

Test Structure 2One Test Structures Used to Deembed Element Values


-1

1

3

5

7

9

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11 Real% Error

Test Structure 2One Test Structures Used to Deembed Element Values


-1

0

1

2

3

4

5

6

7

8

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21 Real% Error

Test Structure 2Two Test Structures Used to Deembed Element Values


-1

0

1

2

3

4

5

6

7

8

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21 Real% Error









Figure #5.3.6-1: Measured vs. Modeled Percent Difference for 32 Fabricated



The fundamental building block models were assembled and used to predict the

behavior of a 9-segment meander resistor using only one test structure to deembed all the

element values of the building block equivalent circuits. An equivalent circuit of the

resistor was constructed by replacing each building block in the structure with its

equivalent extracted circuit, as shown in Fig. 5.3.7-1. Since only first level coupling was

taken into account, each material square in each segment of the resistor was coupled to its

nearest neighbor by a pair of mutual inductances and a coupling capacitance. As inferred

71

from the circuit description, the resulting circuit using 30µm linewidths and 300µm line

lengths per segment 9-segment resistor was a complex, highly interconnected system,

consisting of approximately 700 nodes. The longest path length of the resistor was

approximately 0.175 wavelengths long at 20GHz.

MaterialSquare

R =0 .0 8 O hmL = 1 E- 11 HC = 2.7 e- 15 FCC = 1. 2 e -15 F

L

C

R L R

C C C C

1 2

Pad

R =0 .0 8 O h mL = 1 E- 11 HC = 2.7 e-1 5 FC C = 1. 2e - 15 F

L

C

R L R

C C C C

1 2

Coupled Pair

R=0 . 09 O h mL = 1 E -11 HC = 1 .1 e-1 5 FCM =0 .4 e -1 5 FCC = 1 .4 e -1 5 FL M = 0 .2 0

LC

R L R

C C CC

LC

R L R

C C C C

L M

L MCM

1

2

3

4

.

R= 0 .4 O h mL= 3 .7 E- 1 1 HC = 5.3 e -1 5 FCC=2 7e - 1 5 F

L

C

R L R

CC CC

1 2

CC CC

2L

C

R L R1

Multi-BlockL

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

Pad

Pad

Figure #5.3.7-1: 9-Segment Meander Resistor and Associated Building Blocks

5.4 Results

The AC small signal circuit simulation, as before, was completed in less than 2

minutes. Comparisons in the form of percent differences from the actual measured

device are shown in Fig. 5.4-1.

The improved 1TS modeling methodology clearly outperformed the original and

improved methods, yielding an enhancement of approximately 1.0% at frequencies up to

72

9GHz for S11R. However for S21R, the improved modeling methodology outperformed

the original and improved 1TS at frequencies below 1GHz and equaled the improved 1TS

method at frequencies above 1GHz.

This is mainly due to enhanced matching; i.e. using a dependent “test structure 1

& 2” (contained within test structure 2) in contrast to an independent “test structure 1”

and an independent “test structure 2” located at different sites on the substrate. The

number of frequency points (400) in the measured input file also enabled a successful

extraction of all building block equivalent circuit values.

Measured 9 Segment Meander Resistor vs. Original, Improved, Improved 1 TS, Characterization MethodMeasured 9 Segment Meander Resistor vs. Original, Improved, Improved 1 TS, Characterization Method

S11R % Difference From Measured Device Original vs. Improved vs. Improved 1 TS





Original Method vs. Improved Method vs. Improved 1 TS

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21 RealPercent

Error


Improved Method 1 TS

9 Se gm e nt M ode le d Re sistorO riginal M e thod v s. Im prov e d M e thod v s. Im prov e d 1 TS

-2.00E-01

-1.00E-01

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

8.00E-01

1.E+08 1.E+09 1.E+10 1.E+11

F re que ncy

S 11/S 21Re a l

S11 Meas ured Dev ic e

S11 Original Method

S11 Improv ed Method

S11 Improv ed Method 1 TSS21 Meas ured Dev ic e

S21 Original Method

S21 Improv ed Method

S21 Improv ed Method 1 TS


Original Method vs. Improved Method vs. Improved 1 TS

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11 RealPercent

Error


Improved Method 1 TS

Figure #5.4-1: 9 Segment Meander Resistor

Measured vs. Original, Improved, and Improved 1TS Characterization Methods

73

5.5 Summary

The successful deembedding of 22 element values from 4 building block

equivalent circuits using only one test structure was presented in this chapter. The 9-

segment meander resistor was again used to showcase the improvements made to the

existing methodologies. The percent improvement comparison plots in Fig. 5.5-1 show

that this methodology out performed the original method for S11R and S21R, and

outperformed the improved method for S11R, into the GHz. However, the improved

method for S21R at frequencies less than 3.5GHz out performed the improved 1TS

method.

Percent Comparisons Between The Original, Improved andImproved 1TS Methodologies

Percent Comparisons Between The Original, Improved andImproved 1TS Methodologies

9 Segment Modeled Resistor Calculated Percent Improvement Original Characterization Method vs. Improved Method

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11/S21 RealPercent

Improvement

S11 Original MethodS21 Original MethodS11 Improved MethodS21 Improved Method

Figure #5.5-1: 9 Segment Meander Resistor Percent Comparisons

74

CHAPTER 6

PREDICTIVE STATICTICAL ANALYSIS VIA MEASUREMENT OF

CONONICAL BUILDING BLOCKS

6.1 Introduction

As stated previously, a further developed passive device modeling methodology

that accurately predicts the high frequency electrical behavior, and the statistical variation

of embedded passive devices on a generic substrate, will be presented. This research will

prove that the variation in the complete equivalent circuit models, based only on the

circuit building blocks, can be used to predict such variations in actual fabricated devices.

Process parameters are the optimum basis for statistical circuit modeling of

passive devices [29]. To statistically model a device material properties such as oxide

and metal thickness variations, and physical properties such as dielectric constant and

resistivity variations are required. Process variations equate to element value variations

when using the aforementioned modeling methodology. Therefore to statistically model

a device, measurements of multiple test structures need to be taken and the equivalent

circuit values deembedded. This deembedding process automatically accounts for the

effects of processing fluctuations and nonideal material properties.

75

Mean and relative variations in the deembedded circuit values from all the devices

are then calculated and used in the Hspice MonteCarlo simulator as limits to generate

random circuit values for a defined number of simulations. A predicted range of

operation for the actual modeled device is then generated from the MonteCarlo

simulations. Ideally, the designer would like to demonstrate that the variation in the

equivalent circuit models, based only on the data used to extract them, can be used to

predict variations in the actual fabricated device. Therefore, accurate measured vs.

modeled test structure results, that generate accurate device models, and consequently

realistic range of operation predictions, will be the basis of this process.

This chapter will explain in detail the procedure of predicting passive device (i.e.

meander resistor) performance variations and yield, when provided with a statistical

distribution of the building block models and a circuit schematic.

6.2 Modeling Procedure

The procedure begins by characterizing and modeling the passive device (i.e.

meander resistor) as stated in Chapter 3. A brief description of the various steps is

provided below and shown in Fig. 6.2-1.

1. The first step is to determine what structure geometries to consider, in compiling a set

of building blocks and test structures that will be measured and characterized.

76

2. The second step is to create building block geometries. Current visualization

software is useful in determining areas of constant current flow, which provide the

boundaries for the building blocks.

3. The third step is to design test structures comprised of the building blocks, which will

enable their accurate modeling.

4. At this point, test structures are fabricated, tested, and high frequency network

analysis and DC resistance measurements taken.

5. The measured test structure data and a circuit file description are then used as input to

the optimization tool. PEEC element value initial guesses are based on the measured

results for each structure. Once optimization for one structure is complete, the results

are used, as initial guess, for the remaining optimizations.

6. The deembedded circuit values of the building block equivalent circuits are obtained

through non-linear optimization using the Hspice Optimization Tool. The designer

now has an accurate spice model of each building block, which can then be used to

design any value/configuration meander resistor.

77


Simulate in SPICESimulate in SPICE--type type simulatorsimulator

DRCDRC

Design Desired Passive Design Desired Passive DeviceDevice


Generate Design Rules Generate Design Rules Associated with Each Associated with Each

BlockBlock


PassPassPass

FailFailFail



6.3 Statistical Variations

The ability to predict the performance of an embedded passive device is

fundamental to those attempting to design integrated circuits, modify fabrication

processes, plan production schedules, or specify product-operating characteristics. In

general, a three-step MonteCarlo technique is used to accomplish this prediction: 1.) A

statistical model is formed to characterize the fabrication process used; 2.) A circuit

design is created using a Spice type simulator and applying nominal device values for the

target process; and 3.) Randomly generated element values of the process model are

simulated in a MonteCarlo tool to produce a representative set of output performance

characteristics. The impact of random process variations can be inferred from these

78

simulations, and electrical yield can then be estimated using the percentage of that sample

which meets the performance requirements [29]. Electrical yield refers to the number of

parts that meet the electrical test specifications; whereas, maximizing yield is important

for the overall process efficiency.

6.3.1 Statistical Model

The statistical model used in this research is formed via the passive device

modeling methodology described in Chapters 3, 4, and 5. Test structures are designed

and fabricated in a process of interest and contain the fundamental building blocks of the

larger modeled device. The building blocks, modeled as Spice compatible RLC circuits,

are characterized using actual measured data and a non-linear optimization method.

Circuit models complete with deembedded element values are defined for each building

block, and accurately reproduce the test structure behavior.

Once a defined number of these test structures have been characterized, mean and

relative variations in the deembedded circuit values of the building blocks are then

calculated and used in the Hspice MonteCarlo simulator as limits to generate random

circuit values for a defined number of simulations. Hspice analyzes the yield by using

statistical techniques and observing the effects of element and model parameter variation

[31]. Fig. 6.3.1-1 shows some of the deembedded element values and the syntax used in

the MonteCarlo simulation tool.

79

*************************************** 30 u Line Specs 1**************************************+ c_sq_1 = UNIF(1.262198980e-13, 9.72E-01) + rsq_1 = UNIF(2.510000000e-06, 0.00E+00) + lsq_1 = UNIF(1.816762077e-11, 4.91E-02)+ csq_1 = UNIF(6.520000000e-26, 9.69E-01)+ r_sub_sq = 10mega*************************************** Corner and Coupled Lines Specs 1**************************************.param+ cou_l_1 = UNIF(2.205573309e-01, 2.97E-01) + c_cou_1 = UNIF(6.714851620e-24, 8.15E-01) + r2_1 = UNIF(6.652141530e-01, 4.43E-01)+ c2_1 = UNIF(1.357575649e-15, 2.34E-01)+ l2_1 = UNIF(4.295089543e-12, 3.34E-01)+ c_co2cr_1 = UNIF(8.071681342e-12, 3.66E-01) + r_sub_corrn = 10mega

c_sq csq lsq rsq c_pad rpadlow 3.56172E-15 6.520000E-26 1.7331E-11 0.00000251 1.27E-12 3.754664high 2.49093E-13 4.098440E-24 1.91221E-11 0.00000251 1.88E-12 5.626608median 1.26327E-13 2.081820E-24 1.82265E-11 0.00000251 1.57E-12 4.690636ab var 1.22765E-13 2.016620E-24 8.95575E-13 0 3.06E-13 0.935972rel var 0.971805619 9.686813E-01 0.049135815 0 0.194744 0.19954

cou_1 c_cou l2 r2 c2 c_co2crlow 0.159291519 8.919877E-25 2.76911E-12 0.398398019 1.08E-15 5.56E-12high 0.293682413 8.726346E-24 5.54968E-12 1.032752749 1.75E-15 1.2E-11median 0.226486966 4.809167E-24 4.1594E-12 0.715575384 1.41E-15 8.77E-12ab var 0.067195447 3.917179E-24 1.39028E-12 0.317177365 3.32E-16 3.21E-12rel var 0.296685713 8.145234E-01 0.334250965 0.443248011 0.234432 0.366037

Statistical Variations for Each Element ValueCalculate and Inserted into Hspice MonteCarlo Simulator

Statistical Variations for Each Element ValueCalculate and Inserted into Hspice MonteCarlo Simulator

Figure #6.3.1-1: Mean and Absolute Deviations for Deembedded Element Values

6.3.2 Circuit Design

An equivalent circuit of the modeled device (device under inspection) is

constructed by replacing each building block in the structure with its equivalent extracted

circuit values. To accurately represent the variations in the complete circuit model, the

correlation between the building block element values needs to be investigated.

The correlation between two or more random variables is a question defined as;

does one or more random variable depend in any way on any of the other random

variables [30]. This question was addressed by examining the “Sputtering” technique

that was used to deposit the gold onto the 96% Alumina substrate. Sputtering deposits a

random nonuniform layer of metal onto a substrate surface. This basically means that

one portion of the deposited metal is in no way a function of any other deposited portion;

therefore, the correlation between segments does not exist.

80

Due to the lack of correlation between the building blocks, each element is

basically independent of any other element in the circuit. Therefore, to create an accurate

circuit model file all the elements in the design need to generate a random value for the

defined number of simulations. This circuit file can become quite large, for example a 9-

segment meander resistor needs to generate 1336 random circuit values for each

simulation. Fig. 6.3.2-1 is a block diagram of a 9-segment meander resistor and shows

the amount of random number generations per simulation, for the PEEC element values.

As a comparison, the statistical analysis procedure was also applied to a circuit model file

using the premise that all the building block equivalent circuits were correlated.

MaterialSquare

R=0.08 OhmL= 1E-11 HC = 2.7e-15 FCC = 1.2e-15 F

LC

R L R

CC CC

1 2

Pad

R=0.08 OhmL=1E-11 HC = 2.7e-15 FCC = 1.2e-15 F

LC

R L R

CC CC

1 2

Coupled Pair

R=0.09 OhmL= 1E-11 HC = 1.1e-15 FCM=0.4e-15 FCC = 1.4e-15 FLM = 0.20

LC

R L R

CC CC

LC

R L R

CC CC

L MLM

CM

1

2

3

4

CC

LC

R1

Multi-Block

LC

R L R

CC CC1 2

LC

R L R

CC CC1 2

.

R=0.4 OhmL= 3.7E-11 HC = 5.3e-15 FCC=2 7e-15 F

CC2

L R

LC

R L R

CC CC1 2

LC

R L R

CC CC1 2

Pad

Pad

TOTAL = 1336 Random Generated Element Values

Pads = (2 BBlocks * 8 Elements) = 16 ElementsMaterial Squares = (8 BBlocks * 8 Elements) = 64 ElementsCoupled Squares = [((4 BBlocks * 9 Coupled) * 3) * 8 Elements] + (9 Coupled * 8 Elements) = 936 Elements

Corners = (5 BBlocks * 8 Corners * 8 Elements) = 320 Elements

Figure #6.3.2-1: Number of Random Number Generation for MonteCarlo

Simulation

81

6.3.3 MonteCarlo Analysis

The MonteCarlo analysis uses a random number generator to create a uniform

parameter distribution by means of the relative variation; whereas, the nominal value is

varied by +/- (nominal_value * relative_variation) [31]. A new random variable is

calculated each time a parameter is used. Fig. 6.3.3-1 shows how the relative variation is

calculated.

Uniform DistributionUniform Distribution

Rel_variation = Abs_variation / Nom_valueRel_variation = Abs_variation / Nom_value

Abs_variationAbs_variation

Nom_valueNom_value

PopulationPopulation

Figure #6.3.3-1: Uniform Distribution

Based on the statistics calculated for each element value in the equivalent circuit

model, a MonteCarlo analysis is performed using Hspice. A MonteCarlo simulation with

a uniform parameter distribution is generated for a defined number of sets of circuit

model parameters. Following the simulations, the output response for the modeled

device is compared with actual measured data to determine if the statistical variations of

the model parameter values reside in the range of the MonteCarlo predicted output. To

accurately compare the measured vs. MonteCarlo results, several statistical methods were

used to find a number that represents both sets of data.

82

6.3.4 Statistical Analysis Methods

The field of statistics is concerned with methods of organizing, summarizing, and

interpreting, data [32]. “Data” is defined as information; whereas, as any collection of

information is a collection of data. However for statisticians, “data” means numerical

information. Basically, statistics amounts to a collection of techniques for dealing with

sets of numbers, organizing them, summarizing them, and discovering their meaning. An

excel workbook with a set of worksheets is used to discern all collected data. Mean,

variance, and standard deviation between MonteCarlo generated and actual measured

data are the basis for the techniques described below. Fig. 6.3.4-1 is a portion of the

excel workbook that filters the MonteCarlo and measured data then applies several

statistical techniques to the results.

Excel Spread Sheet Used To Calculate Statistical InformationExcel Spread Sheet Used To Calculate Statistical InformationFrequency MC Meas Filter MC CFD Meas CFD MC CFDN Meas CFDN MC CFDNT Meas CFDNT T-Test

45000000 0.4181 0.399 0.395158 1 0 0.002000 0.000000 0.002000 0.000000 Actual CN T-Score0.4169 0.404 0.398099 2 0 0.004000 0.000000 0.004000 0.000000 -3.013391 0.545626 0.0174250.4114 0.405 0.401041 10 1 0.020000 0.031250 0.020000 0.0000000.4231 0.407 0.403982 24 1 0.048000 0.031250 0.048000 0.031250 Corralation Cofficient0.4253 0.408 0.406924 57 3 0.114000 0.093750 0.114000 0.125000 CN T-Score0.4149 0.408 0.409865 108 7 0.216000 0.218750 0.216000 0.281250 0.974649 0.9972350.4219 0.409 0.412806 179 9 0.358000 0.281250 0.358000 0.3437500.4186 0.41 0.415748 277 11 0.554000 0.343750 0.554000 0.531250 Mean Error0.4089 0.411 0.418689 370 16 0.740000 0.500000 0.740000 0.656250 Actual CN T-Score0.4261 0.414 0.421630 419 20 0.838000 0.625000 0.838000 0.875000 -0.87% 14.36% 0.48%0.4138 0.415 0.424572 464 22 0.928000 0.687500 0.928000 0.9062500.4076 0.417 0.427513 489 27 0.978000 0.843750 0.978000 1.000000 Standard Deviation Error0.4135 0.417 0.430455 497 28 0.994000 0.875000 0.994000 1.000000 Actual CN T-Score0.4179 0.417 0.433396 500 32 1.000000 1.000000 0.998000 1.000000 -43.72% 8.00% -0.20%0.4164 0.418 0.436337 500 32 1.000000 1.000000 1.000000 1.0000000.4216 0.418 0.439279 1.000000 1.000000 1.000000 1.000000 ANOVA0.4025 0.419 Actual CN0.4121 0.42 Filter Adjustments 9.063427 0.2977080.4152 0.421 0.002941 500 320.4162 0.421 4.584301 500 320.4233 0.423 Divider 130.427 0.424

0.4105 0.425 MC = MonteCarlo0.4228 0.426 Meas = Measured0.4195 0.426 CFD = Cumulative Frequency Distribution0.4169 0.426 CFDN = Cumulative Frequency Distribution Normalized0.4124 0.427 CFDNT = Cumulative Frequency Distribution Normalized T-Scores0.4178 0.428

Figure #6.3.4-1: Excel Spread Sheet

83

6.3.5 Mean and Standard Deviation

The mean is a measure of central tendency, “middleness”, or the arithmetic

average. Adding up all the scores and dividing by the number of scores is the method of

calculating the mean. The formula for the mean value of X is:

MxΣXN

Where Mx is the mean of X and N is the number of scores (or numbers in a population).

