Broadband Microwave Negative Group Delay Transmission Line Phase Shifters

Transcription

1 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

2 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

3 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

4 Table of Contents Acknowledgments... iii Table of Contents... iv List of Tables... vi List of Figures... vii 1 Introduction Motivation Background of Dispersion Engineered Metamaterials Negative Refractive Index Transmission Line Metamaterials Negative Group Delay Microwave Applications of Negative Group Delay Proposal Thesis Outline Loaded Transmission Line Metamaterials Generalized Transmission Line Theory Broadband Impedance Matching & the Distortionless Condition Generalized Loaded TL Analysis Condition 1: Host TL Length Condition 2: Broadband Impedance Matching Condition 3: Small Propagation Factor Unit Cell Bloch Impedance and Phase Relationship The Metamaterial TL Unit Cell Foster Reactance Theorem, Energy Storage and Group Delay Lossless MTM-TL Phase shifters Selection of Lossless MTM-TL for Group Delay Minimization iv

5 2.7 Negative Group Delay & Beam Squint The NGD MTM-TL Unit Cell NGD Phase Shifter Design Maximum Return Loss per NGD Unit Cell Simulation & Experimental Results Microstrip NGD Phase Shifters with Ideal Lumped Elements Simulated NGD TL Phase Shifter Simulated Two-Stage NGD TL Phase Shifter Simulated Two-Stage Stagger Tuned NGD Phase Shifter Simulated 0 0 NGD NRI-TL Phase Shifter & Beam Squint Removal Experimental Results Calibration & Measurement Setup Planar Substrate Selection & Fabrication RF Surface Mount Component Selection NGD TL Phase Shifter (3dB loss) NGD TL Phase Shifter (2dB loss) NGD NRI-TL Phase Shifter Conclusions & Future Work Conclusions Future Work References v

6 List of Tables Table 2.1: A comparison of the low-pass and high-pass first-order MTM-TL unit cells vi

7 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 Figure 2.2: Generic Transmission Line Equivalent Circuit Model of arbitrary series impedance per unit length Z and shunt admittance per unit length Y Figure 2.3: Circuit model of the spatially extended unit cell for the generically loaded TL Figure 2.4: Equivalent lumped element L-section circuit models for host TL Figure 2.5: Generic unit cell with host TL sections replaced with lumped element equivalent circuits Figure 2.6: Equivalent Lumped element Generic Unit cell. Host TL effects absorbed by Z and Y Figure 2.7: The phase, gain and delays of the network comprised of cascaded MTM-TL Figure 2.8: MTM-TL s equivalent circuit Figure 2.9: Reactance of a 6 th order network displaying the monotonic increase of reactance with increasing frequency per Foster Reactance Theorem Figure 2.10: S 21 polar plots depicting the phase shift for the low-pass unit cell and the highpass unit cell and the NRI-TL unit cell Figure 2.11: S 21 polar plots over entire frequency range for the low-pass unit cell and the high-pass unit cell and the NRI-TL unit cell vii

8 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 Figure 2.13: Series fed antenna array with elements physically spaced by a distance d E and inter connected through a phase shift φ 0 to produce a main beam angle in the θ direction Figure 2.14: a non-linear phase response that is approximately linear over a bandwidth exhibiting simultaneous superluminal phase and group delay Figure 3.1: Pole-Zero mapping for the 3rd order Butterworth filter responses Figure 3.2: The NGD MTM-TL unit cell Figure 3.3: (a) Polar plot of both S 21 and S 11 (located at the origin) and (b) S 21 phase and magnitude versus frequency for a NGD Unit cell (ω 0 = 1, Q = 2, A = 0.338) Figure 3.5: NGD unit cell S 21 phase and magnitude versus frequency for a NGD Unit cell Figure 3.6: A NGD MTM-TL unit cell symmetrically cascaded with its host TL producing a broadband NGD phase shifter Figure 3.7: NGD phase flattening concept increasing the phase bandwidth Figure 3.8: Return Loss vs. normalized frequency for NGD unit cells with equivalent NGD but with insertion losses ranging from 2dB to 5dB Figure 3.9: Maximum Return Loss as a function of Maximum Insertion Loss, both occurring at the resonance frequency Figure 4.1: Microstrip NGD phase shifter schematic with ideal Lumped element models in Agilent Advanced Design System (ADS) Software Figure 4.2: (a) Group delay and (b) phase of microstrip phase shifter for both a NGD phase shifter (red/solid) and an unloaded TL (blue/dotted) Figure 4.3: (a) Insertion loss of microstrip phase shifter for both the NGD phase shifter and an unloaded TL and (b) the return loss for the NGD phase shifter viii

