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Using Loop-Gap Resonators to Characterize the Permeability of Metamaterials at Microwave Frequencies Cornell, Ava H. 2021-04

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Using Loop-Gap Resonators toCharacterize the Permeabilityof Metamaterials at MicrowaveFrequenciesbyAva H. CornellA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFBACHELOR OF SCIENCE WITH HONOURSinIrving K. Barber Faculty of Science(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)May 2021© Ava H. Cornell 2021AbstractA negative-index material (NIM), a metamaterial with simultaneously negative effectivepermittivity and permeability, was composed from periodic arrays of split-ring resonators(SRRs) and aluminum cut wires. The NIM was loaded into the bore of a loop-gap res-onator (LGR) and reflection coefficient measurements were used to characterize its perme-ability. Unexpectedly, many numerical simulations and experimental measurements havesuggested that the imaginary component of the cut wire permeability can be negativewhich implies power generation rather than dissipation. In this project, the reflectioncoefficient measurements were fit to a model proposed by Pendry and coworkers and usedto determine the resonant frequency, magnetic plasma frequency, and damping constantof the metamaterial’s effective permeability. By comparing these parameters with thosefound for arrays of exclusively SRRs, the presence of cut wires was shown to have almostno effect on the permeability of the NIM when in the presence of a pure magnetic field. Infuture research, similar analysis could be done for measurements taken when an externalsource is used to establish a current in the cut wires.iiTable of ContentsAbstract iiTable of Contents iiiList of Tables vList of Figures viAcknowledgements viii1 Introduction 11.1 Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Split-Ring Resonators (SRRs) . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Dimensions and Materials . . . . . . . . . . . . . . . . . . . . . . . 41.3 Cut Wire Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Loop-Gap Resonators (LGRs) . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Measurement Methods 93 Modelling |S11| of An Empty LGR 103.1 The Effective Impedance of a Loop Gap Resonator (LGR) . . . . . . . . . 103.2 Modelling |S11,e| of an Inductively-Coupled LGR . . . . . . . . . . . . . . . 123.3 Plotting |S11,e| for Varying Coupling Constants . . . . . . . . . . . . . . . . 123.4 Experimentally Determining |S11,e| for Varying Coupling Constants . . . . 134 Permeability of SRR Arrays 154.1 Modelling Permeability of SRR Arrays . . . . . . . . . . . . . . . . . . . . 154.2 The Effective Impedance of an LGR Filled with Magnetic Material . . . . 154.3 Plotting |S11,f | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Plotting |S11,f | for Frequency Dependent Permeability . . . . . . . . . . . . 194.5 Investigating the Effect of Negative µ′′ on |S11,f | . . . . . . . . . . . . . . . 204.6 Experimentally Determining |S11| for an Array of SRRs . . . . . . . . . . . 225 Permittivity of Cut Wire Arrays 245.1 Modelling Effective Permittivity . . . . . . . . . . . . . . . . . . . . . . . . 24iiiTable of Contents5.2 Designing Cut Wire Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.1 Modelling Possible Array Dimensions . . . . . . . . . . . . . . . . . 255.2.2 Modelling Permittivity of Our Array . . . . . . . . . . . . . . . . . 275.3 Composing Cut Wire Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3.1 One-Loop-One-Gap Resonator Arrays . . . . . . . . . . . . . . . . . 285.3.2 Two-Loop-One-Gap Resonator Arrays . . . . . . . . . . . . . . . . 285.4 Measuring |S11| of Cut Wire Arrays . . . . . . . . . . . . . . . . . . . . . . 295.4.1 One-loop-one-gap Loop Gap Resonator (LGR) . . . . . . . . . . . . 295.4.2 Two-loop-one-gap Loop Gap Resonator (LGR) . . . . . . . . . . . . 316 Modelling the Effective Impedance of a Partially Filled LGR 347 Permeability of NIM Arrays 377.1 Composing NIM Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 Measuring |S11| of NIM Arrays . . . . . . . . . . . . . . . . . . . . . . . . 387.2.1 One-loop-one-gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.2.2 Two-loop-one-gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.3 Determining the Permeability of NIM Arrays . . . . . . . . . . . . . . . . . 408 Further Investigations 448.1 Applying Current to Cut Wire Arrays . . . . . . . . . . . . . . . . . . . . 448.2 Incorporating Other Metamaterials . . . . . . . . . . . . . . . . . . . . . . 459 Conclusion 46Bibliography 47AppendicesA |S11| Measurements of 1× 1×N NIM Arrays 49B |S11| Measurements of 2× 2×N NIM Arrays 53C |S11| MATLAB Fit for Arrays of NIMs 57ivList of Tables7.1 Best fit parameters for one-loop-one-gap resonator filled with four SRRsand four 3/16-inch diameter Teflon rods. . . . . . . . . . . . . . . . . . . . 427.2 Best fit parameters for one-loop-one-gap resonator filled with four SRRsand four 3/16-inch diameter aluminum cut wires. . . . . . . . . . . . . . . 42vList of Figures1.1 Angle of Refraction as an EM Wave Travels into an NIM . . . . . . . . . . 11.2 Schematic of SRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Infinite Periodic Cut Wire Array . . . . . . . . . . . . . . . . . . . . . . . 51.4 Cross-Section of One-Loop-One-Gap Resonator . . . . . . . . . . . . . . . 61.5 LRC Circuit Representation of Bore of LGR . . . . . . . . . . . . . . . . . 71.6 Cross-Section of Two-Loop-One-Gap Resonator . . . . . . . . . . . . . . . 83.1 LRC Circuit Representation of LGR with Empty Bore . . . . . . . . . . . 103.2 |S11,e| vs. Frequency for Various Coupling Coefficients . . . . . . . . . . . . 133.3 |S11,e| vs. Frequency for Various Coupling Loop Orientations . . . . . . . . 144.1 Model of Complex Permeability of an SRR Array . . . . . . . . . . . . . . 164.2 LRC Circuit Representation of LGR Filled with Magnetic Material . . . . 164.3 |S11,f | vs. f Modelled for Various µ′ and µ′′ Values . . . . . . . . . . . . . . 194.4 |S11,f | vs. f Modelled for Various fp and γ Values . . . . . . . . . . . . . . 204.5 Model of Complex Permeability of an SRR Array with Negative µ′′ . . . . 214.6 Model of |S11| for an SRR Array with Negative µ′′ . . . . . . . . . . . . . . 214.7 Array of one-dimensional N = 4 SRR Array Loaded into LGR . . . . . . . 224.8 |S11| vs. f for One-Loop-One-Gap LGR Loaded with N SRRs . . . . . . . 235.1 Model of εr vs. ka for Array of Ideally Conducting Cut Wires . . . . . . . 245.2 Comparing εr vs. f for An Array of Ideally Conducting Wires with AnArray of Lossy Wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Model of ε′′ vs. f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Comparing εr vs. f for Our Array of Cut Wires (Neglecting Losses) . . . . 275.5 1× 1×N Array of Cut Wires . . . . . . . . . . . . . . . . . . . . . . . . . 285.6 2× 2×N Array of Cut Wires . . . . . . . . . . . . . . . . . . . . . . . . . 295.7 |S11| vs. f Measured for a 1-D Array of N 3/16-inch Diameter AluminumRods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.8 |S11| vs. f Measured for a 1-D Array of N 1/8-inch Diameter AluminumRods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.9 |S11| vs. f Measured for a 2-D Array of N 3/16-inch Diameter AluminumRods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.10 |S11| vs. (f/f0) − 1 Measured for a 2-D Array of N 3/16-inch DiameterAluminum Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.11 |S11| vs. f Measured for a 2-D Array of N 1/8-inch Diameter AluminumRods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33viList of Figures5.12 |S11| vs. (f/f0) − 1 Measured for a 2-D Array of N 1/8-inch DiameterAluminum Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.1 1× 1×N Array of SRRs and Cut Wires . . . . . . . . . . . . . . . . . . . 377.2 2× 2×N Array of SRRs and Cut Wires . . . . . . . . . . . . . . . . . . . 387.3 NIM Array Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . . 387.4 |S11| vs. f for 1 × 1 × N Arrays of SRRs and 3/16-inch Cut Wires whenN = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.5 |S11| vs. f for 2 × 2 × N Arrays of SRRs and 3/16-inch Cut Wires whenN = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.6 |S11| vs. f fit to Pendry’s model for N = 4 1-D Array . . . . . . . . . . . 417.7 µ′ and µ′′ Plotted for One-dimensional N=4 NIM Arrays . . . . . . . . . . 428.1 Experimental Set-Up for Applying Current to Cut Wire Arrays . . . . . . 44A.1 |S11| vs. f for 1 × 1 × N Arrays of SRRs and 3/16-inch Cut Wires whenN = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.2 |S11| vs. f for 1× 1×N Arrays of SRRs and 1/8-inch Cut Wires when N= 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.3 |S11| vs. f for 1 × 1 × N Arrays of SRRs and 3/16-inch Cut Wires whenN = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.4 |S11| vs. f for 1× 1×N Arrays of SRRs and 1/8-inch Cut Wires when N= 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51A.5 |S11| vs. f for 1 × 1 × N Arrays of SRRs and 3/16-inch Cut Wires whenN = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51A.6 |S11| vs. f for 1× 1×N Arrays of SRRs and 1/8-inch Cut Wires when N= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.7 |S11| vs. f for 1× 1×N Arrays of SRRs and 1/8-inch Cut Wires when N= 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52B.1 |S11| vs. f for 2 × 2 × N Arrays of SRRs and 3/16-inch Cut Wires whenN = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53B.2 |S11| vs. f for 2× 2×N Arrays of SRRs and 1/8-inch Cut Wires when N= 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54B.3 |S11| vs. f for 2 × 2 × N Arrays of SRRs and 3/16-inch Cut Wires whenN = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54B.4 |S11| vs. f for 2× 2×N Arrays of SRRs and 1/8-inch Cut Wires when N= 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55B.5 |S11| vs. f for 2 × 2 × N Arrays of SRRs and 3/16-inch Cut Wires whenN = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55B.6 |S11| vs. f for 2× 2×N Arrays of SRRs and 1/8-inch Cut Wires when N= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56B.7 |S11| vs. f for 2× 2×N Arrays of SRRs and 1/8-inch Cut Wires when N= 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56viiAcknowledgementsFirst and foremost, I would like to thank Dr. Jake Bobowski for the opportunity to workwith him on this project. I cannot adequately express how much I appreciate the supporthe has given me in writing this thesis, throughout this project, and over the course of mydegree. I would also like to thank my family for their encouragement throughout all ofmy endeavours.viiiChapter 1IntroductionIn this thesis we will develop a method to experimentally measure the permeability ofmetamaterials composed of arrays split-ring resonators (SRRs) and cut wires. In orderto do this, we will use loop-gap resonators (LGRs) and employ Pendry et al.’s model tocharacterize these arrays [1].1.1 MetamaterialsMetamaterials are materials engineered to have electromagnetic properties that are notfound in natural materials. Specifically, all naturally-occurring materials have a positiveelectric permittivity, εr, magnetic permeability, µr, and refractive index, n. Whereas formetamaterials, at least one of these properties, given by Eqs. (1.1) to (1.3) respectively,is negative [2].