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Broadband Microwave Negative Group Delay Transmission Line Phase Shifters

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Broadband Microwave Negative Group Delay
Transmission Line Phase Shifters
by
Sinan Keser
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
Copyright © by Sinan Keser 2012
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Abstract
Broadband Microwave Negative Group Delay
Transmission Line Phase Shifters
Sinan Keser
Master of Applied Science
Graduate Department of Electrical and Computer Engineering
2012
The analysis and design of passive broadband negative group delay (NGD) transmission line
phase shifters is presented. By extending the metamaterial transmission line concept to include
loss, a NGD unit cell is proposed. Phase shifters are supplemented with NGD unit cells to
produce a flattened phase response significantly increasing phase bandwidths. The design
methodology of a NGD phase shifter is presented with consideration of nominal phase,
frequency, impedance, maximum insertion loss and bandwidth. The relation between gain,
bandwidth and group delay signifies a fundamental design limitation and tradeoff. A significant
application of NGD phase shifters for removing beam squint in series fed antenna arrays is
discussed. Several NGD phase shifters are fabricated and experimentally verified in the UHF
band upwards of 1 GHz using planar microstrip transmission lines loaded with passive surface
mount RF components with both positive and negative phase shifts.
ii
Acknowledgments
First and foremost I would like to thank my supervisor Professor M. Mojahedi for all his support
and patience during the course of my degree. Without him, this work would not have been
possible.
I’d like to also give thanks to my fellow graduate students with whom I have had many delightful
conversations with over the years and I’m quite grateful to have had an opportunity to meet such
intelligent people.
I’d like to also thank Tse Chan who provided me with a lot of assistance with lab work as he was
always there to help me with failing lab equipment.
Finally, I am very grateful for my friends and families emotional and moral support. Without the
love and support from my mother and my sister’s encouragement to complete this seemingly
insurmountable task, I would not have been able to do it.
iii
Table of Contents
Acknowledgments ........................................................................................................................ iii
Table of Contents ......................................................................................................................... iv
List of Tables ................................................................................................................................ vi
List of Figures .............................................................................................................................. vii
1 Introduction ................................................................................................................................ 1
1.1 Motivation ........................................................................................................................... 3
1.2 Background of Dispersion Engineered Metamaterials ....................................................... 4
1.2.1
Negative Refractive Index Transmission Line Metamaterials ................................ 5
1.2.2
Negative Group Delay ............................................................................................ 5
1.2.3
Microwave Applications of Negative Group Delay ............................................... 6
1.3 Proposal............................................................................................................................... 8
1.4 Thesis Outline ..................................................................................................................... 9
2 Loaded Transmission Line Metamaterials ............................................................................... 11
2.1 Generalized Transmission Line Theory ............................................................................ 12
2.1.1
Broadband Impedance Matching & the Distortionless Condition ........................ 14
2.2 Generalized Loaded TL Analysis ..................................................................................... 16
2.2.1
Condition 1: Host TL Length ................................................................................ 17
2.2.2
Condition 2: Broadband Impedance Matching ..................................................... 20
2.2.3
Condition 3: Small Propagation Factor................................................................. 21
2.2.4
Unit Cell Bloch Impedance and Phase Relationship ............................................ 22
2.3 The Metamaterial TL Unit Cell ........................................................................................ 25
2.4 Foster Reactance Theorem, Energy Storage and Group Delay ........................................ 27
2.5 Lossless MTM-TL Phase shifters ..................................................................................... 29
2.6 Selection of Lossless MTM-TL for Group Delay Minimization ...................................... 32
iv
2.7 Negative Group Delay & Beam Squint ............................................................................ 34
3 The NGD MTM-TL Unit Cell ................................................................................................. 37
3.1 NGD Phase Shifter Design ............................................................................................... 44
3.2 Maximum Return Loss per NGD Unit Cell ...................................................................... 46
4 Simulation & Experimental Results ......................................................................................... 49
4.1 Microstrip NGD Phase Shifters with Ideal Lumped Elements ......................................... 49
4.1.1
Simulated -300 NGD TL Phase Shifter ................................................................. 50
4.1.2
Simulated -900 Two-Stage NGD TL Phase Shifter .............................................. 52
4.1.3
Simulated -450 Two-Stage Stagger Tuned NGD Phase Shifter ............................ 53
4.1.4
Simulated 00 NGD NRI-TL Phase Shifter & Beam Squint Removal ................... 55
4.2 Experimental Results ........................................................................................................ 57
4.2.1
Calibration & Measurement Setup........................................................................ 57
4.2.2
Planar Substrate Selection & Fabrication ............................................................. 57
4.2.3
RF Surface Mount Component Selection ............................................................. 58
4.2.4
-300 NGD TL Phase Shifter (3dB loss)................................................................. 59
4.2.5
-300 NGD TL Phase Shifter (2dB loss)................................................................. 62
4.2.6
00 NGD NRI-TL Phase Shifter ............................................................................. 64
5 Conclusions & Future Work .................................................................................................... 69
5.1 Conclusions ....................................................................................................................... 69
5.2 Future Work ...................................................................................................................... 70
References .................................................................................................................................... 72
v
List of Tables
Table 2.1: A comparison of the low-pass and high-pass first-order MTM-TL unit cells ......... 30
vi
List of Figures
Figure 1.1: A two-port transmission type phase shifter .................................................................. 1
Figure 1.2: Phase response depicting phase bandwidth .................................................................. 2
Figure 1.3: loaded TL unit cell with simultaneous NGD and NRI ................................................ 7
Figure 2.1: two wire transmission line and the equivalent lumped element circuit of the
transmission line. .......................................................................................................................... 12
Figure 2.2: Generic Transmission Line Equivalent Circuit Model of arbitrary series
impedance per unit length Z’ and shunt admittance per unit length Y’. ....................................... 13
Figure 2.3: Circuit model of the spatially extended unit cell for the generically loaded TL ........ 16
Figure 2.4: Equivalent lumped element L-section circuit models for host TL ............................. 17
Figure 2.5: Generic unit cell with host TL sections replaced with lumped element equivalent
circuits ........................................................................................................................................... 18
Figure 2.6: Equivalent Lumped element Generic Unit cell. Host TL effects absorbed by Z
and Y.............................................................................................................................................. 19
Figure 2.7: The phase, gain and delays of the network comprised of cascaded MTM-TL. ......... 26
Figure 2.8: MTM-TL’s equivalent circuit .................................................................................... 26
Figure 2.9: Reactance of a 6th order network displaying the monotonic increase of reactance
with increasing frequency per Foster Reactance Theorem. .......................................................... 27
Figure 2.10: S21 polar plots depicting the phase shift for the low-pass unit cell and the highpass unit cell and the NRI-TL unit cell. ........................................................................................ 31
Figure 2.11: S21 polar plots over entire frequency range for the low-pass unit cell and the
high-pass unit cell and the NRI-TL unit cell ................................................................................ 31
vii
Figure 2.12: the range of possible phase shifts divided between the unloaded TL and the
NRI-TL on the basis of the minization of group delay. ................................................................ 34
Figure 2.13: Series fed antenna array with elements physically spaced by a distance dE and
inter connected through a phase shift 0 to produce a main beam angle in the  direction. ....... 35
Figure 2.14: a non-linear phase response that is approximately linear over a bandwidth
exhibiting simultaneous superluminal phase and group delay .................................................... 36
Figure 3.1: Pole-Zero mapping for the 3rd order Butterworth filter responses .......................... 387
Figure 3.2: The NGD MTM-TL unit cell ..................................................................................... 38
Figure 3.3: (a) Polar plot of both S21 and S11 (located at the origin) and (b) S21 phase and
magnitude versus frequency for a NGD Unit cell (0 = 1 ,  = 2,  = 0.338). ..................... 40
Figure 3.5: NGD unit cell S21 phase and magnitude versus frequency for a NGD Unit cell ....... 42
Figure 3.6: A NGD MTM-TL unit cell symmetrically cascaded with its host TL producing a
broadband NGD phase shifter. ...................................................................................................... 44
Figure 3.7: NGD phase flattening concept increasing the phase bandwidth. ............................... 45
Figure 3.8: Return Loss vs. normalized frequency for NGD unit cells with equivalent NGD
but with insertion losses ranging from 2dB to 5dB. ..................................................................... 47
Figure 3.9: Maximum Return Loss as a function of Maximum Insertion Loss, both occurring
at the resonance frequency. ........................................................................................................... 48
Figure 4.1: Microstrip NGD phase shifter schematic with ideal Lumped element models in
Agilent Advanced Design System (ADS) Software ..................................................................... 50
Figure 4.2: (a) Group delay and (b) phase of -300 microstrip phase shifter for both a NGD
phase shifter (red/solid) and an unloaded TL (blue/dotted). ......................................................... 51
Figure 4.3: (a) Insertion loss of -300 microstrip phase shifter for both the NGD phase shifter
and an unloaded TL and (b) the return loss for the NGD phase shifter ........................................ 51
viii
Figure 4.4: (a) Phase and (b) group delay of -900 microstrip phase shifter for both a NGD
phase shifter and an unloaded TL. ................................................................................................ 52
Figure 4.5: (a) Insertion loss of -300 microstrip phase shifter for both the NGD phase shifter
and an unloaded TL and (b) return loss for the NGD phase shifter (same at both ports). ............ 53
Figure 4.6: (a) Phase and (b) group delay of a 2-stage stagger tuned -300 NGD phase shifter.
In addition to the total phase and group delay, the first and second NGD unit cell and
unloaded -300 ................................................................................................................................ 54
Figure 4.7: (a) Return losses of input and output ports of the combined two-stage NGD
phase shifter and (b) the insertion loss of the NGD phase shifter ................................................ 54
Figure 4.8: NGD NRI-TL phase shifter unit cell .......................................................................... 55
Figure 4.9: Simulated phase response versus frequency for a 00 NRI-TL phase shifter as well
as a NRI-NGD phase shifter ......................................................................................................... 56
Figure 4.10: Main beam angle versus frequency for an unloaded TL, NRI-TL, and NGD
NRI-TL. ........................................................................................................................................ 56
Figure 4.11: NGD TL phase shifter schematic ............................................................................. 59
Figure 4.12: Photograph of the fabricated NGD unit cell ............................................................. 60
Figure 4.13: Experimentally measured and simulated phase shift vs. frequency for a -300
NGD TL Phase shifter compared with unloaded microstrip TL. .................................................. 60
Figure 4.14: Experimentally measured and simulated insertion loss vs. frequency for -300
NGD TL phase shifter ................................................................................................................... 61
Figure 4.15: Experimentally measured and simulated return loss versus frequency for the 300 NGD TL phase shifter ............................................................................................................ 61
Figure 4.16: Photograph of the Phase shifter. ............................................................................... 62
ix
Figure 4.17: Phase response for a -300 NGD TL phase shifter. Experimental results (solid) as
well as simulation results obtained using both ideal and Modelithics component models .......... 63
Figure 4.18: Insertion loss for a -300 NGD TL phase shifter........................................................ 63
Figure 4.19: Return loss for a -300 NGD TL phase shifter. .......................................................... 64
Figure 4.20: (a) 15mm two stage 00 NRI-TL phase shifter circuit (b) 15mm NGD NRI-TL
phase shifter circuit ....................................................................................................................... 65
Figure 4.21: (a) Photograph of the fabricated NGD NRI-TL phase shifter and (b) a magnified
view of the loading elements ........................................................................................................ 65
Figure 4.22: Phase responses of the simulated and experimentally measured NGD NRI-TL
phase shifter and the simulated NRI-TL phase shifter of equal lengths. ...................................... 66
Figure 4.23: Group delays versus frequency of the simulated and experimentally measured
NGD NRI-TL phase shifter and the simulated NRI-TL phase shifter of equal lengths. .............. 67
Figure 4.24: Insertion loss response for the simulated and measured NGD NRI-TL phase
shifter and an equal length NRI-TL. ............................................................................................. 68
Figure 4.25: Return loss response for the simulated and measured NGD NRI-TL phase
shifter and an equal length NRI-TL. ............................................................................................. 68
x
1
1 Introduction
Phase shifters are a ubiquitous device in microwave engineering. They can be found in many RF
and microwave systems such as antennas, radars and in both wired and wireless networks
amongst many others. Indeed, the phase shifter is particularly important as it is a fundamental
constituent component found in many common microwave devices including couplers, power
dividers power combiners, filters, resonators and so forth. When the input of a linear timeinvariant network (i.e. a voltage or current signal) is a time harmonic or sinusoidal signal, the
output is also a sinusoidal signal of the same frequency, but with a different phase and
magnitude. A microwave phase shifter may be generally defined as a linear network whose
output transfer function (as determined from its scattering parameters) produces a specified
frequency dependent phase shift () or alternatively phase delay,  = − ⁄ as illustrated in
Figure 1.1. It is important to note that negative phase shifts produce outputs that lag the input in
time and are considered to have a positive phase delay. Although the phase shifter output
magnitude may also differ from that of the input (i.e. through attenuation or amplification), it is
not within the scope of a phase shifter to do so.
Figure 1.1: A two-port transmission type phase shifter characterized by its scattering parameters (left)
with a sinusoidal input and output waveform and indicated phase delay (right).
2
There are several criteria with which microwave phase shifters may be classified. When the
phase shifter’s input and output occur at the same single port, the phase shifter is considered to
be a reflection type. Alternatively, when the output is located at a distinct secondary port, the
phase shifter is a transmission type. The focus of this thesis is on passive transmission-type
microwave phase shifter with fixed phases (i.e. not variable).
Because the phase shift is generally frequency dependent, its variation over the operating
bandwidth must also be considered as shown in Figure 1.2. In addition to the specified phase
shift 0 , a phase error tolerance  may also be specified, i.e. the tolerable phase deviation
from the specified nominal phase shift. The resulting bandwidth Δ over which the phase is
within this tolerance is referred to as the phase bandwidth as shown in Figure 1.2.
Figure 1.2: Phase response depicting the phase shift () with phase shift  , at frequency  . A
phase error tolerance  and corresponding phase bandwidth  are indicated.
For a monotonically decreasing phase with respect to frequency, as is the always case for passive
lossless phase shifters, the phase bandwidth may be maximized when the slope of the phase
response Δ/Δ is minimized and ideally zero. The concept is similar to the group delay, which
is defined as being the derivative of the phase with respect to frequency
 () = −

.