9 Figure 4.4: (a) Phase and (b) group delay of microstrip phase shifter for both a NGD phase shifter and an unloaded TL Figure 4.5: (a) Insertion loss of 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) Figure 4.6: (a) Phase and (b) group delay of a 2-stage stagger tuned NGD phase shifter. In addition to the total phase and group delay, the first and second NGD unit cell and unloaded 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 Figure 4.8: NGD NRI-TL phase shifter unit cell Figure 4.9: Simulated phase response versus frequency for a 0 0 NRI-TL phase shifter as well as a NRI-NGD phase shifter Figure 4.10: Main beam angle versus frequency for an unloaded TL, NRI-TL, and NGD NRI-TL Figure 4.11: NGD TL phase shifter schematic Figure 4.12: Photograph of the fabricated NGD unit cell Figure 4.13: Experimentally measured and simulated phase shift vs. frequency for a NGD TL Phase shifter compared with unloaded microstrip TL Figure 4.14: Experimentally measured and simulated insertion loss vs. frequency for NGD TL phase shifter Figure 4.15: Experimentally measured and simulated return loss versus frequency for the NGD TL phase shifter Figure 4.16: Photograph of the Phase shifter ix

10 Figure 4.17: Phase response for a 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 NGD TL phase shifter Figure 4.19: Return loss for a NGD TL phase shifter Figure 4.20: (a) 15mm two stage 0 0 NRI-TL phase shifter circuit (b) 15mm NGD NRI-TL phase shifter circuit Figure 4.21: (a) Photograph of the fabricated NGD NRI-TL phase shifter and (b) a magnified view of the loading elements 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 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 Figure 4.24: Insertion loss response for the simulated and measured NGD NRI-TL phase shifter and an equal length NRI-TL Figure 4.25: Return loss response for the simulated and measured NGD NRI-TL phase shifter and an equal length NRI-TL x

11 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 φ(ω) or alternatively phase delay, τ p = φ ω 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).

12 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 φ 0, a phase error tolerance φ tol may also be specified, i.e. the tolerable phase deviation from the specified nominal phase shift. The resulting bandwidth Δω 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 φ(ω) with phase shift φ 0, at frequency ω 0. A phase error tolerance φ tol and corresponding phase bandwidth Δω 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 Δφ/Δω 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 τ g (ω) = φ ω. (1-1) For a linear of quasi linear phase response, the slope Δφ/Δω is given by the group delay (neglecting the sign) and therefore the maximization of bandwidth can be achieved when the

13 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.

14 4 A notable application of the NRI-TL is the 0 0 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 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 0 0 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.

15 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 3- D[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] 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

16 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 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].

17 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

18 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

19 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. NRI- TL 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

20 10 presented. Finally, chapter five provides a summary of conclusions and discusses the potential for future work.

21 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

22 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]).

23 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 V z 2 + γ 2 V = 0, 2 I z 2 + γ 2 I = 0. (2-1a) (2-1b) Where the frequency dependent complex propagation constant γ (where the prime denotes a per unit length quantity) is defined by γ (ω) = α + jβ = Z Y. (2-2) The attenuation constant α [Nepers/m] and the propagation constant β [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 Z = R + jωl, Y = G + jωc. (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 V(z) = V + e γ z + V e +γ z, I(z) = I + e γ z I e +γ z. (2-4a) (2-4b)

24 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. Z 0 = V + I + = V I and is given by Z 0 = Z Y. (2-5) Finally, the phase velocity and phase delay for a segment of length Δz are defined by considering the imaginary part of complex propagation constant The group delay and group velocity may similarly be defined as: Δτ p = Δz = β Δz. (2-6) v p ω Δτ g = Δz = β Δz. (2-7) v g ω 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) 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 Z c of the TL model shown in Figure 2.1