εr =εε0(1.1) µr =µµ0(1.2) n =√εrµr (1.3)Note that ε0, the permittivity of free space, has the approximate value of 8.85×10−12 F/mand µ0, the permeability of free space, has the value 4pi × 10−7 H/m. In this project, wewill be focusing on negative index materials (NIMs) which classify a specific type ofmetamaterial that has a negative index of refraction over part of the electromagneticspectrum. For this to occur, the permeability and permittivity must be simultaneouslynegative [3].Figure 1.1: An EM wave moving from a conventional material (with n1 > 0) into an NIM(n2 < 0, dashed line) and a material with a positive refractive index (n2 > 0, solid line).11.1. MetamaterialsThis effect can be further explained by showing an EM wave moving from a conven-tional material into a NIM compared to when it moves into another conventional materialwith a positive refractive index, as shown in Fig. 1.1. To analyze this situation, we willfollow the work done by V. G. Veselago in his 1968 paper “The Electrodynamics of Sub-stances with Simultaneously Negative Values of ε and µ” [4]. To begin we will definethe boundary conditions for the electromagnetic wave as it moves from material 1 intomaterial 2 to be:~E1 = ~E2 (1.4) ~H1 = ~H2 (1.5)ε1E⊥1 = ε2E⊥2 (1.6) µ1H⊥1 = µ2H⊥2 (1.7)Here, we have denoted the electric field as ~E and the magnetic field intensity as ~H.The superscript is used to describe the component of the waves that is parallel to theboundary between material 1 and material 2, while ⊥ describes the component perpen-dicular to the boundary. Lastly, µ1 and ε1 describe the permeability and permittivityof material 1 and µ2 and ε2 describe the permeability and permittivity of material 2 [5].Before we begin our analysis of the effects of µ2 and ε2 on these boundary conditions,we will define ~E to be travelling in the plane of the paper. This results in ~H travellingperpendicular to the page, and thus ~H has no component perpendicular to the bound-ary between material 1 and 2. This yields the trivial result of zero equals zero for theboundary condition denoted by Eq. (1.7).To begin, we will assume that our EM wave is travelling from material 1 into material2 which are both defined to have positive permittivity and permeability, but, ε2 > ε1. AsEqs. (1.4) and (1.5) do not depend on permittivity or permeability and Eq. (1.7) has atrivial solution, we are only interested in Eq. (1.6). For these conditions, Eq. (1.6) yields,E⊥2 =ε1ε2E⊥1 , (1.8)where, from our initial set-up, we can deduce that E⊥2 < E⊥1 . This is consistent withSnell’s law, n1sinθ = n2sinθ′. Now, we can perform a similar analysis for when material2 is defined to have a negative permeability, µ2, and permittivity, ε2, while µ1 and ε1remain positive. Again, we are only interested in analyzing Eq. (1.6), which again yieldsEq. (1.8). However, for this scenario ε1ε2< 0 so E⊥2 is refracted in a different direction andresults in a negative angle of refraction. This is illustrated by the dashed line in Fig. 1.1.Therefore, for this material, Snell’s law results in n1sinθ = n2sin−θ′ = −n2sinθ′. Thus,for Snell’s law to hold, it is required that the refractive index, n2, is negative for materialswith negative permeability, µ2, and permittivity, ε2.In this thesis, we will construct arrays of cut wires as well as arrays of both SRRs and cutwires and characterize their effective relative permeability, µr, at microwave frequencies.21.2. Split-Ring Resonators (SRRs)In “The Complex Permeability of Split-Ring Resonator Arrays Measured at MicrowaveFrequencies” S.L. Madsen and J.S. Bobowski, measured the permeability of SRR arrays[6]. Now, we will work to see what effect, if any, that addition of cut wires has to thesearrays. Previous numerical simulations and experimental measurements have suggestedthat the imaginary component of µr = µ′ − jµ′′ is negative for arrays of cut wires [7].This is unexpected as it suggests power gain rather than dissipation in the NIM.1.1.1 ApplicationsNIMs have many interesting applications which include the creation of super lenses aswell as cloaking devices [2]. Conventional lenses can only focus light on areas equal toor larger than its wavelength squared. Many simulations have suggested that by utilizingNIMs, a super lens could be created that is not limited by wave optics [8]. Additionally,NIMs have been proposed to be able shield an object from view by controlling its EMradiation [9].1.2 Split-Ring Resonators (SRRs)For the purposes of this project, we will define SRRs as planar structures with negligiblelength, composed of two concentric, conducting rings with slits in each, on opposingsides, as shown in Fig. 1.2. In 1999, Pendry et al. proposed SRR arrays exhibit aFigure 1.2: Schematic depiction of an SRR with radius, r, ring thickness, c, and distancebetween conductors, d. Taken from [6].negative effective permeability just above their resonant frequency [1]. He derived amodel for the real, µ′, and imaginary, µ′′, components of the an SRR arrays effectiverelative permeability, which are given by:µr = µ′ − jµ′′ (1.9)31.3. Cut Wire Arraysµ′ = 1− [1− (ωs/ωp)2][1− (ωs/ω)2][1− (ωs/ω)2]2 + (γ/ω)2] (1.10)µ′′ =(γ/ω)[1− (ωs/ωp)2][1− (ωs/ω)2]2 + (γ/ω)2 . (1.11)Here, ωs denotes the SRRs resonant angular frequency, ωs is the SRRs magnetic plasmaangular frequency, and γ is the damping constant, which characterizes the sharpness ofthe resonance.1.2.1 Dimensions and MaterialsThe SRRs used in this project are depicted in Fig. 1.2. They were made by S. L.Madsen and J. S. Bobowski [6]. The SRR itself is secured on FR-4 printed circuit board,shown in green in Fig. 1.2, which has a dielectric constant of about 4.5 at frequency of 1GHz [10] [6]. The circuit board had thickness t = 1.54 mm and side-length a = 21.0 mmand the SRR has dimensions r = 5.56 mm, c = 1.91 mm. d = 0.15 mm [6]. Additionally,2 × 2 planes of four SRRs were composed on printed circuit board of side-length 2a withevenly spaced SRRs in each quadrant.1.3 Cut Wire ArraysArrays of conducting wires are known to have a negative effective permittivity. Maslovskiet al. derived a model describing the effective permittivity of an array of cut wires underquasi-static conditions [11]. This model considered a infinite periodic array of conductingwires as depicted in Fig. 1.3. It was assumed that in these arrays of wires, only EM planewaves in the fundamental Floquet mode can propagate [12]. This derivation resulted inthe permittivity of the array being defined as,ε = ε0(I − 2pi~z0~z0(ka)2 log( a24r0(a−r0))), (1.12)where I is the identity tensor, and the wave number, k, is given by ω√0µ0, where ω isthe angular frequency. Moreover, ~z0 describes the unit vector in the z direction. Thus,the vector product of z0 with itself results in a matrix which isolates the electric fieldtravelling in the z direction, which, in this model, is along the axis of the cut wire array.The cut-off wavelength, which restricts the modes of the plane wave travelling throughthe cut wire array to the Floquet mode, is therefore given by,λ0 = a√(2pi) log(a24r0(a− r0)). (1.13)This result, which only pertains to ideally conducting wires, was then extended to incor-41.3. Cut Wire ArraysFigure 1.3: An infinite periodic array of cut wires. Array has period a and wires haveradius r0. Figure modelled after Fig. 1.4(a) in [13].porate lossy and loaded wires [11]. These wires were described by having an additionaleffective surface impedance,Zs =1 + j√2√ωµ0σ, (1.14)where σ is the metal conductivity. This impedance is added with the impedance of thearrays inductance to yield the complex permittivity,ε = ε0(I − 2pi~z0~z0(ka)2 log( a24r0(a−r0))− jka ar0 Zsη), (1.15)where η, the free-space impedance is given by√µ0/ε0. Overall, similarly to effectiverelative permeability, effective relative permittivity can be expressed conventionally as acomplex value,εr = ε′ − jε′′, (1.16)where ε′ is the real component of permittivity and ε′′ is the imaginary component ofpermittivity when losses are considered. Unlike the permeability of SRR arrays, which isonly negative right above its resonance frequency, the permittivity of cut wire arrays canbe negative over a broad range of frequencies. This is further explored in Chapters 5.1and 5.2.51.4. Loop-Gap Resonators (LGRs)1.4 Loop-Gap Resonators (LGRs)In order to measure the permeability of our arrays of both cut wires and SRRs, loop-gapresonators (LGRs) will be used. An LGR is composed of a long, hollow bore made ofconducting material with a gap down the entire length of the bore and a coupling loopsuspended in one end. The cross-section of a one-loop-one-gap LGR is shown in Fig.1.4(a), while the actual resonator used is pictured in Fig. 1.4(b).(a) Schematic. Figure modified from [14]. (b) LGR used for measure-ments.Figure 1.4: The cross-section of a one-loop-one-gap resonator with gap thickness t, gapwidth w, and bore side-length x.Figure 1.4 shows that the corners of the LGR’s bore are rounded which was done toavoid large current densities that would be expected at sharp corners [6]. However, theside-length of the bore is still given as x and the cross-sectional area is approximated tobe x2. LGRs is can be easily modelled as LRC circuits. An LRC circuit, such as theone shown in Fig. 1.5, is a simple circuit with an inductor, capacitor, and a resistor inseries. Typically, LGR’s are cylindrical with a circular cross-section of inner radius, r0.The shell of the LGR acts as a single turn inductor with inductance, L ≈ µ0pir02l, wherethe cross-sectional area of the circular bore is given by pir02 and its length is denotedl. The capacitance is given by C0 ≈ ε0wlt and the resistance is R0 ≈ 2ρpir0lδ0 , where thecircumference of the bore in 2pir0 and δ is the skin depth of the material that the LGR ismade from [15]. Skin depth is a frequency dependent quantitative measure of how deepthe AC current flowing in the resonator penetrates into its surface [15]. Specifically, δ0,denotes the skin depth of the resonator at its resonant frequency [14]. These parameterscan be modified for our rectangular LGR with a square cross-section by substituting inx2 as the cross-sectional area and 4x as the bore’s inner perimeter.61.4. Loop-Gap Resonators (LGRs)Figure 1.5: The equivalent circuit model of an LGR bore. Figure modified from [14].The approximate inductance can then be represented as,L0 ≈ µ0x2l. (1.17)The gap of the LGR has the same dimensions, so it provides the same capacitance of,C0 ≈ ε0wlt. (1.18)Lastly, the effective resistance of the LGR at its resonance frequency, ω0, is denoted,R0 ≈ 4ρxlδ0. (1.19)We can calculate the skin depth of an LGR using,δ =√2ρµ0ω, (1.20)where ρ is the resistivity of the material the LGR is composed of [15]. The LGRs used inthis project are made of aluminum which has resistivity ρ = 2.82×10−8 Ωm [16]. Fromthese parameters, the resonance frequency of the LGR is estimated to be,f0 =12pi√LC, (1.21)where L and C are given by L0 and C0 for an empty LGR. This logic can also be extendedto a two-loop-one-gap LGR. However, while the capacitance of the LGR is still determinedby the dimensions of the gap, the inductance and resistance are independent for each bore.These larger LGRs were used to accommodate large 2× 2 arrays of SRRs and cut wires.71.4. Loop-Gap Resonators (LGRs)1.4.1 DimensionsThe one-loop-one-gap LGR we used, as shown in Figs. 1.4(a) and (b), had dimensionsl = 112 mm, x = 21.6 mm, w = 5.0 mm, and t = 1.3 mm [6]. Using these dimensions inconjunction with Eqs. (1.17), (1.18), and (1.21), we estimate the resonance frequency ofthe empty loop gap resonator to be 1 GHz. Additionally, a two-loop-one-gap resonator,as shown in Figs. 1.6(a) and (b), was used. It had dimensions l = 112 mm, x = 42.7 mm,w = 5.0 mm, and t = 2.0 mm [6]. As the bore of this resonator is larger, its inductancewill increase relative to the one-loop-one-gap resonator. This, in turn, will lower theresonance frequency. In order to counteract this change, the capacitance of the LGRwas increased by increasing the gap thickness, t, with respect to the one-loop-one-gapresonator. This yields an expected resonance frequency of approximately 0.7 GHz for theempty two-loop-one-gap LGR.(a) Schematic. Taken from [6]. (b) LGR used for measurements.Figure 1.6: Two-loop-one-gap resonator with gap thickness, t, and gap width, w.8Chapter 2Measurement MethodsIn order to collect data, a vector-network analyzer (VNA) was used. Specifically, theDG8SAQ VNA 3E from SDR-Kits. This VNA is powered from a PC USB-bus and coversfrequencies from 1 kHz to 1.3 GHz. The VNA, which was connected to the LGR’s couplingloop using an SMA cable, allowed us to both introduce and extract signals from our LGR.Specifically, we will be interested in characterizing the reflected signal. To do this, we willmeasure the coefficient of the reflected signal, S11, from our LGR’s coupling loop.Before we could take these measurements, the system had to be calibrated. To do this,we used the Magi-Cal device. Essentially, once the settings for this automatic calibrationsystem were loaded into data collection software, the Magi-Cal device was connected tothe LGR’s coupling loop using an SMA cable. From here, the calibration was run. Asfor this portion of data collection we were only interested in the reflected signal, thecalibration was set to only run through an open circuit, a short circuit, and a circuitwith a 50 Ω load. Additionally, the audio settings of the VNA were adjusted to collectforty samples per IF period, three pre-samples, and 3 post-samples. This resulted in thedesired sinusoidal signal being delivered to the coupling loop. Once the VNA was set up,we were able to acquire S11 sweeps over various ranges of microwave frequencies.The VNA collected S11 data on a logarithmic decibel scale. In order to analyze thedata, we converted it to a linear scale using,|S11|lin = 10|S11|dB/20. (2.1)In Eq. (2.1), the absolute value bars indicate that we are only concerned with the mag-nitude of the S11 signal. Additionally, the factor of twenty in the exponent is due to the|S11|lin being defined in terms of a voltage ratio instead of a power ratio as shown by Eqs.(2.2) and (2.3).|S11|dB = 10 log(V2V1)2(2.2) |S11|lin =V2V1(2.3)Here, V1 denotes the voltage supplied to the coupling loop and V2 denotes the reflectedvoltage. Thus, as the voltage ratio is equivalent to the square root of the power ratio,there is an extra factor of one-half in the exponent as we convert from the decibel scale.9Chapter 3Modelling |S11| of An Empty LGRTo model |S11,e| we must calculate the effective impedance of the empty LGR, Ze, andthen exploit the characteristics of transmission lines to determine the reflection coefficient.3.1 The Effective Impedance of a Loop GapResonator (LGR)In order to find the effective impedance of a one-loop-one-gap resonator inductivelycoupled to a coupling loop of inductance L1, its equivalent circuit model, as depicted inFig. 