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Study of Nonlinear Transmission Lines and their Applications

Kasra Payandehjoo

Department of Electrical & Computer Engineering McGill University Montreal, Canada

October 2006

A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Master of Engineering.

© 2006 Kasra Payandehjoo

2006/10/04

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i

Dedication

1 dedicate this thesis to my parents and my sister in thanks for their endless love,

support, and encouragement.

~-

ii

Acknowledgments

First and foremost 1 would like to express my deepest gratitude to my supervisor

Professor Ramesh Abhari for giving me the privilege of working in her research

group, for her advise and guidance, and most of aIl for her continuous encouragement

throughout the course of my research.

1 would like to express my appreciation to my coIleagues Mf. Asanee Suntives

and Mf. Arash Khajoueizadeh for their helpful discussions and advices.

1 appreciate the assistance of the Multifor employees for their assistance in fab

ricating the boards.

1 would like to thank Mr. Bob Thomson for the meticulously solde ring the

boards.

1 am proud to be a member of the MACS (Microelectronics and Computer Sys

tems) laboratory at McGill University and would like to acknowledge aH the grad

uate students in the laboratory.

1 would like to thank Mrs. Golnaz Motiey for reviewing my thesis for grammat

ical errors.

1 would also like to thank Mr. Sadok Aouini for reviewing french abstract for

grammatical errors.

1 greatly appreciate NSERC Canada's support of my research.

Last but not least 1 would like to thank my family for their constant love, support,

and encouragement.

iii

Abstract

With the increasing market demand for wideband multifunctional electronic sys

tems, real-time broadband measurement systems with few picoseconds switching

rates are essential. Furthermore, stable millimeter wave sources are required to

drive these wideband electronic systems. Nonlinear transmission lines (NLTLs) are

high impedance transmission lines periodically loaded with reverse biased diode

serving as varactors. Extremely high bandwidths are achievable because of the pos

sibility to fabricate these structures monolithicaIly, which is why pulses with ultra

short transitions can be generated using NLTLs. Also, efficient wideband frequency

conversion is made possible by NLTL technology.

In this thesis, a comprehensive study of NLTLs and their applications is pre

sented. Sharpening of the edges of electrical pulses, voltage dependent true time

delay, and harmonic generation in NLTLs are investigated through analytical studies

as weIl as circuit simulations and experimental measurements. Designing the best

possible mixers, frequency doublers, and edge sharpeners and optimizing them are

not the objects of this thesis. The main objective is to study an alternative design

approach by using NLTLs. To this end, analytical solution for the magnitude of the

third harmonic along a nonlinear transmission line is derived for the first time. Also,

for the first time the lowpass nature of the NLTL is combined with the solutions for

the magnitudes of harmonies in order to improve the validity range of the predicted

harmonics. An NLTL harmonic generator is fabricated and measurement results are

reported.

Inspired by the distributed nature of nonlinear transmission lines, a novel filter

ing method is introduced for the suppression of the unwanted signaIs in different

iv

NLTL applications. The filtering method is applied to a nonlinear transmission line

frequency multiplier in order to filter the third harmonie. The distributed filtering

is also used to suppress the image signal in an NLTL mixer. The proposed filtering

method is general and can be applied to other periodic structure as well (such as

distributed amplifiers and distributed mixers). For implementing the filtering, com

pact complementary split ring resonators are proposed and designed for an NLTL

frequency doubler.

v

Abrégé

Avec la demande croissante du marché pour des systèmes électroniques multifonc

tionnels à large bande, les systèmes de mesure en temps réel à bande large ayant

un temps de retournement inférieur à une picoseconde sont essentiels. En outre,

des sources stables d'onde millimétrique sont nécessaires pour exciter ces systèmes

électroniques à bande large. Les lignes de transmission non-linéaires (Nonlinear

Transmission Lines ou NLTLs) sont des lignes de transmission à grande impédance

périodiquement chargées avec des diodes qui agissent comme varactors. Des bandes

passantes extrêmement larges sont réalisables en raison de la possibilité de fabriquer

ces structures monolithiquement permettant ainsi des impulsions avec des transi

tions ultra courtes produites en utilisant NLTLs. En plus, la conversion efficace de

fréquences à bande large est rendue possible par la technologie NLTL.

Dans cette thèse, une étude compréhensive des NLTLs et de leurs applications

est présentée. La compression des bords des impulsions électriques, retard en temps

réel dépendant de la tension et la génération harmonique dans les NLTLs sont in

vestigués par des études analytiques, des simulations des circuits et des mesures

expérimentales. La conception et optimisation des meilleurs mélangeurs, doubleurs

de fréquence et compresseurs de bord ne sont pas les objets de cette thèse. L'objectif

principal est d'étudier une approche alternative de conception en employant des

NLTLs. À cet effet, la solution analytique pour l'amplitude du troisième harmonique

dans une ligne de transmission non-linéaire est dérivée pour la première fois. De

plus, pour la première fois une méthode combinant la nature passe-bas des NLTLs

avec les solutions mathématiques des amplitudes d'harmoniques est proposée afin

d'améliorer l'intervalle de validité des harmoniques prévus. Un générateur har-

vi

monique NLTL est fabriqué et les résultats de mesure sont rapportés.

Inspiré par la nature distribuée des lignes de transmission non-linéaires, une

nouvelle méthode de filtrage est également présentée pour la suppression des sig

naux non désirés dans différentes applications des NLTLs. La méthode de filtrage

est appliquée à un multiplicateur de fréquence NLTL afin de filtrer le troisième

harmonique. Le filtrage distribué est également employé pour supprimer le signal

d'image dans un mélangeur NLTL. La méthode de filtrage proposée est générale et

peut être aussi bien appliquée aux autres structures périodiques (comme les amplifi

cateurs distribués et les mélangeurs distribués). Pour l'application du filtrage, des

résonateurs bagues fendues complémentaires compacts sont proposés et conçus pour

un doubleur de fréquence NLTL.

Contents

1 Introduction

1.1 Thesis Rationale and Contributions

1.2 Thesis Outline ........... .

2 Analysis of N onlinear Transmission Lines

2.1 Time Domain Analysis

2.2 Floquet Analysis . . .

2.3 Classifying NLTL Applications Based on Bragg Frequency

2.4 Conclusions ......................... .

vii

1

4

6

8

9

13

16

20

3 N onlinear Transmission Lines as Edge Sharpeners 21

3.1 Introduction.......... 21

3.2 Compression of the Rise-time 22

3.3 General Guidelines in Designing an NLTL Edge Sharpener 26

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 N onlinear Transmission Lines as N onlinear Delay Lines 29

4.1 Introduction to Nonlinear Delay Lines ........... 29

4.2 Theory of Nonlinear Transmission Lines as Variable Delay Lines 30

Contents

4.3 An NLTL-Based Variable Delay Line

4.4 Conclusion...............

VIlI

33

34

5 Nonlinear Transmission Lines as Harmonie Generators 36

5.1 Introduction.......................... 36

5.2 Harmonie Balance Analysis of a Nonlinear Transmission Line . 37

5.3 Simulation of a Nonlinear Transmission Line as a Harmonie Generator 42

5.3.1 First Estimate of the Harmonie Voltages Without Including

the Filtering Effect of the NLTL . . . . . . . . . . 43

5.3.2 Including the Bragg Filtering Effect of the NLTL 43

5.4 Fabrication and Measurement of a Nonlinear Transmission Line Har-

monie Generator

5.5 Conclusions ...

49

53

6 Distributed Filtering of U nwanted SignaIs in N onlinear Transmis

sion Lines 60

6.1 Introduction................................ 60

6.2 Introducing Distributed Filtering in the Nonlinear Transmission Line 61

6.3 Example 1: Filtering of the 3rd Harmonie in an NLTL Frequency

Doubler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4 Example 2: Filtering of the Image Signal in an NLTL Mixer

6.5 Example 3: Implementing the NLTL Harmonie Generator with Dis-

tributed Filtering .......... .

6.5.1 The Prototype NLTL Circuit

6.5.2

6.5.3

Implementing the Distributed Filtering

Simulation Results After Including the CSRRs in the NLTL

67

69

69

70

72

Contents ix

6.5.4 Sensitivity Analysis of the NLTL ..... . . . . . . . . . . . 74

6.6 ProposaI of a Distributed Filter for the NLTL Frequency Multiplier

Fabricated in Section 5.4

6.7 Conclusions

7 Conclusions

7.1 Future Works

A Large-Signal S-Parameter (LSSP) Simulations

References

75

79

81

............ 82

84

86

x

List of Figures

1.1 Circuit schematic of a Nonlinear Transmission Line (NLTL) 3

2.1 Model for an infinitesimal section of a transmission line . . . 9

2.2 Model for an infinitesimal section of a fully distributed NLTL 11

2.3 A periodic structure ...................... 13

2.4 Unit cell of a periodically loaded nonlinear transmission line 14

2.5 Equivalent LC model for the unit cell of a periodically loaded nonlin-

ear transmission line

2.6 The NLTL prototype

2.7 Input and Output waveforms when fin = 1G Hz, the sol id curves are

15

17

the inputs and the dashed curves are the outputs . . . . . . . . . .. 18

2.8 Input and Output spectrums, the solid curves are the inputs and the

dashed curves are the outputs . . . . . . . . . . . . . . . . . . . . .. 19

2.9 Input and Output waveforms when fin = 6GHz, the solid curves are

the inputs and the dashed curves are the outputs . . . . . . . . . .. 19

3.1 The capacitance profile of the diodes

3.2 Sharpening of the risetime

3.3 The studied NLTL structure

23

23

24

List of Figures xi

3.4 Risetime compression along the NLTL ................. 26

4.1 Unit cell of a nonlinear transmission line 31

4.2 Phase and group veloeities in an NLTL . 32

4.3 Equivalent Le model of the unit cell of an NLTL 32

4.4 The NLTL Variable delay Line ....... 33

4.5 True time delay versus reverse bias voltage 34

5.1 nth stage of an NLTL ...... 38

5.2 Magnitude of the 1 st harmonie . 43

5.3 Magnitude of the 2nd harmonie 44

5.4 Magnitude of the 3rd harmonie. 44

5.5 Bloek diagram for the pro cess of predicting the harmonies 45

5.6 Magnitude of the 1 st harmonie . 46

5.7 Magnitude of the 2nd harmonie 46

5.8 Magnitude of the 3rd harmonie. 47

5.9 Input and output voltage waveforms along the NLTL 48

5.10 Predieted and simulated output voltage waveforms 48

5.11 NLTL Harmonie Generator .......... 49

5.12 The diseontinuity in the eoplanar structure. 51

5.13 Lumped model for the diseontinuity . . . . . 51

5.14 Z21 of the diseontinuity and its first order approximation 52

5.15 The fabrieated NLTL harmonie generator . 54

5.16 The measurement setup .... 55

5.17 Magnitude of the first harmonie 56

/ ~-- 5.18 Magnitude of the second harmonie 57

List of Figures xii

5.19 Magnitude of the third harmonie .................... 58

6.1 (a) The NLTL with periodic filtering (b) Equivalent lumped-element

model of the NLTL with periodic filtering. . . . . . . . . . . . 62

6.2 Dispersion diagram of the NLTL after adding the tank circuits 63

6.3 Dispersion diagram of the NLTL after adding the tank circuits and

satisfying conditions in Equation (6.8) ................. 65

6.4 Transmission coefficients for the first three harmonies before adding

the tank circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.5 Block diagram of the simulation test bench used in the analysis of

the mixer example ............................ 67

6.6 Down conversion loss: (a) before adding the tank circuits (b) after

adding the tank circuits

6.7 NLTL frequency doubler

6.8 Transmission coefficients for the first three harmonies

6.9 The SRR proposed by Pendry

6.10 The CSRR .......... .

68

70

71

72

73

6.11 Transmission coefficient of the new transmission line structure 73

6.12 Transmission coefficient for the first three harmonies in the NLTL

structure loaded with CSRRs ...................... 74

6.13 Effect of input voltage variations on the conversion efficiencies for the

unloaded NLTL: (a) 821 for the second harmonie (b)821 for the third

harmonie ................................. 76

List of Figures

6.14 Effect of input voltage variations on the conversion efficiencies in the

NLTL loaded with CSRRs: (a) 821 for the second harmonie (b)821

for the third harmonie

6.15 Layout of the CSRR .

xiii

77

78

6.16 The transmission coefficient of the CPW unit cell loaded with the

CSRR of Figure 6.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.17 Transmission coefficient for the first three harmonies in the NLTL

structure loaded with CSRRs ...................... 80

xiv

List of Tables

2.1 NLTL Parameter Values . . . . . . . . . . . . . . . . . . . . . . . .. 18

3.1 NLTL Parameter Values ............. 25

5.1 NLTL Harmonie Generator Parameter Values .... . . . . . . . . . 50

6.1 Coplanar NLTL Parameter Values.

6.2 CSRR Dimensions ........ .

69

78

List of Acronyms

CPW

CSRR

FDNLTL

KCL

KVL

LSSP

NLTL

MMIC

PLNLTL

SRD

SRR

Coplanar WaveGuide

Complementary Split Ring Resonator

Fully Distributed Nonlinear Transmission Line

Kirchoff's Current Law

Kirchoff's Voltage Law

Large Signal S-Parameters

N onlinear Transmission Line

Monolithic Microwave Integrated Circuits

Periodically Loaded Nonlinear Transmission Line

Step Recovery Diode

Split Ring Resonator

xv

1

Chapter 1

Introduction

The market demand for wideband multifunctional electronic systems has been grow

ing incessantly thus pushing the system design and device fabrication technologies

towards achievement of few picoseconds switching rates and millimeter-wave band

cutoff frequencies. This has been made possible as transistors with gain cutoff fre

quencies beyond 500 GHz have been fabricated [1] and Schottky diodes with cutoff

frequencies in the THz range are introduced [2]. Characterization ofreal-time broad

band measurement systems requires accurate and reliable measurement instruments

to ensure efficient operation of these state-of-the-art components before their use in

complex systems. Also, stable high frequency sources are required to drive wideband

electronic systems.