Therefore, the mean of variable X (symbolized as Mx ) equals the sum of the X scores

(ΣX) divided by the number of scores (N).

The variance is the most frequently used measure of variability [32]. The formula

for the variance is:

s2 N X2. X( )2

N N 1( )

The standard deviation is derived from the variance and also indicates the relative

variability within a group. The formula for the standard deviation is:

s N X2. X( )2

N N 1( )

To aid in the evaluation of measured vs. modeled data, percent differences in

mean and standard deviation will be used. Fig. 6.3.5-1 displays the Mean Value Percent

Difference results using two different sets of data. The first set compares the raw

MonteCarlo generated (population 500) data vs. the raw Measured (population 32) data.

84

The second set compares the normalized cumulative frequency distribution data

(population 16). Both sets are displaying the magnitude of S11 and S21 using data

generated from a circuit model file constructed using the correlated premise. The results

begin to diverge at 1.84GHz for S11 magnitude and 4.5GHz for S21 magnitude.

Mean Percent Error Comparing Cumulative Frequency Distribution Data For

MonteCarlo Prediction (16 Data Points) vs. Actual Measured Device (16 Data Points)

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error Correlated Blocks

Mean Percent ErrorComparing


-0.10%

0.40%

0.90%

1.40%

1.90%

1.E+08 1.E+09 1.E+10

Frequency




-1.00%

-0.50%

0.00%

0.50%

1.00%

1.50%

2.00%

1.E+08 1.E+09 1.E+10

Frequency




-1.00%

9.00%

19.00%

29.00%

39.00%

49.00%

1.E+08 1.E+09 1.E+10

Frequency


Mean Percent Difference For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration

Mean Percent Difference For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration

1.84GHz1.84GHz 1.84GHz1.84GHz


Figure #6.3.5-1: Mean Value Percent Difference Between MC and Measured Data

For S11/S21 Magnitude

Fig. 6.3.5-2 displays the Standard Deviation Percent Difference results using two

different sets of data. The first set compares the raw MonteCarlo generated (population

500) data vs. the raw Measured (population 32) data. The second set compares the

normalized cumulative frequency distribution data (population 16). Both sets are

displaying the magnitude of S11 and S21 using data generated from a circuit model file

constructed using the correlated premise. The results begin to diverge at 2.5GHz for S11

85

magnitude using the first set of data, and at 1.74GHz for S11 magnitude using the second

set of data. The results begin to diverge at 4.2GHz for S21 magnitude using the first set

of data, and at 7.0GHz for S21 magnitude using the second set of data.

Standard Deviation Percent Error Comparing Cumulative Frequency Distribution Data For


-1.00%

4.00%

9.00%

14.00%

19.00%

24.00%

29.00%

1.E+08 1.E+09 1.E+10

Frequency




-1.00%

4.00%

9.00%

14.00%

19.00%

24.00%

29.00%

1.E+08 1.E+09 1.E+10

Frequency


Standard Deviation Pecent ErrorComparing


0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

1.E+08 1.E+09 1.E+10

Frequency




0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

140.00%

160.00%

180.00%

1.E+08 1.E+09 1.E+10

Frequency


Standard Deviation Percent Difference For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File ConfigurationStandard Deviation Percent Difference For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration



Figure #6.3.5-2: Standard Deviation Value Between MC and Measured Data For

S11/S21 Magnitude

6.3.6 Frequency Distribution

The s-parameters generated from the MonteCarlo results and the actual measured

devices are transposed from rows representing a particular frequency to columns

representing MonteCarlo simulations and measured device data at a particular frequency

point. The range of values from high to low is split up into sixteen linearly place filter

bins. How often a value occurs within a range of values, defined by the filter bins, is

86

calculated and normalized. These normalized cumulative frequency distribution plots are

generated at every frequency point for S11/S21 magnitude up to a certain frequency.

These plots illustrate that the measured data from each fabricated device at a particular

frequency fit within the MonteCarlo predicted range. They also show how the standard

deviations between predicted and measured compare. Trend lines were added to each

plot to simplify the evaluation; blue is the MonteCarlo results and red is the Measured

data. Fig. 6.3.6-1 displays a normalized cumulative frequency distribution plot at several

frequency points for S11. Fig. 6.3.6-2 display a normalized cumulative frequency

distribution plot at several frequency points for S21.

87

Correlated Premise @ 600MHzCorrelated Premise @ 600MHz

Normalized Cumulative Frequency Distribution For S11 Magnitude MonteCarlo Prediction vs. Actual Measured Device at 1GHz

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.397

8

0.400

7

0.403

6

0.406

5

0.409

4

0.412

2

0.415

1

0.418

0

0.420

9

0.423

7

0.426

6

0.429

5

0.432

4

0.435

2

0.438

1

0.441

0

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

Correlated Premise @800MHzCorrelated Premise @800MHz


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.399

9

0.402

8

0.405

6

0.408

4

0.411

2

0.414

1

0.416

9

0.419

7

0.422

5

0.425

4

0.428

2

0.431

0

0.433

9

0.436

7

0.439

5

0.442

3

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.401

9

0.404

7

0.407

4

0.410

2

0.413

0

0.415

8

0.418

6

0.421

3

0.424

1

0.426

9

0.429

7

0.432

5

0.435

2

0.438

0

0.440

8

0.443

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

Correlated Premise @ 1.15GHzCorrelated Premise @ 1.15GHz


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.405

0

0.407

7

0.410

4

0.413

1

0.415

8

0.418

5

0.421

2

0.423

9

0.426

6

0.429

3

0.432

0

0.434

7

0.437

4

0.440

1

0.442

8

0.445

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.412

7

0.415

2

0.417

7

0.420

3

0.422

8

0.425

3

0.427

8

0.430

4

0.432

9

0.435

4

0.437

9

0.440

5

0.443

0

0.445

5

0.448

0

0.450

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.424

9

0.427

2

0.429

4

0.431

7

0.433

9

0.436

2

0.438

4

0.440

7

0.442

9

0.445

2

0.447

4

0.449

7

0.451

9

0.454

2

0.456

4

0.458

7

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

Figure #6.3.6-1: Normalized Cumulative Frequency Distribution Plot

88



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.565

0

0.567

9

0.570

8

0.573

8

0.576

7

0.579

6

0.582

5

0.585

4

0.588

3

0.591

2

0.594

1

0.597

1

0.600

0

0.602

9

0.605

8

0.608

7

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

Correlated Premise @ 1GHzCorrelated Premise @ 1GHz


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.564

5

0.567

4

0.570

3

0.573

2

0.576

1

0.579

0

0.581

9

0.584

8

0.587

7

0.590

6

0.593

4

0.596

3

0.599

2

0.602

1

0.605

0

0.607

9

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.559

9

0.562

6

0.565

4

0.568

1

0.570

9

0.573

6

0.576

4

0.579

2

0.581

9

0.584

7

0.587

4

0.590

2

0.592

9

0.595

7

0.598

4

0.601

2

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.552

9

0.555

4

0.558

0

0.560

5

0.563

1

0.565

6

0.568

1

0.570

7

0.573

2

0.575

8

0.578

3

0.580

9

0.583

4

0.586

0

0.588

5

0.591

1

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.543

8

0.546

0

0.548

3

0.550

6

0.552

9

0.555

2

0.557

5

0.559

8

0.562

1

0.564

3

0.566

6

0.568

9

0.571

2

0.573

5

0.575

8

0.578

1

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.533

1

0.535

7

0.538

3

0.540

9

0.543

5

0.546

1

0.548

7

0.551

3

0.553

9

0.556

5

0.559

1

0.561

7

0.564

3

0.566

9

0.569

5

0.572

1

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

Figure #6.3.6-2: Normalized Cumulative Frequency Distribution Plot

6.3.7 Pearson Product-Moment Correlation Coefficient

Using the normalized distribution information, a measure of linear relationship

between the two sets of data can be obtained by means of the Pearson Product-Moment

Correlation Coefficient (r). A perfect correlation between two variables results in r = 1.0,

89

no correlation at all results in r = 0.0, and any number in-between means that there is

some correlation between the variables however the relationship is not perfect. The

degree of correlation is a function of statistical significance, which is determined using

numerical charts to ascertain the outcome. The formula for the Correlation Coefficient is:

γ N X Y.( ). X Y.

N X2. X( )2 N Y2. Y( )2.

Where N is the number in the population; and X Y, are the elements of the population.

Fig. 6.3.7-1 displays a Correlation Coefficient plot for S11 and S21 magnitude via the

data generated from a circuit model file constructed using the correlated premise. The

correlation between the two sets of data begins to diverge at 2.0GHz for S11 magnitude

and 4.5GHz for S21 magnitude.

Corralation CofficientComparing Cumulative Frequency Distribution Data For


0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.E+08 1.E+09 1.E+10

Frequency

S21 MagCorrelation

Coff.Correlated Blocks

Correlation CofficientComparing Cumulative Frequency Distribution Data For


0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.E+08 1.E+09 1.E+10

Frequency

S11 MagCorrelation

Coff.Correlated Blocks

Correlation Coefficient For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration

Correlation Coefficient For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration


Figure #6.3.7-1: Correlation Coefficient Between MC and Measured Data For

S11/S21 Magnitude

90

6.3.8 T-Test

The “t” test is one of the most commonly used statistical tests [32]. Its primary

purpose is to determine whether the means of two groups of numbers differ to a

statistically significant degree. The larger the absolute value of t, the more likely it is to

reflect a significant difference between the two groups under comparison. Charts that

evaluate statistical significance for the t test have also been developed and are included in

the appendix. The formula for the t test is:

sp2 N1 1 s1

2. N2 1 s22.

N1 N2 2

s M1 M2

sp2

N1

sp2

N2

τ obt

M1 M2

s M1 M2

Where τ obt is the value of t obtained through the data; N1 N2, is the number of elements in

each group; s12 s2

2, the estimates of the variances of the two populations; M1 M2, is the

means of the two groups; sp2 s M1 M2, are the values needed in order to arrive at τ obt

2.

Fig. 6.3.8-1 displays the t-test results using two different sets of data. The first set

compares the raw MonteCarlo generated (population 500) data vs. the raw Measured

91

(population 32) data. The second set compares the normalized cumulative frequency

distribution data (population 16). Both sets are displaying the magnitude of S11 and S21

using data generated from a circuit model file constructed using the correlated premise.

The t-test results begin to diverge at 1.84GHz for S11 magnitude and 4.5GHz for S21

magnitude.

T-Test For Independent SamplesComparing Cumulative Frequency Distribution Data For


0.00

0.50

1.00

1.50

2.00

2.50

1.E+08 1.E+09 1.E+10

Frequency

S21 MagT Value Correlated Blocks

T-Test For Independent SamplesComparing


-0.50

1.50

3.50

5.50

7.50

9.50

1.E+08 1.E+09 1.E+10

Frequency




-0.50

1.50

3.50

5.50

7.50

9.50

1.E+08 1.E+09 1.E+10

Frequency




-0.25

0.25

0.75

1.25

1.75

2.25

1.E+08 1.E+09 1.E+10

Frequency


T-Test For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration

T-Test For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration

1.84GHz1.84GHz1.84GHz1.84GHz

4.5GHz4.5GHz

4.5GHz4.5GHz

Figure #6.3.8-1: T-Test Between MC and Measured Data For S11/S21 Magnitude

6.3.9 ANOVA

The analysis of variance (ANOVA) is a technique used to test for statistical

significance of the differences among the means of two or more groups. The test looks at

the amount of variability (the differences) between the means of the groups, compared

with the amount of variability among the individual scores in each group; that is, the

92

variance between groups versus the variance within groups [32]. The ANOVA starts

with the total amount of variability in the data and divides it up (partitioning) into various

categories. Basically, it compares the variability among the group means with the

variability that occurred just by chance or error. To calculate an ANOVA the total sum

of the squares:

sT2 X MT

2

NT 1

SST

dfT

is broken down into two parts; 1.) A sum of squares within groups, SSW, which reflects

the degree of variability within groups, however is not sensitive to overall differences

between the groups; and 2.) A sum of squares between groups, SSB, which reflects

differences between groups, however is not sensitive to variability within groups. The

total sum of squares is the sum of the sum of squares within and the sum of squares

between:

SST SSW SSB

The total degrees of freedom is:

dfT dfW dfB

Adding up all df’s within all the groups yields dfW. The number of groups minus 1 equal

dfB. The mean square within, a measure of the variability within groups is:

MSW

SSW

dfW

The mean square between, a measure of variability between groups is:

MSB

SSB

dfB

93

To find out if the between-group differences are significantly greater than chance,

comparing the two by computing the ratio:

Fobt

MSB

MSW

Fobt is the ratio of a mean square between groups to a mean square within groups. The

“obtained” (obt) value is compared with a critical value (extracted from tables) to

ascertain its validity. Fig. 6.3.9-1 displays the ANOVA results using two different sets of

data. The first set compares the raw MonteCarlo generated (population 500) data vs. the

raw Measured (population 32) data. The second set compares the normalized cumulative

frequency distribution data (population 16). Both sets are displaying the magnitude of

S11 and S21 using data generated from a circuit model file constructed using the

correlated premise. The ANOVA results begin to diverge at 1.74GHz for S11 magnitude

and 4.5GHz for S21 magnitude.

94

ANOVA AnalysisComparing Cumulative Frequency Distribution Data For


-0.25

-0.05

0.15

0.35

0.55

0.75

0.95

1.15

1.35

1.E+08 1.E+09 1.E+10

Frequency

S21 MagF(obt) Correlated Blocks

ANOVA AnalysisComparing


-1.00

4.00

9.00

14.00

19.00

24.00

29.00

1.E+08 1.E+09 1.E+10

Frequency




-0.02

0.18

0.38

0.58

0.78

0.98

1.E+08 1.E+09 1.E+10

Frequency




-0.25

4.75

9.75

14.75

19.75

24.75

29.75

1.E+08 1.E+09 1.E+10

Frequency


ANOVA For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration

ANOVA For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit File Configuration



Figure #6.3.9-1: ANOVA Between MC and Measured Data For S11/S21 Magnitude

6.4 Summary

This chapter explained in detail the method of predicting passive device

performance variations and yield, when provided with a statistical distribution of the

building block models and a circuit schematic. Correlation between the building block

equivalent circuit values was also examined for use in collating the circuit model file

used in the MonteCarlo simulations. A description of the MonteCarlo simulation process

and several statistical analysis techniques, used to evaluate and render the results, were

also presented. The data used in rendering the example plots was generated from a

circuit model file constructed using the “correlated” premise in contrast to the

95

noncorrelated or “independent” premise. In the following chapters, this predictive

modeling methodology will be applied to a 9-segment meander resistor using the

modeling methodology described in Chapters 4 and 5, and using the independent

premise.

96

CHAPTER 7

PREDICTIVE STATISTICAL ANALYSIS OF 9 SEGMENT

MEANDER RESISTOR

7.1 Introduction

As microelectronic technology continues to progress, there is a constant focus on

higher levels of system integration and miniaturization. For example, in many

applications it’s desirable to package several integrated circuits (ICs) together in

multichip modules (MCMs) to achieve further compactness and higher performance.

Passive components (i.e., capacitors, resistors, and inductors) are an essential

requirement for many MCM applications [35]. A significant advantage of MCM

technology is the ability to embed a large number of passive components directly into the

substrate at low cost; whereas, examination of component value statistical variation is

crucial for designing and characterizing their performance.

Successful design of passive structures requires that accurate models of the

various components exist or can be easily obtained. However, for high frequency designs

passive structures can be electrically long, and due to their full 3-dimensional geometries,

have very complex field patterns. Standard modeling methods for microstrip or stripline

based structures do not apply for these components. Therefore in order to successfully

97

design structures for high frequencies, the behavior of the passive components that

comprise the structure must be characterized accurately at those frequencies. Recently,

computer-aided design tools such as Hspice [31] have become indispensable in IC

design; whereby, accurate circuit simulation using Hspice is dependent on both the

structural validity of the device models and the accuracy of the values used as model

parameters. For that reason, the extraction of an optimum set of device model parameter

values is crucial in characterizing the relationship between the model and the measured

behavior.

In this chapter, the improved building block modeling methodology originally

created by Dr. Poddar [28] is further developed by focusing on the statistical variation of

the deembedded element values of the building block models. It will be demonstrated

that the variation in the complete equivalent circuit models, based only on the circuit

building blocks, can be used to predict such variations in actual fabricated devices. The

9-segment meander resistor fabricated on a 96% Alumina substrate will be the device

under inspection.

Three sets of resistor structures were fabricated on a 96% Alumina substrate: 32

straight metal segments (test structure 1), 32 three-segment resistors (test structure 2), and

32 nine-segment resistors (modeled device). Scattering parameter measurements ranging

in frequencies from 45MHz to 20GHz were taken from all the structures. The measured

s-parameter data from test structure 1 and 2 was used to deembed the element values of

the partial element equivalent circuit (PEEC) for both sets of 32 test structures, using a

non-linear optimization algorithm in Hspice. This process generated a unique equivalent

circuit model for each of the 64 test structures. Mean and relative variation was then

98

calculated for each deembedded component from the 64 unique equivalent circuit

models, and used in a MonteCarlo analysis of the 9-segment modeled device. This

analysis generated 500 s-parameter curves of the modeled device electrical behavior.

The s-parameter measurements from the MonteCarlo analysis and the actual

measured device were compared to determine if the measured data resides in the range of

curves generated by the MonteCarlo analysis. By inspection, it was noted that the

electrical behavior from the fabricated devices was in fact contained within the range

predicted by the MonteCarlo results. Therefore by means of comparison between the

MonteCarlo results and measured data, it was shown that the statistical variations of the

component values provide an accurate representation of the overall modeled device

performance.


A three-step MonteCarlo technique was used to accomplish the statistical

prediction: 1.) A statistical model was formed to characterize the fabrication process

used; 2.) A circuit design was created using a Spice type simulator and applying nominal

device values for the target process; and 3.) Randomly generated element values of the

process model were simulated in a MonteCarlo tool to produce a representative set of

output performance characteristics. The impact of random process variations can be

inferred from these simulations, and electrical yield can then be estimated using the

percentage of that sample which meets the performance requirements [29].

99

7.3 Test Structure Description

Two sets (test structures 1 and 2) of 32 test structures were fabricated by

depositing gold, using a sputtering technique, on a 96% Alumina substrate. The metal

thickness used was 3000 angstroms with 30µ widths and spacings for all connecting

segments. All resistor structures were on one layer, with connections to the devices made

using a ground-signal-ground probe pad pattern.

Test structure 1 consisted of two probe pads connected to a 30µ wide strip of

metal approximately 300µ long. Test structure 2 consisted of two probe pads connected

to five material squares which are connected to nine coupled material squares, and two

corners which are made up of five material squares.

The layout and fabrication mask photos of test structures’ 1 and 2 are shown in

Fig. 7.3-1.

Test StructuresTest Structures

Test Structure 2Test Structure 2Test Structure 1Test Structure 1

LayoutLayout

MaskMask

Figure #7.3-1: Test Structures 1 and 2 Layout and Mask Photos

100

These test structures allowed the modeling of the probe pads, the material squares,

the coupled material squares, and the corner primitives.