25 15 γ TL = Z Y = (R + jωl )(G + jωc ), Z c = Z + jωl ) = (R Y (G + jωc ). (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. γ TL,low loss jω L C R + G Z Z 0. (2-9) 0 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 R G = L C = Z Y = Z 0 2. (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 = Z = Y Z 0, Z 0 Z c = Z 0. (2-11a) (2-11b) For a conventional transmission line with series inductance and resistance then, the propagation constant is given by

26 16 R + jωl γ TL = = R + jω L C. (2-12) Z 0 Z 0 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 Z 0 symmetrically loaded in a T-configuration with an equivalent series impedance Z Δz and shunt admittance Y Δz, 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

27 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. A C B 1 Z Δz cos θ = 2 2 j Z 0sin θ cos θ D j Y sin θ 2 2 cos θ 2 Y Δz 1 jy 0 sin θ 2 jz 0sin θ 2 1 Z Δz 2 cos θ (2-13) 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 γ and Bloch impedance Z BL for forward travelling waves (analogous to the characteristic impedance of a TL) using the following two relations cosh(γ Δz) = Z BL = A + D 2 B A 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 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

28 18 jωl TL = Z 0 θ 2 = jz 0β TL Δz 2, (2-15a) jωc TL = Y 0 θ 2 = jy 0β TL Δz 2. (2-15b) where β TL and Z 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 1 + θ 2 2 θ jy 0 2 θ j Z θ 1 j Z 0 2. (2-16) θ jy θ 2 2 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 Z = Z Δz + jωl TL, Y = Y Δz + jωc TL. (2-17a) (2-17b)

29 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 A C B ZY = Z2 Y + 1. (2-18) D ZY Y Using these ABCD parameters, the propagation factor γ (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 cosh γ = ZY. (2-19) By applying hyperbolic trigonometric identities, (2-19) may be re-expressed as sinh 2 γ 2 = 1 4 ZY. (2-20) Similarly, the Bloch impedance can be found using (2-14b) and is given by Z BL = Z Y 1 + ZY 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

30 20 γ = 2 ln 1 + ZY 2 + ZY + ZY 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 cosh 2 x - sinh 2 x =1 producing a relation given as: Z BL = Z Y cosh γ 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 Z Z 0 = Y Y 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 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 z = Z/Z 0 as sinh γ 2 = 1 2 z, (2-25) Z BL Z 0 = 1 + z 2 2 = cosh γ 2. (2-26)

31 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 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 sinh γ 2 γ ! γ ! γ 2 +, (2-27a) cosh γ ! γ ! γ 2 +. (2-27b) Provided the function s arguments are sufficiently small, the two functions can be replaced by their first order approximations given by sinh γ 2 γ 2 when 1 24 γ 2 1, (2-28a) cosh γ 2 1 when 1 8 γ 2 1. (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 γ Z Z 0 = z, (2-29) Z BL Z 0. (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,

32 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 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 sinh 1 2 (α + jφ) = 1 (r + jx). (2-31) 2 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 α 2 cos φ 2 = r 2, (2-32a) cosh α 2 sin φ 2 = x 2. (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 (x > 1 ): φ = π, α 0 (attenuating mode, stop-band) cosh α 2 = x 2, Z BL = j Z 0 sinh α 2. (2-33a) (2-33b) Case 1 (x < 1 ): α = 0, φ 0 (propagating mode, pass-band)

33 23 sin φ 2 = x 2, (2-34a) Z BL = Z 0 cos φ 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 Z 0. Alternatively, the reflection coefficient of a one or more unit cells cascaded and terminated in the impedance Z 0 can found in terms of the phase using the relation in (2-34b) as Γ = Z 0 Z BL = 1 cos φ 2 Z 0 Z BL 1 + cos φ. (2-35a) 2 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 3 rd 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 [S] GUC = A + B Z 0 + C Y 0 + D 1 B C 2 Z 0 Y 0 B 2 C. (2-36) Z 0 Y 0 Because the network is reciprocal, the scattering parameter matrix is symmetrical as expected. Therefore, we need only consider S 11 and S 21 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