One of the main factors limiting the bandwidth of the real-time sampling oscil

loscopes is the width of the sampling aperture [3]. The sampling aperture is the

time interval during which the switch controlled by the local-oscillator (LO) strobe

signal (strobe drive) opens the path between the RF input and the IF output [2].

The st robe drive, with very sharp transition edges, turns the switches on for the

1 Introduction 2

narrow sampling aperture required in a wideband sampling oscilloscope [2]. The

conventional technology for realization of fast switches and strobe drives utilizes

step recovery diodes (SRDs), i.e. diodes with graded doping which aUows fast re

lease of stored charges in switching from forward to reverse bias. The transition

time of the strobe drive generated by SRDs is on the order of tens of picoseconds

which limits the bandwidth of the sampling circuits [3]. Other electrical pulse gen

erators, such as resonant tunneling diodes, can achieve lower transition times but

their applications are limited to small voltage swings [4].

N onlinear transmission lines have been historicaUy the design configuration of

choice in using a succession of diodes for generation of fast switching pulses [3]. In

these types of switching and sampling applications NLTLs have proved to result

in faster transitions compared to SRDs according to the reports in [4]. In fact,

nonlinear transmission lines (NLTLs) are interconnects, in which nonlinearities are

introduced to provide tunable features for controUing the shape of the signal and

signal propagation characteristics along the line. The latter feature lends NLTLs

to pulse shaping and dispersion compensation applications [5-11]. In modern high

speed systems dispersion and broadening of electrical pulses along the interconnects

is one of the main signal integrity concerns. Therefore, restoration of the degraded

rise and faU times is of great importance in numerous digital and mixed-signal

systems.

NLTLs are implemented either by periodic loading of the line with nonlinear

elements (Periodically Loaded Nonlinear Transmission Lines [12,13]) or by contin

uous distributed doping of nonlinearities within the substrate (Fully Distributed

Nonlinear Transmission Lines [12,13]). In periodicaUy loaded NLTLs the repetitive

loading can be added by mounting discrete diodes on the line or periodic local-

1 Introduction 3

ized doping of the semiconductor substrate of the transmission line. However, in

fully distributed NLTLs the nonlinearities are doped continuously in the substrate.

The most well-known configuration for the NLTL is a transmission line periodically

loaded by reverse biased diodes as shown in Figure 1.1.

Fig. 1.1 Circuit schematic of a Nonlinear Transmission Line (NLTL)

NLTLs contain nonlinear components hence can be used for frequency conversion

[14-19]. They are very suit able for high frequency harmonie generation since they

have an almost real input impedance and can be designed to provide a good match

ing to a resistive impedance (The input impedance is purely resistive for a lossless

NLTL). One of the methods of designing stable sources at microwave and millime

ter wave frequency ranges uses high-frequency frequency multipliers in conjunction

with stable lower frequency sources. Conventional millimeter-wave range frequency

multipliers are diode multipliers which exhibit a reactive input impedance [20] as

opposed to their NLTL counterparts with resistive input impedance. Therefore, the

bandwidth of operation of diode multipliers is limited due to the unavailability of

broadband matching circuits.

Furthermore, the voltage dependent capacitance of the reverse biased diodes

in NLTLs results in voltage dependent propagation velocity which can be used to

sharpen the rising or falling (or both) edge of a pulse depending on the capacitance

1 Introduction 4

profile of the diodes [5-11]. The bandwidth of the NLTL limits the fastest achievable

transitions. The voltage dependent phase velo city in NLTLs also makes them good

candidates for wideband true time delay lines and tunable phase shifters [21-23].

Therefore, they can be used in the design of phased-array antennas. In these struc

tures the phase differences between the signaIs fed to the antennas are changed to

reinforce the radiation pattern of the array in a desired direction. A critical factor

that limits the bandwidth of the phased-array antennas is the bandwidth of the com

ponents providing the phase difference, whether they are phase shifters or true time

delay lines [22](structures that introduce a fixed delay in arrivaI of the output signal

regardless of the frequency content of the input signal). Phase shifters are narrow

band compared to true time delay lines [21], which can be naturaIly implemented

using NLTL technology.

1.1 Thesis Rationale and Contributions

This thesis presents a comprehensive study of the NLTL circuits and their appli

cations. This investigation begins with analytical derivations and CAD simulation

of a generic NLTL circuit foIlowed by covering a number of popular NLTL applica

tions. In most of the analytical derivations and CAD simulations in this thesis the

NLTL is assumed to be lossless. NLTLs are distributed periodic structures exhibit

ing passband and stopband regions in their frequency response. For the frequency

ranges considered in this thesis only the first passband is deemed important. Thus,

the considered NLTLs are assumed to be lowpass structures characterized by their

lowpass cutoff frequencies, often referred to as the Bragg cutoff frequency. The unit

ceIl of an NLTL is shown in Figure 1.1. The Bragg cutoff frequency depends on the

1 Introduction 5

length and configuration of the unit cell and the average reverse-bias capacitance

of the diodes. From the discussion of the previous section it is evident that the

NLTL can be utilized in a variety of applications. Indeed, each application is often

prescribed to a certain frequency range in the passband of the NLTL. In addition

to the Bragg cutoff frequency, the cutoff frequency of the diodes also determines

the maximum frequency of operation of NLTLs. The diodes can be Schottky diodes

with TeraHertz cutoff frequencies which push the operating frequency limit of the

NLTLs far beyond the needs of today's electronics [2]. The NLTL can be fabricated

monolithically offering small unit-ceIllengths and average capacitances. Note that

the large values of the capacitance of discrete diodes limits the Bragg frequency.

Therefore, integrated implementation schemes result in extremely high Bragg cutoff

frequencies. This is why ultra wideband NLTLs can be fabricated.

In summary significant advantages of NLTLs are listed below:

1. They use the fastest semiconductor devices, i.e. diodes

2. They are compatible with various integrated circuit technologies thus

• high bandwidths are achievable

• more compact electronic systems can be realized as NLTLs can be inte

grated with other electronic circuits on the same chip

3. Their input impedance is purely resistive thus wideband mat ching can be

achieved in connection with other circuits

Considering these advantages and the discussed potentials for use of NLTLs

in high frequency applications, different NLTL circuits are studied by analytical

solutions, circuit simulations, and experimental characterization in this thesis. In

1 Introduction 6

this endeavor, contributions have been made to the analysis and design of NLTL

circuits that are described as follows.

Using the harmonic balance technique the conversion gain for the third harmonic

in a NLTL is derived for the first time [24]. Inspired by the periodic nature of

NLTLs, distributed filtering of the unwanted signaIs in different NLTL applications

is proposed for the first time in this thesis [25]. An NLTL harmonic generator is

fabricated in the coplanar waveguide configuration and the measurement results are

reported. Moreover, a novel band reject filter topology by use of complementary

split ring resonators is proposed and designed for filtering the third harmonic in the

NLTL harmonic generator.

1.2 Thesis Outline

Following the introductory and overview material presented in this Chapter, the

rest of the thesis describes analysis, design, simulation and experimental studies.

In Chapter 2, time and frequency domain analyses of an NLTL are discussed and

the frequency range for different applications of NLTLs are introduced. Sharpening

of the rising edge of an electrical pulse in an NLTL has been demonstrated in

Chapter 3. In Chapter 4, the voltage dependent delay along an NLTL has been

investigated and the plot of the delay versus applied voltage to an NLTL is presented.

In Chapter 5, the NLTL is studied in a time harmonic regime and approximate

equations for the magnitudes of the first three harmonics in the NLTL are derived

and verified through simulations conducted by the Agilent ADS software. Also an

NLTL harmonic generator is fabricated and measurement results are reported in

Chapter 5. Chapter 6 discusses the periodic loading of NLTLs with tank circuits

1 Introduction 7

for the pur pose of filtering of unwanted signaIs in different NLTL applications. The

proposed method is applied to three applications. Finally, in Chapter 7, conclusions

and potential future works for further investigation of NLTL circuits are discussed.

Chapter 2

Analysis of N onlinear

'Iransmission Lines

8

In order to properly characterize a nonlinear transmission line, both time domain

and frequency domain analyses of the structure are necessary. In this chapter, the

time domain analysis is conducted for a fully distributed NLTL, by considering an

infinitesimal section of length dx, and the voltage dependent propagation velo city

of the NLTL is introduced. The results of this analysis also apply to a periodically

loaded NLTL if the length of the unit cells are much sm aller than the wavelength.

The time domain analysis does not include the lowpass nature of a periodically

loaded NLTL. In fact, this periodic characteristic is best captured by using Floquet

Theorem ( [26]) which is applied in the time-harmonic regime. This frequency

domain analysis gives the lowpass cutoff frequency of a periodically loaded NLTL,

which is also referred to as the Bragg cutoff frequency. The Bragg cutoff frequency

determines the frequency ranges for different applications of NLTLs [4].

2006/10/04

2 Analysis of Nonlinear Transmission Lines 9

2.1 Time Domain Analysis

A transmission line is characterized by its capacitance and inductance per unit

length. An infinitesimal section of an NLTL can be modeled as shown in Figure 2.l.

The NLTL is extended in the x direction which is also the direction of propagation.

1 Ldx I+dI --.. ..... rY"YY"

l + +

V CdxIV~dV - --

Fig. 2.1 Model for an infinitesimal section of a transmission line

Writing the voltage drop across the inductor and KCL at the output node yields:

al av= -Ldxat

av aI=-Cdxat

(2.1)

(2.2)

Dividing both sides of these equations by dx and differentiating the first equation

by x and the second equation by t leads to the following equation for the voltage

waveform:

(2.3)

2 Analysis of N onlinear Transmission Lines 10

which has a general solution of the form

t t V(x, t) = V+(x - f"T""n) + V-(x + f"T""n)

yLC yLC (2.4)

corresponding to waves traveling in the positive x and negative x directions with a

propagation velocity of:

U=_l_ VLC

The general solution for the current waveform is:

where Zo = ~ is defined to be the characteristic impedance of the NLTL.

(2.5)

(2.6)

A fully distributed nonlinear transmission line is a transmission line in which the

diode nonlinearity is continuously distributed in the substrate of the transmission

line [13]. Applying a reverse bias voltage to this NLTL results in a distributed

variable capacitance along the transmission line. The model for an infinitesimal

section of a fully distributed NLTL is found by replacing the capacitance per unit

length of the model in Figure 2.1 by a variable capacitance per unit length as shown

in Figure 2.2.

If V and lare assumed to be differentiable single - valued functions of x then

the nodal equations can be written as

al + C(V) av = 0 ax at

av LaI = 0 ax + ai

(2.7)

(2.8)


1 --+-

+ V

Ldx

C(V)d

--

I+dI ~

+ V+dV

-Fig. 2.2 Model for an infinitesimal section of a fully distributed NLTL

Il

where C(V) is the variable nonlinear capacitance per unit length. Equations (2.7)

and (2.8) can be solved through applying the method of characteristics by forming

the following pair of linear combinat ion of the two equations as suggested in [27]

and [28]

(2.9)

(2.10)

here À 1 and À2 are the combining multipliers. As explained in [27] by choosing À 1

and À2 to be

(2.11)

and using the following nonlinear mapping

{

Bx = Ba + B{3

Bt = J LC(V)Ba - J LC(V)B{3 (2.12)


or

{

80: = ~(8x + 8t/ y'LC(V))

8{3 = ~(8x - 8t/ y'LC(V))

the following Equations can be found from Equations (2.9) and (2.10)

81 = _JC(V) 8V 80: L ao:

81 = JC(V)8V 8{3 L 8{3

12

(2.13)

(2.14)

(2.15)

One solution to Equations (2.14) and (2.15) is V({3) (forward traveling wave)

and another solution is V(o:) (backward traveling wave) according to [28]. Points

of constant (3 have the same voltage on the forward traveling wave and thus from

Equation (2.13) the propagation velocity of the forward traveling voltage wave is

found to be

U(V) _ 1 - y'LC(V)

(2.16)

Equation (2.16) suggests that different voltages travel at different velocities along

the NLTL and experience different delays. Depending on the capacitance profile this

may lead to the sharpening of one (or both) of the edges as demonstrated in Section

3.2. Equation (2.16) is also valid for a periodically loaded NLTL if the length of the

unit sections is much smaller than the wavelength of the highest operating frequency.