3.1, was analyzed.Figure 3.1: The equivalent circuit model of an LGR inductively coupled to a couplingloop of inductance, L1. Figure modified from [14].In Fig. 3.1, L1 is the effective inductance of the coupling loop, which is assumed tohave no losses. L0 denotes the self inductance of the LGR which is defined as the ratioof magnetic flux to the opposing current in a loop. C0 is the capacitance of the LGR gapand R0 is its effective resistance at its resonance frequency. More generally, the effectiveresistance of the LGR at all frequencies can be denoted R. Lastly, M0 represents themutual inductance between L1 and L0 and is given by k√L0L1, where k is a couplingcoefficient whose value is between zero and one [17].103.1. The Effective Impedance of a Loop Gap Resonator (LGR)In order to analyze this circuit, Kirchhoff’s voltage law is applied. This law statesthat the sum of the potential differences around a loop equals zero [18]. A voltage isapplied to the coupling loop from the VNA which drives a current, i1, across the couplingloop. This, in turn, induces an emf through the resonator bore resulting in a secondcurrent, i2, in the LGR. Additionally, the impedance of the LGR’s capacitance is definedas ZC0 = 1/jωC0 and the impedance of the LGR’s inductance is defined as ZL0 = jωL0.Applying Kirchhoff’s voltage law to the coupling loop we obtainV1 = jωL1i1 + jωM0i2, (3.1)where V1 is the voltage applied to the coupling loop. Now, analyzing the circuit loop ofthe LGR we obtain,0 = jωM0i1 − ji2ωC0+ i2R + jωL0i2. (3.2)Solving Eq. (3.2) for i2 we obtain,i2 = i1−jωM0R + j(ωL0 − 1ωC0 ). (3.3)The effective impedance is defined as Ze = V1/i1. Therefore, by combining Eqs. (3.1) and(3.3) and factoring out and dividing by i1 we find,Ze =V1i1= jωL1 +ω3M20C0RωC0 + j(ω2L0C0 − 1) . (3.4)By multiplying by the complex conjugate of the denominator in Eq. (3.4), the imaginaryand real components of Ze are separated where Ze = Re + jXe. The real component ofthe effective impedance is,Re =Rω2M20R2 + (ωL0 − 1ωC0 )2, (3.5)and the imaginary component of the effective impedance is,Xe = ωL1 − ω2M20ωL0 − 1ωC0R2 + (ωL0 − 1ωC0 )2. (3.6)Now, to obtain the desired form of Ze, we must express Eqs. (3.5) and (3.6) in terms ofω0, Q0, and R0 by re-expressing L0 and C0 as L0 = Q0R0/ω0 and C0 = 1/ω0Q0R0. Addi-tionally, we can write R in terms of R0 by analyzing the skin depth, δ, of the resonator.The skin depth, denoted by Eq. (1.20), is a quantitative measure of how deeply the ACcurrent flowing in the resonator penetrates the surface. Thus, the resistance of the LGRcan be represented by,R =4ρxlδ, (3.7)113.2. Modelling |S11,e| of an Inductively-Coupled LGRwhere l is the length of the resonator, x is the side length, and ρ is the resistivity of thematerial [15]. Now, by combining Eqs. (1.20) and (3.7) and defining R0 = 2√2ρµ0ω0(x/l),where µ0 is the permeability of free space, we obtain R = R0√ω/ω0 [14]. By implementingthese changes, we obtain the real and imaginary components of Ze shown in Eqs. (3.8)and (3.9), respectively.Re =[(ωM0)2/R0]√ω/ω0(ω/ω0) +Q20[(ω/ω0)− (ω0/ω)]2(3.8)Xe = ωL1 − [(ωM0)2/R0]Q0[(ω/ω0)− (ω0/ω)](ω/ω0) +Q20[(ω/ω0)− (ω0/ω)]2(3.9)3.2 Modelling |S11,e| of an Inductively-Coupled LGRNow that we have determined the effective impedance of an empty LGR, Ze, we canconsider the characteristics of transmission lines to model the magnitude of the reflectedsignal, |S11,e|. The LGR is inductively-coupled to a coupling loop connected to a coaxialcable with characteristic impedance Z0. In order to find the reflection coefficient using Zeand Z0 we must exploit the result,S11,e =Ze − Z0Ze + Z0, (3.10)where Ze = Re + jXe and, the characteristic impedance, Z0, of our transmission lineis 50 Ω [19]. Combining this expression for Ze with Eq. (3.10) and multiplying by thecomplex conjugate, we obtain:S11,e =R2e +X2e − Z20(Re + Z0)2 +X2e+ j2Z0Xe(Re + Z0)2 +X2e. (3.11)To solve for the magnitude of the reflected signal |S11,e|, we must multiply Eq. (3.11) bythe complex conjugate and square root the result. This results in the following expressionfor the magnitude of the reflection coefficient for an inductively-coupled empty LGR:|S11,e| =√[(|Ze|2/Z20)− 1]2 + [2(Xe/Z0)]2[(|Ze|2/Z20) + 1] + 2(Re/Z0), (3.12)where |Ze|2 = R2e + X2e .3.3 Plotting |S11,e| for Varying Coupling ConstantsUsing Eq. (3.12) in conjunction with Eqs. (3.8) and (3.9) we can plot |S11,e| as a functionof frequency for different values of mutual inductance, M0. Here, M02/R0 is set equalto k2×(0.1nH2/mΩ), where k is the coupling coefficient. Additionally, we will use test123.4. Experimentally Determining |S11,e| for Varying Coupling Constantsvalues of ω0 = 2pi×(1 GHz), Q0 = 500, L1 = 12 nH, and Z0 = 50 Ω. To vary the mutualinductance, M0, |S11,e| is plotted as a function of frequency for k = 0.1, 0.2, 0.3, 0.4, and0.5. The resulting plots are shown in Fig. 3.2.0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05Frequency (Hz) 10900.10.20.30.40.50.60.70.80.91|S 11,e|k = 0.1k = 0.2k = 0.3k = 0.4k = 0.5Figure 3.2: |S11,e| plotted as a function of frequency for k = 0.1, 0.2, 0.3, 0.4, and 0.5.As shown in Fig. 3.2, when k = 0.2, |S11,e| approaches a value very close to zero at itsresonance peak. For critically coupled systems, the maximum signal is transferred fromthe coupling loop to the LGR, which means Ze is equal to Z0 as shown by Eq. (3.10).Therefore, |S11,e| is equal to zero meaning the LGR and coupling loop are critically coupledwhen k = 0.2. Thus, when k < 0.2, the LGR is undercoupled. The plot resulting froman undercoupled mutual inductance, for example when k = 0.1, is shifted left and has ahigher |S11,e| value at resonance. Lastly, when k > 0.2, the LGR is overcoupled. The plotresulting from an overcoupled mutual inductance, as shown in Fig. 3.2 when k = 0.3, 0.4,and 0.5, is less sharp, shifted right, and has a higher |S11,e| value at resonance [20].3.4 Experimentally Determining |S11,e| for VaryingCoupling Constants|S11,e| as a function of frequency was measured using the two-loop-one-gap resonatorshown in Fig. 1.6. In order to change the coupling constant k, the orientation of thecoupling loop was tuned. To find the orientation that resulted in critical coupling, thecoupling loop was tuned until |S11,e| approached zero. From here, it was adjusted untila characteristic undercoupled peak was seen, and further adjusted until a characteristic133.4. Experimentally Determining |S11,e| for Varying Coupling Constantsovercoupled peak was seen. The data collected was then linearized using Eq. (2.1) andplotted as shown in Fig. 3.3.0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11,e|Critically CoupledUndercoupledOvercoupledFigure 3.3: |S11,e| plotted as a function of frequency for two-loop-one-gap resonator forvarious coupling loop orientations.Comparing Fig. 3.2 and Fig. 3.3, we can see that in both figures the plots producedwhen the LGR and coupling loop are critically coupled approach zero at their resonancefrequencies. On the other hand, the undercoupled and overcoupled plots have greatervalues of |S11,e| for their resonance peaks. As expected, as shown in Fig. 3.3, the un-dercoupled plot has a slightly lower resonance frequency and the overcoupled plot has aslightly higher resonance frequency and less defined peak. Additionally, it is importantto note that the two-loop-one-gap LGR’s resonant frequency, f0, was estimated to be ap-proximately 0.7 GHz. Therefore, while the the model for the one-loop-one-gap resonatorshown in Fig. 3.2 shows resonances occurring at ∼ 1 GHz, it is expected that f0 formeasurements taken using the two-loop-one-gap resonator will be closer to 0.7 GHz, asobserved.14Chapter 4Permeability of SRR Arrays4.1 Modelling Permeability of SRR ArraysIn order to analyze the permeability of SRR arrays, we will be using Pendry et al.’smodel as given in Eqs. (1.9) to (1.11) [1]. Eqs. (1.10) and (1.11) can be re-written interms of frequency by substituting in ω = 2pif . This yields,µ′ = 1− [1− (fs/fp)2][1− (fs/f)2][1− (fs/f)2]2 + [γ/(2pif)]2 (4.1)andµ′′ =[γ/(2pif)][1− (fs/fp)2][1− (fs/f)2]2 + [γ/(2pif)]2 , (4.2)where fs is the resonant frequency, fp is the plasma frequency, and γ is the dampingconstant [1]. Now, as our SRR arrays are expected to have resonance frequencies ofapproximately 1 GHz, we can set fs = 1 GHz, from here we set fp equal to 1.05 GHzand γ/(2pi) to 10 MHz. Using these test values, we can plot µ′ and µ′′ as a function offrequency. The obtained curves are plotted in Fig. 4.1.As depicted in Fig. 4.1, µ′ is negative just above its resonance frequency, fs, until itcrosses back over zero at its plasma frequency, fp. Additionally, µ′′ peaks at fs beforereturning to zero at fp. An important feature of this model is that µ′′ will be greaterthan or equal to zero as long as fp > fs. This is expected as a positive µ′′ implies powerdissipation, which is expected as a signal is supplied and reflected from the metamaterial.4.2 The Effective Impedance of an LGR Filled withMagnetic MaterialAs we load SRRs with a complex effective permeability into the bore of an LGR, therelative effective permeability of the LGRs bore will change. We must account for thiswhen we find the impedance of the coupled LGR. To do this, we will complete a similaranalysis as shown in Chapter 3.1, but, this time, we will account for the magnetic materialfilling the LGRs bore. To begin, we must analyze the equivalent circuit model shown inFig. 4.2.154.2. The Effective Impedance of an LGR Filled with Magnetic Material0.85 0.9 0.95 1 1.05 1.1 1.15Frequency (Hz) 109-4-20246810Permeability'''Figure 4.1: The real, µ′, and imaginary, µ′′, components of the SRR relative perme-ability plotted as a function of frequency when fp = 1.05 GHz, fs = 1 GHz, andγ/(2pi) = 10 MHz.Figure 4.2: The equivalent circuit model of an filled LGR inductively coupled to couplingloop of inductance, L1. Figure modified from [14].L1 denotes the inductance of the coupling loop which is inductively coupled to the LGR,which now has inductance L. For this derivation we will assume the bore of the LGR isfilled with a magnetic material having relative permeability µr = µ′ − jµ′′. This causesthe inductance of the LGR to go from L = L0 to L = (µ′ − jµ′′)L0. Again, the effectivecapacitance and resistance of the LGR are denoted as C0 and R0, respectively. Lastly,the mutual inductance between the coupling loop and the resonator is denoted M . For164.2. The Effective Impedance of an LGR Filled with Magnetic Materialan empty LGR bore, the mutual inductance was given by M0 = k√L0L1. Now, however,as the inductance of the bore is being multiplied by µr, the mutual inductance becomesM = k√µrL0L1. From this, we can make use of the result M2 = k2µrL0L1. Lastly, recallthat the resistance of the LGR bore has frequency dependence given by R = R0√ω/ω0.Similarly to the circuit analysis used to determine the effective impedance of a couplingloop inductively coupled to an empty LGR, Kirchhoff’s voltage law will be exploited inthis problem [17]. Again, there is a current i1 in the coupling loop, and thus, by Faraday’sLaw, a current i2 is induced in the LGR. Applying Kirchhoff’s voltage law to the couplingloop we obtainV1 = jωL1i1 + jωMi2, (4.3)where V1 is the voltage applied to the coupling loop. Now, analyzing the circuit loop ofthe LGR we obtain,0 = jωMi1 − ji2ωC0+ i2R + jωµ′L0i2 + ωµ′′L0i2. (4.4)Solving Eq. (4.4) for i2 we obtain,i2 =−jωMR + ωµ′′L0 + j(ωµ′L0 − 1ωC0 )i1. (4.5)The effective impedance is defined as Zf = V1/i1. Therefore, in order to solve for Zf wecan combine Eqs. (4.3) and (4.5) and factor out i1 as follows,Zf = jωL1 +ω3M2C0RωC0 + ω2C0µ′′L0 + j(ω2µ′L0C0 − 1) . (4.6)Now, in order to remove the imaginary term of Zf from the denominator of the equation,we must multiply the numerator and denominator of Eq. (4.6) by the complex conjugateof the denominator. From this, we obtain,Zf = jωL1 + ω3M2C0RωC0 + ω2C0µ′′L0 − j(ω2µ′L0C0 − 1)(RωC0 + ω2C0µ′′L0)2 + (ω2µ′L0C0 − 1)2 . (4.7)We can now utilize the result, M2 = k2µrL0L1 which, as µr = µ′ − jµ′′, can be writtenas M2 = k2(µ′ − jµ′′)L0L1. Using the result in Eq. (4.7), and separating the effectiveimpedance as Zf = Rf + jXf we obtain,Rf =ω2k2L0L1(µ′RωC0 + µ′′)(R√ωC0 +√ω3C0µ′′L0)2 + (√ω3C0µ′L0 − 1√ωC0 )2, (4.8)174.3. Plotting |S11,f |which denotes the real component of the effective impedance, and,Xf = ωL1 − ω2k2L0L1[µ′′(RωC0 + ω2C0µ′′L) + µ′(ω2µ′L0C0 − 1)](R√ωC0 +√ω3C0µ′′L0)2 + (√ω3C0µ′L0 − 1√ωC0 )2, (4.9)which denotes the imaginary component of the effective impedance. Finally, to obtainthe desired form, we must express Eqs. (4.8) and (4.9) in terms of ω0, Q0, and R0 byre-expressing L0 and C0 as L0 = Q0R0/ω0, C0 = 1/ω0Q0R0, and R = R0√ω/ω0. Thesesubstitutions allow us to obtain the desired form of the real and imaginary componentsof Zf in Eqs. (4.10) and (4.11), respectively.Rf =ωL1(ω/ω0)k2[ µ′Q0√ω/ω0 + µ′′(ω0/ω)][ 1Q0√ω/ω0 + µ′′(ω/ω0)]2 + [µ′(ω/ω0)− (ω0/ω)]2(4.10)Xf = ωL1(1− (ω/ω0)k2[µ′′[ 1Q0√ω/ω0 + µ′′(ω/ω0)] + µ′[µ′(ω/ω0)− (ω0/ω)]][ 1Q0√ω/ω0 + µ′′(ω/ω0)]2 + [µ′(ω/ω0)− (ω0/ω)]2)(4.11)4.3 Plotting |S11,f|Equation (3.12) models the magnitude of the reflection coefficient of the signal from aninductively coupled empty LGR. This model can be expanded to include measurements inwhich the LGR bore is filled with magnetic material. This changes the notation slightlyto,|S11,e/f | =√[(|Ze/f |2/Z20)− 1]2 + [2(Xe/f/Z0)]2[(|Ze/f |2/Z20) + 1] + 2(Re/f/Z0), (4.12)where |Ze/f |2 = R2e/f + X2e/f and Z0 is still the characteristic impedance of the transmissionline. Therefore, using this equation in conjunction with Eqs. (4.10) and (4.11) we can plotthe reflected signal of an inductively coupled LGR, |S11,e/f |, as a function of frequency. Inorder to do this, we will use the following parameters f0 = 1 GHz, Q0 = 500, L1 = 12 nH,and Z0 = 50 Ω, and k = 0.065. Also noting that ω, the angular frequency, is defined as2pif . By defining these fixed values while varying µ′ and µ′′, we were able to explore theeffect that varying the real and imaginary components of the permeability of the magneticmaterial filling the resonator’s bore will have on the resonance peaks.Initially, values of µ′ = 1 and µ′′ = 0 were plotted. This curve corresponds to thenear-critical coupling of an LGR with an empty bore and is displayed in blue in Fig. 4.3.Next, values of µ′ = 1.1 and µ′′ = 0 were plotted. The resulting curve is shown in red inFig. 