Investigation of Magnitude and Phase Errors in Waveguide Samples for the Nicolson-Ross-Weir Permittivity Technique

Transcription

1 Univesity of New Hampshie Univesity of New Hampshie Sholas' Repositoy Maste's Theses and Capstones Student Sholaship Winte 016 Investigation of Magnitude and Phase Eos in Waveguide Samples fo the Niolson-Ross-Wei Pemittivity Tehnique Paul Galvin Univesity of New Hampshie, Duham Follow this and additional woks at: Reommended Citation Galvin, Paul, "Investigation of Magnitude and Phase Eos in Waveguide Samples fo the Niolson-Ross-Wei Pemittivity Tehnique" (016). Maste's Theses and Capstones This Thesis is bought to you fo fee and open aess by the Student Sholaship at Univesity of New Hampshie Sholas' Repositoy. It has been aepted fo inlusion in Maste's Theses and Capstones by an authoied administato of Univesity of New Hampshie Sholas' Repositoy. Fo moe infomation, please ontat

2 Investigation of Magnitude and Phase Eos in Waveguide Samples fo the Niolson-Ross-Wei Pemittivity Tehnique Abstat The Niolson-Ross-Wei (NRW) tehnique is a ommonly-used method fo measuing the eletomagneti popeties of low-loss mateials. The tehnique entails plaing the mateial unde test in a waveguide, and then infeing the eletomagneti popeties of that mateial fom the efletion and tansmission oeffiients measued at the ends of the guide. While NRW tehnique geneally povides eliable and auate esults, thee ae onditions whee eos an aise. Some of the known eos ae attibuted to sample pepaation, although eos have been obseved even with pefetly-pepaed samples. Those eos geneally esult in an oveestimation of loss and unealisti values fo pemeability, and they ae shown to be assoiated with phase eos in the efletion and tansmission oeffiients. The wok epoted hee shows the elationship between measuement phase eos and how they impat estimated eletomagneti popety values. Futhe, an appoah to oet fo those eos when evaluating non-magneti mateials is given. Keywods Eletial engineeing This thesis is available at Univesity of New Hampshie Sholas' Repositoy:

3 INVESTIGATION OF MAGNITUDE AND PHASE ERRORS IN WAVEGUIDE SAMPLES FOR THE NICOLSON-ROSS-WEIR PERMITTIVITY TECHNIQUE BY Paul Galvin B.S.E.E. Univesity of New Hampshie, May 013 THESIS Submitted to the Univesity of New Hampshie In Patial Fulfillment of The Requiements of the Degee of Maste of Siene in Eletial Engineeing Deembe 016

4 This thesis/dissetation has been examined and appoved in patial fulfillment of the equiements fo the degee of Maste of Siene in Eletial Engineeing by: Thesis Dieto, Kent A. Chambelin Pofesso of Eletial & Compute Engineeing Kondagunta Sivapasad Pofesso of Eletial & Compute Engineeing Allen D. Dake Assoiate Pofesso of Eletial & Compute Engineeing On August 9, 016 Oiginal appoval signatues ae on file with the Univesity of New Hampshie Gaduate Shool. iii

5 DEDICATION To my family, fo all the enouagement they ve given ove the yeas iv

6 ACKNOWLEDGEMENTS I would like to fomally thank D. Kent Chambelin, D. Allen Dake, and D. Kondagunta Sivapasad fo all thei guidane both with this pojet and aoss all my studies. I would also like to expess my gatitude fo all the staff at Amphenol-TCS fo the time and esoues they supplied to help me aomplish this eseah. v

7 TABLE OF CONTENTS Dediation... iv Aknowledgements... v Table of Contents... vi List of Figues... viii Abstat... x Chapte 1: Intodution Oveview of the Wok Plane wave Inident on an Infinite Slab Plane Wave Inident on a Dieleti Slab with a Finite Thikness... 6 Chapte : Eletomagneti Wave Model Pemittivity Calulations fom Waveguide Tansmission Pemittivity Coeting Method Eo Modeling Method...4 Chapte 3: Simulations Raw Simulation Data Redution of Pemittivity Eos...33 Chapte 4: Disussion of Results Half-Wavelength Disontinuities in NRW Tehnique Niolson-Ross-Wei Best Paties...44 vi

8 Chapte 5: Conlusions Simulations Niolson-Ross-Wei Methodology Futue Wok...49 Appendies...50 Appendix A: Calulation of Phase Constant and Infinite Slab Refletion Coeffiient fom S- Paametes...51 Appendix B: Calulation of Relative Pemittivity and Pemittivity, Thee Region Plane Wave Exitation...54 Appendix C: Calulation of Relative Pemittivity and Pemittivity, Thee Region TEM o TE wave Exitation...56 Appendix D: Waveguide Mateial Mode...61 Appendix E: Niolson-Ross-Wei Mateial Measuement Tehnique...6 List of Refeenes...63 vii

9 LIST OF FIGURES Figue 1: Plane Wave Inident on an Infinite Length of Mateial... 3 Figue : Multi-Region Dieleti Measuement Setup... 6 Figue 3: Multi-Region Dieleti Measuement Setup Coss-Setion... 7 Figue 4: Finite Thikness Dieleti Slab Inside a Waveguide...14 Figue 5: Infinite Slab of Dieleti Mateial with Non-Nomal Angle of Inidene...16 Figue 6: Oientation of the Phase Constant Tems...19 Figue 7: Teflon, S 11 Phase Eos, Real Pat of Pemittivity...7 Figue 8: Teflon, S 11 Phase Eos, Imaginay Pat of Pemittivity...8 Figue 9: Teflon, Invese S 11 Phase Eos, Real Pat of Pemittivity...8 Figue 10: Teflon, Invese S 11 Phase Eos, Imaginay Pat of Pemittivity...9 Figue 11: Plexiglas, S 11 Phase Eos, Real Pat of Pemittivity...30 Figue 1: Plexiglas, S 11 Phase Eos, Imaginay Pat of Pemittivity...30 Figue 13: Teflon, S 1 Phase Eos, Real Pat of Pemittivity...31 Figue 14: Teflon, S 1 Phase Eos, Imaginay Pat of Pemittivity...3 Figue 15: Plexiglas, S 1 Phase Eos, Real Pat of Pemittivity...3 Figue 16: Plexiglas, S 1 Phase Eos, Imaginay Pat of Pemittivity...33 Figue 17: Teflon, S 11 Phase Eos, Real Pat of Pemittivity with Known Pemeability...35 Figue 18: Teflon, S 11 Phase Eos, Imaginay Pat of Pemittivity with Known Pemeability...35 Figue 19: Plexiglas, S 11 Phase Eos, Real Pat of Pemittivity with Known Pemeability...36 Figue 0: Plexiglas, S 11 Phase Eos, Imaginay Pat of Pemittivity with Known Pemeability 37 Figue 1: Teflon, S 1 Phase Eos, Real Pat of Pemittivity with Known Pemeability...38 Figue : Teflon, S 1 Phase Eos, Imaginay Pat of Pemittivity with Known Pemeability...38 Figue 3: Plexiglas, S 1 Phase Eos, Real Pat of Pemittivity with Known Pemeability...39 Figue 4: Plexiglas, S 1 Phase Eos, Imaginay Pat of Pemittivity with Known Pemeability 39 viii