Also by integrating Equation (2.15) with respect to {3 the voltage-dependent

characteristic impedance of the NLTL is found to be

z~ J CfVl (2.17)


2.2 Floquet Analysis

+ VII

.. d

+ Vn+l

.,

Fig. 2.3 A periodic structure

13

---

In the Floquet analysis of a periodic embodiment, the structure is assumed to

be composed of an infinite number of identical sections often referred to as unit

cells (Figure 2.3) [26]. Knowing the ABCD matrix of the unit cell the relationship

between the voltage and current waveforms before and after the nth unit cell can be

written as:

(2.18)

also assume that voltage and current are traveling waves which propagate along the

line in the +x direction. Assuming a phase reference at x=O and an infinitely long

line, the voltage and current at the nth terminal can only differ from those at the

n + 1 terminal by the propagation factor e-,d [26]:

(2.19)

The following homogeneous matrix equations is obtained from Equations (2.18)

and (2.19):


B ] [Vn+l] [ 0 ] D - e--yd In+l 0

(2.20)

For this homogeneous matrix equation to have non-trivial solutions, the deter-

minant of the 2 by 2 matrix should be equal to zero, which yields the dispersion

equation of the periodic structure:

A+D cosh(,d) = 2

Unit Cell

Id/2 d/2 1 .. ... Zo Zo

L.. __ "' __ ...J

Fig. 2.4 Unit ceIl of a periodicaIly loaded nonlinear transmission line

(2.21)

Figure 2.4 shows the unit cell of a periodically loaded NLTL. If the varactor is

replaced by its average capacitance, the ABCD matrix of the unit cell can be found

from:

[

cos(~) jZOSin(~d)] [ 1 . f3d f3d

10 sin( '"2 ) cos( '"2 ) jwCaverage

0] [ cos(~d) 1 da sin(~d)

jZo sin(~d) ]

cos(~d)

(2.22)


where the first and the third matrices on the right hand side of Equation (2.22)

are each the ABCD matrix of a section of transmission line with characteristic

impedance Zo and length of d/2. (3 is the phase constant along this transmission

li ne section. The second matrix on the right hand side of Equation (2.22) is the

ABCD matrix of the parallei capacitor element at the center. Knowing the ABCD

matrix of the unit cell and using Equation (2.21) the characteristic equation or the

dispersion equation of the NLTL is found to be:

cosh( "id) - cos((3d) + w:o Caverage sin((3d) = 0

,------------1 1 L/2 L/2 1

1

: C+CaveragJ 1

1 1.. 1 I ______ ~ ______ I

Fig. 2.5 Equivalent Le model for the unit cell of a periodically loaded nonlinear transmission Hne

(2.23)

In order to find an approximate closed form formula for the Bragg cutoff fre

quency, the Floquet analysis can be also applied to the LC model of the NLTL

whose unit cell is shown in Figure 2.5. The ABCD matrix of this unit cell is:

[AB 1 [1 jw2

L 1 [ 1 0] [1 jw2L 1

C D 0 1 jwC 1 0 1 (2.24)

which leads to the following dispersion equation. This is an approximation of Equa-

tion (2.23): w2LC

cosh("(d) = 1 - -2- (2.25)


Sinee the right hand side of Equation (2.25) is real, either a = 0 or f3d = (0 or 7r).

a = 0 and f3d # (0 or 7r) corresponds to the nonattenuating propagating wave,

while in the case of a # 0 and f3d = (0 or 7r), there is no propagation and the

wave is attenuated along the line [26]. Solving the dispersion diagram for the cutoff

frequencies (f3d = (0 or 7r)) yields:

2 W Bragg = ---;~:::==::::::::::=====:=

yi L( C + Caverage) (2.26)

which shows that periodically loaded NLTLs are lowpass structures. This cutoff

frequency is usually referred to as the Bragg cutoff frequency and is one of the main

characteristics of NLTLs.

2.3 Classifying NLTL Applications Based on Bragg

Frequency

The frequency range for different applications of nonlinear transmission lines is de-

termined by the Bragg cutoff frequency according to [4].

• Low Dispersion (f < < f Bragg): At this range the dispersion is negligible. The

applications recommended for this band use the voltage dependent propaga

tion velo city and delay characteristic of NLTLs. The main applications at this

frequency range are edge sharpening (large signal application since the sig

nal is intentionally distorted to get faster transitions) and variable delay lines

(small signal application sinee the signal should not get distorted and only the

De bias is changed to control the propagation speed) .

• Intermediate Dispersion (fBragg/5 < f < fBragg/2): At this range the disper-


sion phenomenon can be used to filter higher order harmonies and to increase

the conversion gain of certain harmonies. Frequency multipliers are the main

applications at this frequency range .

• Righ Dispersion (fBragg/2 < f < fBragg): Large Amplitude narrow pulses

often referred to as Solitons are formed [4]. As the input pulse propagates

along the NLTL, its energy is compressed in time. Renee, the pulse width is

decreased and the amplitude is increased.

A simple NLTL structure is simulated to investigate the aforementioned fre

quency ranges for different NLTL applications. The NLTL considered for this study

is shown in Figure 2.6. It is a periodically loaded NLTL with the same values as

those reported in [29]. Each Le network in the model represents a lem-long section

ofthe line. The reverse biased varactor diodes are modeled by Equation (2.27). The

components values and parameters are shown in Table 2.1.

unit cell .-.- .... --- ...

L/2

1 1 1

"='" 1 L.._~ __ _

Fig. 2.6 The NLTL prototype

(2.27)


In Equation (2.27) CjO is the zero-bias junction capacitance, m is the grading

coefficient, Va is the junction voltage, and C is the equivalent capacitance of a lcm

long section of the transmission line.

Table 2.1 NLTL Parameter Values

Parameter Value

L 2.33nH C 229fF

CjO 916fF m 0.5 Vo 0.6

number of stages 10

The circuit of Figure 2.6 was simulated using Agilent ADS. From Equation (2.26)

the Bragg cutoff frequency for this structure is found to be 7.8 CH z for a 0 - 4V

input. Figure 2.7 shows the output of the NLTL together with the lGHz input. It

can be seen that at this frequency the risetime of the single tone input is sharpened.

4.5 4.0 3.5 3.0 2.5

V 2.0 1.5 • 1.0

, J ,

0.5 , .... "' 0.0 .... _-

-0.5 30

... : '\ .. .. *_ ...... _ .. _'-._ ... -, ..

., '" '_ •• 0"

" ~" ~. .. . • , , ... .

'- ... '

31 Time (ns)

r'tt .... 'L ..

" , ... .. ,,~ ..

32

Fig. 2.7 Input and Output waveforms when fin = 1GH z, the solid curves are the inputs and the dashed curves are the outputs

Figure 2.8 shows the spectrum of the NLTL output for a 2.5 CH z input. As

expected, it can be seen that, as expected, strong harmonics are generated at this


frequency. Finally, the output of the NLTL 2.9 is shown for a 6 GHz sinusoidal

input. It can be seen that the width of the sinusoid is compressed and its peak is

amplified.

V

2.5

2.0

1.5

1.0 • • 0.5 •

.. ~ .... • • 1

0.0 0 1 2 3 4 5 6 7 8 9 10

f(GHz)

Fig. 2.8 Input and Output spectrums, the solid curves are the inputs and the dashed curves are the outputs

5.0~--------------~--------------~W-----~

4.5 4.0 3.5 3.0

V 2.5 2.0 1.5 1.0 0.5 0.0

-0.5 +------------------------------------;-----1 48 48.35

Timl' (ns)

Fig. 2.9 Input and Output waveforms when fin = 6GH z, the solid curves are the inputs and the dashed curves are the outputs


2.4 Conclusions

In this chapter time-domain analysis is given to derive the voltage dependent prop

agation speed along a NLTL. In addition, Floquet analysis is applied to periodically

loaded NLTLs to capture the lowpass nature of these structures. Based on the Bragg

cutoff frequency the frequency range for different NLTL applications are introduced.

Few of these applications are investigated in the later chapters.

21

Chapter 3

N onlinear n-ansmission Lines as

Edge Sharpeners

3.1 Introduction

Modern communication circuits operate at millimeter wave frequency range and

digital electronics pro cess 40 Gb/s data rates [4]. The market demands for pushing

the operation limits of the existing electronic circuits even beyond these frequency

ranges. The advances in semiconductor electronics have enabled high-frequency ap

plications such as radar and atmospheric studies. Characterization of these systems

requires high frequency broadband measurement systems. The main factor limiting

the bandwidth of sampling oscilloscopes is the duration of the sampling aperture.

Fast edges for the strobe drive are required for a very narrow sampling aperture [2].

Short pulses are also required in ultrashort pulse plasma reflectometry and short

pulse radars [6]. The voltage dependent delay that is inherent in signal propagation

in a nonlinear transmission line can be used to generate ultra sharp electric pulses.


In fact, by using NLTL technology Lecroy Corp. has recently come up with a digital

sampling oscilloscope with a sampling bandwidth of 100 GHz [2].

In this chapter, the voltage-dependent delay in an NLTL is used to compress

the risetime of an electrical pulse from 250ps to 155ps. Aiso general guidelines in

designing an NLTL to operate as an edge sharpener are introduced.

3.2 Compression of the Rise-time

The voltage dependent propagation speed along a nonlinear transmission line (Equa

tion (2.16)) suggests that different voltage levels travel at different velocities, thus

experiencing different delays:

7 = ll~e = lline yi LC(V) (3.1)

Depending on the capacitance profile of the reverse biased diodes the rising or

falling edge of the electrical pulses can be compressed. For example, if the capaci

tance of the varactor decreases as the reverse voltage increases, as shown in Figure

3.1, Equation (2.16) implies that higher voltages travel faster and experience less de

lay. Therefore, the rising edge of the pulses is sharpened as depicted hypothetically

in Fig 3.2.

Assuming that no dispersion is present, the amount of compression, denoted by

b.7 = t~nput - t':;'tPUt can be found from:

3 N onlinear Transmission Lines as Edge Sharpeners

c

v

Fig. 3.1 The capacitance profile of the diodes

V2

NLTL

.. v,

Fig. 3.2 Sharpening of the risetime

23

Equation (3.3) suggest that the amount of compression can be increased by

choosing a longer NLTL since it linearly increases with lline. However, a periodically

loaded NLTL is intrinsically a lowpass structure with the Bragg cutoff frequency

found from Equation (2.26). This lowpass behavior affects the compression of the

ri se (or fall) time and transition edges are rounded as they propagate in the NLTL.

Equation (3.3) only gives the amount of compression, when the frequency content


of the signal is mueh sm aller than the eut off frequeney of the NLTL (i.e. fmax < <

fBragg)' In using Equation (3.3) to estimate the fastest possible risetime of the

output pulse, one should verify that an output pulse with sueh a risetime eould

propagate without degradation in the lowpass NLTL.

unit cell ... ------,.

Lt2

1 1 1 --,-_....;. ___ 1

Fig. 3.3 The studied NLTL structure

The NLTL eonsidered here is shown in Figure 3.3. It is a periodieally loaded with

varaetors of the values reported in [29]. Each LC network in the model represents

a lem-long section of the line. The reverse biased varaetor diodes are modeled by

Equation (3.4). The eomponents values and parameters are shown in Table 3.1.

Q(V) = J (C;o/ VI +;;,)dV + CV (3.4)

In Equation (3.4) CjO is the zero-bias junction eapaeitance, m is the grading

coefficient, Va is the junetion voltage, and C is the equivalent eapaeitance of a lem

long section of the transmission line.

The input to the NLTL is a trapezoidal pulse with a magnitude of 4V and rise

and fall times of 250ps (0% - 100%). Considering the NLTL parameters listed in

Table 3.1, the Bragg eutoff frequeney is:


Table 3.1 NLTL Parameter Values

Parameter Value

L 2.33nH C 229fF

CjO 916fF m 0.5 Vo 0.6

number of stages 10

1 fBragg = = 7.8 GH z

-/LCaverage (3.5)

The frequency content of the input pulse can be computed by using the formula

given in [30]: 1

fmax = (o/c Oo/c) = 1.27 GHz 7rtr 0 0 - 10 0 (3.6)

which is much sm aller than fBragg. This assures that sorne amount of compression

can be expected. Equation (3.3) predicts a risetime compression of 155 ps, which

corresponds to an output risetime of 95 ps. The frequency content of such an output

is: 1

f - = 3.35 GHz Jmax - 7rtr (O% - 100%) (3.7)

which is still sm aller than the Bragg cutoff frequency. The circuit of Figure 3.3 was

ported to Agilent ADS and Figure 3.4 shows the simulated voltage waveforms at

different observation points along the NLTL. The compression of the risetime can

be monitored as the signal propagates along the NLTL. The 10% - 90% risetime

of the output signal is about 90 ps and corresponds to a 0% - 100% risetime of

112ps, which is slightly larger than the predicted output risetime probably due to

dispersion.