The 96% Alumina structure was physically designed using integrated circuit

design tools within the Avanti design environment. Rena Huang PhD. Candidate in the

School of Electrical and Computer Engineering at the Georgia Institute of Technology

fabricated the design in Georgia Tech’s MiRC cleanroom. The size of the completed

coupon was approximately 2.25" x 2.25". Metal lines were drawn 30µ wide. The

embedded structures were interfaced accessed using ground-signal-ground probe pads.

The complete test structure coupon is shown in Fig. 7.3.1-1.

Test Structures & Measured DevicesTest Structures & Measured Devices

Figure #7.3.1-1: Test Structure Coupon

The test structures were measured using network analysis techniques. Since very

low loss metal was used in the manufacturing process, DC resistance measurements were

unreliable and not used. For high frequency measurements, an HP 8510C network

101

analyzer was used in conjunction with a Cascade Microtech probe station and ground-

signal-ground configuration probes. Calibration was accomplished using a supplied

substrate and the application of the line-reflect-match (LRM) calibration method. Data

was gathered for each of the test structures at 400 frequency points between 45MHz and

20GHz and stored by means of data acquisition software.

7.4 Modeling Scheme

7.4.1 9-Segment Meander Resistor Modeling Procedure

The improved modeling methodology detailed in Chapter 4 will be utilized to

model the 9-segment meander resistor. The fundamental idea behind this modeling

procedure is that passive structures are comprised of several key geometrical building

blocks. The building block equivalent circuits are derived from fabricated test structures

and measurements using optimization and extraction routines. Passive RLC models for

each embedded building block are extracted and take into account effects of processing

fluctuations and nonideal material properties. The objective here is to predict the

electrical behavior of arbitrary geometry passive devices in a standard circuit simulator;

therefore, providing a major speedup over methods that do not utilize lumped elements.

The steps involved in the modeling procedure for the 9-segment meander resistor

are described in a brief outline below:

102

1. The 64 test structures and 32 9-segment meander resistors were physically designed

(96% Alumina) and fabricated. Rena Huang PhD. Candidate fabricated the designs in

the MiRC cleanroom at Georgia Tech. High frequency s-parameter measurements of

the devices were taken by on-wafer ground-signal-ground probing.

2. The measured data was used in a circuit optimization input file to deembed the

element values for the 2 sets of 32 structures. Initial guesses used in the Hspice

Optimizer were made based on s-parameter measurements for one of each structure.

Once optimization for one structure was completed, the results were used for the

remaining optimizations.

3. 32 unique equivalent circuit models of the two test structures were obtained.

Fig. 7.4.1-1 shows the building blocks, the equivalent circuits, and some of the

element values generated from the Hspice Optimizer.





Pad PrimitivePad PrimitiveMaterial Square PrimitiveMaterial Square Primitive



1 3

2 4

R =0 . 09 Oh mL = 1 E -1 1 HC = 1 .1 e -1 5 FC M =0 .4 e -1 5 FCC = 1 .4 e -1 5 FL M = 0 .2 0

LC

R L R

C C CC

LC

R L R

C C C C

L M

L MCM

1

2

3

4

1 2

R = 0 . 08 O h mL = 1 E- 11 HC = 2.7 e- 1 5 FC C = 1. 2e - 15 F

L

C

R L R

C C C C

1 2

1 2

R =0 .0 8 O hmL = 1 E- 11 HC = 2.7 e- 15 FC C = 1. 2e - 15 F

L

C

R L R

C C C C

1 2

.

R= 0 .4 O h mL= 3 .7 E- 1 1 HC = 5.3 e -1 5 FCC= 2 7e - 1 5 F

L

C

R L R

CC CC

1 2

CC CC

2L

C

R L R1

1

2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

Figure #7.4.1-1: Building Blocks and Equivalent Circuits

103


Two sets of 32 test structures (1 and 2) were characterized. Mean and absolute

deviations in the deembedded circuit values of the building blocks were then calculated

and used in the Hspice MonteCarlo simulator as limits to generate random circuit values

for 500 simulations. Fig. 7.4.2-1 shows the deembedded element values from test

structure 1 & 2 and the relative variations used in the MonteCarlo simulation tool.

Equivalent Circuit Element ValuesStatistical Variations

Equivalent Circuit Element ValuesStatistical Variations

TS1 Sum of Sq.s' c_sq csq lsq rsqlow 1.11576 3.36E-15 6.52E-26 1.73E-11 2.51E-06high 29.9662 2.49E-13 6.52E-26 1.91E-11 2.51E-06median 2.745285 1.08E-13 6.52E-26 1.84E-11 2.51E-06ab var 1.629525 1.05E-13 0 1.03E-12 0rel var 0.593572252 0.968897 0 0.056241 0

TS1 c_pad rpad lpad cpadlow 1.2655E-12 3.75288 2.79E-11 2.21E-14high 1.87813E-12 5.625027 3.73E-11 2.59E-14median 1.58577E-12 4.37748 3.05E-11 2.44E-14ab var 3.20265E-13 0.6246 2.68E-12 2.27E-15rel var 0.2019619 0.142685 0.087727 0.092907

TS2 Sum of Sq.s' cou_1 c_cou l2 r2low 0.71189 0.159073 9.2E-25 1.27E-13 0.212918high 440.398 0.324847 2.74E-23 8.36E-12 1.589938median 1.690735 0.218506 6.36E-24 4.07E-12 0.753571ab var 0.978845 0.059433 5.44E-24 3.95E-12 0.540653rel var 0.578946435 0.271997 0.855404 0.968861 0.717455

TS2 c2 c_co2cr l2_2 r2_2 c2_2low 2.72533E-16 1.65E-15 1.51E-12 0.097137 2.8E-16high 2.79231E-15 3.38E-11 9.79E-12 1.115453 3.48E-15median 1.32663E-15 7.69E-12 4.16E-12 0.755395 1.42E-15ab var 1.0541E-15 7.68E-12 2.64E-12 0.658258 1.14E-15rel var 0.794567513 0.999786 0.635701 0.871409 0.803456

TS2 c_co2cr_2 l2_3 r2_3 c2_3 c_co2cr_3low 2.35967E-13 8.79E-13 0.278774 1.37E-18 2.77E-12high 1.14784E-11 1.21E-11 1.109648 4.25E-15 2.6E-11median 7.00171E-12 4.42E-12 0.705214 1.41E-15 7.81E-12ab var 6.76575E-12 3.54E-12 0.426439 1.41E-15 5.03E-12rel var 0.966298662 0.801238 0.604695 0.999028 0.644819

Figure #7.4.2-1: Mean and Absolute Deviations for Deembedded Element Values

104


An equivalent circuit of the 9-segment meander resistor was constructed by

replacing each building block in the structure with its equivalent extracted circuit values.

Due to the lack of correlation between the building blocks, each element is independent

of any other element in the circuit. Therefore, to create an accurate circuit design all the

elements in the design need to generate a random value for the defined number of

simulations. The 9-segment meander resistor generated 1336 random circuit values for

each simulation. This circuit file was automatically generated using Unix shell scripts

detailed in Chapter 9. Fig. 7.4.3-1 is a block diagram of a 9-segment meander resistor

and shows the amount of random number generations per simulation, for the PEEC

element values.

MaterialSquare

R=0.08 OhmL= 1E-11 HC = 2.7e-15 FCC = 1.2e-15 F

LC

R L R

CC CC1 2

Pad

R=0.08 OhmL=1E-11 HC = 2.7e- 15 FCC = 1.2e-15 F

LC

R L R

CC CC1 2

Coupled Pair


LC

R L R

CC CC

LC

R L R

CC CC

L MLM

CM

1

2

3

4

CC

LC

R1

Multi-Block

LC

R L R

CC CC1 2

LC

R L R

CC CC1 2

.

R=0.4 OhmL= 3.7E-11 HC = 5.3e-15 FCC=2 7e-15 F

CC2

L R

LC

R L R

CC CC1 2

LC

R L R

CC CC1 2

Pad

Pad





Simulation

105


Based on the relative variation calculated for each element value in the

equivalent circuit models, a MonteCarlo analysis was performed using Hspice. A

MonteCarlo analysis with a uniform parameter distribution simulated 500 sets of circuit

model parameters as shown in Fig. 7.4.4-1.

MonteCarlo Analysis Results Using Independent PremiseMonteCarlo Analysis Results Using Independent Premise

S21 RealS21 Real

S11 RealS11 Real

S21 ImaginaryS21 Imaginary


Figure #7.4.4-1: MonteCarlo Analysis Generated 500 Simulation

S-parameter data was generated for each of the 500 simulations, and extracted

from the output file using Unix scripts (detailed in Chapter 9). Due to a Microsoft Excel

limitation of 255 series plots per graph, all the data could not be rendered together.

Therefore, the MonteCarlo maximum and minimum were found for S11/S21

106

magnitude/phase and used as upper and lower limits (red) with the actual measured data

contained within the predicted range. This plot is shown in Fig. 7.4.4-2.

Following the simulations, the output response for the modeled device was

compared to actual measured data from the 32 fabricated 9-segment structures. It was

established that the electrical behavior from the 32 actual measured devices, was

predicted by the MonteCarlo analysis using the statistical variations of the equivalent

circuit element values deembedded from the test structures, as shown in Fig. 7.4.4-2.

S21 Magnitude MonteCarlo Range (Red) vs. Actual Measured Devices

0.36

0.41

0.46

0.51

0.56

0.61

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21Mag

S11 Phase MonteCarlo Range (Red) vs. Actual Measured Devices

-90

-70-50

-30

-1010

30

5070

90

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11Phase


0.380.430.480.530.580.630.680.730.78

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11Mag

MonteCarlo Predicted Range vs. Actual Measured 9-Segment Resistors MonteCarlo Predicted Range vs. Actual Measured 9-Segment Resistors


-90

-70-50

-30-10

1030

5070

90

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21Phase

Figure #7.4.4-2: MonteCarlo Predicted vs. Actual Measured

A close-up of the MonteCarlo vs. measured S11 magnitude plot (Fig. 7.4.4-3)

shows that the measured data stays within the MonteCarlo predicted range up to 2GHz.

By inspection, it appears that the parasitics begin to take effect at approximately 1GHz;

107

therefore, the deembedded model for the 9-segment resistor accurately predict the

measured electrical behavior range past the first resonance for S11 magnitude.

A close-up of the MonteCarlo vs. measured S21 magnitude plot (Fig.7.4.4-3)

shows that the measure data stays within the MonteCarlo predicted range up to 5GHz.

Again by inspection, it appears that the parasitics begin to take effect at 1GHz and at

3GHz; therefore, the deembedded model accurately predicts the measured behavior range

for S21 magnitude past the first and second resonance.

The MonteCarlo vs. measured phase plots were not as impressive as the

magnitude plots, because the measured data fell outside the MonteCarlo predicted range;

however, the trends were identical. Looking at the scale of +/- 90 degrees in Fig. 7.4.4-2,

it appears that the error between the MonteCarlo vs. measured is trivial compared to an

inductive 90 degrees and a capacitive –90 degrees of phase shift. The error actually

seems to be some DC offset.

MonteCarlo Predicted Range vs. Actual Measured 9-Segment ResistorsFor S11/S21 Magnitude



0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

1.E+08 1.E+09 1.E+10

Frequency

S11Mag


0.46

0.48

0.5

0.52

0.540.56

0.58

0.6

0.62

1.E+08 1.E+09 1.E+10

Frequency

S21Mag

2GHz2GHz

5GHz5GHz


108


To accurately compare the MonteCarlo range vs. measured results, several

statistical methods were used to find a number that represented both sets of data at each

frequency point.




calculating the mean. The variance is the most frequently used measure of variability

[32]. The standard deviation, derived from the variance, also indicates the relative

variability within a group and is calculated by taking the square root of the variance.






(population 16). Both sets are displaying the mean percent difference in magnitude of

S11 and S21, and are comparing data generated from a circuit model file constructed

using the correlated premise (Blue Plot), and the independent premise (Yellow Plot).

The results for both types of premise begin to diverge at 1.84GHz for S11

magnitude and 4.5GHz for S21 magnitude, however; the results using the correlated

premise have a lower percent difference at low frequencies (less than 1GHz) and a higher

percent difference beyond 1GHz.

109



-1.00%

9.00%

19.00%

29.00%

39.00%

49.00%

1.E+08 1.E+09 1.E+10

Frequency

S21 Mag% Error

Independent Blocks

Correlated Blocks



-1.00%

-0.50%

0.00%

0.50%

1.00%

1.50%

2.00%

1.E+08 1.E+09 1.E+10

Frequency

S21 Mag% Error

Independent Blocks

Correlated Blocks



-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error

Independent Blocks

Correlated Blocks



-0.10%

0.40%

0.90%

1.40%

1.90%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error

Independent Blocks

Correlated Blocks

Mean Percent Difference For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration

Mean Percent Difference For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration



Figure #7.4.6-1: Mean Value Percent Difference Between MonteCarlo and

Measured Data For S11/S21 Magnitude





displaying the standard deviation percent difference of the magnitude of S11 and S21,

and are comparing data generated from a circuit model file constructed using the

correlated premise (Blue Plot), and the independent premise (Yellow Plot).




110

of data, and at 7.0GHz for S21 magnitude using the second set of data. However; the

results using the independent premise have a much lower percent difference at all

frequencies for S11 and S21 magnitude.



-1.00%

4.00%

9.00%

14.00%

19.00%

24.00%

29.00%

1.E+08 1.E+09 1.E+10

Frequency

S21 Mag% Error

Independent Blocks

Correlated Blocks



0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

1.E+08 1.E+09 1.E+10

Frequency

S21 Mag% Error

Independent Blocks

Correlated Blocks



-1.00%

4.00%

9.00%

14.00%

19.00%

24.00%

29.00%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error

Independent Blocks

Correlated Blocks



0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

140.00%

160.00%

180.00%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error

Independent Blocks

Correlated Blocks

Standard Deviation Percent Difference For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration


2.5GHz2.5GHz

1.74GHz1.74GHz


Figure #7.4.6-2: Standard Deviation Percent Difference Between MonteCarlo and


7.4.7 Frequency Distribution Plots


devices were transposed from rows representing a particular frequency to columns


point. The range of values from high to low was split up into sixteen linearly place filter

111

bins. The frequency of occurrence was calculated and normalized to the number of data

points (500 for MonteCarlo, and 32 for actual measured). These normalized cumulative

frequency distribution (NCFD) plots were generated at every frequency point for

S11/S21 magnitude up to 7.5GHz. These plots illustrate that the measured data from

each fabricated device at a particular frequency fits, or doesn’t fit, within the MonteCarlo

predicted range. The trend lines generated within the NCFD plots also show how the

standard deviation between predicted and measured compare.

Fig. 7.4.7-1 and Fig. 7.4.7-2 display NCFD plots at several frequency points for

S11 magnitude, and Fig. 7.4.7-3 and Fig. 7.4.7-4 display NCFD plots at several

frequency points for S21 magnitude. Data generated via the correlated premise vs. the

independent premise is compared in all plots. By inspection, the standard deviation

match between the MonteCarlo predicted vs. measured, using the independent premise,

clearly outperforms the correlated premise at both frequencies.

112

MonteCarlo Prediction vs. Actual Measured Device NCFD Plot S11 Magnitude @ 600MHz Using Independent Premise

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.392

0

0.395

9

0.399

8

0.403

8

0.407

7

0.411

6

0.415

5

0.419

4

0.423

4

0.427

3

0.431

2

0.435

1

0.439

0

0.443

0

0.446

9

0.450

8

Filter Bins

NCFD

MonteCarloMeasuredMonteCarlo TrendMeasured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.394

4

0.398

2

0.402

1

0.405

9

0.409

7

0.413

6

0.417

4

0.421

3

0.425

1

0.428

9

0.432

8

0.436

6

0.440

5

0.444

3

0.448

1

0.452

0

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.396

6

0.400

4

0.404

1

0.407

9

0.411

7

0.415

4

0.419

2

0.423

0

0.426

7

0.430

5

0.434

3

0.438

0

0.441

8

0.445

6

0.449

3

0.453

1

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

600MHz600MHz

800MHz800MHz

950MHz950MHz

MonteCarlo Prediction vs. Actual Measured Device NCFD PlotS11 Magnitude @ 600MHz Using Correlated Premise

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.397

8

0.400

7

0.403

6

0.406

5

0.409

4

0.412

2

0.415

1

0.418

0

0.420

9

0.423

7

0.426

6

0.429

5

0.432

4

0.435

2

0.438

1

0.441

0

Filter Bins

NCFD



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.399

9

0.402

8

0.405

6

0.408

4

0.411

2

0.414

1

0.416

9

0.419

7

0.422

5

0.425

4

0.428

2

0.431

0

0.433

9

0.436

7

0.439

5

0.442

3

Filter Bins

NCFD



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.401

9

0.404

7

0.407

4

0.410

2

0.413

0

0.415

8

0.418

6

0.421

3

0.424

1

0.426

9

0.429

7

0.432

5

0.435

2

0.438

0

0.440

8

0.443

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

600MHz600MHz

800MHz800MHz

950MHz950MHz

Figure #7.4.7-1: NCFD Plot Comparisons Between Correlated Premise and

Independent Premise For S11

113

MonteCarlo Prediction vs. Actual Measured Device NCFD Plot S11 Magnitude @ 1.15GHz Using Independent Premise

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.400

1

0.403

7

0.407

4

0.411

0

0.414

7

0.418

4

0.422

0

0.425

7

0.429

3

0.433

0

0.436

6

0.440

3

0.443

9

0.447

6

0.451

2

0.454

9

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.408

7

0.412

1

0.415

5

0.418

8

0.422

2

0.425

6

0.429

0

0.432

4

0.435

7

0.439

1

0.442

5

0.445

9

0.449

3

0.452

6

0.456

0

0.459

4

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.421

1

0.424

2

0.427

3

0.430

3

0.433

4

0.436

5

0.439

6

0.442

6

0.445

7

0.448

8

0.451

9

0.455

0

0.458

0

0.461

1

0.464

2

0.467

3

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

1.15GHz1.15GHz

1.5GHz1.5GHz

2.0GHz2.0GHz

MonteCarlo Prediction vs. Actual Measured Device NCFD PlotS11 Magnitude @ 1.15GHz Using Correlated Premise

