OA18843A - Site specification for directional guided surface wave transmission in a lossy media. - Google Patents

Abstract

Various examples are provided for site specification for directional guided surface wave transmission in a lossy media. In one example, a probe site includes a propagation interface including first and second regions comprising different lossy conducting mediums. A guided surface waveguide probe positioned adjacent to the first and second regions can generate at least one electric field to launch a guided surface wave along the propagation interface in a radial direction defined by the first region and restricted by the second region. The propagation interface can also include additional regions comprising the same or différent lossy conducting mediums. One or more of the régions can be prepared regions.In some cases, the regions can correspond to a terrestrial medium (e.g., a shoreline) and water (e.g., seawater along the shoreline).

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

This application daims priority to. and the benefit of. co-pending U S. provisionai application entitled -Site Spécification for Dlrectional Guided Surface Wave Transmission .n a Lossy Media having séria, no. 62/305.910. filed March 9. 2016. which is hereby incorporated by reference in its entirety.

This application is related to co-pending U.S. Non-provisionai Patent Application entitled Excitation and Use of Guided Surface Wave Modes on Lossy Media. which was _filed on March 7. 2013 and assigned Appiication Number 13/789,536. and was pubiished on September 11, 2014 as Publication Number US2014/0252886 A1, and which ,s incorporated _{herein by} reference in its entirety. This appiication is also reiated to co-pending U.S. Nonprovisionai Patent Application entitled Excitation and Use of Guided Surface Wave Mod _on Lossy Media. which was filed on March 7. 2013 and assigned Application Number 13/789 525. and was pubiished on September 11. 2014 as Publication Number US2014/0252865 A1. and which is incorporated herein by reference in its entirety. Th,s application is aise reiated to co-pending U.S. Non-provisionai Patent Appiication entrtied

September 10. 2014 and assigned Application Number 14/483,089. and which rs incorporated herein by reference in its entirety. This application is aiso related to co-pendrng U S Non-provisionai Patent Application entitled Excitation and Use of Guided Surface Waves. which was filed on dune 2. 2015 and assigned Appiication Number 14/728.507. and _which is incorporated herein by reference ,n its entirety. This appiication is aiso reiated to co-pending U.S. Non-provisionai Patent Appiication entitled Excitatron and Use of Gu.de Surface Waves. which was filed on dune 2. 2015 and assigned Application Number

Ί 4/728 492. and which is incorporated herein by reference in its entirety. This appircat.cn .s _als0 reiated to co-pending U.S. Non-provisionai Patent Appiication entitied Site Préparation for Guided Surface Wave Transmission in a Lossy Media.’ which was filed on September 9 2015 and assigned Application Number 14/848.413, and which is incorporated herein by reference in its entirety.

background

For over a century. signais transmitted by radio waves involved radiation fields launched using conventions! antenna structures. In contrast to radio science, eiectricai power distribution Systems in the iast century involved the transmission of energy guided along electrical conductors. This understanding of the distinction between radio frequency ₁₀ (RF) and power transmission has existed since the early 1900’s.

SUMMARY

Aspects of the present disclosure are related to site spécification for directional guided surface wave transmission in a lossy media.

In one aspect, among others, a probe site comprises a propagation interface including a first région and a second région adjacent to the first région, the first région comprising a first iossy conducting medium and the second région comprising a second lossy conducting medium; and a guided surface waveguide probe positioned adjacent to the first région and lhe second région, the guided surface waveguide probe configured to generale at least one electrical field that synthesizes a wave front having a complex Brewster angie of incidence corresponding to the first lossy conducting medium wben excited by an excitation source, where the wave front launches a guided surface wave along _the propagation interface in a radiai direction that is defined by the first région and restricted _by the second région. In one or more aspects of the present disclosure, the wave front can intersect with the propagation interface at the complex Brewster angle of incidence at a crossover distance from the guided surface waveguide probe. The first région and the second région can extend aiong the propagation interface from adjacent to the guided surface waveguide probe to beyond the crossover distance. The first lossy conducting medium can be a terrestrial medium (e.g.. Earth). The second région can extend into the terrestrial medium at least to a depth of a complex image of the guided surface wavegu.de probe or at ieast to a depth of a complex image plane of the guided surface wavegu.de probe.

_In one or more aspects of the present disclosure. the second lossy conducting one or more aspects of the present disciosure. .be second lossy conducting medium can be _a terrestrial medium. The first région can be a prepared région having a conducW and permittivity that is different from the terrestria. medium. The prepared région can comprise an excavation containing water. The water can be seawater. The prepared région can comprise an excavation containing an aggregate composition of the terrestria! medium and an added material. In one or more aspects of the present disclosure. the second reg.on can extend around the guided surface waveguide probe from a first side of the first région to a second side of trie f.rst région. The propagation interface can inciude a third région adiacent to the first région opposite the second région. The third reg.on can comprise the seco lossy conducting medium or can comprise a third iossy conducting medium.

other Systems, methods. features. and advantages ot the present disciosure wii! be or become apparent to one with .......the art upon examination of trie foüowing drawings _a„d detailed description. It is intended that ai. such additional Systems, methods. features. and advantages be induded within this description, be within the scope of the present disclosure, and be protected by trie accompanying ciaims. in addition, al. optionai and preferred features and modifications of trie described embodiments are usabie m a.i aspec s ₀₍ the disclosure taught herein. Furthermore. the individual features of the dépendent claims, as well as ail optionai and preferred features and modifications of the descnbed embodiments are combinable and interchangeabie with one another.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disciosure can be better understood with reference to

The components in the drawings are not necessarily to scale, the following drawings.

emphasis instead being placed upon clearly illustrating the principles of the disclosure Moreover, in the drawings. like reference humerais designate corresponding parts throughout the several views.

FIG. 1 is a chart that depicts field strength as a function of distance for a guided electromagnetic field and a radiated electromagnetic held.

FIG. 2 is a drawing that illustrâtes a propagation interface with two régions employed ₁₀ for transmission of a guided surface wave according to various embodiments of the present disclosure.

_F|_G. 3 is a drawing that illustrâtes a guided surface waveguide probe disposed with respect to a propagation interface of FIG. 2 according to various embodiments of the present disclosure.

₁₅ FIG. 4 is a plot of an example of the magnitudes of dose-in and far-out asymptotes of first order Hankel functions according to various embodiments of the present disdosure.

FIGS. 5A and SB are drawings that illustrate a complex angle of incidence of an electric field synthesized by a guided surface waveguide probe according to various embodiments of the present disclosure.

₂₀ FIG. 6 is a graphical représentation illustrating the effect of élévation of a charge terminal on the location where the electric field of FIG. SA intersects with the lossy conducting medium at a Brewster angle according to various embodiments of the present disclosure.

FIGS. 7A through 7C are graphical représentations of examples of guided surface ₂₅ waveguide probes according to various embodiments of the present disclosure.

FIGS. 8A through SC are graphical représentations illustrating examples of équivalent image plane models of the guided surface waveguide probe of FIGS. 3 and 7A· 7C according to various embodiments of the present disclosure.

FIGS. 9A through 9C are graphical représentations illustrating exemples of singlewire transmission line and classic transmission Une models of the équivalent image p.ane models of FIGS. SB and SC according to various embodiments of the present disclosure.

FIG. 9D is a plot illustrating an example of the reactance variation of a lumped ₅ element tank circuit with respect to operating frequency according to various embodiments of the present disclosure.

FIG. 10 is a flow chart illustrating an example of adjusting a guided surface waveguide probe of FIGS. 3 and 7A-7C to launch a guided surface wave along the surface of a lossy conducting medium according to various embodiments of the present disciosure.

₁₀ FIG. 11 is a plot illustrating an example of the relationship between a wave tltt angle and the phase de.ay of a guided surface waveguide probe of FIGS. 3 and 7A-7C according to various embodiments of the present disclosure.

FIG. 12 is a drawing that illustrâtes an example of a guided surface waveguide probe according to various embodiments of the present disclosure.

₁₅ FIG 13 is a graphical représentation illustrating the incidence of a synthesized electric field at a comptex Brewster angle to match the guided surface waveguide mode at the Hankel crossover distance according to various embodiments of the present drsdosure.

fig. 14 is a graphical représentation of an exemple of a guided surface waveguide probe of FIG. 12 according to various embodiments of the present disclosure.

₂₀ FIG 15A includes plots of an example of the imaginary and real parts of a phase delay (Φ„) of a charge terminai T, of a guided surface waveguide probe according to various embodiments of the present disclosure.

fig. 15B is a schematic diagram of the guided surface waveguide probe of FIG. 14 according to various embodiments of the present disclosure.

₂₅ fig. 16 is a drawing that illustrâtes an exampte of a guided surface waveguide probe according to various embodiments of the present disclosure.

FIG. 17 is a graphical représentation of an example of a guided surface waveguide probe of FIG. 16 according to various embodiments of the present disclosure.

FIGS. 18A through 18C depict examples of receiving structures that can be employed to reçoive energy transmitted in the form of a guided surface wave iaunched by a guided surface waveguide probe according to the various embodiments of the present disclosure.

FIG. 18D is a flow chart illustrating an example of adjusting a receiving structure according to various embodiments of the present disclosure.

FIG. 19 depicts an example of an additional receiving structure that can be employed to receive energy transmitted in the form of a guided surface wave iaunched by a guided surface waveguide probe according to the various embodiments of the present disclosure.

FIG. 20 illustrâtes a cross sectionai view of an example guided surface waveguide probe site including a propagation interface according to various embodiments of the present disclosure.

FIG. 21A illustrâtes a cross sectionai view of another example guided surface waveguide probe site in which a portion of a région of ihe propagation interface in FIG. 20 is prepared to more efficiently launch a guided surface wave according to various embodiments of the present disclosure.

fig. 21B illustrâtes a top down view of the guided surface waveguide probe site in FIG. 21A according to various embodiments of the present disclosure.

FIG. 22 illustrâtes a stage in the préparation of the portion of the région of the propagation interface in FIG. 21A according to various embodiments of the present disclosure.

FIGS. 23A-23D illustrais top down views of examples of guided surface waveguide probe sites prepared for directional guided surface wave transmission according to various embodiments of the present disclosure.

FIGS. 24A and 24B «lustrale top down views of examples of guided surface waveguide probe sites prepared for directional guided surface wave transmission according to various embodiments of the present disclosure.

FIGS. 25A and 25B illustrate top down views of examples of guided surface waveguide probe sites utilizing géographie features of the iandscape for directionai guided surface wave transmission according to various embodiments of the present disclosure.

DETAILED DESCRIPTION

TO begin, some terminology shall be established to provide clarity in the discussion of concepts to foiiow. First, as contemplated herein. a formai distinction is drawn between radiated electromagnetic fields and guided electromagnetic fields.

As contemplated herein. a radiated electromagnetic field comprises electromagnetic ₁₀ energy that is emitted from a source structure in the form of waves that are not bound to a waveguide. For example, a radiated eiectromagnetic field is generaiiy a beid that leaves an electric structure such as an antenna and propagates through the atmosphère or other medium and is not bound to any waveguide structure. Once radiated electromagnetic waves leave an electric structure such as an antenna. they continue to propagate in the med.um of ₁₅ propagation (such as air) Independent of their source until they dlsslpate regardless of whether the source continues to operate. Once electromagnetic waves are radiated. they are not recoverabie unless intercepted. and. if not intercepted. the energy inhérent in the radiated eiectromagnetic waves is lest forever. Eieclricai structures such as antennas are designed to radiate electromagnetic fields by maximizing the ratio of the radiation résistance ₂₀ to the structure loss résistance. Radiated energy spreads oui in space and is lest regardless of whether a receiver is present. The energy density of the radiated fields is a funotion of distance due to géométrie spreading. Accordingly. the term radiate- in ali its forms as used herein refers to this form of electromagnetic propagation.

A guided eiectromagnetic field is a propagating eiectromagnetic wave whose energy is concentrated within or near boundaries between media having different eiectromagnetic properties. in this sense. a guided eiectromagnetic field is one that is bound to a waveguide and may be characterized as being conveyed by the current flowing in the waveguide. If there is no load to receive and/or dissipate the energy conveyed in a guided electromagnetic wave, then no energy is iost except for that which is dissipated in the conductivity of the guiding medium. Stated another way, if there is no load for a guided eiectromagnetic wave, then no energy is consumed. Thus. a generator or other source generating a guided end. such a generator or other source essentially runs idie until a ioad is presented. Th.s .s akin to running a generator to generate a 60 Hertz electromagnetic wave that is transmitted over power ünes where there is no electrical load. it should be noted that a guided eiectromagnetic field or wave is the équivalent to what is termed a transmission line mode. This contrasts with radiated eiectromagnetic waves in which real power is supplied at ail times in order to generate radiated waves. Uniike radiated eiectromagnetic waves, guided eiectromagnetic energy does not continue to propagée aiong a finite iength wavegu.de after me energy source is turned off. Accordingly. the term guide in ail its forms as used herein refers to this transmission mode of electromagnetic propagation.

Referring now to FIG. 1, shown is a graph 100 of fieid strength in décibels (dB) above an arbitrary reference in volts per meter as a function of distance in kilometers on a iog-dB _ptot to further iiiustrate the distinction between radiated and guided eleetromagnetio (raids. The graph 100 of FIG. 1 depicts a guided field strength curve 103 that shows the field strength of a guided eiectromagnetic field as a function of distance. This guided fieid strength curve 103 is essentiel., the same as a transmission line mode. Aise, the graph 100 , of FIG. 1 depicts a radiated field strength curve 106 that shows the field strength of a radiated electromagnetic field as a function of distance.

Of interest are the shapes of the curves 103 and 106 for guided wave and for rad/afion propagation, respective!,. The radiated fieid strength curve 106 falls off geometrically (1/d. where d is distance), which is depicted as a straight line on the iog-iog scale. The guided field strength curve 103. on the other band, has a characteristic exponentiel decay of e'-ZÆ and exhibits a distinctive knee 109 on the log-log scale. The guided fieid strength curve 103 and the radiated field strength curve 106 intersecl at point

112. which occurs at a Crossing distance. At distances less than the Crossing distance at intersection point 112. the field strength of a guided electromagnetic field is significantly greater at most locations than the field strength of a radiated electromagnetic field. At distances greater than the Crossing distance, the opposite is true. Thus, the guided and radiated field strength curves 103 and 106 further illustrate the fondamental propagation différence between guided and radiated electromagnetic fields. For an informai discussion of the différence between guided and radiated electromagnetic fields, reference is made to Mllligan, T. Modem Antenna Design, McGraw-Hill, 1» Edition, 1985. pp.8-9, which is incorporated herein by reference in its entirety.

The distinction between radiated and guided electromagnetic waves, made above. is readily expressed formaily and placed on a rigorous basis. That two such diverse solutions could émargé from one and the same linear partial differential équation, the wave équation, analytically follows from the boundary conditions imposed on the problem. The Green function for the wave équation, «self, contains the distinction between the nature of radiation and guided waves.

In empty space, the wave équation is a differential operator whose eigenfunctions possess a continuous spectrum of eigenvalues on the complex wave-number plane. This transverse electro-magnetic (TEM) field is called the radiation field. and those propagatmg fields are called Hertzien waves. However. in the presence of a conducting boundary. the 20 wave équation plus boundary conditions mathematically lead to a spectral représentation of wave-numbers composed of a continuous spectrum plus a sum of discrets spectra. To this end, reference is made to Sommerfeld. A.. liber die Ausbreitung der Wellen in der Drahtlosen Télégraphié, Annalen der Physlk. Vol. 28.1909, pp. 665-736. Also see Sommerfeld. A.. Problems of Radio, published as Chapter 6 in Partial Differential ₂₅ Fonntinns in Phvsics - Lectures on Theoreticai Physjçs^^umeXd, Academie Press, 1949 pp. 236-289, 295-296; Collin, R. E„ Hertzian Dipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20” Century Controversies, IFEE Antennas and Propagation Magazine. Vol. 46. No. 2, April 2004. pp. 64-79; and Reich, H. J., Ordnung, PF. Krauss,

H.L.. and Skalnik. J.G., MgowayeTheory and Teçhniflues. Van Nostrand, 1953. pp. 291293. each of these references being incorporated herein by reference in its entirety.

The terms -groundwave'and Surface waveidentify two distinctly different physical propagation phenomena. A surface wave arises analyticaliy from a distinct po/e yieiding a discrets component in the plane wave spectrum. See. e.g.. -The Excitation of Plane Surface Waves by Cuiien. A.L.. (^ng-M (Britleh). Vol. 10f. Part iV, August 1954, pp. 225-235). in this context. a surface wave is considered to be a guided surface wave. The surface wave (in the Zenneck-Sommerfeld guided wave sense) is, physically and mathematically, not the same as the ground wave (in the Weyl-Norton-FCC sense) that is now so familier from radio broadcasting. These two propagation mechanlsms arise from the excitation of different types of eigenvalue spectra (continuum or discrète) on the compiex plane. The field strength of lhe guided surface wave decays exponentially with distance as illustrated by guided field strength curve 103 of FIG. 1 (much like propagation in a lossy waveguide) and resembles propagation in a radial transmission line, as opposed to lhe dassical Hertzien radiation of the ground wave, which propagates spherically. possesses a continuum of eigenvaiues. falis off geometricaliy as illustrated by radiated held strength curve 106 of FIG. 1. and results from branch-cut intégrais. As experimentaiiy demonstrated _by C R Burrows in The Surface Wave in Radio Propagation over Plane Earth _(e—os Of the IRE. Vol. 25. No. 2. February. 1937. pp. 219-229) and The Surface Wave in Radio Transmission (BelU^boratonesRecard, Vol. 15. June 1937. pp. 321-324). vertical antennes radiale ground waves but do not launch guided surface waves.

To summarize the above, first. the connus part of the wave-number eigenvaiue spectrum. corresponding to Oranch-cul intégrais, produces the radiation field. and second, the discrète spectra. and corresponding residue sum arising from the potes enciosed by the contour of intégration, resuit in non-TEM traveling surface waves that are exponenftaiiy damped in the direction transverse to the propagation. Such surface waves are gu.ded transmission line modes. For further explanation, reference is made to Friedman, B..

Princinles and Techniqu*jo^^ ^Wi,e*’ ¹⁹⁵⁶' ^PP' ^PP' ^{283 286}’

298-300.

_In free space. antennes excite the continuum eigenvaiues of the wave equat.on, _Which is a radiation fieid. where the outwardiy propagating RF energy with E, and Η_ψ inphase is iost forever. On the other hand. waveguide probes excite discrète eigenvaiues. which results in transmission line propagation. See Collin, R. E., FieldlteayofGuided waves McGraw-Hill, 1960, pp. 453. 474-477. While such theoreticai analyses hâve held out the hypothetical possibi.ity of iaunching open surface guided waves over planar or spherical surfaces of iossy, bomogeneous media, for more than a century no known structures in the engineering arts hâve existed for accomplishing this with any practical

,. tko oariv iQOO's the theoretical analysis set efficiency. Unfortunately, since it emerged m the early 1900 s, forth above has essentially remained a theory and there hâve been no known structures for practically accomplishing the iaunching of open surface guided waves over planar or spherical surfaces of lossy, homogeneous media.

According to the various embodiments of the present disclosure, various guided surface waveguide probes are described that are configured to excite eiectric fieids that couple into a guided surface waveguide mode along the surface of a lossy conduct.ng medium. Such guided e.ectromagnetic fieids are substantialiy mode-matched in magnitude , rxn tho irface of the lossy conducting medium, end phase to a guided surface wave mode on the surface

Such a guided surface wave mode can aise be termed a Zenneck waveguide mode. By virtue of the tact that the restant fieids excited by the guided surface waveguide probes described herein are substantiel mode-matched to a guided surface waveguide mode on _the surface of the iossy conducting medium, a guided eiectromagnetic fieid in the form of a guided surface wave is launched along the surface of the lossy conducting medium.

According to one embodiment. the lossy conducting medium comprises a terrestnai med.um such as the Earth.

Referring to FIG. 2. shown is a propagation interface that provides for an exam.nat.on of the boundary value solutions to Maxwell's équations derived in 1907 by Jonathan

Zenneck as set forth in Ns paper Zenneck. J.. 'On the Propagation of Piane Beetrontagnetic Waves Along a Fiat Conducting Surface and their Relation to Wireless Teiegraphy. Annaien der Physik, Séria, 4, Vol. 23. September 20. 1907. pp. 846-866. FIG. 2 deplcts cyiindnca! coordinates for radiaiiy propagating waves along the interface between a tossy conduchng medium specified as Région 1 and an insulator specified as Région 2. Région 1 can comprise, for exampie, any lossy conducting medium. In one example, such a iossy conducting medium can comprise a terrestriai medium such as the Êarfh or other med.um. Région 2 is a second medium that shares a boundary interface with Région 1 and has different constitutive parameters relative to Région 1 Région 2 can comprise, for exampie. any insulator such as the atmosphère or other medium. The reflection coefficient for such a boundary interface goes to zéro only for incidence at a complex Brewster angie. See Stratton J.A.. Beçtromasnetiçjïem. McGraw-Hill. 1941, p. 516.