3 N onlinear 'Iransmission Lines as Edge Sharpeners 26

~Lf\ u \.~ .. " u. U· . .o.t.'~ '6-.. ··M·· •• ·u '~., U -u ...... u u 'ù, ù .. t,.,· .. ,'ui·U" 'p ' .. if"u".,· t:. t:f"(,,'T.j"',;4 t'Ao --

Fig. 3.4 Risetime compression along the NLTL

3.3 General Guidelines in Designing an NLTL Edge

Sharpener

The following general guidelines can be followed in designing NLTL edge sharpeners

using discrete diodes and assuming that the input and output waveforms are given:

• Step 1: Calculate the frequency content of the output pulse with the desired

risetime.

• Step 2: Choose a Bragg cutoff frequency much higher than the bandwidth of

the desired output pulse (note that choosing a high cutoff frequency would

result in small unit cell lengths and varactor capacitances. The designer must

also consider these issues and fabrication constraints when choosing fBragg).

• Step 3: Choose a varactor diode with a wide range of capacitance variation

with the applied voltage. Calculate Caverage.

• Step 4: Use the value of fBragg chosen in Step 2 to calculate L (At this stage,

as an engineering approximation, ignore the capacitance of the transmission

line sections).


• Step 5: Choose a high impedance transmission line and calculate the required

length of the unit section to result in the inductance calculated in Step 4. If

the length is too small (big) go back to Step 3 and choose a varactor with a

larger (smaller) capacitance.

• Step 6: Find the equivalent capacitance of the unit cell of the transmission

line and recalculate the Bragg frequency.

• Step 7: If the new Bragg is not large enough to allow the desired output pulse

to propagate along the NLTL go back to Step 3 and choose a varactor with

a larger capacitance or go back to Step five and choose another transmission

line with lower capacitance (The capacitance per unit length of a transmission

line is dependent on the dimensions of the transmission line and also on the

type of the transmission line, i.e. coplanar, stripline, microstripline, .. ).

• Step 8: Determine the number of stages required to obtain the desired com

preSSlOn.

In this guideline, no consideration is made for the input impedance. If matching

to a specific impedance is of great importance, Step 4 should be replaced by:

• Step 4*: Use the desired input impedance and Caver age calculated in Step 3 to

find the value of Land at the same time make sure that the new fBragg is still

high enough.

3.4 Conclusions

In this chapter the application of an NLTL in sharpening of the risetime is demon

strated. The described method is implemented in compressing the risetime of a


trapezoidal pulse from 250ps to 112ps. Aiso general guidelines for designing NLTL

edge sharpeners using discrete diodes are presented and can be modified for on-chip

designs. For example simple steps can be added to determine the area and the

doping profile of the diodes in order to get a desirable variable capacitance.

29

Chapter 4

N onlinear n-ansmission Lines as

N onlinear Delay Lines

4.1 Introduction to N onlinear Delay Lines

With the growing commercial radar applications and smart telecommunication sys

tems, the demand for antenna arrays with beam steering capabilities is increasing

significantly. In many of these applications the beam is expected to move between

two positions in a few microseconds, which makes beam steering antenna arrays the

only option [22]. In pha:sed array antennas, the phase differenee between the input

signaIs to different array elements is varied to change the beam angle and shape.

The phase differenee is produced either by a phase shifter or a true time delay de

viee. The bandwidth of the phased array antenna is limited by the bandwidths

of the antenna elements, power divider, and the elements that provide the phase

differenee. Inherently, the phase angle in a phase shifter is frequency dependent

which limits the bandwidth of the phased array antenna [22]. In contrast, true time


delay elements are extremely wide band. Many difIerent true time delay elements

have been studied so far, such as digital delay lines [31], optical delay lines [32], di

electric delay lines [33], nonlinear delay lines [21], and piezoelectric delay lines [34J.

Nonlinear delay lines have the advantage that they are small and have very high

bandwidths.

Nonlinear transmission lines are by nature nonlinear delay lines if operating in

the small signal regime. The DC voltage level con troIs the capacitance of the reverse

biased diode and thus the phase velocity of the small signal input. The bandwidth

of the true time delay line is limited by the Bragg cutofI frequency and the cutofI

frequency of the diodes. Extremely high cutofI frequencies are obtainable if the

nonlinear transmission line is fabricated monolithically.

In Section 4.2, the theory behind the application of NLTLs as variable delay lines

is presented. An NLTL variable delay line is designed and simulated and the plot

of the delay versus the bias voltage is presented. This is followed by conclusions at

Section 4.3.

4.2 Theory of Nonlinear Transmission Lines as Variable

Delay Lines

Figure 4.1 shows the unit cell of a nonlinear transmission line. In the small sig

nal regime the capacitance of the reverse biased diode is almost constant and is a

function of the DC bias voltage (Cd(VDC )). Applying the Bloch-Floquet analysis to

this periodic structure, as described in Section 2.2, the dispersion equation of the


Unit Cell

IdÎ2 d/2 1 (II ..

'- ____ ..J

Fig. 4.1 Unit ceIl of a nonlinear transmission line

nonlinear transmission line is found to be:

cos(f3d) - cos(f3od) + w:o Cd (VDC ) sin(fJod)) = 0 (4.1)

In Equation (4.1), f30 = wy'ïif is the phase constant along the unloaded transmission

line, Zo is the characteristic impedance of the unloaded transmission line, d is the

length of each unit cell, and (Cd(VDC)) is the capacitance of the reverse biased diode

at the De bias voltage (VDC). The phase and group velocities can be found from

Equation (4.1):

w wd Vp = f3 = cos-1(cos(wdy'ïif) _ ~Cd(VDC) sin(wdy'ïif)) (4.2)

Figure 4.2 shows the phase and group velocities versus frequency. It can be

seen that at frequencies far below the Bragg frequency (f < f Bragg /5) the phase

velo city is almost constant (frequency independent) and phase and group velocities

4 N onlinear Transmission Lines as N onlinear Delay Lines

are equal.

'" ~ ·C ..5i .. ... "0 ... • !:l -; Ë 0 z

1.1

1

0.9

0.8

0.7

0.6

0.5[

0.40 1

-.............. ...... ............. ...... ..., .... ,

"-" ,

: Phase Velocity " ,. : Group Velocity '\

2 3 4 f(GHz)

5

'\. , 6

Fig. 4.2 Phase and group velocities in an NLTL

At low frequencies (f < fBragg/5) Equation (4.2) can be approximated as:

1 vp = --fïïÇ-+----;:;zo--::c::-

d7.(Vc;-D-C7)

y f.1é 2d

32

(4.4)

Equation (4.4) shows the dependence of the phase velocity on the capacitance of

the reverse biased diode. Therefore, by changing the DC bias voltage the phase

velocity and thus the time delay change. Also note that at low frequencies the

transmission line sections can be replaced by equivalent LC sections (Figure 4.3).

r-----~ 1 Ll/2 LIll, , 1 1 1 1 Ca q 1 1 1

Fig. 4.3 Equivalent Le model of the unit cell of an NLTL


Using this simplification and considering that at frequencies far below the Bragg

cutoff frequency the phase velocity is almost constant, another equivalent of equation

(4.4) can be found: 1

v P = -ylr:L:=07( C:::::o=+==:::C d:=;('="'V D:=C~)=;=/ d=:::::) (4.5)

where Lo and Co are the inductance and capacitance of the unloaded transmission

line per unit length respectively.

4.3 An NLTL-Based Variable Delay Line

t.~------------------------~ ......... _ ................ - ....... 110-

G W

Fig. 4.4 The NLTL Variable delay Line

Figure 4.4 shows the diagram of the NLTL considered for simulations. The

unloaded transmission line is a coplanar waveguide with characteristic impedance

of 82 n, center conductor width of 406.4 J-l m, and gap spacing of 889 J-l m. The

substrate is a O.64mm thick Rogers R0301O. The coplanar waveguide is loaded

with SMV1232 diodes (which are varactor diodes) at every 2.489 mm. The NLTL

consists of 50 sections. The small-signal input has a magnitude of -80 dBs. Figure

4.5 shows the plot of the delay versus the bias voltage obtained from ADS transient


simulations and calculation of Equation (4.5). It can be observed that, the delay

decreases with increasing bias voltage since the capacitance of the reverse biased

diode reduces as predicted by Equation (4.5).

Delay (no;;) 3

2.5

2

• : T.·ansient Simulations

-- : Equation 4.5

• 1.50~-""1 ----:-2---::------'-----:::---l:--=----!:---:'-9----"lO

Renrse Bills Yoltage (V)

Fig. 4.5 True time delay versus reverse bias voltage

4.4 Conclusion

The nonlinear transmission line can be used as a controllable true time delay line in

the feeding network of phased array antennas to provide means of beam steering and

shaping the radiation pattern. The required delay for this pur pose can be controlled

by changing the bias voltage as depicted in Figure 4.5. Since the nonlinear trans-

mission line is an analog device the precision of the delay line depends only on the

precision of the control De voltage. The possibility to make the NLTL monolith-

ically can make the structure extremely compact and also can increase the Bragg


cutoff frequency, sin ce it allows small unit cell lengths and variable capacitances.

In this manner, extremely wide bandwidths of operation are achievable, since the

bandwidth is only limited by the diode cutoff frequency and the Bragg frequency.

36

Chapter 5

N onlinear Transmission Lines as

Harmonie Generators

5.1 Introduction

High frequency multipliers are widely used in microwave signal generators due to

the lack of efficient and stable sources at microwave frequencies. These microwave

multipliers are driven by lower frequency stable sources to generate low phase-noise

signaIs [18]. Schottky diodes were traditionally used as the basic nonlinear com

ponents in microwave frequency multipliers because of their fast switching capa

bility [18]. Common diode multipliers often have single-diode arrangements and

exhibit reactive input impedances which lead to narrow bandwidth. As weIl, high

conversion efficiencies are not obtainable using single-diode multipliers [20]. Another

approach in implementation of frequency multipliers utilizes nonlinear transmission

lines in either fully distributed or periodically loaded configurations. The average

input impedance of a nonlinear transmission line is almost real (becomes resistive if

5 N onlinear Transmission Lines as Harmonie Generators 37

losses are ignored) and is determined by Caverage as derived in Section 2.1. Also, it

has been shown that high conversion efficiency (low conversion loss) is feasible using

nonlinear transmission lines [20].

The lowpass behavior of periodically loaded nonlinear transmission lines, which

is due to their distributed nature and is determined by the Bragg cutoff frequency

(as shown in Chapter 2), can be used to filter higher order harmonies and to in

crease the conversion efficiency of lower order harmonies [19], which is desirable in

frequency multipliers. If the Bragg eut off frequency is much larger than the input

frequency, the nonlinear transmission line can be also utilized as a comb generator.

Traditionally step recovery diodes (SRDs) were used as comb generators, however,

they are subject to recombination noise as weIl as shot noise and introduce unde

sirable phase noise [35]. On the other hand NLTLs employa different mechanism

for harmonie generation based on the variable capacitance of the diodes and have

much better phase noise characteristics [35].

5.2 Harmonie Balance Analysis of a Nonlinear Transmission

Line

In order to find the equations for different harmonies generated in a nonlinear trans

mission line, harmonie balance analysis can be performed. Harmonie balance is a

frequency domain method, in which aIl steady state waveforms are included in cir

cuit solution in a generalized Fourier series format. Often numerical techniques

are used to solve the resultant nonlinear equations in order to find the vector of

Fourier series expansion coefficients [36J. The nonlinear interconnect considered in

the following analytical derivations is a periodically-ioaded transmission line. The


periodic loads are varactor diodes and the transmission line is considered lossless.

Figure 5.1 illustrates the lumped component model for the nth stage of the dis-

crete NLTL. Each section comprises a series inductance L and a shunt capacitance

C. Q(V(x)) represents the total charge stored by the equivalent capacitance per

section in the transmission line model and the nonlinear capacitance of the diode.

Writing Kirchoff's voltage law results in the following partial differential equations:

V(x-d) U2

Unit Ce)) r------------------Il .. I~ ..

U2 V(x) L/2

• • • • • • • • • • • ________ =r ________ •

Fig. 5.1 nth stage of an NLTL

dh L- = V(x - d) - V(x)

dt

dh Lili = V(x) - V(x + d)

also writing Kirchoff's current law at no de n results in:

L/2 V(x+d)

--o ....... (V(x+d)

(5.1)

(5.2)

(5.3)


combining (5.1), (5.2), and (5.3) leads to the following partial differential equation:

L d2~~X) = V(x - d) + V(x + d) - 2V(x) (5.4)

also by writing the Taylor expansion of V(x) and by ignoring terms of orders above

two, equation (5.4) simplifies to:

(5.5)

Q(x,t) is the charge stored in the nth node. It is in general in the form of:

v v

Q(V) = J C(V)dV = J (CjO / VI + ~ )dV + CV, (5.6)

where the first term represents the charge due to the nonlinear capacitance of the

reverse biased diode and the second term is the charge due to the equivalent ca

pacitance of one section of the transmission line. Equation (5.5) can be simplified

to

(5.7)

by dividing the lefthand and righthand si des by d2 • Lo is the per unit length induc-

tance of the transmission line and Qo is the average stored-charge per unit length.