10 Figue 5: Impedane Diagam fo Lossless Tansmission Line with Two Refletions...41 Figue 6: Ripple as a Funtion of d / λ...4 Figue 7: 0.5 Inh Teflon Sample, S 1 Phase Eos, Real Pat of Pemittivity...43 Figue 8: 0.5 Inh Teflon Sample, S 1 Phase Eos, Imaginay Pat of Pemittivity,...44 Figue 9: Teflon, Real Pat of Pemittivity with S 11 Phase and Mateial Thikness Eos...45 Figue 30: Teflon, Imaginay Pat of Pemittivity Phase and Mateial Thikness Eos...46 Figue 31: MATLAB Code fo Ceating S-Paametes...61 Figue 3: NRW MATLAB Code With Pemeability Fixing Ability...6 ix

11 ABSTRACT Investigation of Magnitude and Phase Eos in Waveguide Samples fo the Niolson-Ross-Wei Pemittivity Tehnique By Paul Galvin Univesity of New Hampshie, Deembe 016 The Niolson-Ross-Wei (NRW) tehnique is a ommonly-used method fo measuing the eletomagneti popeties of low-loss mateials. The tehnique entails plaing the mateial unde test in a waveguide, and then infeing the eletomagneti popeties of that mateial fom the efletion and tansmission oeffiients measued at the ends of the guide. While NRW tehnique geneally povides eliable and auate esults, thee ae onditions whee eos an aise. Some of the known eos ae attibuted to sample pepaation, although eos have been obseved even with pefetly-pepaed samples. Those eos geneally esult in an oveestimation of loss and unealisti values fo pemeability, and they ae shown to be assoiated with phase eos in the efletion and tansmission oeffiients. The wok epoted hee shows the elationship between measuement phase eos and how they impat estimated eletomagneti popety values. Futhe, an appoah to oet fo those eos when evaluating non-magneti mateials is given. x

12 CHAPTER 1: INTRODUCTION Chapte 1 seves to outline the oganiation fo the emainde of the eseah. This hapte details the poess to alulate the elative pemittivity fom plane-wave efletions, whih is used as the building blok fo the full development of the Niolson-Ross-Wei equations in Chapte. Additionally, this hapte will define tems, geometies, and onepts that will be efeened thoughout the emainde of the wok. 1.1 Oveview of the Wok The Niolson-Ross-Wei (NRW) tehnique is a method of alulating the elative pemittivity and elative pemeability of a mateial by measuing the efletions and tansmission fom an inident eletomagneti wave on a slab of the mateial. While the NRW tehnique povides auate and eliable esults in most ases, thee ae identifiable onditions whee measuements of the pemittivity of ommon dieletis povide values fo pemittivity and loss that ae unealisti. In most ases, it is the imaginay potion of the omplex pemittivity that is suseptible to these eos. This wok demonstates that these eos ae atifats of phase eos intodued in the measuement poess, and a simple mathematial edution an impove esults suh that they pove useful. Setions 1. and 1.3 begin desibing the NRW tehnique by analying plane-wave efletion and tansmission fo a dieleti sample. Sine any taveling wave inident upon a disontinuity will ause both a efleted and tansmitted wave fom the plane of inidene, the magnitude of the efletion and tansmission oeffiients is elated to the eletomagneti popeties of that mateial. Fo a plane-wave with nomal inidene upon an infinite slab of a dieleti, the atio of the efleted to inident waves is suffiient to alulate the elative 1

13 pemittivity of the mateial in a staightfowad manne. Setion 1.3 extends the equations fo this infinite-slab inidene to a ase whee the mateial unde test has a fixed, known, thikness. Fo patial onsideations, NRW measuements typially take plae in a etangula waveguide. Setion.1 desibes the equations neessay to ompute the elative pemittivity of the mateial in a waveguide, whih is ompliated by non-nomal angles of inidene. The equations given in Setion.1 ae the full omplement of equations assoiated with the NRW tehnique, and they an be used to alulate the efletion and tansmission oeffiients within a waveguide. The wok pesented hee uses a mathematial model, sepaate fom the NRW, to onstut and alulate the efletion and tansmission oeffiients aused by a known mateial inside an X-Band waveguide. Setion. details the model eated to detemine the efletion and tansmission oeffiients, inset known phase eos into these tems, and then use the NRW tehnique to solve fo the eo influened pemittivity and pemeability tems. Additionally, Setion.3 desibes the method used to edue the effet the phase eos have on the final pemittivity alulation. The esults fom the modeling ae inluded in Chapte 3, with Chaptes 4 and 5 dediated to disussion of the esults, in addition to onlusions and futue wok in this field. 1. Plane wave Inident on an Infinite Slab Figue 1 shows a plane wave inident upon an infinite slab of a dieleti mateial. The fields ae oiented fo onveniene, suh that the eleti field E is a veto exlusively in the x- dietion ( E E ), the magneti field H is exlusively in the y-dietion ( x y H H ), and the Poynting veto S is exlusively in the -dietion ( S S ). Due to this simplifiation of the geomety, the eleti field will be denoted simply as E, whee a positive value indiates an eleti field popagating in the positive x-dietion, and a negative value indiates an eleti