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.405

0

0.407

7

0.410

4

0.413

1

0.415

8

0.418

5

0.421

2

0.423

9

0.426

6

0.429

3

0.432

0

0.434

7

0.437

4

0.440

1

0.442

8

0.445

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.412

7

0.415

2

0.417

7

0.420

3

0.422

8

0.425

3

0.427

8

0.430

4

0.432

9

0.435

4

0.437

9

0.440

5

0.443

0

0.445

5

0.448

0

0.450

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.424

9

0.427

2

0.429

4

0.431

7

0.433

9

0.436

2

0.438

4

0.440

7

0.442

9

0.445

2

0.447

4

0.449

7

0.451

9

0.454

2

0.456

4

0.458

7

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

1.15GHz1.15GHz

1.5GHz1.5GHz

2.0GHz2.0GHz



114


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.557

5

0.561

5

0.565

4

0.569

3

0.573

3

0.577

2

0.581

1

0.585

1

0.589

0

0.593

0

0.596

9

0.600

8

0.604

8

0.608

7

0.612

6

0.616

6

Filter Bins

NCFD

MonteCarloMeasured

MonteCarlo TrendMeasured Trend


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.558

0

0.561

9

0.565

9

0.569

9

0.573

8

0.577

8

0.581

8

0.585

7

0.589

7

0.593

7

0.597

6

0.601

6

0.605

6

0.609

6

0.613

5

0.617

5

Filter Bins

NCFD

MonteCarloMeasured



0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.553

6

0.557

2

0.560

9

0.564

6

0.568

2

0.571

9

0.575

6

0.579

3

0.582

9

0.586

6

0.590

3

0.593

9

0.597

6

0.601

3

0.604

9

0.608

6

Filter Bins

NCFD

MonteCarlo

MeasuredMonteCarlo TrendMeasured Trend

800MHz800MHz

1.0GHz1.0GHz

2.0GHz2.0GHz


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.565

0

0.567

9

0.570

8

0.573

8

0.576

7

0.579

6

0.582

5

0.585

4

0.588

3

0.591

2

0.594

1

0.597

1

0.600

0

0.602

9

0.605

8

0.608

7

Filter Bins

NCFD



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.564

5

0.567

4

0.570

3

0.573

2

0.576

1

0.579

0

0.581

9

0.584

8

0.587

7

0.590

6

0.593

4

0.596

3

0.599

2

0.602

1

0.605

0

0.607

9

Filter Bins

NCFD



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.559

9

0.562

6

0.565

4

0.568

1

0.570

9

0.573

6

0.576

4

0.579

2

0.581

9

0.584

7

0.587

4

0.590

2

0.592

9

0.595

7

0.598

4

0.601

2

Filter Bins

NCFD


800MHz800MHz

1.0GHz1.0GHz

2.0GHz2.0GHz



115


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.547

1

0.550

5

0.553

9

0.557

4

0.560

8

0.564

2

0.567

6

0.571

0

0.574

4

0.577

8

0.581

3

0.584

7

0.588

1

0.591

5

0.594

9

0.598

3

Filter Bins

NCFD

MonteCarloMeasured



0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.538

6

0.541

7

0.544

8

0.547

9

0.551

1

0.554

2

0.557

3

0.560

4

0.563

6

0.566

7

0.569

8

0.572

9

0.576

0

0.579

2

0.582

3

0.585

4

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.517

0

0.519

9

0.522

8

0.525

7

0.528

6

0.531

5

0.534

4

0.537

3

0.540

2

0.543

1

0.546

0

0.548

9

0.551

8

0.554

7

0.557

6

0.560

5

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

3.0GHz3.0GHz

4.0GHz4.0GHz

6.0GHz6.0GHz


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.552

9

0.555

4

0.558

0

0.560

5

0.563

1

0.565

6

0.568

1

0.570

7

0.573

2

0.575

8

0.578

3

0.580

9

0.583

4

0.586

0

0.588

5

0.591

1

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.543

8

0.546

0

0.548

3

0.550

6

0.552

9

0.555

2

0.557

5

0.559

8

0.562

1

0.564

3

0.566

6

0.568

9

0.571

2

0.573

5

0.575

8

0.578

1

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.521

4

0.524

3

0.527

1

0.530

0

0.532

9

0.535

8

0.538

6

0.541

5

0.544

4

0.547

2

0.550

1

0.553

0

0.555

9

0.558

7

0.561

6

0.564

5

Filter Bins

NCFD

MonteCarlo

Measured


3.0GHz3.0GHz

4.0GHz4.0GHz

6.0GHz6.0GHz




Using the normalized cumulative distribution information, a measure of linear

relationship between the MonteCarlo predicted and actual measured data was calculated

using the correlation coefficient. A perfect correlation between two variables results in r

= 1.0, no correlation at all results in r = 0.0, and any number in-between means that there

is some correlation between the variables however the relationship is not perfect.

116

Fig. 7.4.8-1 displays the Correlation Coefficient results using the normalized

cumulative frequency distribution data (population 16) generated from the magnitude of

S11 and S2. The Correlation Coefficients are compared to data generated via the

correlated premise (Blue Plot) vs. the independent premise (Yellow Plot).


magnitude and 4.5GHz for S21 magnitude. The results using the independent premise

have a higher Correlation Coefficient than the results using the correlated premise at all

the frequencies.



0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.E+08 1.E+09 1.E+10

Frequency

S11 MagCorrelation

Coff.

Independent Blocks

Correlated Blocks



0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.E+08 1.E+09 1.E+10

Frequency

S21 MagCorrelation

Coff.

Independent Blocks

Correlated Blocks

Correlation Coefficient For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration



Figure #7.4.8-1: Pearson Product-Moment Correlation Coefficient for MonteCarlo

Predicted and Actual Measured Data

7.4.9 T-Test

The primary purpose of the “t” test is to determine whether the means of two

groups of numbers differ to a statistically significant degree. The larger the absolute

value of t, the more likely it is to reflect a significant difference between the two groups

under comparison.

117




distribution data (population 16). Both sets are displaying the t-test values using the

magnitude of S11 and S21, and are comparing data generated via the correlated premise

(Blue Plot) vs. the independent premise (Yellow Plot).


magnitude and 4.5GHz for S21 magnitude, however; the results using the independent

premise have a lower t-test value at higher frequencies (greater than 1.14GHz for S11,

and greater then 2.5GHz for S21) and a slightly lower value below 1.14GHz and 2.5GHz

for S11 and S21 respectively.



0.00

0.50

1.00

1.50

2.00

2.50

1.E+08 1.E+09 1.E+10

Frequency

S21 MagT Value

Independent Blocks

Correlated Blocks



-0.50

1.50

3.50

5.50

7.50

9.50

1.E+08 1.E+09 1.E+10

Frequency

S21 MagT Value

Independent Blocks

Correlated Blocks



-0.25

0.25

0.75

1.25

1.75

2.25

1.E+08 1.E+09 1.E+10

Frequency

S11 MagT Value

Independent Blocks

Correlated Blocks



-0.50

1.50

3.50

5.50

7.50

9.50

1.E+08 1.E+09 1.E+10

Frequency

S11 MagT Value

Independent Blocks

Correlated Blocks

T-Test For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration

T-Test For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration


4.5GHz4.5GHz

4.5GHz4.5GHz

Figure #7.4.9-1: T-Test for MonteCarlo Predicted and Actual Measured Data

118

7.4.10 ANOVA





variance between groups versus the variance within groups.[terrified] The ANOVA

starts with the total amount of variability in the data and divides it up (partitioning) into

various categories. Basically, it compares the variability among the group means with

the variability that occurred just by chance or error.

Fig. 7.4.10-1 displays the ANOVA results using two different sets of data. The

first set compares the raw MonteCarlo generated (population 500) data vs. the raw

Measured (population 32) data. The second set compares the normalized cumulative

frequency distribution data (population 16). Both sets are displaying the ANOVA values

using the magnitude of S11 and S21, and are comparing data generated via the correlated

premise (Blue Plot) vs. the independent premise (Yellow Plot).



premise have a lower ANOVA value at higher frequencies (greater than 1.0GHz for S11,

and greater then 2.0GHz for S21).

119



-0.25

-0.05

0.15

0.35

0.55

0.75

0.95

1.15

1.35

1.E+08 1.E+09 1.E+10

Frequency

S21 MagF(obt)

Independent Blocks

Correlated Blocks



-1.00

4.00

9.00

14.00

19.00

24.00

29.00

1.E+08 1.E+09 1.E+10

Frequency

S21 MagF(obt)

Independent Blocks

Correlated Blocks



-0.02

0.18

0.38

0.58

0.78

0.98

1.E+08 1.E+09 1.E+10

Frequency

S11 MagF(obt)

Independent Blocks

Correlated Blocks



-0.25

4.75

9.75

14.75

19.75

24.75

29.75

1.E+08 1.E+09 1.E+10

Frequency

S11 MagF(obt)

Independent Blocks

Correlated Blocks

ANOVA For S11/S21 Magnitude Using Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration

ANOVA For S11/S21 Magnitude Using Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration



Figure #7.4.10-1: ANOVA for MonteCarlo Predicted and Actual Measured Data

7.5 Summary

This chapter explained in detail the application of the predictive statistical

analysis methodology, described in Chapter 6, to a 9-segment meander resistor.

Modeling the 9-segment resistor, successfully proved the method of predicting passive

device performance variations and yield, when provided with a statistical distribution of

the building block models and a circuit schematic.

Using the improved method from Chapter 4 and the independent premise when

collating the circuit model file, results from the 9-segment meander resistor analysis were

120

compared to the analysis of the same 9-segment meander resistor; using the same

modeling methodology; and, using the correlated premise when collating the circuit

model file.

The results of the Mean Percent Difference, the t-test, and the ANOVA showed

that both types of premise produced the same behavior at lower frequencies (1GHz for

S11 and 3GHz for S21), however at higher frequencies the independent premise out

preformed the correlated premise.

The results of the Standard Deviation Percent Difference, and the Correlation

Coefficient showed an overall improvement at all frequencies using the independent

premise. The NCFD plots best exemplified the contrast in the data generated by the two

types of premise, with the independent premise outperforming the correlated premise.

121

CHAPTER 8

PREDICTIVE STATISTICAL ANALYSIS OF 9 SEGMENT

MEANDER RESISTOR MODELED USING ONLY 1 TEST

STRUCTURE

8.1 Introduction

The predictive statistical analysis of the 9-segment meander resistor detailed in

Chapter 7 will now be accomplished using only one test structure to deembed all the

building block equivalent circuit element values. The modeling methodology will follow

the procedure described in Chapter 4 except for the elimination of test structure one. The

methodology presented in this chapter will also demonstrate that the variation in the

complete equivalent circuit models, based only on the circuit building blocks, can be used

to predict such variations in actual fabricated devices. The 9-segment meander resistor

fabricated on a 96% Alumina substrate will be the device under inspection.

Two sets of resistor structures were fabricated on a 96% Alumina substrate: 32

three-segment resistors (test structure 2), and 32 nine-segment resistors (modeled device).

Scattering parameter measurements ranging in frequencies from 45MHz to 20GHz were

taken from all the structures. The measured s-parameter data from test structure 2 was

122

used to deembed the element values of the partial element equivalent circuit (PEEC) for

the 32 test structures, using a non-linear optimization algorithm in Hspice. This process

generated a unique equivalent circuit model for each of the 32 test structure devices.

Mean and relative variation was then calculated for each component from the 32 unique

equivalent circuit models, and used in a MonteCarlo analysis of the 9-segment modeled

device. This analysis generated 500 s-parameter curves of the modeled device electrical

behavior.

The s-parameter measurements from the MonteCarlo analysis and the actual

measured device were compared to determine if the measured data resides in the range of

curves generated by the MonteCarlo analysis. By inspection, it was noted that the

electrical behavior from the fabricated devices was in fact contained within the range

predicted by the MonteCarlo results. Therefore by means of comparison between the

MonteCarlo results and measured data, it was shown that the statistical variations of the

component values provide an accurate representation of the overall modeled device

performance.


The same three-step MonteCarlo technique, as in Chapter 7, was used to

accomplish the statistical prediction: 1.) A statistical model was formed to characterize

the fabrication process used; 2.) A circuit design was created using a Spice type

simulator and applying nominal device values for the target process; and 3.) Randomly

generated element values of the process model were simulated in a MonteCarlo tool to

123

produce a representative set of output performance characteristics. The impact of random

process variations can be inferred from these simulations, and electrical yield can then be

estimated using the percentage of that sample which meets the performance requirements

[29].

8.3 Test Structure Description

One set (test structures 2) of 32 test structures were fabricated by depositing gold

on a 96% Alumina substrate. The metal thickness used was 3000 angstroms with 30µ

widths and spacings for all connecting segments. All resistor structures were on one

layer, with connections to the devices made using a ground-signal-ground probe pad

pattern.

Test structure 2 consisted of two probe pads connected to five material squares

which are connected to nine coupled material squares, and two corners made up of five

material squares.

The layout and fabrication mask photos of test structure 2 and the modeled 9-

segment meander resistor are shown in Fig. 8.3-1.

124


9-SegmentMeander Resistor

9-SegmentMeander Resistor

Figure #8.3-1: Test Structure 2 and 9-Segment Resistor Layout and Mask Photos

These test structure 2 allowed the modeling of probe pads, material square,

coupled material squares, and corner primitives.


The 96% Alumina structure was physically designed using integrated circuit

design tools within the Avanti design environment. Rena Huang PhD. Candidate in the

School of Electrical and Computer Engineering at the Georgia Institute of Technology

fabricated the design in Georgia Tech’s MiRC cleanroom. The size of the completed

coupon was approximately 2.25" x 2.25". Metal lines were drawn 30µ wide. The

embedded structures were interfaced accessed using ground-signal-ground probe pads.

The complete test structure coupon is shown in Figure 8.3.1-1.

125

Figure #8.3.1-1: Test Structure Coupon

All the test structure two devices were measured using network analysis

techniques. Since very low loss metal was used in the manufacturing process, DC

resistance measurements were unreliable and not used. For high frequency

measurements, an HP 8510C network analyzer was used in conjunction with a Cascade


accomplished using a supplied substrate and the application of the line-reflect-match

(LRM) calibration method. Data was gathered for each of the test structures at over 400

frequency points between 45MHz and 20GHz and stored by means of data acquisition

software.

126

8.4 Modeling Scheme

8.4.1 9-Segment Meander Resistor Modeling Procedure

The improved modeling methodology using only one test structure, detailed in

Chapter 5, will be utilized to model the 9-segment meander resistor. The fundamental

idea behind this modeling procedure is that passive structures are comprised of several

key geometrical building blocks. The building block equivalent circuits are now derived

from only one fabricated test structure and measurements using optimization and

extraction routines. Passive RLC models for each embedded building block are extracted

and take into account effects of processing fluctuations and nonideal material properties.

The objective here is to predict the electrical behavior of arbitrary geometry passive

devices in a standard circuit simulator; therefore, providing a major speedup over

methods that do not utilize lumped elements.

The steps involved in the modeling procedure for the 9-segment meander resistor

are described in a brief outline below:

1. The 32 test structures and 32 9-segment meander resistors were physically designed

(96% Alumina) and fabricated. Rena Huang PhD. Candidate fabricated the designs in

the MiRC cleanroom at Georgia Tech. High frequency s-parameter measurements of

the devices were taken by on-wafer ground-signal-ground probing.

2. The measured data was used in a circuit optimization input file to deembed the

element values for the 32 structures. Initial guesses used in the Hspice Optimizer

127

were made based on s-parameter measurements for one of each structure. Once

optimization for one structure was completed, the results were used for the remaining

optimizations.

3. 32 unique equivalent circuit models of test structure two (the only test structure used)

were obtained.

Fig. 8.4.1-1 shows the building blocks, the equivalent circuits, and some of the

element values generated from the Hspice Optimizer.


1 3

2 4

R =0 .09 Oh mL = 1 E -1 1 HC = 1.1 e -1 5 FC M =0 .4e - 1 5 FCC = 1 .4 e-1 5 FL M = 0 .2 0

LC

R L R

C C CC

LC

R L R

C C C C

L M

L MCM

1

2

3

4

1 2

R =0. 08 O hmL = 1 E- 11 HC = 2.7 e- 1 5 FC C = 1.2e - 15 F

L

C

R L R

C C C C

1 2

1 2

R =0 .08 O h mL = 1 E- 11 HC = 2.7 e- 1 5 FC C = 1.2e- 15 F

L

C

R L R

C C C C

1 2

.

R= 0 .4 O hmL= 3 .7 E- 11 HC = 5.3 e -1 5 FCC = 2 7e - 1 5 F

L

C

R L R

CC CC

1 2

CC CC

2L

C

R L R1

1

2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2

L

C

R L R

CC CC

1 2



Primitives



Figure #8.4.1-1: Building Blocks & Equivalent Circuits Using Only Test Structure 2

128


Thirty-two test structures (test structure 2) were characterized. Mean and relative

variation in the deembedded circuit values of the building blocks was calculated and used

in the Hspice MonteCarlo simulator as limits to generate random circuit values for 500

simulations. Fig. 8.4.2-1 shows the deembedded element values from test structure 2 and

the relative variations used in the MonteCarlo simulation tool.


Statistical Variations of Equivalent Circuit Element ValuesStatistical Variations of Equivalent Circuit Element Values

Simulation Sum of Squares c_sq csq lsq rsq c_pad rpad lpad cpadlow 0.558429 3.96E-14 3.04E-27 1.74E-11 2.11E-06 1.37E-12 3.213313 2.52E-11 2.1E-14high 63.4942 1.65E-13 2.12E-25 1.95E-11 3.87E-06 2.78E-12 5.772112 4.38E-11 2.66E-14median 1.53612 1.02E-13 6.21E-26 1.84E-11 2.47E-06 1.75E-12 4.452542 3.23E-11 2.46E-14ab var 0.977691 6.21E-14 5.91E-26 9.67E-13 3.59E-07 3.83E-13 1.23923 7.09E-12 3.66E-15rel var 0.636467854 0.610554 0.95101 0.052618 0.145617 0.218885 0.27832 0.219803 0.148648

Simulation Sum of Squares cou_1 c_cou l2 r2 c2 c_co2cr l2_2 r2_2 c2_2 c_co2cr_2 l2_3 r2_3 c2_3low 0.558429 0.167882 3.25E-24 3.24E-12 0.283159 8.62E-16 5.13E-12 2.83E-12 0.364671 5.2E-17 4.71E-12 5.43E-13 0.405914 6.73E-16high 63.4942 0.256087 1.19E-23 7.58E-12 1.292548 2.19E-15 1.28E-11 7.61E-12 0.904278 2.55E-15 1.06E-11 5.92E-12 0.965507 2.57E-15median 1.53612 0.211063 6.98E-24 4.52E-12 0.743332 1.36E-15 7.78E-12 4.83E-12 0.72981 1.48E-15 6.77E-12 4.4E-12 0.706911 1.36E-15ab var 0.977691 0.043182 3.73E-24 1.28E-12 0.460173 4.93E-16 2.65E-12 1.99E-12 0.365139 1.43E-15 2.06E-12 3.86E-12 0.300997 6.85E-16rel var 0.636467854 0.204592 0.534402 0.282134 0.619067 0.36382 0.340874 0.412802 0.50032 0.964868 0.30473 0.876576 0.425792 0.504412

Figure #8.4.2-1: Mean and Relative Variations for Deembedded Element Values

From Test Structure Two


An equivalent circuit of the 9-segment meander resistor was constructed by

replacing each building block in the structure with its equivalent extracted circuit values.

Due to the lack of correlation between the building blocks, each element is independent

of any other element in the circuit. Therefore, to create an accurate circuit representation

all the elements in the design need to generate a random value for the defined number of

MonteCarlo simulations. The 9-segment meander resistor generated 1336 random circuit

values for each simulation. This circuit file was automatically generated using Unix shell

129

scripts detailed in Chapter 9. Fig. 8.4.3-1 is a block diagram of a 9-segment meander

resistor and shows the amount of random number generations per simulation, for the

PEEC element values.