(1-1)
For a linear of quasi linear phase response, the slope Δ/Δ is given by the group delay
(neglecting the sign) and therefore the maximization of bandwidth can be achieved when the
3
magnitude of the group delay is minimized. It is this concept that forms the basis for phase
bandwidth improvement by compensating the positive group delay of passive lossless phase
shifters with negative group delay (NGD).
In addition to the phase delay, phase tolerance and phase bandwidth specifications mentioned,
a phase shifter should also consider return loss, insertion loss, as well as the variation of the
insertion loss over the operating bandwidth. These factors should typically be minimized to
prevent pulse shape distortion and signal degradation. It is worth noting that additional factors
including size, weight, cost, reciprocity, power handling capability, and noise figure should also
be considered to make appropriate design tradeoffs.
1.1 Motivation
The most conventional implementation of a passive transmission-type phase shifter is the
transmission line (TL) with an electrical length selected to produce the required phase delay. The
TL offers a simple low loss solution, however certain phase shift and minimum port to port
length specifications may require prohibitively long TL lengths, an example being the interelement phase shifters in a series-fed antenna array. As a result, the size, loss and phase
bandwidth may become unfavourable.
Novel solutions using dispersion engineering concepts and TL metamaterials to reduce the
group delay and thus increase the phase bandwidth have been proposed in the past. By loading a
TL with discrete or distributed lumped elements, often in a periodic fashion, the resulting
structure’s effective phase delay and group delay may be designed for a specific need while still
maintaining wideband impedance matching. For instance, when a TL is loaded with series
capacitors and shunt inductors (i.e. the dual configuration of the circuit equivalent model of a TL
section), the resulting loaded TL is referred to as a negative refractive index TL (NRI-TL) and
may exhibit a zero or negative phase delay over a wide band, while maintaining low group delay
variation and with broadband impedance matching. The NRI-TL then behaves much like an
unloaded TL insofar as it exhibits constant input and output impedances, low group delay
dispersion and low insertion loss, but is capable of producing novel phase delays not possible
with unloaded TLs.
4
A notable application of the NRI-TL is the 00 phase shifter. An unloaded TL can trivially
accomplish this by having a zero length, however this is not of much practical use, so the
alternative is to realize a -3600 phase shift with a TL one guided wavelength long (or some
integer multiple given the ambiguity of phase). Alternatively, the NRI-TL phase shifter can
achieve a 00 phase shift in a far shorter length than the unloaded TL and with substantially less
group delay and thus over a wider phase bandwidth. Indeed, it can be shown that the resulting
increased phase bandwidth of the NRI-TL is a consequence of its reduced group delay in
comparison to the significantly longer unloaded TL. However, as it is often the case, a minimum
port-to-port distance of the phase shifter may already be specified, in which case the NRI-TL
may not necessarily offer a lower group delay (and thus wider phase bandwidth) alternative after
all. The NRI-TL, like the unloaded TL always has a positive group delay and although the
loading reactances produce negative phase delay, they also further increase the positive group
delay of the host TL. The resulting increase in group delay is unavoidable for passive, lossless
media and reactances, but is not necessarily the case for Non-Foster media (i.e. active and/or
lossy media).
The motivation of this thesis is to extend the loaded TL metamaterials concept of the NRI-TL
to the analysis and design of a negative group delay (NGD) transmission line phase shifter. The
proposed NGD unit cell is a passive, symmetrical two-port network that produces negative group
delay over an appreciably wide bandwidth while also maintaining a low return loss and insertion
loss. The proposed NGD unit cell also achieves a lower variation in both the insertion loss and
group delay when compared to other passive NGD implementations. The NGD unit cell then
may be cascaded with an equivalently matched network with a positive group delay (e.g. an
unloaded TL or NRI-TL), thereby achieving an effectively zero group delay. The result is a
broadband matched phase shifter with a significantly wider phase bandwidth as a result of the
phase flattening with respect to frequency as discussed earlier.
1.2 Background of Dispersion Engineered Metamaterials
The analysis and design of the proposed negative group delay metamaterial TL (MTM-TL)
necessitates a brief review of metamaterial TLs as well as other relevant microwave NGD
networks.
5
1.2.1
Negative Refractive Index Transmission Line Metamaterials
A transmission line realized with conventional dielectric materials has material parameters (i.e.
the electric permittivity and magnetic permeability) that are positive, resulting in positive phase
and group delays. By introducing periodic scatterers into the medium and ensuring a sufficiently
small period size, the resulting structure may be modeled as having effectively new and
potentially negative material parameters over an appreciable bandwidth. This effective medium
concept was demonstrated using one dimensional (1-D) structures[1], as well as 2-D [2] and 3D[3] structures exhibiting simultaneously negative permittivity and permeability and thus
achieving an experimentally verified negative refractive index. Of particular interest is the 1-D
case where the NRI-TL is used to produce passive, compact and broadband phase shifters[4].
Because the phase shifter is a component often found in common microwave devices and
applications, the NRI-TL phase shifter has been used to also improve the performance of several
other applications including power dividers[5], couplers[6], rat race couplers[7], filters[8], sub
wavelength focusing[9][10] and Wilkinson baluns[5]. The NRI-TL concept has also been shown
to improve radiative applications including leaky wave antennas as well as antenna feed
networks [11][12]. There have been numerous publications [13–15] speculating about the
potential uses and interest continues to grow in the research of metamaterials and dispersion
engineering[16–18].
1.2.2
Negative Group Delay
Although the group delay may be defined as the derivative of a linear time invariant system’s
phase response with respect to the frequency and applies to both spatially negligible (electronic)
or spatially extended systems (microwave or optical devices), the latter may also be
characterized by its phase and group velocity (i.e. delay per unit length). Group velocity is the
velocity by which the peak of a well-behaved electromagnetic pulse travels through a medium.
Although for many years superluminal (faster than speed of light in free space) or negative group
velocities were considered nonphysical, in more recent years many experiments have
demonstrated their existence and hence their physical nature [19][20]. In fact, not only has it
been shown that both superluminal and negative group velocities do not violate causality, but
that all linear and causal media must exhibit these velocities over some frequency bands (for
6
example, frequencies at which the maximum loss occurs) as required by the Kramers-Kronig
relations [21].
The group velocity is not strictly associated with the information or signal velocity as
previously thought. The propagation of information is better understood by considering the
propagation of “fronts” or forerunners of a pulse defined as a low energy and high frequency
transient component of a signal. A finite duration signal requires a time at which it is turned on
(a discontinuity in the signal or its higher order derivatives), which when modeled in the
frequency domain requires an infinite frequency spectral component. It is these fronts that must
always propagate slower than the speed of light and that are associated with true information
propagation, not an arbitrary point on a pulse’s envelope (e.g. the peak of the pulse). At these
high frequencies, the oscillating charges cannot accelerate fast enough to respond to the
excitation and thus the polarization is zero and the effective permittivity at these frequencies is
that of the free space.
Negative group delay and time-domain pulse advancement have been demonstrated many
times at low frequency (i.e. less than 1 MHz) using simple electronic amplifier circuits[22] [23].
Any stable linear time invariant system may realize NGD by introducing left-hand plane zeroes
in its transfer function. The use of finite Q (and thus non-zero bandwidth) left-hand plane
transmission zeroes in the transfer function response may be implemented in several ways such
as an active bi-quadratic filter amplifier[24] or by using lossy resonators in conjunction with
operational amplifiers [25–27], to name two common approaches.
1.2.3
Microwave Applications of Negative Group Delay
Because NGD can be used to increase phase bandwidth, it too has several microwave
applications. A microwave phase shifter demonstrated by Ravelo et al [28] has an increased
phase bandwidth by compensating for the positive group delay of a host TL by introducing lossy
resonators as well as gain [29]. Using several of these NGD phase shifters, a broadband
balun[30] was designed demonstrating enhanced performance. The phase flattening concept
leading to a higher phase bandwidth as discussed in section 1.1 was further extended to produce
an ultrawideband phase shifter [31].
7
Simultaneous negative phase delay and NGD was exhibited in a loaded transmission line
metamaterial[32] that loaded a TL with both high-pass and lossy resonators as shown in Figure
1.3. This particular implementation however suffered from poor return loss and narrow
bandwidth, the reasons for which will be addressed in this thesis.
Figure 1.3: An example of a loaded TL unit cell with simultaneous NGD and NRI [1]
Furthermore, a natural application of NGD is the reduction of signal latency, which is useful in
communication systems. Although information cannot propagate faster than the speed of light in
a vacuum, it has been shown that reducing the group delay of a communication channel can also
have potential applications by reducing latency. If a communication channel interconnecting the
transmitter and receiver exhibits superluminal or negative group velocities, the signal latency can
be reduced or even eliminated[33]. It has also been shown that NGD may have practical use in
digital electronics by shaping a clock signal better. By incorporating NGD at baseband
frequencies, a square wave’s rise and fall times may be considerably reduced. This was
demonstrated for a R-C interconnect TL model, where a significant improvement to the rise and
fall times of square waves was shown [34–36].
Perhaps the most naturally suited application of NGD and dispersion engineering is channel
equalization [37]. This was demonstrated using a monolithic microwave integrated circuit
(MMIC) where an amplifier was designed to reduce the group delay variation for UWB
applications [38] [39]. Unlike other equalization techniques, NGD was used to reduce the group
delay ripple without having to increase the average group delay. Finally, because NGD tends to
produce pulse compression [40], NGD may find applications for pulse shaping applications as
well [41], [42]. An important microwave application of NGD is the reduction or complete
elimination of beam squint in a series fed antenna array [43] when incorporated into the interelement feed network. This has been demonstrated in [44] and [45], where the main beam scan
8
angle switched directions as frequency was swept, a feature not seen in subluminal group delay
feed structures. In fact, NGD is a necessary requirement for this effect and beam squint may only
be removed entirely with the addition of NGD. Indeed, Sievenpiper presented simulations
demonstrated a microstrip leaky wave antenna with a constant main beam angle over a decade
bandwidth when a substrate with superluminal group velocity was used. This is accomplished by
the use of non-Foster negative capacitors[46]. Another interesting approach to reducing beam
squint using non-foster negative capacitances is also presented in [47].
Although many NGD implementations have gain, most are implemented at lower frequencies
using electronic operational amplifiers and are generally not valid solutions at microwave or
higher frequencies, where particular active technologies do not scale up and the constraints of
spatially extended systems are not taken into account. The active NGD implementations at RF
frequencies compensate loss associated with the lossy resonators by simultaneously amplifying
the NGD frequencies as well as frequencies adjacent to the NGD bandwidth, such that the
relative gain of the NGD band remains relatively unchanged. Attempts to selectively amplify
only the NGD band will perturb the phase response and thus the group delay. Furthermore, most
of the NGD implementations with gain at microwave frequencies do so with very poor power
added efficiency and noise figures, essentially sinking all the incident power into the resistive
loss through a matching resistor. In fact, the efficiency is further reduced by allowing the active
elements (i.e. transistors) to drive the output power into the output matching resistors and the
resistors in the NGD resonators. Although they do produce NGD with moderate gain, there is no
discernible motivation, beyond compactness, to combining the NGD and wideband amplification
stages.
1.3 Proposal
The motivation of this thesis is to provide an analysis and design of a broadband, compact,
passive two-port microwave phase shifters exhibiting NGD. Unlike previous NGD
implementations, the proposed NGD unit cell is frequency and impedance scalable (analogous to
a filter prototype) and is designed to produce a specified NGD while simultaneously minimizing
return loss, gain variation and group delay variation over an appreciably large bandwidth. This is
achieved by extending the concept of metamaterial TLs and loading a TL with lossy resonators
9
in a band-stop configuration. By loading the TL with a balanced configuration similar to the
NRI-TL with low Q resonators, the inevitable variations of loss and group delay are mitigated.
A loaded TL unit cell with arbitrary loading and loss is analyzed using periodic dispersion
analysis in conjunction with conventional linear filter theory. Like any filter prototype response,
the proposed NGD unit cell is easily scaled in both impedance and frequency. A fundamental
limitation relating the insertion loss, bandwidth and maximum NGD achievable by the NGD unit
cell is established indicating an inescapable tradeoff between the three. As a result, an increase in
NGD, bandwidth or gain (reciprocal of insertion loss) requires a proportional decrease in the
others. This suggests a maximum limit on the achievable performance, much like the gainbandwidth product of an amplifier, but a gain-bandwidth-NGD product is required instead.
The design of NGD phase shifters is presented whereby the NGD unit cell is used in
conjunction with other positive group delay phase shifters (i.e. a simple TL segment or even a
NRI-TL with negative phase delay). The design methodology considers the specified nominal
phase and frequency, maximum insertion loss and minimum bandwidth, while minimizing phase
and gain variation. The potential to significantly reduce the group delay and gain variations of
the NGD unit cell even further by use of cascaded stagger-tuned cells is briefly explored as well.
The selection between positive or negative phase delays on the basis of group delay
minimization while accounting for the minimum port-to-port distance is considered. The
significance of NGD in the complete removal of beam squint in series fed antenna arrays is also
discussed. Finally, NGD phase shifters with both positive and negative phase delays (i.e. NRITL with NGD) are fabricated and experimentally verified in the UHF band upwards of 1 GHz by
using a microstrip transmission line loaded with passive surface mount RF components.
1.4 Thesis Outline
This thesis is composed of 5 chapters. Following the introduction, chapter two introduces the
theoretical analysis of loaded transmission lines and extends the metamaterial TL concept to
develop the NGD unit cell. Chapter three proceeds with the design methodology and
considerations of NGD phase shifter design, including a cursory analysis of beam squint. In
chapter four, simulations and additional design criteria are discussed and experimental results are
10
presented. Finally, chapter five provides a summary of conclusions and discusses the potential
for future work.
11
2 Loaded Transmission Line Metamaterials
By loading a host transmission line with lumped elements in a periodic fashion (i.e. cascading
unit cells), the resulting structure may be conceptualized as still being a broadband matched TL
but with novel properties, referred to as a metamaterial TL (MTM-TL). In transmission line
theory, a TL is modeled as an infinitely divisible cascade of incremental circuits consisting of
series inductance per unit length and shunt capacitance per unit length. By applying Kirchhoff’s
voltage and current laws, the Telegrapher’s equations are derived and the TL properties are
characterized. This approach can be extended to include the reactance of the loading elements in
addition to the host TL section’s per unit inductance and capacitance. The validity of this
assumption however is subject to the length of the periodic loading, and is only valid over a
finite frequency band. By applying periodic dispersion analysis (Floquet theorem) to a unit cell
of the periodically loaded TL, the resulting propagation constant and characteristic impedance of
an infinitely long structure comprised of these unit cells can be deduced. It is worth noting that
the network need not be infinitely long for the analysis to hold, rather the network need only be
terminated in its characteristic impedance – much like the antiquated image parameter method or
constant-k filter design.
This section begins by first reviewing transmission line theory and the derivation of
Telegrapher’s equations but replaces the incremental low-pass circuit model with a generic or
arbitrary impedance and admittance per unit length. The corresponding propagation constant and
impedance is then determined. The distortionless or matching condition and the effects of loss
are also considered. Next, the physically realizable generically loaded transmission line is
considered and it is shown how after imposing three conditions on the unit cell, the loaded TL
behaves similar to the unloaded TL, albeit with novel characteristics. The first condition limits
the electrical size of the host TL such that its lumped element equivalent circuits may be used
instead. The second condition is analogous to the distortionless condition of TL theory and
12
relates the series and shunt loading networks. Finally, the third condition – limiting the
magnitude of the propagation factor per unit cell – can be invoked to further emulate broadband
TL behavior.
2.1 Generalized Transmission Line Theory
This section considers the equivalent incremental circuit model of a transmission line, however
the series inductance and shunt capacitance of a lossless TL is instead replaced with arbitrary
impedances. Since the proposed NGD unit cell consists of lossy resonators, the effect of loss
cannot be neglected and the arbitrary impedances are complex quantities.
The classical lumped element incremental circuit model for a lossy transmission line is shown
in Figure 2.1. It is comprised of reactive elements that store energy and resistive elements that
dissipate energy. The inductance models the energy stored in the magnetic field induced by
current in the conducting wires and is proportional to the effective permeability of medium. The
capacitance models the energy stored in the electric field and is proportional to the effective
permittivity of the medium.
Figure 2.1: (a) A two wire transmission line of length ∆z and (b) the equivalent lumped element circuit
for an incremental length ∆z of the transmission line.
This method may also be applied to the unit cell shown in Figure 2.2 with arbitrary series
impedance and shunt admittance instead. The Telegrapher’s equations are found by applying
Kirchhoff’s current law at the output terminal and Kirchhoff’s voltage laws around the closed
loop as depicted in Figure 2.2 (in the manner described in [48]).
13
Figure 2.2: Generic Transmission Line Equivalent Circuit Model of arbitrary series impedance per unit
length Z’ and shunt admittance per unit length Y’.
The Telegrapher’s equations then can be readily combined to produce wave propagation
equations describing the spatially dependent time harmonic voltages and currents supported on
the TL waveguide structure:
 2
+ ′2  = 0 ,
 2
2 
 2
+ ′2  = 0.
(2-1a)
(2-1b)
Where the frequency dependent complex propagation constant ′ (where the prime denotes a per
unit length quantity) is defined by
′() =  +  = √′′.
(2-2)
The attenuation constant  [Nepers/m] and the propagation constant  [radians/m] defined in
(2-2) and are given by the real and imaginary parts of the complex propagation constant,
respectively. Recall that for a conventional TL as shown in Figure 2.1, the complex values for Z
and Y for time harmonic excitations are given by
′ = ′ + ′,
′ = ′ + ′.
(2-3a)
(2-3b)
Equations (2-1a) and (2-1b) can be simultaneously solved to produce separated equations for
both the voltages and currents as a function of position along the TL as
() =  +  −′ +  −  +′ ,
() =  +  −′ −  −  +′ .
(2-4a)
(2-4b)
14
The voltages and currents given above represent a superposition of travelling waves, one in the
positive +z and the other in the contra directional -z direction. The ratio between the voltage and
current travelling wave pair is referred to as the characteristic impedance (i.e. 0 =  + ⁄ + =
 − ⁄− − and is given by
′
0 = � .
′
(2-5)
Finally, the phase velocity and phase delay for a segment of length Δ are defined by considering
the imaginary part of complex propagation constant
Δ =
Δ 
= Δ .
 