Considering a time harmonie regime and a sinusoidal input of radian frequency w,

V and Qo can be expanded in complex Fourier series, as suggested in [37]:

00

V(x, t) = Vde + L {Vn(x)ejnwt + V';(x)e- jnwt } (5.8) n=l


DO

Qo(x, t) = QOdc + L {Qn(x)e?ru.Jt + Q~(x)e-jru.Jt} (5.9) n=l

Substituting (5.8) and (5.9) into (5.7) results in the following differential equa-

tion:

(5.10)

On the other hand, Qo is a nonlinear funetion of voltage and the relationship

shown in Equation (5.6) ean be expanded in terms of Taylor's series:

Sinee for passive mixers the magnitudes of higher order harmonies are mueh

smaller than the magnitude of the first harmonie, we ean assume that V;, V3 « Vi.

Also, for the eapaeitance profile of a reverse biased diode (Equation (5.6)) it ean

be eonsidered that Q~(Vdc), Q~' (Vdc ) «Q~(Vdc). Substituting (5.8) and (5.9) into

(5.11) and assuming that V;, V3 « Vi and Q~(Vdc),Q~'(Vdc) « Q~(Vdc) , yields

the following approximate equations:

(5.12)

(5.13)

(5.14)

Approximate differential equations for the first and second harmonies can be

5 N onlinear Thansmission Lines as Harmonie Generators 41

found by substituting (5.12) and (5.13) into (5.10), as reported in [37]:

(5.15)

(5.16)

whieh lead to the following closed-form solutions for VI and V2

(5.17)

Il

1'i~ Vl(0)ze-'Y2X e-(2'Yl-'Y2)X - 1

112 = ;1'1 + 1'2 [X(21'1 - 1'2) ] (5.18)

where 1'1 = jwJLOQ~(Vdc) and 1'2 = 2jwJLoQ~(Vdc)' The same method

suggested in [37] is employed in this thesis to derive a closed-form equation for the

third harmonie voltage. By substituting (5.14) into (5.10), the partial differential

equation deseribing the third harmonie voltage waveform is derived:

(5.19)

In the above differential equation,1'3 = 3jwJ LOQ~(Vdc) . Considering the faet

that all losses are ignored, it ean be assumed that 1'1 = 1'2/2 = 1'3/3 . Knowing

the approximate solutions for VI and 112 and eonsidering the boundary eondition

(\13(0)=0), we ean find a propagating wave solution for \13. The amplitude of the

third harmonie is found from [24]:

(5.20)


The conversion efficiency for the first three harmonics is calculated by dividing

the magnitudes of these harmonics ( Equations (5.17), (5.18), and (5.20)) by the

magnitude of the first harmonic.

Note that according to Equation (5.17), the first harmonic propagates at an

frequencies. Therefore, the effect of the Bragg cutoff frequency (which is imposed

by the periodic loading with varactors) is not explicitly included in the formulations.

5.3 Simulation of a Nonlinear Transmission Line as a

Harmonie Generator

In this section the derivations of Section 5.2 for the conversion gain of the first three

harmonics are compared with simulation results. The NLTL structure considered

here is the same structure as that of Section 3.2. The input voltage to this circuit

is a 2. 16Vp-p sinusoid with a De offset of 2V. Therefore, large signal S-parameters

(LSSP) simulations (See the Appendix) are conducted to observe the effect of non

linearity in frequency domain and to determine the conversion gain for the 2nd and

3rd harmonics. The 821 parameters found in this manner for different harmonies are

in fact the conversion gains for these harmonies. Thus, the simulated large-signal

821 parameters are utilized to find the magnitudes of the harmonics.

In Section 5.3.1 the derivations of Section 5.2 are compared with simulation

results for the above NLTL structure without including the filtering effect of the

NLTL. In Section 5.3.2 the derivations of Section 5.2 are modified to include the

lowpass behavior of the NLTL and to predict the magnitudes of the first three

harmonics.


5.3.1 First Estimate of the Harmonie Voltages Without Including the

Filtering Effect of the NLTL

Figures 5.2, 5.3, and 5.4 show the plot of the magnitudes of the first three harmonies

for the above NLTL structure obtained from simulations and derivations of Section

5.2. The horizontal axis in Figures 5.2, 5.3, and 5.4 shows the input frequency

normalized to the Bragg cutoff frequency (which is calculated to be 8 GHz). It can

be seen that the lowpass nature of the NLTL is not included in the derivations and

the magnitude of the error increases as the frequency increases. Thus, for a good

prediction of the magnitude of harmonies in a wider frequency range the lowpass

nature of the NLTL should be considered in derivations of Section 5.2.

1.1 Fi:i~=;;;;;;;;;;;;;""'~:Z======='--' 1.0 • ...... - ... J •• ''1''. 0.9 1 0.8 1. 0.7

: 0.6 \

: l'i 1 0.5 :

1- : Analyncal Solution 1

0.4 ......... : Circuit Simulation 0.3

0.2 0.1 L---'-_-'-----'-_--'--_L-----'-_..i..-----'-_--'----'

o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Fig. 5.2 Magnitude of the 1 st harmonie

5.3.2 Including the Bragg Filtering Effeet of the NLTL

The unit cell of a NLTL (marked in Figure 5.1) is aT-type LC network which is

composed of two series inductors with the value L/2 and the parallel capacitor C.

5 N onlinear Transmission Lines as Harmonie Generators

o.5o.--------,-----..,....---,--------,

0.45

0.40

0.35

0.30

0.25

1~10.20 0.15

0.10

, i ....•.....

\ . '1 i" . ,:: ...... ... , ...... v'

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fig. 5.3 Magnitude of the 2nd harmonie

0.20.-------,------------,-----,

0.18

0.16

0.14

0.12

0.10

1~10.08 0.06

0.04

0.02

0 0

\ L.. ' .-........ ....

. ..... \. .1> •

\1\.. ......... " ••. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

3 finI fBragg

Fig. 5.4 Magnitude of the 3rd harmonie

44


A finite length NLTL formed by cascading a number of these unit cells inherently

exhibits a lowpass filtering characteristic. The cutoff frequency of this lowpass filter

is in another terms, the Bragg frequency of the periodic structure. To predict the

magnitude of the first three harmonics, the transfer function of this lowpass filter is

calculated for a finite length NLTL and is included in calculating Equations (5.17),

(5.18), and (5.20). Figure 5.5 shows the block diagram of the process of predicting

the magnitudes of the first three harmonics by taking into account the lowpass

nature of the NLTL.

r----------------------------, 'LIl 1

~----------------------------~ 1 H( ·m) = an (jmt + ... + a1(jm)+ ao

] bm(jm)m + ... + ~ (jm)+ bo

l i i

i [

VI, Vl, and V3 1. Filtering .1 Final Prediction from the DE

1

! !

Fig. 5.5 Black diagram for the pro cess of predicting the harmonies

Figures 5.6, 5.7, and 5.8 show the plot of the magnitudes of the first three har-

monics for the above NLTL structure obtained from simulations and by modifying

the derivations of Section 5.2 by including the transfer function of the lowpass NLTL.

These plots show that the predictions are valid in a wider frequency range compared

to the predictions obtained without including the filtering (see Figures 5.2, 5.3, and

5 N onlinear Thansmission Lines as Harmonie Generators

5.4).

0.35

0.3

0.25

IVl 10.2

0.15

0.1

Fig. 5.6 Magnitude of the 1 st harmonie

Fig. 5.7 Magnitude of the 2nd harmonie

46

Note that certain approximations were made in predieting the magnitudes of

the first three harmonies shown in Figures 5.2, 5.3, and 5.4. First, it was assumed

that the second and third harmonics are much smaller than the first harmonic. In

Figure 5.3, it can be seen that at fini fBragg = 0.8 the magnitude of the second


........ . ..~ ...... ~

0.0

1 1.1 1.2

Fig. 5.8 Magnitude of the 3rd harmonie

harmonie is about one third of the magnitude of the first harmonie. Furthermore,

the nonlinear and lowpass funetionalities of the NLTL were eonsidered separately

(Figure ??). However, sinee the NLTL is a distributed structure harmonie generation

and filtering are in effeet in a distributed manner and eannot be separated. A more

precise approaeh would be to divide the NLTL into sm aller subseetions and repeat

this analysis for eaeh section as proposed in Chapter 7.

Subsequently, Agilent ADS was used to simulate the NLTL structure with a

single tone 1.6 GHz input voltage of IVp-p and a DC offset of 2V. The voltage

waveforms at the input and output of the NLTL obtained from the simulations

are shown in Figure 5.9. The frequeney speetrum of the output voltage waveform,

shown in Figure 5.9, shows that the first three harmonies are the dominant output

harmonies.

Figures 5.2, 5.3, and 5.4 show that for an input frequeney of 1.6GHz there is a

good agreement between LSSP simulations and predicted harmonie amplitudes. The

method deseribed herein was used to prediet the first three harmonies and to obtain


output input (f-=1.6 GHz)

"f\j\. v~.... •

.=V"v. . '~ f.*,:

S 4 S • 7 • .L .1' ~'tû ... .,'~" ",J"a. -.."ti. ~fGlb) .... _

,~

... . -1. 'a,. ~ JIll .. _<II ~" .'· ... '-IU· .. I. --

Fig. 5.9 Input and output voltage waveforms along the NLTL

the output voltage waveform. Figure 5.10 shows the predieted output by adding

harmonie voltages given in Equations (5.17), (5.18), and (5.20) and simulated output

voltage waveform using Agilent ADS. It ean be seen that the voltage waveforms show

excellent agreement.

3.5,------------i

3.0

2.5

~ut(V) 2.0

1.5

1.0

0.5 CIl ? 0

CIl CIl 0 0 ... N

CIl CIl CIl CIl 0 0 0 0 Co) ~ U. en

Time (os)

-- : Analytical Solutions

-- : CU'cuit Simulation

CIl CIl CIl CIl 0 0 0 ~

...... QI) te 0

Fig. 5.10 Predicted and simulated output voltage waveforms


5.4 Fabrication and Measurement of a Nonlinear

Transmission Line Harmonie Generator

In this section a NLTL harmonic generator is designed for prototyping and experi

mental evaluations. The structure is a 100n coplanar waveguide periodically loaded

by silicon hyperabrupt junction varactor diodes (a special group of varactors whose

reverse bias capacitances are very sensitive to voltage variations [38]) as shown in

Figure 5.11. There are four diodes per stage which result in a total capacitance of

a single diode per stage, however, using one diode per stage would result in a non

symmetric coplanar structure that adds to unwanted disturbances and using two

diodes per stage would increase the average capacitance per stage and thus reduce

the Bragg frequency. The dimensions of the structure as weIl as the capacitance

model are given in Table 5.1.

t.L.....---------------411 ..... __ ....... -411 ..... --.........

G W

Fig. 5.11 NLTL Harmonie Generator

The parasitics of the diode package according to the manufacturer's datasheet

[39] are given as a package capacitance of OF and a package inductance of 0.7nH.


Table 5.1 NLTL Harmonie Generator Parameter Values

Parameter Value

W 9mm G 5mm d 1 mm t 1.575 mm

substrate FR4 number of stages 10

diode model SMV1232

However, when the package is soldered, the copper contacts and solder add more

parasitics at the location of the periodic discontinuities. Therefore, fullwave simu-

lations were conducted to extract the parasitic model for the discontinuity at the

location of the diodes as shown in Figure 5.12. The model shown in Figure 5.13 is

suggested to represent the discontinuity in the structure. The values for the Land

C parameters were found as follows: First, by using Ansoft HFSS the value of C

is obtained by mat ching the impedance of a parallel capacitor with the impedance

extracted from fullwave simulations.

Figure 5.14 shows the magnitude of the C component in the parasitic model

of the discontinuity (Z21) obtained from fullwave simulation. It can be observed

that the impedance profile of the discontinuity is very similar to that of a 0.65 pF

capacitance.

Aiso the inductance of the metal contacts in the discontinuity were found using

the formula for the inductance of a wire of rectangular cross section with sides B

and C, and length l [40]:

[ 21 1 ] L = 0.0021 loge B + C + 2 - loge e (5.21)

~-


Fig. 5.12 The discontinuity in the coplanar structure

Fig. 5.13 Lumped model for the discontinuity

5 N onlinear Transmission Lines as Harmonie Generators

1

0.1 ...... : Fullwave Simulations ... .