MaterialSquare

R=0.08 OhmL= 1E-11 HC = 2.7e-15 FCC = 1.2e-15 F

LC

R L R

CC C C

1 2

Pad

R=0.08 OhmL=1E-11 HC = 2.7e-15 FCC = 1.2e-15 F

LC

R L R

CC C C

1 2

Coupled Pair


LC

R L R

CC CC

LC

R L R

CC CC

LMLM

CM

1

2

3

4

CC

LC

R1

Multi-Block

LC

R L R

CC CC1 2

LC

R L R

CC CC1 2

.

R=0.4 OhmL= 3.7E-11 HC = 5.3e-15 FCC=2 7e-15 F

CC2

L R

LC

R L R

CC CC1 2

LC

R L R

CC CC1 2

Pad

Pad





Simulation


Based on the relative variation calculated for each element value in the

equivalent circuit models, a MonteCarlo analysis was performed using Hspice. A

MonteCarlo analysis with a uniform parameter distribution simulated 500 sets of circuit

model parameters, as shown in Fig.8.4.4-1.

130

MonteCarlo Analysis Results Using OnlyOne Test Structure To Deembed

Building Block Equivalent Circuit Values

MonteCarlo Analysis Results Using OnlyOne Test Structure To Deembed

Building Block Equivalent Circuit Values

S21 RealS21 Real

S11 RealS11 Real



Figure #8.4.4-1: MonteCarlo Analysis Generated 500 Simulations

S-parameter data was generated for each of the 500 simulations, and extracted

from the output file using Unix scripts (detailed in Chapter 9). Due to a Microsoft Excel

limitation of 255 series plots per graph, all the data could not be rendered together.

Therefore, the MonteCarlo maximum and minimum were found for S11/S21

magnitude/phase and used as upper and lower limits (red) with the actual measured data

contained within the predicted range. This plot is shown in Fig. 8.4.4-2.

Following the simulations, the output response for the modeled device was

compared to actual measured data from the 32 fabricated 9-segment structures. It was

131

established that the electrical behavior from the 32 actual measured devices, was

predicted by the MonteCarlo analysis using the statistical variations of the equivalent

circuit element values deembedded from the test structure, as shown in Fig. 8.4.4-2.

MonteCarlo Predicted Range vs. Actual Measured 9-Segment Resistors MonteCarlo Predicted Range vs. Actual Measured 9-Segment Resistors


0.38

0.43

0.48

0.53

0.58

0.63

0.68

0.73

0.78

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11Mag


0.36

0.41

0.46

0.51

0.56

0.61

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21Mag


-90

-70

-50

-30

-10

10

30

50

70

90

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21Phase


-90

-70

-50

-30

-10

10

30

50

70

90

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11Phase



shows the measured data stays within the MonteCarlo predicted range up to 1GHz. By

inspection, it appears that the parasitics begin to take effect at approximately 1GHz;

therefore, the deembedded model for the 9-segment resistor accurately predict the

measured electrical behavior range up to the first resonance for S11 magnitude.


shows the measure data stays within the MonteCarlo predicted range up to 2.7GHz.

132

Again by inspection, it appears that the parasitics begin to take effect at 1GHz and at

3GHz; therefore, the deembedded model accurately predicts the measured behavior range

for S21 magnitude past the first and up to the second resonance.

The MonteCarlo vs. measured phase plots were not as impressive as the

magnitude plots, because the measured data fell outside the MonteCarlo predicted range;

however, the trends were identical. Looking at the scale of +/- 90 degrees in Fig. 8.4.4-2,

it appears that the error between the MonteCarlo vs. measured is trivial compared to an

inductive 90 degrees and a capacitive –90 degrees of phase shift. The error actually

seems to be some DC offset.


0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S21Mag


0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

1.E+08 1.E+09 1.E+10 1.E+11

Frequency

S11Mag



1GHz1GHz

2.7GHz2.7GHz



To accurately compare the MonteCarlo range vs. measured results, several

statistical methods were used to find a number that represented both sets of data at each

frequency point.

133




calculating the mean. The variance is the most frequently used measure of variability

[32]. The standard deviation, derived from the variance, also indicates the relative

variability within a group and is calculated by taking the square root of the variance.






(population 16). Both sets are displaying the mean percent difference in magnitude of

S11 and S21, and are comparing data generated from a circuit model file constructed

using the correlated premise (Blue Plot), the independent premise (Yellow Plot), and the

independent premise using only 1 test structure (independent 1TS, Red Plot).

The results for all three types of premise begin to diverge at 1.84GHz for S11


1TS premise have a lower percent difference at lower frequencies (less than 1GHz) and a

higher percent difference beyond 1GHz for S11 magnitude. The independent 1TS

premise also has a lower percent difference at frequencies lower than 2.5GHz for S21

magnitude, and basically equal the other two types of premise at frequencies greater than

2.5GHz.

134



-1.00%

9.00%

19.00%

29.00%

39.00%

49.00%

1.E+08 1.E+09 1.E+10

Frequency

S21 Mag% Error

Independent Blocks

Correlated Blocks

Independent Blocks 1TS



-1.00%

-0.50%

0.00%

0.50%

1.00%

1.50%

2.00%

1.E+08 1.E+09 1.E+10

Frequency

S21 Mag% Error

Independent BlocksCorrelated Blocks




-0.10%

0.40%

0.90%

1.40%

1.90%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error

Independent Blocks

Correlated Blocks




-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error

Independent Blocks

Correlated Blocks


Mean Percent Difference For S11/S21 Magnitude Using Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration

Mean Percent Difference For S11/S21 Magnitude Using Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration



Figure #8.4.6-1: Mean Value Percent Difference Between MonteCarlo and






displaying the standard deviation percent difference of the magnitude of S11 and S21,

and are comparing data generated from a circuit model file constructed using the

correlated premise (Blue Plot), the independent premise (Yellow Plot), and the

independent 1TS premise (Red Plot).



135


of data, and at 7.0GHz for S21 magnitude using the second set of data. However; the

results using the independent and independent 1TS types of premise have a much lower

percent difference at all frequencies than the correlated premise for S11 and S21

magnitude.



-1.00%

4.00%

9.00%

14.00%

19.00%

24.00%

29.00%

1.E+08 1.E+09 1.E+10

Frequency

S21 Mag% Error

Independent Blocks

Correlated Blocks




0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

1.E+08 1.E+09 1.E+10

Frequency

S21 Mag% Error

Independent Blocks

Correlated Blocks




0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

140.00%

160.00%

180.00%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error

Independent Blocks

Correlated Blocks




-1.00%

4.00%

9.00%

14.00%

19.00%

24.00%

29.00%

1.E+08 1.E+09 1.E+10

Frequency

S11 Mag% Error

Independent Blocks

Correlated Blocks






Figure #8.4.6-2: Standard Deviation Percent Difference Between MonteCarlo and


8.4.7 Frequency Distribution Plots


devices were transposed from rows representing a particular frequency to columns

136


point. The range of values from high to low was split up into sixteen linearly place filter

bins. The frequency of occurrence was calculated and normalized to the number of data

points (500 for MonteCarlo, and 32 for actual measured). These normalized cumulative

frequency distribution (NCFD) plots were generated at every frequency point for

S11/S21 magnitude up to 7.5GHz. These plots illustrate that the measured data from

each fabricated device at a particular frequency fits, or doesn’t fit, within the MonteCarlo

predicted range. The trend lines generated within the NCFD plots also show how the

standard deviation between predicted and measured compare.

NCFD plots at several different frequencies for S11 and S21 magnitude are shown

in Fig. 8.4.7-1 through Fig. 8.4.7-7 and in Fig. 8.4.7-8 through Fig. 8.4.7-14 respectively.

Data generated via the correlated vs. the independent vs. the independent 1TS premise

are compared in these NCFD plots. By inspection, the standard deviation match between

the MonteCarlo predicted vs. measured for the independent premise clearly outperforms

the correlated and the independent 1TS premise at all frequencies.

137


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.392

0

0.395

9

0.399

8

0.403

8

0.407

7

0.411

6

0.415

5

0.419

4

0.423

4

0.427

3

0.431

2

0.435

1

0.439

0

0.443

0

0.446

9

0.450

8

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.396

6

0.399

8

0.403

0

0.406

2

0.409

4

0.412

6

0.415

8

0.419

0

0.422

2

0.425

4

0.428

6

0.431

8

0.434

9

0.438

1

0.441

3

0.444

5

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

Correlated PremiseCorrelated Premise Independent 1TS PremiseIndependent 1TS Premise

Independent PremiseIndependent Premise

Data Extracted at:600MHz



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.397

8

0.400

7

0.403

6

0.406

5

0.409

4

0.412

2

0.415

1

0.418

0

0.420

9

0.423

7

0.426

6

0.429

5

0.432

4

0.435

2

0.438

1

0.441

0

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend

Figure #8.4.7-1: NCFD Plot Comparisons Between Correlated, Independent, and

Independent 1TS Premise For S11 at 600MHz




-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.394

4

0.398

2

0.402

1

0.405

9

0.409

7

0.413

6

0.417

4

0.421

3

0.425

1

0.428

9

0.432

8

0.436

6

0.440

5

0.444

3

0.448

1

0.452

0

Filter Bins

NCFD



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.398

8

0.402

0

0.405

1

0.408

2

0.411

4

0.414

5

0.417

7

0.420

8

0.423

9

0.427

1

0.430

2

0.433

4

0.436

5

0.439

6

0.442

8

0.445

9

Filter Bins

NCFD





-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.399

9

0.402

8

0.405

6

0.408

4

0.411

2

0.414

1

0.416

9

0.419

7

0.422

5

0.425

4

0.428

2

0.431

0

0.433

9

0.436

7

0.439

5

0.442

3

Filter Bins

NCFD




138




-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.396

6

0.400

4

0.404

1

0.407

9

0.411

7

0.415

4

0.419

2

0.423

0

0.426

7

0.430

5

0.434

3

0.438

0

0.441

8

0.445

6

0.449

3

0.453

1

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.400

9

0.404

0

0.407

0

0.410

1

0.413

2

0.416

3

0.419

4

0.422

5

0.425

6

0.428

7

0.431

8

0.434

8

0.437

9

0.441

0

0.444

1

0.447

2

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend




-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.401

9

0.404

7

0.407

4

0.410

2

0.413

0

0.415

8

0.418

6

0.421

3

0.424

1

0.426

9

0.429

7

0.432

5

0.435

2

0.438

0

0.440

8

0.443

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend






-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.400

1

0.403

7

0.407

4

0.411

0

0.414

7

0.418

4

0.422

0

0.425

7

0.429

3

0.433

0

0.436

6

0.440

3

0.443

9

0.447

6

0.451

2

0.454

9

Filter Bins

NCFD

MonteCarloMeasured



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.404

1

0.407

1

0.410

1

0.413

1

0.416

1

0.419

1

0.422

1

0.425

2

0.428

2

0.431

2

0.434

2

0.437

2

0.440

2

0.443

2

0.446

3

0.449

3

Filter Bins

NCFD

MonteCarlo


Data Extracted at:1.15GHz



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.405

0

0.407

7

0.410

4

0.413

1

0.415

8

0.418

5

0.421

2

0.423

9

0.426

6

0.429

3

0.432

0

0.434

7

0.437

4

0.440

1

0.442

8

0.445

6

Filter Bins

NCFD

MonteCarlo

MeasuredMonteCarlo Trend

Measured Trend


Independent 1TS Premise For S11 at 1.15GHz

139




-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.408

7

0.412

1

0.415

5

0.418

8

0.422

2

0.425

6

0.429

0

0.432

4

0.435

7

0.439

1

0.442

5

0.445

9

0.449

3

0.452

6

0.456

0

0.459

4

Filter Bins

NCFD

MonteCarlo


Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.412

1

0.414

9

0.417

7

0.420

5

0.423

4

0.426

2

0.429

0

0.431

8

0.434

7

0.437

5

0.440

3

0.443

1

0.446

0

0.448

8

0.451

6

0.454

5

Filter Bins

NCFD

MonteCarloMeasuredMonteCarlo Trend

Measured Trend




-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.412

7

0.415

2

0.417

7

0.420

3

0.422

8

0.425

3

0.427

8

0.430

4

0.432

9

0.435

4

0.437

9

0.440

5

0.443

0

0.445

5

0.448

0

0.450

6

Filter Bins

NCFD

MonteCarlo


Measured Trend






-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.421

1

0.424

2

0.427

3

0.430

3

0.433

4

0.436

5

0.439

6

0.442

6

0.445

7

0.448

8

0.451

9

0.455

0

0.458

0

0.461

1

0.464

2

0.467

3

Filter Bins

NCFD


Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.424

5

0.427

1

0.429

6

0.432

2

0.434

7

0.437

3

0.439

9

0.442

4

0.445

0

0.447

5

0.450

1

0.452

6

0.455

2

0.457

7

0.460

3

0.462

9

Filter Bins

NCFD


Measured Trend




-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.424

9

0.427

2

0.429

4

0.431

7

0.433

9

0.436

2

0.438

4

0.440

7

0.442

9

0.445

2

0.447

4

0.449

7

0.451

9

0.454

2

0.456

4

0.458

7

Filter Bins

NCFD

MonteCarlo


Measured Trend



140




-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.434

4

0.437

2

0.440

0

0.442

7

0.445

5

0.448

3

0.451

0

0.453

8

0.456

5

0.459

3

0.462

1

0.464

8

0.467

6

0.470

4

0.473

1

0.475

9

Filter Bins

NCFD



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.437

8

0.440

1

0.442

4

0.444

7

0.446

9

0.449

2

0.451

5

0.453

8

0.456

1

0.458

3

0.460

6

0.462

9

0.465

2

0.467

5

0.469

7

0.472

0

Filter Bins

NCFD





-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.437

9

0.439

9

0.441

9

0.443

8

0.445

8

0.447

8

0.449

8

0.451

7

0.453

7

0.455

7

0.457

6

0.459

6

0.461

6

0.463

6

0.465

5

0.467

5

Filter Bins

NCFD

MonteCarlo


Measured Trend








-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.565

0

0.567

9

0.570

8

0.573

8

0.576

7

0.579

6

0.582

5

0.585

4

0.588

3

0.591

2

0.594

1

0.597

1

0.600

0

0.602

9

0.605

8

0.608

7

Filter Bins

NCFD

MonteCarlo


Measured Trend


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.558

0

0.561

9

0.565

9

0.569

9

0.573

8

0.577

8

0.581

8

0.585

7

0.589

7

0.593

7

0.597

6

0.601

6

0.605

6

0.609

6

0.613

5

0.617

5

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.562

8

0.566

0

0.569

3

0.572

5

0.575

7

0.579

0

0.582

2

0.585

4

0.588

7

0.591

9

0.595

1

0.598

4

0.601

6

0.604

9

0.608

1

0.611

3

Filter Bins

NCFD

MonteCarloMeasured




141






-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.564

5

0.567

4

0.570

3

0.573

2

0.576

1

0.579

0

0.581

9

0.584

8

0.587

7

0.590

6

0.593

4

0.596

3

0.599

2

0.602

1

0.605

0

0.607

9

Filter Bins

NCFD

MonteCarlo


Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.557

5

0.561

5

0.565

4

0.569

3

0.573

3

0.577

2

0.581

1

0.585

1

0.589

0

0.593

0

0.596

9

0.600

8

0.604

8

0.608

7

0.612

6

0.616

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.562

3

0.565

5

0.568

7

0.571

9

0.575

1

0.578

3

0.581

6

0.584

8

0.588

0

0.591

2

0.594

4

0.597

6

0.600

8

0.604

1

0.607

3

0.610

5

Filter Bins

NCFD

MonteCarlo


Measured Trend








-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.559

9

0.562

6

0.565

4

0.568

1

0.570

9

0.573

6

0.576

4

0.579

2

0.581

9

0.584

7

0.587

4

0.590

2

0.592

9

0.595

7

0.598

4

0.601

2

Filter Bins

NCFD

MonteCarlo


Measured Trend


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.553

6

0.557

2

0.560

9

0.564

6

0.568

2

0.571

9

0.575

6

0.579

3

0.582

9

0.586

6

0.590

3

0.593

9

0.597

6

0.601

3

0.604

9

0.608

6

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.557

5

0.560

5

0.563

6

0.566

6

0.569

7

0.572

8

0.575

8

0.578

9

0.581

9

0.585

0

0.588

0

0.591

1

0.594

1

0.597

2

0.600

3

0.603

3

Filter Bins

NCFD

MonteCarlo


Measured Trend



142






-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.552

9

0.555

4

0.558

0

0.560

5

0.563

1

0.565

6

0.568

1

0.570

7

0.573

2

0.575

8

0.578

3

0.580

9

0.583

4

0.586

0

0.588

5

0.591

1

Filter Bins

NCFD

MonteCarlo


Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.547

1

0.550

5

0.553

9

0.557

4

0.560

8

0.564

2

0.567

6

0.571

0

0.574

4

0.577

8

0.581

3

0.584

7

0.588

1

0.591

5

0.594

9

0.598

3

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.550

2

0.553

1

0.556

0

0.558

8

0.561

7

0.564

6

0.567

4

0.570

3

0.573

2

0.576

0

0.578

9

0.581

8

0.584

6

0.587

5

0.590

4

0.593

3

Filter Bins

NCFD

MonteCarlo


Measured Trend








-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.543

8

0.546

0

0.548

3

0.550

6

0.552

9

0.555

2

0.557

5

0.559

8

0.562

1

0.564

3

0.566

6

0.568

9

0.571

2

0.573

5

0.575

8

0.578

1

Filter Bins

NCFD

MonteCarlo


Measured Trend


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.538

6

0.541

7

0.544

8

0.547

9

0.551

1

0.554

2

0.557

3

0.560

4

0.563

6

0.566

7

0.569

8

0.572

9

0.576

0

0.579

2

0.582

3

0.585

4

Filter Bins

NCFD

MonteCarlo

Measured



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.540

8

0.543

5

0.546

1

0.548

7

0.551

3

0.553

9

0.556

6

0.559

2

0.561

8

0.564

4

0.567

0

0.569

7

0.572

3

0.574

9

0.577

5

0.580

2

Filter Bins

NCFD

MonteCarlo


Measured Trend



143






-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.533

1

0.535

7

0.538

3

0.540

9

0.543

5

0.546

1

0.548

7

0.551

3

0.553

9

0.556

5

0.559

1

0.561

7

0.564

3

0.566

9

0.569

5

0.572

1

Filter Bins

NCFD

MonteCarlo



-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.528

3

0.531

1

0.533

9

0.536

7

0.539

5

0.542

3

0.545

0

0.547

8

0.550

6

0.553

4

0.556

2

0.559

0

0.561

8

0.564

6

0.567

3

0.570

1

Filter Bins

NCFD

MonteCarlo

Measured

MonteCarlo Trend

Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.529

8

0.532

9

0.535

9

0.539

0

0.542

0

0.545

1

0.548

1

0.551

1

0.554

2

0.557

2

0.560

3

0.563

3

0.566

4

0.569

4

0.572

4

0.575

5

Filter Bins

NCFD

MonteCarlo


Measured Trend








-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.521

4

0.524

3

0.527

1

0.530

0

0.532

9

0.535

8

0.538

6

0.541

5

0.544

4

0.547

2

0.550

1

0.553

0

0.555

9

0.558

7

0.561

6

0.564

5

Filter Bins

NCFD


Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.517

0

0.519

9

0.522

8

0.525

7

0.528

6

0.531

5

0.534

4

0.537

3

0.540

2

0.543

1

0.546

0

0.548

9

0.551

8

0.554

7

0.557

6

0.560

5

Filter Bins

NCFD

MonteCarlo


Measured Trend


-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.517

8

0.521

2

0.524

6

0.528

0

0.531

5

0.534

9

0.538

3

0.541

7

0.545

1

0.548

5

0.552

0

0.555

4

0.558

8

0.562

2

0.565

6

0.569

0

Filter Bins

NCFD

MonteCarlo




144


Using the normalized cumulative distribution information, a measure of linear

relationship between the MonteCarlo predicted and actual measured data was calculated

using the correlation coefficient. A perfect correlation between two variables results in r

= 1.0, no correlation at all results in r = 0.0, and any number in-between means that there

is some correlation between the variables however the relationship is not perfect.