According to various embodiments. the present disdosure sets forth various guided surface waveguide probes .bat generate eiectromagnetic fieids thaf are substantially modematched to a guided surface waveguide mode on the surface of the iossy conducting medium comprising Région 1. According to various embodiments. such electromagnetrc fieids substantially synthesize a wave front incident at a complex Brewster angie of the tossy conducting medium that can resuit in zéro reflection.

To explain further. in Région 2. where an e'' field variation is assumed and where 0 _p * o and z > 0 (with z being the vertical coordinate normal to the surface of Regio „ being the radiai dimension in cylindricai coordinates). Zenneck’s ciosed-form exact solution of Maxwell's équations satisfying the boundary conditions along the interface are expressed by the following electric field and magnetic field components·.

Ηζψ = Ae'™ H^X-jYP'). ⁽¹⁾ p = 4 η!^Ό(-/7Ρ)· ^and °²Ρ \)ωε_α1

E_2I = ^{(3) 2Z} \ωε₀/

In Région 1, where the variation is assumed and where p 0 and z < 0.

Zenneck's closed-form exaot solution ot Maxwell équations satistying the boundary conditions a.ong the interface is expressed by the foilowing electric field and magnat,c t,eld components:

Η_ίφ = Ae^u'^z h!²⁾(-//P)p _{= A} L _e“i^z w5²⁾(-jyp). and ^clp kffi+jû)£i/ (4) (5) (6) ^u>^z Ho²⁾(-/yp)· _ln these expressions, z is the verticai coordinate norma. to the surface of Région 1 and p is the radial coordinate. f^WrP) * a «Wlex argument Hankel function of the second kind and order... u, is the propagation constant in the positive verflca! (z) direction ,n Région 1 u₂ is the propagation constant In the vertical (z) direction in Région 2. a, .s the conductivity of Région 1. . is equal to 2„f. where f is a frequency of excitation. z„ ,s the permittivity of free space. c. is the permittivity of Région 1, 4 is a source constant .rnposed _by the source, and y is a surface wave radial propagation constant

The propagation constants in the ±z directions are determined by separating the boundary conditions. This exercise gives, in Région 2, __ -Jfcp p - A ^Κσι+,ωεχ (7) and gives, in Région 1,

Ui — —U2^r ~ j^·

The radial propagation constant y is given by

I rfi , /c (J Tt y = j/k² ₀ + U₂ = J T^î· (8) (9) _whicb is a complex expression where n is the complex index of refraction given by /-----T (¹°)

In ail of the above Equations, = —, and ω£₀ (12)

I----_ % k₀ — μ₀ε_{0 2π}>

_where _£r comprises the relative permittivity of Région 1. u, is the conductivity of Région 1. c. _is the permittivity of free space. and _(I. comprises the permeabiiity of free space. Thus. the generated surface wave propagates parallel to the interface and exponentialty decays vertical to it. This is known as evanescence.

Thus, Equations (1)-(3) can be consldered to be a cylIndricaUy-symmetric. radialtypropagating waveguide mode. See Barlow, H. M., and Brown. J.. Radio Surface Waves. Oxford University Press. 1962. pp. 10-12. 29-33. The present disctosure details structures that excite this open boundary waveguide mode. Specificaiiy. according to vanous embodiments. a guided surface waveguide probe is provided with a charge termina, of appropriate size that is fed with voltage and/or current and is positioned relative to the boundary interface between Région 2 and Région 1 This may be better understood with reference to FIG. 3. which sbows an example of a guided surface waveguide probe 200a that includes a charge terminal T, e.evated above a lossy conducting medium 203 (e.g.. the Earth) along a vertica. axis zthat is norma. to a plane presented by the lossy conducting medium 203. The lossy conducting medium 203 maXes up Région 1. and a second medium 206 makes up Région 2 and shares a boundary interface with the iossy conducting medium

203.

According to one embodiment. the tossy conducting medium 203 can comprise a terrestrïal medium such as the planet Earth. To this end. such a terrestrial medium comprises ati structures or formations inciuded thereon whether naturel or man-made. For example, such a terrestrial medium can comprise natural éléments such as rock, soil, sand, fresh water, sea water, trees, végétation, and ail other naturai e.ements that make u_P our planet. In addition, such a terrestrial medium can comprise man-made éléments concrète, asphalt, building materials, and other man-made materials. In other e _the iossy conducting medium 203 can comprise some medium other than the Earth. whether naturally occurring or man-made. In other embodiments. the iossy conducting medium 203 can comprise other media such as man-made surfaces and structures such as automob,les. aircraft man-made materiais (such as piywood. plastic sbeeting. or other materia.s) or other media.

,n the case where the tossy conducting medium 203 comprises a terrestriai medium or Earth, the second medium 206 can comprise the atmosphère above the ground. As such, the atmosphère can be termed an atmospheric medium that comprises air and other éléments that make up the atmosphère of the Earth. in addition, it is possible that the second medium 206 can comprise other media reiative to the lossy conducting medium 203.

The guided surface waveguide probe 200a Includes a feed network 209 that couples an excitation source 212 to the charge terminal T, via. e.g., a vertical feed «ne conductor. The excitation source 212 may comprise, for example, an Alternating Current (AC) source or some other source. As contemplai herein, an excitation source can comprise an AC source or other type of source. According to various embodiments, a charge Q, .s .mposed on the charge terminai T, to synthesize an eiectric t.eld based upon the voltage apphed to terminai T, at any given instant. Oepending on the angte of incidence _W of the eiectnc heid _(E) it is possible to substantially mode-match the electric field to a guided surface waveguide mode on the surface of the iossy conducting medium 203 comprising Région 1.

By considering the Zenneck closed-form solutions of Equations (1)-(6), the Leontovich impédance bcundary condition between Région 1 and Région 2 can be stated as a r (13) z x Η₂(ρ,<ρ.θ) ~Js’ where 1 is a unit normal in the positive verflcai (₊z) direction and is the magnetic ffefd strength in Regien 2 expressed by Equation (1) above. Equation (13) implies that the electric and magnetic fields specified in Equations (1)-(3) may resuit in a radia, surface specified by (14) (15) (16) where A is a constant. Further, it should be noted that ciose-in to the guided surface waveguide probe 200 (for p « Λ), Equation (14) above has the behavior , _ -4G2)__u ₌ _ _A_

Jclose^P J πί-ίγρ'Ί $ ^2πρ>

The négative sign means that when source current (I.) flows verticalty upward as illustrated in FIG. 3, the -close-in'· ground current flows radially inward. By field matching on Η_φ -closein,” it can be determined that l_oY _ MCliY ^A “ 4 4 where q,= C,V„ in Equations (1)-(6) and (14). Therefore. the radial surface current density of Equation (14) can be restated as

The fields expressed by Equations (1)-(6) and (17) hâve the nature of a transmission line _morfe bound to a lossy interface, notion fields that are associated with groundwave propagation. See Barfow, H. M. and Brown, d.. Oxford Oniversity

Press, 1962, pp. 1-5.

At this point, a review of the nature of the Hankei fonctions used in Equations (1)-(6) and (17) is provided (or these sortions of the wave équation. One might observe that the Hankei fonctions of the first and second kind and order n are defined as complex combinations of the standard Bessel fonctions of the first and second kinds ^^O(x)=/nU)+J^WnW.^and

Hf\x) = /nW -7^WnW·

These fonctions represent cylindrical waves propagating radia.iy inward (rff«) and ootward (H®), respective^. The définition is anaiogoos to the relationship = cosx ± i sfnx. See, for example. Harrington, R.F.. Tlme-Harmonic Fields. McGraw-Hill, 1961, pp. 460 463.

(17) (18) (19)

That //,!(/<,?) Is an oulgoing wave can be recognized from its large argument asymptotic behavior that Is obtained directly from the sériés définitions of JM and N„(x). Far-out from the guided surface waveguide probe:

^(20a) which. when multiplied by is an outward propagating cylindrica! wave of the form _el^ with a 1/# spatial variation. The first order (n = 1) solution can be determined from Equation (20a) to be

(20b)

Close-in to the guided surface waveguide probe (for p « Λ). the Hankei function of first order (21) and the second kind behaves as

Note that these asymptotic expressions are complex quantities. When x is a real quantity.

Equations (20b) and (21) differ in phase by Jj, which corresponds to an extra phase advance or phase boost of 45· or. equivalently, A/8. The close-in and far-out asymptotes of the first order Hankel function of the second kind hâve a Hankel crossover or transition point where they are of equal magnitude at a distance of p = R_x.

Thus. beyond the Hankel crossover point the far ouf représentation prédominâtes _Over the dose-in représentation of the Hankel function. The distance to the Hankel crossover point (or Hankel crossover distance) can be found by equating Equations (20b) and (21) for -Jyp. and solving for R_x. With x = <τ/ω_Εο, it can be seen that the far-out and dose-in Hankel function asymptotes are frequency dépendent, with the Hankel crossover point moving out as the frequency is lowered. It should also be noted that the Hankel function asymptotes may aiso vary as the conductivity (,) of the iossy conducting medium changes. For example, the conductivity of the soi. can vary with changes in weather conditions.

Referring to FIG. 4, shown is an example of a plot of the magnitudes of the first order Hankel functions of Equations (20b) and (21) for a Région 1 conductivity of = 0.010 mhos/m and relative permittivity ε, = 15. at an operating frequency of 1850 kHz.

Curve 115 is the magnitude of the far-out asymptote of Equation (20b) and curve 118 is the magnitude of the ciose-in asymptote of Equation (21). with the Hankel crossover point 121 occurring at a distance of = 54 feet. While the magnitudes are equal. a phase offset exists between the two asymptotes at the Hankel crossover point 121. It can also be seen that the Hankel crossover distance is much less than a wavelength of the operation frequency.

,o Considering the electric field components given by Equations (2) and (3) of the

Zenneck dosed-form solution in Région 2. it can be seen that the ratio of E_z and E„ asymptotically passes to e, r-/y\ Ho^Z)C-JrP> ___> L = _7l = tan0_ir (²²) e_p V «2 / HÎ^z)(-jyp) N ^ωε° where n is the complex index of refraction of Equation (10) and 8, is the angle of incidence ₁₅ of the electric field. In addition, the vertical component of the mode-matched electric field of

Equation (3) asymptotically passes to p _ E _e-^ ⁽²³⁾ ^^2z p-00 k £₀ / N Βπ Vp which is linearly proportional to free charge on the isolated component of the eievated charge terminal's capacitance at the terminal voltage, q_r„. = C_fr„ x V_T.

₂₀ For example, the height H, of the eievated charge terminal T, in FIG. 3 affects the amount of free charge on the charge terminai T„ When the charge terminal T, is near the ground plane of Région 1, most of the charge Q, on the terminal is -bound. As the charge terminal T, is eievated, the bound charge is lessened untii the charge terminal T, reaches a height at which substantially ail of the isolated charge is free.

The advantage of an increased capacitive élévation for the charge terminal T, is that the charge on the eievated charge terminal T, is further removed from the ground plane, resulting in an increased amount of free charge ,,_r„ to couple energy into the guided surface waveguide mode. As the charge terminal T, is moved away from the ground plane, the charge distribution becomes more uniformly distributed about the surface of the terminal. The amount of free charge is related to the self-capacitance of the charge terminal T„ s For example, the capacitance of a spherical terminal can be expressed as a function of physical height above the ground plane. The capacitance of a sphere at a physical height of h above a perfect ground is given by

C^aspher. = 4«_οα(1 ₊ M + M² + M’ ₊ 2M< _{+ +} ···), (24) where the diameter of the sphere is 2a. and where M = α/2/r with h being the height of the ₁₀ spherical terminal. As can be seen. an increase in the terminal height h reduces the capacitance C of the charge terminal. It can be shown that for élévations of the charge terminal T, that are at a height of about four times the diameter (40 = 8a) or greater, the charge distribution Is approxlmately uniform about the spherical terminal, which can improve the coupling into the guided surface waveguide mode.

₁₅ m the case of a sufficiently isolated terminal, the self-capacitance of a conductive sphere can be approximated by C = 4rr_£<,a, where a is the radius of the sphere in meters, and the self-capacitance of a disk can be approximated by C = 8c_oa, where a is the radius of the disk in meters. The charge terminal T, can include any shape such as a sphere, a disk, a cylinder. a cône, a torus. a hood, one or more rings, or any other randomized shape or combination of shapes. An équivalent spherical diameter can be determined and used for positioning of the charge terminal T-j.

This may be further understood with reference to the example of FIG. 3, where the charge terminal T, is eievated at a physical height of = H, above the lossy conducting medium 203. To reduce the effects of the -bound- charge, the charge terminal T, can be ₂₅ positioned at a physical height that is at least four times the spherical diameter (or équivalent spherical diameter) of the charge terminal T, to reduce the bounded charge effects.

Referring next to FIG. 5A, shown is a ray optics interprétation of the electric field produced by the elevated charge Q, on charge terminal T, of FIG. 3. As in optics, minimizing the reflection of the incident electric field can improve and/or maximize the energy coupled into the guided surface waveguide mode of the lossy conducting medium s 203. For an electric field (E,) that is polarized par*I to the plane of incidence (not the boundary interface), the amount of reflection of the incident electric field may be delermined using the Fresnel reflection coefficient, which can be expressed as _E||„ _ V(E_F.-jx)-_Sin²^-(^r-P^osfli, (25)

- _{E|| (} 7(E_r-;x)-sin² flj+(e_r-fx) cos fl/ where 9, is the conventîonal angle of incidence measured with respect to the surface normal.

In the example of FIG. 5A. the ray optic Interprétation shows the incident field polarized parallel to the plane of incidence having an angle of incidence of <f„ which is measured with respect to the surface normal (z). There will be no reflection of the incident electric field when Γ,(_β() = 0 and thus the incident electric field wi» be completely coupled ₁₅ into a guided surface waveguide mode along the surface of the lossy conducting medium 203. It can be seen that the numerator of Equation (25) goes to zéro when the angle of incidence is = arctan(7£r À) ⁼ θ'.0· where x = σ/ωζ„. This complex angle of incidence (S,.,) is referred to as the Brewster angle. Referring back to Equation (22). it can be seen that the same complex Brewster angle (e,._B) relationship is present in both Equations (22) and (26).

As illustrated in FIG. 5A. the electric field vector E can be depicted as an incoming non-uniform plane wave, polarized parallel to the plane of incidence. The electric field vector E can be created from indépendant horizontal and vertical components as

Ε(θ,) = E_p p + E_z z.

Geometrically, the illustration in FIG. 5A suggests that the electric field vector B can be given by

E_p(p,z) = E(p,z) cos0i, and £_z(p,z) = E(p.z) cos(j- e_;) = E(p,z)smO_lt (28a) (28b) which means that the field ratio is (29) _ -J— = tan ψί.

E_z tan fl,

A generalized parameter W, called wave tilt. is noted herein as the ratio of the horizontal electric field component to the vertical electric field component given by

W = Si. ₌ |ιν|ε'^ψ, or

Εζ (30b) λ = tan0i =

W Epl^l which is complex and has both magnitude and phase. For an electromagnetic wave in Région 2 (FIG. 2). the wave tilt angle (Ψ) is equal to the angle between the normal of the wave-front at the boundary interface with Région 1 (FIG. 2) and the tangent to the boundary interface. This may be easier to see in FIG. 5B, which illustrâtes equi-phase surfaces of an electromagnetic wave and their normale for a radial cylindrical guided surface wave. At the boundary interface (z = 0) with a perfecl conductor. the wave-front normal is parallel to the tangent of the boundary interface, resulting in W = 0. However, in the case of a lossy dielectric, a wave tilt W exists because the wave-front normal is not parallel with the tangent of the boundary interface at z = 0.

Applying Equation (30b) to a guided surface wave gives tan 0; _B - — y — y/^£r -*7* ~ ^{n —} w ~ |w[ ' ttp r

With the angle of incidence equal to the complex Brewster angle («,.,). the Fresnel reflection coefficient of Equation (25) vanishes, as shown by . Jx)-sln^z cos θί

ΠΐίΑ,β) J(e_r-/x)-sin² βί+(ε_Γ-Α) cos θ( _Θ^_{Θ{ Β} (31) (32) = 0.

By adjusting the complex field ratio of Equation (22), an incident field can be synthesized to be incident at a complex angle at which the reflection Is reduced or eliminated. Establishing this ratio as n = ^{resulls in ,he} ^thesized ^{electric ,ield bei 9 inCid}®ⁿ‘ ^a‘ ’θ complex Brewster angle, making the reflections vamsh.

The concept of an electrical effective height can provide further insight into synthesizing an electric field with a complex angle of incidence with a guided surface waveguide probe 200. The electrical effective height (lr_e„) has been defmed as ⁽³³⁾ for a monopole with a physical height (or length) of Slnce the expression dépends upon ₁₀ the magnitude and phase of the source distribution along the structure, the effective height (or length) is complex in general. The intégration of the distributed current l(z) of the structure is performed over the physicai height of the structure (_M. and normalized to the ground current (/„) flowing upward through the base (or input) of the structure. The distributed current along the structure can be expressed by ₁₅ /(z) =/_ccos(0_oz), ⁽³⁴⁾ where is the propagation factor for current propagating on the structure. In the example of FIG. 3, f_c is the current that is distributed along the vertical structure of the guided surface waveguide probe 200a.

For example, consider a feed network 209 that includes a low loss coil (e.g., a helical ₂₀ coil) at the bottom of the structure and a vertical feed line conductor connected between the coil and the charge terminal T„ The phase delay due to the coil (or helical deiay line) is _θ = β l_r With a physical length of l_c and a propagation factor of _ 2π _ 2π (35)

FP “ A_p ” V_rA₀ ' where V, is the velocity factor on the structure. Λ„ is the wavelength at the supplied zs frequency, and Λ,, is the propagation wavelength resulting from the velocity factor Vf. The phase delay is measured relative to the ground (stake or System) current

In addition, the spatial phase delay along the length of the vertical feed line conductor can be given by θ„ = ΛΧ where h * the propagation phase constant for the vertical feed line conductor. In some implémentations, the spatial phase delay may be approximated by since the différence between the physical height of the ₅ guided surface waveguide probe 200a and the vertical feed Une conductor length is much less than a wavelength at the supplied frequency (Λ,). As a resuit, the total phase delay through the coii and vertica! feed line conductor is Φ = 6_C + Θ,. and the current fed to the top of the coil from the bottom of the physical structure is f_c(9_c + e_y) = <³⁶>

With the total phase delay Φ measured reiative to the ground (stake or System) current /, Consequently, the electrical effective height of a guided surface waveguide probe 200 can be approximated by h_efr = 1 cos(ftz) dz = M'’ ⁽³⁷⁾ (or the case where the physical height tr„ « Λ.. The complex effective height of a monopole _{15 Λ},„ = at an angte (or phase detay) of Φ. may be adjusted to cause the source fields to match a guided surface waveguide mode and cause a guided surface wave te be Launched on the lossy conducting medium 203.

In the example of FIG. 5A. ray optics are used to illustrais the complex angle trigonometry of the incident electric field (E) having a complex Brewster angle of incidence _M at the Hankel crossover distance («J 121. Recall from Equation (26) that. for a lossy conducting medium, the Brewster angle is complex and specified by tan Θ₍.Β - ) „_£o ” (38)

Electrically. the géométrie parameters are related by the electrical effective height (h.,,) of the charge terminal T by

R_x ΐΆηψίβ = x VI/ - h_eff · (39) where = (./2) - »,_e is the Brewster angte measured from the surface of the iossy conducting medium. To couple into the guided surface waveguide mode, the wave tilt of the electric field at the Hankel crossover distance can be expressed as the ratio of the electrical effective height and the Hankel crossover distance _B = W_Rx.

-* Rx

Since both the physical height (h_p) and the Hankel crossover distance (R«) are real quantities, the angle (Ψ) of the desired guided surface wave tilt at the Hankel crossover distance (R,) is equal to the phase (Φ) of the complex effective height (W This implies that by varying the phase at the supply point of the coil. and thus the phase delay m to Equation (37). the phase, Φ. of lhe complex effective height can be manipulated to match the angle of the wave tilt. Ψ. of the guided surface waveguide mode at the Hankel crossover point 121; Φ - T.

in FIG. 5A, a right triangle is depicted having an adjacent side of length R, along the lossy conducting medium surface and a complex Brewster angle ψ,„ measured between a ₁₅ ray 124 extending between the Hankel crossover point 121 at R, and the center of lhe charge terminal T„ and lhe lossy conducting medium surface 127 between the Hankel crossover point 121 and lhe charge terminal T,. With the charge terminal T, positioned at physical height and excited with a charge having the appropriate phase detay Φ. the resulting electric field is incident with the lossy conducting medium boundary interface at the 20 Hankel crossover distance R„ and at the Brewster angle. Under these conditions, the guided surface waveguide mode can be excited without reflection or substantially negligible reflection.