-: O.65pF Capacitor

0.01-+---...... --.......... -. ....... ..0..+----....... - ............................. 0.1 1

f(GHz)

Fig. 5.14 Z21 of the discontinuity and its first order approximation

10

52


loge e is a parameter depending on Band C and is found from the lookup tables

given in [40]. AlI the dimensions in Equation (5.21) should be in millimeters and the

resulting value for the inductance is in f.-LH. Using this formula the total effective

inductance of the contacts in the discontinuity was found to be 0.7nH per stage

which cornes in series with the parasitic inductance of the diode package.

The picture of the fabricated NLTL is shown in Figure 5.15. Measurements were

performed using an Anritsu MG3696B signal generator and an Anritsu MS2665C

spectrum analyzer. The measurement setup is shown in Figure 5.16. The input is

a 5Vp-p sinusoid with a DC offset of 2.5V. Figures 5.17, 5.18, and 5.19 show the

simulation and measurement results for the magnitudes of the first, second, and

third harmonics at the output of the NLTL, respectively.

5.5 Conclusions

NLTL frequency multipliers have the advantage of provide wider bandwidth and

lower conversion loss compared to single-stage multipliers. In this chapter, the dif

ferential equations for the first three harmonic voltages in an NLTL were derived

in the time harmonic regime. In general, these differential equations are a set of

dependent equations. Thus, finding descriptive formulas for different harmonics re

quires solving a number of dependent differential equations. However, a simplifying

approximation can be made by assuming that the magnitudes of the higher order

harmonics are much sm aller than those of lower order harmonics and also by ignor

ing harmonies of orders four and above. By using this assumption the differential

equation for the first harmonic is approximated to be independent of the other har

monics and the equation for the first harmonic is found. By knowing the solution


Fig. 5.15 The fabrieated NLTL harmonie generator


Fig. 5.16 The measurement setup


20

10

o Amplitude

(dBm) -10

-20

-30

-40 o 0.5

: Measurement

: Simulation

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

f(GHz)

Fig. 5.17 Magnitude of the first harmonie


20

10

o Amplitude

(dBm) -10

-20

-30

-40 o

... -, ,.: ., : ... ,...... .'. ,,' ,'4> .... .... ... :.

\ ~ • ~ • • .... • • • • • • • • ..... »

• • • • • • • • • •

: Measurement

: Simulation

~~ ...... ' ". : ... . • • · .. : • : '. :: ...•...

• 1 .. .' 1 ···1·····

1 •

1

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

f(GHz)

Fig. 5.18 Magnitude of the second harmonie


20

10

o Amplitude

(dBm) -10

-20

-30

-40 o 0.5

: Me~lsurement

...... : Simulation

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

f(GHz)

Fig. 5.19 Magnitude of the third harmonie


for the first harmonie and by eonsidering that the approximate differential equation

for the second harmonie is a funetion of the first and the second harmonies, the

solution for the second harmonie is found. In the same manner by plugging in the

solutions for the first and second harmonies in the differential equation of the third

harmonie, the solution for the third harmonies is found. The lowpass nature of the

NLTL is not included in these derivations and ean be eounted for by filtering the

solutions with the transfer function of the Le ladder representing the NLTL. An

NLTL harmonic generator was also fabricated. The measurement results show good

agreement with simulation results in identifying the Bragg eutoff frequeney and the

frequencies where the conversion gains for the second and third harmonics peak.

60

Chapter 6

Distributed Filtering of U nwanted

Signais in N onlinear Transmission

Lines

6.1 Introduction

Many applications of NLTLs require suppression of certain unwanted signaIs. For

example one of the main applications of high frequency harmonic generators is in

designing high frequency sources. In this application, suppression of the unwanted

harmonics is an important factor in improving the conversion efficiency of the desired

harmonics [41]. Also in mixer circuits, filtering of the image signal is extremely

important in heterodyne receivers, since after the downconversion of the input signal

the unwanted image signal and the RF signal both lie at the IF frequency [42]. The

objective of this chapter is to introduce a novel method for implementing these filters

in NLTL harmonic generators and mixers. The challenging task in accomplishment

6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 61

of this goal is to ensure that the dispersion characteristics is not altered across the

rest of the passband. The filtering method proposed here uses tank circuits that

are periodically inserted in the signal path of the transmission line. In Section 6.2,

the Floquet analysis is applied to study the effect of ad ding these tank circuits on

the performance of the NLTL. Four sample applications of distributed filtering are

reported in Sections 6.3, 6.4, 6.5, and 6.6.

6.2 Introducing Distributed Filtering in the Nonlinear

Transmission Line

In order to remove the unwanted signaIs in NLTL applications the use of distributed

filtering is proposed. The reasons for implementing the filter in a distributed manner

are the following added advantages: Firstly, many application of NLTLs may include

a number of output points tapped out at different stages along the NLTL (e.g. a

feed network of an antenna array). Secondly, at microwave frequencies lumped

components cannot be used in filter design and the dimensions of microstrip (or

stripline) filters become comparable to the wavelength. Therefore, providing a means

of more compact filter design would save a lot of precious chip and board's area.

Finally, the design process is very simple and straight forward. Figure 6.1 shows

the nonlinear transmission li ne with the periodic loadings and its equivalent lumped

circuit model. Cl and L 1 are the equivalent lumped capacitance and inductance

of the transmission line section with length d. C2 and L2 are the capacitance and

inductance of the tank circuit.

In order to apply Floquet analysis to this periodic structure, the ABCD matrix

6 Distributed Filtering of Unwanted Signals in Nonlinear Transmission Lines 62

unit cell (a)

2C2 12C2 Cl 1

1 1 1 _ 1 L... ____ ..:" ____ J

(b)

Fig. 6.1 (a) The NLTL with periodic filtering (b) Equivalent lumpedelement model of the NLTL with periodic filtering.

of the unit cell should be derived:

where

is the ABCD matrix of the tank circuit and

. L jWLrcav] JW 1 - 4

1 - w2L!Cav 2

(6.1)

(6.2)

(6.3)

is the ABCD matrix of the lumped model for the NLTL unit cell. Applying Floquet

Theorem to the boundaries of the unit cell represented by the ABCD matrix of


Equation (6.1) , the dispersion equation is found as foUows:

A D L C 2L2Cav

h( d) - + - 1 _ 2~ _ W 2 cos 'Y - 2 - W 2 1 - L2C2W2

------------------------------------. ..•... --_ .. _-~ .--

..-" ;'

;" ;"

/",,,

" .' ,"

"", ----------r------:..::-;;..--;:;.;-;;.;-=-_-----;

,/

"",,/,'

,./

-----: Witbout tank circuit

--: Witb tank circuit

Fig. 6.2 Dispersion diagram of the NLTL after adding the tank circuits

(6.4)

Since aU losses are ignored in this analysis, 'Y = j f3. Plotting the dispersion

diagram shows that (see Figure 6.2) the tank circuits introduces a bandgap in the

dispersion diagram of the NLTL. In this bandgap the propagation constant assumes

negative real values. The boundary frequencies of the stopbands can be identi

fied from the dispersion equation. The phase condition for the occurrence of the

stopband for this lowpass periodic structure are given as foUows:

1. f3d = 7r:

(6.5)

Solutions: Wl and W2


2. f3d = 0:

(6.6)

(6.7)

Note that because of the loading effect of the tank circuit, the Bragg frequency

is shifted to a new frequency. If the values of the components in the equivalent Le

model for the tank circuit satisfy

and (6.8)

W2 and W3 merge and the bandwidth of the bandgap is reduced as shown in Figure

6.3. Under these conditions the tank circuit no longer loads the NLTL except at the

resonance frequency of the tank circuit.

6.3 Example 1: Filtering of the 3rd Harmonie in an NLTL

Frequeney Doubler

In applying distributed filtering to a frequency multiplier circuit certain consider

ations should be made since the approximate differential equations for the second

and third harmonics (Equations 5.16 and 5.19) indicate that the lower order har

monics contribute to the magnitude of the higher harmonics. Thus, the lower order

harmonics cannot be filtered in a distributed manner in order to get a clean higher

order harmonic. For example, by filtering the second harmonic the magnitude of


2 -----------------------------------_ .....

1

-~-- -: Witbout tank circuit

--: With tllnk cirmit

Fig. 6.3 Dispersion diagram of the NLTL after adding the tank circuits and satisfying conditions in Equation (6.8)

the third harmonie would be reduced as weIl.

The NLTL structure studied here is the same structure as the one described in

Section 3.2. The input is a 2Vp-p sinusoid with a De offset of 2V. Agilent ADS

was used to find the transmission coefficient for the first three harmonics as shown

in Figure 6.4.

The objective is to filter the 3r d harmonic at a frequency where it is maximum

(i.e. at an input frequency of 2.4GHz as marked in Figure 6.4 (a)) to get a cleaner

2nd harmonic. Thus, the tank circuit was designed to resonate at 7.2 GHz and to

satisfy the conditions in (6.8).

{

Ltank = lOpH f = 7.2GHz

Ctank = 48.9pF (6.9)

Figure 6.4 (b) shows the transmission coefficients for the first three harmonics

after including the tank circuits. It can be seen that the isolation between the second


lO-r--------------_r---r--~--_r--~

O~------~--------------~------~ -10

-4

-50

-60

1 1.5 2 2.5 3 3.5 4 frequeD~y (GHz)

(a)

frequen~y (GHz)

(h)

4.5 1

5

Fig. 6.4 Transmission coefficients for the first three harmonies before adding the tank circuits


and third harmonics has been improved by about 30 dBs.

6.4 Example 2: Filtering of the Image Signal in an NLTL

Mixer

Efficient rejection of the image signal is a critical issue in the performance of elec

tronic mixers. The NLTL structure of Section 6.3 is also considered here. Figure 6.5

shows a block diagram of the circuit that was simulated using Agilent ADS's LSSP

simulation engine. The LO frequency is 5GHz and has a power of lOdBm, while the

IF frequency is 400 MHZ. The RF signal is at 5.4 GHz with a power of -30 dBm.

Therefore, the image signal is at 4.6 GHz (fimage = ho - fIF). Figure 6.6 (a) is the

plot of the down conversion loss versus the LO frequency.

Power Combiner LOoutput

,.....-~N~L~TL!'""--......

Fig. 6.5 Block diagram of the simulation test bench used in the analysis of the mixer example

----

6 Distributed Filtering of U nwanted SignaIs in N onlinear Transmission Lines 68

--= "C --l'Il l'Il Q ~

= Q ..... l'Il .. ~ Q

U = ~ ~

-. = "C --l'Il l'Il Q ~

= Q ..... l'Il .. ~ >-= Q

U = ~ ~

80

60

40

20

0 0.5

80

60

40

20

0 0.5

1.5

1.5

2.5 3.5 4.5 5.5 LO Frequency (GHz)

(a)

2~ 3~ 4~ 5~ LO Frequency(GHz)

(b)

6.5

6.5

Fig.6.6 Down conversion 10ss: (a) before adding the tank circuits (b) after adding the tank circuits


The tank circuit was designed to filter the incoming signaIs at 4.6 GHz.

{

Ltank = 50pH f = 4.6GHz

Ctank = 23.94pF (6.10)

Figure 6.6 (b) shows that the contribution of the image signal to the output

power is reduced by about 60 dBs after including the tank circuits. Note that the

down conversion loss is filtered at two frequencies; i.e. when RF frequency is 4.6

GHz and when LO frequency is 4.6 GHz.

6.5 Example 3: Implementing the NLTL Harmonie

Generator with Distributed Filtering

6.5.1 The Prototype NLTL Circuit

An NLTL is designed to operate as a frequency doubler and is shown in Figure 6.7.

The structure is a coplanar waveguide periodically loaded by reverse biased diodes

on an FR4 substrate. The parameters of the coplanar waveguide as well as the diode

model are reported in table 6.1.

Table 6.1 Coplanar NLTL Parameter Values

Parameter Value

W 9mm

G 5mm d 10mm t 1.575 mm

substrate FR4 number of stages 10

diode model SMV1232

6 Distributed Filtering of U nwanted SignaIs in N onlinear Transmission Lines 70

t~~------------~~----

Fig. 6.7 NLTL frequency doubler

Ansoft HFSS was used to conduct fullwave simulations to model the coplanar

waveguide sections. The S-parameters obtained from fullwave simulations were then

imported into Agilent ADS to simulate the complete NLTL circuit. Figure 6.8 shows

the transmission coefficients for the first three harmonics obtained from Large Signal

S-Parameter simulations. The input voltage is a 5Vp-p sinusoid with a De offset

of 2.5V, which results in an input impedance of 50n (due to Equation (2.17)). The

objective is to increase the isolation between the second and third harmonics at an

input frequency of 1GHz where the third harmonic is maximum (see Figure 6.8).

6.5.2 Implementing the Distributed Filtering

Split Ring Resonators are small resonant elements with a high quality factor at mi-

crowave frequencies [43] which were originally proposed by Pendry [44]. The SRR

proposed by Pendry is shown in Figure 6.9. Although originally split ring resonators

were proposed to construct left-handed materials with negative refractive indexes,

they are becoming very popular in microwave filter design due to many advantages.