14 field popagating in the negative x-dietion. Similaly, the magneti field and Poynting veto will be denoted simply as H and S espetively. Notation extends to inlude subsipts indiating the measuement egion the field exists in, as well as a supesipt indiating inident waves (i), efleted waves (), o tansmitted waves (t). Figue 1: Plane Wave Inident on an Infinite Length of Mateial By measuing the popagating waves inident and efleted fom the infinite dieleti slab, the input efletion oeffiient fo the system an be detemined. It is known that the efletion oeffiient fo a two-egion onfiguation an be alulated using the intinsi impedanes of eah egion. The intinsi impedane, η, is defined as the atio of eleti field stength to magneti field stength. Fo a plane wave, the following equation defines the intinsi impedane E µ µ 1 0 (1) 0 H 3

15 The dependene of the pemittivity of the dieleti in detemining its intinsi impedane is what enables using the efletion oeffiient to alulate the elative pemittivity. In this example, thee ae two egions. Region 1, fee-spae, has an intinsi impedane 1 0 Ω. Fo this 1 wok, the infinite-slab efletion oeffiient denotes the efletion oeffiient fom an mn inident wave in mateial m, inident upon an infinite slab of mateial n. In this patiula example, is theefoe the efletion oeffiient fo a wave in fee-spae inident upon the 1 infinite dieleti slab in egion. Additionally, fo this example, in, the measued efletion 1 oeffiient fom a efleted wave in egion 1, is equivalent to. Sine with 1 and being known ε in an be alulated assuming the mateial is non-magneti (theefoe µ, an be foed to 1+j0). 1 in µ µ µ µ 1 1 µ µ 1 1 () Next, multiplying both sides by the denominato simplifies the equation. µ µ (3) µ µ Next, eaanging the tems to isolate the elative pemittivity to the same side of the equation enables solving fo pemittivity. 4

16 µ µ µ ( 1) (4) Dividing both sides of the equation by ( 1) finally isolates the 1 µ tem, and multiplying both the top and bottom of the ight side of the equation by -1 etuns a simple fomat fo the equation. µ (5) With the equation solved in tems of both the elative pemittivity and the elative pemeability, it is now possible to solve fo only the elative pemittivity. By squaing both sides of the equation, as well as substituting the value of 1 fo, the equation is solved in tems of only the elative pemittivity (6) Finally, taking the invese of both sides of the equation esults in an equation fully solved fo the elative pemittivity. 1 Γ 1 Γ 1 1 (7) This final equation shows that fo this infinite slab example, the efletion oeffiient is a suffiient measue to alulate the elative pemittivity of the mateial, povided it is known to be a non-magneti mateial. If the mateial wee magneti, with an unknown μ, the atio of 5

17 elative pemeability and elative pemittivity ould be alulated, but they ould not be sepaated fom one anothe without futhe infomation. 1.3 Plane Wave Inident on a Dieleti Slab with a Finite Thikness The pevious example is unealisti sine an infinite slab of dieleti is unealiable, although it does demonstate the elationship between efletion oeffiient and the eletomagneti popeties of a mateial. When the -dimension of the dieleti slab is edued fom being infinitely thik to having a set thikness, d, a thid egion is intodued. Fo this example, this thid egion is fee-spae, just like egion 1. Figue and Figue 3 illustate a ealiable measuement setup, wheein thee is a known thikness to the mateial being tested. Figue : Multi-Region Dieleti Measuement Setup 6

18 Figue 3: Multi-Region Dieleti Measuement Setup Coss-Setion The goal of this setion is to deive a new equation elating the fields efleted fom the finite-slab to the infinite-slab efletion oeffiient desibed in Setion 1.. That elationship will enable the pemittivity to be alulated fom finite-slab efletion oeffiient. As in the infinite-slab ase, the input efletion oeffiient is detemined by measuing the atio of the efleted wave to the inident wave at the bounday of the dieleti slab. The fist efletion soue is the bounday between egions 1 and, and the seond is the bounday between egions and 3. The infinite-slab efletion oeffiient between the fee-spae in egion 1 and the dieleti in egion would be Γ 1, while the infinite-slab efletion oeffiient fo between the dieleti in egion and the fee-spae in egion 3 would be Γ 1. These two efletion oeffiients ae both based upon the same two intinsi impedanes, with the only 7

19 diffeene is whih mateial the inident wave is tavelling in. As suh, the only diffeene between the two is the sign, theefoe Γ Γ. 1 1 Sine the two boundaies ae not isolated fom the est of the system, the input efletion oeffiient is neithe Γ no Γ. Despite that, these tems emain useful duing the alulation 1 1 poess. Sine Γ is a funtion of the dieleti pemittivity and pemeability alone, by solving 1 fo Γ 1 oeffiient pemittivity. pemittivity an be alulated. Theefoe, the goal is to alulate the input efletion Γ in tems of the infinite slab efletion oeffiient Γ to enable solving fo the in 1 The input efletion oeffiient is the efletion oeffiient seen at the bounday between egions 1 and. It is dependent upon the intinsi impedane of egion 1, whih is fee-spae, and the impedane seen looking into the system fom that bounday. In the pevious example, sine thee was dieleti fo an infinite distane in the -dietion, this was simply the intinsi impedane of the dieleti mateial. In this ase, the fee-spae in egion 3 ats as a load to the system, eating a standing wave intenal to egion. Theefoe, the ealied impedane intenal to egion vaies along the -dietion. The input efletion oeffiient is based upon this impedane at distane d fom the egion to egion 3 bounday. d L 0 1 in d L 0 1 (8) The next step is to detemine the funtion whih desibes the impedane at a point inside egion at a distane,, fom the egion to egion 3 bounday. Note that stats at eo at the ight bounday, and ineases to the ight. Impedane is a onept whih desibes the elationship between the eleti and magneti fields at a given point. Inside 8

20 t egion, two taveling waves exist, the wave tansmitted fom egion 1, S, and the wave efleted fom the egion to egion 3 bounday, S. These need to be added. E E E t H H H t (9) In ode to ontinue solving, the waves need to be put in tems of one anothe. In this ase, the two waves ae sepaated by a distane of d. Theefoe, the efleted wave, S is t dependent on the wave tansmitted into egion S popagating a distane of d, then being efleted by the infinite slab efletion oeffiient 1 exist:. Theefoe the following elationships E Γ 1 1 jβ d H Γ H e t E e t jβ d (10) Note that the efleted magneti field is negative sine popagation fo the efleted wave is in the opposite dietion as the inident wave, E H must point in the negative -dietion. Additionally, β is the phase onstant fo the -oiented potion of the tavelling wave. Fo this example, sine the tavelling wave is entiely in the -dietion, β. The fields E and H ae defined at the egion to egion 3 bounday, while the fields E t and t H ae both defined at the egion 1 to egion bounday. Theefoe, to detemine the supeposition of the two fields at distane, popagation fatos must be applied to eah tem. The efleted fields both must tavel a distane in the negative dietion, thus it has a popagation fato of jβ e. The fields tansmitted into egion both must tavel a distane d+, jβ theefoe have a popagation fato of d e. 9