If the physical height of the charge terminal T, is decreased without changing the phase delay Φ of the effective height (h_eZ;), the resulting electric field intersects the lossy ₂₅ conducting medium 203 at the Brewster angie at a reduced distance from the guided surface waveguide probe 200. FIG. 6 graphically illustrâtes the effect of decreasing the physical height of the charge terminal T, on the distance where the electric field is incident at lhe

Brewster angle. As the height is decreased from h, through h₂ to h„ the point where the electric field intersects with the lossy conducting medium (e.g.. the Earth) at the Brewster angle moves doser to the charge terminal position. However, as Equation (39) indicates, the height H, (FIG. 3) of the charge terminal T, should be at or higher than the physical height (/>„) in order to excite the far-out component of the Hankel function. With the charge terminal T, positioned at or above the effective height (h_e„), the lossy conducting medium 203 can be illuminated at the Brewster angle of incidence (ψ,._Β = (rr/2) - β,._Β) at or beyond the Hankel crossover distance (_W 121 as illustrated in FIG. 5A. To reduce or minimize the bound charge on the charge terminal T„ the height should be at least four times the ₁₀ spherical diameter (or équivalent sphericai diameter) of the charge terminal T, as mentioned above.

A guided surface waveguide probe 200 can be configured to establish an electric field having a wave tilt that corresponds to a wave illuminating the surface of the lossy conducting medium 203 at a complex Brewster angle, thereby exciting radial surface 15 currents by substantially mode-matching to a guided surface wave mode at (or beyond) the

Hankel crossover point 121 at R_x.

Referring to FIG. 7A. shown is a graphical représentation of an example of a guided surface waveguide probe 200b that includes a charge terminal T,. As shown in FIG. 7A, an excitation source 212 such as an AC source acts as the excitation source for the charge 20 terminal T,, which is coupled to the guided surface waveguide probe 200b through a feed network 209 (FIG. 3) comprising a coil 215 such as, e.g . a helical coil. In other implémentations, the excitation source 212 can be inductively coupled to the coil 215 through a primary coil. In some embodiments. an impédance matching network may be included to improve and/or maximize coupling of the excitation source 212 to the coil 215.

₂₅ As shown in FIG. 7A, the guided surface waveguide probe 200b can include the upper charge terminal T, (e.g., a sphere at height h„) that is positioned along a vertical axis s that is substantially normal to the plane presented by the lossy conducting medium 203, A second medium 206 is located above the lossy conducting medium 203. The charge terminal T, has a self-capacitance C_T. During operation, charge Q, is imposed on the terminal T, depending on the voltage applied to the terminal T, at any given instant.

In the example of FIG. TA. the coil 215 is coupled to a ground stake (or groundmg 5 system) 218 at a first end and to the charge terminal T, via a vertical feed line conductor

221. In some implémentations, the coil connection to the charge terminal T, can be adjusted usina a tap 224 of the coil 215 as shown in FIG. TA. The coil 215 can be energized at an operating frequency by the excitation source 212 comprising, for example, an excitation source through a tap 22T at a lower portion of the coil 215. In other impiementations. the ₁₀ excitation source 212 can be inductively coupled to the coil 215 through a primary coil. The charge terminal T, can be configured to adjust its load impédance seen by the vertical feed line conductor 221, which can be used to adjust the probe impédance.

FIG. TB shows a graphical représentation of another example of a guided surface waveguide probe 200c that inciudes a charge terminal T.. As in FIG. TA. the guided surface 15 waveguide probe 200c can include the upper charge terminal T, positioned over the lossy conducting medium 203 (e.g.. at height in the exemple of FIG. TB. the phasing coii 215 is coupled at a first end to a ground stake (or grounding System) 218 via a lumped element tank circuit 260 and to the charge terminal T, al a second end via a vertical feed line conductor 221. The phasing coil 215 can be energized at an operating frequency by the excitation source 212 through, e.g, a tap 22T at a lower portion of the coi! 215. as shown in _F,G. TA. to other implémentations, the excitation source 212 can be inductively coupled to the phasing coil 215 or an inductive coil 263 of a tank circuit 260 through a pnrnary coil 269. The inductive coil 263 may a!so be called a -lumped element- coil as it behaves as a lumped element or inductor. In the example of FIG. TB, the phasing coil 215 is energized by the excitation source 212 through inductive coupling with the inductive coil 263 of the lumped element tank circuit 260. The lumped element tank circuit 260 comprises the inductive coil 263 and a capacitor 266. The inductive coil 263 and/or the capacitor 266 can be fixed or variable to allow for adjustment of the tank circuit résonance, and thus the probe impédance

FIG. 70 shows a graphical représentation of another example of a guided surface waveguide probe 200d that includes a charge terminal T„ As in FIG. 7A, the guided surface waveguide probe 200d can include the upper charge terminal T, positioned over the lossy conducting medium 203 (e.g.. at height /.„). The feed network 209 can comprise a plurality of phasing coils (e.g.. helical coils) instead of a single phasing coil 215 as illustrated in FIGS. 7A and 7B. The plurality of phasing coils can include a combination of helical colis to provide the appropriate phase delay (e.g.. = e„ + Sa. where β„ and Sa correspond to the phase delays of coils 215a and 215b, respectively) to launch a guided surface wave. In the example of FIG. 7C, the feed network includes two phasing coils 215a and 215b connected in sériés with the lower coil 215b coupled to a ground stake (or grounding System) 218 via a lumped element tank circuit 260 and the upper coil 215a coupled to the charge terminal T, via a vertical feed fine conductor 221. The phasing coils 215a and 215b can be energized at an operating frequency by the excitation source 212 through. e.g.. inductive coupling via a primary coil 269 with, e.g., the upper phasing coil 215a, the lower !5 phasing coil 215b, and/or an inductive coil 263 of the tank circuit 260. For example, as shown in FIG. 7C. the coil 215 can be energized by the excitation source 212 through inductive coupling from the primary coil 269 to the lower phasing coil 215b. Alternatively. as in the example shown in FIG. 7B. the coil 215 can be energized by the excitation source 212 through inductive coupling from the primary coil 269 to the inductive coil 263 of the lumped element tank circuit 260. The inductive coil 263 and/or the capacitor 266 of the lumped element tank circuit 260 can be fixed or variable to allow for adjustment of the tank circuit résonance, and thus the probe impédance.

At this point, it should be pointed out that there is a distinction between phase delays for traveling waves and phase shifts for standing waves. Phase delays for traveiing waves, ₂₅ Θ = βί, are due to propagation time delays on distributed element wave guiding structures such as, e.g.. the coil(s) 215 and vertical feed line conductor 221. A phase delay is not experienced as the traveling wave passes through the lumped element tank circuit 260. As a resuit, the total traveling wave phase delay through. e.g., the guided surface waveguide probes 200c and 200d is still Φ = 0_C + 9_y.

However. the position dépendent phase shifts of standing waves, which comprise forward and backward propagating waves, and load dépendent phase shifts dépend on both s the line-length propagation delay and at transitions between line sections of different characteristic impédances. It should be noted that phase shifts do occur in lumped element circuits. Phase shifts also occur at the impédance discontinuities between transmission line segments and between line segments and loads. This cornes from the complex reflection coefficient, Γ = | r|e'*. arising from the impédance discontinuities. and results m standing 10 waves (wave interférence patterns of forward and backward propagating waves) on the distributed element structures. As a resuit, the total standing wave phase shift of the guided surface waveguide probes 200c and 200d includes the phase shift produced by the lumped element tank circuit 260.

Accordingly, it should be noted that coils that produce both a phase delay for a ₁₅ traveling wave and a phase shift for standing waves can be referred to herein as ‘phasing coils.· The coils 215 are examples of phasing coils. It should be further noted that coils in a tank circuit, such as the lumped element tank circuit 260 as described above, act as a lumped element and an inductor. where the tank circuit produces a phase shift for standing waves without a corresponding phase delay for traveling waves. Such coils acting as 20 lumped éléments or inductors can be referred to herein as ‘inductor coils or lumped element coils. Inductive coil 263 is an example of such an inductor coi, or lumped element coil. Such inductor coils or lumped element coils are assumed to hâve a uniform current distribution throughout the coil. and are electrically small relative to the wavelength of operation of the guided surface waveguide probe 200 such that they produce a negligible delay of a traveling wave.

The construction and adjustment of the guided surface waveguide probe 200 is based upon various operating conditions, such as the transmission frequency, conditions of the lossy conducting medium (e.g., soil conductivity σ and relative permittivity e_r), and size of the charge terminal T_b The index of refraction can be calculated from Equations (10) and (11)as n = jz_r-jx, (⁴¹ >

where x = σ/ωε₀ with ω = 2nf. The conductivity σ and relative permittivity ε_Γ can be determined through test measurements of the lossy conducting medium 203. The complex Brewster angle (θ_ίιΒ) measured from the surface normal can also be determined from

Equation (26) as _B = arctan(7^fr ~ A)> (⁴^) or measured from the surface as shown in FIG. 5A as

Ψί,Β =f- ^θί.Β· (⁴³)

The wave tilt at the Hankel crossover distance (h/_Kx) can also be found using Equation (40).

The Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21) for and solving for R_x as illustrated by FIG. 4. The electrical 15 effective height can then be determined from Equation (39) using the Hankel crossover distance and the complex Brewster angle as h_eff = ^tan Ψί,β· (⁴⁴)

As can be seen from Equation (44), the complex effective height (h_eff) includes a magnitude that is associated with the physical height (h_p) ofthe charge terminal T, and a phase delay 20 (Φ) that is to be associated with the angle (Ψ ) of the wave tilt at the Hankel crossover distance (R_x). With these variables and the selected charge terminal configuration, it is possible to détermine the configuration of a guided surface waveguide probe 200.

With the charge terminal Tî positioned at or above the physical height (h_p), the feed network 209 (FIG. 3) and/or the vertical feed line connecting the feed network to the charge 25 terminal Τ_Ί can be adjusted to match the phase delay ( Φ ) of the charge Qi on the charge terminal T to the angle (Ψ ) of the wave tilt (VF). The size of the charge terminal Ti can be chosen to provide a sufficiently large surface for the charge Qi imposed on the terminais. In general, it is désirable to make the charge terminal Ti as large as practical. The size of the charge terminal T, should be large enough to avoid ionization of the surrounding air, which can resuit in electrical discharge or sparking around the charge terminal.

The phase delay of a helically-wound coil can be determined from Maxwell s équations as has been discussed by Corum, K.L. and J.F. Corum, RF Coils, Helîcal Resonators and Voltage Magnification by Cohérent Spatial Modes,” Microwave Review, Vol. 7, No. 2, September 2001, pp. 36-45., which is incorporated herein by reference in its entirety. For a helical coil with H/D > 1, the ratio of the velocity of propagation (v) of a wave along the coil's longitudinal axis to the speed of light (c), or the “velocity factor,” is given by

where H is the axial length of the solenoidal hélix, D is the coil diameter, N is the number of turns of the coil, s = H/N is the turn-to-turn spacing (or hélix pitch) of the coil, and λ₀ is the free-space wavelength. Based upon this relationship, the electrical length, or phase delay, of the helical coil is given by ^(4e>

The principle is the same if the hélix is wound spirally or is short and fat, but V_f and e_c are easier to obtain by experimental measurement. The expression for the characteristic (wave) impédance of a helical transmission line has also been derived as _{Zc =} £2 ^)- 1.027]. (47)

The spatial phase delay 9_y of the structure can be determined using the traveling wave phase delay of the vertical feed line conductor 221 (FIGS. 7A-7C). The capacitance of a cylindrical vertical conductor above a prefect ground plane can be expressed as

C_A = Farads, (48)

O-¹ where h_w is the vertical length (or height) of the conductor and a is the radius (in mks umts). As with the helical coil, the traveling wave phase delay of the vertical feed line conductor can be given by = = = (49) ^z *Λν^Λϋ where p_w is the propagation phase constant for the vertical feed line conductor, h_w is the vertical length (or height) of the vertical feed line conductor. V_w is the velocity factor on the wire, A_o is the wavelength at the supplied frequency, and Â_w is the propagation wavelength resulting from the velocity factor For a uniform cylindrical conductor, the velocity factor is a constant with l<_v = 0.94, or in a range from about 0.93 to about 0.98. If the mast is considered to be a uniform transmission line, its average characteristic impédance can be approximated by

-4 ⁽⁵⁰⁾ where V_w ~ 0.94 for a uniform cylindrical conductor and a is the radius of the conductor. An alternative expression that has been employed in amateur radio literature for the characteristic impédance of a single-wire feed line can be given by

Z_w = 138 (51)

Equation (51) implies that Z_w for a single-wire feeder varies with frequency. The phase delay can be determined based upon the capacitance and characteristic impédance.

With a charge terminal T positioned over the lossy conducting medium 203 as shown in FIG. 3, the feed network 209 can be adjusted to excite the charge terminal T, with the phase delay ( Φ ) of the complex effective height (h_eff) equal to the angle (Ψ) of the wave tilt at the Hankel crossover distance, or Φ = Ψ. When this condition is met, the electric field produced by the charge oscillating Qi on the charge terminal Ti is coupled into a guided surface waveguide mode traveling along the surface of a lossy conducting medium 203. For example, if the Brewster angle (e_iiB), the phase delay (θ_γ) associated with the vertical feed line conductor 221 (FIGS. 7A-7C), and the configuration of the coil(s) 215 (FIGS. 7A-7C) are known, then the position of the tap 224 (FIGS. 7A-7C) can be determined and adjusted to impose an oscillating charge Q, on the charge terminal T, with phase Φ = Ψ. The position of the tap 224 may be adjusted to maximize coupling the traveling surface waves into the guided surface waveguide mode. Excess coil length beyond the position of the tap 224 can be removed to reduce the capacitive effects. The vertical wire height and/or the geometrical parameters ofthe helical coil may also be varied.

The coupling to the guided surface waveguide mode on the surface of the lossy conducting medium 203 can be improved and/or optimized by tuning the guided surface waveguide probe 200 for standing wave résonance with respect to a complex image plane associated with the charge Q) on the charge terminal Τ_Ί. By doing this, the performance of the guided surface waveguide probe 200 can be adjusted for increased and/or maximum voltage (and thus charge Qi) on the charge terminal T,. Referring back to FIG. 3, the effect of the lossy conducting medium 203 in Région 1 can be examined using image theory analysis.

Physically, an elevated charge Qi placed over a perfectly conducting plane attracts the f_ree charge on the perfectly conducting plane, which then “piles up in the région under the elevated charge Qi. The resulting distribution of “bound electricity on the perfectly conducting plane is similar to a bell-shaped curve. The superposition of the potential of the elevated charge Q_b plus the potential of the induced “piled up” charge beneath it, forces a zéro equipotential surface for the perfectly conducting plane. The boundary value problem solution that describes the fields in the région above the perfectly conducting plane may be obtained using the classical notion of image charges, where the field from the elevated charge is superimposed with the field from a corresponding “image charge below the perfectly conducting plane.

This analysis may also be used with respect to a lossy conducting medium 203 by assuming the presence of an effective image charge Q/ beneath the guided surface waveguide probe 200. The effective image charge Q/ coïncides with the charge Q, on the charge terminal T, about a conducting image ground plane 130, as illustrated in FIG. 3.

However, the image charge Q/ is not merely located at some real depth and 180° out of phase with the primary source charge Ch on the charge terminal Ti, as they would be in the case of a perfect conductor. Rather, the lossy conducting medium 203 (e.g,, a terrestrial medium) présents a phase shifted image. That is to say, the image charge Qi is at a complex depth below the surface (or physical boundary) of the lossy conducting medium 203. For a discussion of complex image depth, reference is made to Wait, J. R., Complex Image Theory—Revisited, IEEE Antennas and Propagation Magazine, Vol. 33, No. 4, August 1991, pp. 27-29, which is incorporated herein by reference in its entirety.

Instead of the image charge Q/ being at a depth that is equal to the physical height (Hi) of the charge Qi, the conducting image ground plane 130 (representing a perfect conductor) is located at a complex depth of z = -d/2 and the image charge Q/ appears at a complex depth (i.e., the “depth has both magnitude and phase), given by -Dj = —(d/2 + d/2 + Hj) Φ Hp For vertically polarized sources over the Earth,

-_{d |d|iÇi}(52)

Ye Ye where y² = }ωμ_γσ_Ύ - and(53) k_o = ω^μ_οε_ο,( as indicated in Equation (12). The complex spacing of the image charge, in turn, implies that the external field will expérience extra phase shifts not encountered when the interface is either a dielectric or a perfect conductor. In the lossy conducting medium, the wave front normal is paraIlei to the tangent of the conducting image ground plane 130 at z = - d/2, and not at the boundary interface between Régions 1 and 2.

Consider the case illustrated in FIG. 8A where the lossy conducting medium 203 is a finitely conducting Earth 133 with a physical boundary 136. The finitely conducting Earth 133 may be replaced by a perfectly conducting image ground plane 139 as shown in FIG.8B, which is located at a complex depth z_x below the physical boundary 136. This équivalent représentation exhibits the same impédance when looking down into the interface at the physical boundary 136. The équivalent représentation of FIG. 8B can be modeled as an équivalent transmission line, as shown in FIG. 80. The cross-section of the équivalent structure is represented as a (z-directed) end-loaded transmission line, with the impédance of the perfectly conducting image plane being a short circuit (z_s = 0). The depth Zj can be determined by equating the TEM wave impédance looking down at the Earth to an image ground plane impédance z_in seen looking into the transmission line of FIG. 8C.

In the case of FIG. 8A, the propagation constant and wave intrinsic impédance in the upper région (air) 142 are

Y_o = jUy/g₀£₀ = 0 + ]β_ο , and (55) ₍₅₆₎

Yo y

In the lossy Earth 133, the propagation constant and wave intrinsic impédance are

Y_e = +ϊωε^ , and (57)

Z_e=^

Ye (58)

For normal incidence, the équivalent représentation of FIG. 8B is équivalent to a TEM transmission line whose characteristic impédance is that of air (z₀), with propagation constant of y₀, and whose length is ζ_τ. As such, the image ground plane impédance Z_ln seen at the interface for the shorted transmission line of FIG. 8C is given by

Z_in = Z_o tanhC/oZi). (59)

Equating the image ground plane impédance Z_in associated with the équivalent model of FIG. 8C to the normal incidence wave impédance of FIG. 8A and solving for z_r gives the distance to a short circuit (the perfectly conducting image ground plane 139) as

Zi

(60) where only the first term of the sériés expansion for the inverse hyperbolic tangent is considered for this approximation. Note that in the air région 142, the propagation constant is γ₀ = }βο< so Z_in = JZ₀ tanβ_ηζ_γ (which is a purely imaginary quantity for a real zj, but z_e is a complex value if σ * 0. Therefore, Z_in = Z_e only when z_r is a complex distance.

Since the équivalent représentation of FIG. 8B includes a perfectly conducting image ground plane 139, the image depth for a charge or current lying at the surface of the Earth (physical boundary 136) is equal to distance z_i on the other side of the image ground plane 139, or d = 2 X Zi beneath the Earth's surface (which is located at z = 0). Thus, the distance to the perfectly conducting image ground plane 139 can be approximated by d = 2_Z1«^. (θ1)

Additionally, the “image charge will be equal and opposite to the real charge, so the potential of the perfectly conducting image ground plane 139 at depth z_x = - d/2 will be zéro.

If a charge Qi is elevated a distance Hj above the surface of the Earth as illustrated in FIG. 3, then the image charge Q/ résides at a complex distance of Di = d + Ht below the surface, or a complex distance of d/2 + Ifl below the image ground plane 130. The guided surface waveguide probes 200 of FIGS. 7A-7C can be modeled as an équivalent single-wire transmission line image plane model that can be based upon the perfectly conducting image ground plane 139 of FIG. 8B.

FIG. 9A shows an example of the équivalent single-wire transmission line image plane model, and FIG. 9B illustrâtes an example of the équivalent classic transmission line model, including the shorted transmission line of FIG. 8C. FIG. 9C illustrâtes an example of the équivalent classic transmission line model including the lumped element tank circuit 260.

In the équivalent image plane models of FIGS. 9A-9C, Φ = 6_y + d_c is the traveling wave phase delay of the guided surface waveguide probe 200 referenced to Earth 133 (or the lossy conducting medium 203), Ô_c = β_ρΗ is the electrical length of the coil or coils 215 (FIGS. 7A-7C), of physical length H, expressed in degrees, 0_y = £U^lw is the electrical length of the vertical feed line conductor 221 (FIGS. 7A-7C), of physical length h_w, expressed in degrees. In addition, Θ₍₁ = β_ο d/2 is the phase shift between the image ground plane 139 and the physical boundary 136 of the Earth 133 (or lossy conducting medium 203). In the example of FIGS. 9A-9C, Z_w is the characteristic impédance of the eievated vertical feed line conductor 221 in ohms, Z_c is the characteristic impédance of the coil(s) 215 in ohms, and Z_o is the characteristic impédance of free space. In the example of FIG. 9C, Z_t is the characteristic impédance of the lumped element tank circuit 260 in ohms and 0_t is the corresponding phase shift at the operating frequency.