4.3. As shown, when µ′ is increased above 1, the resonant frequency is lowered by afactor equal to√1/µ′. Lastly, values of µ′ = 1 and µ′′ = 0.01 were plotted. This result isshown in cyan in Fig. 4.3 and illustrates that when µ′′ > 0, the produced curve is muchbroader and the resonance is no longer critically coupled.184.4. Plotting |S11,f | for Frequency Dependent Permeability9.2 9.4 9.6 9.8 10 10.2 10.4Frequency (Hz) 10800.10.20.30.40.50.60.70.80.91|S 11|' = 1, '' = 0' = 1.1, '' = 0' = 1, '' = 0.01Figure 4.3: |S11,f | plotted as a function of frequency for various permeability values.4.4 Plotting |S11,f| for Frequency DependentPermeabilityWe will also explore the effect the frequency dependent permeability given in Eqs. (4.1)and (4.2) has on the magnitude of the reflection coefficient, |S11,f |. In order to do this, wewill again plot Eq. (4.12) in conjunction with Eqs. (4.10) and (4.11). However, this time,the real, µ′, and imaginary, µ′′, components of the relative permeability will be defined byEqs. (4.1) and (4.2), thus making these values frequency dependent. Similarly to in theplots shown in Figs. 4.1 and 4.3, the following parameters are set: f0 = 1 GHz, Q0 = 500,L1 = 12 nH, and Z0 = 50 Ω, k = 0.065, and fs = 1 GHz. This time we will be varyingfp and γ to explore the effect they have on the produced resonance curves. Using theoutlined values and equations, |S11,f | is plotted as a function of frequency and displayedin Fig. 4.4.As expected, a double resonance is seen. |S11,f | was originally plotted with fp = 1.05 GHzand γ/(2pi) = 10 MHz. These values were then varied to explore their effect on the pro-duced resonance curves. As shown by the red curve, when fp was decreased to 1.03 MHz,the double resonance peaks shifted closer together. Additionally, the primary resonancepeak, while still higher in magnitude than the secondary peak, decreased in magnitude.Moreover, the secondary peak increased in magnitude. Then, as shown by the magentacurve, when fp was increased to 1.07 MHz, the double resonance peaks shifted furtherapart and the primary peak increased in magnitude while the secondary peak decreasedin magnitude. Next, the damping constant, γ, was varied while fp stayed constant at1.05 GHz. As expected, when γ was decreased, as illustrated by the cyan curve, the194.5. Investigating the Effect of Negative µ′′ on |S11,f |Figure 4.4: |S11,f | plotted as a function of frequency for various fp and γ values.resonance peaks became sharper and increased in magnitude. On the other hand, whenγ was increased, the resonance peaks flattened.4.5 Investigating the Effect of Negative µ′′ on |S11,f |Finally, we can investigate what effect a negative imaginary component of the perme-ability, µ′′, would have on the expected |S11,f | plot. From Eq. (4.2) we can deduce that ifthe magnetic plasma frequency, fp, of the metamaterial is less than its resonant frequency,fs, µ′′ will be negative. We can plot the resulting permeability using Eqs. (4.1) and (4.2)and the following test parameters: fs = 1 GHz, fp = 0.95 GHz, and γ/(2pi) = 10 MHz.The resulting plots are displayed in Fig. 4.5. As shown in Fig. 4.5, the imaginary compo-nent of the permeability, as shown in red, becomes negative at the resonance frequency.This is opposite to the effect seen in Fig. 4.1 when fp > fs. The real component of thepermeability, as plotted in blue, crosses over zero and becomes negative at its magneticplasma frequency, before spiking back up to a positive value at the resonant frequency.Thus, the negative portion of µ′ occurs just below the resonant frequency instead of justabove, as seen in Fig. 4.1. Using these values, and including test values of f0 = 1 GHz,Q0 = 500, L1 = 12 nH, and Z0 = 50 Ω, and k = 0.065, we can plot the expected |S11,f |curve produced from an LGR filled with a magnetic material with fs > fp, and thusnegative µ′′. To do this, we will again use Eqs. (4.1), (4.2), (4.10), and (4.11) to plot Eq.(4.12). The resulting plot of |S11,f | as a function of frequency is displayed in Fig. 4.6.204.5. Investigating the Effect of Negative µ′′ on |S11,f |0.85 0.9 0.95 1 1.05 1.1 1.15Frequency (Hz) 109-40-30-20-1001020Permeability'''Figure 4.5: The real, µ′, and imaginary, µ′′, components of the SRR relative perme-ability plotted as a function of frequency when fp = 0.95 GHz, fs = 1 GHz, andγ/(2pi) = 10 MHz.0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25Frequency (Hz) 10911.00011.00021.00031.0004|S 11|Figure 4.6: |S11| as a function of frequency when fp = 0.95 GHz, fs = 1 GHz, andγ/(2pi) = 10 MHz.As plotted in Fig. 4.6, the expected |S11| curve for an LGR filled with a magneticmaterial with a negative imaginary component of permeability is positive. This is anunphysical result as it suggests that the reflected voltage is greater than the voltagesupplied to the LGR.214.6. Experimentally Determining |S11| for an Array of SRRs4.6 Experimentally Determining |S11| for an Arrayof SRRsIn this section, we will experimentally measure the |S11| signals over a range of mi-crowave frequencies for SRR arrays using a LGR. We will load arrays of N = 1, 2, 3, and4 SRRs into the one-loop-one-gap resonator. To begin, the coupling loop was oriented inorder to acquire near-critical coupling. This position was found to be when the plane ofthe coupling loop was perpendicular to the bore axis. The coupling loop was then securedin this position using a set screw. Then, SRRs were placed as far as possible from thecoupling loop as possible with their outermost slit opposite to the LGR’s gap. As thearray increased from one SRR, a 21 mm spacer was placed in between the SRRs. TheLGR loaded with four SRRs is pictured in Fig. 4.7. Sweeps of the reflection coefficientFigure 4.7: Array of four SRRs loaded into the one-loop-one-gap LGR spaced at approx-imately 21 mm apart.over frequencies from 0.7 GHz to 1.3 GHz were taken for arrays of N = 1, 2, 3, and 4SRRs. These signals were then converted to a linear scale using Eq. (2.1) and the resultingplots are shown in Fig. 4.8.224.6. Experimentally Determining |S11| for an Array of SRRs0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)-0.200.20.40.60.811.2|S 11|N=0N=1N=2N=3N=4Figure 4.8: |S11| as a function of frequency for the one-loop-one-gap LGR when loadedwith N SRRs spaced at approximately 21 mm apart.Figure 4.8 shows |S11| plotted as a function of frequency for the five data sets collected.As expected, for an empty LGR a single resonance peak at about 0.9 GHz was measured.Then, when the bore of the LGR was filled with one SRR, a second resonance peakappeared at just below 1.1 GHz. As more SRRs were added, the peaks spread furtherapart. And, when three SRRs and four SRRs were loaded, a third dip appeared at afrequency of about 1 GHz. Finally, away from the resonances, |S11| was observed todecrease as frequency increased due to losses in the coaxial cable leading to the couplingloop. These observations are all consistent with what was shown in Fig. 3(a) of S. L.Madsen and J.S. Bobowski’s paper for a similar experimental set-up [6].23Chapter 5Permittivity of Cut Wire Arrays5.1 Modelling Effective PermittivityJust as SRR arrays have a negative permeability over a small range of microwavefrequencies, cut wire arrays are known to display a negative permittivity at microwavefrequencies. Using Maslovski et al.’s model describing the effective permittivity of anarray of cut wires we were able to explore the permittivity of cut wire arrays of differentdimensions [11]. To begin, we considered their model for ideally conducting wires, asdenoted in Eq. (1.12). We considered an array of wires aligned with the z-axis with anelectric field parallel to these wires. With these conditions, Eq. (1.12) reduced to,εr = 1− 2pi(ka)2 log( a24r0(a−r0)), (5.1)where εr is the relative effective permittivity, k is the wavenumber, a is the grid period,and r0 is the wire radius. Note that k = 2pi/λ where λ denotes wavelength. Using thisequation we were able to reproduce the solid lines plotted in Fig. 3 of Maslovski et al.’spaper. To do this, we set r0/a = 0.01 and 0.1 in Eq. (5.1) and plotted εr as a function ofka for both ratios. The resulting plots are displayed in Fig. 5.1.0.2 0.25 0.3 0.35 0.4 0.45 0.5ka-160-140-120-100-80-60-40-200Relative Effective Permittivityr0= 0.01ar0= 0.1aFigure 5.1: The effective permittivity of an array of ideally conducting wires as a functionof ka. The plot produced when r0/a = 0.01 is shown in red and the plot produced whenr0/a = 0.1 is shown in blue.245.2. Designing Cut Wire Arrays5.2 Designing Cut Wire Arrays5.2.1 Modelling Possible Array DimensionsNow, using this model, we designed a model for our cut wire array. Based on theSRRs and LGRs that we will be using, we have a predetermined grid period, a0, of21.6 mm. Using this value our goal is to determine suitable radii, r, for our cut wires toapproximately satisfy the r0/a = 0.1 ratio. This ratio was chosen as using this ratio weobserved in Fig. 5.1 a larger effect on the permittivity. Using the selection of aluminumrod sizes from the McMaster Carr website, aluminum rods with diameters of 3/16-inchesand 4.00 mm were chosen. This gives r0 values of 2.38 mm and 2.00 mm, respectively.Next, in order to plot relative effective permittivity in terms of frequency by using of Eq.(5.1), wavenumber k was represented in terms of frequency using k = 2pif /c. This givesthe following,εr = 1− c22pi(fa)2 log( a24r0(a−r0)), (5.2)where c is the speed of light in a vacuum. Eq. (5.2) was then plotted using r0 values of2.38 mm and 2.00 mm, shown in cyan and magenta respectively. The resonant frequenciesof the SRRs and LGRs to be used alongside these cut wire arrays range from 0.6 to 1.2 GHzso the horizontal axis was set to plot this frequency range.0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)-90-80-70-60-50-40-30-20Relative Effective Permittivityr0= 2.38 mm (with real component of losses)r0= 2.00 mm (with real component of losses)r0= 2.38 mm (ideally conducting wires)r0= 2.00 mm (ideally conducting wires)Figure 5.2: εr as a function of frequency. The plot produced for ideal conductors whenr0 = 2.38 mm is shown in cyan and when r0 = 2.00 mm is shown in magenta. The realcomponent of εr when losses are considered is plotted in blue (for r0 = 2.38 mm) and red(for r0 = 2.00 mm).255.2. Designing Cut Wire ArraysEquations (5.1) and (5.2) assume the cut wires used are ideally conducting and thusexperience no losses. Maslovski et al. also proposed a correction to this model for lossyor loaded wires, which experience some surface impedance Zs [20]. This impedance isgiven by Eq. (1.14) and the corrected permittivity is denoted by Eq. (1.15). Again, aswe considered an array with wires aligned with the z-axis with an electric field parallel tothese wires. Eq. (1.15) can be reduced to,εr = 1− 2pi(ka)2 log( a24r0(a−r0))− jka ar0 Zsη. (5.3)Substituting Eq. (1.14) into Eq. (5.3) we obtain a complex effective relative permittivitywhich can be expressed as εr = ε′ − jε′′ where ε′, the real component of permittivity, isgiven by,ε′ =(B + b)2 + b2 − 2pi(B + b)(B + b)2 + b2, (5.4)and ε′′, the imaginary component of permittivity, is given by,ε′′ =2pib(B + b)2 + b2. (5.5)For both Eqs. (5.4) and (5.5),b ≡ ka2r0√ωε02σ(5.6)and,B ≡ (ka)2 log(a24r0(a− r0)). (5.7)The real part of the permittivity, as denoted in Eq. (5.4), was plotted as a function offrequency using r0 values of 2.38 mm and 2.00 mm. The produced functions are plotted inblue and red on Fig. 5.2. This plot shows that when the real component of the conductor’slosses is taken into account, the relative effective permittivity shifts very slightly upwards,towards the horizontal axis. However, the effect of losses is nearly negligible for goodconductors.The imaginary part of the permittivity was also plotted as a function of frequency usingr0 values of 2.38 mm and 2.00 mm. This plot is shown in Fig. 5.3. This figure shows thatε′′ asymptotically approaches zero as frequency increases.265.2. Designing Cut Wire Arrays0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)0.20.30.40.50.60.70.80.9r0 = 2.38 mmr0 = 2.00 mmFigure 5.3: The imaginary component of εr, ε′′, plotted as a function of frequency.5.2.2 Modelling Permittivity of Our ArrayDue to supply availability, our actual cut wire arrays were composed of aluminum wiresof both 1/8-inch and 3/16-inch diameters. These result in radii of 1.59 mm and 2.38 mm.These are very similar and identical to 2.00 mm and 2.38 mm and, as such, are expectedto yield similar results when Maslovski et al.’s models are plotted. To further exploreour exact array dimensions, εr was plotted as a function of frequency using the ideallyconducting wire model, denoted in Eq. (5.2). These plots are shown in Fig. 5.4. Aspredicted by Fig. 5.2, over the range of 0.6 GHz to 1.2 GHz, εr is modelled to be negativeand increasing logarithmically as frequency increases.0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)-90-80-70-60-50-40-30-20Relative Effective Permittivityr0= 2.38 mmr0= 1.59 mmFigure 5.4: εr as a function of frequency modelled for ideal conductors.275.3. Composing Cut Wire Arrays5.3 Composing Cut Wire Arrays5.3.1 One-Loop-One-Gap Resonator ArraysUsing aluminum wires of both 1/8-inch and 3/16-inch diameter, cut wire arrays wereassembled. These arrays were then loaded into an LGR bore and their effect of the borespermeability was analyzed. In order to create secure arrays of cut wires with consistentdimensions, Teflon sheets were used. Teflon was used as it is expected to have no effecton the magnetic properties of the array. To begin, we created a 1 × 1 × N array of cutwires for measurements using the one-loop-one-gap resonator. This array consisted offour cut wires secured between two Teflon sheets with 1/8-inch thickness. The holdershad approximate width and length of 16.4 mm and 85 mm, respectively. Similarly to theSRR arrays previously described, a grid period of a ≈ 21.6 mm was used. This set-up isdepicted in Fig. 5.5.