21 jβ d jβ t t E E e E e E e E e e jβ d jβ j d t jβ d j 1 t j d t jβ d j t H H e H e H e H e e 1 j d j d t e e j d j d 1 1 t j j d j d j d d e e 1 1 E e e H e e (11) With an equation fo obseved impedane as a funtion of distane in tems of the popagation onstant, the infinite slab efletion oeffiient, and the intinsi impedane of egion, the input efletion oeffiient an now be solved by substituting in the obseved impedane at distane d. d 1 in d 1 e e e e j ( d d ) j d d j d d j d d e e e e j d d j d d j d d j d d 1 1 j d 1 e 1 j 1 d 1 e 1 j d 1 e 1 j 1 d 1 e 1 (1) Fom hee, two additional substitutions an be made. Sine egion 1 is fee-spae,. 1 0 Additionally, the intinsi impedane in egion an be eplaed by (1). Those two substitutions allow the following edutions: j d µ 1 e 1 0 j 0 d e 1 e 1 in j d µ 1 e 1 0 j 0 d e 1 e µ 1 e e 1 µ 1 e e 1 j d 1 j d e 1 j d 1 j d e (13) Γ in µ 1 e e 1 µ 1 e e 1 j d 1 j d e 1 j d 1 j d e

22 In an attempt to onsolidate tems (5) is used to emove the pemittivity and pemeability tems. Additionally, is eplaed with. 1 1 j d j d j d e 1 1 e e j d j d j d e 1 1 e e in j d j d j d 1 1 e 1 e e j d j d j d 1 1 e 1 e e (14) The ightmost tem was obtained by expanding the fatoiation. The final edution step is to multiply the top and bottom of the equation by 1 j d j d e e in ode to edue the equation to a single fation. S 11 in (1 1) j d j d j ( d j d e e e e ( ) ( ) ) (1 1) ( ) 1 1 S 11 Γ in j d j d e e ) 1 1 ( e e j d j d 1 e 1 1 e j d j d e e 1 1 ( ) 1 j d j d (15) The above esult is the most edued fom fo the input efletion oeffiient in tems of the infinite slab efletion oeffiient 1. Unlike peviously, when the equations wee entiely in tems of known o measued vaiables, this time thee ae two unknown quantities, and. 1 Futhe, both tems ae dependent upon both the elative pemittivity and the elative pemeability. By assuming the elative pemeability is 1+j0, as was done in the pevious example, both tems beome funtions of only the elative pemittivity. Despite this, thee is still no analyti method to solve fo the pemittivity, and an iteative appoximation method would be equied. Instead, a diffeent appoah is taken. Sine the envionment fo egion 1 is the same as egion 3, it is patial to measue the total tansmission though the system. Following the 11

23 same appoah as was used to alulate the efletion oeffiient, the net tansmission oeffiient is alulated to be the following: S E 3 T t 1 i j d E 1 e e j d (16) With both the tansmission and efleted fields measued, and both equations in tems only of the unknowns and, thee is suffiient infomation to solve analytially fo eah tem 1 independently. Sine these ae non-linea elationships, solving fo the unknowns is not a staightfowad poblem. A tehnique fo solving this system is shown in Appendix A. pemittivity: One is obtained, the same poess as befoe an be used to alulate the 1 1 Γ μ 1 Γ 1 1 (17) One additional benefit, howeve, is the infomation obtained by solving fo. The following equation desibes the phase onstant fo a wave tavelling in a mateial: (18) A pimay advantage of this method is that due to having two sepaate equations entiely in tems of the elative pemeability and the elative pemittivity, it is now possible to solve fo both values simultaneously. Appendix B details the mathematis neessay to onvet the phase onstant and the infinite slab efletion oeffiients to the elative pemittivity and elative pemeability values. 1

24 CHAPTER : ELECTROMAGNETIC WAVE MODEL Chapte povides the bakgound on the model of the waveguide used to measue efletion and tansmission oeffiients. This Chapte also shows how the Niolson-Ross-Wei (NRW) tehnique of alulating pemittivity and pemeability values fom the efletion and tansmission oeffiients measued in a hollow, etangula waveguide. Additionally, it desibes the geneation of the S-Paametes fom the basi pemittivity, sample sie, and waveguide sie paametes. Finally, it outlines the theoy being tested in this wok to assess the eliability of the Niolson-Ross-Wei tehnique in the pesene of measued phase eos..1 Pemittivity Calulations fom Waveguide Tansmission The examples used thus fa to illustate the elationship between eletomagneti popeties and the efletion and tansmission oeffiients have been fo plane waves nomally inident on a bounday. Howeve, beause of patial onsideations, measuements ae pefomed in waveguide stutues, whee wave popagation is not TEM. Consequently, the plane-wave elationships given above must be modified to aommodate the TE o TM waves popagating in a waveguide. Retangula waveguides ae used almost exlusively to measue efletion and tansmission oeffiients, and the waveguides ae opeated using the lowest-ode mode, the TE 10 mode. The advantages hee ae that it is known that aoss a given bandwidth, the medium suppots only a single mode of opeation, and thus all measued waves an be assumed to be TE 10 waves. Additionally, the etangula geomety allows fo vey simple pepaation of samples, whih an be easily ontolled to vey peise speifiations. Figue 4 shows the system as it applies in a waveguide. 13

25 Figue 4: Finite Thikness Dieleti Slab Inside a Waveguide This figue shows that the pimay disadvantage of this method is the inident waves no longe have nomal inidene to the bounday. The goal fo this setion is to expand the equations fom Setion 1.3 to wok fo any TEM o TE wave. With this goal, the full Niolson- Ross-Wei method will be shown. Additionally, it will be shown that these new equations an be used fo plane-wave exitation as well as the waveguide modes, and theefoe ae not new, but athe a moe omplete vesion of the govening equations. The following equation details the angle of an inident wave in the y-plane elative to the fowad -dietion. i 1 f o s 1 f (19) 14