At the base of the guided surface waveguide probe 200, the impédance seen “looking up” into the structure is Z_T = Z_base. With a load impédance of:

where C_T is the self-capacitance of the charge terminal Τ_Ίι the impédance seen “looking up into the vertical feed line conductor 221 (FIGS. 7A-7C) is given by:

_ „ ZL+ZyptanhCj^v/i^) _ y Ζι,+Αν tanh(;0_y) (ββ) ² Zur+Z/.tanhtj/îwhw) ^W Z_w+Z_b tanh(jfly) and the impédance seen “looking up into the coil 215 (FIGS. 7A and 7B) is given by:

_ Z₂+Z_c tanh(//?_pH) _ Z₂4-Z_e tanh(;g_c) (β4)

Zbase ~ Z_{c Zc+Z2 tanh}(jp_pH) ^c z_c+Z₂ tanh(je_c) ' _Where the feed network 209 includes a plurality of coils 215 (e.g., FIG. 7C), the impédance seen at the base of each coil 215 can be sequentially determined using Equation (64). For example, the impédance seen “looking up” into the upper coil 215a of FIG. 7C is given by:

Z₂+Z_cn tanh(j/îpH) _ y z₂+z_cn tanh(;e_ca) (64.1)

Z_C0i\ - Z_{ca Zca}+2₂ tanhO/ïpH) ^ca Z_ca+Z₂ tanh(ye_cfl)’ and the impédance seen “looking up into the lower coil 215b of FIG. 7C can be given by.

Zcoit+Zçb tanh(j/îpH) _ „ Z_coil+z_cb tanhÇ/e_cl)) (64.2)

Zbase - Z_cb z_cb+z_coiltanh(jPpH) ^cb Z_cb+Z_coit tanh(.jO_cb)’ where Z_ca and Z_cb are the characteristic impédances of the upper and lower coils. This can be extended to account for additional coils 215 as needed. At the base of the guided surface waveguide probe 200, the impédance seen “looking down into the lossy conducting medium 203 is = Z_in, which is given by:

Ζ,,Ζ, Uni. _{= z h(/0} j (65) ^{Z,n Δ}° Z_o+Z_s tanh[j/?₀(d/2)j ⁰ where Z_s = 0.

Neglecting losses, the équivalent image plane model can be tuned to résonance when Zt + Z_T = 0 at the physical boundary 136. Or, in the low loss case, Xj + Aj = 0 at the physical boundary 136, where X is the corresponding reactive component. Thus, the impédance at the physical boundary 136 “looking up” into the guided surface waveguide probe 200 is the conjugate of the impédance at the physical boundary 136 “looking down” into the lossy conducting medium 203. By adjusting the probe impédance via the load impédance Z_L of the charge terminal T, while maintaining the traveling wave phase delay Φ equal to the angle of the media’s wave tilt Ψ, so that Φ = Ψ, which improves and/or maximizes coupling of the probe’s electric field to a guided surface waveguide mode along the surface ofthe lossy conducting medium 203 (e.g., Earth), the équivalent image plane models of FIGS. 9A and 9B can be tuned to résonance with respect to the image ground plane 139. In this way, the impédance of the équivalent complex image plane model is purely résistive, which maintains a superposed standing wave on the probe structure that maximizes the voltage and elevated charge on terminal Τ_Ί , and by Equations (1)-(3) and (16) maximizes the propagating surface wave.

While the load impédance Z_L of the charge terminal Tf can be adjusted to tune the probe 200 for standing wave résonance with respect to the image ground plane 139, in some embodiments a lumped element tank circuit 260 located between the coil(s) 215 (FIGS. 7B and 7C) and the ground stake (orgrounding system) 218 can be adjusted to tune the probe 200 for standing wave résonance with respect to the image ground plane 139 as illustrated in FIG. 9C. A phase delay is not experienced as the traveling wave passes through the lumped element tank circuit 260. As a resuit, the total traveling wave phase delay through, e.g., the guided surface waveguide probes 200c and 200d is still Φ = 9_C + g However, it should be noted that phase shifts do occur in lumped element circuits. Phase shifts also occur at impédance discontinuities between transmission line segments and between line segments and loads. Thus, the tank circuit 260 may also be referred to as a “phase shift circuit.

With the lumped element tank circuit 260 coupled to the base of the guided surface waveguide probe 200, the impédance seen “looking up” into the tank circuit 260 is Z_y = Z_ttininfl, which ^{can be} 9^iven by:

Ztuning ~ ^base ~ %t>

where Z_t is the characterîstic impédance of the tank circuit 260 and Zi,_ase is the impédance seen “looking up” into the coil(s) as given in, e.g., Equations (64) or (64.2). FIG. 9D illustrâtes the variation of the impédance of the lumped element tank circuit 260 with respect to operating frequency (f₀) based upon the résonant frequency (f_p) of the tank circuit 260. As shown in FIG. 9D, the impédance of the lumped element tank 260 can be inductive or capacitive depending on the tuned self-resonant frequency of the tank circuit. When operating the tank circuit 260 at a frequency below its self-resonant frequency (f_p). its terminal point impédance is inductive, and for operation above f_p the terminal point impédance is capacitive. Adjusting either the inductance 263 or the capacitance 266 of the tank circuit 260 changes f_p and shifts the impédance curve in FIG. 9D, which affects the terminal point impédance seen at a given operating frequency f₀.

Neglecting losses, the équivalent image plane model with the tank circuit 260 can be tuned to résonance when Zj + Z_T = 0 at the physical boundary 136. Or, in the low loss case, = o at the physical boundary 136, where X is the corresponding reactive component. Thus, the impédance at the physical boundary 136 “looking up into the lumped element tank circuit 260 is the conjugate of the impédance at the physical boundary 136 “looking down” into the lossy conducting medium 203. By adjusting the lumped element tank circuit 260 while maintaining the traveling wave phase delay Φ equal to the angle of the media’s wave tilt Ψ, so that Φ = Ψ, the équivalent image plane models can be tuned to résonance with respect to the image ground plane 139. In this way, the impédance of the équivalent complex image plane model is purely résistive, which maintains a superposed standing wave on the probe structure that maximizes the voltage and elevated charge on terminal T_b and Improves and/or maximizes coupling of the probe’s electric field to a guided surface waveguide mode along the surface ofthe lossy conducting medium 203 (e.g., earth).

It follows from the Hankel solutions, that the guided surface wave excited by the guided surface waveguide probe 200 is an outward propagating travelinq wave. The source distribution along the feed network 209 between the charge terminal T and the ground stake (or grounding system) 218 of the guided surface waveguide probe 200 (FIGS. 3 and 7A-7C) is actually composed of a superposition of a traveling wave plus a standinq wave on the structure. With the charge terminal T, positioned at or above the physical height h_p, the phase delay of the traveling wave moving through the feed network 209 is matched to the angle of the wave tilt associated with the lossy conducting medium 203. This modematching allows the traveling wave to be launched along the lossy conducting medium 203. Once the phase delay has been established for the traveling wave, the load impédance Z_L of the charge terminal T, and/or the lumped element tank circuit 260 can be adjusted to bring the probe structure into standing wave résonance with respect to the image ground plane (130 of FIG. 3 or 139 of FIG. 8), which is at a complex depth of - d/2. In that case, the impédance seen from the image ground plane has zero reactance and the charge on the charge terminal Ti is maximized.

The distinction between the traveling wave phenomenon and standing wave phenomena is that (1 ) the phase delay of traveling waves (θ = βά) on a section of transmission line of length d (sometimes called a delay line”) is due to propagation time delays; whereas (2) the position-dependent phase of standing waves (which are composed of forward and backward propagating waves) dépends on both the line length propagation time delay and impédance transitions at interfaces between line sections of different characteristic impédances. In addition to the phase delay that arises due to the physical length of a section of transmission line operating in sinusoïdal steady-state, there is an extra reflection coefficient phase at impédance discontinuities that is due to the ratio of Z_oa/Z_ob, where Z_oa and Z_ob are the characteristic impédances of two sections of a transmission line such as, e.g., a helical coil section of characteristic impédance Z_oa = Z_c (FIG. 9B) and a straight section of vertical feed line conductor of characteristic impédance Z_ob — Z_w (FIG.

9B).

As a resuit of this phenomenon, two relatively short transmission line sections of widely differing characteristic impédance may be used to provide a very large phase shift. For example, a probe structure composed of two sections of transmission line, one of low impédance and one of high impédance, together totaling a physical length of, say, 0.05 Λ, may be fabricated to provide a phase shift of 90°, which is équivalent to a 0.25 λ résonance. This is due to the large jump in characteristic impédances. In this way, a physically short probe structure can be electrically longer than the two physical lengths combined. This is illustrated in FIGS. 9A and 9B, where the discontinuitîes in the impédance ratios provide large jumps in phase. The impédance discontinuity provides a substantial phase shift where the sections are joined together.

Referring to FIG. 10, shown is a flow chart 150 illustrating an example of adjusting a guided surface waveguide probe 200 (FIGS. 3 and 7A-7C) to substantially mode-match to a guided surface waveguide mode on the surface of the lossy conducting medium, which launches a guided surface traveling wave along the surface of a lossy conducting medium 203 (FIGS. 3 and 7A-7C). Beginning with 153, the charge terminal T-ι of the guided surface waveguide probe 200 is positioned at a defined height above a lossy conducting medium 203. Utilizing the characteristics of the lossy conducting medium 203 and the operating frequency of the guided surface waveguide probe 200, the Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21 ) for -jyp, and solving for R_x as illustrated by FIG. 4. The complex index of refraction (n) can be determined using Equation (41), and the complex Brewster angle (d_iJ}) can then be determined from Equation (42). The physical height (h_p) of the charge terminal Tj can then be determined from Equation (44). The charge terminal Ti should be at or higher than the physical height (h_p) in order to excite the far-out component of the Hankel function. This height relationship is initially considered when launching surface waves. To reduce or minimize the bound charge on the charge terminal T_b the height should be at least four times the spherical diameter (or équivalent spherical diameter) of the charge terminal T_v

At 156, the electrical phase delay Φ of the elevated charge Qi on the charge terminal T₁ Is matched to the complex wave tilt angle Ψ. The phase delay (0_C) of the helical coil(s) and/or the phase delay (0_y) of the vertical feed line conductor can be adjusted to make Φ equal to the angle (M^J) of the wave tilt (W). Based on Equation (31 ), the angle (Ψ) of the wave tilt can be determined from:

= —-— = i = |l4/|e^yM\ (θθ)

E_z tan fl; s n

The electrical phase delay Φ can then be matched to the angle of the wave tilt. This angular (or phase) relationship is next considered when launching surface waves. For example, the electrical phase delay Φ = 6_C + R_y can be adjusted by varying the geometrical parameters of the coil(s) 215 (FIGS. 7A-7C) and/or the length (or height) of the vertical feed line conductor 221 (FIGS. 7A-7C). By matching Φ = Ψ, an electric field can be established at or beyond the Hankel crossover distance (R_x) with a complex Brewster angle at the boundary interface to excite the surface waveguide mode and launch a traveling wave along the lossy conducting medium 203.

Next at 159, the impédance of the charge terminal T, and/or the lumped element tank circuit 260 can be tuned to resonate the équivalent image plane model of the guided surface waveguide probe 200. The depth (d/2) of the conducting image ground plane 139 of FIG. 9A and 9B (or 130 of FIG. 3) can be determined using Equations (52), (53) and (54) and the values of the lossy conducting medium 203 (e.g., the Earth), which can be measured. Using that depth, the phase shift (0_d) between the image ground plane 139 and the physical boundary 136 of the lossy conducting medium 203 can be determined using θ_α _ p_{g d}/₂. The impédance (Z_in) as seen looking down into the lossy conducting medium

203 can then be determined using Equation (65). This résonance relationship can be considered to maximize the Iaunched surface waves.

Based upon the adjusted parameters of the coil(s) 215 and the length of the vertical feed line conductor 221, the velocity factor, phase delay, and impédance of the coil(s) 215 and vertical feed line conductor 221 can be determined using Equations (45) through (51). In addition, the self-capacitance (C_T) of the charge terminal T₁ can be determined using, e.g., Equation (24). The propagation factor (β_ρ) of the coil(s) 215 can be determined using Equation (35) and the propagation phase constant (β_ιν) for the vertical feed line conductor 221 can be determined using Equation (49). Using the self-capacitance and the determined values of the coil(s) 215 and vertical feed line conductor 221, the impédance (Z_base) of the guided surface waveguide probe 200 as seen “looking up” into the coil(s) 215 can be determined using Equations (62), (63), (64), (64.1) and/or (64.2).

The équivalent image plane model of the guided surface waveguide probe 200 can be tuned to résonance by, e.g., adjusting the load impédance Z_L such that the reactance component X_base of Z_base cancels out the reactance component X_in of Z_in, or X_base + X_in = 0. Thus, the impédance at the physical boundary 136 “looking up into the guided surface waveguide probe 200 is the conjugale ofthe impédance atthe physical boundary 136 “looking down into the lossy conducting medium 203. The load impédance Z_L can be adjusted by varying the capacitance (C_T) of the charge terminal T-, without changing the electrical phase delay Φ = of the charge terminal T_v An itérative approach may be taken to tune the load impédance Z_L for résonance of the équivalent image plane model with respect to the conducting image ground plane 139 (or 130). In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., Earth) can be improved and/or maximized.

The équivalent image plane model of the guided surface waveguide probe 200 can also be tuned to résonance by, e.g., adjusting the lumped element tank circuit 260 such that the reactance component X_tunfw of Z_tuning, cancels out the reactance component X_in of Z_in, _or x_tuni + x_in = 0. Consider the paraliel résonance curve in FIG. 9D, whose terminal point impédance at some operating frequency (f₀) is given by ^T (j2nfLp) + (j2KfCp) ¹ l-(2nfL_p) L_pC_p

As C_p (or L_p) is varied, the self-resonant frequency (f_p) of the paraliel tank circuit 260 changes and the terminal point reactance X_T(f₀) at the frequency of operation varies from inductive (+) to capacitive (-) depending on whether f₀ < f_p or f_p < f_o. By adjusting f_p, a wide range of reactance at f₀ (e.g., a large inductance = Χτ(ίο)/^{ω or a} small capacitance C_eq(f₀) = -1/ωΧ_τ(Ο can be seen at the terminais of the tank circuit 260.

To obtain the electrical phase delay (Φ) for coupling into the guided surface waveguide mode, the coil(s) 215 and vertical feed line conductor 221 are usually less than a quarter wavelength. For this, an inductive reactance can be added by the lumped element tank circuit 260 so that the impédance at the physical boundary 136 “looking up” into the lumped element tank circuit 260 is the conjugale of the impédance at the physical boundary 136 “looking down” into the lossy conducting medium 203.

As seen in FIG. 9D, adjusting f_p of the tank circuit 260 (FIG. 7C) above the operating frequency (/₀) can provide the needed impédance, without changing the electrical phase delay Φ = 9_C + 6_y of the charge terminal T_b to tune for résonance of the équivalent image plane model with respect to the conducting image ground plane 139 (or 130). In some cases, a capacitive reactance may be needed and can be provided by adjusting f_p of the tank circuit 260 below the operating frequency. In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., earth) can be improved and/or maximized.

This may be better understood by illustrating the situation with a numerical example. Consider a guided surface waveguide probe 200b (FIG. 7A) comprising a top-loaded vertical stub of physical height h_p with a charge terminal T, at the top, where the charge terminal T, is excited through a helical coil and vertical feed line conductor at an operational frequency (f₀) of 1.85 MHz. With a height (H,) of 16 feet and the lossy conducting medium 203 (e.g., Earth) having a relative permittivity of ε_Γ = 15 and a conductivity of = 0.010 mhos/m, several surface wave propagation parameters can be calculated for f₀ = 1.850 MHz. Under these conditions, the Hankel crossover distance can be found to be R_x = 54.5 feet with a physical height of h_p = 5.5 feet, which is well below the actual height of the charge terminal Tl While a charge terminal height of H·, = 5.5 feet could hâve been used, the taller probe structure reduced the bound capacitance, permitting a greater percentage of free charge on the charge terminal providing greater field strength and excitation of the traveling wave.

The wave length can be determined as:

= - = 162.162 meters,( ⁰ fo where c is the speed of light. The complex index of refraction is: n = y/e_r-Jx = 7.529 - j 6.546,(68) from Equation (41), where x = a_r/me₀ with ω = 2nf_o, and the complex Brewster angle is.

0_iB = arctan(7^- ~ A) ~ 85.6 -j 3.74-4°.(69) from Equation (42). Using Equation (66), the wave tilt values can be determined to be.

14/ = —2— = 1 = IW | = 0.101e^;4O,614°.( tan θ_ίιΒn

Thus, the helical coil can be adjusted to match Φ = Ψ = 40.614°

The velocity factor of the vertical feed line conductor (approximated as a uniform cylindrical conductor with a diameter of 0.27 inches) can be given as V_w ~ 0.93. Since h_p « λ₀, the propagation phase constant for the vertical feed line conductor can be approximated as:

2π ₌_^ ₌ 0.042 m^-1. (⁷¹) ^Pw A_w v_wA_a

From Equation (49) the phase delay of the vertical feed line conductor is:

= 11.640. (72)

By adjusting the phase delay of the helical coil so that 0_C = 28.974° = 40.614° - 11.640°, Φ will equal Ψ to match the guided surface waveguide mode. To illustrate the relationship between Φ and M', FIG. 11 shows a plot of both over a range of frequencies. As both Φ and Ψ are frequency dépendent, it can be seen that their respective curves cross over each other at approximately 1.85 MHz.

For a helical coil having a conductor diameter of 0.0881 inches, a coil diameter (D) of 30 inches and a turn-to-turn spacing (s) of 4 inches, the velocity factor for the coil can be determined using Equation (45) as:

and the propagation factor from Equation (35) is:

R = — = 0.564 m-¹. (⁷⁴)

V/-Â0

With 6_C — 28.974°, the axial length of the solenoidal hélix (H) can be determined using

Equation (46) such that:

H = — - 35.2732 inches . (⁷⁵)

P_P

This height détermines the location on the helical coil where the vertical feed line conductor is connected, resulting in a coil with 8.818 turns (N = H/s).

With the traveling wave phase delay of the coil and vertical feed line conductor adjusted to match the wave tilt angle (Φ = e_c + 0_y = Ψ). the load impédance (Z_L) of the charge terminal Ti can be adjusted for standing wave résonance of the équivalent image plane model ofthe guided surface waveguide probe 200. From the measured permittivity, conductivity and permeability of the Earth, the radial propagation constant can be determined using Equation (57) y_e = -Jj(j0Ui(di + Jûje^) = 0.25 + j 0.292 m , (⁷θ) and the complex depth of the conducting image ground plane can be approximated from Equation (52) as:

d » - = 3.364 + j 3.963 meters ,(77)

Ye with a corresponding phase shift between the conducting image ground plane and the physical boundary of the Earth given by:

= β₀(ά/2) = 4.015 -j 4.73°.(78)

Using Equation (65), the impédance seen “looking down into the lossy conducting medium 203 (i.e., Earth) can be determined as:

Z_in = Z_o tanh(/0_d) = «in + 7¾ = 31.191 +j 26.27 ohms.(79)

By matching the reactive component (X_in) seen “looking down” into the lossy conducting medium 203 with the reactive component (X_Ms_e) seen looking up into the guided surface waveguide probe 200, the coupling into the guided surface waveguide mode may be maximized. This can be accomplished by adjusting the capacitance of the charge terminal T, without changing the traveling wave phase delays of the coil and vertical feed line conductor. For example, by adjusting the charge terminal capacitance (C_T) to 61.8126 pF, the load impédance from Equation (62) is:

Z. = = -j 1392 ohms, (80) juC_r and the reactive components at the boundary are matched.

Using Equation (51), the impédance of the vertical feed line conductor (having a diameter (2a) of 0.27 inches) is given as

Z_w = 138 log ZZaZ ⁼ 537.534 ohms,(81) and the impédance seen “looking up” into the vertical feed line conductor is given by Equation (63) as:

₌ z_L+z_wtanhUe_y) _{= 835 438 ohms}(82) ^{2 w} z^+ZttanhO'Ély) ⁷

Using Equation (47), the characteristic impédance of the helical coil is given as ₌ Ê2 _ 1.027] = 1446 ohms,(83) and the impédance seen looking up into the coil at the base is given by Equation (64) as:

z₂+z_ct_al1hü'fl_c) ₌ _. 26.271 ohms. (84) ^base ⁰c_Zc+z₂tanh(je_c) ⁷

When compared to the solution of Equation (79), it can be seen that the reactive components are opposite and approximately equal, and thus are conjugales of each other. Thus, the impédance (Z_ip) seen “looking up into the équivalent image plane model of FIGS. 9A and 9B from the perfectly conducting image ground plane is only résistive or Z_ip = R + /0.