(2-6)
The group delay and group velocity may similarly be defined as:
Δ =
Δ 
=
Δ .
 
(2-7)
As compared to the velocities (phase or group velocities), delay terms (phase or group delays)
are of particular interest because they apply equally well to spatially extended (e.g. TLs or
waveguides) or spatially negligible systems (e.g. electronic circuits).
2.1.1
Broadband Impedance Matching & the Distortionless Condition
Because the impedance Z, and admittance Y are generally frequency dependent and complex
valued, both the propagation constant and the characteristic impedance are also frequency
dependent and complex valued. This is generally an undesirable characteristic of a TL as it gives
rise to variations in output signal power for different frequencies, the results of which are:
reflection of power from the source, pulse distortion, and loss – all of which are ultimately
problematic for a waveguide or a phase shifter.
Consider the complex propagation constant γ′TL and characteristic impedance  of the TL
model shown in Figure 2.1
15
γ′TL = √′′ = �(′ + ′)(′ + ′) ,
 = �
(′ + ′)
′
=�
.
(′ + ′)
′
(2-8a)
(2-8b)
Under low loss conditions, the real valued impedance R’ and conductance G’ are much smaller
than the reactive elements L’ and C’ and can thus be neglected altogether (i.e. R’=G’=0). Upon
doing so, the resulting characteristic impedance given in (2-8b) is approximately constant and
equivalent to the characteristic impedance of a lossless TL. The propagation constant in equation
(2-8a) can be similarly simplified using a first order Taylor series approximation. The result
shows an propagation constant equivalent to lossless case (i.e. with a constant phase delay and
constant group delay), but with an additional attenuation term.
1 ′
γ′TL,l  ≈ �′′ + � + ′0 � .
2 0
(2-9)
Alternatively, the Distortionless condition (also referred to as the Heaviside condition), which
does not require low loss, may be imposed to ensure the characteristic impedance remains
constant with respect to frequency. Essentially, the ratio of the series impedance and shunt
admittance is required to be equal for all frequencies. This condition may be generally stated as
′ ′ ′
= = = 02 .
′ ′ ′
(2-10)
This condition is of course already true for the lossless TL. When arbitrary impedances and
admittances are considered, this condition is satisfied by ensuring that the per unit length
admittance Y’ and impedance Z’ are dual networks of one another and are scaled using equation
(2-10) accordingly. For example, a parallel G-L combination in either Z’ or Y’ requires a series
R-C in the other to ensure for broadband impedance matching. As a result, the propagation
constant and characteristic impedance can be simplified as:
γ′TL
′
=
=  ′0 ,
0
 = 0 .
(2-11a)
(2-11b)
For a conventional transmission line with series inductance and resistance then, the propagation
constant is given by
16
γ′TL =
′ + ′ ′
=
+ �′′ .
0
0
(2-12)
which is the expected result for distortionless TL with loss. Therefore, by carefully selecting the
loading impedance and admittance, the propagation constant can be specifically designed while
maintaining impedance matching. This matching concept is critical to the design of metamaterial
TLs including the NRI-TL, also referred to as the composite right/left handed TL (CRLH-TL),
the lossy CRLH TL[13][49], the dual CRLH-TL, Extended CRLH-TL [50] and a generalized
model of metamaterial TLs[51][52], all of which emphasize phase delay design. These concepts
however can similarly be applied to the analysis and design of a loaded TL with a specified
group delay (positive or negative).
The following section provides a cursory examination of applying periodic dispersion analysis
to a spatially extended TL. Upon invoking certain conditions, it is shown how the resulting
analysis converges to that of the generic TL detailed in this section and by proper selection of the
loading elements, broadband matched NGD phase shifters can be readily designed.
2.2 Generalized Loaded TL Analysis
The spatially extended loaded TL unit cell analyzed in this section is shown in Figure 2.3. It
consists of a host TL of electrical length θ and characteristic impedance Z0 symmetrically loaded
in a T-configuration with an equivalent series impedance ′Δ and shunt admittance ′Δ,
where the primed variables represent per unit length quantities.
Figure 2.3: Circuit model of the spatially extended unit cell for the generically loaded TL
17
A common approach to analyze and obtain the resulting network response of several cascaded
networks as depicted in Figure 2.3 is by multiplying the constituent networks’ ABCD matrices.
This is readily determined by matrix multiplication as shown below.
�




�=�
1
0

 ′ Δ
 � �
2
2 ��

 0  � �
1
2

1
 0  � �
2 ��

 � �
 ′ Δ
2


 � �
0  � � 1
2
2 ��
��


 � �
1 0  �2�
0
2
0
 ′ Δ
2 �
(2-13)
1
Upon performing the matrix multiplication to obtain the resultant ABCD network parameters in
(2-13), Floquet theorem may be applied to determine the complex propagation constant ′ and
Bloch impedance BL for forward travelling waves (analogous to the characteristic impedance of
a TL) using the following two relations
cosh(′Δ) =
BL =
+
,
2

√2 − 1
.
(2-14a)
(2-14b)
Significant simplification can be made if the electrical length of the host TL is sufficiently small
such that the trigonometric terms are replaced by their first order Taylor series approximation or
alternatively, by replacing the TL segments with equivalent lumped element circuit models.
2.2.1
Condition 1: Host TL Length
By replacing the TL sections with their lumped element equivalent L-section circuit models
depicted in Figure 2.4, the spatially extended unit cell is then entirely comprised of lumped
elements.
Figure 2.4: Equivalent lumped element L-section circuit models for host TL
The equivalent inductance and capacitance of the L-section circuits are determined using the host
TLs electrical length and characteristic impedance and are given by
18
TL = 0
TL = 0


= 0 TL
,
2
2
(2-15a)


= 0 TL
.
2
2
(2-15b)
where TL and 0 represent the propagation constant and characteristic impedance of the host
TL, respectively. The resulting ABCD matrices for the two L sections in Figure 2.4 are given by
 2
⎡1 + � �
2
⎢
⎢

⎢

0
⎣
2

 0 ⎤
2⎥
⎥
⎥
1 ⎦
⎡ 1
⎢
⎢
⎢ 
⎣0 2

⎤
2 ⎥
⎥.
 2⎥
1+� � ⎦
2
 0
(2-16)
Indeed, the same result may be reproduced when small argument approximations are made for
the trigonometric terms in (2-13), and are valid when the host TL electrical length is sufficiently
short. As indicated in Figure 2.5, the L-sections are oriented to maintain symmetry in the unit
cell. This allows the series loading impedance and shunt admittance to effectively absorb the
transmission line models.
Figure 2.5: Generic unit cell with host TL sections replaced with lumped element equivalent circuits
Therefore, the spatially extended unit cell shown in Figure 2.3 can be reformulated into a simpler
unit cell with no associated length as shown Figure 2.6, where the effective loading impedances
Z and Y, which effective absorb the host TL and thus are no longer per unit length quantities can
be expressed as
 =  ′ Δ + TL ,
 =  ′ Δ + TL .
(2-17a)
(2-17b)
19
Figure 2.6: Equivalent lumped element generic unit cell. Host TL effects absorbed by Z and Y
The ABCD parameters of the unit cell in Figure 2.6 are given by

�


⎡
+1
2
⎢
�=⎢
⎢

′
⎣

1 2
 +1
4

+1
2
⎤
⎥.
⎥
⎥
⎦
(2-18)
Using these ABCD parameters, the propagation factor ′ (the prime denoting it is not a per unit
length quantity like the complex propagation constant) of the unit cell can be found using (2-14a)
and is given by
1
cosh  = 1 +  .
2
(2-19)
By applying hyperbolic trigonometric identities, (2-19) may be re-expressed as

1
sinh2 � � =  .
2
4
(2-20)
Similarly, the Bloch impedance can be found using (2-14b) and is given by


BL = � �1 +
.

4
(2-21)
The Bloch impedance expressed in equation (2-21) is indeed identical to the image impedance of
a T-network found using image parameter method of filter design. Similarly, the propagation
factor given in equation (2-20) is also equivalent to that of the image parameter method and can
be verified by re-expressing equation (2-19) using the inverse hyperbolic cosine function
definition to produce the equivalent result using the image parameter method given by
20
 = 2 ln �1 +

 2
+ � + � � � .
2
2
(2-22)
It is interesting to note that propagation factor in (2-20) and the Bloch impedance in (2-21) may
be related to one another directly through the hyperbolic trigonometric identity cosh2x - sinh2x =1
producing a relation given as:


BL = � cosh � � .