26 In this equation, f is the utoff fequeny fo an empty waveguide fo the mode being used, and f is the fequeny of the input wave. The equations in Setion 1.3 wee designed with patiula equiements in ode to be salable to this moe omplex situation. Thanks to this effot, many of the equations aleady developed ae still valid. In patiula, the equations shown as well as in Appendix A used to alulate Γ and fom S and S emain unhanged. The diffeene due to the oblique inidene is in the alulation of elative pemittivity and elative pemeability fom Γ and 1. The issue with the waveguide tansmission is that the phase onstant,, and the infinite-slab efletion oeffiient, Γ ae ompliated by the intodution of a non-nomal angle 1 of inidene. Figue 5 shows the petinent fields fo analying the new efletion oeffiient fo the infinite dieleti slab ase. 15

27 Figue 5: Infinite Slab of Dieleti Mateial with Non-Nomal Angle of Inidene This figue details two diffeent angles fo the taveling wave, an inident angle in the fee-spae ( i ), and a seond fo the tansmitted wave inside the dieleti ( t ). Snell s law detemines the elationship between these two vaiables. n n (0) sin sin 1 i t whee n is the index of efation, whih is defined as n. Snell s law is useful fo the NRW poess, sine all diet wave measuements ae in fee-spae, it is impossible to dietly measue the angle of tansmission intenal to the medium, thus Snell s law allows us to put it tems of othe known infomation. 16

28 The eleti field in this TE oientation is pependiula to the plane of inidene. Theefoe, this is efeed to as a pependiula polaiation. The Fesnel efletion oeffiient fo pependiula polaiation is defined as the following: Γ 1 o s o s i 1 t o s o s i 1 t (1) By dividing the top and bottom of the equation by, the equation an be put in tems of the elative pemittivity and pemeability, sine : 0 0 Γ 1 1 o s o s i t o s o s i t 1 o s o s i t o s o s i t () Next, the must be eliminated. Rewiting Snell s law, the following an be alulated fo : t t t n n s in s in 1 i n s in s in t n s in 1 i s in s in s in i n i t (3) The angle of the tansmitted wave only ous in (3) inside a osine tem. The tigonometi identity x o s sin 1 1 x is used to edue the tigonometi opeations equied. s in s in s in 1 o s ( ) o s s in 1 1 t i i i (4) 17

29 (4) is now substituted bak into () in ode to emove the unknown value of. t Γ 1 s in i o s o s o s 1 i t i s in i o s o s i t o s 1 i (5) Next, (19) is substituted into (5) to emove any efeene to the angle. Γ 1 o s o s o s 1 o s 1 1 s in f o s 1 f f 1 1 f 1 s in f o s 1 f f 1 1 f (6) Using the tigonometi identity x sin o s 1 1 x again also edues the equation fo the infinite slab efletion oeffiient. Γ 1 f 1 1 f f f f f f f f f 1 1 f f f 1 1 f f 1 1 f (7) 18

30 Γ 1 f f f 1 1 f f f f 1 1 f (8) The phase onstant is simple to adjust fo the waveguide opeation. The equations fo the plane-wave inidene wee all defined using the -dietional phase onstant,. In that example, the Poynting veto was entiely in the -dietion, so that was equivalent to the phase onstant. In this example, sine the Poynting veto is at an angle in the y-plane, must be alulated. Figue 6 illustates the elationship between and. Figue 6: Oientation of the Phase Constant Tems Theefoe, the following mathematial elationship is shown. (9) o s t (4) is substituted into (9) in ode to emove the dependene on the tansmitted angle. o s 1 s in t i (30) 19

31 0 Finally, (19) is used to eplae the inidene angle values with known fequeny values. Additionally, the tigonometi identity 1 sin o s 1 x x futhe edues the equation fo the phase onstant s in o s i f f f f 1 f f f f (31) The equation fo the phase onstant an be futhe edued by squaing both sides of the equation, followed by multiplying both sides of the equation by the denominato. 1 1 f f f f f f (3)

32 The goal behind all of these equations is to eventually alulate the elative pemeability and elative pemittivity. Theefoe, any attempt to onvet between the phase onstant and the - dietion oiented phase onstant must not ely on these values, sine they will be unknown until the end of the alulation poess. By substituting the definition of (18) into the equation, these tems an be emoved. f f f f f f f f (33) f The final substitution omes fom the identity. f (34) Whee is the utoff wavelength. Finally, taking the squae oot of both sides solves fo the tem. (35) Additionally, the equation ould be solved fo : (36) 1

33 This equation is now entiely in tems of known onstants,, and. Theefoe, in onjuntion with the equation fo the efletion oeffiient, thee is suffiient infomation to solve fo both the elative pemittivity and pemeability, as was done in Setion 1.3 Despite the added omplexity aused by the waveguide equiing oblique angles of inidene, the equations developed above ae fa moe obust. Due to the eliane on the utoff fequeny, these equations ae valid fo all TEM and TE mode waves, in any tansmission medium, as long as the utoff fequeny an be alulated. Fo plane-waves and TEM mode waves, the utoff fequeny is set to 0. It an be seen, that fo utoff fequenies of 0 (infinite utoff wavelength), the equations fo phase onstant and efletion oeffiient edue to thei epesentations fom Setion 1.3. Γ f 0 f 0 f 1 0 f * 1 1 (37) (38) Appendix C details the mathematis involved in using the alulated and Γ tems in 1 ode to solve fo the elative pemittivity and elative pemeability. The following two equations show the final esults.

