study of nonlinear transmission lines and their applicationspeople.virginia.edu/~rmw5w/nonlinear...
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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)
2 Analysis of N onlinear Transmission Lines
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)
2 Analysis of N onlinear Transmission Lines
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 Analysis of N onlinear Transmission Lines
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):
2 Analysis of N onlinear Transmission Lines 14
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)
2 Analysis of Nonlinear Transmission Lines 15
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)
2 Analysis of N onlinear Transmission Lines 16
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-
2 Analysis of N onlinear Transmission Lines 17
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)
2 Analysis of N onlinear Transmission Lines 18
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
2 Analysis of N onlinear Transmission Lines 19
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 Analysis of Nonlinear Transmission Lines 20
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.
3 N onlinear Transmission Lines as Edge Sharpeners 22
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
3 N onlinear Transmission Lines as Edge Sharpeners 24
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:
3 N onlinear Transmission Lines as Edge Sharpeners 25
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).
3 N onlinear Transmission Lines as Edge Sharpeners 27
• 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
3 N onlinear Transmission Lines as Edge Sharpeners 28
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
4 N onlinear Transmission Lines as N onlinear Delay Lines 30
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
4 N onlinear Transmission Lines as N onlinear Delay Lines 31
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
4 N onlinear Transmission Lines as N onlinear Delay Lines 33
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
4 N onlinear Transmission Lines as N onlinear Delay Lines 34
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
4 N onlinear Transmission Lines as N onlinear Delay Lines 35
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
5 N onlinear Transmission Lines as Harmonie Generators 38
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)
5 N onlinear Transmission Lines as Harmonie Generators 39
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
5 N onlinear Transmission Lines as Harmonie Generators 40
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)
5 N onlinear Transmission Lines as Harmonie Generators 42
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 N onlinear Transmission Lines as Harmonie Generators 43
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
5 N onlinear Transmission Lines as Harmonie Generators 45
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
5 N onlinear Transmission Lines as Harmonie Generators 47
........ . ..~ ...... ~
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
5 N onlinear Transmission Lines as Harmonie Generators 48
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 N onlinear Transmission Lines as Harmonie Generators 49
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.
5 N onlinear Transmission Lines as Harmonie Generators 50
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)
~-
5 N onlinear Transmission Lines as Harmonie Generators 51
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
5 N onlinear Transmission Lines as Harmonie Generators 53
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
5 N onlinear Transmission Lines as Harmonie Generators 54
Fig. 5.15 The fabrieated NLTL harmonie generator
5 N onlinear Transmission Lines as Harmonie Generators 55
Fig. 5.16 The measurement setup
5 N onlinear Transmission Lines as Harmonie Generators 56
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
5 N onlinear Transmission Lines as Harmonie Generators 57
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
5 N onlinear Transmission Lines as Harmonie Generators 58
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
5 N onlinear Transmission Lines as Harmonie Generators 59
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 63
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 64
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 65
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 66
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 67
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 69
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-
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 71
-: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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 74
--== "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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 75
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 77
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 78
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 79
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
6 Distributed Filtering of Unwanted SignaIs in Nonlinear Transmission Lines 80
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
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