First of an in contrast to many conventional resonator components in microwave fil-


-:S21for the lst

harmonic

0-,. ... ;;;;;;;;;;::::::-----:-----;-----; ----: Sllfor the ~~harmOnic _.- : S21for the 3 harmonie

-10

-== -20 "0 -~ 001'1 -30

-40

-50-h~~~~~~~~~~~~~~~

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f (GHz)

Fig. 6.8 Transmission coefficients for the first three harmonies

ters (half wavelength short circuit stubs, and quarter wavelength transformers) they

have subwavelength thus resulting in a miniaturized structure. As well, because

of their planar geometry they are very easy to fabricate using PCB and MMIC

technologies. Finally, for each SRR, there is a dual counterpart or so-called Com

plementary Split Ring Resonator (CSRR) [45] which is made by etching the same

SRR shape but in a slot pattern on the conductor surface. The equivalent circuit

models for few SRRs and CSRRs in microstrip and coplanar waveguide structures

are given in [45]. Approximate formulas are also provided in [45] that can be used

for synthesis of the resonators for a desired frequency. Fine-tunings can be done

through multiple full-wave simulations. Because of these advantages and the fact

that NLTL is usually implemented by a coplanar waveguide geometry, CSRRs were

chosen to configure the tank circuits.

A complementary split ring resonator, which is composed of a single slot ring

6 Distributed Filtering of Unwanted SignaIs in Nonlinear Thansmission Lines 72

Fig. 6.9 The SRR proposed by Pendry

(shown in Figure 6.10) was designed to be incorporated periodically in the signal

line of the coplanar waveguide structure. Note that this is prior to adding the CSRR

in the NLTL.

Fullwave simulations were conducted using Ansoft HFSS to find the S-parameters

of this new unit cell. Figure 6.11 shows the transmission coefficient of the unit cell.

It can be seen that the structure resonates at 3GHz.

6.5.3 Simulation Results After Induding the CSRRs in the NLTL

The S-parameters obtained from fullwave simulations were imported in Agilent ADS

where harmonic balance simulation was conducted to analyze the NLTL. Figure

6.12 shows the transmission coefficients for the first three harmonics after loading

the NLTL with the CSRRs. It can be seen that the isolation between the second

and third harmonics at an input frequency of 830MHz has been improved by about

16dBs. Note that this input frequency is almost one third of the resonance frequency

of the CSRR that was designed to filter the third harmonic at 3G Hz.

6 Distributed Filtering of Unwanted Signals in Nonlinear Transmission Lines 73

o -2

-8

-10

-12

.

.

1

o

0.2 mm

Fig.6.10 The CSRR

,r ,/ V

-,

j

1 2 4 6 8 10

f(GHz)

Fig. 6.11 Transmission coefficient of the new transmission Hne structure


--== "0 --... ~ 00

-: Szlor the lst harmonie

0 ----: SZJforthe t'dhannonie ..... ;;;::::-----------; --- ~ Sllfor the 3rdhanDonie

-10

-20

-4

-50~~~~~~~~~~~~~~~~

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f (GHz)

Fig. 6.12 Transmission coefficient for the first three harmonies in the NLTL structure loaded with CSRRs

Aiso note that the indueed stop bands due to the CSRRs ean be reeognized in

the plot of the transmission coefficient of aU three harmonies.

6.5.4 Sensitivity Analysis of the NLTL

Harmonie generation is the most efficient, when Bragg frequeney prevents power

spreading to the speetrum above the desired harmonie [18]. In other words, more

power is injeeted into the desirable harmonies if other harmonies are suppresses.

This is why the conversion gain for the nth harmonie peaks slightly below the Bragg

eut off frequeney where an the higher order harmonies (n + 1 and higher) are sup

pressed by the NLTL yet the nth harmonie has not experieneed the lowpass behavior

of the NLTL. Thus, the Bragg eut off frequeney determines the frequeney at which

the response of different harmonies peak. However, the Bragg eut off frequeney is


dependent on the average capacitance of the unit cell (as derived in Section 2.2) and

is sensitive to voltage fluctuations. Figure 6.13 shows how the frequency of the peak

of the conversion gain for the 2nd and 3rd harmonics changes with input voltage vari

ations in the unloaded NLTL(for three different input voltage (indicated by Vreverse

in Figure 6.13 ranges: OV-2.5V, OV-5V, and OV-7.5V, the input is a sinusoide).

Conducted simulations for different NLTL structures loaded with tank circuits

demonstrate that if the induced stop band by this added filter is wide enough and

close to fBragg of NLTL, then the conversion gains for different harmonics peak

right below the lower cutoff frequency of the filter stop band (W2 in Equation (6.5))

instead of fBragg. Figure 6.14 shows how the frequency of the peak of the conversion

gain for the 2nd and 3rd harmonics changes with input voltage variations in the

NLTL loaded with CSRRs (the plots are for three different input voltage ranges:

OV-2.5V, OV-5V, and OV-7.5V). It can be seen that the frequency of the peak is

almost unchanged with input voltage variations. As a result the sensitivity of the

harmonic generator to input voltage variations is reduced by adding CSRRs, which

proves another important advantage of using the proposed method of distributed

filtering.

6.6 ProposaI of a Distributed Filter for the NLTL

Frequency Multiplier Fabricated in Section 5.4

In this section, a complementary split ring resonator is proposed for including in

the NLTL harmonic generator of Section 5.4, in order to increase the isolation be-

tween the second and third harmonies at the input frequency of 500 MHz. Figure

6.15 shows the layout of the proposed complementary split ring resonator, while its

6 Distributed Fiitering of Unwanted SignaIs in Nonlinear Transmission Lines 76

,.-,

= "0 --""" M 00

,-... = -c --""" M 00

0

-10

-20

-30

-40

-50

0

-10

-20

~30

-40

l, ! 1

, 1

: 1 T_; io ! -r-·--:iiliA~tfIF~···t-··········· T·--·H:I*RaS~ng--r--·· __ ··

! : 1 1

0.0 0.5 1.0

.yr~Y~r~~-·-···l-_·-··_· : !

. O-l.~ ! o-sv b-7.5V

1.5 2.0 2.5 3.0 3.5 4.0 f (GHz)

(a)

Inc~easi~g 1 : ....... V·············_-t--···_·_+·---···~··

jrevers, 1 1

.. j·.··--·--t----+--··-1"-· .. ··.-.··· O-l.SV 1 o-sv l,.

07.SV : .. _ .... _ ............. - ._ .. _._-_...... -_. _._. __ ...... _ ... - .. _ .. _._._--_ ... _- - --!.

-50~~~~~~~~~~~~~~~~~~

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f (GHz)

(b)

Fig. 6.13 Effect of input voltage variations on the conversion efficiencies for the unloaded NLTL: (a) 8 21 for the second harmonie (b)821 for the third harmonie


O~-----'-----c----~-'----r-----~----. i i 1 1

.•....................... ·+·4 .--I~cr-eas~g--10

-. IVrevers~ = -20-Y~--~-----+~·~~~~~,4-----~----~ "C -.-!

OON -30~------~~------~-------~'H

-40

0.0 0.5 1.0 1.5 f (GHz)

(a)

2.0 2.5 3.0

O~------~----~----~----~------------~

_I·-+---.---.--~-A,·---·---r-------+---·-·-·--··+-··---···----·--~-·-·-·----I

-40

-50-r~~~~~~~~rT~~~~~~~~

0.0 0.5 1.0 1.5 2.0 2.5 3.0

f (GHz)

(b)

Fig. 6.14 Effect of input voltage variations on the conversion efficienci es in the NLTL loaded with CSRRs: (a) 8 21 for the second harmonic (b) 8 21 for the third harmonic


dimensions are given in Table 6.2. The structure contains three slot rings to achieve

the desired low frequency resonance without without dedicating a large area for its

implementation.

~.4'----------~----------~.~: ~~ ~ !.-G--t--

t..d

Fig. 6.15 Layout of the CSRR

Table 6.2 CSRR Dimensions

Parameter Value

W 9mm G 5mm d 0.2 mm L 10 mm

LI 9.2 mm L 2 8.4 mm

L3 7.6 mm W I 6.6 mm t 1.575 mm

substrate FR4

l


Figure 6.16 shows the fullwave simulation results for the unit cell of the CPW

with the CSRR. Agilent ADS's Momentum was used for these simulations. It can

be seen that the first resonance of the structure occurs at 1.715 GHz.

-= "'0 '-" ....

M 00

0

-2

-4

-6

-8

-10

-12 0 2 4 6 8 10

f(GHz)

Fig. 6.16 The transmission coefficient of the CPW unit cell loaded with the CSRR of Figure 6.15

The S-parameter files of the CPW NLTL frequency multiplier with the new

CSRR design was ported to Agilent ADS. Figure 6.17 shows the magnitudes of the

conversion gains for the first three harmonies after including the CSRR of Figure

6.15. It can be seen that the isolation between the second and third harmonies at

an input frequency of 500MHz (where the third harmonie used to be maximum) has

been improved by about 7 dBs after including the distributed filtering.

6.7 Conclusions

In this chapter, a novel method based on distributed filtering is proposed for the

suppression of the unwanted signaIs for various NLTL applications. The distributed


0

-10 --= "C -20 '-' .... I.f.r~

-30

0.5 1.0 1.5 2.0 f (GHz)

Fig. 6.17 Transmission coefficient for the first three harmonies in the NLTL structure loaded with CSRRs

filtering is implemented by loading the NLTL circuit with Le resonator circuits.

Floquet analysis is used to study the effect of adding the distributed filtering on

the performance of the NLTL. These studies show that a bandgap is introduced in

the dispersion diagram of the NLTL that can be designed to be very narrowband

or wideband. The application of a narrowband stopband is in filtering of the third

harmonie and the image signal in NLTL frequency doublers and mixers, respectively.

Implementation of the tank circuits by using eSRRs is proposed and the design

process is presented in a couple of examples. eSRRs are used as distributed filtering

in a epw NLTL to increase the isolation from the third harmonie in a frequency

doubler and also to reduce the sensitivity of the doubler to variations of the input

voltage. Aiso a eSRR is proposed for the harmonie generator of Section 5.4 to

increase the isolation between the second and third harmonies at the input frequency

of 500 MHz.

81

Chapter 7

Conclusions

Nonlinear transmission lines' wide bandwidth and their ability to generate pulses

with few picosecond transitions makes them excellent candidates for millimeter wave

range frequency conversion and ultra fast pulse shaping. In this thesis, nonlinear

transmission lines and their various high frequency applications have been studied.

Applications such as edge sharpening, frequency conversion, and true time delay

lines are investigated by using analytical approaches and circuit simulations. Aiso

an NLTL circuit was fabricated and evaluated experimentally. However, the object

of this study was not designing the best possible mixers, frequency doublers, edge

sharpeners, and etc. but inspection of an alternative design approach which proves

to be very advantageous. In Chapter 3, a risetime compression of about 50% was

achieved for an input with a risetime of 250ps. The general guideline for designing

NLTL edge sharpeners using discrete diodes was also presented. It was also pointed

out that the efficiency of edge sharpener can be improved wh en the NLTL is fab

ricated monolithically. In this manner wider operation bandwidth can be achieved

and sharper pulses can be generated.

7 Conclusions 82

An NLTL-based variable delay line that takes advantage of the voltage dependent

phase velocity in a NLTL was presented in Chapter 4 and a 50% change in delay

over a 5V DC bias range was reported. In Chapter 5, a harmonic balance analysis

of periodically loaded NLTLs was presented and an approximate formula for the

conversion gain for third harmonic was derived for the first time. Furthermore, a

harmonic generator with a second harmonic conversion efficiency of -IOdE at 700

MHz was fabricated and tested.

In Chapter 6, a novel method based on distributed filtering was proposed for the

suppression of the unwanted signaIs in NLTL frequency multipliers and mixers. The

application of this method in filtering of the third harmonic in an NLTL frequency

doubler was demonstrated. This method was also applied to an NLTL mixer to

filter the image signal. CSRRs were suggested for implementing the distributed

filtering method. A compact CSRR was designed to increase the isolation between

the second and third harmonics at an input frequency of 500M H z for the harmonic

generator that was fabricated in Chapter 5.

The main advantage of these NLTL structures is that they can have very high

bandwidths and operating frequencies since they can be fabricated monolithically

which also enables very compact NLTL structures. They also save power since they

are passive structures. Furthermore, due to their distributed nature they can be

designed to provide a wideband matching thus offering power efficient performance.

7.1 Future Works

In Section 5.3.2, the validity range for the equations predicting the magnitudes of

the harmonics in an NLTL vas improved by including the lowpass effect of the

7 Conclusions 83

NLTL. In this method, the harmonic magnitudes, predicted based on the deriva

tions of Section 5.2, are filtered by the transfer function of the NLTL. In other words,

the harmonic generation and lowpass filtering tasks of the NLTL were separated.