34 1 Γ 1 1 Γ 1 f 1 f (39) 1 0 (40) Using these equations to take a measued set of Satteing Paametes and using them to alulate the elative pemittivity and pemeability is efeed to as the Niolson-Ross-Wei method. In most texts, the equations ae povided. It ontinues to be notewothy that these equations apply to both TEM and TE waves, as TEM mode waves will have a utoff fequeny of 0, and the equations poeed to edue to the plane-wave equations detailed in example.. Pemittivity Coeting Method The objetive of this wok is to develop a method to eliminate, o at least minimie, eos when using the NRW tehnique to estimate the pemittivity of low-loss, non-magneti mateials. Fo non-magneti mateials, the elative pemeability should always have a value of 1 + j0. Sine in (71) the elative pemittivity value is dependent upon the value alulated fo elative pemeability in (69) any eo aused by S-Paamete measuements, sample pepaation, sample thikness values, o waveguide dimensions, will affet both the pemittivity and the pemeability. Sine the pemittivity is alulated fom both the S-Paametes and the pemeability value, thee ae two vaiables in the pemittivity alulation poess that will ontain this eo. By foing the pemeability to the known oet value (i.e., µ = 1+j0), the numbe of eant tems in the pemittivity alulation an be edued. 3

35 Mathematially foing the pemeability to a fixed value is not in itself new to the NRW tehnique. This wok aims to pove that the small phase eos that ou in VNA measuements ae a signifiant ause of eo in the imaginay potion of the elative pemittivity, and that though this oetion a moe auate alulation is obtained. In addition, an effot to quantie this eo oetion is poposed. Finally, the best paties fo the NRW as a whole ae ompaed to the ideal onditions fo this wok, to detemine what the ideal measuement situation fo non-magneti mateials tuly is..3 Eo Modeling Method This wok utilies the equations pesented in Setions.1 and., as well as Appendies A, B, and C in ode to eate a full model whih uses pemittivity, thikness, and waveguide dimension inputs to alulate S-Paametes, and then uses those S-Paametes to alulate the pemittivity and pemeability of the sample. The model will impose ontolled eos into the S-Paamete alulation poess so that the effet of known eos on the final pemittivity esults. The eos intodued will be phase eos on S 11 and S 1, as S-typial measued phase eos ae moe unstable than magnitude eos The most impotant pat of this wok uses the method in Setion.3 to foe the pemeability to 1+j0 in ode to edue the eos in alulation of the pemittivity. The equations in Setion.1 show the elationship between S-Paametes and eletomagneti popeties fo a wave tavelling in a NRW test setup. Setion.1 also detailed the NRW tehnique to onvet these S-Paametes and physial system popeties bak into eletial popeties. The modeling method is implemented in Mathwoks MATLAB softwae pakage. The fist stage of the modeling method uses the wok detailed in Setion.1 to alulate S-Paametes. This method uses Equations (15,16,18,19,8,35) to onvet the input 4

36 paametes to a set of equivalent S-Paametes. This funtion allows the model to vay and speify the mehanial length of the sample (d), the elative omplex pemittivity of the sample (ε ), the y-dimensional sie of the waveguide opening (a), and the fequeny ange to be simulated. This method assumes that the waveguide is fully filled so that the distane fom the measuement pot to the ai-to-dieleti bounday is eo. The S-Paametes etuned ae fom the ai-filled waveguide end piees, measued fom a distane of eo to the sample unde test. The elevant ode may be found in Appendix D. The final stage of the modeling method uses the wok detailed in Setion.1 to alulate the pemittivity and pemeability using the Niolson-Ross-Wei tehnique. It is assumed that thee ae no eos in measuements, and the NRW tehnique is followed exatly though Equations (45,48,50-5,69,71). This funtion allows the model to vay and povide the S 11 and S 1 paametes, the sample thikness (d), the y-dimensional sie of the waveguide opening (a), and the fequeny ange used. Additionally, the method allows the model to speify whethe the elative pemeability should be alulated as defined in (69), o if it should be foed to be 1. The funtion etuns the omplex elative pemittivity and the omplex elative pemeability. The funtion may be found in Appendix E. Between these setions, eos ae intodued in a ontolled manne. This model allows any of the paametes used by the NRW tehnique in Appendix E to be vaied to detemine the ole they play in total eo. Fo this eseah, only the phase of S 11 and S 1 ae vaied. The intent of the eseah is to show that the limitations imposed by uent VNA tehnology and abling intodues enough phase eo that a peise value of omplex pemittivity fo low-loss mateials annot be obtained without utiliing the pemeability fix desibed in Setion.. 5

37 CHAPTER 3: SIMULATIONS Chapte 3 povides thee-dimensional simulated data fo Niolson-Ross-Wei waveguide measuements of a vaiety of metal walls and filling mateials. Advantages of simulated data inlude the ability to eate idealied situations whee thee ae no ai gaps, mateial defomities, o non-homogenous mateials. The esults in this hapte ae intended to be used in onjuntion with the measued esults, and not as a eplaement fo physial measuements. 3.1 Raw Simulation Data The NRW tehnique has unaeptable auay eos in low-loss mateial measuements [5]. The goal of this setion is to attibute these eos to the phase stability of VNA measuements. To demonstate the impat of measuement eos on estimated omplex pemittivity, two ommon dieleti mateials, Teflon and Plexiglas, will be used as examples in this setion. Teflon is ommonly used as the insulato in able design, and Plexiglas is the pimay insulato in pinted iuit boads. Teflon is vey low loss, with a omplex pemittivity of.1 - j at 10 GH [6]; Plexiglas has highe loss, with omplex pemittivity of 3.45 j0.138 at 10 GH [6]. A Veto Netwok Analye is a peision piee of equipment whih measues the amplitude and phase of signals in a two-pot system. In ode to use a Veto Netwok Analye, oaxial ables must be attahed to the pots. These ables have phase instabilities due to tempeatue vaiations and due to mehanial bends. Fo phase-sensitive measuements, suh as the measuements that will be used fo NRW alulations, phase-stable ables ae typially used. Phase-stable ables ae moe esistant to phase hanges aused by flexing ables o tempeatue hanges. Typial phase stability fo ables in the X-Band fequeny ange of GH is ±-3 [7]. This wok will analye fo a maximum deviation of, in ode to estimate the eos in a system whee the use is using the best equipment possible. 6

38 All simulations in this hapte inlude the dieleti losses fom the low-loss mateials, but do not aount fo waveguide onduto losses. With the length of the waveguides being used hee, the onduto losses ae low enough that they an be ignoed without intoduing appeiable measuement inauaies []. The fist two simulations show the esults of the NRW tehnique measuing 0.3 inhes of Teflon when the phase eo of S 11 is vaied fom 0 to, and when the phase eo is vaied fom - to 0. Figue 7: Teflon, S11 Phase Eos, Real Pat of Pemittivity 7

39 Figue 8: Teflon, S11 Phase Eos, Imaginay Pat of Pemittivity Figue 9: Teflon, Invese S11 Phase Eos, Real Pat of Pemittivity 8