34 24 (Z 2 4Z 2 0 )Y + 4Z (2Z 0 + Z)(ZY + 2Z 0 Y + 4) [S] = 8Z 0 (2Z 0 + Z)(ZY + 2Z 0 Y + 4) 8Z 0 (2Z 0 + Z)(ZY + 2Z 0 Y + 4) (Z 2 4Z 2. (2-37) 0 )Y + 4Z (2Z 0 + Z)(ZY + 2Z 0 Y + 4) 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. yz 2 + 4z 4y yz 2 + 4yz + 4(z + y) + 8 [S] = 8 yz 2 + 4yz + 4(z + y) yz 2 + 4yz + 4(z + y) + 8 yz 2 + 4z 4y yz 2 + 4yz + 4(z + y) + 8 (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 z 3 z 3 + 4z 2 + 8z + 8 [S] = 8 z 3 + 4z 2 + 8z z 3 + 4z 2 + 8z + 8. (2-39) z 3 z 3 + 4z 2 + 8z + 8 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

35 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: S 11 S 12 S 21 S 22 MTM 0 e z. (2-40) e z 0 This form is very convenient as the phase and magnitude of the transmission scattering parameters, S 21 and S 12, are given by the real and imaginary parts of the normalized series impedance, z ln S 21 = r, φ 21 = x. (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.

36 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. γ net = γ A + γ B φ net = φ A + φ B (2-42) α net = α A + α B. 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

37 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 dx dω > 0, db dω > 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 6 th order network displaying the monotonic increase of reactance with increasing frequency per Foster Reactance Theorem.

38 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 dx dω = 2W av V 2 0, db dω = 2W av I 2 0. (2-44) where W av 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 S 21 with respect to frequency. This condition is may be expressed as W av = a 1 2 φ 21 ω = a 1 2 τ g. (2-45) where W av is the total time averaged stored energy of the network and a 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: τ g = φ S21 ω = x ω > 0. (2-46)

39 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.

40 30 Table 2.1: A comparison of the low-pass and high-pass first-order MTM-TL unit cells Low-pass High-pass circuit reactance, X susceptance, B frequency limitation phase φ = X/Z 0 phase delay τ p = φ/ω group delay τ g = φ ω ωl z ωc y ω ω lp ω ω lp, ω lp = 1 L z C y τ lp = + 1 ω lp + 1 ω lp = τ lp 1/ωC z 1/ωL z ω ω hp ω hp ω, ω hp = 1 L y C z τ hp = ω hp ω 2 ω hp ω 2 = τ hp The phase for the low-pass, high-pass and the combined NRI-TL unit cells are shown below in Figure 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.

41 31 Figure 2.10: S 21 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 As the phase increases, the magnitude of S 21 decreases as a result of the return loss increasing, where the 3dB cutoff frequencies occurring at a phase of ±135 0 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 Figure 2.11: S 21 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

42 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 φ 0 at a centre frequency ω 0. This can also be understood as a requirement of an equivalent phase delay (i.e. τ p = φ 0 /ω 0 ), however the phase specification can be satisfied by a adding a multiple of the wavelength at the given frequency. For example, a phase shift can be realized with a transmission line of one guided wavelength (or an integer multiple thereof) or by using a 0 0 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 0 0. That being said, a zero length TL is the ideal realization of a 0 0 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 ε r 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.

43 33 Consider the design of a MTM-TL phase shifter where a frequency ω 0, phase φ 0 (in the range [0, 2π]), minimum port-to-port distance d E and dielectric permittivity ε r are all specified. The minimum phase delay specification is given by τ min = ε r d E /c (expressed in [0, 2π]). The group delay for the positive phase delay (low-pass MTM-TL) and the negative phase delay NRI- TL phase shifter can then be expressed as: τ gtl (ω) = 2π φ 0 ω 0 τ min > φ 0 ω 0 φ 0 ω τ min < φ, 0 ω 0 (2-47a) φ 0 2φ min ω τ gnri (ω) = 2π + 2φ min φ 0 ω τ min > φ 0 ω 0 τ min < φ. 0 ω 0 (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 π + φ min < φ 0 < φ min. (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 φ min 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 φ min 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.

44 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.

45 35 Figure 2.13: Series fed antenna array with elements physically spaced by a distance d E and inter connected through a phase shift φ 0 to produce a main beam angle in the θ direction. By using the array factor approach, the main beam angle as depicted above is given by θ(ω) = sin 1 φ 0 + 2mπ ω c d. (2-49) E By differentiating the main beam angle with respect to frequency, the beam squint is identified and given by θ ω = τ 2mπ g τ p + ω. (2-50) ωd E c cos θ 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

46 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 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 ε r and inter-element spacing d E 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: λ 2d E > ε r sin θ. (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.