(a) Schematic model of cut wire array. (b) N = 4 array loaded intoone-loop-one-gap LGR.Figure 5.5: 1× 1×N array of cut wires with grid period a ≈ 21.6 mmThe counterbores in the a Teflon sheets had 1/16-inch depth and were created usingboth 1/8-inch and 3/16-inch diameter end mills for their respective arrays. The cut wireswere cut to by a height of approximately 18.5 mm so as to fit tightly into the LGR borewhile in their holders. Additionally, Teflon rods of both 1/8-inch and 3/16-inch diameterand a height of approximately 18.5 mm were also made. These were used to replicatethe geometry of our cut wire set-up while taking measurements with no cut wires. Thisenabled us to accurately determine the effect of the addition of aluminum wires. TheseTeflon rods were used as place holders for arrays of N = 1, 2, and 3.5.3.2 Two-Loop-One-Gap Resonator ArraysSimilar cut wire arrays were constructed to be used for measurements with the two-loop-one-gap resonator. However, this time 2×2×N arrays were made. Thus, N = 1, 2, 3 and 4correspond to four, eight, twelve, and sixteen cut wires, respectively. The wires used had285.4. Measuring |S11| of Cut Wire Arraysthe same dimensions as for the 2×2×N arrays. A side-view of the array is shown in Fig.5.6(a) and the array loaded into the two-loop-one-gap LGR is depicted in Fig. 5.6(b).(a) Schematic model of cut wire array. (b) N = 4 array loaded into two-loop-one-gapLGR.Figure 5.6: 2× 2×N array of cut wires with grid period a ≈ 21.6 mmIt is important to note that there are sixteen cut wires in total. Moreover, the bottomat top Teflon sheets have 1/8-inch thickness and the middle sheet has 1/4-inch thicknessand the grid period of the cut wire array is a ≈ 21.6 mm. The holders had approximatewidth and length of 39 mm and 85 mm, respectively.5.4 Measuring |S11| of Cut Wire Arrays5.4.1 One-loop-one-gap Loop Gap Resonator (LGR)3/16-inch Diameter Aluminum RodsIn order to explore the permeability of cut wire arrays, we measured the reflectioncoefficient of the signal, |S11|, for arrays of different numbers of cut wires. Again, thecoupling loop of the one-loop-one-gap resonator was oriented perpendicular to the boreaxis in order to acquire near-critical coupling and secured. We began by measuring thereflection coefficient, |S11|, of the empty LGR. We then repeated this measurement forarrays of four (N = 4), three (N = 3), two (N = 2), one (N = 1), and zero (N = 0)aluminum cut wires of 3/16-inch diameter. The LGR loaded with four aluminum cutwires is displayed in Fig. 5.5(b). As an aluminum cut wire was removed, a Teflon rod ofthe same dimensions filled its place between the two Teflon sheets. For consistency, theseTeflon rods were placed closest to the coupling loop to keep the aluminum cut wires as faras possible from the coupling loop. Thus, the N = 0 case describes a situation in whichfour Teflon rods were placed between the Teflon sheets. The resulting |S11| curves wereconverted to a linear scale using Eq. (2.1) and plotted as shown in Fig. 5.7(a).295.4. Measuring |S11| of Cut Wire Arrays0.905 0.91 0.915 0.92 0.925 0.93 0.935 0.94 0.945Frequency (GHz)00.10.20.30.40.50.60.7|S 11|Empty BoreN = 4N = 3N = 2N = 1N = 0(a) All Arrays0.905 0.91 0.915 0.92 0.925 0.93 0.935 0.94 0.945Frequency (GHz)00.10.20.30.40.50.60.7|S 11|N = 0N = 4(b) Only N = 0 and N = 4Figure 5.7: |S11| as a function of frequency measured for an empty bore and for a 1-Darray of N 3/16-inch diameter aluminum rods.From Fig. 5.7(a) we can qualitatively conclude that the addition of cut wires to thebore of the resonator has a very minimal and non-systematic effect on the measured |S11|curves. Each resonance occurs at a frequency of approximately 0.92 GHz and the dipshave similar width and depth. This effect is further exemplified in Fig. 5.7(b) in whichonly arrays of N = 0 and N = 4 are overlayed. This |S11| curves are nearly identicaldespite the large difference in array composition. This was not the result we expected. Aswe add aluminum cut wires to the bore, the open volume of the bore will decrease. Thus,as the inductance of the LGR is proportional to the volume, we expected the inductanceof the LGR to decrease as arrays of cut wires are loaded into the resonator’s bore. This,in turn, would be expected to increase the resonance frequency denoted by Eq. (1.21).However, for the one-loop-one-gap resonator, we observed no systematic changes as thebore of the LGR is filled with cut wires.1/8-inch Diameter Aluminum RodsThese measurements were replicated with arrays N 1/8-inch aluminum cut wires. Theresulting reflection coefficients, |S11|, were plotted as a function of frequency and aredisplayed in Fig. 5.8. Again, the addition of the aluminum cut wires had nearly no effecton the produced |S11| curves.305.4. Measuring |S11| of Cut Wire Arrays0.9 0.905 0.91 0.915 0.92 0.925 0.93 0.935 0.94 0.945Frequency (GHz)00.10.20.30.40.50.60.7|S 11|Empty BoreN = 4N = 3N = 2N = 1N = 0Figure 5.8: |S11| as a function of frequency measured for an empty bore and for a 1-Darray of N 1/8-inch diameter aluminum rods.5.4.2 Two-loop-one-gap Loop Gap Resonator (LGR)3/16-inch Diameter Aluminum RodsThis experimental procedure was then replicated using the two-loop-one-gap resonator.Again, to begin, we oriented the coupling loop in order to critically couple the LGR withthe coupling loop. Once this orientation, in which the plane of the coupling loop wasnearly parallel with the bore’s axis, was found, the coupling loop was secured with a setscrew and arrays of 3/16-inch diameter aluminum rods were loaded into the bore of theLGR. This time, however, we were working with 2× 2×N arrays of cut wires. The arrayset-up displayed in Fig. 5.6 was used. This array was centered in the bore of the resonatornot containing the coupling loop. This experimental set-up is depicted for an N = 4 arrayof cut wires in Fig. 5.6(b). Similarly to in the one-loop-one-gap case, when the number ofaluminum rods was reduced, they were removed from the side closest to the coupling loopfirst and replaced with Teflon rods. The resulting linearized plots of |S11| as a function offrequency for an empty resonator and for when the bore was filled with various arrays ofaluminum wires of 3/16-inch diameter are displayed in Fig. 5.9.Unlike with the one-loop-one-gap resonator, these measurements show that the reso-nance frequency decreases as N increases. This does not agree with what is expectedfrom Eq. (1.21). To further explore how the resonance peaks change with arrays of cutwires, we plotted |S11| as a function of (f/f0)− 1 as shown in Fig. 5.10 . While the plotsvary slightly in depth and width, there are no observed systematic changes as N changes.315.4. Measuring |S11| of Cut Wire Arrays0.7 0.705 0.71 0.715 0.72 0.725 0.73 0.735 0.74 0.745 0.75Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11|Empty BoreN = 4N = 3N = 2N = 1N = 0Figure 5.9: |S11| as a function of frequency for 2× 2×N arrays of N 3/16-inch diameteraluminum rods.-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025(f/f0) - 100.10.20.30.40.50.60.70.80.91|S 11|Empty BoreN = 4N = 3N = 2N = 1N = 0Figure 5.10: |S11| as a function as a function of (f/f0) − 1 for 2 × 2 × N arrays of N3/16-inch diameter aluminum rods.1/8-inch Diameter Aluminum RodsLastly, this procedure was reproduced using 1/8-inch diameter aluminum cut wires.Similarly to what we observed with the 3/16-inch diameter rods, the resonance frequencydecreases as N increases. This result is displayed in Fig. 5.11. When the reflectioncoefficient of the signal, |S11|, is plotted as a function of (f/f0)− 1, as shown in Fig. 5.12,no systematic changes are observed as N is varied.325.4. Measuring |S11| of Cut Wire Arrays0.7 0.705 0.71 0.715 0.72 0.725 0.73 0.735 0.74 0.745 0.75Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11|Empty BoreN = 16N = 12N = 8N = 4N = 0Figure 5.11: |S11| as a function of frequency for 2× 2×N arrays of N 1/8-inch diameteraluminum rods.-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025(f/f0) - 100.10.20.30.40.50.60.70.80.91|S 11|Empty BoreN = 4N = 3N = 2N = 1N = 0Figure 5.12: |S11| as a function as a function of (f/f0) − 1 for 2 × 2 × N arrays of N1/8-inch diameter aluminum rods.Overall, as we add cut wires to the bore of the LGR, we qualitatively observed veryminimal changes in the produced |S11| reflections. No additional resonance peaks areobserved and the width and depth of the LGR’s resonance is nearly independent of cutwire array size. This suggests that, as |S11| depends on permeability of the materialinside of the LGR’s bore, the addition of cut wires has a very minimal effect on the bore’spermeability.33Chapter 6Modelling the Effective Impedanceof a Partially Filled LGRIn Chapter 4.2, we found the effective impedance of the LGR with a completely filledbore, Zf , and represented it in terms of its real, Eq. (4.10), and imaginary, Eq. (4.11),components as Zf = Rf + jXf . Now, we will explore how this effective impedance varieswhen the resonator bore is only partially filled with the magnetic material. First, we willfind the inductance of the empty-bore resonator. To do this, we must define our resonatorlength as s and its cross-sectional area as A. Then, using the formula for inductance of asolenoid, L = µ0N2As, substituting the number of turns, N = 1, we obtain the inductanceof the empty-bore resonator:L0 =µ0As. (6.1)We will now suppose the LGR has been partially filled with the magnetic material withrelative permeability µr = µ′ − jµ′′. This material has length x < s and a filling factor,η, which is defined as η = x/s. In order to find the inductance of the empty portion ofthis LGR, Le, we must redefine the length as s − x. This gives us the inductance of theempty portion of the bore to beLe =µ0As− x. (6.2)To simplify this equation, we can factor out an s in the denominator and then substitutein Eq. (6.1) to obtainLe =L01− η . (6.3)We can do a similar analysis for the inductance of the filled portion of the bore. However,this time we must change the length to x as well as include the relative permeability ofthe magnetic material filling the bore. This gives us the inductance of the filled portionof the bore,Lf =µ0µrAx. (6.4)Similarly, in order to put this equation in terms of L0 we can substitute in x = ηs andthen define µ0A/s as L0 using Eq. (6.1). This gives us the desired result for the impedanceof the filled portion of the bore,Lf =µrL0η. (6.5)34Chapter 6. Modelling the Effective Impedance of a Partially Filled LGRNow, we will argue that the effective inductance, L, of the partially-filled LGR can bedetermined by the parallel combination of Le and L0. We can imagine that if there was acurrent applied through a coupling loop on one side of the bore, a magnetic field, ~B, wouldpermeate the entire bore of the LGR. Thus, the magnetic flux, Φ would be equal throughall cross-sections of the LGR – independent of whether there was magnetic materialfilling that area. This means that the induced emf, ε, is equal for the region containingthe magnetic material and for the empty region as ε = −dΦ/dt and the magnetic fluxthrough the entire resonator bore is unchanged. Thus, as the induced emf is the samethrough each region, we can combine Le and L0 as though they were in parallel in a simplecircuit. This set-up is analogous as the voltage change over two inductors in parallel isequal, similarly to the induced emf through each region of the LGR. Using this model weobtain,1L=1Le+1Lf. (6.6)From here, we can obtain a common denominator, take the reciprocal of each side, andsubstitute in Eqs. (6.3) and (6.5) to obtainL =µrL20η(1− η)×η(1− η)L0η + µrL0(1− η) . (6.7)We can now cancel terms and substitute in µr = µ′ − jµ′′. This gives us,L = L0(µ′ − jµ′′)(µ′ − jµ′′)− η(µ′ − jµ′′) + η . (6.8)Next, we can collect like terms and multiply by the complex conjugate of the denominatorin order to remove imaginary terms from the denominator. This