2
2.2.2
(2-23)
Condition 2: Broadband Impedance Matching
Comparison with equation (2-23) above indicates that a constant value for the Bloch impedance
may be achieved when both the ratio of Z and Y and the propagation constant are constant and
may be expressed as


= .
0 0
(2-24)
This condition may be satisfied over a large bandwidth when the admittance network Y,
comprised of lumped elements, is the dual of the impedance Z network as previously discussed
in the section 2.1.1. For example, if the Z network includes a resonance circuit, the Y network
must also include the corresponding dual (anti-resonance) circuit of an equivalent resonance
frequency and quality factor. This is referred to as the balanced condition for the NRI-TL and is
required to ensure no stop-bands arise at frequencies where the Bloch impedance is imaginary.
By imposing the condition in (2-24), either Z or Y remains free to be chosen, but not both.
Therefore, the equations in (2-20) and (2-21) may be re-written in terms of the normalized series
impedance  = /0 as

1
sinh � � =  ,
2
2
BL
 2

= �1 + � � = cosh � � .
0
2
2
(2-25)
(2-26)
21
The third and final condition in the following section is needed to obtain the initially sought after
transmission like equations discussed in section 2.1, which restricts the magnitude of the
propagation factor, similar to condition one. It is important to note that the term propagation
factor is used and not propagation constant, which is a per unit length quantity.
2.2.3
Condition 3: Small Propagation Factor
The result of the two conditions of the preceding sections − the minimization of a single unit
cell’s host TL length and the broadband matching condition − the resulting propagation factor
and Bloch impedance reduce greatly as shown in (2-25) and (2-26). In fact, the aforementioned
expressions may be simplified further by invoking Taylor series approximations on the
hyperbolic functions, which are also valid for complex arguments. To illustrate this, the first
three terms of the Taylor series’ for both hyperbolic sine and cosine functions are shown below

 1  3 1  5
sinh � � ≈ + � � + � � + ⋯,
2
2 3! 2
5! 2

1  2 1  4
cosh � � ≈ 1 + � � + � � + ⋯ .
2
2! 2
4! 2
(2-27a)
(2-27b)
Provided the function’s arguments are sufficiently small, the two functions can be replaced by
their first order approximations given by


sinh � � ≈
2
2
when
1
||2 ≪ 1 ,
24
γ
1
cosh � � ≈ 1 when ||2 ≪ 1 .
2
8
(2-28a)
(2-28b)
The resulting approximations when applied to (2-25) and (2-26) produce a simple and familiar
form for the propagation factor and characteristic impedance as
≈

=,
0
 ≈ 0 .
(2-29)
(2-30)
This third condition in addition to the previous two ensures that the characteristic impedance as
given by (2-30) is constant and thus broadband impedance matching is achieved. The
propagation factor is also reduced to just being the normalized series impedance. Therefore,
22
condition 3 is satisfied when the normalized impedance is sufficiently small. Special care then
must be taken when complex quantities are considered. As a result, instead of assuming
condition three remains valid, an alternative approach may be taken whereby the scattering
parameters are directly computed and the resulting return loss is obtained directly.
The
significance of this will become clear when resistors are permitted in the loading networks and
must be sufficiently small to ensure a high return loss.
2.2.4
Unit Cell Bloch Impedance and Phase Relationship
Consider the broadband impedance matched propagation constant expressed in equation (2-25),
but expressed in terms of its real and imaginary part and with the normalized series impedance z
broken down into its real and imaginary parts, i.e. its normalized resistance r and reactance x
shown as
1
1
sinh � ( + )� = ( + ) .
2
2
(2-31)
The hyperbolic sine function can be decomposed into its real and imaginary parts and equated to
the respective real and imaginary parts of the right hand side of (2-31) as follows

 
sinh cos = ,
2
2 2

 
cosh sin = .
2
2 2
(2-32a)
(2-32b)
If both the series impedance and shunt admittance are lossless (i.e. r=0), the first term in (2-32a)
vanishes, which requires that at least one of the hyperbolic functions on the right hand side must
also vanish. This condition can be decomposed into two cases and the resulting (2-32b) can be
simplified and the Bloch impedance given in (2-26)) can similarly be simplified.
Case 1 ( > 1 ):
Case 1 ( < 1 ):
 = ,  ≠ 0 (attenuating mode, stop-band)
cosh
 
= ,
2 2
BL =  0 sinh

.
2
 = 0,  ≠ 0 (propagating mode, pass-band)
(2-33a)
(2-33b)
23
sin
 
= ,
2 2
BL = 0 cos
(2-34a)

.
2
(2-34b)
The first case produces only attenuation and no phase progression, with a purely imaginary
Bloch impedance and thus no real power flow (for an infinite periodic structure). The second
case is of greater interest as the expression in (2-34b) indicates a simple relationship between the
phase shift and Bloch impedance. It suggests a clear trade-off between impedance matching,
which requires the cosine term to be equal to unity thus the phase shift per unit cell must be
limited to prevent the Bloch impedance from diverging from the desired characteristic
impedance 0 . Alternatively, the reflection coefficient of a one or more unit cells cascaded and
terminated in the impedance 0 can found in terms of the phase using the relation in (2-34b) as

0 − BL 1 − cos 2
Γ=
=
.
0 − BL 1 + cos 
2
(2-35a)
In section 2.2.3, the propagation factor of a loaded TL unit cell was found by applying the
Floquet theorem for periodic structures using the generically loaded unit cell’s ABCD
parameters, given in equation (2-13). Instead of invoking the 3rd condition however, the
scattering parameters can be obtained directly using relations in [48] without requiring a small
propagation factor. Since the ABCD parameters for the network shown in Figure 2.6 are known,
the scattering parameters are found as
[]


⎡ −
−1 0
0


⎢
= � + + + � ⎢
0 0
⎢ 2
⎣
⎤
⎥
⎥.

⎥
−
0 0 ⎦
2
(2-36)
Because the network is reciprocal, the scattering parameter matrix is symmetrical as expected.
Therefore, we need only consider S11 and S21 to know the full response. The scattering
parameters of the generic unit cell can then be found by substituting the values from the ABCD
matrix found in (2-18) and produces
24
( 2 − 402 ) + 4
⎡
⎢(20 + )( + 20  + 4)
[] = ⎢
80
⎢
⎣ (20 + )( + 20  + 4)
80
⎤
(20 + )( + 20  + 4) ⎥
⎥.
2
2
( − 40 ) + 4
⎥
(20 + )( + 20  + 4) ⎦
(2-37)
Equation (2-37) can be re-expressed in terms of the normalized impedance z and admittance y
thereby eliminating the characteristic impedance entirely as shown below.
 2 + 4 − 4
⎡
2
⎢ + 4 + 4( + ) + 8
[] = ⎢
8
⎢
⎣ 2 + 4 + 4( + ) + 8
8
⎤
+ 4 + 4( + ) + 8 ⎥
⎥
 2 + 4 − 4
⎥
2
 + 4 + 4( + ) + 8⎦
 2
(2-38)
The impedance matching condition may be invoked to further reduce the scattering parameters.
That is, when y and z are equivalent, per the matching condition given in (2-24), the scattering
matrix may be re-expressed in terms of z only as
3
⎡
3
2
⎢ + 4 + 8 + 8
[] = ⎢
8
⎢
⎣  3 + 4 2 + 8 + 8
8
⎤
+
+ 8 + 8 ⎥
⎥.
3
⎥
 3 + 4 2 + 8 + 8 ⎦
3
4 2
(2-39)
The scattering parameters of the generic unit cell only uses the first two conditions in section 2.2
and does not require that the propagation factor or impedances to be small as is the case with the
preceding section. It can therefore provide a more robust, albeit less immediately intuitive
understanding of the generic unit cell. More importantly, it specifies the reflection coefficient of
the unit cell and so a return loss can be computed, which was not the case for the periodic
dispersion analysis. In general, the reflection coefficients given in (2-39) indicate that as the z
approaches 0, the return loss vanishes. Similarly, it can be shown that the transmission scattering
parameters also converge to the results from the section 2.2.3.
25
2.3 The Metamaterial TL Unit Cell
With the three conditions in sections 2.2 satisfied, the resulting MTM-TLs propagation factor is
given by the normalized impedance z, which is the equivalent form given in (2-11a) for the
distortionless TL. The characteristic impedance is constant and thus provides broadband input
and output impedance matching at frequencies where the aforementioned conditions remain
valid. Therefore the scattering parameters are analogous to that of a TL and can be written as:
11
�
21
12
22
�

≈�
0

−
 −
0
�.
(2-40)
This form is very convenient as the phase and magnitude of the transmission scattering
parameters, S21 and S12, are given by the real and imaginary parts of the normalized series
impedance, z
ln|21 | = − ,
21 = − .
(2-41)
where r and x correspond to real and imaginary part of z. The phase and magnitude then may be
designed by an appropriate selection of the resistance and reactance of the normalized impedance
z, respectively. If there is no series resistance (and shunt conductance) then the magnitude of
transmission parameters is 0dB as expected for a lossless network.
There are two important properties of the MTM-TL. First, by cascading MTM-TLs of the
same characteristic impedance or other equivalently matched networks, the resulting network is
also broadband impedance matched and the resulting phase and gain are equal to the sum of the
constituent networks’ phases and gains. This is akin to connecting two transmission lines of the
same characteristic impedance but of different phase velocities. This is described mathematically
in (2-42) below.
26
Figure 2.7: The phase, gain and delays of the network comprised of cascaded MTM-TL are the sum of its
constituent networks’ phase, gain and delays, respectively.
 =  + 
 =  + 
(2-42)
 =  +  .
The second functional property of the MTM-TL is the decoupling of both the host TLs and the
loading elements contributions to the resulting magnitude and phase response. That is, the
propagation factor in (2-29) being given by z can be decomposed into the sum of the loading
impedance z and the inductance of the host TL. Combined with the property given in (2-42), the
two networks can be decoupled into two distinct cascaded networks. The networks shown in
Figure 2.3 and Figure 2.5 can be re-expressed as shown in Figure 2.8 below, where the effects of
the host TL and loading are now split into two distinct cells, the ordering of which is
inconsequential. The analysis can proceed by effectively neglecting the host TL from the unit
cell and treat it as an external cascaded TL section modeled by its lumped element equivalent
circuit.
Figure 2.8: The MTM-TL’s equivalent circuit depicting the de-coupling of the host TL’s lumped element
equivalent circuit cascaded with a network comprised of only the loading elements.
It is worth noting that the TL section depicted in Figure 2.3 only accounts for the TL length
between the two identical series loading elements of the unit cell. The total length of the host TL
27
being loaded is inconsequential with respect to validity of the small TL assumption. Only the
length between the two series loading elements must be sufficiently small to satisfy the
assumption. It is also worth noting that it is physically impossible to completely eliminate the
effects of the host TL for a unit cell of non-zero length or alternatively, it is impossible to not
have an equivalent series inductance and shunt capacitance to account for the unavoidable
electric and magnetic fields that exist in the TL waveguide structure and in any physically
realizable lumped element components. As a result, any host TL of non-zero length necessarily
introduces both positive phase delay and group delay.
2.4 Foster Reactance Theorem, Energy Storage and Group Delay
Because the phase shift of a MTM-TL is given by normalized reactance x (or normalized
susceptance b), the corresponding group delay can be ascertained by differentiation of the
reactance with respect to frequency. The Foster Reactance Theorem states that the frequency
derivative of the input impedance X or admittance B of a lossless one-port network must always
be positive and can be stated as

> 0,


>0.

(2-43)
To illustrate this property, consider the reactance of a network whose transfer function is a 6th
order polynomial shown in Figure 2.9. It is clear that the reactance X monotonically increases as
frequency increases with its poles and zeroes alternating. It is clear that its frequency derivative
or slope is always be positive in accordance with Foster Reactance Theorem.
Figure 2.9: Reactance of a 6th order network displaying the monotonic increase of reactance with
increasing frequency per Foster Reactance Theorem.
28
A more generalized form of the Foster reactance theorem given in (2-43) may be obtained when
the energy storage of the lossless one-port network is considered. The relationship between the
aforementioned reactance derivative and total stored energy is given in [53] and expressed as
 2
=
≥ 0,
||2

 2
=
≥ 0.
||2

(2-44)
where  is the total time averaged stored energy in the network and V and I are the voltage and
current at the port terminals, respectively. This result was extended to periodic structures whose
unit cell need only satisfy the general constraints of losslessness, linearity, time invariance, and
passivity. When the structure operates in a pass band (i.e. at frequencies with real Bloch
impedance with only a phase shift per unit cell and no attenuation), the group delay per cell in
the direction of power flow is the average stored energy per cell per unit total power and must be
positive.
These results are extended in [54] for two-port networks where it is shown that the time
averaged stored energy in a passive lossless two-port network can be derived from a network’s
scattering parameters. In particular, it is shown that the stored energy in a passive lossless
reciprocal symmetrical or anti-symmetrical two-port network is proportional to the group delay,
which is found by differentiating the phase of the transmission scattering parameter S21 with
respect to frequency. This condition is may be expressed as
 = −|1 |2
21
= |1 |2 ∙  .

(2-45)
where  is the total time averaged stored energy of the network and 1 is the available power
from the source at port one. These results are easily extended to the MTM-TL whose propagation
factor is given by the normalized impedance z or y. For a lossless MTM-TL, the impedance
consists of only a reactance and thus the phase is given by the normalized reactance, x. The
MTM-TLs group delay then is found by differentiating the normalized reactance and according
to (2-43), must also be positive at all frequencies as shown:
 = −
21 
=
> 0.