47 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 S 21 ) 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 3 rd 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 3 rd 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

48 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

49 39 Z 1 = R z + jωl z + 1 jωc z, Y 1 = 1 + jωc R y + 1 jωl y. y (3-1a) (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 Z 0 ) 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 A γ = z = y = 1 + jq ω ω ω, 0 0 ω (3-2a) A = R z Z 0 = Z 0 R y, Q = 1 R z L z C z = R y C y L y, ω 0 = 1 = 1. L z C z L y C y (3-2b) Although the NGD unit cell contains three resonators (and is therefore a 6 th 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 ω 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 S 21 response of the NGD unit cell can be found using (2-40) and (3-2a) and is given by S 21 (ω) = e γ = exp A 1 + jq ω ω ω. (3-3) 0 0 ω

50 40 The S 21 response of a unit cell with a normalized resonance frequency (ω 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 S 21, 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 S 21 as a function of frequency. ω = 0, ω = 1 Figure 3.3: (a) Polar plot of both S 21 and S 11 (located at the origin) and (b) S 21 phase and magnitude versus frequency for a NGD Unit cell (ω 0 = 1, Q = 2, A = 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 S 11 of the NGD unit cell can still be computed prior to invoking condition three of section In fact, the S 11 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.

51 41 The transmission phase φ 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 IL = S 21 (ω) = 20 ln 10 A 1 + Q 2 ω ω 0 ω 0 ω 2, (3-4) AQω 2 φ 21 = arg[s 21 (ω)] 0 (ω 2 ω 2 0 ) = Q 2 (ω 2 ω 2 0 ) 2 + ω 2 2 ω. (3-5) 0 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: τ g = Aω 0 (ω2 + ω 2 0 ) ω 2 ω 0 Q Q ω ω 0 2 ω 2 + ω 0 Q ω ω 0 2 ω Q 2 2 ω 0 2 ω 2 + ω 2 2. (3-6) 0 The preceding equations appear to be complicated, so to better understand the relationship between the variables A, Q and ω 0, critical points of the NGD unit cell s phase and magnitude response are identified and illustrated in Figure 3.3 below.

52 42 IL max Δφ pp τ g,max Δω NGD Figure 3.4: NGD unit cell S 21 phase and magnitude versus frequency for a NGD Unit cell (ω 0 = 1, Q = 2, A = 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 ω 0 and as a result, the relations given in (3-4) and (3-6) greatly reduce to φ(ω 0 ) = 0, (3-7) IL max = IL(ω 0 ) = A, (3-8) τ g,max = τ g (ω 0 ) = 2AQ ω 0. (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 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 ω ± ZGD, with the ± indicating the upper and

53 43 lower frequencies. These frequencies as well as the NGD bandwidth Δω NGD (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 ± ω ZGD = ω Q 2 ± ω 0 2Q, (3-10a) + Δω NGD = ω ZGD ω ZGD = ω 0 Q, ω ZGD + 2 ω ZGD = ω 0. (3-10b) These relations can be used to determine the peak to peak phase difference Δφ pp 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: Δφ pp = φ max φ min = A, (3-11) Δφ pp = AQ = τ g,max. (3-12) Δω NGD ω 0 2 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 τ g,max Δω NGD = 1. (3-13) 2 IL max 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.

54 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 φ TL with a phase bandwidth BW1, a NGD phase response φ NGD and the resultant phase response φ LTL, which is the combination of the two and exhibits a wider phase bandwidth BW2.

55 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 Z o, centre frequency ω 0, and positive group delay τ ps 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 max or a bandwidth Δω ps 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 τ ps, the zero group delay condition is simply given by: 2AQ ω 0 = τ ps. (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. A = IL max, Q = 2 ω 0 τ ps, Δω IL ps = k max 2 τ ps IL max. (3-15a)

56 46 Q = k ω 0 Δω ps, A = 2 τ ps Δω ps, IL k max = 2 τ ps Δω ps. (3-15b) k 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 (ω ± ZGD ). The phase is very nonlinear 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 k 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, Z 0 and ω 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, Z 0 and ω 0 instead. Rearranging and inverting the six descriptive formulas in (3-1b) produces the six design formulas R z = AZ 0, C z = Q Z 0 ω 0, L z = Z 0 Qω 0. R y = Z 0 A, C y = 1 QZ 0 ω 0, L y = QZ 0 ω 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 S 11 in depicted in the Figure 3.2 polar plot. The return loss can be computed by not invoking the 3rd

57 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. RL max = 20 log S 11 (ω 0 ) = 20 log A 3 + 4A 2. (3-17) + 8A + 8 A 3

58 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 τ g,max Δω NGD IL max = 3A2 + 8A + 8 A 3 + 4A 2 + 8A + 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.