yields,L = L0(µ′ − jµ′′)(µ′(1− η) + η)− jµ′′(1− η)×(µ′(1− η) + η) + jµ′′(1− η)(µ′(1− η) + η) + jµ′′(1− η) . (6.9)From here, the denominator and numerator are multiplied out and the imaginary and realcomponents from the numerator are separated in order to write the effective inductanceof the partially filled LGR as:L = L0( |µr|2(1− η) + µ′η[η + µ′(1− η)]2 + [µ′′(1− η)]2 − jµ′′η[η + µ′(1− η)]2 + [µ′′(1− η)]2), (6.10)where |µr|2 = (µ′)2 + (µ′′)2. Equation (6.10) can be expressed as L = `L0 where` = `′ - j`′′. Here, the real component of `, `′, is denoted,`′ =|µr|2(1− η) + µ′η[η + µ′(1− η)]2 + [µ′′(1− η)]2 , (6.11)35Chapter 6. Modelling the Effective Impedance of a Partially Filled LGRand its imaginary component, `′′ is denoted,`′′ =µ′′η[η + µ′(1− η)]2 + [µ′′(1− η)]2 . (6.12)Before moving on, we will show that the results we obtained for the effective inductanceof the partially filled LGR bore, L, make sense for when η = 0, η = 1, and µr = 1. First,when η = 0 we obtain L = L0. This makes sense as if the filling factor is zero, the LGR isempty and thus should have the inductance given for an empty bore. Next, when η = 1we obtain L = L0(µ′ − jµ′′). This denotes the inductance of the LGR when its bore iscompletely filled with a magnetic material of relative permeability µr = µ′ − jµ′′. Again,this makes sense, as when the filling factor is one, the LGR bore is completely filled thushas the inductance expected for an LGR with a filled bore. Lastly, when µr = 1, L = L0.This makes sense as µr =µµ0. So, if µr = 1, the permeability of the magnetic material, µis equal to the permeability of free space, µ0. Thus, it follows that the inductance wouldbe equal to the inductance of an LGR with an empty bore.Now, we can modify Eqs. (4.10) and (4.11) so they represent the effective impedance ofan LGR with a partially filled bore. Previously, as the bore was completely filled with themagnetic material, its inductance was given by L = (µ′−jµ′′)L0. Now, we have determinethe partially filled bore’s inductance is given as L = (`′ − j`′′)L0 where `′ is denoted inEq. (6.11) and `′′ is given by Eq. (6.12). This is the only change we made to this modelto represent a partially filled bore. Thus, Eqs. (4.10) and (4.11) can easily be modified torepresent an LGR with its bore partially filled with a magnetic material by substitutingµ′ with `′ and µ′′ with `′′. This results in the effective impedance of the partially filledbore, Z, to be given by Z = R+ jX, where the real component of the effective impedanceis,R =ωL1(ω/ω0)k2[ `′Q0√ω/ω0 + `′′(ω0/ω)][ 1Q0√ω/ω0 + `′′(ω/ω0)]2 + [`′(ω/ω0)− (ω0/ω)]2, (6.13)and the imaginary component is,X = ωL1(1− (ω/ω0)k2[`′′[ 1Q0√ω/ω0 + `′′(ω/ω0)] + `′[`′(ω/ω0)− (ω0/ω)]][ 1Q0√ω/ω0 + `′′(ω/ω0)]2 + [`′(ω/ω0)− (ω0/ω)]2). (6.14)36Chapter 7Permeability of NIM Arrays7.1 Composing NIM ArraysIn this section, we will explore the permeability of arrays of both SRRs and aluminumcut wires. In order to do this, the array configurations in Figs. 5.5 and 5.6 were modifiedto include SRRs. The 1× 1×N array composition is shown on the left of Fig. 7.1 and anSRR is shown on the right. Again, Teflon sheets of 1/8-inch thickness were used to secureFigure 7.1: 1× 1×N array of SRRs and cut wires with grid period a ≈ 21.6 mm besidesingle SRR.the wires in the array. To ensure a distance of a/2 between the center of the cut wireand the SRR, the cut wire indents could not be centred on the Teflon holders. We had toaccount for the fact that the SRR was attached to a substrate with a total thickness of1.54 mm. Thus, to create this consistent a/2 distance, the Teflon holders had length ofapproximately 10.4 mm from one edge to the centre of the cut wire indent, and 8.2 mmfrom here to the opposite edge. They had an approximate width of 16.5 mm. In Fig. 7.1,the black side depicts where the SRR is secured on its green substrate.A similar modification, which is shown in Fig. 7.2, was used to modify the 2 × 2 × Narrays of cut wires to also incorporate SRRs. Comparably to the arrays of solely cutwires, the top Teflon sheets used to secure the array in place had 1/8-inch thickness whilethe centre sheet had 1/4-inch thickness. The 1/16-inch counterbores used to hold the cutwires were also slightly off-centred as for the one dimensional arrays to ensure a consistentdistance of a/2 between the cut wires and the SRRs. However, the width of these Teflonsheets was approximately 39 mm. The arrays displayed in Fig. 7.1 and 7.2 are shownloaded into the bores of their respective LGR in Fig. 7.3(a) and (b), respectively.377.2. Measuring |S11| of NIM ArraysFigure 7.2: 2× 2×N array of SRRs and cut wires with grid period a ≈ 21.6 mm beside2× 2 plane of SRRs.(a) 1×1×N array in one-loop-one-gapLGR(b) 2× 2×N array in two-loop-one-gap LGRFigure 7.3: Experimental set-up fpr N = 4 arrays of SRRs and aluminum cut wires loadedinto LGRs.7.2 Measuring |S11| of NIM Arrays7.2.1 One-loop-one-gapTo begin, we explored the 1×1×N array using the one-loop-one-gap LGR. As previouslydescribed, the coupling loop was set perpendicular to the bore axis to acquire near-criticalcoupling and secured. Then, a single SRR and a single Teflon rod (3/16-inch diameter)were loaded into the bore as far from the coupling loop as possible. In addition, the SRRwas loaded in first followed by the Teflon rod to reduce the end effects on the SRR, whichcan occur as the induced magnetic field loops around the edges and into the bore of theLGR. This process was repeated using a single SRR and a single aluminum rod (3/16-inch diameter) and subsequently carried out for arrays of N = 2, 3, and 4 SRRs andboth aluminum and Teflon rods of both 1/8-inch and 3/16-inch diameter. The loadedLGR used for these measurements is pictured in Fig. 7.3(a). The plots of |S11| as a387.2. Measuring |S11| of NIM Arraysfunction of frequency, shown in Appendix A, show little variation when aluminum rodsare used instead of Teflon rods. The only minimal change observed was, as the volumeof aluminum of the bore increased, the resonance peaks shifted slightly closer together.Moreover, the resonance curves show similar qualitative features to those displayed in Fig.4.8, in which the reflection coefficient SRR arrays was measured. The most significantshift was observed between the N = 4 arrays of 3/16-inch diameter wires, as shown inFig. 7.4. However, the variation in |S11| curve is still very minimal. This suggests thatthe aluminum rods have an insignificant effect on the permeability of the NIM.0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure 7.4: |S11| as a function of frequency when four SRRs and four 3/16-inch diameteraluminum cut wires (shown in red) or four 3/16-inch diameter Teflon rods (shown in blue)are loaded in the bore of the LGR.7.2.2 Two-loop-one-gapSimilar measurements were taken using the 2×2×N NIM array using the two-loop-onegap LGR using the experimental set-up depicted in Fig. 7.3(b). Again, we loaded in theNIM in the bore opposite the coupling loop and did not place the SRRs at the edge ofthe bore to reduce end effects. The obtained |S11| curves are displayed in Appendix B.Once again, the addition of cut wires had a very minimal effect on the produced resonancepeaks. The only minimal change observed was when aluminum cut wires were added tothe array, the resonant frequencies of |S11| were slightly lower. This effect becomes slightlymore prominent as the volume of aluminum loaded into the bore of the LGR increasesand is most pronounced when comparing the N = 4 arrays of 3/16-inch diameter wires,as shown in Fig. 7.5.This result is consistent with what we observed when we filled the bore of the two-loop-one-gap resonator with only aluminum cut wires (no SRRs) and compared the results to397.3. Determining the Permeability of NIM Arrays0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure 7.5: |S11| as a function of frequency when sixteen SRRs and sixteen 3/16-inchdiameter aluminum cut wires (shown in red) or sixteen 3/16-inch diameter Teflon rods(shown in blue) are loaded in the bore of the LGR.when the bore was filled with Teflon rods. For these measurements, we observed thatas the volume of aluminum in the bore increased, the resonance frequency of the singleresonance dip decreased. However, despite the small decrease in resonance frequencyobserved as the volume of aluminum in the bore increases, the shape and depth of theresonance curves do not change with the addition of the aluminum rods. So, in conclusion,similarly to for the 1× 1×N NIM arrays, the addition of cut wires has little to no effecton the produced resonance curves and thus an insignificant effect on the permeability ofthe NIM.7.3 Determining the Permeability of NIM ArraysIn order to determine the permeability of the 1-D array of SRRs and cut wires, we canfit the |S11| curves using Pendry et al.’s model [1] which we stated in Eqs. (4.1) and (4.2).However, this model must be modified to fit data with three resonance peaks. Thus,Eqs. (4.1) and (4.2) will be extended to Eqs. (7.1) and (7.2), respectively. It has beenshown that an NIM with an |S11| curve with three defined resonant dips will observe apermeability with two resonances [6].µ′ = 1− [1− (fs,1/fp,1)2][1− (fs,1/f)2][1− (fs,1/f)2]2 + [γ1/(2pif)]2 −[1− (fs,2/fp,2)2][1− (fs,2/f)2][1− (fs,2/f)2]2 + [γ2/(2pif)]2 (7.1)µ′′ =[γ1/(2pif)][1− (fs,1/fp,1)2][1− (fs,1/f)2]2 + [γ1/(2pif)]2 +[γ2/(2pif)][1− (fs,2/fp,2)2][1− (fs,2/f)2]2 + [γ2/(2pif)]2 (7.2)407.3. Determining the Permeability of NIM ArraysBy combining these equations with the model we have for the effective impedance of LGRpartially filled with magnetic material, as denoted in Eqs. (6.11) to (6.14), we are ableto fit the data we have collected for a partially filled resonator using the MATLAB codeshown in Appendix C. From this fit we will extract parameters for resonant frequency, fs,magnetic plasma frequency, fp, and γs/(2pi), for both resonances expected for the NIMpermeability. Using the fit values found in Table 2 in “The Complex Permeability of Split-Ring Resonator Arrays Measured at Microwave Frequencies” by S.L. Madsen and J.S.Bobowski [6], we set the following initial parameters: fp,1 = 0.930 GHz, fs,1 = 0.850 GHz,γ = γs,1√f/fs,1, γs,1/(2pi) = 33 MHz, fp,2 = 1.031 GHz, fs,2 = 1.015 GHz, γ = γs,2√f/fs,2,and γs,2/(2pi) = 11 MHz. These parameters provide only a starting point for the fit andwill be modified for our data set. Additionally, k = 0.3ksc where ksc is initially set to 1 butwill be specified to each data set as it depends on the specific coupling loop orientation.Additionally, L1=42.54 nH L1,sc where L1,sc is initially set to 1.75. In our first fit, wewill determine ksc and L1,sc which will then be used for all subsequent fits. Now, we canplot |S11| as a function of frequency for the data we have collected and fit it to |S11| asdetermined by the following model,|S11| =√[(|Z|2/Z20)− 1]2 + [2(X/Z0)]2[(|Z|2/Z20) + 1] + 2(R/Z0), (7.3)where |Z|2 = R2+X2 and R and X are given by Eqs. (6.13) and (6.14), respectively. Thefollowing values are assumed Z0 = 50Ω, f0 = 857.7 MHz, and Q0 = 49.08. The first dataset we will fit was collected while an N = 4 array of SRRs and 3/16-inch Teflon rods wasloaded into the LGR bore. This data set and fit is shown in Fig. 7.6(a).0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (Hz) 10900.10.20.30.40.50.60.70.80.91|S 11|DataFit(a) Teflon0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (Hz) 10900.10.20.30.40.50.60.70.80.91|S 11|DataFit(b) AluminumFigure 7.6: |S11| as a function of frequency measured for one-loop-one-gap resonator filledwith four SRRs and four 3/16-inch diameter rods and fit using Pendry et al.’s modelmodified for a partially filled LGR.From here, we determined and set ksc to 1.54 and L1,sc to 1.1339. The obtained parametersfor fp,1, fs,1, γs,1/(2pi), fp,2, fs,2, and γs,2/(2pi) are presented in Table 7.1. The same fit was417.3. Determining the Permeability of NIM ArraysTable 7.1: Best fit parameters for one-loop-one-gap resonator filled with four SRRs andfour 3/16-inch diameter Teflon rods.fs,1 (GHz) fp,1 (GHz) γs,1/(2pi) (MHz) fs,2 (GHz) fp,2 (GHz) γs,2/(2pi) (MHz)0.8445±0.0001 0.9272±0.0003 29.0±0.2 1.0211±0.0003 1.0325±0.0003 24.0±0.5then repeated for the |S11| curve measured from the N = 4 array of SRRs and 3/16-inchaluminum cut wires. For this fit, we fixed the values of ksc and L1,sc to be 1.54 and 1.1339,respectively. The measured |S11| curve and obtained fit for this NIM are presented in Fig.7.6(b). Additionally, the obtained best fit parameters are shown in Table 7.2.Table 7.2: Best fit parameters for one-loop-one-gap resonator filled with four SRRs andfour 3/16-inch diameter aluminum cut wires.fs,1 (GHz) fp,1 (GHz) γs,1/(2pi) (MHz) fs,2 (GHz) fp,2 (GHz) γs,2/(2pi) (MHz)0.8520±0.0002 0.9298±0.0003 30.2±0.3 1.0254±0.0003 1.0365±0.0004 23.2±0.5From the parameters obtained from the fits, we are able to find the relative effectivepermeability of our array by plotting Eqs. (7.1) and (7.2). The resulting plots of bothµ′, the permeability’s real component, and µ′′, the permeability’s imaginary component,are shown for both N = 4 arrays of Teflon rods with SRRs and aluminum cut wires withSRRs in Fig. 7.7. By analyzing Fig. 7.7, we can see that the addition of aluminum (as6 7 8 9 10 11 12 13Frequency (Hz) 108-2-1012345Permeability' (Teflon)'' (Teflon)' (Aluminum)'' (Aluminum)Figure 