(2-46)
29
This condition applies to most passive filters including common filters such the Butterworth and
Chebyshev prototype filters. The inequality conditions in (2-46) does not hold for networks with
either loss or gain. Although not sufficient by itself, the introduction of insertion loss or gain is
required for a zero group delay or NGD.
The relationship between phase, reactance, energy storage and the group delay suggests that
the total stored energy in reactive elements should be minimized in order to minimize the
resulting group delay. Therefore, given a required phase delay for a unit cell, the series
impedance should be selected to have only positive or negative reactances, but not both as this
needlessly adds group delay. Recall however that the host TL of a MTM-TL unavoidably adds
positive phase and group delays and therefore negative phase delay MTM-TLs (i.e. the NRI-TL)
should seek to minimize this host TL for group delay minimization.
2.5 Lossless MTM-TL Phase shifters
There are fundamentally only two distinct types of first-order lossless MTM-TL unit cells − the
low-pass unit cell (i.e. a series inductance and shunt capacitance) and the dual high-pass case
(i.e. a series capacitance and shunt inductance). The low-pass MTM-TL is equivalent to the
otherwise unloaded TL, however the loading effectively decreases the phase velocity of the host
TL (or increases the phase index) thereby increasing the phase delay. The loading may introduce
undesirable effects such as increased loss and added cost and complexity, but may be preferable
when the physically extension or meandering of the unloaded TL alternative becomes
prohibitively large. The popular broadband impedance matched NRI-TL can equivalently be
viewed as a high-pass unit cell cascaded with a low-pass unit cell. As described in section 2.3, an
isolated high-pass unit cell is not physically realizable and a low-pass MTM-TL unit cell must
also be included to account for the host TL.
The two first order unit cells and their respective properties are listed in Table 2.1 below. The
frequency limitation shown indicates when the reactances and phases are sufficiently small for
the MTM-TL conditions of section 2.3 to remain valid. The host TLs lumped element
approximation places an upper bound on the frequency of the low-pass section. The phases and
delays for both unit cells are also shown. The low-pass and high-pass unit cells have respectively
negative and positive phase delays, but both have positive group delays.
30
Table 2.1: A comparison of the low-pass and high-pass first-order MTM-TL unit cells
Low-pass
High-pass

−1/
 ≪ 
 ≫ ℎ
circuit
reactance, X
susceptance, B

frequency
limitation
phase
 = /0
phase delay
 = −/
group delay
 = −⁄
−

1
,  =

� 
 = +
+
1

1

= 
−1/
−
ℎ
1
, ℎ =

� 
ℎ = −
ℎ
2
ℎ
2
= −ℎ
The phase for the low-pass, high-pass and the combined NRI-TL unit cells are shown below in
Figure 2.10. The phases for the low-pass and high-pass unit cells are located in the lower-half
and upper-half planes, respectively. The directional arrows indicate that both phases decrease
(become more negative) and move clockwise around the unit circle as frequency is increased −
an indication of positive group delay for all cases shown.
31
Figure 2.10: S21 polar plots depicting the phase shift for (a) the low-pass unit cell (blue, bottom half
plane) and the high-pass unit cell (red, upper half plane) and (b) the NRI-TL unit cell, which is effectively
the sum of both the low-pass and high-pass unit cells and capable of both positive and negative phase
shifts.
The phases shown in Figure 2.10 are only for single unit cells and only illustrate the phase over a
narrow range or bandwidth where equation (2-41) is valid – in accordance with the relations
given in section 2.2.4. As the phase increases, the magnitude of S21 decreases as a result of the
return loss increasing, where the 3dB cutoff frequencies occurring at a phase of ±1350 indicating
by the intersection with the dotted inner circles. The phase range can be increased while also
maintaining the Bloch impedance and thus matching by increasing the number of unit cells. By
using the more precise scattering parameter equations given in (2-39), the phase and magnitudes
over the whole frequency range are shown in Figure 2.11.
Figure 2.11: S21 polar plots over entire frequency range for (a) the low-pass unit cell (bottom half plane)
and the high-pass unit cell (upper half plane) and (b) the NRI-TL unit cell, which is effectively the sum of
both the low-pass and high-pass unit cell responses and capable of both positive and negative phase shifts
32
2.6 Selection of Lossless MTM-TL for Group Delay Minimization
The design of a MTM-TL phase shifter with increased phase bandwidth can be accomplished by
reducing the group delay at the operating frequency. Therefore, it is instructive to consider the
selection between either a positive or negative phase delay MTM-TL on the basis of minimizing
the group delay. The specifications of the phase shifter required for comparison include the
specified phase shift 0 at a centre frequency 0 . This can also be understood as a requirement
of an equivalent phase delay (i.e.  = −0 /0 ), however the phase specification can be
satisfied by a adding a multiple of the wavelength at the given frequency. For example, a -3600
phase shift can be realized with a transmission line of one guided wavelength (or an integer
multiple thereof) or by using a 00 NRI-TL. In this example, the NRI-TL is clearly superior in that
it is much more compact and more importantly, has far less group delay and thus far greater
bandwidth over which the phase is within a given phase error tolerance of 00. That being said, a
zero length TL is the ideal realization of a 00 phase shifter, however this solution is trivial and is
of little practical use. Therefore, it is instructive to consider phase shifters of an equal or
comparable length, or more accurately to consider a specified minimum port-to-port length.
A series fed antenna array has its radiating elements inter-connected by phase shifters that
must physically span the length that the radiating elements are physically separated by. This
length affects the antenna radiation pattern and should not be so large as to capture grating lobes
in the visible region of the array factor and destroying the main beam. On the other hand, it
should not be too small thus producing unwanted inter-element coupling between radiating
elements. The inter-element phase shifters may be implemented using a specified dielectric
medium, such as printed TLs on a dielectric substrate with a specified permittivity εr and thus
phase velocity. The length and phase velocity can be combined to identify the minimum phase
delay and group delay of the host TL of any MTM-TL phase shifter used. This minimum phase
delay can be mitigated with the use of NRI-TL phase shifters, but the group delay will remain
positive. In fact, the high-pass loading of the NRI-TL introduces additional positive group delay
to the otherwise unloaded TL and can therefore have a smaller phase bandwidth under certain
specifications. Therefore, it is instructive to compare the unloaded TL (or the low-pass loaded
TL) against the NRI-TL and determine which of the two produces a lower group delay given the
design criteria.
33
Consider the design of a MTM-TL phase shifter where a frequency 0 , phase 0 (in the range
[0, −2]), minimum port-to-port distance  and dielectric permittivity  are all specified. The
minimum phase delay specification is given by  = √  / (expressed in [0, −2]). The
group delay for the positive phase delay (low-pass MTM-TL) and the negative phase delay NRITL phase shifter can then be expressed as:
2 − 0
0
 () =
⎨ 0
⎩ 
⎧
0
0
,
0
<−
0
 > −

0 − 2

 () =
2
+
2
 − 0
⎨
⎩

⎧
0
0
.
0
<−
0
 > −

(2-47a)
(2-47b)
It can seen (2-47a) and (2-47b) that there are two cases to consider arising from the fact that if
the minimum phase delay may be too large such that the shortest transmission line connection
still produces too much of a phase delay and must therefore be extended further or meandered an
additional wavelength. By comparing the group delays of the positive and negative phase delay
MTM-TL phase shifters, a simple inequality is produced that indicates under what conditions the
NRI-TL has the lower group delay and is given by
 +  < 0 <  .
(2-48)
The result is depicted in Figure 2.12 indicating the ranges for the lower group delay option. It is
worth noting that the condition in (2-48) assumes that the NRI-TL host TL would not have to be
physically extended beyond the minimum phase  to accommodate the number of unit cells
required, which is generally a valid assumption as NRI-TL unit cell can be implemented in a
very short length (i.e. on the order of millimetres). This must be taken into account however
when the minimum distance is not specified, in which case the distance would be minimized and
 becomes the minimum length with which the NRI-TL can be realized. When the minimum
phase is zero, the NRI-TL is then preferable for phases in the upper half plane and the low-pass
MTM-TL is preferable for phases in the lower half plane.
34
Figure 2.12: An illustration of the range of possible phase shifts divided between the unloaded TL
(positive phase delay) and the NRI-TL (negative phase delay) on the basis of the minization of group
delay.
Finally, it is important to note that there are factors in addition to minimizing group delay to
consider, including the variation of group delay (generally referred to as group delay dispersion),
insertion loss and insertion loss variation or ripple, as well as fabrication cost and complexity,
just to name a few. As a result, phase shifts marginally in the NRI-TL domain near the cut-off
phase as depicted in Figure 2.12 may still be better suited for a conventional TL given its
simplicity.
2.7 Negative Group Delay & Beam Squint
An interesting application for the minimization of a phase shifter’s group delay is the reduction
of a linear series-fed antenna array’s beam scan angle variation with respect to frequency. This
variation of an antenna array’s scan angle as depicted in Figure 2.13 is referred to as beam squint
and is typically an undesirable behaviour. It will be shown how beam squint can be greatly
suppressed and even entirely eliminated only with the addition of NGD to the inter-element
phase shifters.
35
Figure 2.13: Series fed antenna array with elements physically spaced by a distance dE and inter
connected through a phase shift  to produce a main beam angle in the  direction.
By using the array factor approach, the main beam angle as depicted above is given by
0 + 2
() = sin−1 � 
�.



(2-49)
By differentiating the main beam angle with respect to frequency, the beam squint is identified
and given by
2
  −  + 
=
.


cos


(2-50)
From the above equation, beam squint can be eliminated if and only if the numerator vanishes.
This requires that the m=0 main lobe be captured in the visible region of the array factor. This
cannot be achieved by a conventional transmission line phase shifter. The magnitude of the interelement phase shift in (2-49) must necessarily be larger than the denominator, which is the freespace phase delay, since a conventional phase shifter cannot have a phase velocity that exceeds
the speed of light or a distance less than the physical distance between the elements. The m=0
main lobe can however be captured if and only if negative phase delay is used and thus eliminate
the terms with ‘m’ in (2-50). The total phase delay need not be negative, but rather must only be
less than the phase delay associated with free space as given in the denominator of (2-49). This
ensures that the arcsine function’s argument magnitude in (2-49) is less than one and the
resulting main beam angle is real valued. However, the numerator is still present if the group
36
delay positive. Therefore, beam squint can be eliminated with simultaneous use of superluminal
phase and group delay. This can be summarized as such: to eliminate beam squint, the interelement phase delay as a function of frequency must be a true-time delay less than that of the
delay in an equivalent length of free space. That is, the phase shift must be within the light-line
and be linear as depicted in Figure 2.14. This necessitates the use of NGD to fully eliminate
beam squint. Because the main beam angle monotonically increases as frequency is increased for
all positive group delay phase shifters, NGD enables the beam angle direction to reverse. If the
main lobe’s angle is permitted to oscillate about a nominal beam angle as frequency in swept, the
phase shift need not be linear. The tolerable beam angle range would ostensibly be subject to the
beam width of the main lobe, where a wider beam width would tolerate more beam squinting.
Figure 2.14: A non-linear phase response exhibiting bandwidth of squint free operation
Without NGD however, the analysis of section 2.7 can be used to select between positive and
negative phase delay on the basis of beam squint minimization instead. The group delay
minimization expression in equation (2-48) can be adapted to a beam squint minimization
expression when a substrate with permittivity εr and inter-element spacing dE and main beam
angle are specified. Therefore, the negative phase or NRI-TL phase shifter produces less beam
squint when the following inequality is satisfied:

> � − sin  .
2
(2-51)
If this inequality is not satisfied, then a conventional unloaded TL should be used or if space is of
concern, a low-pass loaded MTM-TL phase shifter.
37
3
The NGD MTM-TL Unit Cell
A MTM-TL unit cell exhibits NGD when the transfer function of the impedance network Z (and
thus the transmission scattering parameter S21) has left half plane zeroes, which may be
accomplished with the addition of anti-resonance circuits. However, zeroes without some
accompanying loss will have an infinite quality factor and thus lie on the jω-axis, producing a
nominally infinite NGD over a zero bandwidth. A practical physical realization will of course
have some loss and thus a large but finite NGD, albeit over a very narrow bandwidth. Instead,
resistors are added to produce lower quality factor zeroes which produce an appreciably wide
bandwidth over which NGD is present. This is can be seen by considering several filter
transformations of the 3rd order Butterworth filter prototype depicted in Figure 3.1. The low-pass
case has no zeroes and thus cannot have NGD. The high-pass case does have zeroes, but they are
all located at the origin. The low-pass to band-stop frequency transformation shown in Figure
3.1(c) has zeroes located at non-zero frequencies, but they are all infinite quality factor (since
they lie on the jω-axis). The addition of resistance into the unit cell can, in effect, shift all poles
and zeroes further into the left-hand plane, thereby decreasing their respective quality factors.
Because all zeroes are closer to the jω-axis than the poles are, the zeroes effect on the transfer
function are greater producing a reduction in both gain and group delay about the centre
frequency.
Figure 3.1: Pole-Zero mapping for the 3rd order Butterworth maximally flat filter prototype for the (a)
low-pass configuration, (b) high-pass configuration, (c) band-stop configuration and (d) band-stop
configuration with shifted poles and zeroes by adding resistors.
The proposed NGD MTM-TL unit cell is shown below in Figure 3.1 with its corresponding
series impedance shunt admittance expressed in (3-1a) and (3-1b), respectively. The NGD unit
38
cell topology resembles that of a band-stop filter with the addition of resistors as previously
described. Because of the addition of resistors, the impedance of the NGD unit cell always
remains finite even when the susceptance of the parallel L-C becomes infinite at its resonance
frequency. To understand how NGD is produced, consider how the phase shift produced by the
MTM-TL is related to the reactance. The series impedance of the NRI-TL is dominated by the
capacitor at low frequency as it behaves as an open circuit and conversely, is dominated by the
inductor at high frequencies where it again behaves as an open circuit. As the frequency
increases past the resonance frequency, the series reactance vanishes and the reactance goes from
being negative (capacitive) to positive (inductive). The NGD unit cell behaves in an opposite
manner. Its series impedance is dominated by the inductor at low frequencies where it behaves as
a short circuit, negating the impedance of the resistor and capacitor in parallel. Conversely, the
capacitor dominates at high frequencies where it then behaves as a short. As the frequency
increases towards the resonance frequency, the parallel L-C susceptance diverges to positive
infinity and produces a zero at the resonance frequency. Without the parallel resistor, the total
series reactance behaves in a manner similar to Figure 2.9, where the reactance is asymptotic on
negative infinity above the resonance frequency and would monotonically increase. However the
reactance remains finite and bounded because of the parallel resistor. Indeed, the network
behaves as a purely resistive attenuator at the resonance frequency with the peak insertion loss
also occurring. Therefore, the reactance now goes from being inductive to capacitive and thus the
phase is effectively increasing over a non-zero bandwidth, producing NGD. This will become
more evident upon inspecting Figure 3.2(b).
Figure 3.2: The NGD MTM-TL unit cell
39
 −1 =  +  + 1⁄ ,
 −1 =
(3-1a)
1
+  + 1⁄ .