When the electric fields produced by a guided surface waveguide probe 200 (FIG. 3) are established by matching the traveling wave phase delay of the feed network to the wave tilt angle and the probe structure is resonated with respect to the perfectly conducting image ground plane at complex depth z = -d/2. the fields are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium, a guided surface traveling wave is launched along the surface of the lossy conducting medium. As illustrated in FIG. 1, the guided field strength curve 103 of the guided electromagnetic field has a characteristic exponential decay of e~^ad/Vd and exhibits a distinctive knee 109 on the log-log scale.

If the reactive components of the impédance seen “looking up” into the coil and “looking down” into the lossy conducting medium are not opposite and approximately equal, then a lumped element tank circuit 260 (FIG. 7C) can be included between the coil 215 (FIG. 7A) and ground stake 218 (FIGS. 7A) (or grounding system). The self-resonant frequency of the lumped element tank circuit can then be adjusted so that the reactive components looking up” into the tank circuit of the guided surface waveguide probe and “looking down” into the into the lossy conducting medium are opposite and approximately equal. Under that condition, by adjusting the impédance (Z_ip) seen “looking up” into the équivalent image plane model of FIG. 9C from the perfectly conducting image ground plane is only résistive or Z_fp =K+;0.

In summary, both analytically and experimentally, the traveling wave component on the structure of the guided surface waveguide probe 200 has a phase delay (Φ) at its upper terminal that matches the angle (Ψ) of the wave tilt of the surface traveling wave (Φ Ψ). Under this condition, the surface waveguide may be considered to be “mode-matched”. Furthermore, the résonant standing wave component on the structure of the guided surface waveguide probe 200 has a Vmax at the charge terminal T, and a V_Min down at the image plane 139 (FIG. 8B) where Z_ip = R_lp + j 0 at a complex depth of z = - d/2, not at the connection at the physical boundary 136 ofthe lossy conducting medium 203 (FIG. 8B). Lastly, the charge terminal T) is of sufficient height Ht of FIG. 3 (h > R_x tan0_{Î B}) so that eîectromagnetic waves incident onto the lossy conducting medium 203 at the complex Brewster angle do so out at a distance (> R_x) where the 1 /VF term is prédominant. Receive circuits can be utilized with one or more guided surface waveguide probes to facilitate wireless transmission and/or power delivery Systems.

Referring back to FIG. 3, operation of a guided surface waveguide probe 200 may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, an adaptive probe control system 230 can be used to control the feed network 209 and/or the charge terminal T_t to control the operation of the guided surface waveguide probe 200. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 203 (e.g., conductivity σ and relative permittivity ε_Γ), variations in field strength and/or variations in loading of the guided surface waveguide probe 200. As can be seen from Equations (31), (41) and (42), the index of refraction (n), the complex Brewster angle (θ_{ί Β}), and the wave tilt (|Ι4/|β^;ψ) can be affected by changes in soil conductivity and permittivity resulting from, e.g., weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivity sensors, ground parameter meters, field meters, current monitors and/or load receivers can be used to monitor for changes in the operational conditions and provide information about current operational conditions to the adaptive probe control system 230. The probe control system 230 can then make one or more adjustments to the guided surface waveguide probe 200 to maîntain specified operational conditions for the guided surface waveguide probe 200. For instance, as the moisture and température vary, the conductivity of the soil will also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe 200. Generally, it would be désirable to monitor the conductivity and/or permittivity at or about the Hankel crossover distance R_x for the operational frequency. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations (e.g., in each quadrant) around the guided surface waveguide probe 200.

The conductivity measurement probes and/or permittivity sensors can be configured to evaluate the conductivity and/or permittivity on a periodic basis and communicate the information to the probe control system 230. The information may be communicated to the probe control system 230 through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate wired or wireless communication network. Based upon the monitored conductivity and/or permittivity, the probe control system 230 may evaluate the variation in the index of refraction (n), the complex Brewster angle (θ_ίΒ), and/or the wave tilt (|Μφ'^ψ) and adjust the guided surface waveguide probe 200 to maîntain the phase delay (Φ) of the feed network 209 equal to the wave tilt angle (Ψ) and/or maîntain résonance of the équivalent image plane model of the guided surface waveguide probe 200. This can be accomplished by adjusting, e.g., 0_y, 0_c and/or C_T. For instance, the probe control system 230 can adjust the self-capacitance of the charge terminal T, and/or the phase delay (0_y. 0J applied to the charge terminal to maîntain the electrical launching efficiency of the guided surface wave at or near its maximum. For example, the self-capacitance of the charge terminal Τ_Ί can be varied by changing the size of the terminal. The charge distribution can also be improved by increasing the size of the charge terminal Ti, which can reduce the chance of an electrical discharge from the charge terminal T_v In other embodiments, the charge terminal T, can include a variable inductance that can be adjusted to change the load impédance Z_L. The phase applied to the charge terminal T, can be adjusted by varying the tap position on the coil 215 (FIGS. 7A-7C), and/or by including a plurality of predefined taps along the coil 215 and switching between the different predefined tap locations to maximize the launching efficiency.

Field or field strength (FS) meters may also be distributed about the guided surface waveguide probe 200 to measure field strength of fields associated with the guided surface wave. The field or FS meters can be configured to detect the field strength and/or changes în the field strength (e.g., electric field strength) and communicate that information to the probe control system 230. The information may be communicated to the probe control system 230 through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. As the load and/or environmental conditions change or vary during operation, the guided surface waveguide probe 200 may be adjusted to maintain specified field strength(s) at the FS meter locations to ensure appropriate power transmission to the receivers and the loads they supply.

For example, the phase delay (Φ = + 9_C) applied to the charge terminal T₁ can be adjusted to match the wave tilt angle (Ψ). By adjusting one or both phase delays, the guided surface waveguide probe 200 can be adjusted to ensure the wave tilt corresponds to the complex Brewster angle. This can be accomplished by adjusting a tap position on the coil(s) 215 (FIGS. 7A-7C) to change the phase delay supplied to the charge terminal T,. The voltage level supplied to the charge terminal Τ_Ί can also be increased or decreased to adjust the electric field strength. This may be accomplished by adjusting the output voltage of the excitation source 212 or by adjusting or reconfiguring the feed network 209. For instance, the position of the tap 227 (FIG. 7A) for the excitation source 212 can be adjusted to increase the voltage seen by the charge terminal T₁₍ where the excitation source 212 comprises, for example, an AC source as mentioned above. Maintaining field strength levels within predefined ranges can improve coupling by the receivers, reduce ground current losses, and avoid interférence with transmissions from other guided surface waveguide probes 200.

The probe control system 230 can be implemented with hardware, firmware, software executed by hardware, or a combination thereof. For example, the probe control system 230 can include processing cîrcuitry including a processor and a memory, both of which can be coupled to a local interface such as, for example, a data bus with an accompanying control/address bus as can be appreciated by those with ordinary skill in the art. A probe control application may be executed by the processor to adjust the operation of the guided surface waveguide probe 200 based upon monitored conditions. The probe control system 230 can also include one or more network interfaces for communicating with the various monitoring devices. Communications can be through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. The probe control system 230 may comprise, for example, a computer system such as a server, desktop computer, laptop, or other system with like capability.

Referring back to the example of FIG. 5A, the complex angle trigonometry is shown for the ray optic interprétation of the incident electric field (E) of the charge terminal T, with a complex Brewster angle (θ_ίιΒ) at the Hankel crossover distance (R_x). Recall that, for a lossy conducting medium, the Brewster angle is complex and specified by Equation (38). Electrically, the géométrie parameters are related by the electrical effective height (h_eff ) of the charge terminal T_t by Equation (39). Since both the physical height (h_p) and the Hankel crossover distance (R_x) are real quantities, the angle of the desired guided surface wave tilt at the Hankel crossover distance (V/_fix) is equal to the phase (Φ) of the complex effective height (/i_e/y). With the charge terminal T) positioned at the physical height h_p and excited with a charge having the appropriate phase delay Φ, the resulting electric field is incident with the lossy conducting medium boundary interface at the Hankel crossover distance R_x, and at the Brewster angle. Under these conditions, the guided surface waveguide mode can be excited without reflection or substantially negligible reflection.

However, Equation (39) means that the physical height of the guided surface waveguide probe 200 can be relatively small. While this will excite the guided surface waveguide mode, this can resuit in an unduly large bound charge with httle free charge. To compensate, the charge terminal T₁ can be raised to an appropriate élévation to increase the amount of free charge. As one example rule of thumb, the charge terminal T can be positioned at an élévation of about 4-5 times (or more) the effective diameter of the charge terminal T_v FIG. 6 illustrâtes the effect of raising the charge terminal Τ_Ί above the physical height (h_p) shown in FIG. 5A. The increased élévation causes the distance at which the wave tilt is incident with the lossy conductive medium to move beyond the Hankel crossover point 121 (FIG. 5A). To improve coupling in the guided surface waveguide mode, and thus provide for a greater launching efficiency of the guided surface wave, a lower compensation terminal T₂ can be used to adjust the total effective height (h_TE) of the charge terminal T, such that the wave tilt at the Hankel crossover distance is at the Brewster angle.

Referring to FIG, 12, shown is an example of a guided surface waveguide probe 200e that includes an elevated charge terminal T, and a lower compensation terminal T_z that are arranged along a vertical axis z that is normal to a plane presented by the lossy conducting medium 203. In this respect, the charge terminal T, is placed directly above the compensation terminal T₂ although it is possible that some other arrangement of two or more charge and/or compensation terminais T_N can be used. The guided surface waveguide probe 200e is disposed above a lossy conducting medium 203 according to an embodiment of the present disclosure. The lossy conducting medium 203 makes up Région 1 with a second medium 206 that makes up Région 2 sharing a boundary interface with the lossy conducting medium 203.

The guided surface waveguide probe 200e includes a feed network 209 that couples an excitation source 212 to the charge terminal T, and the compensation terminal T₂. According to various embodiments, charges Qi and Q₂ can be imposed on the respective charge and compensation terminais T, and T₂, depending on the voltages applied to terminais T, and T₂ at any given instant, h is the conduction current feeding the charge Q, on the charge terminal Tj via the terminal lead, and l₂ is the conduction current feeding the charge Q₂ on the compensation terminal T₂ via the terminal lead.

According to the embodiment of FIG. 12, the charge terminal T, is positioned over the lossy conducting medium 203 at a physical height Hn and the compensation terminal T₂ is positioned directly below T_t along the vertical axis z at a physical height H_2l where H₂ is less than Η_Ί. The height h of the transmission structure may be calculated as h = Hi - H_z, The charge terminal T, has an isolated (or self) capacitance C₁₍ and the compensation terminal T₂ has an isolated (or self) capacitance C₂. A mutual capacitance C_M can also exist between the terminais T, and T₂ depending on the distance therebetween. During operation, charges Q, and Q₂ are imposed on the charge terminal T1 and the compensation terminal T₂, respectively, depending on the voltages applied to the charge terminal T, and the compensation terminal T₂ at any given instant.

Referring next to FIG. 13, shown is a ray optics interprétation of the effects produced by the elevated charge Ch on charge terminal T( and compensation terminal T₂ of FIG. 12. With the charge terminal Τ_Ί elevated to a height where the ray intersects with the lossy conductive medium at the Brewster angle at a distance greater than the Hankel crossover point 121 as illustrated by line 163, the compensation terminal T₂ can be used to adjust h_TE by compensating for the increased height. The effect of the compensation terminal T₂ is to reduce the electrical effective height of the guided surface waveguide probe (or effectively raise the lossy medium interface) such that the wave tilt at the Hankel crossover distance is at the Brewster angle as illustrated by line 166.

The total effective height can be written as the superposition of an upper effective height (h_UE) associated with the charge terminal T, and a lower effective height (h_LE) associated with the compensation terminal T₂ such that h_TE = h_{UE +} h_LE = h_pe^⁺^ + h_ae^i{phd+<pL) = Rx x <⁸⁵) where Φ_σ is the phase delay applied to the upper charge terminal T_n Φ/„ is the phase delay applied to the lower compensation terminal T₂, β = 2π/λ_ρ is the propagation factor from

Equation (35), h_p is the physical height of the charge terminal T, and h_d is the physical height of the compensation terminal T₂. If extra lead lengths are taken into considération, they can be accounted for by adding the charge terminal lead length z to the physical height h of the charge terminal T, and the compensation terminal lead length y to the physical height h_d of the compensation terminal T₂ as shown in h_TE = (h_p + _Ζ)_ε>^'^^+Φυ>> + Çhd + γ)β^^,ι^^+Φ'·^} = RX*W. (86) The lower effective height can be used to adjust the total effective height (/i_TE) to equal the complex effective height (h_eff) of FIG. 5A.

Equations (85) or (86) can be used to détermine the physical height of the lower disk of the compensation terminal T₂ and the phase angles to feed the terminais in order to obtain the desired wave tilt at the Hankel crossover distance. For example, Equation (86) can be rewritten as the phase delay applied to the charge terminal D as a function of the compensation terminal height (/i_d) to give

A ÎR_ry.W-(h_d+y)e^^Îlh<i'ⁱ'Py⁺'^t’!j\ <ΜΜ = + ^ln -------(ÎÇ^) ) ⁽⁸⁷⁾

To détermine the positioning of the compensation terminal T₂, the relationships discussed above can be utilized. First, the total effective height (h_TE) is the superposition of the complex effective height (h_UE) of the upper charge terminal T, and the complex effective height (h_LE) of the lower compensation terminal T₂ as expressed in Equation (86). Next, the tangent of the angle of incidence can be expressed geometrically as «ηψ_Ε=!ρ. (88) which is equal to the définition of the wave tilt, W. Finally, given the desired Hankel crossover distance R_x, the h_TE can be adjusted to make the wave tilt of the incident ray match the complex Brewster angle at the Hankel crossover point 121. This can be accomplished by adjusting h_p, Φ_υ( and/or h_d.

These concepts may be better understood when discussed in the context of an example of a guided surface waveguide probe. Referring to FIG. 14, shown is a graphical représentation of an example of a guided surface waveguide probe 200f including an upper charge terminal Τ_Ί (e.g., a sphere at height h_T) and a lower compensation terminal T₂ (e.g, a disk at height h_d) that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. During operation, charges Qi and Q₂ are imposed on the charge and compensation terminais Ti and T₂, respectively, depending on the voltages applied to the terminais T_t and T₂ at any given instant.

An AC source can act as the excitation source 212 for the charge terminal T_b which is coupled to the guided surface waveguide probe 200f through a feed network 209 comprising a phasing coil 215 such as, e.g, a helical coil. The excitation source 212 can be connected across a lower portion of the coil 215 through a tap 227, as shown in FIG. 14, or can be inductively coupled to the coil 215 by way of a primary coil. The coil 215 can be coupled to a ground stake (or grounding system) 218 at a first end and the charge terminal Ti at a second end. In some implémentations, the connection to the charge terminal T, can be adjusted using a tap 224 at the second end of the coil 215. The compensation terminal T₂ is positioned above and substantially parallel with the lossy conducting medium 203 (e.g, the ground or Earth), and energized through a tap 233 coupled to the coil 215. An ammeter 236 located between the coil 215 and ground stake (or grounding system) 218 can be used to provide an indication of the magnitude of the current flow (/₀) at the base of the guided surface waveguide probe. Alternatively, a current clamp may be used around the conductor coupled to the ground stake (or grounding system) 218 to obtain an indication of the magnitude of the current flow (/₀).

In the example of FIG. 14, the coil 215 is coupled to a ground stake (or grounding system) 218 at a first end and the charge terminal T at a second end via a vertical feed line conductor 221. In some implémentations, the connection to the charge terminal T) can be adjusted using a tap 224 at the second end of the coil 215 as shown in FIG. 14. The coil 215 can be energized at an operating frequency by the excitation source 212 through a tap 227 at a lower portion of the coil 215. In other implémentations, the excitation source 212 can be inductively coupled to the coil 215 through a primary coil. The compensation terminal

T₂ is energized through a tap 233 coupled to the coil 215, An ammeter 236 located between the coil 215 and ground stake (or grounding system) 218 can be used to provide an indication of the magnitude of the current flow at the base of the guided surface waveguide probe 200f. Alternatively, a current clamp may be used around the conductor coupled to the ground stake (or grounding system) 218 to obtain an indication of the magnitude of the current flow. The compensation terminal T₂ is positioned above and substantially parallel with the lossy conducting medium 203 (e.g., the ground).

In the example of FIG. 14, the connection to the charge terminal T_t is located on the coil 215 above the connection point of tap 233 for the compensation terminal T₂. Such an adjustment allows an increased voltage (and thus a higher charge Qi) to be applied to the upper charge terminal Τ_υ In other embodiments, the connection points for the charge terminal Ti and the compensation terminal T₂ can be reversed. It is possible to adjust the total effective height (h_TE) of the guided surface waveguide probe 200f to excite an electric field having a guided surface wave tilt at the Hankel crossover distance R_x. The Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21 ) for -jyp, ^and solving for Rx as illustrated by FIG. 4. The index of refraction (n), the complex Brewster angle (θί>β and ^fl), the wave tilt (|14φ'^ψ) and the complex effective height (/ie// = ^{can de} determined as described with respect to Equations (41) - (44) above.

With the selected charge terminal Τί configuration, a spherical diameter (or the effective spherical diameter) can be determined. For example, if the charge terminal Τ_Ί is not configured as a sphere, then the terminal configuration may be modeled as a spherical capacitance having an effective spherical diameter. The size of the charge terminal T, can be chosen to provide a sufficiently large surface for the charge Ch imposed on the terminais. In general, it is désirable to make the charge terminal Ti as large as practical. The size of the charge terminal should be large enough to avoid ionization of the surroundmg air, which can resuit in electrical discharge or sparking around the charge terminal. To reduce the amount of bound charge on the charge terminal T_b the desired élévation to provide free charge on the charge terminal T, for launching a guided surface wave should be at least 4-5 times the effective spherical diameter above the lossy conductive medium (e.g., the Earth). The compensation terminal T₂ can be used to adjust the total effective height (h_TE) of the guided surface waveguide probe 200f to excite an electric field having a guided surface wave tilt at R_x. The compensation terminal T₂ can be positioned below the charge terminal Ti at = h_T ~ h_p, where h_T is the total physical height of the charge terminal T,. With the position of the compensation terminal T₂ fixed and the phase delay 4¼ applied to the upper charge terminal T_b the phase delay 4>_t applied to the lower compensation terminal T₂ can be determined using the relationships of Equation (86), such that.

Φυ(Μ = -P(h_d + y) - )ln

R_;fxW-(h_ÎÎ+_Z)e^^,'P^+/Îz+l“^f-⁾ (h_d+y) (89)

In alternative embodiments, the compensation terminal T₂ can be positioned at a height h_(l where = 0. This is graphically illustrated in FIG. 15A, which shows plots 172 and 175 of the imaginary and real parts of Φ_ϋ( respectively. The compensation terminal T₂ is positioned at a height h_d where Im^} = 0, as graphically illustrated in plot 172. At this fixed height, the coil phase Φ₍₇ can be determined from Re^}, as graphically illustrated in plot 175.

With the excitation source 212 coupled to the coil 215 (e.g., at the 50Ω point to maximize coupling), the position of tap 233 may be adjusted for parallel résonance of the compensation terminal T₂ with at least a portion of the coil at the frequency of operation.

FIG. 15B shows a schematic diagram of the general electrical hookup of FIG. 14 in which V! is the voltage applied to the lower portion of the coil 215 from the excitation source 212 through tap 227, V₂ is the voltage at tap 224 that is supplied to the upper charge terminal T_b and V₃ is the voltage applied to the lower compensation terminal T₂ through tap 233. The résistances R_p and R_d represent the ground return résistances of the charge terminal T) and compensation terminal T₂, respectively. The charge and compensation terminais T and T₂ may be configured as spheres, cylinders, toroids, rings, hoods, or any other combination of capacitive structures. The size of the charge and compensation terminais T, and T₂ can be chosen to provide a sufficiently large surface for the charges Qi and Q₂ imposed on the terminais. In general, it is désirable to make the charge terminal T, as large as practical. The size of the charge terminal T_t should be large enough to avoid ionization of the surrounding air, which can resuit in electrical discharge or sparking around the charge terminal. The self-capacitance C_p and C_d of the charge and compensation terminais T, and T₂ respectively, can be determined using, for example, Equation (24).

As can be seen in FIG. 15B, a résonant circuit is formed by at least a portion of the inductance ofthe coil 215, the self-capacitance C_d of the compensation terminal T₂, and the ground return résistance R_d associated with the compensation terminal T₂. The parallel résonance can be established by adjusting the voltage V₃ applied to the compensation terminal T₂ (e.g., by adjusting a tap 233 position on the coil 215) or by adjusting the height and/or size of the compensation terminal T₂ to adjust C_d. The position of the coil tap 233 can be adjusted for parallel résonance, which will resuit in the ground current through the ground stake (or grounding system) 218 and through the ammeter 236 reaching a maximum point. After parallel résonance of the compensation terminal T₂ has been established, the position of the tap 227 for the excitation source 212 can be adjusted to the 50Ω point on the coil 215.