40 Figue 10: Teflon, Invese S11 Phase Eos, Imaginay Pat of Pemittivity Figue 7 - Figue 10 show that the oientation of the phase eo ontols whethe the elative pemittivity eo is a positive o negative dietion, but not the magnitude of the eo. Inteestingly, the eal potion of elative pemittivity has ineasing eo with positive phase eo, and deeasing eo with negative phase hange, while the imaginay potion has deeasing eo with positive phase eo, and ineasing eo with positive phase hange. Sine the magnitude of the eos aused does not vay based on positive o negative phase eos, only the ase of negative phase hange will be highlighted fo the emainde of this wok. Figue 7 shows that the eal potion of the pemittivity is affeted in a mino way by these phase issues. Figue 8 shows that the imaginay potion is muh moe affeted, with substantial eos. The maximum peentage eo aoss all simulations in Figue 8 is 7000% ompaed to the 0 phase eo ase, whih is a deviation fom the known imaginay potion of pemittivity by

41 The next simulation is fo Plexiglas also using a 0.3 inh sample and vaying S 11 fom 0 to. Figue 11: Plexiglas, S11 Phase Eos, Real Pat of Pemittivity Figue 1: Plexiglas, S11 Phase Eos, Imaginay Pat of Pemittivity 30

42 Figue 11 and Figue 1 show that the S 11 phase eos ause imaginay pemittivity eos fo the highe loss Plexiglas as well as the low-loss Teflon shown in Figue 7 and Figue 8. The maximum eo on the imaginay potion of pemittivity is moe with a maximum deviation fom the known imaginay potion of pemittivity by appoximately 0.15, but the maximum peentage eo is lowe at 108%. Fo appliations equiing peise ontol of the total netwok losses, this measuement eo is still too geat to aept. The next simulations ae 0.3 inhes of Teflon and Plexiglas with the phase eo of S 1 vaying fom 0 to. Figue 13: Teflon, S1 Phase Eos, Real Pat of Pemittivity 31

43 Figue 14: Teflon, S1 Phase Eos, Imaginay Pat of Pemittivity Figue 15: Plexiglas, S1 Phase Eos, Real Pat of Pemittivity 3

44 Figue 16: Plexiglas, S1 Phase Eos, Imaginay Pat of Pemittivity Aside fom the issue at GH, it is seen that the imaginay potion of pemittivity has appoximately the same total eos with S 1 phase eos as with S 11 phase eos. The eal potion has slightly highe eo with S 1 phase eos than with S 11 phase eos. 3. Redution of Pemittivity Eos Taditionally, thee exists two main identifiable soues of eo in the NRW tehnique, whih use diffeent methods to ompensate. The fist is the peviously mentioned halfwavelength sample issue. Typially, thee exists two main stategies to deal with this issue. The fist is uve fitting the pemittivity esults, in ode to ompensate fo a naow-band issue. The pimay issue hee is this is taking an othewise omplete set of equations and eplaing it with an estimation based on aveaging. The seond is to intentionally keep samples shote than one half-wavelength in the mateial, even at the highest fequeny. This method woks well, but does equie a deent estimate of the mateial s pemittivity ahead of sample pepaation, as well as the ability to popely pepae potentially small samples. 33

45 The seond known issue is sample pepaation. When the sample is loaded into the waveguide, if it is not popely ut o molded, thee will be some amount of gap aound the edges. In these egions, the model fo a filled waveguide will be inoet, and the esults ae jeopadied. Additionally, fo small samples, suh as those fo highe pemittivity mateials attempting to emain eletially shote than one half-wavelength, small thikness eos an be poblemati fo auate esults. This soue is the pimay blame fo most wide-band eos in the NRW tehnique. Fo non-magneti mateials, the pimay method to inease auay is to foe the pemeability to be one. This ineases the auay by emoving a known soue of eo in the alulation fo pemittivity. This wok shows that most eos ae in fat S 11 and S 1 phase eos, not fom poo sample pepaation, but athe inadequate fixtuing. While poo sample pepaation may ause simila issues, it is in most situations easie to ontol the sample pepaation to have less than of phase eo than it would be to ontol the fixtuing to that toleane. With this goal, the wok sets out to pove that the same tehnique of foing the pemeability to one that has been poven effetive fo sample eos will wok fo phase eos. The following simulations epliate the S 11 phase eos fom Setion 3.1, but with the NRW tehnique foing the pemeability to one with the goal of eduing aw eos. The fist simulation is 0.3 inhes of Teflon with S 11 phase eos anging fom 0 to. 34

46 Figue 17: Teflon, S11 Phase Eos, Real Pat of Pemittivity with Known Pemeability Figue 18: Teflon, S11 Phase Eos, Imaginay Pat of Pemittivity with Known Pemeability 35

47 Immediately it an be seen that the maximum deviation fom the known value fo the imaginay potion of pemittivity has been edued fom nealy to appoximately This edues the peentage eo fom being 7000% to only 1000%. While still vey high, this is a damati inease in pefomane. The next simulation is 0.3 inhes of Plexiglas with S 11 phase eos anging fom 0 to. Figue 19: Plexiglas, S11 Phase Eos, Real Pat of Pemittivity with Known Pemeability 36

48 Figue 0: Plexiglas, S11 Phase Eos, Imaginay Pat of Pemittivity with Known Pemeability Again the maximum deviation fom the known value fo imaginay pemittivity dopped fom 0.15 to only 0.037, whih epesents a dop fom 108% eo to 7%. This eo is fa moe manageable, and while still high, is likely within useable anges fo many appliations. The final two simulations ae fo 0.3 inhes of Teflon and Plexiglas with S 1 phase eos anging fom 0 to. 37

49 Figue 1: Teflon, S1 Phase Eos, Real Pat of Pemittivity with Known Pemeability Figue : Teflon, S1 Phase Eos, Imaginay Pat of Pemittivity with Known Pemeability 38

50 Figue 3: Plexiglas, S1 Phase Eos, Real Pat of Pemittivity with Known Pemeability Figue 4: Plexiglas, S1 Phase Eos, Imaginay Pat of Pemittivity with Known Pemeability 39

51 These simulations ontinue to suppot the hypothesis that phase eos make up a lage potion of NRW tehnique eos, and that by foing the elative pemeability to one, most of these eos ae signifiantly edued. This follows fom (71), whih shows that the pemeability and the popagation onstant ae the two soues in the alulation of the pemittivity whih ae affeted by the S-Paamete phase eo. By emoving the eo in pemeability by using the known value, the eo is edued. 40