(3-1b)
As required for a MTM-TL unit cell, the broadband matching condition ensures that the
normalized series impedance z and shunt admittance y (normalized to a characteristic impedance
Z0) are equivalent and also represent the propagation factor. As a result, all three resonators in
the NGD unit cell share the same resonance frequency and quality factor. The impedance given
in (3-1a) can then be re-written in a more compact form to express the propagation factor
expressed as
===
=
 0
=
,
0 
=

  ,
1 +  � − 0 �
0

1 
�
=  �
,
 

0 =
1
� 
(3-2a)
=
1
� 
.
(3-2b)
Although the NGD unit cell contains three resonators (and is therefore a 6th order network), the
MTM-TL conditions in section 2.2 reduce the equations to that of a second order RLC resonance
circuit. It is also important to note the scalability of the NGD unit cell. Similar to a filter
prototype response like the Butterworth filter for example, the NGD unit cell is scalable in both
characteristic impedance and frequency. The variables used in (3-2a) are chosen to correspond to
typical resonator parameters − the resonance frequency 0 and quality factor Q – and have the
same properties. The variable A is used to represent the normalized resistance and conductance
and will be discussed further later. At resonance frequency, the imaginary portion of the
denominator of (3-2a) vanishes producing a purely real propagation factor and so the NGD unit
cell behaves as a purely resistive attenuator producing the maximum attenuation. It will be
shown that the maximum NGD also occurs at this frequency. The S21 response of the NGD unit
cell can be found using (2-40) and (3-2a) and is given by
21 () =  − = exp �−

  �.
1 +  � − 0 �
0
(3-3)
40
The S21 response of a unit cell with a normalized resonance frequency (0 = 1) and 3dB
maximum insertion loss is depicted in a polar plot as well as the corresponding phase and
magnitude plots in Figure 3.2 below. The left figure is a polar plot indicating the locus of S21,
which almost forms a perfect circle. The right figure below is the corresponding magnitude and
phase plot, which is related to the polar plot by tracing the imaginary and real parts S21 as a
function of frequency.
ω = 0, ∞
ω=1
Figure 3.3: (a) Polar plot of both S21 and S11 (located at the origin) and (b) S21 phase and magnitude
versus frequency for a NGD Unit cell (0 = 1 ,  = 2,  = 0.338).
The phase response exhibits the upward slope indicative of NGD near the resonance frequency.
Although the overall phase response is highly non-linear towards the edge frequencies, the
bandwidth centred at the resonance frequency is relatively linear and can be used in conjunction
with a positive group delay phase shifter to produce a resulting zero group delay phase shifter
with increased phase bandwidth.
Although the analysis of the MTM-TL is contingent on broadband impedance matching, the
reflection coefficient S11 of the NGD unit cell can still be computed prior to invoking condition
three of section 2.2.3. In fact, the S11 locus is also included in the polar plot in Figure 3.2. It is
located in the vicinity of the origin but is difficult to see as it is almost identically zero at all
frequencies. This indicates that the NGD unit cell’s return loss is very low at all frequencies,
unlike the lossless MTM-TL unit cells. It will be shown that the maximum return loss is actually
only dependent on the maximum insertion loss A. Therefore, provided A is sufficiently small, the
phase and magnitude given (3-3) is valid at all frequencies.
41
The transmission phase 21 and magnitude or insertion loss (IL) of the NGD unit cell can be
obtained by applying the logarithm to (3-3) and separating the real and imaginary parts. The
insertion loss and phase may be simply re-expressed by
 = |21 ()| =
21
20

∙
,
ln 10 1 +  2  − 0 2
�
�

0
02 (2 − 02 )
= arg[21 ()] = − 2 2
.
 ( − 02 )2 +  2 02
(3-4)
(3-5)
where the gain is expressed in decibels (hence the additional constant in (3-4) to convert from
Nepers) and the phase is expressed in radians. The group delay can readily be computed by
differentiating the phase in (3-5) with respect to frequency and is given by:
0
0
2
2
2
2
2
2
0 ( + 0 ) � −   − 0 � � +   − 0 �
 =
∙
.
2

1
2 2
2
4
� + � 2 − 2� 0  + 0 �

(3-6)
The preceding equations appear to be complicated, so to better understand the relationship
between the variables A, Q and 0 , critical points of the NGD unit cell’s phase and magnitude
response are identified and illustrated in Figure 3.3 below.
42
ILmax
Δ
,
Δ
Figure 3.4: NGD unit cell S21 phase and magnitude versus frequency for a NGD Unit cell (0 = 1 ,  =
2,  = 0.338), depicting several properties and relationships between the group delay, loss and
bandwidth.
By identifying the absolute maxima of the relations given in (3-4) and (3-6) (by finding the roots
of the derivative with respect to frequency), the maximum NGD (i.e. the absolute minimum
group delay) and the maximum insertion loss are found to simultaneously occur at the resonance
frequency 0 and as a result, the relations given in (3-4) and (3-6) greatly reduce to
(0 ) = 0 ,
 = (0 ) =  ,
, = − (0 ) =
2
.
0
(3-7)
(3-8)
(3-9)
The maximum insertion loss given by (3-4) occurs at the resonance frequency and is simply
given by A when expressed in Nepers. Of course, the insertion loss can simply be converted
from Nepers to decibels by simply multiplying by 8.686. The frequencies at which the insertion
loss reduces to half of the maximum insertion loss (i.e. the 3dB cut-off frequencies) coincide
with the frequencies at which the phase is maximum and minimum (i.e. when the phase slope or
±
group delay is zero). These frequencies are denoted as 
, with the ± indicating the upper and
43
lower frequencies. These frequencies as well as the NGD bandwidth Δ (i.e. the bandwidth
over which the group delay is negative) expressed in (3-10) and are annotated on Figure 3.3 and
can be derived from either (3-4) or (3-6) as such
±

= 0 �1 +
+
−
Δ = 
− 
=
1
0
±
,
4 2
2
0
,

+
−


= 02 .
(3-10a)
(3-10b)
These relations can be used to determine the peak to peak phase difference Δ as well as the
average group delay in the NGD bandwidth − the ratio the peak to peak phase and the NGD
bandwidth – and are given by:
Δ =  −  =  ,
Δ
 ,
=
=
.
Δ 0
2
(3-11)
(3-12)
The result suggests that the NGD unit cell phase response can effectively be stretched vertically
by increasing A, which also amplifies the insertion loss by the same factor. Similarly, the NGD
bandwidth can be stretched by reducing Q, however unless the insertion loss is increased by the
same factor, the maximum NGD occurring at the resonance frequency (and the average NGD
indicated by equation (3-12)) will necessarily be reduced by the same factor. Therefore, an
important conclusion can be deduced from equations (3-8), (3-9) and (3-10b) relating the NGD
bandwidth, maximum NGD and the minimum insertion loss (expressed in terms of Nepers and
not decibels for simplicity) and is given as
1 , ∙ Δ
= 1.
2

(3-13)
This result indicates that neither the maximum NGD or NGD bandwidth can be increased
without decreasing the other or further increasing the maximum insertion loss (and loss at all
frequencies for that matter). Similarly, the loss cannot decrease without having to reduce the
NGD bandwidth or maximum NGD at the resonance frequency. This relationship is similar to
the gain-bandwidth product of an amplifier. However a gain-bandwidth-NGD product is needed
instead.
44
3.1 NGD Phase Shifter Design
The design of a phase shifter with increased phase bandwidth is achieved through phase
flattening (i.e. zero group delay). This is achieved when a positive group delay phase shifter
satisfying the phase, frequency and length specifications, is combined with an impedance
matched NGD unit cell with an equal but negative group delay producing a net zero group delay
over an appreciably large bandwidth. This concept is illustrated in the Figure 3.4, where a host
TL is symmetrically loaded to realize a NGD unit cell producing a broadband phase shifter with
a wider phase bandwidth, albeit with additional insertion loss and insertion loss variation. It is
important to note that the adjoining TL segments indicated in Figure 3.4 may be other MTM-TLs
such as the NRI-TL or any other similarly matched phase shifters for that matter.
Figure 3.5: A NGD MTM-TL unit cell symmetrically cascaded with its host TL producing a broadband
NGD phase shifter.
Figure 3.5 shows a positive group delay linear phase response  with a phase bandwidth BW1,
a NGD phase response  and the resultant phase response  , which is the combination of
the two and exhibits a wider phase bandwidth BW2.
45
Figure 3.6: NGD phase flattening concept increasing the phase bandwidth.
To obtain a flat phase response and increase the phase bandwidth, a zero group delay is ideal.
This section considers the design of a NGD unit cell for the purpose of compensating the positive
group delay of a host phase shifter (i.e. a TL or NRI-TL). The specifications to consider are the
phase shifter’s input impedance Zo, centre frequency ω0, and positive group delay  at the
centre frequency (which is assumed to be relatively constant and not vary excessively in the
operating frequency range). Finally, given the relation in equation (3-13), either a maximum
insertion loss IL or a bandwidth Δ may be specified, but not both. The condition for zero
group delay then can be found by equating the maximum NGD in (3-9) with the sum of both the
group delay of the host phase shifter as well as the group delay of the host TL used to realize the
NGD MTM-TL (which may easily be computed using equation (2-7)). When this sum is denoted
′
by 
, the zero group delay condition is simply given by:
2
′
= 
.
0
(3-14)
This leaves only A and Q to be determined before all the NGD unit cell component values are
known. If the maximum insertion loss (expressed in decibels) or the bandwidth (in radians per
second) is specified, the A and Q values can be determined by (3-15a) or (3-15b), respectively.
 = IL ,
′
0 ∙ 
=2
,
IL
Δ
′
 
=
.
2 IL
(3-15a)
46
=
0
,
Δ
=2
′
′

∙ Δ

∙ Δ
, IL = 2
.


(3-15b)
The newly introduced variable k in (3-15a) and (3-15b) is a correction factor relating the phase
shifter bandwidth and quality factor. The group delay and insertion loss are not constant over the
±
NGD bandwidth and both decrease towards the edge frequencies (
). The phase is very non-
linear at these frequencies as seen in Figure 3.3 (i.e. the group delay rapidly changes from
negative to positive at these frequencies) and signals with significant spectral power density at
these frequencies may undergo significant pulse shape distortion. As a result, the phase shifter
should not operate at these frequencies and so the phase shifter bandwidth cannot operate over
the entire NGD bandwidth. The additional factor k is introduced to effectively decrease the
quality factor thereby increasing the NGD bandwidth. This is accomplished by selecting k such
that 0.2 ≤  ≤ 1, where a lower k value produces a larger NGD bandwidth (avoiding the
potentially problematic edge frequencies), but also increases the insertion loss by a factor of 1/k
as shown in (3-15b.
The expressions for A, Q, 0 and 0 in (3-1b) for the NGD unit cell in Figure 3.1 can be
inverted such that the R, L, and C lumped element component values in the unit cell can be
expressed in terms of A, Q, 0 and 0 instead. Rearranging and inverting the six descriptive
formulas in (3-1b) produces the six design formulas
 = 0 ,
0
 =
,

 =

,
0 0
1
 =
,
0 0
 =
0
.
0
0
 =
.
0
(3-16)
3.2 Maximum Return Loss per NGD Unit Cell
The validity of the MTM-TL analysis is contingent on the third condition described in section
2.2.3, which is necessary to ensure a sufficiently low return loss for the MTM-TL unit cell. For
lossless MTM-TLs, a relationship between the Bloch impedance and phase shift per unit cell was
identified, which also suggests the limitation. This limitation however does not apply when loss
is introduced and was briefly alluded to while discussing the NGD unit cells return loss S11 in
depicted in the Figure 3.2 polar plot. The return loss can be computed by not invoking the 3rd
47
condition of section 2.2.3, but using the relations in (2-39) instead. This is illustrated in Figure
3.6 where the return Loss as a function of frequency for NGD unit cells with equivalent
maximum NGD but with insertion losses ranging from 2dB to 5dB. The return loss clearly
increases as the insertion loss or NGD unit cell variable A is increased. It is also clear that the
maximum return loss occurs at the resonance frequency in all cases.
Figure 3.7: Return Loss vs. normalized frequency for NGD unit cells with equivalent NGD but with
insertion losses ranging from 2dB to 5dB. As the insertion loss increases, so too does the return loss. It is
also evident that the maximum return loss also occurs at the resonance frequency.
At the resonance frequency, the NGD unit cell behaves as a purely resistive attenuator, however
unlike a resistive ‘T’-type attenuator which can be perfectly impedance matched for large
attenuations, the NGD unit cell is not. Using the scattering parameters found in (2-39), the
maximum return loss for the NGD unit cell is determined in (3-17) and is plotted as a function of
the maximum insertion loss A in decibels shown in Figure 3.7.
 = 20 log|11 (0 )| = 20 log �
3
�.
3 + 42 + 8 + 8
(3-17)
48
Figure 3.8: Maximum Return Loss as a function of Maximum Insertion Loss, both occurring at the
resonance frequency.
As the insertion loss per unit cell increases, the return loss similarly increases monotonically. As
a result, the validity of the NGD MTM-TL unit cell analysis and design equations of section 0
equations begin to diminish. Referring to Figure 3.7, it is recommended that the insertion loss per
unit cell be conservatively below 10dB, which should not be an issue. It is worth noting that the
gain-bandwidth-NGD product described in (3-13) also begins to diminish as the insertion loss
increased. Using the more precise scattering parameters, the product in (3-13) can be reexpressed in a more accurate form valid for high insertion losses too and is given by
, ∙ Δ
32 + 8 + 8
= 3
.