Voltage V₂ from the coil 215 can be applied to the charge terminal T_1t and the position of tap 224 can be adjusted such that the phase delay (Φ) of the total effective height (h_TE) approximately equals the angle of the guided surface wave tilt (IV_Rx) at the Hankel crossover distance (RJ. The position of the coil tap 224 can be adjusted until this operating point is reached, which results in the ground current through the ammeter 236 increasing to a maximum. At this point, the résultant fields excited by the guided surface waveguide probe 200f are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium 203, resulting in the launching of a guided surface wave along the surface of the lossy conducting medium 203. This can be venfied by measuring field strength along a radial extending from the guided surface waveguide probe 200.

Résonance of the circuit including the compensation terminal T₂ may change with the attachment of the charge terminal T, and/or with adjustment of the voltage applied to the charge terminal T) through tap 224. While adjusting the compensation terminal circuit for résonance aids the subséquent adjustment of the charge terminal connection, it is not necessary to establish the guided surface wave tilt (W_Rx) at the Hankel crossover distance (R_x). The system may be further adjusted to improve coupling by iteratively adjusting the position of the tap 227 for the excitation source 212 to be at the 50Ω point on the coil 215 and adjusting the position of tap 233 to maximize the ground current through the ammeter 236. Résonance of the circuit including the compensation terminal T₂ may drift as the positions of taps 227 and 233 are adjusted, or when other components are attached to the coil 215.

In other implémentations, the voltage V₂ from the coil 215 can be applied to the charge terminal Τ_Ί, and the position of tap 233 can be adjusted such that the phase delay (Φ) of the total effective height (/i_TE) approximately equals the angle (Ψ) ofthe guided surface wave tilt at R_x. The position of the coil tap 224 can be adjusted until the operating point is reached, resulting in the ground current through the ammeter 236 substantially reaching a maximum. The résultant fields are substantially mode-matched to a guided surface waveguide mode on the surface ofthe lossy conducting medium 203, and a guided surface wave is Iaunched along the surface of the lossy conducting medium 203. This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe 200. The system may be further adjusted to improve coupling by iteratively adjusting the position of the tap 227 for the excitation source 212 to be at the 50Ω point on the coil 215 and adjusting the position of tap 224 and/or 233 to maximize the ground current through the ammeter 236.

Referring back to FIG. 12, operation of a guided surface waveguide probe 200 may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, a probe control system 230 can be used to control the feed network 209 and/or positioning of the charge terminal T, and/or compensation terminal T₂ to control the operation of the guided surface waveguide probe 200. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 203 (e.g., conductivity σ and relative permittivity ε_Γ), variations in field strength and/or variations in loading of the guided surface waveguide probe 200. As can be seen from Equations (41) - (44), the index of refraction (71), the complex Brewster angle (0_tB and Ψ,-,β), the wave tilt (|IV) and the complex effective height (h_eff = /ι_ρβ'^φ) can be affected by changes in soil conductivity and permittivity resulting from, e.g., weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivity sensors, ground parameter meters, field meters, current monitors and/or load receivers can be used to monitor for changes in the operational conditions and provide information about current operational conditions to the probe control system 230. The probe control system 230 can then make one or more adjustments to the guided surface waveguide probe 200 to maintain specified operational conditions for the guided surface waveguide probe 200. For instance, as the moisture and température vary, the conductivity of the soil will also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe 200. Generally, it would be désirable to monitor the conductivity and/or permittivity at or about the Hankel crossover distance R_x for the operational frequency. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations (e.g., in each quadrant) around the guided surface waveguide probe 200.

With reference then to FIG. 16, shown is an example of a guided surface waveguide probe 200g that includes a charge terminal and a charge terminal T₂ that are arranged along a vertical axis z. The guided surface waveguide probe 200g is disposed above a lossy conducting medium 203, which makes up Région 1. In addition, a second medium 206 shares a boundary interface with the lossy conducting medium 203 and makes up Région 2. The charge terminais T, and T₂ are positioned over the lossy conducting medium 203. The charge terminal 1Ί is positioned at height H_1t and the charge terminal T₂ is positioned directly below T, along the vertical axis z at height H₂, where H₂ is less than H,. The height h of the transmission structure presented by the guided surface waveguide probe 200g is h = H, H₂. The guided surface waveguide probe 200g includes a feed network 209 that couples an excitation source 212 such as an AC source, for example, to the charge terminais T, and T₂.

The charge terminais T-, and/or T₂ include a conductive mass that can hold an electrical charge, which may be sized to hold as much charge as practically possible. The charge terminal Ti has a self-capacitance Ci, and the charge terminal T₂ has a selfcapacitance C_2t which can be determined using, for example, Equation (24). By virtue ofthe placement of the charge terminal T! directly above the charge terminal T₂, a mutual capacitance C_M is created between the charge terminais Tj and T₂. Note that the charge terminais Tj and T₂ need not be identical, but each can hâve a separate size and shape, and can include different conducting materials. Ultimately, the field strength of a guided surface wave launched by a guided surface waveguide probe 200g is directly proportional to the quantity of charge on the terminal T_v The charge Q, is, in turn, proportional to the selfcapacitance C, associated with the charge terminal T_t since Qi = C,V, where V is the voltage imposed on the charge terminal Ti.

When properly adjusted to operate at a predefined operating frequency, the guided surface waveguide probe 200g generates a guided surface wave along the surface of the lossy conducting medium 203. The excitation source 212 can generate electrical energy at the predefined frequency that is applied to the guided surface waveguide probe 200g to excite the structure. When the electromagnetic fields generated by the guided surface waveguide probe 200g are substantially mode-matched with the lossy conducting medium 203, the electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle that results in little or no reflection. Thus, the surface waveguide probe 200g does not produce a radiated wave, but launches a guided surface traveling wave along the surface of a lossy conducting medium 203. The energy from the excitation source 212 can be transmitted as Zenneck surface currents to one or more receivers that are located within an effective transmission range of the guided surface waveguide probe 200g.

One can détermine asymptotes of the radial Zenneck surface current }_p(p) on the surface of the lossy conducting medium 203 to be Jflp) close-in and /₂(p) far-out, where

Close-in (p < λ/8): _W ~ A = and (90)

Far-out (p » Λ/8): /_P(p) ~ h = ^{z x} (91 ) where f is the conduction current feeding the charge Q_t on the first charge terminal Τ_Ί, and l₂ is the conduction current feeding the charge Q₂ on the second charge terminal T₂. The charge Qi on the upper charge terminal T, is determined by Qi - CiV_b where C, is the isolated capacitance of the charge terminal Τ_Ί. Note that there is a third component to/j set forth above given by (β^)/Ζ_ρ, which follows from the Leontovich boundary condition and is the radial current contribution in the lossy conducting medium 203 pumped by the quasistatic field of the elevated oscillating charge on the first charge terminal Qi. The quantity Ζ_ρ=/ωμ₀/^ >s the radial impédance of the lossy conducting medium, where y_e = (jwPiffi -ω²ρι£ι)^1/2·

The asymptotes representing the radial current close-in and far-out as set forth by Equations (90) and (91) are complex quantifies. According to various embodiments. a physical surface current J(p) is synthesized to match as close as possible the current asymptotes in magnitude and phase. That is to say close-in, |)(p)| is to be tangent to |Al, and far-out |J(p)| is to be tangent to |/₂|. Also, according to the various embodiments, the phase of J(p) should transition from the phase of A close-in to the phase of/₂ far-out.

In order to match the guided surface wave mode at the site of transmission to launch a guided surface wave, the phase of the surface current |/₂|far-out should differ from the phase of the surface current |AI close-in by the propagation phase corresponding to _e-y/?(P2-Pi) plus a constant of approximately 45 degrees or 225 degrees. This is because there are two roots for Vÿ, one near π/4 and one near 5π/4. The properly adjusted synthetic radial surface current is /„(M,0) ^Η^-ίΥΡ).<

Note that this is consistent with équation (17). By Maxwell's équations, such a J(p) surface current automatically créâtes fields that conform to

Η_φ=^ε-^Ιί22( p ₌ e^-lt2Z and(94)

E - (LL) _e-u₂z (_yyp).(95) ^z 4 \ωε₀/

Thus, the différence in phase between the surface current |/₂| far-out and the surface current |/₁1 close-in for the guided surface wave mode that is to be matched is due to the characteristics of the Hankel functions in Equations (93)-(95). which are consistent with Equations (1)-(3)- It is of significance to recognize that the fields expressed by Equations (1)-(6) and (17) and Equations (92)-(95) hâve the nature of a transmission line mode bound to a lossy interface, not radiation fields that are associated with groundwave propagation.

In order to obtain the appropriate voltage magnitudes and phases for a given design of a guided surface waveguide probe 200g at a given location, an itérative approach may be used. Specifically, analysis may be performed of a given excitation and configuration of a guided surface waveguide probe 200g taking into account the feed currents to the terminais T₁ and T₂, the charges on the charge terminais Ti and T₂₁ and their images in the lossy conducting medium 203 in order to determine the radial surface current density generated. This process may be performed iteratively until an optimal configuration and excitation for a given guided surface waveguide probe 200g is determined based on desired parameters. To aid in determining whether a given guided surface waveguide probe 200g is operating at an optimal level, a guided field strength curve 103 (FIG. 1) may be generated using Equations (1)-(12) based on values for the conductîvity of Région 1 (aj and the permittivity of Région 1 (ej at the location of the guided surface waveguide probe 200g. Such a guided field strength curve 103 can provide a benchmark for operation such that measured field strengths can be compared with the magnitudes indicated by the guided field strength curve 103 to détermine if optimal transmission has been achieved.

In order to arrive at an optimized condition, various parameters associated with the guided surface waveguide probe 200g may be adjusted. One parameter that may be vaned to adjust the guided surface waveguide probe 200g is the height of one or both of the charge terminais T, and/or T₂ relative to the surface of the lossy conducting medium 203. In addition, the distance or spacing between the charge terminais Ti and T₂ may also be adjusted. In doing so, one may minimize or otherwîse alter the mutual capacitance C_u or any bound capacitances between the charge terminais T and T₂ and the lossy conducting medium 203 as can be appreciated. The size of the respective charge terminais T, and/or T₂ can also be adjusted. By changing the size of the charge terminais T-, and/or T₂, one will alter the respective self-capacitances C₁ and/or C₂, and the mutual capacitance C_M as can be appreciated.

Still further, another parameter that can be adjusted is the feed network 209 associated with the guided surface waveguide probe 200g. This may be accomplished by adjusting the size of the inductive and/or capacitive réactances that make up the feed network 209. For example, where such inductive réactances comprise coils, the number of turns on such coils may be adjusted. Ultimately. the adjustments to the feed network 209 can be made to alter the electrical length of the feed network 209, thereby affecting the voltage magnitudes and phases on the charge terminais Τ_Ί and T₂.

Note that the itérations of transmission performed by making the various adjustments may be implemented by using computer models or by adjusting physical structures as can be appreciated. By making the above adjustments, one can create corresponding close-in surface current h and “far-out” surface current }₂ that approximate the same currents ](p) of the guided surface wave mode specified in Equations (90) and (91) set forth above. In doing so, the resulting electromagnetic fields would be substantially or approximately mode matched to a guided surface wave mode on the surface of the lossy conducting medium 203.

While not shown in the example of FIG. 16, operation of the guided surface waveguide probe 200g may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, a probe control system 230 shown in FIG. 12 can be used to control the feed network 209 and/or positioning and/or size of the charge terminais T, and/or T₂ to control the operation of the guided surface waveguide probe 200g. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 203 (e.g.. conductivity σ and relative permittivity c_r), variations in field strength and/or variations in loading of the guided surface waveguide probe 200g.

Referring now to FIG. 17, shown is an example of the guided surface waveguide probe 200g of FIG. 16, denoted herein as guided surface waveguide probe 200h. The guided surface waveguide probe 200h includes the charge terminais Tt and T₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203 (e.g., the Earth). The second medium 206 is above the lossy conducting medium 203. The charge terminal T, has a self-capacitance C₁₍ and the charge terminal T₂ has a self-capacitance C_z. During operation, charges Q, and Q₂ are imposed on the charge terminais T, and T₂, respectively, depending on the voltages applied to the charge terminais and T₂ at any given instant. A mutual capacitance C_M may exist between the charge terminais T, and T₂ depending on the distance therebetween. In addition, bound capacitances may exist between the respective charge terminais T, and T₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminais Ti and T₂ with respect to the lossy conducting medium 203.

The guided surface waveguide probe 200h includes a feed network 209 that comprises an inductive impédance comprising a coil L_1a having a pair of leads that are coupled to respective ones of the charge terminais Ti and T₂. In one embodiment, the coil

L_1a is specified to hâve an electrical length that is one-half (½) of the wavelength at the operating frequency of the guided surface waveguide probe 200h.

While the electrical length of the coil L_la is specified as approximately one-half (1/2) the wavelength at the operating frequency, it is understood that the coil Lu may be specified with an electrical length at other values. According to one embodiment, the fact that the coil Li_a has an electrical length of approximately one-half (1/2) the wavelength at the operating frequency provides for an advantage in that a maximum voltage différentiel is created on the charge terminais Τ_Ί and T₂. Nonetheless, the length or diameter of the coil L_1a may be increased or decreased when adjusting the guided surface waveguide probe 200h to obtain optimal excitation of a guided surface wave mode. Adjustment of the coil length may be provided by taps located at one or both ends of the coil. In other embodiments, it may be the case that the inductive impédance is specified to hâve an electrical length that is significantly less than or greater than one-half (1/2) the wavelength at the operating frequency of the guided surface waveguide probe 200h.

The excitation source 212 can be coupled to the feed network 209 by way of magnetic coupling. Specifically, the excitation source 212 is coupled to a coil L_P that is inductively coupled to the coil L_la. This may be done by link coupling, a tapped coil, a variable reactance, or other coupling approach as can be appreciated. To this end, the coil Lp acts as a primary, and the coil L_1a acts as a secondary as can be appreciated.

In order to adjust the guided surface waveguide probe 200h for the transmission of a desired guided surface wave, the heights of the respective charge terminais Ti and T₂ may p_e altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminais T and T₂ may be altered. In addition, the size of the coil L_1a may be altered by adding or eliminating turns or by changing some other dimension of the coil Li_a. The coil L_1a can also include one or more taps for adjusting the electrical length as shown in FIG. 17. The position of a tap connected to either charge terminal Τ_Ί or T₂ can also be adjusted.

Referring next to FIGS. 18A, 18B, 18C and 19, shown are examples of generalized receive circuits for using the surface-guided waves in wireless power delivery Systems. FIG. 18A depict a linear probe 303, and FIGS. 18B and 18C depict tuned resonators 306a and 306b, respectively. FIG. 19 is a magnetic coil 309 according to various embodiments of the present disclosure. According to various embodiments, each one of the linear probe 303, the tuned resonators 306a/b, and the magnetic coil 309 may be employed to receive power transmitted in the form of a guided surface wave on the surface of a lossy conducting medium 203 according to various embodiments. As mentioned above, in one embodiment the lossy conducting medium 203 comprises a terrestrial medium (or Earth).

With spécifie reference to FIG. 18A, the open-circuit terminal voltage at the output terminais 312 of the linear probe 303 dépends upon the effective height of the linear probe 303. To this end, the terminal point voltage may be calculated as

Vr = C’ E<nc · dl, (9θ) where E_inc is the strength of the incident electric field induced on the linear probe 303 m Volts per meter, dl is an element of intégration along the direction of the linear probe 303, and h_e is the effective height of the linear probe 303. An electrical load 315 is coupled to the output terminais 312 through an impédance matching network 318.

When the linear probe 303 is subjected to a guided surface wave as described above, a voltage is developed across the output terminais 312 that may be applied to the electrical load 315 through a conjugale impédance matching network 318 as the case may be. In order to facilîtate the flow of power to the electrical load 315, the electrical load 315 should be substantially impédance matched to the linear probe 303 as will be described below.

Referring to FIG. 18B, a ground current excited coil L_R possessing a phase delay equal to the wave tilt of the guided surface wave includes a charge terminal T_R that is elevated (or suspended) above the lossy conducting medium 203. The charge terminal T_R has a self-capacitance C_R. In addition, there may also be a bound capacitance (not shown) between the charge terminal T_R and the lossy conducting medium 203 depending on the height of the charge terminal T_R above the lossy conducting medium 203. The bound capacitance should preferably be minimized as much as is practicable. although this may not be entirely necessary in every instance.

The tuned resonator 306a also includes a receiver network comprising a coil L_R having a phase delay Φ. One end of the coil L_R is coupled to the charge terminal T_R, and the other end ofthe coil L_R is coupled to the lossy conducting medium 203. The receiver network can include a vertical supply line conductor that couples the coil L_R to the charge terminal T_R. To this end, the coil L_R (which may also be referred to as tuned resonator l__RC_R) comprises a series-adjusted resonator as the charge terminal C_R and the coil L_R are situated in sériés. The phase delay of the coil L_R can be adjusted by changing the size and/or height of the charge terminal T_R, and/or adjusting the size of the coil L_R so that the phase delay Φ of the structure Is made substantially equal to the angle of the wave tilt Ψ. The phase delay of the vertical supply line can also be adjusted by, e.g., changing length of the conductor.

For example, the reactance presented by the self-capacitance C_R is calculated as 1/)ω6_Λ. Note that the total capacitance of the tuned resonator 306a may also include capacitance between the charge terminal T_R and the lossy conducting medium 203, where the total capacitance of the tuned resonator 306a may be calculated from both the selfcapacitance C_R and any bound capacitance as can be appreciated. According to one embodiment, the charge terminal T_R may be raised to a height so as to substantially reduce or eliminate any bound capacitance. The existence of a bound capacitance may be determined from capacitance measurements between the charge terminal T_R and the lossy conducting medium 203 as previously discussed.

The inductive reactance presented by a discrete-element coil L_R may be calculated as ;ωί, where L is the lumped-element inductance of the coil l__R. If the coil l__R is a distributed element, its équivalent terminal-point inductive reactance may be determined by conventional approaches. To tune the tuned resonator 306a, one would make adjustments so that the phase delay is equal to the wave tilt for the purpose of mode-matching to the surface waveguide at the frequency of operation. Under this condition, the receiving structure may be considered to be “mode-matched” with the surface waveguide. A transformer link around the structure and/or an impédance matching network 324 may be inserted between the probe and the electrical load 327 in order to couple power to the load. Inserting the impédance matching network 324 between the probe terminais 321 and the electrical load 327 can effect a conjugate-match condition for maximum power transfer to the electrical load 327.

When placed in the presence of surface currents at the operating frequencies power will be delivered from the surface guided wave to the electrical load 327. To this end, an electrical load 327 may be coupled to the tuned resonator 306a by way of magnetic coupling, capacitive coupling, or conductive (direct tap) coupling. The éléments of the coupling network may be lumped components or distributed éléments as can be appreciated.

In the embodiment shown in FIG. 18B, magnetic coupling is employed where a coil L_s is positioned as a secondary relative to the coil L_R that acts as a transformer primary. The coil L_s may be link-coupled to the coil L_R by geometrically winding it around the same core structure and adjusting the coupled magnetic flux as can be appreciated. In addition, while the tuned resonator 306a comprises a serles-tuned resonator, a parallel-tuned resonator or even a distributed-element resonator of the appropriate phase delay may also be used.

While a receiving structure immersed in an electromagnetic field may couple energy from the field, it can be appreciated that polarization-matched structures work best by maximizing the coupling, and conventional rules for probe-coupling to waveguide modes should be observed. For example, a TE₂₀ (transverse electric mode) waveguide probe may be optimal for extracting energy from a conventional waveguide excited in the TE₂₀ mode. Similarly, in these cases, a mode-matched and phase-matched receiving structure can be optimized for coupling power from a surface-guided wave. The guided surface wave excited by a guided surface waveguide probe 200 on the surface ofthe lossy conducting medium

203 can be considered a waveguide mode of an open waveguide. Excluding waveguide losses, the source energy can be completely recovered. Useful receiving structures may be E-field coupled, H-field coupled, or surface-current excited.

The receiving structure can be adjusted to increase or maximize coupling with the guided surface wave based upon the local characteristics of the lossy conducting medium 203 in the vicinity of the receiving structure. To accomplish this, the phase delay (Φ) ofthe receiving structure can be adjusted to match the angle (Ψ) of the wave tilt of the surface traveling wave at the receiving structure. If configured appropriately, the receiving structure may then be tuned for résonance with respect to the perfectly conducting image ground plane at complex depth z = -d/2.