However, because of the distributed nature of NLTLs the filtering and harmonic

solution should be incorporated in a distributed manner and separating these func

tionalities introduces errors. One way to improve the predictions would be to divide

the NLTL into sm aller sections (each composed of a few unit cells) and to perform

the harmonic analysis and filtering for each section.

A CSRR was proposed in Section 6.6 to improve the performance of the NLTL

frequency doubler of Section 5.4. Fabricating this new NLTL frequency doubler and

doing the measurements would be the next step.

It is also of great importance to investigate modern electronic systems to see

if they can benefit from the advantages of these potentially compact, wideband,

low-cost, and low-power NLTL structures. Also comparison of NLTLs with their

non-distributed counterparts is an essential task in proving their competency.

Appendix A

Large-Signal S-Parameter (LSSP)

Simulations

84

Scattering parameters are widely used to characterize microwave networks. One

drawback of S-parameters is that they apply to a linear and small-signal analysis [46}.

The concept of Large-Signal S-Parameters is very useful in representing nonlinear

networks [47}. Unlike small-signal S-parameters, for which small-signal response of a

linear approximation of the circuit is derived, large-signal S-parameters use harmonic

balance simulations of a nonlinear network for its characterization [48}. Because of

the nonlinearity ofthe network, large-signal S-parameters are power dependent [46}.

Like small-signal S-parameters, large-signal S-parameters are defined as the ra

tios of the reflected and incident waves at different ports when other ports are

properly terminated. However, there are different methods to find large-signal S

parameters. In a two port network where PartI is the active port (the input port)

with a signal source of power Po at frequency f and Port2 is the passive port (the

output port), 5 11 and 5 21 are found by using harmonic balance techniques and by

2006/10/04

A Large-Signal S-Parameter (LSSP) Simulations 85

terminating Port2 with the complex conjugate of its reference impedance and by

finding the following wave ratios:

S11 = ~If

S21 = ~If (A.I)

According to [48], S22 and S12 are found by terminating PartI with the complex

conjugate of its reference impedance and driving Port2 with a source of power P'

(which is equal to the power delivered to the passive output port in the nonlinear

circuit) and finding:

S22 = ~If

S12 = ~If (A.2)

However, since in the absence of the input power the nonlinear network would

work at a different regime, S22 and S12 found from this method may be affected

by large errors [47]. Other approaches are suggested in [46], [47], and [49] to count

for this problem. For example, PartI may be driven with a source of power Po at

frequency f while Port2 is driven with a source of power P' at frequency f + of

(where of «f). S11 and S21 are computed from (A.I), while S22 and S12 are

found from [47]:

s -~I ~~I 22 - a2 f+l1f - a2 f (A.3)

S - hl e;,:.hl 12 - a2 f+l1f - a2 f

Agilent ADS used the method suggested in [48].

86

References

[1] Y. Yamashita, A. lndoh, K. Shinohara, K. Hikosaka, T. Matsui, S. Hiyamizu, and T. Mimura, "Pseudomorphic l nO.52Alo.48As / l nO.7GaO.3As HEMT's with ultrahigh Ir of 562 GHz," IEEE Electron Devices Lett., vol. 23, pp. 573-575, October 2002.

[2] S. H. Pepper and K. Schoen, "NLTLs push sampler products past 100 GHz," Microwaves and RF Online Journal, October 2005.

[3] C. J. Madden, Picosecond and Subpicosecond Electrical Shock- Wave Generation on a Gallium Arsenide Nonlinear Transmission Line. PhD thesis, Stanford University, August 1990.

[4] M. G. Case, Nonlinear Transmission Lines for Picosecond Pulse, Impulse and Millimeter- Wave Harmonie Generation. PhD thesis, University of California Santa Barbara, July 1993.

[5] E. Afshari and A. Hajimiri, "Non-linear transmission lines for pulse shaping in silicon," IEEE Custom Integrated Circuits Conference, pp. 91-94, September 2003.

[6] W. Zhang, Nonlinear Transmission Line Technology and Applications. PhD thesis, University of California Los Angeles, 1996.

[7] M. Birk, D. Behammer, and H. Schumacher, "All-silicon nonlinear transmission line integrated into a Si/SiGe heterostructure bipolar transistor process," Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems, pp. 75-78, April 2000.

[8] U. Bhattacharya, S. Allen, and M. Rodwell, "DC-725 GHz sampling circuits and subpicosecond nonlinear transmission lines using elevated coplanar waveguide," IEEE Microwave and Guided Wave Letters, vol. 5, pp. 50-52, February 1995.

[9] M. Birk, H. Kibbel, C. Warns, A. Trasser, and H. Schumacher, "Efficient transient compression using an all-silicon nonlinear transmission line," IEEE Microwave and Guided Wave Letters, vol. 8, pp. 196-198, May 1998.

References 87

[10] S. Ibuka, K. Abe, T. Miyazawa, A. Ishii, and S. Ishii, "Fast high-voltage pulse generator with nonlinear transmission line for high repetiion rate operation," IEEE Trans. on Plasma Science, vol. 25, pp. 266-271, April 1997.

[11] M. Falah and D. Linton, "High data rate pulse regeneration using non-linear transmission line technology (NLTL)," 6th IEEE High Frequency Postgraduate Student Colloquium, pp. 136-141, September 2001.

[12] M. Li, K. Krishnamurthi, and R. G. Harrison, "A fully distributed heterostructure-barrier varactor nonlinear transmission-line frequency multiplier and pulse sharpener," IEEE Trans. Microwave Theory and Tech., vol. 46, pp. 2295-2301, December 1998.

[13] J. Duchamp, P. Ferrari, M. Fernandez, A. Jrad, X. Melique, J. Tao, A. Arscott, D. Lippens, and R. Harrison, "Comparison of fully distributed and periodically loaded nonlinear transmission lines," IEEE Trans. Microwave Theory and Tech., vol. 51, pp. 1105-1116, April 2003.

[14] D. Salameh and D. Linton, "Microstrip GaAs nonlinear transmission line (NLTL) harmonie and pulse generators," IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1118-1121, July 1999.

[15] D. W. V. D. Weide, "A YIG-tuned nonlinear transmission line multiplier," IEEE MTT-S Int. Digest, vol. 2, pp. 557-560, June 1993.

[16] R. Marsland, M. Shakour, and D. Bloom, "Millimeter-wave second harmonie generation on a nonlinear transmission line," Electronics Letters, vol. 26, pp. 1235-1237, August 1990.

[17] M. Fernandez, E. Delos, X. Melique, S. Arscott, and D. Lippens, "Monolithic coplanar transmission lines loaded by heterostructure barrier varactors for a 60 GHz tripler," IEEE Microwave and Wireless Components Letters, vol. 11, pp. 498-500, December 2001.

[18] E. Carman, M. Case, M. Kamegawa, R. Yu, K. Giboney, and M. J. W. Rodweil, "V-band and W-band broad-band monolithic distributed frequency multipliers," IEEE Microwave and Guided Wave Letters, vol. 2, pp. 253-254, June 1992.

[19] N. Paravatsu, Novel Frequency Multiplier Architectures for Millimeter Wave Applications. PhD thesis, University of Virginia, January 2005.

References 88

[20] E. Carman, K. Giboney, M. Case, M. Kamegawa, K. A. R. Yu, M. J. Rodwell, and J. Franklin, "23-39 GHz distributed harmonie generation on a soliton nonlinear transmission line," IEEE Microwave and Guided Wave Letters, vol. 1, pp. 28-31, February 1991.

[21] W. M. Zhang, R. P. Hsia, C. Liang, G. Song, C. W. Domier, and N. C. Luhmann, "Novellow-Ioss delay line for broadband phased antenna array applications," IEEE Microwave and Guided Wave Letters, vol. 6, pp. 395-397, November 1996.

[22] C.-C. Chang, Microwave and Millimeter Wave Beam SteeringjShaping Phased Antenna Arrays and Related Applications. PhD thesis, University of California Davis, 2003.

[23] N. Paravatsu, Distributed Broadband Frequency Translator and its use in Coherent Microwave Measurement. PhD thesis, University of Delaware, 1998.

[24] K. Payandehjoo and R. Abhari, "A study of signal propagation in nonlinear interconnects," IEEE 14th Topical Meeting on Electrical Performance of Electronie Packaging, pp. 195-198, October 2005.

[25] K. Payandehjoo and R. Abhari, "Distributed filtering of unwanted signaIs in nonlinear transmission line harmonie generators and mixers," 4th International IEEE-NEWCAS Conference, pp. 225-228, June 2006.

[26] D. M. Pozar, Microwave Engineering. New York: John Wiley and Sons, third ed., 2005.

[27] R. Courant and D. Hilbert, Methods of Mathematical Physics. New York: John Wiley and Sons, 1962.

[28] M. Rodwell, Picosecond Electrical Wavefront Generation and Picosecond and Optoelectronic Instrumentation. PhD thesis, Stanford University, 1988.

[29] J. R. Burger, "Modeling solitary waves on nonlinear transmission lines," IEEE Trans. Circuits Systems, vol. 42, pp. 34-36, January 1995.

[30] M. 1. Montrose, EMC and the Printed Circuit Board: Design, Theory, and Layout Made Simple. New York: IEEE Press, 1999.

[31] K. Balasubramanian and V. Rajaravivarma, "Electronic scanning of sonar beam from a phased array of acoustic transducers controlled through programmable digital delay lines," Proceedings of the Thirty-Fourth Southeastern Symposium on System Theory, pp. 366-370, March 2002.

References 89

[32] B. Vidal, J. Corral, and J. Marti, "Optical delay line employing an arrayed waveguide grating in fold-back configuration," Microwave and Wireless Components Letters, vol. 13, pp. 238-240, June 2003.

[33] D. Dolfi, M. Labeyrie, P. Joffre, and J. Huignard, "Liquid crystal microwave phase shifters," Electronics Letters, vol. 29, pp. 926-928, May 1993.

[34] T. Yun and K. Chang, "A low-cost 8 to 26.5 GHz phased array antenna using a piezoelectric transducer controUed phase shifter," IEEE Trans. on Antenna and Propagation, vol. 49, pp. 1290-1298, September 200l.

[35] C. Picosecond Pulse Labs, Boulder, "A new breed of comb generators featuring low phase noise and low input power," Microwaves Journal: MTT-S IMS Show Issue, vol. 49, p. 278, May 2006.

[36] P. J. Rodrigues, Computer-Aided Analysis of Nonlinear Microwave Circuits. Boston: Artech House, 1998.

[37] K. S. Champlin and D. R. Singh, "SmaU signal second harmonie generation by a nonlinear transmission line," IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 351-353, March 1986.

[38] K. Lundien, R. Mattauch, J. Archer, and R. Malik, "Hyperabrupt junction varactor diodes for millimeter-wavelength harmonic generators," IEEE Trans. Microwave Theory Tech., vol. 31, pp. 235-238, February 1983.

[39] www.ortodoxism.rojdatasheetsj AlphalndustrieslncjmXyzrtsr.pdf.

[40] F. W. Grover, Inductance Calculations: Working Formulas and Tables. New York: D. Van Nostrand Company, 1946.

[41] F. Martin, F. Falcone, J. Bonache, T. Lopetegi, M. Laso, and M. SoroUa, "New PBG nonlinear distributed structures: application to the optimization of millimeter wave frequency multipliers," 27th International Conference on Infrared and Millimeter Waves, pp. 59-60, September 2002.

[42] A. Pham, C. Kim, M. Kang, M. Chong, S. Lee, and C. Su, "A novel image rejection CMOS mixer using T-structure filter," 7th International Conference on Solid-State and Integrated Circuits Technology, vol. 2, pp. 1299-1302, October 2004.

[43] F. Martin, F. Falcone, J. Bonache, R. Marques, and M. SoroUa, "Miniaturized coplanar waveguide stop band filters based on multiple tuned split ring resonators," IEEE Microwave and Wireless Components Letters, vol. 13, pp. 511-513, December 2003.

References 90

[44] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech., vol. 47, pp. 2075-2084, November 1999.

[45] J. D. Baena, J. Bonache, F. Martin, R. Marquez, F. Falcone, T. Loptegei, M. A. G. Laso, J. Garcia, 1. Gil, M. F. Portillo, and M. SoroHa, "Equivalentcircuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech., vol. 53, pp. 1451-1461, April 2005.

[46] V. Rizzoli, A. Lipparini, and F. Mastri, "Computation of large-signal Sparameters by harmonie-balance techniques," Electronic Letters, vol. 24, pp. 329-330, March 1988.

[47] G. Xuebang, "Large-signal s-parameters simulation of microwave power circuits," International Conference on Microwave and Millimeter Wave Technology Proceedings, pp. 1061-1063, August 1998.

[48] Agilent-Technologies, "Large-signal S-parameter simulation," September 2004.

[49] S. Mazumder and P. van der Pujie, "Two-signal method for measuring the largesignal S-parameters of transistors," IEEE Trans. Microwave Theory Tech., vol. MTT26, pp. 417-420, June 1978.

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