52 CHAPTER 4: DISCUSSION OF RESULTS This hapte aims to explain and disuss the esults fom the waveguide model obtained in Chapte 3. It attempts to daw onnetions with aepted best paties fo using the Niolson- Ross-Wei tehnique, to impove eal-wold measuements with the best effiieny. 4.1 Half-Wavelength Disontinuities in NRW Tehnique When measuing S-Paametes peiodi ipple with espet to fequeny in the efletion and insetion paametes due to a mismathed line is ommon. To illustate this effet, the lossless tansmission line shown in Figue 5 will be used. Figue 5: Impedane Diagam fo Lossless Tansmission Line with Two Refletions When stimulating the tansmission line fom pot 1, efletions ou at two loations, one with efletion oeffiient Γ, and the seond with efletion oeffiient Γ. Any efletions aused at the seond disontinuity have a hane to be e-efleted at the fist disontinuity, ausing standing waves to be pesent along the line [3]. At pot 1, the elative phase of the two efleted signals is detemined by the length of the distane d. Theefoe, fo wavelengths whee d is an 41

53 intege numbe of half-wavelengths long, the two efletions ae 180 out-of-phase, and theefoe the standing waves seen at pot 1 ae minimal. When d is an intege numbe of quate-wavelengths but not half-wavelengths, the two efletion ae in-phase, and eate a maximum standing wave. (15) in Chapte 1 shows the effetive input efletion oeffiient to the input of Figue 5. Fo a lossless line, e d is simplified futhe to e d j 4 [3]. Figue 6 shows the S 11 paamete as the value of d is adjusted fom 0 to 1 fo Γ = 0.5. Figue 6: Ripple as a Funtion of d / λ Figue shows that as the distane between the two disontinuities is an intege numbe of half-wavelengths long, the magnitude of efletions (S 11) is at a minimum and the magnitude of tansmission (S 1) is at a maximum [3]. Convesely, at one-quate wavelength away fom 4

54 these loations, efletions ae at a maximum while tansmission loss is at a minimum [3]. At loations whee S 11 is at a minimum, (48) has auay limitations due to the denominato of (45) appoahing eo. Due to limited peision fom measuements, ommon patie ditates maintaining that sample thikness emain less than one-half wavelength in the mateial at the highest fequeny of inteest. These issues wee seen in Figue 15 and Figue 16 at about GH is due to the eletial length of the mateial being exatly ½ wavelength long at that fequeny. This is tue also fo the Teflon ase, if the sample length was ineased. See Figue 7 and Figue 8. Figue 7: 0.5 Inh Teflon Sample, S1 Phase Eos, Real Pat of Pemittivity 43

55 Figue 8: 0.5 Inh Teflon Sample, S1 Phase Eos, Imaginay Pat of Pemittivity, The eos fo Teflon ae moe notable due to the lowe loss though the dieleti. Sine Teflon is vey low loss, the efletions anel almost entiely, ausing the esult of the division to be vey unstable with small S 1 phase eos. This instability with low-loss mateial with half-wavelength sample lengths is a known issue, whih this wok assoiates with the measuements seen in Figue 15 and Figue Niolson-Ross-Wei Best Paties The main NRW best paties an be summaied by the following thee items: keep the sample eletially thin, ensue that the sample fully fills the waveguide holde, and foe the elative pemeability to unity fo non-magneti mateials. This eseah agees with these pevious assessments, but assets that the easoning to foe the pemeability to a fixed value is not to potet fom use eo, but athe phase instability due to instumentation eos. The Veto Netwok Analye has vey low phase eos, but the abling used to onnet the 44

56 analye to the fixtue an have signifiant phase eos, even afte alibation. This wok shows that these eos ae ausing a signifiant potion of eo in the alulation of the imaginay pemittivity. Simulations show that eos fo the imaginay pat of the elative pemittivity ae loosely elated to the eo in thikness, but vey lage eos in the measuement of the sample thikness ae neessay to eate small hanges in the imaginay pat of the elative pemittivity. On the othe hand, thikness eo is vey tightly oupled to the eal pat of the elative pemittivity. See Figue 9 and Figue 30 fo a ompaison between thikness eos and phase eos. Note that the sale equied fo Figue 30 shows the no eo ase being appoximately equivalent to the ase whee the measued thikness of the sample is 10% lage than the atual thikness. Figue 9: Teflon, Real Pat of Pemittivity with S11 Phase and Mateial Thikness Eos 45

57 Figue 30: Teflon, Imaginay Pat of Pemittivity Phase and Mateial Thikness Eos These figues show that the imaginay potion of pemittivity is not tightly elated to eos in the measuement of the sample thikness. On the othe hand, the eal potion of pemittivity is vey sensitive to these thikness measuement eos. This suggests that the soue of the obseved eo is not fom sample pepaation, as is often suggested, but athe instumentation eos elated to the phase of the measuement, as was shown in Chapte 3. With this in mind, ae may be taken to edue these phase eos as muh as possible, by using phase stable ables, avoiding bending o heating ables between alibation steps and measuements, and using othe measuement best paties. These steps will edue the measuement phase eos, whih will inease the oveall measuement quality. Remaining eos an be futhe edued by foing the elative pemeability to one as was shown in Chapte 3. 46

58 CHAPTER 5: CONCLUSIONS Chapte 5 ombines the esults gatheed in pevious haptes in ode to povide a top-level analysis of the ompehensive eseah. Compaison is made to the mathematial theoies to how the measued and simulated data math the mathematial model. Potential suggestions will be outlined fo futue wok to edue the emaining eo soues. 5.1 Simulations Simulations show a lea patten whee the phase eos that ae ommon in VNA measuements eate notable eos in the apability to alulate the imaginay potion of the elative pemittivity, as was suggested by the hypothesis of this wok. This eo was able to be edued by an ode of magnitude by foing the elative pemeability to one, whih an be done fo all non-magneti mateials. In theoy, this method will wok fo magneti mateials with known pemeability values as well, by setting the pemeability to its known value. Simulations additionally showed phase eos on S 11 and S 1 tems ontibuted equally to measuement inauaies. This is elevant, sine phase eos on both pots 1 and an ause eos, and annot be edued independently. This equies that phase stability be ensued fo both VNA pots used in measuements. The following table shows the phase eos both using the diet NRW tehnique and with the pemeability fixing method. 47