Fig. 8.4.8-1 displays the Correlation Coefficient results using the normalized

cumulative frequency distribution data (population 16) generated from the magnitude of

S11 and S2. The Correlation Coefficients are compared to data generated via the

correlated premise (Blue Plot) vs. the independent premise (Yellow Plot) vs. the

independent 1TS premise (Red Plot).


magnitude and 4.5GHz for S21 magnitude. The results using the independent and

independent 1TS types of premise have a higher Correlation Coefficient than the results

using the correlated premise at all frequencies below 5GHz.

145



0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.E+08 1.E+09 1.E+10

Frequency

S21 MagCorrelation

Coff.

Independent Blocks

Correlated Blocks




0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.E+08 1.E+09 1.E+10

Frequency

S11 MagCorrelation

Coff.

Independent Blocks

Correlated Blocks





Figure #8.4.8-1: Pearson Product-Moment Correlation Coefficient for MonteCarlo

Predicted and Actual Measured Data

8.4.9 T-Test

The primary purpose of the “t” test is to determine whether the means of two

groups of numbers differ to a statistically significant degree. The larger the absolute

value of t, the more likely it is to reflect a significant difference between the two groups

under comparison.




distribution data (population 16). Both sets are displaying the t-test values using the

magnitude of S11 and S21, and are comparing data generated via the correlated premise

(Blue Plot) vs. the independent premise (Yellow Plot) vs. the independent 1TS premise

(Red Plot).

146



1TS premise has a lower t-test value at frequencies less than 1.0GHz for S11, and less

than 2.5GHz for S21.



-0.50

0.00

0.50

1.00

1.50

2.00

2.50

1.E+08 1.E+09 1.E+10

Frequency

S21 MagT Value

Independent Blocks

Correlated Blocks




-0.50

1.50

3.50

5.50

7.50

9.50

1.E+08 1.E+09 1.E+10

Frequency

S21 MagT Value

Independent Blocks

Correlated Blocks




-0.25

0.25

0.75

1.25

1.75

2.25

1.E+08 1.E+09 1.E+10

Frequency

S11 MagT Value

Independent Blocks

Correlated Blocks




-0.50

1.50

3.50

5.50

7.50

9.50

1.E+08 1.E+09 1.E+10

Frequency

S11 MagT Value

Independent Blocks

Correlated Blocks


T-Test For S11/S21 Magnitude Using Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration

T-Test For S11/S21 Magnitude Using Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration


4.5GHz4.5GHz

4.5GHz4.5GHz

Figure #8.4.9-1: T-Test for MonteCarlo Predicted and Actual Measured Data

8.4.10 ANOVA





147

variance between groups versus the variance within groups [32]. The ANOVA starts

with the total amount of variability in the data and divides it up (partitioning) into various

categories. Basically, it compares the variability among the group means with the

variability that occurred just by chance or error.

Fig. 8.4.10-1 displays the ANOVA results using two different sets of data. The

first set compares the raw MonteCarlo generated (population 500) data vs. the raw

Measured (population 32) data. The second set compares the normalized cumulative

frequency distribution data (population 16). Both sets are displaying the ANOVA values

using the magnitude of S11 and S21, and are comparing data generated via the correlated

premise (Blue Plot) vs. the independent premise (Yellow Plot) vs. the independent 1TS

premise (Red Plot).



1TS premise have a lower ANOVA value at frequencies less than 1.0GHz for S11, and

less than 2.5GHz for S21.

148



-0.25

-0.05

0.15

0.35

0.55

0.75

0.95

1.15

1.35

1.E+08 1.E+09 1.E+10

Frequency

S21 MagF(obt)

Independent Blocks

Correlated Blocks




-1.00

4.00

9.00

14.00

19.00

24.00

29.00

1.E+08 1.E+09 1.E+10

Frequency

S21 MagF(obt)

Independent Blocks

Correlated BlocksIndependent Blocks 1TS



-0.02

0.18

0.38

0.58

0.78

0.98

1.E+08 1.E+09 1.E+10

Frequency

S11 MagF(obt)

Independent Blocks

Correlated Blocks




-0.25

4.75

9.75

14.75

19.75

24.75

29.75

1.E+08 1.E+09 1.E+10

Frequency

S11 MagF(obt)

Independent BlocksCorrelated Blocks


ANOVA For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration

ANOVA For S11/S21 MagnitudeUsing Data Generated Via Correlated Circuit Configuration vs. Independent Circuit Configuration

1.84GHz1.84GHz

1.84GHz1.84GHz


Figure #8.4.10-1: ANOVA for MonteCarlo Predicted and Actual Measured Data

8.5 Summary

This chapter explained in detail the application of the predictive statistical

analysis methodology, described in Chapter 6, to a 9-segment meander resistor.

Modeling the 9-segment meander resistor using only one test structure, successfully

proved the method of predicting passive device performance variations and yield, when

provided with a statistical distribution of the building block models and a circuit

schematic.

149

Using the improved method from Chapter 5 and the independent premise when

collating the circuit model file, results from the 9-segment meander resistor analysis

(using only one test structure to deembed and model the device) were compared to the

analysis of the same 9-segment meander resistor; using the modeling methodology in

Chapter 4; and, using both the correlated and independent premise when collating the

circuit model file.

The results generated using the raw data vs. the normalized filtered data for all the

statistical analysis techniques were found to mirror the behavior of one another, which

was expected. The Mean Percent Difference, the t-test, and the ANOVA, all rendered the

independent 1TS premise superior for S11 and S21 magnitude up to 1GHz and 2.5GHz

respectively. The Standard Deviation Percent Difference displayed a similar behavior for

the independent and independent 1TS types of premise for S11 and S21 magnitude up to

the same frequencies respectively. The NCFD plots best exemplified the contrast in the

data generated by all three types of premise, with the independent premise outperforming

the independent 1TS premise, and correlated premise.

150

CHAPTER 9

RESEARCH AUTOMATION USING UNIX SHELL SCRIPTS

9.1 Introduction

This research further develops a passive device modeling methodology that

accurately predicts the high frequency electrical behavior, and the statistical variation of

embedded passive devices on a generic substrate. By utilizing a small number of test

structures, to deembed the lumped element circuit values of canonical building blocks,

accurate statistical models of new passive devices can be created using this methodology.

In summary, a nonlinear optimizer is used to find the optimal building block

models, by fitting extensive high frequency measurements of the test structures behavior.

This process is repeated on numerous fabricated test structures producing a relative

variation in the lumped element circuit values of the building block models. A predicted

range of operation for the actual modeled device is then generated using a MonteCarlo

simulation tool, and the relative variations of the building block models.

To begin this research, two sets of thirty-two test structures are fabricated by

depositing gold on a 96 % alumina substrate, and s-parameter measurements taken. The

Hspice Optimization tool uses these measurements to deembed the lumped element

circuit values of the building block models, which are contained within the fabricated test

151

structures. Relative variations of the deembedded element values are calculated and

added to the modeled device circuit file, for simulation in the Hspice MonteCarlo tool.

By means of comparison, between the range predicted by the Monte Carlo simulations

and the measured data, the statistical variations of the component values were shown to

provided an accurate representation of the overall passive device performance.

The goal of this research is stated above, however investigation into the accuracy

of “measured vs. modeled” needed to be accomplish using the premise; the more accurate

the measured vs. modeled test structure results, the more accurate the modeled device,

and consequently the tighter and more realistic the predicted range of operation. This

investigation encompassed test structure circuit file examination along with statistical

analysis comparisons. The aforementioned generated a huge amount of data that needed

to be synthesized; therefore, Unix scripts were developed to act as interface between user

and computer to automate the entire analysis process. The scope of the automation

includes; 1.) Hspice Optimization simulations; 2.) Extracting and calculating percent

difference in measured vs. modeled s-parameters; 3.) Statistical analysis calculations

such as the relative variations of the deembedded element values; 4.) Generating the

device model circuit file; 5.) Hspice MonteCarlo analysis; 6.) Extracting and converting

s-parameters to polar coordinates; and 7.) Saving and localizing all circuit and

optimization files. A block diagram of the basic components of the automation flow is

shown in Fig. 9.1-1.

152

HspiceHspiceMonteCarloMonteCarlo

AnalysisAnalysis

HspiceHspiceOptimization Optimization

AnalysisAnalysis

Output Data Output Data Synthesis And Synthesis And

StorageStorage

Plotting Plotting FormatFormat

Analysis Analysis InputInput

Statistical Analysis Automation FlowStatistical Analysis Automation Flow

Figure #9.1-1: Main Components of Automation Flow

The automation detailed in this chapter allowed this research to successfully

complete in an elegant timely fashion. The statistical prediction of a passive device using

this methodology was experimentally verified for a 9-segment meander resistor, by

demonstrating that the variation in the complete equivalent circuit models, based only on

the circuit building blocks, can be used to predict such variations in the actual fabricated

device.

This chapter will act as a tutorial by explaining the functionality of each block,

and how to successfully complete an analysis. Messages detailing the progression of the

automated flow have been designed into the analysis scripts, and are visible in the Unix

shell. The Analysis Input shown in Fig. 9.1-2 will be the first block presented.

153

HspiceHspice MonteCarloMonteCarloAnalysisAnalysis

HspiceHspice Optimization Optimization AnalysisAnalysis

Output Data Synthesis Output Data Synthesis And StorageAnd Storage

Plotting FormatPlotting Format

Analysis Analysis InputInput

Input DataUsed in Statistical Automation Flow

Input DataUsed in Statistical Automation Flow

Figure #9.1-2: Analysis Input Block

9.2 Statistical Analysis Flow Input

The first step in using the analysis flow is to place the measured device data in a

predefined location. The s-parameter measurements of test structure one, test structure

two, and the 9-segment meander resistors were placed in the r1_txt, r2_txt, and r9_txt,

directories respectively. The naming convention for the 32 measured device files are as

follows; R1_<number>.txt, R2_<number>.txt, and R9_<number>.txt.

To start the analysis open a Unix shell and type in "sh hspice_runs". The message

“Hspice Optimization Analysis” will appear in the Unix shell and the window shown in

Fig. 9.2-1 will emerge. As stated previously, an investigation into the measured vs.

modeled accuracy was required which included test structure circuit file examination,

154

along with statistical analysis comparisons. The window in Fig. 9.2-1 asks the user to

choose which test structure circuit file configuration to analyze.

Circuit Configuration AnalysisCircuit Configuration Analysis

Figure #9.2-1: Choose Circuit Topology

After clicking on a choice, the user will then press “enter” the message “Pick Test

Structure” will appear in the Unix shell, and the window shown in Fig. 9.2-2 will emerge.

The deembedding procedure dictates which test structure will be analyzed first,

depending on a one (Chapter 5) or two (Chapter 4) test structure analysis.

155

Test Structure AnalysisTest Structure Analysis

Figure #9.2-2: Number of Test Structures Characterized

After clicking on a choice, the user will press “enter” the message “Insert Number

of Measured Test Structures” will appear in the Unix shell, and the window shown in Fig.

9.2-3 will emerge. The user will enter the number of measured test structure (1 and 2)

that have been placed in the r1_txt and r2_txt directories.

Measured Test StructuresMeasured Test Structures

Figure #9.2-3: Number of Measured Devices

The user will then press “enter” the message “Insert Initial Guesses” will appear

in the Unix shell, and the window shown in Fig. 9.2-4 will emerge. The user will insert

on the appropriate line, the initial guess values for each deembedded element value in test

156

structure one and test structure two. These values will be initially used to set up the first

extraction of optimized values from the output file.

Element Value Initial GuessElement Value Initial Guess

Figure #9.2-4: Initial Guess Values

The user will then press “enter” the message “Removing Previous .out File” will

appear in the Unix shell. This message tells the user that the optimization analysis has

begun and any previously generated .out file will be removed to avoid a conflict within

157

Hspice. The next block presented will be the “Hspice Optimization Analysis” block, and

is shown in Fig. 9.2-5.


HspiceHspiceOptimization Optimization

AnalysisAnalysis


Analysis InputAnalysis Input

Hspice Optimization AnalysisUsing Automation Flow

Hspice Optimization AnalysisUsing Automation Flow


Figure #9.2-5: Hspice Optimization Analysis Block

9.3 Hspice Optimization Analysis

The modeling and statistical analysis of the 9-segment meander resistor

has begun. The building block characterization portion of the analysis will execute 1152

individual optimizations to arrive at 32 unique equivalent circuit models of test structure

one, and 32 unique equivalent circuit models of test structure two. The automated flow

will then calculate the relative variation in each element value, construct and collate the

.cir file, and run a MonteCarlo analysis consisting of 500 simulations. The output data

will be synthesized, converted to polar coordinates, compared, and formatted for

Microsoft Excel rendering.

158

This section will demonstrate an actual analysis while detailing the messages

generated in the Unix shell as the analysis proceeds. The first message “Updating

Resistor File Number To 1” means that the first set of s-parameters from the r1_txt

directory (R1_1.txt) will be used as input for the optimization of test structure one.

The next message “Generating .out File Using Initial Guesses” means that Hspice

is using the initial guess values to deembed the first set of optimized element values using

a very low-resolution setting, a very wide range of possible values, and the “steepest

decent” curve fit algorithm.

At this point, a summary of the Hspice optimization tool and an explanation of the

Gauss-Newton Levernber-Marguardt algorithm, used to perform the optimizations, need

to be conveyed.

9.3.1 Hspice Optimization Analysis Tool

The Hspice Optimization Tool uses nonlinear estimation to fit element parameters

of a circuit file to measured s-parameter data. Nonlinear estimation, or nonlinear

optimization, is the process of fitting a mathematical model to experimental data to

determine unknown parameters in the model. The optimization process is often nonlinear

because the observed data does not vary in direct proportion to the parameters in

question; whereas, parameters are chosen so that the output of the model is the best match

to the observed data. Having found accurate values of the estimated parameters,

designers can then make accurate predictions of electrical behavior from the underlying

model. These results are an integral part of circuit design, process refinement, and many

other aspects of the design process.

159

Nonlinear optimization is an ever-present tool of modern technology, and the

Gauss-Newton Levenberg-Marquardt algorithm is the breakthrough that accounts for

much of its commonplace use [40].

9.3.2 Gauss-Newton Levenberg-Marquardt Algorithm

The Gauss-Newton Levenberg-Marquardt algorithm (GNLM), which combines

the robustness of a steepest descent method with the efficiency of a Gauss-Newton step,

is used to accomplish the nonlinear estimation in the Hspice Optimization Tool [31]. The

GNLM method switches continuously from a gradient method (large lambda) far from

the minimum, to a Gauss-Newton step as the minimum is approached and lambda is

reduced. Basically this is a compromise between steepest descent (very slow

convergence, but converges from anywhere) and the Gauss-Newton method (very fast

convergence, but converges only close to the optimum). To further understand the

GNLM method, a closer look at Gauss-Newton component of the algorithm is necessary.

Consider that pbar is the parameter vector and the components p1 p2 pn.. Tneed to

be estimated. The Gauss-Newton optimization method is specifically designed for

minimizing the objective function, which has the form of “the sum of squares” functions,

such as the one given in equation (1) [40].

1

M

m

Wm2 H

mc pbar Hmobs pbar

2.

=

(1)

or

160

E pbar

1

M

m

fm pbar2

=

(2)

where

fm pbar Wm Hmc pbar H

mobs pbar. (3)

where Hmc pbar is the calculated head corresponding to measurement H

mobs pbar ,and Wm

is the weighting coefficient of measurement m. The first order derivatives of E pbar can

be obtained from (2)

pi

Edd

2

1

M

m

fm pi

fmdd.

=

. i 1 N..( ) (4)

where N is the number of parameters to be estimated. The second derivatives are

d2 Edpi dpj

2

1

M

m

dfm

dpi

dfm

dpj

fm

d2 fm

dpi dpj

.

=

. i 1 N..( ) ; j 1 N..( ) (5)

In equation (5) fm pbar is residual, when pbar is not too far from the optimum

value, it can be assumed that the value of fm pbar is small and the second order terms of

the right-hand side of (5) can be ignored, thus

d2 Edpi dpj

2

1

M

m

dfm

dpi

dfm

dpj=

. (6)

161

It is possible to define a new matrix

A

df1

dp1

df2

dp1

.

.

.

dfM

dp1

df1

dp2

df2

dp2

.

.

.

dfM

dp2

.....

.....

.

.

.

.....

df1

dpN

df2

dpN

.

.

.

dfM

dpN

(7)

which consist of derivatives of functions f1 pbar f2 pbar fM pbar.., with respect to the

variation of each parameter component p1 p2 pN.., . Usually M is much larger than N,

therefore A is not a square matrix. Using matrix A and considering (4), gradient ∆ E can

now be represented in matrix form

∆ E 2 AT. f. (8)

where

f f1 f2 fM.. T (9)

Considering also (6) and matrix A, the Hessian matrix G can be replaced

approximately by

162

G 2 AT. A. (10)

Now (9) and (10) can be substituted by Newton’s equation given in equation (11)

pk 1 pk G 1 gk. (11)

yielding

pk 1 pk AkT Ak

.1

AkT fk

.. (12)

where the subscript k in (12) indicates that Ak and fk are evaluated at p bark. Equation

(12) can also be written in the form

AkT Ak

. ∆ pk AkT fk (13)

which is a linear system of equations where AkT Ak

. is an N by N square matrix and AkT Ak

.

is an N-dimensional vector and ∆ pk pk 1 pk is the unknown vector, which can be solved

by Gaussian elimination or decomposition methods from (13) [40].

The Gauss-Newton method is different from Newton’s method because the

Hessian matrix is calculated using only the first derivatives. In the generation of a

Gaussian-Newton sequence for inverse solutions, several problems may be encountered.

First, it is possible that the search sequence does not converge, i.e. E pk 1 E pk> for

163

some k. Second, matrix AkT Ak is near singular (elements very close to zero), and

solution of (13) cannot be obtained. Third, the displacement vector ∆ pk obtained for (13)

is so large that parameter values are not in the admissible region. In order to avoid the

above-mentioned difficulties, it was necessary to develop the modified Gauss-Newton

algorithm by incorporating the Levenberg-Marquardt method.