For example, consider a receiving structure comprising the tuned resonator 306a of FIG. 18B, including a coil L_R and a vertical supply line connected between the coil L_R and a charge terminal T_R. With the charge terminal T_R positioned at a defined height above the lossy conducting medium 203, the total phase delay Φ of the coil L_R and vertical supply line can be matched with the angle (Ψ) of the wave tilt at the location of the tuned resonator 306a. From Equation (22), it can be seen that the wave tilt asymptoticaily passes to where ε_Γ comprises the relative permittivity and σι is the conductivity of the lossy conducting medium 203 at the location of the receiving structure, ε₀ is the permittivity of free space, and ω = 2ττ/, where f is the frequency of excitation. Thus, the wave tilt angle (Ψ) can be determined from Equation (97).

The total phase delay (Φ = + θ_γ) of the tuned resonator 306a includes both the phase delay (0_C) through the coil L_R and the phase delay of the vertical supply line (θ_γ). The spatial phase delay along the conductor length l_w ofthe vertical supply line can be given by = P_wl_w, where p_w is the propagation phase constant for the vertical supply line conductor. The phase delay due to the coil (or helical delay line) is 9_C - β_ρΙ_Ε, with a physical length of l_c and a propagation factor of ₌ 2π ₌ 2Π (98)

Λ_ρ Ι//λο ’ where is the velocity factor on the structure, A_o is the wavelength at the supplied frequency, and A_p is the propagation wavelength resulting from the velocity factor V_f. One or both of the phase delays (0_C + 0_y) can be adjusted to match the phase delay Φ to the angle (Ψ) of the wave tilt. For example, a tap position may be adjusted on the coil L_R of FIG. 18B to adjust the coil phase delay (0J to match the total phase delay to the wave tilt angle (Φ _ qq p_or example, a portion of the coil can be bypassed by the tap connection as illustrated in FIG. 18B. The vertical supply line conductor can also be connected to the coil Lr via a tap, whose position on the coil may be adjusted to match the total phase delay to the angle of the wave tilt.

Once the phase delay (Φ) of the tuned resonator 306a has been adjusted, the impédance of the charge terminal T_R can then be adjusted to tune to résonance with respect to the perfectly conducting image ground plane at complex depth z = -d/2. This can be accomplished by adjusting the capacitance of the charge terminal T, without changing the traveling wave phase delays of the coil L_R and vertical supply line. In some embodiments, a lumped element tuning circuit can be included between the lossy conducting medium 203 and the coil L_R to allow for résonant tuning of the tuned resonator 306a with respect to the complex image plane as discussed above with respect to the guided surface waveguide probe 200. The adjustments are similar to those described with respect to FIGS. 9A-9C.

The impédance seen “looking down into the lossy conducting medium 203 to the complex image plane is given by:

z in = Λ in + jXin = ?₀ tanh(j/ï₀(d/2)), (99) where β_ο = ω/μ^~ο· For vertically polarized sources over the Earth, the depth of the complex image plane can be given by:

d/2 « , (¹⁰⁰) where is the permeability of the lossy conducting medium 203 and = ε,.ε₀.

At the base of the tuned resonator 306a, the impédance seen “looking up” into the receiving structure is Z_T = Z_hase as illustrated in FIG. 9A or Z_T = Z_tunlng as illustrated in FIG.

90. With a terminal impédance of:

= (¹⁰¹) ^R JùjCr where C_R is the self-capacitance of the charge terminal T_R, the impédance seen looking up” into the vertical supply line conductor of the tuned resonator 306a is given by.

Z_R+Z_wtanh(j^_tvh_w) _ „ Z_n+Z_W tanhQfly) (102) ^Z2 ~ Zw+Zn tanh(jfiwhw) ^W ZW+Z_R tanh^) ’ and the impédance seen “looking up into the coil l__R of the tuned resonator 306a is given by:

Z_z+Z_R tanh(y/îpH) _ Z₂+Z_R tanhQ'gç) (103)

Zbase “ Rbase T jXbase ~ _Zr+2₂ tanh(j/?_pH) ^c Zr+^zz tanh(/0_c)

By matching the reactive component (X_in) seen “looking down” into the lossy conducting medium 203 with the reactive component (Xbase) ^seen “looking up” into the tuned resonator 306a, the coupling into the guided surface waveguide mode may be maximized.

Where a lumped element tank circuit is included at the base of the tuned resonator 306a, the self-resonant frequency of the tank circuit can be tuned to add positive or négative impédance to bring the tuned resonator 306b into standing wave résonance by matching the reactive component (X_(n) seen “looking down” into the lossy conducting medium 203 with the reactive component seen “looking up” into the lumped element tank circuit.

Referring next to FIG. 18C, shown is an example of a tuned resonator 306b that does not include a charge terminal T_R at the top of the receiving structure. In this embodiment, the tuned resonator 306b does not include a vertical supply line coupled between the coil L_R and the charge terminal T_R. Thus, the total phase delay (Φ) of the tuned resonator 306b includes only the phase delay (0_C) through the coil L_R. As with the tuned resonator 306a of FIG. 18B, the coil phase delay 0_ccan be adjusted to match the angle (Ψ) of the wave tilt determined from Equation (97), which results in Φ = Ψ. While power extraction is possible with the receiving structure coupled into the surface waveguide mode, it is difficult to adjust the receiving structure to maximize coupling with the guided surface wave without the variable reactive load provided by the charge terminal T_R. Including a lumped element tank circuit at the base of the tuned resonator 306b provides a convenient way to bring the tuned resonator 306b into standing wave résonance with respect to the complex image plane.

Referring to FIG. 18D, shown is a flow chart 180 illustrating an example of adjusting a receiving structure to substantially mode-match to a guided surface waveguide mode on the surface of the lossy conducting medium 203. Beginning with 181, if the receiving structure includes a charge terminal T_R (e.g., of the tuned resonator 306a of FIG. 18B), then the charge terminal T_R is positioned at a defined height above a lossy conducting medium 203 at 184. As the surface guided wave has been established by a guided surface waveguide probe 200, the physical height (h_p) of the charge terminal T_R may be below that of the effective height. The physical height may be selected to reduce or minimize the bound charge on the charge terminal T_R (e.g., four times the spherical diameter of the charge terminal). If the receiving structure does not include a charge terminal T_R (e.g., of the tuned resonator 306b of FIG. 18C), then the flow proceeds to 187.

At 187, the electrical phase delay Φ of the receiving structure is matched to the complex wave tilt angle Ψ defined by the local characteristics of the lossy conducting 20 medium 203. The phase delay (0J of the helical coil and/or the phase delay (Θ_γ) of the vertical supply line can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (W). The angle (Ψ) of the wave tilt can be determined from Equation (86). The electrical phase delay Φ can then be matched to the angle of the wave tilt. For example, the electrical phase delay ¢ = + can be adjusted by varying the geometrical parameters of the coil L_R and/or the length (or height) of the vertical supply line conductor.

Next at 190, the resonator impédance can be tuned via the load impédance of the charge terminal T_R and/or the impédance of a lumped element tank circuit to resonate the équivalent image plane model of the tuned resonator 306a. The depth (d/2) of the conducting image ground plane 139 (FIGS. 9A-9C) below the receiving structure can be determined using Equation (100) and the values of the lossy conducting medium 203 (e.g., the Earth) at the receiving structure, which can be locally measured. Using that complex depth, the phase shift (0_d) between the image ground plane 139 and the physical boundary 136 (FIGS. 9A-9C) of the lossy conducting medium 203 can be determined using if — β₀ d/2. The impédance (Z_in) as seen “looking down into the lossy conducting medium 203 can then be determined using Equation (99). This résonance relationship can be considered to maximize coupling with the guided surface waves.

Based upon the adjusted parameters of the coil L_R and the length of the vertical supply line conductor, the velocity factor, phase delay, and impédance of the coil L_R and vertical supply line can be determined. In addition, the self-capacitance (C_R) of the charge terminal T_R can be determined using, e.g., Equation (24). The propagation factor (/J_p) of the coil L_r can be determined using Equation (98), and the propagation phase constant (f_w) for the vertical supply line can be determined using Equation (49). Using the self-capacitance and the determined values of the coil L_R and vertical supply line, the impédance (Z_hase) of the tuned resonator 306 as seen “looking up” into the coil L_R can be determined using Equations (101), (102), and (103).

The équivalent image plane model of FIGS. 9A-9C also apply to the tuned resonator 306a of FIG. 18B. The tuned resonator 306a can be tuned to résonance with respect to the complex image plane by adjusting the load impédance Z_R of the charge terminal T_R such that the reactance component X_base of Z_base cancels out the reactance component of X_in of Z_in, or X_base + Χ_ίη = θ· Where the tuned resonator 306 of FIGS. 18B and 18C includes a lumped element tank circuit, the self-resonant frequency of the parallel circuit can be adjusted such that the reactance component Ztuntng cancels out the reactance component of X_ln of Z,„. or = 0. Thus, the Impédance at the physical boundary

136 (FIG. 9A) “looking up” into the coil of the tuned resonator 306 is the conjugale of the impédance at the physical boundary 136 “looking down into the lossy conducting medium 203. The load impédance Z_R can be adjusted by varying the capacitance (C_R) of the charge terminal T_B without changing the electrical phase delay Φ = e_c + θ_γ seen by the charge terminal T_R. The impédance of the lumped element tank circuit can be adjusted by varying the self-resonant frequency (f_p) as described with respect to FIG. 9D. An itérative approach may be taken to tune the resonator impédance for résonance of the équivalent image plane model with respect to the conducting image ground plane 139. In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., Earth) can be improved and/or maximized.

Referring to FIG. 19, the magnetic coil 309 comprises a receive circuit that is coupled through an impédance matching network 333 to an electrical load 336. In order to facilitate réception and/or extraction of electrical power from a guided surface wave, the magnetic coil 309 may be positioned so that the magnetic flux of the guided surface wave, Η_ψ, passes through the magnetic coil 309, thereby inducing a current in the magnetic coil 309 and producing a terminal point voltage at its output terminais 330. The magnetic flux of the guided surface wave coupled to a single turn coil is expressed by

J⁷ = fiAcsl-^lrFoiï (¹⁰⁴>

where F is the coupled magnetic flux, p_r is the effective relative permeability of the core of the magnetic coil 309, μ₀ is the permeability of free space, H is the incident magnetic field strength vector, n is a unit vector normal to the cross-sectional area of the turns, and A_cs is the area enclosed by each loop. For an N-turn magnetic coil 309 oriented for maximum coupling to an incident magnetic field that is uniform over the cross-sectional area of the magnetic coil 309, the open-circuit induced voltage appearing at the output terminais 330 of the magnetic coil 309 is

V = -N—~ -j(op_rp_QNHA_cs, (¹⁰⁵>

at where the variables are defined above. The magnetic coil 309 may be tuned to the guided surface wave frequency either as a distributed resonator or with an extemal capacitor across its output terminais 330, as the case may be, and then impedance-matched to an external electrical load 336 through a conjugate impédance matching network 333.

Assuming that the resulting circuit presented by the magnetic coil 309 and the electrical load 336 are properly adjusted and conjugate impédance matched, via impédance matching network 333, then the current induced in the magnetic coil 309 may be employed to optimally power the electrical load 336. The receive circuit presented by the magnetic coil 309 provides an advantage in that it does not hâve to be physically connected to the ground.

With reference to FIGS. 18A, 18B, 18C and 19, the receive circuits presented by the linear probe 303, the tuned resonator 306, and the magnetic coil 309 each facilitate receiving electrical power transmitted from any one of the embodiments of guided surface waveguide probes 200 described above. To this end, the energy received may be used to supply power to an electrical load 315/327/336 via a conjugate matching network as can be appreciated. This contraste with the signais that may be received in a receiver that were transmitted m the form of a radiated electromagnetic field. Such signais hâve very low available power, and receivers of such signais do not load the transmitters.

It is also characteristic of the present guided surface waves generated using the guided surface waveguide probes 200 described above that the receive circuits presented by the linear probe 303, the tuned resonator 306, and the magnetic coil 309 will load the excitation source 212 (e.g., FIGS. 3, 12 and 16) that is applied to the guided surface waveguide probe 200, thereby generating the guided surface wave to which such receive circuits are subjected. This reflects the fact that the guided surface wave generated by a given guided surface waveguide probe 200 described above comprises a transmission line mode. By way of contrast, a power source that drives a radiating antenna that generates a radiated electromagnetic wave is not loaded by the receivers, regardless of the number of receivers employed.

Thus, together one or more guided surface waveguide probes 200 and one or more receive circuits in the form of the linear probe 303, the tuned resonator 306a/b, and/or the magnetic coil 309 can make up a wireless distribution system. Given that the distance of transmission of a guided surface wave using a guided surface waveguide probe 200 as set forth above dépends upon the frequency, it is possible that wireless power distribution can be achieved across wide areas and even globally.

The conventional wireless-power transmission/distribution Systems extensively investigated today include “energy harvesting from radiation fields and also sensor coupling to inductive or reactive near-fields. In contrast, the présent wireless-power system does not waste power in the form of radiation which, if not intercepted, is lost forever. Nor is the presently disclosed wireless-power system limited to extremely short ranges as with conventional mutual-reactance coupled near-field Systems. The wireless-power system disclosed herein probe-couples to the novel surface-guided transmission line mode, which is équivalent to delivering power to a load by a wave-guide or a load directly wired to the distant power generator. Not counting the power required to maintain transmission field strength plus that dissipated in the surface waveguide, which at extremely low frequencies is insignificant relative to the transmission losses in conventional high-tension power hnes at 60 Hz, ail of the generator power goes only to the desired electrical load. When the electrical load demand is terminated, the source power génération is relatively idle.

FIG. 20 illustrâtes a cross sectional view of an example guided surface waveguide probe site 2100 including a boundary or propagation interface 2102 between the lossy conducting medium 203 and the second medium 206, and a probe 2110. The illustration of the guided surface waveguide probe site 2100 is provided as a représentative example and is not drawn to scale. Other probe sites consistent with the concepts described herein can include additional probes similar to the probe 2110, various guided surface wave receive structures, and other equipment.

The probe 2110 may be embodied as any of the guided surface waveguide probes 200a, 200b, 200c, 200d, 200e, 200f, 200g or 200h described herein, or variations thereof.

At the guided surface waveguide probe site 2100, the probe 2110 is configured to launch a guided surface wave (or waves) along the interface 2102 between the lossy conducting medium 203 and the second medium 206. Consistent with the concepts described above, the probe 2110 is configured to provide a phase delay that matches a wave tilt angle associated with a complex Brewster angle of incidence associated with the lossy conducting medium 203 in the vicinity of the probe 2110. Thus, the probe 2110 is configured to launch a guided surface wave along the propagation interface 2102 by generating an electric field having a predetermined complex wave tilt angle at or beyond a crossover distance R_x from the guided surface waveguide probe. To achieve the desired complex wave tilt angle at the crossover distance R_x. the probe 2110 can be embodied having certain structural and electrical characteristics, consistent with the description provided above. In one embodiment, for example, the probe 2110 can include a charge terminal elevated at a height over a lossy conducting medium 203, a ground stake or grounding system, and a feed network coupled between the charge terminal and the ground stake or grounding system.

In FIG. 20, the lossy conducting medium 203 may include any lossy conducting medium, such as the Earth, for example, and the second medium 206 may include the atmosphère of the Earth. It should be appreciated that the conductivity σ! and permittivity of the lossy conducting medium 203 at the site 2100 dépends upon various factors, such as the géographie location of the site 2100, the surrounding land geography (e.g., hills, mountains, rock formations, lakes, rivers, etc.) atthe site 2100, the surrounding phytogeography (e.g., trees, plants, etc.) at the site 2100, the current température of at the site 2100, the relative humidity at the site 2100, the water content of the lossy conducting medium 203 at the site 2100, etc. Generally, conductivity is a measure of ability to conduct electricity, and permittivity is a measure of résistance encountered when forming an electric field in a medium.

Because the composition of matter in the lithosphère (e.g., relative outermost surface) of the Earth varies by géographie location, the conductivity and permittivity of the Earth’s surface varies by géographie location. In the context of variations in conductivity on the surface of the Earth, a map of the estimated effective ground conductivity in the United States may be found at http://www.fcc.gov/encyclopedia/m3-map-effective-groundconductivity-united-states-wall-sized-map-am-broadcast-stations. The information provided in the map may be used to predict the propagation of amplitude modulated (AM) signais across the United States. For example, a higher ground conductivity indicates better AM propagation characteristics. The map shows that the conductivity in the United States ranges between about 0.5 and 30 millimhos per meter. Thus, in FIG. 20, it should be appreciated that the conductivity of the lossy conducting medium 203 may vary, as the site 2100 may vary in location across the Earth.

The permittivity of a région is related to the amount of electric field generated per unit charge in that région. A higher electric flux exists in a région having high permittivity because of at least polarization effects. Likewise, a lower electric flux exists in a région having low permittivity. Permittivity is measured in farads per meter. The response of various materials to an electromagnetic field may dépend, at least in part, on the frequency of the field. Thus, permittivity may be considered a complex function of the angular frequency of an applied field.

The response to static fields is described as the low-frequency or static limit of permittivity. While static permittivity may be a fair approximation for alternating fields of low frequencies, a measurable phase différence or shift may emerge for fields of higher frequencies. The frequency at which the phase différence or shift occurs may dépend, at least in part, on the température the medium. Thus, in FIG. 20, it should be appreciated that the permittivity of the lossy conducting medium 203 may vary over time based on the moisture content in or température of the lossy conducting medium 203 at the site 2100, for example.

As noted above, the probe 2110 is configured to launch a guided surface wave along the propagation interface 2102 by generating an electric field having a predetermined complex wave tilt angle at or beyond the crossover distance R_x from the guided surface waveguide probe 2110. Thus, the probe 2110 may be designed to launch a guided surface wave for the nominal conductivity and permittivity conditions of the lossy conducting medium 203 at the site 2100. In other words, the probe 2110 may be designed to launch a guided surface wave for an average annual température, relative moisture content, etc., of the Earth at the site 2100. In this case, because the nominal conductivity σ₁ and permittivity _£1 conditions of the Earth at the site 2100 may vary over time (i.e., vary from the nominal or annual average values) based at least upon changes in whether, for example, the embodiments described herein include various ways to preserve or maintain the nominal (or other désirable) conductivity and permittivity ε_τ conditions in the lossy conducting medium 203 at the site 2100.

In one embodiment, if the nominal conductivity σ. and permittivity _E1 conditions of the lossy conducting medium 203 at the site 2100 are, at least to some extent, undesirable or unsuitable, at least a portion of the lossy conducting medium 203 may be prepared to more efficiently or effectively launch the guided surface wave. The prepared portion of the lossy conducting medium 203 may extend to or beyond the propagation interface between the lossy conducting medium 203 and the second medium 206 at the crossover distance R_x.

FIG 21A illustrâtes a cross sectional view of another example of a guided surface waveguide probe site 2200 in which a portion 2220 of the lossy conducting medium 203 is prepared to more efficiently launch a guided surface wave according to various embodiments of the present disclosure. In FIG. 21 A, the portion 2220 of the lossy conducting medium 203 has been prepared, at least to some extent, to more efficiently launch a guided surface wave from the probe 2110. Here, it is noted that efficiency in launching a guided surface wave may be related to a ratio of the amount of energy provided to the probe 2110 and the level of energy in the guided surface wave launched along the propagation interface 2102. Efficiency in launching a guided surface wave may additionally or alternatively be related to the propensity for a guided surface wave to launch along the propagation interface 2102. With regard to the manner in which the portion 2220 of the lossy conducting medium 203 is prepared, the portion 2220 of the Earth may be excavated and mixed with various types of materials or replaced with a different material, as described in further detail below with reference to FIG. 22.

With reference to FIG. 21 A, it can be seen that the portion 2220 of the lossy conducting medium 203 extends beyond the crossover distance R_x along the propagation 5 interface 2102. It is noted that, among embodiments, the portion 2220 of the lossy conducting medium 203 may extend along the propagation interface 2102 at least to or, in preferred embodiments, at some distance beyond the crossover distance R_x. In some embodiments, however, the portion 2220 of the lossy conducting medium 203 may not fully extend to the crossover distance R_x. Further, in various embodiments, the portion 2220 may 10 extend down into the lossy conducting medium 203 at least to the depth “A” of the complex image of the guided surface waveguide probe 2110. The depth, size or extent of the complex image can dépend in part upon the height “h of the probe 2110 as described herein. As has been described, the complex image is located at a complex image depth. Thus, the depth A can correspond to the real portion of the complex image depth. In other 15 embodiments, the portion 2220 of the lossy conducting medium 203 may extend down to other, shallower depths. For example, the depth “A can correspond to the depth (or the real portion of the complex depth) of the complex image plane 130 illustrated in FIG. 3

FIG. 21B illustrâtes a top down view of the guided surface waveguide probe site 2200 in FIG. 21A according to various embodiments of the present disclosure. In FIG. 21 B, 20 it can be appreciated that the portion 2220 of the lossy conducting medium 203 includes an area that extends across the propagation interface 2102, defined circularly or radially beyond the crossover distance R_x, measured from about the center of the guided surface waveguide probe 2110. Stated another way, the circulât or radial distance to the edge of the portion 2220 is greater than the crossover distance. As should be appreciated, however, that the 25 portion 2220 of the lossy conducting medium 203 need not be circulât. Instead, the portion 2220 of the lossy conducting medium 203 may be formed in various sizes and shapes, preferably extending in any embodiment beyond the crossover distance R_x from the guided surface waveguide probe 2110. For example, the portion 2220 can comprise a ring encircling the guided surface waveguide probe 2110 that has an inner radius that is less than the crossover distance and an outer radius that is greater than the crossover distance.