 + 42 + 8 + 8
(3-18)
The more accurate gain-bandwidth-NGD product shown (3-18) is approximately equal to one
and monotonically decreases from unity as A increases, therefore the product is optimized when
the conditions of the MTM-TL unit cell are enforced.
49
4 Simulation & Experimental Results
To experimentally verify and demonstrate the use of NGD to produce phase shifters with wider
phase bandwidth at microwave frequencies, planar microstrip TLs are loaded with lumped
elements. The host TL length is selected to produce the required phase shift or phase delay and is
loaded with surface mount lumped elements (i.e. fixed value RF chip inductors, capacitors and
resistors) which realize lossy band stop resonators producing NGD . In addition to NGD,
negative phase delay was also added by loading the TL with additional lumped elements in a
high-pass configuration producing a NRI-TL with NGD. The frequency range selected was
chosen to be in the 0.5 GHz to 1 GHz range. This frequency range was chosen to be in the
microwave regime but low enough such that deleterious parasitic effects in surface RF mount
components were not so large so as to overwhelm the nominal reactances in the lumped element
resonators.
4.1 Microstrip NGD Phase Shifters with Ideal Lumped Elements
All microstrip TL phase shifters are simulated and fabricated on Rogers RT Duroid 5880
substrates with dielectric permittivity, εr = 2.2, dissipation factor δ = 0.0009, copper thickness
T=17μm and conductivity σ =5.8 x 107 S/m. The low permittivity value was selected to
minimize the substrate phase index and thus minimize the amount of the positive group delay of
the host TL. The dielectric heights selected were h=0.787mm as well as h=1.57 mm in different
implementations.
The simulated NGD unit cell schematic used in Agilent ADS design software is shown below
in Figure 4.1. It is comprised of a microstrip TL that is symmetrically loaded. The resonators are
composed of ideal lumped elements. The effects of the microstrip discontinuities (to
accommodate the loading elements) are taken into account by using the microstrip gap (MGAP)
50
components in the Agilent ADS design software. The width of the shunt branch was selected to
be 1mm, a reasonable value to accommodate standard 0603 surface mount components and also
small enough to maintain the small host TL condition. Additionally, the shunt branch contains a
microstrip ‘T’ junction (MTEE) component as well as a microstrip via to the ground, modeling
the ground connection.
Figure 4.1: Microstrip NGD phase shifter schematic with ideal Lumped element models in Agilent
Advanced Design System (ADS) Software
Using this unit cell schematic, several microstrip NGD TL phase shifters are designed and
simulated.
4.1.1
Simulated -300 NGD TL Phase Shifter
A -300 phase shifter with a maximum insertion loss of 3dB at a centre frequency of 1GHz is
designed. Using the equations discussed in chapter 3, with very minor tuning to account for
phase deviation resulting from the non-idealities associated with the microstrip setup (i.e. the
gaps and via), the TL length and lumped element component values were determined to be
d=18mm, Ry=43.4Ω, Rz=57.6Ω, Ly=4.72nH, Lz=13.4nH, Cz=1.89pF and Cy=5.37pF. The results
are shown in Figure 4.2 and Figure 4.3 below.
51
(a)
(b)
Figure 4.2: (a) Group delay and (b) phase of -300 microstrip phase shifter for both a NGD phase shifter
(red/solid) and an unloaded TL (blue/dotted).
(a)
(b)
Figure 4.3: (a) Insertion loss of -300 microstrip phase shifter for both the NGD phase shifter (red/solid)
and an unloaded TL (blue/dotted) and (b) the return loss for the NGD phase shifter (same at both ports).
The unloaded TL return loss below 70dB and not indicated.
The NGD phase response depicted in Figure 4.2(b) shows a significant improvement of the
phase bandwidth over the unloaded TL. The ±20 phase bandwidth for the unloaded TL was
50MHz whereas the ±20 phase bandwidth for the NGD phase shifter was increased to 650MHz.
The phase flattening is a consequence of the added NGD, which is visible from Figure 4.2(b).
The maximum insertion loss shown is 3dB and occurs at 1GHz as expected. Because of the use
of low quality factor resonators for a wideband resonance, the gain ripple is quite low. The return
loss indicated in Figure 4.3(b) remains below 40dB over the entire displayed frequency range.
An unexpected but insignificant resonance in the return loss occurs at 700MHz resulting from
the microstrip gap, junction and via models.
52
4.1.2
Simulated -900 Two-Stage NGD TL Phase Shifter
A -900 phase shifter with a maximum insertion loss of 8dB at a centre frequency of 1GHz is
designed by cascading two NGD unit cells shown in Figure 4.1. Each unit cell was designed to
produce slightly more than half of the required NGD and half of the total loss (i.e. 4dB loss per
cell), but each unit cell has the same bandwidth. When the phase error tolerance is relaxed, the
NGD unit cell may be designed to overcompensate the positive group delay of the host TL
producing an upward phase slope (i.e. NGD) increasing the phase bandwidth further. Using the
design equations in chapter 3 with very minor tuning to account for phase deviation resulting
from the non-idealities associated with the microstrip setup (i.e. the gaps and via), the TL length
and lumped element component values were determined to be d=54mm, Ry=108.6Ω, Rz=23.0Ω,
Ly=19.5nH, Lz=3.3nH, Cz=7.78pF and Cy=1.30pF. The results are shown in Figure 4.4 and Figure
4.5 below.
(a)
(b)
Figure 4.4: (a) Phase and (b) group delay of -900 microstrip phase shifter for both a NGD phase shifter
(red/solid) and an unloaded TL (blue/dot).
53
(a)
(b)
Figure 4.5: (a) Insertion loss of -300 microstrip phase shifter for both the NGD phase shifter (red/solid)
and an unloaded TL (blue/dot) and (b) return loss for the NGD phase shifter (same at both ports). The
unloaded TL return loss below 70dB and not indicated.
The NGD phase response depicted in Figure 4.4(b) shows a significant improvement to the phase
bandwidth over the unloaded TL. The ±50 phase bandwidth for the unloaded TL was 49MHz
whereas the ±50 phase bandwidth for the NGD phase shifter was increased to 650MHz. The
maximum insertion loss shown in Figure 4.5(a) is 8dB and occurs at 1GHz as expected. The
return loss indicated in Figure 4.5(b) remains below 36dB over the entire displayed frequency
range.
4.1.3
Simulated -450 Two-Stage Stagger Tuned NGD Phase Shifter
A demonstration of using two non-identical NGD unit cells that are stagger tuned is presented.
The unit cell schematic is shown in Figure 4.1 and is equivalent to those used in sections 4.1.1
and 4.1.2, but with two cascaded NGD unit cells of different values instead. By using design
equations (3-14)-(3-16), each NGD unit cell is designed with approximately the same bandwidth
and loss of 3dB, however their resonance frequencies are staggered such that net phase and gain
response are significantly flattened in the frequency region between the two respective unit cell
frequencies. The phase and group delay are shown in Figure 4.9. Each NGD unit cell has also
been manually tuned slightly to allow for some group delay ripple. Specifically, the resultant
group delay shown in (b) oscillates around 0 ns, increasing the phase bandwidth compared to a
single NGD unit cell.
54
unit cell 1
unit cell 2
two-stage
unit cell 1
unit cell 2
two-stage
(a)
(b)
Figure 4.6: (a) Phase and (b) group delay of a 2-stage stagger tuned -300 NGD phase shifter. In addition
to the total phase and group delay (red/solid), the first and second NGD unit cell and unloaded -300
(grey/dot-dash) responses are also shown.
input port
output port
unit cell 1
unit cell 2
two-stage
(a)
(b)
Figure 4.7: (a) Return losses of input (blue/dash) and output (red/solid) ports of the combined two-stage
NGD phase shifter and (b) the insertion loss of the NGD phase shifter (red/solid) as well as the individual
first (blue/dotted) and second (purple/dash) NGD unit cells
When each unit cell has approximately the same maximum loss, NGD and the same bandwidth
(i.e. 1/Q), moreover at offset frequencies, the effect is significant. The resulting NGD bandwidth
is greatly increased while only marginally increasing the loss, but the gain variation and group
delay variation over this bandwidth are decreased significantly.
The phase response shown in Figure 4.6(a) maintains an approximately constant phase shift over
the frequency range of 500MHz to 1400MHz. This represents a arithmetic centre frequency and
55
bandwidth of 950MHz and 900MHz, respectively, a 95% relative bandwidth. The maximum loss
of each of the two constituent NGD cells is 3dB, but the total peak loss is only about 4dB as
depicted in Figure 4.9(b). Between approximately 600MHz and 1200MHz, the gain ripple
remains within 0.5dB, a result not seen with one stage. In fact, if phase ripple is further tolerated,
the gain ripple can be further minimized through optimization.
4.1.4
Simulated 00 NGD NRI-TL Phase Shifter & Beam Squint Removal
Using the Modelithics surface mount component models (discussed in section 4.2.3), a hybrid
NRI-TL and NGD unit cell phase shifter was designed and simulated to demonstrate the removal
of beam squint in a series fed antenna array. The response of the NGD NRI-TL is compared
against a simulated NRI-TL phase shifter designed by using 3 surface-mount components in a
symmetric T-configuration (shown in Figure 2.3) with the length d=17mm, series capacitance
Cnri=33pF and shunt inductance Lnri=150nH.
Figure 4.8: NGD NRI-TL phase shifter unit cell
The NGD NRI-TL phase shifter shown in is a hybrid unit cell with both the NRI and NGD
loading components placed in one unit cell. Its component values are given by: Ry=100Ω,
Rz=15Ω, Ly=121nH, Lz=6.2nH, Lnri=51.1nH, Cz=12pF, Cy=0.75pF and Cnri=27pF. The phase
response versus frequency is shown in . The ±20 phase bandwidths of the NRI-TL and NRI-NGD
TL phase shifters are 79 MHz and 553 MHz, respectively. This represents a 700% improvement
of the phase bandwidth.
56
Figure 4.9: Simulated phase response versus frequency for a 00 NRI-TL phase shifter as well as a NRINGD phase shifter
Both phase shifters have minimum return losses of 13dB over the frequency range from 0.6 GHz
to 1.2 GHz. The NRI-TL has an approximately constant insertion loss of 0.13dB, whereas the
NGD NRI-TL has an insertion loss of 2.6dB with a 0.05dB ripple from approximately 0.8GHz to
1.1GHz. This small gain ripple over a wideband is attributed to multiple resonances,
characteristic of parasitic effects resulting from surface mount components.
Figure 4.10: Main beam angle versus frequency for an unloaded TL, NRI-TL, and NGD NRI-TL. The
bandwidths over which the main beam angle remains within ±50 for the conventional TL, NRI-TL, and
NGD NRI-TL are given by 27MHz, 122MHz and 607MHz, respectively.
57
The main beam angles given by (2-49), using the phase shifts produce by the two simulated
phase shifters, in addition to a -3600 meandered TL, are shown in Figure 4.10. The beam angle
for the NGD NRI- TL phase shifter is evidently far flatter than its counterparts. The bandwidths
over which the main beam angle remains within ±50 for the conventional -3600 TL, NRI-TL, and
NGD NRI-TL are given by 27MHz, 122MHz and 607MHz, respectively. Therefore, the NGD
provides a 5 fold improvement for this figure of merit. Finally, if insertion loss coupled with
NGD is of concern, power amplification in conjunction with NGD can be used.
4.2 Experimental Results
4.2.1
Calibration & Measurement Setup
Experimental measurements were obtained using a Vector Network Analyzer (VNA) interfacing
with the microstrip TL through 50Ω Sub-Miniature A (SMA) printed circuit board connectors.
The centre pin of the connectors were soldered to the microstrip TL and the ground plane
soldered to the SMA connector’s exterior to ensure a low resistance connection. Both the thrureflect-line (TRL) calibration techniques and the short-open-load-through (SOLT) calibration
technique were used to calibrate the VNA and remove the unwanted measurements errors.
4.2.2
Planar Substrate Selection & Fabrication
All microstrip TL phase shifters were simulated and fabricated on Rogers RT Duroid 5880
substrates with dielectric permittivity, εr = 2.2, dissipation factor δ = 0.0009, copper thickness
T=17μm and conductivity σ =5.8 x 107 S/m. The low permittivity value was selected to
minimize the substrate phase index and thus to minimize the positive group delay of the host TL.
The dielectric heights selected were h=0.787mm and h=1.57 mm in different implementations.
The microstrip TL and surface mount component pads were fabricated using a photolithographic
wet etching process, whereby the substrate is laminated with a photoresist, which is then exposed
to UV light through a pattern mask. The non-exposed photoresist is then removed by a chemical
solution and the substrate finally placed in a ferric chloride solution to remove the excess copper.
58
4.2.3
RF Surface Mount Component Selection
Although NGD can be demonstrated more easily at low frequencies, frequencies upwards of
1GHz were selected to demonstrate NGD phase shifters in the microwave regime. As a result,
ideal lumped element circuit models are not sufficiently accurate and do not account for parasitic
effects associated with RF surface mount chip components. An alternative solution is to use
vendor provided data files of the component’s scattering parameters in Agilent ADS which can
greatly improve accuracy. These models often do not take the substrate and solder pad
dimensions into account and are thus limited in their accuracy. After several failed attempts at
correctly fabricating phase shifters that agreed with the theoretical design, an alternative solution
was sought using Modelithics enhanced RF surface mount component model library. Each
library contains measurement-based empirical data models obtained with a vector network
analyzer using microstrip interconnects. The models also account for the substrate permittivity
and height as well as the soldering pad dimensions (including gap widths, which ranged from
0.5mm to 0.9mm) on which the surface mount components are placed. The components selected
based on model availability and performance were Coilcraft 0603CS and 0603HP chip inductors
with a 2% tolerance, Murata 0603 GQM1885 and GRM1885 series ceramic chip capacitors for
high frequency applications with varying tolerances (typically less than 10%) and Vishay thick
film chip resistors with 1% tolerance.
After determining the substrate, ideal component values and the available chip components to
realize the lumped element design values, a more comprehensive simulation was necessary.
Modelithics component values were chosen as close to the ideal values as possible as determined
in chapter 3. First, the resistors were selected since they are not as frequency dependent and are
critical to obtaining the proper NGD, insertion loss and impedance match. Optimization
techniques available in Agilent ADS design software was used to select a pair of resistors (the
series and shunt resistors) by verifying that simulation results produce sufficiently low return
loss and the correct insertion loss. The inductor and capacitor components were then determined.
Starting with nominal design values, gradient optimization techniques were used where the
optimization goals were a combination of return loss minimization, typically below 20dB as well
as obtaining the correct NGD. The optimized component values however are not necessarily
commercially available for purchase and must be verified.
59
4.2.4
-300 NGD TL Phase Shifter (3dB loss)