The contribution of the Levenberg-Marquardt method was to guarantee that

E pk 1 E pk> . The method then took on the name “Gauss-Newton-Levenberg-

Marquardt” algorithms whose modification included and additional term added to AkT Ak

to avoid the singularity

AkT Ak λ M I ∆ pk

. AkT fk (14)

where λ M is a coefficient and I is the unit matrix. When λ M 0, the method reduced to

Gauss-Newton, and if λ M tends to infinity, ∆ pkturns to so called steepest descent direction

and the size ∆ pktends to zero. Therefore, E pk 1 E pk< can always be expected by

increasing he value of λ M.

The procedure for selecting λ M is as follows:

1. Compute E pk

2. Pick a modest value for λ M, say λ M 0.001

3. Solve the linear system of equations (14) to find ∆ pk and calculate E pk ∆ pk

164

4. If E pk ∆ pk E> pk , increase λ M by a factor of 10, and go back to 3

5. If E pk ∆ pk E pk< , decrease λ M by a factor of 10 update the trial solution by

replacing pk with pk ∆ pk and go to the next time step

9.3.3 Optimization Analysis Flow For Test Structure One

Continuing on with the automated analysis, the next message “Inserting CLOSE,

RELIN, RELOUT, and Resistor File Number: 1 Into .cir File” appearing in the Unix shell

mean the settings that control the accuracy, the GNLM optimization component used, and

the measured data used as input, are being updated in the circuit file. In this statement

the optimization model name is “converge”, and convergence success is defined when the

input/output parameters change is less than 1% between successive iterations

(relin/relout). The “close” parameter corresponds to the “lambda” mentioned above. A

high lambda value initiates only the steepest decent component of the GNLM method.

When the gradient of the results functions are less than “cendif” the more time

consuming derivative methods are used. Due to the inaccuracy of the initial guesses, the

goal of the first optimization is to just complete a successful convergence.

Optimization Parameter Settings .model converge opt relin=1e-2 relout=1e-2 cendif=0.1 close=400 itropt=1500

file= '../r1_txt/R1_1.txt'

Table #9.3.3-1: Optimization Settings

165

The next message “Extracting Optimization Information and Element Values

From .out File” appearing in the Unix shell means that the optimization log information

and the most recent optimized element values are being extracted from the .out file.

Optimization Log .model converge opt relin=1e-2 relout=1e-2 cendif=0.1 close=500 itropt=1500 grad= 1.00u close= 500.00 cut= 2.00 residual sum of squares = 5.52464 norm of the gradient = 100.625 marquardt scaling parameter = 500.000 no. of function evaluations = 10 no. of iterations = 0

Optimized Element Values optimized parameters opt1 %norm-sen %change .param c_sq = 1.079756913e-13 $ 7.213396783e-01 -2.251310361e-02 .param rsq = 2.510000000e-06 $ 0. 0. .param lsq = 1.839982236e-11 $ 4.544907570e+01 -9.654611944e-04 .param csq = 6.520000000e-26 $ 0. 0. .param c_pad = 1.590160258e-12 $ 1.099024270e+01 1.007808764e-02 .param rpad = 4.378772295e+00 $ 1.785141128e+01 -5.200190276e-03 .param lpad = 3.049392072e-11 $ 8.867906848e+00 -1.993604484e-02 .param cpad = 2.439993256e-14 $ 1.612002378e+01 -2.764101106e-04

Table #9.3.3-2: Optimization Log and Values

The next message “Checking For Element Values = Zero” will appear in the Unix

shell. Because some of the optimized capacitance values are extremely small,

occasionally the optimizer will return a zero instead of a very small number. This will

create a conflict within Hspice when the next optimization runs. Therefore, the scripts

check for zero values and if found replace the zero with the last optimized non-zero

value. Due to the progressive resolution adjustments shown in Table 9.3.3-5, optimized

values rarely change by any great amount from one run to another.

166

Optimized Element Value Check (rm element_check_c_sq) awk 'if ( $1 == "0." ) print "(cd runs; cp ../spare_1a temp_1a)" ' runs/temp_1a > element_check_c_sq (chmod 777 element_check_c_sq; sh element_check_c_sq) (cd runs; cp temp_1a ../spare_1a)

Table #9.3.3-3: Element Value Check

The next messages “Inserting New Element Values Into .cir File,” “Calculating

The High and Low Ranges Using: 1000,” and “Inserting The High and Low Ranges Into

The .cir File” will appear in the Unix shell. The newly optimized element values are

extracted from the .out file and inserted into the .cir file along with the calculated high

and low element value range. As the optimizations proceed, for a particular measured s-

parameter set, the relin/relout, the close, and the possible range are adjusted to ultimately

converge to the optimum. The range of possible element values is calculated by dividing

and multiplying the optimized element value for a particular run by 10000, 1000, 100, 50,

25, 10, and 5 as the optimization proceed.

167

Optimized Element Value Insertion .param ****************************************** * 30 u line specs ***************************************** + c_sq = opt1(VX_C_SQ,c_sq_LOW,c_sq_HIGH) + rsq = opt1(VX_RSQ,rsq_LOW,rsq_HIGH) + lsq = opt1(VX_LSQ,lsq_LOW,lsq_HIGH) + csq = opt1(VX_CSQ,csq_LOW,csq_HIGH) + r_sub_sq = 10g ************************************** * pad specs ************************************** + c_pad = opt1(VX_C_PAD,c_pad_LOW,c_pad_HIGH) + rpad = opt1(VX_RPAD,rpad_LOW,rpad_HIGH) + lpad = opt1(VX_LPAD,lpad_LOW,lpad_HIGH) + cpad = opt1(VX_CPAD,cpad_LOW,cpad_HIGH) + r_sub_pad = 10g ************************************** .param ****************************************** * 30 u line specs ***************************************** + c_sq = opt1(1.080e-13,1.08e-17,1.08e-09) + rsq = opt1(2.510e-06,2.51e-10,0.0251) + lsq = opt1(1.840e-11,1.84e-15,1.84e-07) + csq = opt1(6.520e-26,6.52e-30,6.52e-22) + r_sub_sq = 10g ************************************** * pad specs ************************************** + c_pad = opt1(1.590e-12,1.59e-16,1.59e-08) + rpad = opt1(4.379e+00,0.0004379,43790) + lpad = opt1(3.050e-11,3.05e-15,3.05e-07) + cpad = opt1(2.440e-14,2.44e-18,2.44e-10) + r_sub_pad = 10g *****************************

Table #9.3.3-4: Element Value Insertion

The next messages “Removing .temp Files,” “Saving and Appending

Optimization Results To Log File,” and “Running TS1 Hspice Optimization Auto Run

Number: 1” will appear in the Unix shell. This step cleans up the previous optimization

run by deleting all .temp files, and by appending the newly generated optimization

information to the master file. The scripts then kick off another optimization, using the

168

previous optimized element values as initial guesses, and adjusting the resolution and

convergence settings for a more accurate analysis. The script lines in Table 9.3.3-5 show

the close, the relin, and the relout settings as the optimizations proceed.

Optimizations Resolution Settings 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

relin=1e-2 relout=1e-2 cendif=0.1 close=300 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=200 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=100 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=50 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=25 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=10 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=5 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=1 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=0.1 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=0.01 itropt=1500 relin=1e-2 relout=1e-2 cendif=0.1 close=0.001 itropt=1500 relin=1e-3 relout=1e-2 cendif=0.1 close=0.001 itropt=1500 relin=1e-4 relout=1e-2 cendif=0.1 close=0.001 itropt=1500 relin=1e-5 relout=1e-2 cendif=0.1 close=0.001 itropt=1500 relin=1e-6 relout=1e-2 cendif=0.1 close=0.001 itropt=1500 relin=1e-7 relout=1e-2 cendif=0.1 close=0.001 itropt=1500

Table #9.3.3-5: Resolution Settings

All the optimization log information generated from each of the 18 analyses is

extracted from the .out file and appended into one file. Table 9.9.3-6 shows the

optimization log information generated from three optimizations using one set of

measured data.

169

Optimizations Optimization Log Information 1

2

3

.model converge opt relin=1e-5 relout=1e-5 cendif=0.1 close=0.001 itropt=1500 grad= 1.00u close= 1.00m cut= 2.00 residual sum of squares = 2.97267 norm of the gradient = 2.090913E-02 marquardt scaling parameter = 67108.9 no. of function evaluations = 57 no. of iterations = 2 .model converge opt relin=1e-6 relout=1e-6 cendif=0.1 close=0.001 itropt=1500 grad= 1.00u close= 1.00m cut= 2.00 residual sum of squares = 2.97267 norm of the gradient = .817667 marquardt scaling parameter = 268435. no. of function evaluations = 33 no. of iterations = 1 .model converge opt relin=1e-7 relout=1e-7 cendif=0.1 close=0.001 itropt=1500 grad= 1.00u close= 1.00m cut= 2.00 residual sum of squares = 2.97267 norm of the gradient = 8.64253 marquardt scaling parameter = 268435. no. of function evaluations = 33 no. of iterations = 1

Table #9.3.3-6: Optimization Log Information

After all 18 analyses have completed for one set of measured data, the “optimal”

optimized element values are inserted into the test structure one simulation file. The

message “Inserting Optimized Element Values Into Simulation File” appears in the Unix

shell window along with the message “Running Simulation File and Extracting S-

Parameter Values.” Table 9.3.3-7 shows the optimized element values calculated after

running 18 optimization simulations.

170

Optimized Element Values In Simulation .cir File .param **************************** * 30 u line specs **************************** + c_sq = 1.230444586e-13 + rsq = 2.510000000e-06 + lsq = 1.831326317e-11 + csq = 6.520000000e-26 + r_sub_sq = 10g **************************** * pad specs **************************** + c_pad = 1.485754792e-12 + rpad = 4.734107069e+00 + lpad = 3.282431715e-11 + cpad = 2.378349977e-14 + r_sub_pad = 10g ****************************

Table #9.3.3-7: Optimized Element Values

The next messages “Extracting TS1 s-parameters from .out File and Calculating Percent

Difference” will appear in the Unix shell. The automated script flow extracts the s-

parameter data from the simulation .out file and compares the results with the actual

measured data used to deembed the element values. Table 9.3.3-8 shows some of the

Unix scripts used in calculating the percent difference between measured vs. modeled.

Percent Difference Between Measured vs. Modeled (awk 'print (100 * (sqrt((1 - ($1 / $2)) * (1 - ($1 / $2))))) ' compare_s11_real_2 > s11_real_compare_2) (awk 'print (100 * (sqrt((1 - ($1 / $2)) * (1 - ($1 / $2))))) ' compare_s11_imag_2 > s11_imag_compare_2) (awk 'print (100 * (sqrt((1 - ($1 / $2)) * (1 - ($1 / $2))))) ' compare_s21_real_2 > s21_real_compare_2) (awk 'print (100 * (sqrt((1 - ($1 / $2)) * (1 - ($1 / $2))))) ' compare_s21_imag_2 > s21_imag_compare_2) (paste '-d \t' done_temp_freq s11_real_compare_2 s11_imag_compare_2 s21_real_compare_2 s21_imag_compare_2 > TS1_data_compared_`cat r_file_number`)

Table #9.3.3-8: Percent Difference In Measured vs. Modeled

171

The next messages “Saving and Appending Optimized Element Values For Run

Number: 1 in Master File” will appear in the Unix shell. The final optimized element

values from each set of measured s-parameters are extracted from optimization run 18

(optimal) and appended into a master file. The test structure two scripts will access this

file during their optimization runs, and the MonteCarlo scripts will access this file to

calculate relative variations in the deembedded element values. Table 9.3.3-9 displays

part of this file.

Optimization Run Number TS1 Deembedded Element Values 1

2

3

residual sum of squares = 2.97264 1.226167537e-13 6.520000000e-26 1.831703392e-11 2.510000000e-06 1.485697892e-12 4.735672319e+00 3.283106370e-11 2.378566732e-14 residual sum of squares = 16.5210 1.019855779e-13 6.520000000e-26 1.842597617e-11 2.510000000e-06 1.582988094e-12 4.384412533e+00 3.027150692e-11 2.486747526e-14 residual sum of squares = 14.0001 1.219019979e-13 6.520000000e-26 1.801390055e-11 2.510000000e-06 1.659259059e-12 4.087316426e+00 2.795404944e-11 2.486223391e-14

Table #9.3.3-9: Deembedded Element Values

The next few messages advise that the simulation of test structure one using the

optimum deembedded element values had been run and the s-parameters, optimization

log, and the simulation circuit file, are being extracted from the .out file and saved. The

messages “Saving s-parameters For Run Number: 1,” “Saving Optimization Run Log

File (18 optimizations) For Run Number: 1,” “Saving Sim File For Run Number: 1” will

appear in the Unix window.

172

The optimization process will now be repeated until all 32 sets of measured s-

parameters have been processed and element values deembedded. The message “End of

TS1 Element Extraction” will appear when the flow scripts have completed test structure

one.

9.3.4 Optimization Analysis Flow Test Structure Two

The automation flow will now move to test structure two and display the message

“Moving TS1 Deembedded Elements File to TS2 Directory.” The optimization of all the

test structure two sets of measured s-parameters will mirror the process completed on test

structure one. The only difference is at the end of each optimization routine (18 runs) the

deembedded element values from the corresponding test structure one and test structure

two optimizations (optimal) will be inserted into the 9-segment meander resistor

simulation file and analyzed. The s-parameter results will also be extracted and percent

difference calculated for each of the 32 measured vs. modeled sets of data. Table 9.3.4-1

shows the analysis flow by displaying the actual messages illustrated in the Unix shell.

173

Steps Test Structure Two Optimizations 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 . . . .

24 25 26 27 28 29 30 31 32 33 34 . . . .

35 36

Beginning TS2 Deembedding Using Resistor File: 1 Removing Previous .out File Inserting Initial Guesses and Optimized TS1 Circuit Values Into TS2 .cir file Inserting Opt. TS1 Circuit Values Into TS2 .sim file, and 9-Segment .cir File Generating 1st. .out File Using Initial Guesses Inserting CLOSE, RELIN, RELOUT, and Resistor File #: 1 Into .cir File Extracting & Appending Opt. Info and Element Values From .out File Checking For Element Values = Zero Calculating The High and Low Ranges Using: 1000 Inserting The High and Low Ranges Into The .cir File Removing .temp Files Saving and Appending Optimization Results To Log File Running TS2 Hspice Optimization Auto Run Number: 1 Inserting CLOSE, RELIN, RELOUT, and Resistor File #: 1 Into .cir File Extracting Optimization Information and Element Values From .out File Checking For Element Values = Zero Inserting New Element Values Into .cir File Calculating The High and Low Ranges Using: 1000 Inserting The High and Low Ranges Into The .cir File Removing .temp Files Saving Optimization Results Running TS2 Hspice Optimization Auto Run Number: 2 ........... Running TS2 Hspice Optimization Auto Run Number: 3 ........... ........... ........... ........... ........... Running TS2 Hspice Optimization Auto Run Number: 17 Removing .log Files Inserting Optimized (Optimal) Element Values Into Simulation File Running Simulation File and Extracting S-Parameter Values Extracting 9Seg & TS2 s-parameters and Calculating Percent Difference Saving and Appending Optimized Element Values For Run Number: 1 Saving s-parameters For Run Number: 1 Saving Sim File For Run Number: 1 Saving Runs Log File For Run Number: 1 Compiling and Localizing All Data ........... Beginning TS2 Deembedding Using Resistor File: 2 ........... ........... ........... ........... ........... Beginning TS2 Deembedding Using Resistor File: 32 End of TS2 Element Extraction

Table #9.3.4-1: Test Structure Two Optimizations

At this point, the analysis has generated 32 deembedded building block equivalent

circuit values for test structure one, and generated 32 deembedded building block

174

equivalent circuit values for test structure two. The next block to be presented will be the

“Hspice MonteCarlo Analysis” block, and is shown in Fig. 9.3.4-1.

HspiceHspiceMonteCarloMonteCarlo

AnalysisAnalysis





Hspice MonteCarlo AnalysisUsing Automation Flow

Hspice MonteCarlo AnalysisUsing Automation Flow

Figure #9.3.4-1: Hspice MonteCarlo Analysis Block

9.4 Hspice MonteCarlo Analysis

As soon as the Hspice Optimization Analysis has successfully completed the

message “Start MonteCarlo Circuit Construction” will appear in the Unix shell and the

window shown in Fig. 9.4-1 will emerge. The user will insert the number of pads,

number of material squares, number of coupled material squares, and the number of

corners, then the Unix scripts will generate the entire MonteCarlo circuit file. The 9-

segment meander resistor configuration treating each element value independently (i.e.

no correlation) will generate a 2,827 line, 343KB file. Each element value is set up to

generate a new random number on each simulation.

175

9-Segment Meander ResistorMonteCarlo Circuit File Construction

9-Segment Meander ResistorMonteCarlo Circuit File Construction

Figure #9.4-1: MonteCarlo Circuit File Construction

During the construction of the circuit file, messages will appear in the Unix shell advising

the user of the progress. These messages are shown in Table 9.4-1.

Steps 9-Segment Meander Resistor MonteCarlo Circuit File Construction 1 2 3 4 5 6 . . . 7 8 9

10 11 . . .

12 13 14 15 16 . . .

17 18 19 20 21

Inserting .OPTIONS Command Inserting PAD 1 Inserting PAD 2 Inserting Pad .PARAMS Into .Cir File Inserting Substrate Resistance Value Inserting Material Square 1 . . . Inserting Material Square 8 Inserting Material Square .PARAMS Into .Cir File Inserting Substrate Resistance Value End of Material Square Sub-Circuit Construction Inserting Coupled Material Square 1 . . . Inserting Coupled Material Square 9 Inserting Coupled Material Square .PARAMS Into .Cir File Inserting Substrate Resistance Value End of Coupled Square Sub-Circuit Construction Inserting Corner 1 . . . Inserting Corner 8 Inserting Corner .PARAMS Into .Cir File Inserting Substrate Resistance Value End of Corner Sub-Circuit Construction End of Circuit Construction

Table #9.4-1: Circuit File Construction

176

Once the MonteCarlo circuit file has been collated, the 32 sets of element values

from test structure one and test structure two will be analyzed statistically yielding mean

and relative variation values for each deembedded element value. During this process,

messages will appear in the Unix shell advising the user of the progress. These messages

are shown in Table 9.4-2.

Steps Statistical Analysis of Deembedded Element Values 1 2 3 4 5 6 7 8 9

10 11 12

Combining TS1 and TS2 Deembedded Element Values Into One File Sorting The File (Ascending) Calculating The Median Calculating The Absolute Variation Calculating The Relative Variation Calculating The Mean Calculating The Sum of Squares Calculating The Sum Squared Calculating The Variance Calculating The Standard Deviation Calculating The Standard Deviation/Mean Inserting Relative Variations Into 9-Segment MonteCarlo .cir File

Table #9.4-2: Statistical Analysis of Deembedded Element Values

The 9-segment meander resistor MonteCarlo circuit file is now complete. The

message “Running 9Seg MonteCarlo Analysis” will appear in the Unix shell advising the

user that the MonteCarlo analysis has begun. The MonteCarlo analysis generated 500

simulations of the 9-segment meander resistor circuit file while varying each

deembedded element value on each simulation. The Hspice MonteCarlo analysis was run

using a four-processor HP model 1100 computer. The analysis run time was

approximately 2 hours. At the successful completion of the analysis the message “End of

9Seg MonteCarlo Analysis” will appear in the Unix shell. The last block to be presented

will be the “Output Data Synthesis and Storage” block, and is shown in Fig. 9.4-2.