FIG. 22 illustrâtes a stage in the préparation of the portion 2200 of the lossy conducting medium 203 in FIG. 21A according to various embodiments of the present disclosure. The portion 2200 of the lossy conducting medium 203 can be excavated in any suitable manner. The matter 2310 excavated from the portion 2200 can be mixed with a composition 2320 of other matter, and the aggregate composition provided back as the portion 2200 of the lossy conducting medium 203. The composition 2320 can include various compositions of matter among embodiments. For example, the composition 2320 may include a predetermined amount of sait, gypsum, sand, or gravel, for example, among other compositions of matter. According to aspects of the embodiments, the composition 2320 may be relied upon to vary the nominal conductivity and permittivity conditions of the Earth in the lossy conducting medium 203. In some embodiments, the excavated matter 2310 may be replaced entirely by (rather than being mixed with) the composition 2320. In various embodiments, the excavated matter 2310 can be replaced with another material or fluid (e.g., sea water or other liquid composition) that has different characteristics that the lossy conducting medium 203, The dimensions of the excavation can dépend upon the frequency of the guided surface wave and/or the characteristics of the lossy conducting medium 203 and/or replacement material or fluid.

The depth of the excavation, and thus the depth of the prepared région 2220, can vary with different embodiments of the present disclosure. For example, the depth may dépend on the excitation frequency and the characteristics of the lossy conducting medium 203 and/or the material or fluid added to prepared région 2220. In some embodiments, the depth of the excavation (or prepared région 2220) can extend a portion of the distance to the complex image plane. For instance, the excavation may be shallow or much less than the depth ofthe complex image plane. In other embodiments, the depth of the excavation (or prepared région 2220) can extend to the complex image plane. And in some embodiments, the depth of the excavation (or prepared région 2220) can extend beyond the complex image plane. The depth of the prepared région 2220, and the characteristics of the material or fluid it contains, can hâve a varying effect on whether a guided surface wave is launched in the respective direction intended.

Préparation of the boundary or propagation interface 2102 around the guided surface waveguide probes 2110 can also be used to direct the guided surface wave (or waves) launched by the probe 2110. The probe 2110 (e.g., guided surface waveguide probes 200a, 200b, 200c, 200d, 200e, 200f, 200g or 200h) can be configured to provide a wave tilt angle (Ψ) associated with a complex Brewster angle of incidence ) associated with e.g., the conductivity σ_ρ and permittivity ε_ρ of a prepared région 2220 or the lossy conducting medium 203 in the vicinity of the probe 2110. By preparîng a portion of the area surrounding the probe 2110, it is possible for the electric fields to produce a predetermined complex wave tilt angle at a crossover distance R_x that corresponds to the excitation frequency and the characteristics of the prepared région 2220 while not corresponding to the characteristics of the lossy conducting medium 203. In this way, the wave tilt is substantially mode-matched with the prepared région 2220 to launch a guided surface wave away from the probe 2110, while the wave tilt is not mode-matched with the lossy conducting medium 203. In alternative embodiments, the electric fields can produce a predetermined complex wave tilt angle at a crossover distance R_x that couples with the lossy conducting medium 203 while not coupling with the prepared région 2220. To achieve the desired complex wave tilt angle at the crossover distance R_x, the probe 2110 can be embodied having certain structural and electrical characteristics, consistent with the description provided above. The appropriate complex wave tilt angle allows the guided surface wave to be launched m a direction dictated by the prepared région or régions 2220. The effectiveness of the directional coupling will be affected by the différence in the characteristics (e.g., conductivity and permittivity as discussed above) between the prepared région 2220 and the unprepared région 2230, or between differently prepared régions.

Referring to FIGS. 23A-23D, shown are top down views illustrating examples of the guided surface waveguide probe sites 2200 with portions of the area surrounding the probe 2110 being prepared. In FIG. 23A, a prepared région 2220 is in the form of a wedge that extends radially outward from the probe 2110. By exciting electric fieids that produce a complex wave tilt angle at the crossover distance R_x that matches the characteristics of the prepared région 2220, the incident electric field can be completely coupled into a guided surface waveguide mode in the direction (as indicated by arrows 2300) of the prepared région 2220. Guided surface waves that are launched by a probe 2110 will radially propagate away from the probe 2110 as illustrated by the arrows 2300. If the characteristics of the remaining area surrounding the probe 2110 are sufficiently different from those of the prepared région 2220, the guided surface wave propagates in the direction or range of directions defined by the prepared région 2220 (as indicated by the arrows 2300).

The range in which the guided surface waves can propagate along the surface of the lossy conducting medium 203 can be determined by the shape of the prepared région 2220. Since the incident electric fieids provide a wave tilt angle associated with the complex Brewster angle of incidence associated with the prepared région 2220, they can completely couple into the guided surface waveguide mode around the prepared région 2220 and launch guided surface waves in the radial directions defined by the prepared région 2220. Because the incident electric field(s) do not provide a wave tilt angle associated with the complex Brewster angle of incidence associated with an unprepared région 2230 of the lossy conducting medium 203, they do not couple (or only partially couple) into the guided surface waveguide mode ofthe lossy conducting medium 203 and thus do not effectively launch guided surface waves in the radial directions defined by the unprepared région 2230.

Appropriate excitation ofthe probe 2110 can produce fieids that are substantially mode-matched to a guided surface waveguide mode on the surface of the prepared région 2220, resulting in the launching of a guided surface wave along the surface of the propagation interface in a radial direction as illustrated by the arrows 2300. The degree of coupling may be verified by measuring field strength along a radial extending from the guided surface waveguide probe 2110. The degree of coupling into the guided surface waveguide mode may be expressed based upon the ratio of the measured field strength to a baseline field strength, which can be calculated based upon the transmission frequency and the characteristics of the prepared région 2220. For example, the wave tilt may be considered effectively coupled when the measured field strength is greater than or equal to 90%, 95%, 97%, 98%, or 99% of the baseline field strength or other defined threshold. Similarly, the wave tilt may be considered to be negligible or effectively uncoupled when the measured filed strength is less than or equal to 5%, 4%, 3%, 2%, or 1% of the baseline field strength or other defined threshold. Depending on the characteristics of the lossy conducting medium 203 and prepared régions 2220, the wave tilt may be partially coupled when the measured field strength faits between the two defined thresholds. Other methods for determining the degree of coupling are also possible. For example, in some embodiments, the degree of coupling may be based upon the energy transferred to a load via the guided surface wave based upon the excitation energy of the probe 2110. Defined thresholds as described above can be used to specify the degree of coupling.

The characteristics of the lossy conducting medium 203 and the prepared région 2220 can affect the directionality of the guided surface waves being launched by the probe 2110. The larger the différences between the conductivity σ_ρ and permittivity ε_ρ of the prepared région 2220 and the conductivity σ_χ and permittivity of the unprepared région 2230 of the lossy conducting medium 203, the larger the différence in degree of coupling between the prepared and unprepared régions 2220 and 2230. By reducing the degree of coupling of the wave tilt in the unprepared région 2230 relative to the prepared région 2220, the better the direction of the launched guided surface waves can be controlled.

While FIGS. 23A and 23B illustrate two examples of site préparation for controlling the direction of the guided surface waves generated by a probe 2110, other angular segments and/or geometries of the prepared région 2220 may be utilized. By increasmg or decreasing the angle over which the prepared région 2220 extends around the probe 2110, the direction of the guided surface wave (as îndicated by arrows 2300) can be broadened as illustrated by FIG. 23B, or can be narrowed (or further focused). For example, a radially distributed prepared région 2220 as illustrated in FIGS. 23A and 23B can hâve an angular distribution that ranges from a few degrees (e.g., 5-10 degrees wide) to 180 degrees (as shown in FIG. 23B) or more. In some implémentations, such as the example of FIG. 23C, the prepared région 2220 can substantially surround the probe 2110, with a small area that is unprepared région 2230 to prevent or reduce propagation of the guided surface wave in the direction (as îndicated by arrows 2300) of the unprepared région 2230.

The shape of the prepared région 2220 can also be varied to control the direction of the coupled guided surface wave. For example, the prepared région 2220 can hâve a rectangular or other géométrie shape that extends outward from the probe 2110 to direct the launched guided surface waves. In addition, the number of prepared régions 2220 can be one or more. For example, two or more angular segments can be prepared to direct the guided surface waves in corresponding radial directions (arrows 2300), as shown in FIG. 23D. Other variations in geometry and number of prepared régions 2220 are possible, as can be understood.

Referring next to FIGS. 24A and 24B, shown are top down views illustrating examples of the guided surface waveguide probe sites 2200 with portions of the area surrounding the probe 2110 being prepared. In these examples, the probe 2110 is configured to generate incident electric field(s) that provide a wave tilt angle associated with the complex Brewster angle of incidence associated with the lossy conducting medium 203. The prepared régions 2220 can be constructed with a conductivity σ_ρ and permittivity ε_ρ such that the wave tilt angle associated with the complex Brewster angle of incidence associated with the lossy conducting medium 203 does not effectively couple with the prepared région 2220. Thus, the angular range in which the guided surface waves can propagate along the surface of the lossy conducting medium 203 can be determined by the shape of the unprepared région 2230, as illustrated by the arrows 2300 in FIGS. 24A and 24B.

Since the incident electric fields provide a wave tilt angle associated with the complex Brewster angle of incidence associated with the unprepared région 2230 of the lossy conducting medium 203, they can completely couple into the guided surface waveguide mode around the unprepared région 2230 and launch guided surface waves in the radial directions (as indicated by arrows 2300) defined by the unprepared région 2230. Because the incident electric fields do not provide a wave tilt angle associated with the complex Brewster angle of incidence associated with the prepared région 2220, they do not couple (or only partially couple) into the guided surface waveguide mode of the prepared région 2220 and thus may not effectively launch guided surface waves in the radial directions defined by the prepared région 2230. For example, the prepared région 2220 can comprise an excavated area filled with water or sea water which has a conductivity σ_ρ and permittivity _£p that is sufficiently different from the conductivity and permittivity of the lossy conducting medium 203 to allow the guided surface waves to be directed by the lossy conducting medium 203.

Appropriate excitation of the probe 2110 can produce fields that are substantially mode-matched to a guided surface waveguide mode on the surface of the unprepared région 2230, resulting in the launching of a guided surface wave along the surface of the propagation interface in a radial direction as illustrated by the arrows 2300. The degree of coupling may be verified by measuring field strength along a radial extending from the guided surface waveguide probe 2110. The degree of coupling into the guided surface waveguide mode may be expressed based upon the ratio of the measured field strength to a baseline field strength, which can be calculated based upon the transmission frequency and the characteristics of the unprepared région 2230. For example, the wave tilt may be considered effectively coupled when the measured field strength is greater than or equal to 90%, 95%, 97%, 98%, or 99% of the baseline field strength or other defined threshold. Similarly, the wave tilt may be considered to be negligible or effectively uncoupled when the measured filed strength is less than or equal to 5%, 4%, 3%, 2%, or 1% of the baseline field strength or other defined threshold. Depending on the characteristics of the lossy conducting medium 203 and prepared régions 2220, the wave tilt may be partially coupled when the measured field strength falls between the two defined thresholds. Other methods for determining the degree of coupling are also possible. For example, m some embodiments, the degree of coupling may be based upon the energy transferred to a load via the guided surface wave based upon the excitation energy of the probe 2110. Defined thresholds as described above can be used to specify the degree of coupling.

The characteristics of the lossy conducting medium 203 and the prepared région 2220 can affect the directionality of the guided surface waves being launched by the probe 10 2110. The larger the différences between the conductivity σ_ρ and permittivity ε_ρ of the prepared région 2220 and the conductivity σ_χ and permittivity £_x of the unprepared région 2230 of the lossy conducting medium 203, the larger the différence in degree of coupling between the prepared and unprepared régions 2220 and 2230. By reducing the degree of coupling of the wave tilt in the prepared région 2220 relative to the unprepared région 2230, 15 the better the direction of the launched guided surface waves can be controlled.

While FIGS. 24A and 24B illustrate two examples of site préparation for controlling the direction of the guided surface waves generated by a probe 2110, other angular segments and/or geometries of the prepared région 2220 may be utilized. By increasing or decreasing the angle over which the prepared région 2220 extends around the probe 2110, 20 the direction of the guided surface wave can be broadened as illustrated by FIG. 24B, or can be narrowed (or further focused). For example, a radially distributed unprepared région 2230 can hâve an angular distribution that ranges from a few degrees (e.g., 5-10 degrees wide) to 180 degrees (see, e.g., the prepared région 2220 of FIG. 23A) or more. In some implémentations, the unprepared région 2230 can substantially surround the probe 2110, 25 with a small area that is a prepared région 2220 to prevent or reduce propagation of the guided surface wave in the direction of the prepared région 2220.

The shape of the unprepared région 2230 can also be varied to control the direction of the coupled guided surface wave (arrows 2300). For example, the unprepared région 2230 can hâve a rectangular or other géométrie shape that extends outward from the probe 2110 to direct the launched guided surface waves. In addition, the number of unprepared régions 2230 can be one or more. For example, two or more angular segments can be prepared to direct the guided surface waves in radial directions (as indicated by arrows 2300) corresponding to the unprepared régions 2230. Other variations in geometry and number of unprepared régions 2230 are possible, as can be understood. It may be possible to enhance the directionality of the guided surface waves by extending the prepared régions 2220 beyond the crossover distance.

In some embodiments, the probe 2110 can be configured to launch different guided surface waves in different directions (as indicated by arrows 2300) based upon the prepared régions 2220 and the unprepared régions 2230. For example, in the examples of FIGS. 23B and 24B, the probe 2110 can be configured to launch guided surface waves at a first frequency in the directions defined by the prepared région 2220 and to launch guided surface waves at a second frequency in the directions defined by the unprepared région 2230. In this case, the probe 2110 can be adjusted to excite electric fields at the first frequency such that they produce a complex wave tilt angle at the crossover distance R_x that matches the characteristics of the prepared région 2220. The incident electric field can then be completely coupled into a guided surface waveguide mode in the direction of the prepared région 2220, while not coupling (or only partially coupling) with the unprepared région 2230. The probe 2110 can also be adjusted to excite electric fields at the second frequency such that they produce a complex wave tilt angle at the crossover distance R_x that matches the characteristics of the unprepared région 2230 and completely couple into a guided surface waveguide mode in the direction of the unprepared région 2230, while not coupling (or only partially coupling) with the prepared région 2220. In some embodiments, there may be any number of prepared régions 2220 only limited by spacing around the probe 2110 that can be prepared such that each of the régions has different characteristics that correspond to different guided surface waveguide modes. In this way, a plurality of different guided surface waves can be launched in different directions by a single probe 2110. For example, FIG. 23D illustrâtes an example with two prepared régions 2220, which can be prepared with different characteristics associated with different operational frequencies. The unprepared région 2230 can also allow for coupling at a third operational frequency.

Referring next to FIGS. 25A and 25B, shown are examples of site spécification for controlling the direction of the guided surface waves (arrows 2300) generated by a probe 2110 utilizïng géographie features of the landscape. In the examples of FIGS. 25A and 25B, the lossy conducting medium 203 is a terrestrial medium 2503 (e.g., earth) which has a border that is defined by a body of water 2506 such as, e.g., fresh water or sea water or other treated water. In the example of FIG. 25A, a probe 2110 is positioned on a point or peninsula that is surrounded by water 2506. The water 2506 can act in the same way as the prepared régions 2220 shown in FIG. 24A. Since the incident electric fields generated by the probe 2110 provide a wave tilt angle associated with the complex Brewster angle of incidence of the terrestrial medium 2503, they can completely couple into the guided surface waveguide mode along the peninsula and launch guided surface waves in the radial directions defined by the surrounding water 2506 (as indicated by arrows 2300). By positioning the probe 2110 on the point or peninsula, the guided surface waves that are launched are directed based upon the geography of the point or peninsula.

In the example of FIG. 25B, the probe 2110 is positioned on a coast line 2500 along the water 2506. The water 2506 can act in the same way as the prepared régions 2220 shown in FIG. 24B. The wave tilt angle produced by the electric fields of the probe 2110 allow guided surface waves to be launched in radial directions defined by the shoreline 2500 (as indicated by arrows 2300). While the examples of FIGS. 25A and 25B disclose directmg guided surface waves along the surface of the terrestrial medium 2503, the probe 2110 can be configured to couple into the guided surface waveguide mode of the water 2506 to launch guided surface waves through the water 2506, while not allowing the coupling with the terrestrial medium 2503. For example, the guided surface waveguide probe 2110 in FIG. 25B can also be adjusted to excite electric fields that produce a complex wave tilt angle at the crossover distance R_x that matches the characteristics of the water 2506 and completely couple into a guided surface waveguide mode in the direction of the water 2506, while not coupling (or only partially coupling) with the terrestrial medium 2503. By positioning the probe 2110 at the inside of a bay, the guided surface waves to be launched in radial directions defined by the shorelines extending along either side of the bay. It may be possible to use other géologie features such as naturally occurring bodies of water (e.g., océans, seas, lakes. rivers. etc.) to direct the launched guided surface waves in a desired direction.

It should be noted that as the height of the probe 2110 increases, the crossover distance R_x moves further away from the probe 2110, as illustrated in FIG. 6. As a resuit, as the height of the probe 2110 increases, the excavation area increases as the crossover distance R_x moves further outward.

It should be emphasized that the above-described embodiments of the présent disclosure are merely possible examples of implémentations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. AH such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims. In addition, ail optional and preferred features and modifications of the described embodiments and dépendent claims are usable in ail aspects of the disclosure taught herein. Furthermore, the individual features of the dépendent claims, as well as ail optional and preferred features and modifications of the described embodiments are combinable and interchangeable with one another.

Claims (18)

1. Therefore, the following is claimed:

   1. A probe site, comprising:

   a propagation interface in a surface of a lossy conducting medium, the propagation interface including a first région and a second région adjacent to the first région, the first région comprising a first lossy conducting medium and the second région comprising a second lossy conducting medium; and a guided surface waveguide probe positioned over the propagation interface, and adjacent to the first région and the second région, the guided surface waveguide probe configured to generate at least one electrical field that synthesizes a wave front having a complex Brewster angle of incidence corresponding to the first lossy conducting medium when excited by an excitation source, where the wave front launches a guided surface wave along the propagation interface and the surface of the lossy conducting medium in a radial direction that is defined by the first région and restricted by the second région.
2. 2. The probe site according to claim 1, wherein the wave front intersects with the propagation interface at the complex Brewster angle of incidence at a crossover distance from the guided surface waveguide probe.
3. 3. The probe site according to claim 1, wherein the first région and the second région extend along the propagation interface from adjacent to the guided surface waveguide probe to beyond the crossover distance.
4. 4. The probe site according to claim 1, wherein the lossy conducting medium is a terrestrial medium, and the first lossy conducting medium is the terrestrial medium.
5. 5. The probe site according to claim 4, wherein the second région extends into the terrestrial medium at least to a depth of a complex image of the guided surface waveguide probe.
6. 6. The probe site according to claim 4, wherein the second région extends into the terrestrial medium at least to a depth of a complex image plane of the guided surface waveguide probe.
7. 7 The probe site according to claim 1, wherein the second lossy conducting medium is water.
8. 8 The probe site according to claim 7, wherein the second région is a naturally occurring body of water.
9. 9, The probe site according to claim 1, wherein the lossy conducting medium is a terrestrial medium, and the second lossy conducting medium is the terrestrial medium.
10. 10. The probe site according to claim 9, wherein the first région is a prepared région having a conductivity and permittivity that is different from the terrestrial medium.
11. 11 The probe site according to claim 10, wherein the prepared région comprises an excavation containing water.
12. 12. The probe site according to claim 11, wherein the water is seawater.
13. 13. The probe site according to claim 10, wherein the prepared région comprises an excavation containing an aggregate composition of the terrestrial medium and an added material.
14. 14. The probe site according to claim 1, wherein the second région extends around the guided surface waveguide probe from a first side of the first région to a second side of the first région.
15. 15. The probe site according to claim 1, wherein the propagation interface includes a third région adjacent to the first région opposite the second région.
16. 16. The probe site according to claim 15, wherein the third région comprises the second lossy conducting medium.
17. 17. The probe site according to claim 15, wherein the third région comprises a third lossy conducting medium.
18. 18. The probe site according to claim 1, wherein the lossy conducting medium is a terrestrial medium.