59 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, ε r = 2.2, dissipation factor δ = , copper thickness T=17μm and conductivity σ =5.8 x 10 7 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)

60 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 Simulated NGD TL Phase Shifter A 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.

61 51 (a) (b) Figure 4.2: (a) Group delay and (b) phase of 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 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 ±2 0 phase bandwidth for the unloaded TL was 50MHz whereas the ±2 0 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.

62 Simulated Two-Stage NGD TL Phase Shifter A 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 microstrip phase shifter for both a NGD phase shifter (red/solid) and an unloaded TL (blue/dot).

63 53 (a) (b) Figure 4.5: (a) Insertion loss of 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 ±5 0 phase bandwidth for the unloaded TL was 49MHz whereas the ±5 0 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 Simulated 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 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.

64 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 NGD phase shifter. In addition to the total phase and group delay (red/solid), the first and second NGD unit cell and unloaded (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

65 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 Simulated 0 0 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.

66 56 Figure 4.9: Simulated phase response versus frequency for a 0 0 NRI-TL phase shifter as well as a NRI- NGD 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.

67 57 The main beam angles given by (2-49), using the phase shifts produce by the two simulated phase shifters, in addition to a meandered TL, are shown in Figure 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 ±5 0 for the conventional 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 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 Planar Substrate Selection & Fabrication All microstrip TL phase shifters were simulated and fabricated on Rogers RT Duroid 5880 substrates with dielectric permittivity, ε r = 2.2, dissipation factor δ = , copper thickness T=17μm and conductivity σ =5.8 x 10 7 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.

68 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.

69 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 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, ε r =2.2 and height, h=1.57mm and is shown in Figure 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: R y =156Ω, R z =16Ω, L y =20nH, L z =3nH, C y =1.2pF and C z =32pF. Figure 4.11: NGD TL phase shifter schematic

70 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 NGD TL Phase shifter compared with unloaded microstrip TL. The respective phase bandwidths are indicated.

71 61 Figure 4.14: Experimentally measured and simulated insertion loss vs. frequency for NGD TL phase shifter Figure 4.15: Experimentally measured and simulated return loss versus frequency for the NGD TL phase shifter The phase and insertion loss responses are shown in Figure 4.13 and Figure 4.14, respectively. The ±2 0 phase bandwidth, determined by the frequency range over which the phase is within to -32 0, 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,

72 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 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 phase shift operating at 951 MHz and have a maximum insertion loss of approximately 2dB, nominally producing a smaller phase bandwidth than the 30 0 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, ε r =2.2 and height, h=0.787mm shown in Figure 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 The fabricated phase shifters component values were determined using the optimization techniques outlined in section and are given by: R y =210Ω, R z =5.9Ω, L y =33nH, L z =1.0nH, C y =0.8pF and C z =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.

73 63 Figure 4.17: Phase response for a 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 NGD TL phase shifter. Experimental results and simulation results obtained using both ideal and Modelithics component models

74 64 Figure 4.19: Return loss for a NGD TL phase shifter. Experimental results both simulation results obtained using both ideal and Modelithics component models The ±2 0 phase bandwidth, determined by the frequency range over which the phase is within to -32 0, 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 NGD NRI-TL Phase Shifter The phase shifter demonstrated in sections and 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 0 0 NGD NRI-TL Phase Shifter based on the circuit shown in Figure 4.20 was experimentally demonstrated with the fabricated circuits

75 65 photograph shown in Figure 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 0 0 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 L NRI =56nH, and C NRI =27pF. The schematics for both the NGD NRI-TL and NRI-TL phase shifters are shown below in Figure (a) Figure 4.20: (a) 15mm two stage 0 0 NRI-TL phase shifter circuit (b) 15mm NGD NRI-TL phase shifter circuit (b) (a) (b) Figure 4.21: (a) Photograph of the fabricated NGD NRI-TL phase shifter and (b) a magnified view of the loading elements