177



Output Data Output Data Synthesis Synthesis

And StorageAnd Storage



Output Data Synthesis and Storage ForStatistical Analysis Automation Flow

Output Data Synthesis and Storage ForStatistical Analysis Automation Flow

Figure #9.4-2: Output Data Synthesis

9.5 Output Data Synthesis And Storage

After the entire analysis has completed there is still an issue concerning the

management of all the data generated during the statistical analysis process. Files need to

be saved, parameters need to be converted, and results need to be compared. Portions of

the data synthesis were preformed as they were generated. Such as the s-parameters

results from test structure one, test structure two, and the 9-segment resistor. During the

optimization process the messages “Extracting TS1 s-parameters from .out File and

Calculating Percent Difference,” and “Extracting 9Seg & TS2 s-parameters and

Calculating Percent Difference” were seen scrolling by in the Unix shell. However, most

of the data management is accomplished at the end of the analysis. For example the

results from both sets of 32 deembedded element values were automatically inserted into

178

a 9-segment resistor Hspice circuit file and analyzed. The results were extracted from the

.out file converted to polar coordinates, and put in separate files (i.e. magnitude/phase,

magnitude, phase) and saved. Another option inserts the element values from a particular

deembedded analysis run into a 9-segment Hspice circuit file, simulates the design then

compares the results with all 32 measured 9-segment devices. This is done for each set

of deembedded element values. Table 9.5-1 shows the extent of the data generated and

saved. All the data generated and documented below is saved on a CD, using

approximately 400Meg its available space.

Steps Data Synthesis And Storage 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

Extracting TS1 s-parameters and Calculating Percent Difference Extracting TS2 s-parameters and Calculating Percent Difference Extracting 9Seg s-parameters and Calculating Percent Difference Saving All Percent Difference Files Extracting TS1 Optim. Information and Element Values From .out File Extracting TS2 Optim. Information and Element Values From .out File Saving All Optimization Runs Log File For Run Number: 1 Saving and Appending Optimized TS1 Element Values For Run Number: 1 Saving and Appending Optimized TS2 Element Values For Run Number: 1 Saving TS1 s-parameters For Run Number: 1 Saving TS2 s-parameters For Run Number: 1 Saving 9-Seg s-parameters For Run Number: 1 Saving TS1 Circuit File For Run Number: 1 Saving TS2 Circuit File For Run Number: 1 Saving 9-Seg Circuit File For Run Number: 1 Extracting s-parameters Data For Each (500) MonteCarlo Run Converting All TS1, TS2, 9-Seg S-parameter Data to Polar Coordinates Separating All s-parameters files Into Real, Imag. & Magnitude, Phase Running 9Seg Sims’ Using Deembedded Circuit Values From All Analysis Compiling and Localizing All Data

Table #9.5-1: Data Synthesis and Storage

179

9.6 Conclusion

The Statistical Analysis Automation Flow completed all aspects of this research

except for rendering the plots. This was accomplished using Microsoft Excel mainly due

to the exceptional graphic quality. After learning the mechanics behind the Gauss-

Newton Levenberg-Marquardt Algorithm, running iterative optimizations while adjusting

the resolution, convergence parameters, and range of possible element values, was a

process that had to be preformed. In the quest of statistical prediction, measured vs.

modeled accuracy yields a more realistic outcome. The generation and application of the

Unix shell scripts aided this research immensely. The picture in Fig. 9.6-1 shows the full

scope of the Statistical Analysis Automation Flow.

Convert MC & Model Resistor S-Parameter To Polar Coordinates

HSPICESIMULATIONTS1 & TS2 &

MODEL RESISTOR

Calculate % ErrorMeasured vs. Modeled For TS1 & TS2

Deembedded ElementValues For TS1

HSPICE

OPTIMIZATION

TS1 & TS2

HSPICEMONTECARLO

ANALYSIS

PLOTPLOT

Model Circuit Description

Extract S-ParametersFrom .out files &

Separate Into SingleReal/Imag, Real, Imag

Master Files

Deembedded ElementValues For TS2

Model Circuit Generation

Statistical AnalysisIn Excel PLOTPLOT

PLOTPLOT

Model ResistorMeasured S-Parameters

Save Optimization Files

Save .cif Files

LegendLegendInput Parameters

Hspice Analysis

.cir Files & Measurements

Output File Information

Deembedded Elements

TS1 Initial GuessTS2 Initial Guess

TS1 & TS2 .cir Files

Relative Variation of Element Values

TS1 & TS2 Measured S-Parameters

Number of TS1 & TS2 Measured Devices

Automation of Predictive Passive Device Modeling Methodology Using Unix Shell ScriptsAutomation of Predictive Passive Device Modeling Methodology Using Unix Shell Scripts

Figure #9.6-1: Statistical Analysis Automation Flow

180

CHAPTER 10

CONCLUSION AND RECOMENDATIONS

10.1 Summary of Research and General Conclusions

A further developed passive device modeling methodology that accurately

predicts the high frequency electrical behavior, and the statistical variation of embedded

passive devices on a generic substrate, has been presented. This methodology accurately

modeled new passive devices by utilizing a small number of test structures to deembed

the lumped element circuit values of canonical building blocks. The fundamental idea

behind the modeling procedure is that most designed passive structures are comprised of

several key geometrical building blocks, that is, they can be constructed from several

building block cells representing individual parts of the structure.

The objective of this research is to predict the electrical behavior and statistical

variation of a 9-segment meander resistor in a standard circuit simulator. The number of

test structures needed to accurately model all of the required building blocks must be as

few as possible for the method to be practical; therefore, deembedding 22 element values

from four canonical building blocks using only one test structure was attempted and

successfully achieved, along with rendering a substantial increase in measured vs.

181

modeled accuracy for the test structures as well as the modeled device. This proved that

with enough frequency points and sufficient sensitivity, accurate equivalent circuit

extraction of individual building blocks from multi building block test structures is

possible.

Other improvements to the overall existing methodology such as enhancing the

corner primitive, not deembedding the substrate resistance, and running consecutive

simultaneous optimizations per measured device, also resulted in a substantial increase in

measured vs. modeled accuracy for the test structures as well as the modeled device.

These improvements were fueled by the theory; the more accurate the measured vs.

modeled results of the test structures, the more accurate the modeled device, and

consequently the tighter and more realistic the predicted range of operation.

In an attempt to ease research effort and to gain wider acceptance, the entire

Optimization/MonteCarlo methodology was completely automated. Unix scripts were

developed which act as interface between user and computer by; automatically, running

the complete Hspice Optimization simulations, calculating the relative variations of the

deembedded element values, generating the device model circuit file, running the Hspice

MonteCarlo analysis, extracting and converting all s-parameters to polar coordinates, and

saving and localizing all circuit and optimization files. This automation will go a long

way towards easing the difficulty in using the Hspice Tool, and making the entire

methodology less cumbersome.

182

10.2 Application

The method of predicting passive device performance variations and yield, when

provided with a statistical distribution of the building block models and a circuit

schematic was successfully proven using the 9-segment meander resistor. Randomly

generated element values of the 9-segment meander resistor were simulated in a

MonteCarlo tool to produce a representative set of output performance characteristics.

The impact of random process variations was inferred from these simulations, and

electrical yield was estimated using the percentage of that sample which met the

performance requirement.

Electrical yield refers to the number of parts that meet the electrical test

specifications; whereas, maximizing yield is important for the overall process efficiency.

Fig. 10.2-1 shows the Gaussian distribution curve representing 500 MonteCarlo produced

simulations in terms of mean and standard deviation. The proposition is that 95.44% (2

standard deviations) of the actual 32 measured devices will be contained within the

MonteCarlo predicted range.

183

Mean and Standard DeviationsMean and Standard Deviations

MonteCarlo MeanMonteCarlo Mean

34.13%34.13%34.13%34.13%

13.59%13.59% 13.59%13.59%

00 11-1-1 22-2-2

95.44%95.44%

Standard DeviationsStandard Deviations

Figure #10.2-1: MonteCarlo Predicted Mean and Standard Deviation

10.2.1 Statistical Analysis Results From Chapter 7

The optimum modeling method for the 9-segment meander resistor was presented

in Chapter 7, and the optimum MonteCarlo analysis was generated using the method in

Chapter 7 coupled with the independent premise used in collating the circuit model file.

The MonteCarlo data generated by this method was investigated at two frequency points,

45MHz(DC) & 1.7GHz for S11, and 45MHz(DC) & 4.0GHz for S21. Using the

MonteCarlo generated data; the standard deviation was calculated, multiplied by two, and

added/subtracted to the mean value at each frequency. This created a high and low range

(horizontal red lines) designating the range of possible values and the predicted yield for

the 9-segment meander resistor. The measured data from the 9-segment resistor, at the

corresponding frequency points was inspected to ascertain what percentage of the 32

measured devices fell within the MonteCarlo (two standard deviation) predicted range,

and is shown in Fig. 10.2.1-1Excel plots.

184

MonteCarlo Predicted Two Standard Deviations OutComparison Of Actual Measured Devices At 4.0GHz

0.53

0.535

0.540.545

0.55

0.555

0.56

0.5650.57

0.575

0.58

0 5 10 15 20 25 30

Number of Measured Devices

S21Mag.

Measured Devices

Predicted High Range

Predicted Low Range

MonteCarlo Predicted Two Standard Deviations OutComparison Of Actual Measured Devices At 45MHz

0.5550.56

0.5650.57

0.5750.58

0.5850.59

0.5950.6

0.6050.61

0 5 10 15 20 25 30


S21Mag.

Measured Devices


Predicted Low Range


0.390.395

0.40.4050.41

0.4150.42

0.4250.43

0.4350.44

0.445

0 5 10 15 20 25 30


S11Mag.

Measured DevicesPredicted High Range

Predicted Low Range


0.41

0.415

0.42

0.425

0.43

0.435

0.44

0.445

0.45

0.455

0 5 10 15 20 25 30


S11Mag.

Measured Devices


Predicted Low Range

MonteCarlo Predicted Yield and Element Value Variation96.88% Of Actual Measured Devices (31 out of 32) Fell Within Predicted Range

Using Independent Premise


Using Independent Premise

Not IncludedNot Included

Not IncludedNot Included

Not IncludedNot IncludedNot IncludedNot Included

Figure #10.2.1-1: MonteCarlo Predicted Range vs. Actual Measured Devices Using

Independent Premise

By inspection, 96.88% (1.44% more) of the actual measured devices fell within

the MonteCarlo predicted range, generating a rectangular area of possible values and a

predicted yield. Fig. 10.2.1-2 shows the MonteCarlo (two standard deviation) predicted

range (Red), the actual MonteCarlo predicted range (Blue Dots), and the actual measured

data.

185

9-Segment Meander ResistorPredicted Range of Values and Statistical Yield

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

1.E+08 1.E+09 1.E+10

Frequency

S11Mag


0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

0.61

0.62

1.E+08 1.E+09 1.E+10

Frequency

S21Mag

MonteCarlo Predicted 95.44% of Measured Devices Actual Measured 9-Segment Resistors Totaled 96.88% Using Independent Premise Method

MonteCarlo Predicted 95.44% of Measured Devices Actual Measured 9-Segment Resistors Totaled 96.88% Using Independent Premise Method

1.7GHz1.7GHz

4.0GHz4.0GHz


Independent Premise

10.2.2 Statistical Analysis Results From Chapter 8

The modeling method for the 9-segment meander resistor presented in Chapter 8

coupled with the independent premise used in collating the circuit model file was use to

generated the MonteCarlo simulations. The MonteCarlo data generated by this method

was investigated at two frequency points, 45MHz(DC) & 1.04GHz for S11, and

45MHz(DC) & 2.0GHz for S21. Using the MonteCarlo generated data; the standard

deviation was calculated, multiplied by two, and added/subtracted to the mean value at

each frequency. This created a high and low range (horizontal red lines) designating the

range of possible values and the predicted yield for the 9-segment meander resistor. The

measured data from the 9-segment resistor, at the corresponding frequency points was

inspected to ascertain what percentage of the 32 measured devices fell within the

MonteCarlo (two standard deviation) predicted range, and is shown in Fig. 10.2.2-1Excel

plots.

186


0.555

0.56

0.565

0.57

0.575

0.58

0.585

0.59

0.595

0.6

0 5 10 15 20 25 30


S21Mag.

Measured Devices


Predicted Low Range


0.565

0.57

0.575

0.58

0.585

0.59

0.595

0.6

0.605

0 5 10 15 20 25 30


S21Mag.

Measured Devices


Predicted Low Range


0.395

0.4

0.405

0.41

0.415

0.42

0.425

0.43

0.435

0.44

0 5 10 15 20 25 30


S11Mag.

Measured Devices


Predicted Low Range


0.39

0.395

0.4

0.405

0.41

0.415

0.42

0.425

0.43

0.435

0 5 10 15 20 25 30


S11Mag.

Measured Devices


Predicted Low Range


Using Independent 1TS Premise


Using Independent 1TS Premise

Not IncludedNot Included Not IncludedNot Included

Not IncludedNot Included Not IncludedNot Included


Independent 1TS Premise

By inspection, 96.88% (1.44% more) of the actual measured devices fell within

the MonteCarlo predicted range at all four frequencies, generating a rectangular area of

possible values and a predicted yield. Fig. 10.2.2-2 shows the MonteCarlo (two standard

deviation) predicted range (Red), the actual MonteCarlo predicted range (Blue Dots), and

the actual measured data.

187


0.38

0.39

0.40

0.41

0.42

0.43

0.44

0.45

1.E+08 1.E+09 1.E+10

Frequency

S11Mag


0.55

0.56

0.57

0.58

0.59

0.6

0.61

0.62

1.E+08 1.E+09 1.E+10

Frequency

S21Mag

MonteCarlo Predicted 95.44% of Measured Devices Actual Measured 9-Segment Resistors Totaled 96.88% Using Independent 1TS Premise Method

MonteCarlo Predicted 95.44% of Measured Devices Actual Measured 9-Segment Resistors Totaled 96.88% Using Independent 1TS Premise Method



Independent 1TS Premise

10.2.3 Statistical Analysis Results From Chapter 7 Using Correlated Premise

The modeling method for the 9-segment meander resistor presented in Chapter 7

coupled with the correlated premise used in collating the circuit model file was use to

generated the MonteCarlo simulations. The MonteCarlo data generated by this method

was investigated at two frequency points, 45MHz(DC) & 1.0GHz for S11, and

45MHz(DC) & 2.0GHz for S21. Using the MonteCarlo generated data; the standard

deviation was calculated, multiplied by two, and added/subtracted to the mean value at

each frequency. This created a high and low range (horizontal red lines) designating the

range of possible values and the predicted yield for the 9-segment meander resistor. The

measured data from the 9-segment resistor, at the corresponding frequency points was

inspected to ascertain what percentage of the 32 measured devices fell within the

MonteCarlo (two standard deviation) predicted range, and is shown in Fig. 10.2.3-1Excel

plots.

188


0.56

0.565

0.57

0.575

0.58

0.585

0.59

0.595

0.6

0 5 10 15 20 25 30


S21Mag.

Measured Devices


Predicted Low Range


0.565

0.57

0.575

0.58

0.585

0.59

0.595

0.6

0.605

0 5 10 15 20 25 30


S21Mag.

Measured Devices


Predicted Low Range


0.395

0.4

0.405

0.41

0.415

0.42

0.425

0.43

0.435

0.44

0 5 10 15 20 25 30


S11Mag.

Measured Devices


Predicted Low Range


0.39

0.395

0.4

0.405

0.41

0.415

0.42

0.425

0.43

0.435

0 5 10 15 20 25 30


S11Mag.

Measured Devices


Predicted Low Range


Using Correlated Premise


Using Correlated Premise

2 Not Included2 Not Included 2 Not Included2 Not Included





Correlated Premise

By inspection, 81.25% (14.19% less) of the actual measured devices fell within

the MonteCarlo predicted range at all four frequencies, generating a rectangular area of

possible values and a predicted yield. Fig. 10.2.3-2 shows the MonteCarlo 2 standard

deviation predicted range (Red), the actual MonteCarlo predicted range (Blue Dots), and

the actual measured data.

189


0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

1.E+08 1.E+09 1.E+10

Frequency

S11Mag


0.55

0.56

0.57

0.58

0.59

0.6

0.61

0.62

1.E+08 1.E+09 1.E+10

Frequency

S21Mag

MonteCarlo Predicted 95.44% of Measured Devices Actual Measured 9-Segment Resistors Totaled 81.25% Using Correlated Premise Method

MonteCarlo Predicted 95.44% of Measured Devices Actual Measured 9-Segment Resistors Totaled 81.25% Using Correlated Premise Method



Correlated Premise

10.3 Discussion

An accurate range of possible element values variations and an accurate yield

prediction was generated for the 9-segment meander resistor using the statistical analysis

procedure, and the improved modeling methodology in Chapter 4. A description of the

MonteCarlo simulation process and several statistical analysis techniques, used to

evaluate and render the results, were also presented. The MonteCarlo data generated

from a circuit model file constructed using the “correlated” premise vs. the noncorrelated

or “independent” premise was investigated, and found that no correlation existed between

any of the building block equivalent circuit elements. This was shown to be true in the

poor yield and element value predictions in Figures 10.2.3-1 and 10.2.3-2, and is mainly

due to the non-uniformity of the “Sputtering” technique used to deposit the gold onto the

Alumina substrate. A successful statistical analysis prediction of the same 9-segment

190

meander resistor was also generated using only one test structure and the independent

premise.

10.4 Recommendations

To further refine the measured vs. modeled accuracy of the test structures, higher

order coupling will need to be modeled as designs move further into the giga hertz. With

the automated Optimizatio/MonteCarlo analysis in place, fabricated structures and

measured data is all that is needed to effectively model any type of passive device.

Because no assumptions are made regarding material properties, layered dielectrics, or

imperfect substrates, when using measurement to model a device; a much greater

understanding of device/substrate effects can be extracted by using the automated flow to

its potential.

10.5 Final Conclusion

This research detailed the method of predicting passive device performance

variations and yield, when provided with a statistical distribution of the building block

models and a circuit schematic. A comparison of the MonteCarlo simulation results to

actual measured data of a 9-segment meander resistor revealed that the behavior

variations in the fabricated structures were a subset of the predicted variations obtained

from the MonteCarlo results. This means that the building block equivalent circuits

191

formerly proven to model the entire resistor structure are also accurate enough to model

the variations, created by process fluctuations and non-ideal properties, between devices.

It has therefore been determined that the statistical variation of the component values

provides and accurate representation of the overall resistor performance. This modeling

methodology’s potential was shown in Chapter 7 where performance and parametric

yield of the 9-segment meander resistor was predicted.

Dr. Ravi Poddar’s statement concerning the basis of his novel modeling

methodology was quoted as:

“If accurate models for each of the building blocks along with interaction

information can be obtained, then any arbitrary structure comprised of those

building blocks can be modeled accurately using the individual block models.”

This research has expanded on that quote to now include:

“The variation in the building block equivalent circuit models, based only on

the data used to extract them, can be used to predict the variations in actual

fabricated devices; therefore, providing an overall accurate representation of

the device performance.”

222

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