7.7: The relative effective permeability for one-dimensional arrays of four Teflonrods and four SRRs compared with four aluminum cut wires and four SRRs.427.3. Determining the Permeability of NIM Arraysshown by the solid line plots) had little effect on the permeability of arrays of just SRRs(as shown by the dashed line plots). In fact, the permeability of our NIM is almost entirelydue to SRRs and that the cut wires provide only small perturbation, which is essentiallynegligible. Permeability defines the magnetization a given material gains from a magneticfield [5]. Thus, as cut wire arrays are only expected to produce an electric response, andthus modify the permittivity, it was expected their addition to the SRR array would havea minimal effect on its permeability. Another important result from Fig. 7.7 is that µ′′is found to be always positive. This implies power loss, rather than the power gain someprevious research has predicted, in the NIM [14].43Chapter 8Further Investigations8.1 Applying Current to Cut Wire ArraysThroughout all of of these measurements, the LGR’s bore has only been exposed to theuniform magnetic field induced from the AC signal supplied to the coupling loop whilethe electric field has remained at approximately zero. An area for further research is toinvestigate the effect of supplying the cut wires within the LGR’s bore with current. This,in turn, is expected to vary the LGR bore’s magnetic field. In principle, this allows us toartificially mimic the effect of an electric field within the cut wires, while the LGR’s borestill experiences a purely magnetic field. For the preliminary steps of this investigation,we have designed the experimental set-up displayed in Fig. 8.1.Figure 8.1: Initial experimental set-up for applying current to cut wire arrays. Thefollowing equipment is labelled: 1 is an EM shield containing the LGR; 2,6, and 10 arecirculators; 3 and 7 are power splitters, 4 is an attenuator; 5 is a VNA; 8 and 9 are phaseshifters.Figure 8.1 shows an EM shield (1) which contains a one-loop-one-gap resonator. Inorder to apply current to the cut wires within the resonator’s bore, the wires were cutto be long enough to be accessible from outside of the EM shield. To secure these wires,three holes, one for each wire, were drilled through both sides of the LGR and EM shield.Then, we secured copper electrodes to each end of the wires using a clamp. In order toavoid direct contact with the aluminum LGR, we wrapped masking tape around each cutwire where it passed through the LGR’s shell. The EM shield was used to prevent the448.2. Incorporating Other MetamaterialsLGR from interacting with the outer portion of the cut wires and also limit other radiativelosses due to the laboratory surroundings [21]. Again, the VNA (5) was used as a powersource. The transmitting port of the VNA supplied an AC current to a power splitter(3), which directed half of the signal to a circulator (2) and the other half through anattenuator (4). The signal supplied to port 1 of the circulator (2) was transmitted to thecoupling loop of the LGR via port 2. Then, the signal reflected back to port 2 was passedto the receiving port of the VNA via port 3 of the circulator. The other half of the signaltravelled through the attenuator (4) and was supplied to a phase shifter (8). From here,the signal was split again at the second power splitter (6). Then half of the remainingsignal was directed through a circulator (6) and set to power the copper electrode onthe right hand side of the EM shield. The other half of the signal was directed throughanother phase shifter (9) and circulator (10) before it powered the left copper electrode.The first phase shifter (8) functioned to adjust the phase of the current in the wiresrelative to the oscillating magnetic field in the bore of the resonator. The second phaseshifter (9), was used to set the desired phase of the signals supplied to the copper elec-trodes. The signal supplied to the left electrode was set to be 180◦s out of phase with theright electrodes signal. Thus, while one AC signal was at its maximum, the other was atits minimum, and they crossed zero simultaneously. This experimental set-up allowed usto control the phase of the signals being delivered to the copper electrodes and thus tothe cut wire array.By supplying a current through the cut wires, a magnetic field will be induced whichcircles the wires. This is expected to result in a non-uniform magnetic field within theLGR bore which will allow us to further explore the permeability of the cut wire array.Ultimately, we predict that we may observe a negative imaginary component of the effec-tive relative permeability, µ′′, for specific phases of the current in the cut wires relativeto the oscillating magnetic field.8.2 Incorporating Other MetamaterialsAnother aspect a future project could investigate is the incorporation of other metama-terials. Specifically, we could make use of complementary split-ring resonators (CSRRs),the dual of the SRR [22]. This structure was originally proposed in 2004 by Falcone etal. and was created to provide the negative permittivity component of an NIM array.These CSRRs essentially act as a single electric dipole with negative polarizability. Theycomposed their CSRRs by etching out SRR like dimensions into a planar circuit boardand a thin metallic plate [23]. CSRRs are quite compact and thus may be easier to workwith than cut wires for higher dimension NIM arrays.45Chapter 9ConclusionThe complex permeability, µr = µ′ − jµ′′, of metamaterials was characterized usingLGRs. The metamaterials we worked with included SRR arrays, which have a negativepermeability over a small range of microwave frequencies just above their resonant fre-quency, as well as cut wire arrays, which are designed to have a negative permittivityat microwave frequencies. To characterize these arrays, the experimental procedure pre-sented by S. L. Madsen and J. S. Bobowski to measure the permeability of SRR arrayswas adapted to incorporate arrays of cut wires as well as arrays of SRRs and cut wires[6]. The motivation behind this project was to explore the effect the addition of cut wireshas to the permeability of an array of SRRs. Previous investigations, including numericalsimulations and experimental measurements, have suggested that the imaginary compo-nent of the permeability is negative for cut wire arrays [6] [7]. This is an unphysical resultas it describes power gain in the metamaterial.To determine the permeability of these metamaterials, LGRs were used. An SMA cableconnected the LGR’s coupling loop to a VNA. This allowed us to supply a signal to andextract the reflection coefficient, |S11|, from the LGR’s bore. In order to model |S11|, werepresented the LGR as an LRC circuit. From here, Kirchhoff’s voltage law was applied tofind the effective impedance of the LGR which was used to model the expected |S11| curve[5] [19]. However, as the LGR bore was loaded with magnetic material, the expected |S11|curve varied as the bore’s inductance varied. To account for this, we used Pendry et al.’smodel for the permeability of SRR arrays which accounts for the frequency dependenceof permeability [1].Initially, we measured the |S11| curves for arrays of aluminum cut wires. Overall, wequalitatively observed minimal and non-systematic changes in the produced |S11| reflec-tions for arrays comprised of various numbers of cut wires. Thus, as |S11| depends on thepermeability of the material loaded inside the LGR’s bore, the addition of cut wires hada very small effect on the bore’s permeability. Lastly, we measured |S11| for arrays con-taining both SRRs and cut wires. Using the model we derived for |S11|, we extracted theresonant frequency, magnetic plasma frequency, and damping constant for arrays of SRRsand arrays of both SRRs and cut wires. From these parameters, we plotted the relativeeffective permeability of the arrays and found that the addition of aluminum cut wireshad little effect on the permeability of arrays of solely SRRs. In fact, the permeability ifthe metamaterial was almost entirely due to SRRs. This was anticipated as the cut wiresare expected to only provide an electric response. Moreover, the imaginary component,µ′′, of the permeability of the array was found to be positive over the entire frequencyrange which is expected for power dissipation in the metamaterial.46Bibliography[1] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism fromconductors and enhanced nonlinear phenomena,” IEEE Transactions on MicrowaveTheory and Techniques, vol. 47, no. 11, pp. 2075–2084, 1999.[2] R. S. Kshetrimayum, “A brief introduction to metamaterials,” IEEE Potentials,vol. 23, no. 5, pp. 44–46, 2004.[3] J. B. Pendry, “Negative refraction,” Contemporary Physics, vol. 45, no. 3, pp. 191–202, 2004.[4] V. G. Veselago, “The electrodynamics of substances with simultaneously negativevalues of ε and µ,” Soviet Physics Uspekhi, vol. 10, no. 4, pp. 509–514, 1968.[5] D. J. 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Sorolla Ayza, “Babinet principle applied to the design of meta-surfaces and metamaterials,” Physical review letters, vol. 93, no. 19, pp. 1–4, 2004.48Appendix A|S11| Measurements of 1× 1×N NIMArraysAppendix A contains the remaining experimental |S11| curves plotted as a function offrequency for 1× 1×N arrays of SRRs and 3/16-inch or 1/8-inch aluminum cut wires orTeflon rods. Measurements were taken using the one-loop-one-gap resonator.0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure A.1: |S11| as a function of frequency when a single SRR and one 3/16-inch diameteraluminum cut wire (shown in red) or one 3/16-inch diameter Teflon rod (shown in blue)are loaded in the bore of the LGR.49Appendix A. |S11| Measurements of 1× 1×N NIM Arrays0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure A.2: |S11| as a function of frequency when a single SRR and one 1/8-inch diameteraluminum cut wire (shown in magenta) or one 1/8-inch diameter Teflon rod (shown inblack) are loaded in the bore of the LGR.0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure A.3: |S11| as a function of frequency when two SRRs and two 3/16-inch diameteraluminum cut wires (shown in red) or two 3/16-inch diameter Teflon rods (shown in blue)are loaded in the bore of the LGR.50Appendix A. |S11| Measurements of 1× 1×N NIM Arrays0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure A.4: |S11| as a function of frequency when two SRRs and two 1/8-inch diameteraluminum cut wires (shown in magenta) or two 1/8-inch diameter Teflon rods (shown inblack) are loaded in the bore of the LGR.0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure A.5: |S11| as a function of frequency when three SRRs and three 3/16-inch diameteraluminum cut wires (shown in red) or two 3/16-inch diameter Teflon rods (shown in blue)are loaded in the bore of the LGR.51Appendix A. |S11| Measurements of 1× 1×N NIM Arrays0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure A.6: |S11| as a function of frequency when three SRRs and three 1/8-inch diameteraluminum cut wires (shown in magenta) or three 1/8-inch diameter Teflon rods (shownin black) are loaded in the bore of the LGR.0.7 0.8 0.9 1 1.1 1.2 1.3Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure A.7: |S11| as a function of frequency when four SRRs and four 1/8-inch diameteraluminum cut wires (shown in magenta) or four 1/8-inch diameter Teflon rods (shown inblack) are loaded in the bore of the LGR.52Appendix B|S11| Measurements of 2× 2×N NIMArraysAppendix B contains the remaining experimental |S11| curves plotted as a function offrequency for 2× 2×N arrays of SRRs and 3/16-inch or 1/8-inch aluminum cut wires orTeflon rods. Measurements were taken using the two-loop-one-gap resonator.0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)0.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure B.1: |S11| as a function of frequency when four SRRs and four 1/8-inch diameteraluminum cut wires (shown in magenta) or four 1/8-inch diameter Teflon rods (shown inblack) are loaded in the bore of the LGR.53Appendix B. |S11| Measurements of 2× 2×N NIM Arrays0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure B.2: |S11| as a function of frequency when a four SRR and four 1/8-inch diameteraluminum cut wires (shown in magenta) or four 1/8-inch diameter Teflon rods (shown inblack) are loaded in the bore of the LGR.0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure B.3: |S11| as a function of frequency when eight SRRs and eight 3/16-inch diameteraluminum cut wires (shown in red) or eight 3/16-inch diameter Teflon rods (shown in blue)are loaded in the bore of the LGR.54Appendix B. |S11| Measurements of 2× 2×N NIM Arrays0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)0.