An initial NGD TL phase shifter design based on the circuit shown in Figure 4.11 was
experimentally verified by fabricating a -300 NGD microstrip TL phase shifter operating at 1.0
GHz, with a maximum insertion loss of 2.78 dB. The phase shifter was realized by loading a
50Ω microstrip TL on a Rogers RT Duroid 5880 substrate with relative dielectric permittivity,
εr=2.2 and height, h=1.57mm and is shown in Figure 4.12. A relatively large dielectric height
was selected such that the corresponding 50Ω microstrip TL width would be wider as a result.
The purpose being that the series surface mount components could be soldered directly on the
TL without a need for individual solder pads. The microstrip TL width however had to be
slightly tapered towards the SMA centre pin connection to prevent the microstrip conductor
coming in contact with the SMA external cladding. The shunt components were soldered on
individual copper pads using manufacturer suggested pad dimensions and the via was realized by
drilling a hole through the substrate and inserting a wire to the ground plane. The fabricated
phase shifters component values are given by: Ry=156Ω, Rz=16Ω, Ly=20nH, Lz=3nH, Cy=1.2pF
and Cz=32pF.
Figure 4.11: NGD TL phase shifter schematic
60
Figure 4.12: Photograph of the fabricated NGD unit cell illustrating the T-configuration with 11 lumped
elements present, two additional elements to realize the required series capacitance. Each 0603 SMT
component is approximately 1.5mm in length.
Figure 4.13: Experimentally measured and simulated phase shift vs. frequency for a -300 NGD TL Phase
shifter compared with unloaded microstrip TL. The respective phase bandwidths are indicated.
61
Figure 4.14: Experimentally measured and simulated insertion loss vs. frequency for -300 NGD TL phase
shifter
Figure 4.15: Experimentally measured and simulated return loss versus frequency for the -300 NGD TL
phase shifter
The phase and insertion loss responses are shown in Figure 4.13 and Figure 4.14, respectively.
The ±20 phase bandwidth, determined by the frequency range over which the phase is within -280
to -320, is experimentally found to be 0.61GHz to 1.24GHz. This represents a 0.63GHz
bandwidth. Using the 1.02GHz centre frequency, the relative bandwidth is 63%. In comparison,
the phase bandwidth for the unloaded microstrip TL is between 0.94GHz to 1.08GHz, which
represents only a 14% relative bandwidth. Thus, bandwidth is increased by 450%. The measured
return loss shown in Figure 4.15 was found to be 20dB or less for both input and output ports at
all frequencies between 0.2GHz to 1.35GHz. The measured maximum insertion loss was 3.11dB
and occurred at a lower frequency than the simulated response as shown in Figure 4.14,
62
consequently the experimental phase response differs slightly from the simulated response at
frequencies above 1.2GHz. This discrepancy may be attributed to fabrication errors and more
likely the surface mount component tolerances and parasitic effects, but adequate demonstrates
the broadband phase flattening phase shifter concept.
4.2.5
-300 NGD TL Phase Shifter (2dB loss)
A second NGD microstrip TL phase shifter based on the circuit shown in Figure 4.11 was
designed using Modelithics component models to achieve better design accuracy. The phase
shifter was designed to produce a -300 phase shift operating at 951 MHz and have a maximum
insertion loss of approximately 2dB, nominally producing a smaller phase bandwidth than the
300 NGD TL phase shifter of the preceding section. The phase shifter was experimentally
verified by fabricating a loaded microstrip TL by loading a 50Ω microstrip TL on a Rogers RT
Duroid 5880 substrate with relative dielectric permittivity, εr=2.2 and height, h=0.787mm shown
in Figure 4.16. Because of the 50Ω microstrip TL width of 2.37mm is not wide enough to
accommodate all series surface mount components, the TL width was flared out to accommodate
them, which included an additional capacitor (necessary to realize the required series
capacitance) as shown in Figure 4.16. The fabricated phase shifters component values were
determined using the optimization techniques outlined in section 4.2.3 and are given by:
Ry=210Ω, Rz=5.9Ω, Ly=33nH, Lz=1.0nH, Cy=0.8pF and Cz=28pF (which was realized using two
parallel capacitors instead of one 28pF capacitor, due to availability).
Figure 4.16: Photograph of the Phase shifter. Note that the series capacitance was realized by two
equivalent surface mount chip capacitors in parallel.
63
Figure 4.17: Phase response for a -300 NGD TL phase shifter. Experimental results (solid) as well as
simulation results obtained using both ideal and Modelithics component models
Figure 4.18: Insertion loss for a -300 NGD TL phase shifter. Experimental results and simulation results
obtained using both ideal and Modelithics component models
64
Figure 4.19: Return loss for a -300 NGD TL phase shifter. Experimental results both simulation results
obtained using both ideal and Modelithics component models
The ±20 phase bandwidth, determined by the frequency range over which the phase is within -280
to -320, is experimentally found to be 0.68 GHz to 1.16GHz. This represents a 0.48GHz
bandwidth or a 51% relative bandwidth. This produces a 364% relative bandwidth improvement
compared to the unloaded TL, which is not as large as the 3dB NGD TL phase shifter of the
preceding section. The maximum simulated (using Modelithics models)) and measured insertion
losses were found to be 2.06dB and 2.12dB. It is evident from the Figure 4.18 that the simulated
and measured results all have their maximum insertion losses occurring at different frequencies,
and may be attributed to surface mount component value tolerances or fabrication errors. Finally,
the return loss was both simulated and experimentally verified to be less than 20dB for both
input and output ports (as expected due to symmetry) for all frequencies between 0.2GHz to
1.35GHz. This suggests that the NGD TL phase shifter may be cascaded with itself to produce
the additive response expected of a MTM-TL described in section 2.3.
4.2.6
00 NGD NRI-TL Phase Shifter
The phase shifter demonstrated in sections 4.2.4 and 4.2.5 demonstrate the use of NGD with
positive phase delay microstrip TLs, however a NGD unit cell can be also be integrated with a
NRI-TL with negative phase delay. A 700 MHz 00 NGD NRI-TL Phase Shifter based on the
circuit shown in Figure 4.20 was experimentally demonstrated with the fabricated circuits
65
photograph shown in Figure 4.21. The increase of phase bandwidth of a NRI-TL by adding NGD
is illustrated by comparing a NGD NRI-TL phase shifter against an equivalent length NRI-TL
without NGD. The 00 NRI-TL phase shifter without NGD was designed with an equivalent
15mm length by using two NRI-TL unit cells with NRI component values of LNRI=56nH, and
CNRI=27pF. The schematics for both the NGD NRI-TL and NRI-TL phase shifters are shown
below in Figure 4.20.
(a)
(b)
Figure 4.20: (a) 15mm two stage 00 NRI-TL phase shifter circuit (b) 15mm NGD NRI-TL phase shifter
circuit
(a)
(b)
Figure 4.21: (a) Photograph of the fabricated NGD NRI-TL phase shifter and (b) a magnified view of the
loading elements
66
Like the preceding phase shifters, surface mount components are loaded on a 50Ω microstrip TL
built on a Rogers RT Duroid 5880 substrate with relative dielectric permittivity, εr=2.2 and
height, h=0.787mm shown in Figure 4.21. By limiting the insertion loss between 3 and 4dB and
using the component selection and design process discussed in section 4.2.3, the phase shifter
surface mount component values were determined to be Ry=100Ω, Rz=10Ω, Ly=30nH, Lz=3.6nH,
LNRI=56nH, Cy=2.7pF, Cz=11pF and CNRI=27pF.
The experimentally measured and simulated phase shifts for the NGD NRI-TL phase shifter is
shown in Figure 4.22, including the two-stage NRI-TL simulated phase response for comparison.
It is clear that the NGD NRI-TL phase response is flattened between 400MHz and 100MHz due
to the addition of NGD, however the simulated NGD NRI-TL is flatter. The flatness of the
simulations results from surface mount component non idealities captured by the Modelithics
models but not physically realized in the fabricated circuit. The group delay as a function of
frequency is Figure 4.23. Although the simulated NGD NRI-TL achieves a zero group delay, the
measured group delay although lower than the NRI-TL was unable to achieve NGD. This is
easily seen in Figure 4.3 which depicts the respective phase shifters’ group delays as a function
of frequency.
Simulated NRI-TL
Simulated NGD NRI-TL
Measured NGD NRI-TL
Figure 4.22: Phase responses of the simulated and experimentally measured NGD NRI-TL phase shifter
and the simulated NRI-TL phase shifter of equal lengths.
67
Simulated NRI-TL
Simulated NGD NRI-TL
Measured NGD NRI-TL
Figure 4.23: Group delays versus frequency of the simulated and experimentally measured NGD NRI-TL
phase shifter and the simulated NRI-TL phase shifter of equal lengths.
The insertion loss responses are shown in Figure 4.24. The maximum measured and simulated
NGD NRI-TL insertion losses are 3.37dB and 3.7dB, respectively. The simulated NRI-TL
insertion loss was about 0.1dB, as expected from an ideally lossless design. The simulated NGD
NRI-TL insertion loss response is considerably flat between 525MHz and 830MHz where it
varies by 0.1dB only. The need to achieve the right combination of commercially available
components made the design of a phase shifter simultaneously meeting return loss and group
delay specifications more difficult than those the preceding two sections. The maximum
measured return loss for both ports of the NGD NRI-TL was found to be 14.1dB, significantly
higher than that of the simulation, which had a peak return off of 18dB as indicated Figure 4.25.
Because the NGD NRI-TL has 13 surface mount components, each with its own component
value tolerance and frequency variance, obtaining broadband impedance matching can be
difficult. Although the NGD NRI-TL circuit is symmetric, a discrepancy between the return
losses for each port is seen in Figure 4.25.
68
Simulated NRI-TL
Simulated NGD NRI-TL
Measured NGD NRI-TL
Figure 4.24: Insertion loss response for the simulated and measured NGD NRI-TL phase shifter and an
equal length NRI-TL.
Simulated NRI-TL
Simulated NGD NRI-TL
Measured NGD NRI-TL
Figure 4.25: Return loss response for the simulated and measured NGD NRI-TL phase shifter and an
equal length NRI-TL. Two different port return losses for the measured NGD NRI-TL are shown.
69
5 Conclusions & Future Work
5.1 Conclusions
A novel type of passive microwave phase shifter using lossy resonators loaded on a transmission
line (TL) to produce negative group delay (NGD) thereby increasing the phase bandwidth was
proposed in this thesis. The concept of a loaded TL metamaterials that gave rise to the negative
refractive index TL (NRI-TL) was extended to produce NGD. Unlike the NRI-TL which loads a
TL with lumped elements in a balanced high-pass configuration producing negative phase delay,
the NGD unit cell loads a TL in a band-stop configuration with loss, maintaining wideband
impedance matching.
A loaded TL with arbitrary loading impedances including loss was analyzed using periodic
dispersion analysis in conjunction with conventional linear filter theory. The resulting
expressions were used to characterize the broadband matched NGD unit cell and allow for both
impedance and frequency scalability. A fundamental limitation relating the insertion loss,
bandwidth and maximum NGD achievable by the NGD unit cell was established indicating an
inescapable tradeoff between the three. As a result, an increase in NGD, bandwidth or gain
(reciprocal of insertion loss) requires a proportional decrease in the others. This suggests a
maximum limit on the achievable performance, much like the gain-bandwidth product of an
amplifier, but a gain-bandwidth-NGD product is required instead.
The design of NGD phase shifters was presented whereby a NGD unit cell may be used in
conjunction with other positive group delay phase shifters (i.e. a simple TL segment or even a
NRI-TL with negative phase delay). The design methodology considers the specified nominal
phase and frequency, maximum insertion loss and minimum bandwidth, while minimizing phase
and gain variation. The potential to significantly reduce the group delay and gain variations of
70
the NGD unit cell even further by use of cascaded stagger-tuned cells was briefly explored as
well. The selection between either positive or negative phase delay on the basis of group delay
minimization was considered. The significance of NGD in the reduction and complete
elimination of beam squint in series fed antenna arrays was also discussed.
Several NGD phase shifters have been successfully designed, fabricated and experimentally
verified in the UHF band upwards of 1 GHz. Printed microstrip TLs were loaded with passive
RF surface mount inductors, capacitors and resistors to realize NGD unit cells. An initial NGD
TL phase shifter with positive phase delay was designed and fabricated demonstrating a phase
bandwidth improvement upwards of 450% over the unloaded microstrip TL phase shifter. A
second NGD TL phase shifter with built to demonstrate the trade-off between insertion loss and
phase bandwidth. Finally, NGD was incorporated with a NRI-TL to simultaneously produce
positive phase delay and NGD over a wide bandwidth. Measured data shows good agreement
with simulated data, with discrepancies attributed to RF surface mount components’ non-ideal
parasitic effects and tolerances. Potential for improvement
5.2 Future Work
The use of NGD in microwave applications is promising. Although previous demonstrations are
narrowband, dispersive and suffer high losses, careful design can mitigate such issues. The use
of multiple stages of wideband passive NGD unit cells in addition to gain stages in a distributed
amplification topology holds promise to alleviate the loss problems. The stagger tuning filter
design concept can be used to alleviate the problems of gain and group delay ripple and further
increase the amount of NGD. Tuneability may also be an avenue for further research and
development, whereby the component values are no longer fixed allowing for tuning of the
frequency, delay or loss.
Although NGD may increase a phase shifter’s bandwidth, its use in power dividing and
combining applications (i.e. Hybrid coupler, Wilkinson power divider and combiner, etc) as a
means of increasing their bandwidths is not suitable given the accompanied insertion loss.
Although it has been demonstrated in the past, NGD can reduce or eliminate beam squint in a
series fed antenna array at broadside angles. At non broadside beam angles, NGD is still
beneficial for reducing beam squint since it allows the scan angle to increase with increasing
71
frequency, a characteristic only present in NGD phase shifters. Therefore, if the main beam angle
is permitted to oscillate within a given range (the degree to which would ostensibly be subject to
the main lobe’s beam width), the beam squint need not be identically zero over the entire
required bandwidth, but can oscillate around zero. A planar antenna array could be fabricated
using the proposed NGD and NRI phase shifters (with amplification if necessary) to produce a
novel series-fed array antenna with no beam squint over an appreciably large bandwidth.
Although it is Furthermore, the NGD unit cell components may be fabricated with distributed
elements, which may be more suitable for planar implementation and may also lend to
controlling and mitigating parasitic effects.
72
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