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure B.4: |S11| as a function of frequency when eight SRRs and eight 1/8-inch diameteraluminum cut wires (shown in magenta) or eight 1/8-inch diameter Teflon rods (shownin black) are loaded in the bore of the LGR.0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure B.5: |S11| as a function of frequency when twelve SRRs and twelve 3/16-inchdiameter aluminum cut wires (shown in red) or twelve 3/16-inch diameter Teflon rods(shown in blue) are loaded in the bore of the LGR.55Appendix B. |S11| Measurements of 2× 2×N NIM Arrays0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure B.6: |S11| as a function of frequency when twelve SRRs and twelve 1/8-inch di-ameter aluminum cut wires (shown in magenta) or twelve 1/8-inch diameter Teflon rods(shown in black) are loaded in the bore of the LGR.0.6 0.7 0.8 0.9 1 1.1 1.2Frequency (GHz)00.10.20.30.40.50.60.70.80.91|S 11|AluminumTeflonFigure B.7: |S11| as a function of frequency when sixteen SRRs and sixteen 1/8-inchdiameter aluminum cut wires (shown in magenta) or sixteen 1/8-inch diameter Teflonrods (shown in black) are loaded in the bore of the LGR.56Appendix C|S11| MATLAB Fit for Arrays ofNIMsAppendix C contains the MATLAB code used to fit the |S11| curves measured from a one-loop-one-gap LGR partially filled with magnetic material. This fit makes use of Pendryet al.’s model for SRR permeability, as given by Eqs. (7.1) and (7.2), as well as the modelderived for effective impedance of a partially filled bore, as given by Eqs. (6.11) to (6.14).These equations are used in conjunction with Eq. (4.12) to fit the experimentally mea-sured |S11| curves. This specific fit was done for an N = 4 1-D array of 3/16-inch diameteraluminum cut wires and SRRs.1 % Ava Corne l l2 % February 15 , 20213 % Modif ied from :4 % Jake Bobowski5 % January 12 , 20216 % Try en t e r i ng the f i t f unc t i on f o r a LGR with i t s bore7 % p a r t i a l l y loaded with a s i n g l e SRR and cut wire .8 c l e a r v a r s ;9 format longE ;1011 % Fir s t , en te r the r e a l and imaginary par t s o f the r e l a t i v epe rmeab i l i t y .12 % Use the model proposed by Pendry . However , i t i s p o s s i b l e13 % to a l s o use the Lorenzt ian model . Note a l s o that one can sumc r e a t e a14 % s u p e r p o s i t i o n o f the se r e l a t i v e p e r m e a b i l i t i e s to caputrea d d i t i o n a l15 % f e a t u r e s pre sent in the data . This s u p e r p o s i t i o n should s t i l lobey the16 % Kramers−Kronig r e l a t i o n between the r e a l and imaginary par t s .1718 % Star t w i l l a range o f f r e q u e n c i e s19 f s t a r t = 700 e6 ; % Hz20 f s t o p = 1300 e6 ; % Hz21 f = l i n s p a c e ( f s t a r t , f s top , 5000) ;2257Appendix C. |S11| MATLAB Fit for Arrays of NIMs23 % Enter some parameters f o r Pendry ’ s pe rmeab i l i t y model . Valuesfrom Table 2 o f ”The24 % Complex Permeab i l i ty o f Sp l i t−Ring Resonator Arrays Measuredat Microwave Frequenc ie s ” by Madsen and Bobowski .25 f s 1 = 850 e6 ; % Hz26 fp1 = 930 e6 ; % Hz27 gS1 = 33∗1 e6 ; % Hz28 g1 = gS1 ∗( f / f s 1 ) . ˆ ( 0 . 5 ) ;29 f s 2 = 1015 e6 ; % Hz30 fp2 = 1031 e6 ; % Hz31 gS2 = 11∗1 e6 ; % Hz32 g2 = gS2 ∗( f / f s 2 ) . ˆ ( 0 . 5 ) ;33 mu1 = 1 − (1 − ( f s 1 / fp1 ) ˆ2) ∗(1 − ( f s 1 . / f ) . ˆ 2 ) . / ( ( 1 − ( f s 1 . / f ). ˆ 2 ) . ˆ2 + ( g1 . / f ) . ˆ 2 )− (1 − ( f s 2 / fp2 ) ˆ2) ∗(1 − ( f s 2 . / f ) . ˆ 2 ). / ( ( 1 − ( f s 2 . / f ) . ˆ 2 ) . ˆ2 + ( g2 . / f ) . ˆ 2 ) ;34 mu2 = ( g1 . / f ) ∗(1 − ( f s 1 / fp1 ) ˆ2) . / ( ( 1 − ( f s 1 . / f ) . ˆ 2 ) . ˆ2 + ( g2 . / f ). ˆ 2 ) +(g2 . / f ) ∗(1 − ( f s 2 / fp2 ) ˆ2) . / ( ( 1 − ( f s 2 . / f ) . ˆ 2 ) . ˆ2 + ( g2 . /f ) . ˆ 2 ) ;3536 % Plot the r e l a t i v e pe rmeab i l i t y components37 %plo t ( f , mu1 , ’ r ’ ) ;38 %hold on ;39 %plo t ( f , mu2 , ’b ’ ) ;40 %hold o f f ;41 %f i g u r e ;4243 % Now ente r the s c a l e d inductance (\ e l l ) o f the p a r t i a l l y − f i l l e dLGR bore .44 % These e x p r e s s i o n s should correspond Eqs . ( 6 . 1 1 ) and ( 6 . 1 2 ) .45 N = 4 ;46 spacer = 2 0 . 3 2 ;47 t h i c k n e s s = 1 . 5 4 ;48 LGRlength = 1 1 2 . 0 0 ;49 x = N∗( spacer + t h i c k n e s s ) /( LGRlength ) ; % x i s the f i l l i n gf a c t o r50 murSq = mu1.ˆ2 + mu2 . ˆ 2 ;51 l 1 = (mu1∗x + murSq∗(1 − x ) ) . / ( ( x + mu1∗(1 − x ) ) . ˆ2 + mu2.ˆ2∗ ( 1− x ) ˆ2) ;52 l 2 = mu2∗x . / ( ( x + mu1∗(1 − x ) ) . ˆ2 + mu2.ˆ2∗ ( 1 − x ) ˆ2) ;5354 % Plot the s c a l e d inductance components55 %f i g u r e ;56 %plo t ( f , l1 , ’ r ’ ) ;57 %hold on ;58Appendix C. |S11| MATLAB Fit for Arrays of NIMs58 %plo t ( f , l2 , ’b ’ ) ;59 %plo t ( f , lmag , ’ g ’ ) ;60 %hold o f f ;6162 % OK, now ente r the e x p r e s s i o n s needed to cons t ruc t theimpedance o f the63 % coupled LGR with i t s bore p a r t i a l l y −loaded with a SRR/ cut wire64 % array .65 f 0 = 857 .7 e6 ; % Hz66 Q0 = 4 9 . 0 8 ;6768 % Here , then , i s the impedance o f an i n d u c t i v e l y couple two−loop, one−gap69 % LGR with one o f i t s bores p a r t i a l l y loaded with a SRR array .These70 % e x p r e s s i o n s should correspond to Eqs . ( 6 . 1 3 ) and ( 6 . 1 4 ) .71 L1 = 42.54 e−9; % Henr ies72 k = 0 . 3 ;73 R1 = (2∗ pi ∗ f ∗L1) . ∗ ( f / f 0 ) .∗ k ˆ 2 .∗ ( ( l 1 /Q0) . ∗ ( f / f 0 ) . ˆ 0 . 5 + l 2 . ∗ ( f 0 . /f ) ) . / ( ( ( 1 /Q0) ∗( f / f 0 ) . ˆ 0 . 5 + ( f / f 0 ) .∗ l 2 ) . ˆ2 + ( ( f / f0 ) .∗ l 1 − f 0. / f ) . ˆ 2 ) ;74 X1 = 2∗ pi ∗ f ∗L1 .∗ ( 1 − ( f / f 0 ) ∗k ˆ2 .∗ ( l 2 .∗ ( ( 1 /Q0) ∗( f / f 0 ) . ˆ 0 . 5 + l 2. ∗ ( f / f 0 ) ) + l 1 . ∗ ( l 1 . ∗ ( f / f 0 ) − f 0 . / f ) ) . / ( ( ( 1 /Q0) ∗( f / f 0 ) . ˆ 0 . 5 +( f / f 0 ) .∗ l 2 ) . ˆ2 + ( ( f / f 0 ) .∗ l 1 − f 0 . / f ) . ˆ 2 ) ) ;75 Z1mag = (R1.ˆ2 + X1. ˆ 2 ) . ˆ 0 . 5 ;7677 % Plot the impedance components78 %f i g u r e ;79 %plo t ( f , R1 , ’ r ’ ) ;80 %hold on ;81 %plo t ( f , X1 , ’b ’ ) ;82 %plo t ( f , Z1mag , ’ g ’ ) ;83 %hold o f f ;8485 % Now c a l c u l a t e the r e f l e c t i o n c o e f f i c i e n t S1186 Z0 = 50 ;87 o f f S l o p e = 40 .8 e9 ;8889 ReS11 = ( ( Z1mag/Z0) . ˆ2 − 1) . / ( ( ( Z1mag/Z0) . ˆ2 + 1) + 2∗R1/Z0) ;90 ImS11 = 2∗(X1/Z0) . / ( ( ( Z1mag/Z0) . ˆ2 + 1) + 2∗R1/Z0) ;91 S11mag = ( ReS11 .ˆ2 + ImS11 . ˆ 2 ) . ˆ 0 . 5 − f / o f f S l o p e ;9293 % Plot the r e f l e c t i o n c o e f f i c e n t s94 %f i g u r e ;59Appendix C. |S11| MATLAB Fit for Arrays of NIMs95 %plo t ( f , ReS11 , ’ r ’ ) ;96 %hold on ;97 %plo t ( f , ImS11 , ’b ’ ) ;98 %plo t ( f , S11mag , ’ g ’ ) ;99 %hold o f f ;100101102 % Import the data and p lo t i t with the c a l c u l a t e d r e f l e c t i o nc o e f f i c i e n t . Try103 % adju s t i ng the pe rmeab i l i t y parameters u n t i l the c a l c u l a t e d S11104 % approximately matches the measured S11 .105 M = dlmread ( ’ one−loop one−gap LGR − 4 SRRs − 4 Aluminum rods . txt’ ) ;106 fdata = M( : , 4 ) ’ ;107 S11 = M( : , 5 ) ’ ;108109 f i g u r e ;110 p lo t ( f , S11mag , ’ r ’ ) ;111 hold on ;112 p lo t ( fdata , S11 , ’ ko ’ ) ;113 hold o f f ;114115116 % This stop i s used to stop the program . I f you are j u s tad ju s t i ng117 % parameter va lues to t ry to f i n d good s t a r t i n g values ,uncomment the stop118 % below . I f you ’ d l i k e to a c t u a l l y do the best−f i t , thencomment out the119 % stop .120121 %stop ;122 c l e a r v a r s ;123124 % Enter the value o f the f i l l i n g f a c t o r .125 N = 4 ;126 spacer = 2 0 . 3 2 ;127 t h i c k n e s s = 1 . 5 4 ;128 LGRlength = 1 1 2 . 0 0 ;129 x = N∗( spacer + t h i c k n e s s ) /( LGRlength ) ;130131132 % Import the data .60Appendix C. |S11| MATLAB Fit for Arrays of NIMs133 M = dlmread ( ’ one−loop one−gap LGR − 4 SRRs − 4 Aluminum rods . txt’ ) ;134 fdata = M( : , 4 ) ’ ;135 S11 = M( : , 5 ) ’ ;136137138 % The f i t parameters w i l l be b (1 ) , b (2 ) , b (3 ) , . . .139 syms f140 b = sym( ’b ’ , [ 1 6 ] ) ;141 % b (1) = f s 1 (GHz)142 % b (2) = fp1 (GHz)143 % b (3) = gS1 (MHz)144 % b (4) = f s 2 (GHz)145 % b (5) = fp2 (GHz)146 % b (6) = gS2 (MHz)147 % b (7) = o f f S l o p e (GHz)148149 % Now ente r the e x p r e s s i o n s f o r mu1 ( r e a l part ) and mu2 (imaginary part ) .150 % I n s e r t the appropr ia t e f i t parameter symbols in p lace o f fsmfpm and g .151 g1 = s q r t (b (3 ) ˆ2) ∗1 e6 ∗( f /(b (1 ) ∗1 e9 ) ) . ˆ ( 0 . 5 ) ;152 g2 = s q r t (b (6 ) ˆ2) ∗1 e6 ∗( f /(b (4 ) ∗1 e9 ) ) . ˆ ( 0 . 5 ) ;153 mu1 = 1 − ((1−(b (1 ) . / ( b (2 ) ) ) . ˆ 2 ) .∗(1 −((b (1 ) ∗1 e9 ) . / f ) . ˆ 2 ) )./((1 − ( ( b (1 ) ∗1 e9 ) . / f ) . ˆ 2 ) .ˆ2+( g1 . / f ) . ˆ 2 ) −((1−(b (4 ) . / ( b (5 ) ) ). ˆ 2 ) .∗(1 −((b (4 ) ∗1 e9 ) . / f ) . ˆ 2 ) ) ./((1 − ( ( b (4 ) ∗1 e9 ) . / f ) . ˆ 2 ) .ˆ2+( g2. / f ) . ˆ 2 ) ;154 mu2 = ( ( g1 . / f ) .∗(1−(b (1 ) . / ( b (2 ) ) ) . ˆ 2 ) ) ./((1 − ( ( b (1 ) ∗1 e9 ) . / f ) . ˆ 2 ).ˆ2 .+( g1 . / f ) . ˆ 2 ) +((g2 . / f ) .∗(1−(b (4 ) . / ( b (5 ) ) ) . ˆ 2 ) ) ./((1 − ( ( b (4 )∗1 e9 ) . / f ) . ˆ 2 ) .ˆ2 .+( g2 . / f ) . ˆ 2 ) ;155156 % Now ente r the e x p r e s s i o n s f o r l 1 ( r e a l part ) and l 2 ( imaginarypart ) .157 l 1 = (mu1 . ∗ ( x + mu1∗(1 − x ) ) + mu2.ˆ2∗ ( 1 − x ) ) . / ( ( x + mu1∗(1 − x) ) . ˆ2 + mu2.ˆ2∗ ( 1 − x ) ˆ2) ;158 l 2 = mu2∗x . / ( ( x + mu1∗(1 − x ) ) . ˆ2 + mu2.ˆ2∗ ( 1 − x ) ˆ2) ;159 lmag = ( l 1 . ˆ2 + l 2 . ˆ 2 ) . ˆ 0 . 5 ;160161 % Enter va lues f o r L1 , f0 , Q0 , and k p r e v i o u s l y determined formf i t s to the162 % empty−bore LGR. L1 and k est imated f o r t h i s s p e c i f i c f i t .163 L1 = 42.54∗1 e−9∗1.1339;164 f 0 = 0.8577∗1 e9 ;165 Q0 = 4 9 . 0 8 ;61Appendix C. |S11| MATLAB Fit for Arrays of NIMs166 k = 0 . 3 ∗ 1 . 5 4 ;167168 % Now ente r the e x p r e s s i o n s f o r R and X, the r e a l and imaginarycomponents o f the impedance o f the coupled LGR169 % with a p a r t i a l l y − f i l l e d bore .170 R1 = (2∗ pi ∗ f ∗L1) . ∗ ( f / f 0 ) .∗ k ˆ 2 .∗ ( ( l 1 /Q0) . ∗ ( f / f 0 ) . ˆ 0 . 5 + l 2 . ∗ ( f 0 . /f ) ) . / ( ( ( 1 /Q0) ∗( f / f 0 ) . ˆ 0 . 5 + ( f / f 0 ) .∗ l 2 ) . ˆ2 + ( ( f / f0 ) .∗ l 1 − f 0. / f ) . ˆ 2 ) ;171 X1 = 2∗ pi ∗ f ∗L1 .∗ ( 1 − ( f / f 0 ) ∗k ˆ2 .∗ ( l 2 .∗ ( ( 1 /Q0) ∗( f / f 0 ) . ˆ 0 . 5 + l 2. ∗ ( f / f 0 ) ) + l 1 . ∗ ( l 1 . ∗ ( f / f 0 ) − f 0 . / f ) ) . / ( ( ( 1 /Q0) ∗( f / f 0 ) . ˆ 0 . 5 +( f / f 0 ) .∗ l 2 ) . ˆ2 + ( ( f / f 0 ) .∗ l 1 − f 0 . / f ) . ˆ 2 ) ) ;172 Z1mag = (R1.ˆ2 + X1. ˆ 2 ) . ˆ 0 . 5 ;173174 % Here are the exp r e s s i on f o r c a l c u l a t i n g S11 from R and X. theo f f S l o p e175 % parameter i s used to take in to account l o s s e s a s s o c i a t e d withthe l eng th s176 % of c o a x i a l cab l e that l ead up to the coup l ing loop . Thesel o s s e s grow as177 % frequnecy i n c r e a s e s .178 Z0 = 50 ;179 o f f S l o p e = 40.8∗1 e9 ;180 ReS11 = ( ( Z1mag/Z0) . ˆ2 − 1) . / ( ( ( Z1mag/Z0) . ˆ2 + 1) + 2∗R1/Z0) ;181 ImS11 = 2∗(X1/Z0) . / ( ( ( Z1mag/Z0) . ˆ2 + 1) + 2∗R1/Z0) ;182 S11mag = ( ReS11 .ˆ2 + ImS11 . ˆ 2 ) . ˆ 0 . 5 − f / o f f S l o p e ;183184 % We now d e f i n e the model func t i on r equ i r ed f o r MATLAB’ s f i t t i n grou t in e185 % f i tn lm .186 S11fcn = matlabFunction (S11mag) ;187 mdlS11 = @(b , f ) S11fcn (b (1 ) , b (2 ) , b (3 ) , b (4 ) , b (5 ) , b (6 ) , f ) ;188189 % Here , we c a l l the f i t t i n g rou t in e and prov ide some i n t i t i a lparameter190 % est imate s .191 r e s = f i tn lm ( fdata , S11 , mdlS11 , [ 0 . 8 5 , . 930 , 33 , 1 . 015 , 1 . 031 ,1 1 ] )192193 % Extract the best− f i t paramters .194 f s 1 = r e s . C o e f f i c i e n t s . Estimate (1 ) ;195 fp1 = r e s . C o e f f i c i e n t s . Estimate (2 ) ;196 gS1 = r e s . C o e f f i c i e n t s . Estimate (3 ) ;197 f s 2 = r e s . C o e f f i c i e n t s . Estimate (4 ) ;198 fp2 = r e s . C o e f f i c i e n t s . Estimate (5 ) ;62Appendix C. |S11| MATLAB Fit for Arrays of NIMs199 gS2 = r e s . C o e f f i c i e n t s . Estimate (6 ) ;200 %o f f S l o p e = r e s . C o e f f i c i e n t s . Estimate (7 ) ;201202 % Put the paramter va lue s in to a l i s t .203 param = [ f s1 , fp1 , gS1 , f s2 , fp2 , gS2 ] ;204 param ’205206 % A bunch o f f requency va lue s207 xx = l i n s p a c e (min ( fdata ) − 10e7 , max( fdata ) , 5000) ;208209 % Plot the S11 data and the f i t f unc t i on toge the r .210 f i g u r e ;211 p lo t ( fdata , S11 , ’ or ’ )212 hold on ;213 p lo t ( fdata , mdlS11 (param , fdata ) , ’ Color ’ , ’ k ’ , ’ LineWidth ’ ,2) ;214 hold o f f ;215216 % Plot the r e a l and imaginary components o f the r e l a t i v epe rmeab i l i t y .217 g1 = s q r t ( gS1 ˆ2) ∗1 e6 ∗( xx /( f s 1 ∗1 e9 ) ) . ˆ ( 0 . 5 ) ;218 g2 = s q r t ( gS2 ˆ2) ∗1 e6 ∗( xx /( f s 2 ∗1 e9 ) ) . ˆ ( 0 . 5 ) ;219 mu1 = 1 − ((1−( f s 1 . / fp1 ) . ˆ 2 ) .∗(1−( f s 1 ∗1 e9 . / xx ) . ˆ 2 ) ) ./((1 −( f s 1 ∗1e9 . / xx ) . ˆ 2 ) .ˆ2+( g1 . / xx ) . ˆ 2 )− ((1−( f s 2 . / fp2 ) . ˆ 2 ) .∗(1−( f s 2 ∗1 e9. / xx ) . ˆ 2 ) ) ./((1 −( f s 2 ∗1 e9 . / xx ) . ˆ 2 ) .ˆ2+( g2 . / xx ) . ˆ 2 ) ;220 f i g u r e ;221 p lo t ( xx , mu1 , ’−−b ’ ) ;222 hold on ;223224 mu2 = ( ( g1 . / xx ) .∗(1−( f s 1 . / fp1 ) . ˆ 2 ) ) ./((1 −( f s 1 ∗1 e9 . / xx ) . ˆ 2 ) .ˆ2 .+(g1 . / xx ) . ˆ 2 ) +((g2 . / xx ) .∗(1−( f s 2 . / fp2 ) . ˆ 2 ) ) ./((1 −( f s 2 ∗1 e9 . / xx ). ˆ 2 ) .ˆ2 .+( g2 . / xx ) . ˆ 2 ) ;225 p lo t ( xx , mu2 , ’−−r ’ ) ;226 hold o f f ;227228 % Write the best i f t parameters and the best− f i t | S11 | curve ,mu1 , and mu2 to txt f i l e s .229 M = [ xx ; mdlS11 (param , xx ) ; mu1 ; mu2 ] ’ ;230 dlmwrite ( ’ one−loop one−gap LGR − 4 SRRs − 4 Aluminum rods − bestf i t parameters . txt ’ , param ’ , ’ d e l i m i t e r ’ , ’\ t ’ , ’ p r e c i s i o n ’ ,9)231 dlmwrite ( ’ one−loop one−gap LGR − 4 SRRs − 4 Aluminum rods − bestf i t data . txt ’ , M, ’ d e l i m i t e r ’ , ’\ t ’ , ’ p r e c i s i o n ’ , 9)63

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