OA18686A - Excitation and Use of Guided Surface Waves - Google Patents

Abstract

Disclosed are various embodiments for transmitting and receiving energy conveyed in the form of a guided surface-waveguide mode along the surface of a lossy medium such as, e.g., a terrestrial medium excited by a guided surface waveguide probe.

Description

EXCITATION AND USE OF GUIDED SURFACE WAVES

CROSS REFERENCE TO RELATED APPLICATIONS

This application daims priority to, and the benefit of, co-pending U.S. Non-provisional Patent Application entitled “Excitation and Use of Guided Surface Waves,” which was filed on June 2, 2015 and assigned Application Number 14/728,492, and co-pending U.S. Non-provisional Patent Application entitled “Excitation and Use of Guided Surface Waves,” which was filed on June 2, 2015 and assigned Application Number 14/728,507, both of which are incorporated herein by reference in their entireties.

This application is related to co-pending U.S. Non-provisional Patent Application entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on March 7, 2013 and assigned Application Number 13/789,538, and was published on September 11, 2014 as Publication Number US2014/0252886 A1, and which is incorporated herein by reference in its entirety. This application is also related to co-pending U.S. Nonprovisional Patent Application entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on March 7, 2013 and assigned Application Number 13/789,525, and was published on September 11, 2014 as Publication Number US2014/0252865 A1, and which is incorporated herein by reference in its entirety. This application is further related to co-pending U.S. Non-provisional Patent Application entitled Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on September 10, 2014 and assigned Application Number 14/483,089, 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 conventional antenna structures. In contrast to radio science, electrical power distribution Systems in the last century involved the transmission of energy guided along electrical conductors. This understanding of the distinction between radio frequency (RF) and power transmission has existed sincethe early 1900’s.

SUMMARY

Embodiments of the present disclosure are related to excitation and use of guided surface waves.

In one embodiment, among others, a method, comprises positioning a charge terminal at a defined height over a lossy conducting medium; adjusting a phase delay (Φ) of a feed network connected to the charge terminal to match a wave tilt angle (Ψ) corresponding to a complex Brewster angle of incidence (θ_ίΒ) associated with the lossy conducting medium; adjusting a load impédance (Z_L) of the charge terminal based upon an image ground plane impédance (Z_in) associated with the lossy conducting medium; and exciting the charge terminal with an excitation voltage via the feed network, where the excitation voltage establishes an electric field that couples into a guided surface waveguide mode along a surface of the lossy conducting medium.

In one or more aspects of these embodiments, the feed network can comprise a feed line conductor coupled to the charge terminal and a coil coupled between the lossy conducting medium and the feed line conductor, where the phase delay (Φ) of the feed network includes a phase delay (0_y) associated with the feed line conductor and a phase delay (0_C) associated with the coil. Adjusting the phase delay (Φ) can comprise adjusting the phase delay (6_C) associated with the coil. A connection of the feed line conductor can be repositioned on the coil to adjust the phase delay (6_C) associated with the coil. The connection of the feed line conductor can be repositioned on the coil via a variable tap. The complex Brewster angle of incidence (θ_ίΒ) associated with the lossy conducting medium can be based upon an operational frequency of the excitation voltage and characteristics of the lossy conducting medium. The characteristics of the lossy conducting medium can include conductivity and permittivity.

In one or more aspects of these embodiments, the image ground plane impédance (Z_in) can be based at least in part upon a phase shift (0_d) between a physical boundary of the lossy conducting medium and a conducting image ground plane. The physical boundary of the lossy conducting medium and the conducting image ground plane can be separated by a complex depth. The load impédance (Z_L) of the charge terminal can be adjusted based upon a reactive component of the image ground plane impédance (Z_in). The load impédance (Z_L) of the charge terminal can be adjusted to match the reactive component of the image ground plane impédance (Z_in) with a structure impédance (Z_base) associated with the feed network and the charge terminal. The phase delay (Φ) of the feed network can be fixed while the load impédance (Z_L) of the charge terminal is adjusted. The charge terminal can hâve an effective spherical diameter, and the defined height of the charge terminal can be at least four times an effective spherical diameter to reduce the bound capacitance. The charge terminal can be coupled to an excitation source via a coil.

In one or more aspects of these embodiments, a change in a characteristic of the lossy conducting medium can be sensed; and the phase delay (Φ) of the feed network connected to the charge terminal can be adjusted to match a modified wave tilt angle in response to the change in the characteristic of the lossy conducting medium, the modified wave tilt angle corresponding to a complex Brewster angle of incidence associated with the lossy conducting medium having the changed characteristic. The method can further comprise adjusting the load impédance (Z_L) of the charge terminal based upon a new image ground plane impédance based upon the lossy conducting medium having the changed characteristic. The lossy conducting medium can be a terrestrial medium.

In another embodiment, a guided surface waveguide probe comprises a charge terminal elevated over a lossy conducting medium; and a feed network configured to couple an excitation source to the charge terminal, the feed network configured to provide a voltage to the charge terminal with a phase delay (Φ) that matches a wave tilt angle (Ψ) associated with a complex Brewster angle of incidence (0_is) associated with the lossy conducting medium, and the charge terminal having a load impédance (Z_L) that is determined based upon an image ground plane impédance (Z_in) associated with the lossy conducting medium.

In one or more aspects of these embodiments, the feed network can comprise a feed line conductor coupled to the charge terminal and a coil coupled between the lossy conducting medium and the feed line conductor, where the phase delay (Φ) of the feed network includes a phase delay (0_y) associated with the feed line conductor and a phase delay (9_C) associated with the coil. The coil can be a helical coil. The excitation source can be coupled to the coil via a tap connection. An impédance matching network can be coupled between the excitation source and the tap connection on the coil. The excitation source can be magnetically coupled to the coil or coupled to the coil via a tap connection. The feed network can be configured to vary the phase delay (Φ) to match the wave tilt angle (Ψ).

In one or more aspects of these embodiments, a probe control System can be configured to adjust the feed network based at least in part upon characteristics of the lossy conducting medium. The feed network can comprise a coil coupled between the excitation source and the charge terminal, where the charge terminal can be coupled to the coil via a variable tap. The probe control System can adjust a position of the variable tap in response to a change in the characteristics of the lossy conducting medium.

In another embodiment, a method comprises positioning a charge terminal of a guided surface waveguide probe at a defined height over a lossy conducting medium; adjusting a traveling wave phase delay (Φ) of the guided surface waveguide probe to match a wave-tilt angle (Ψ) of a surface wave of the lossy conducting medium; simultaneously exciting a superposed standing wave on the guided surface waveguide probe by exploiting phase delays from transmission line sections of the guided surface waveguide probe plus phase jumps arising from discontinuities in characteristic impédances of the transmission line sections, the superposed standing wave based upon a complex image plane located at a complex depth from a base of the guided surface waveguide probe; and exciting the charge terminal with an excitation voltage via the transmission line sections, where an excitation charge distribution establishes an electric field that couples into a guided surface-wave waveguide mode along a surface of the lossy conducting medium.

In another embodiment, a method, comprises coupling a receiving structure to a lossy conducting medium; and mode-matching with a guided surface wave established on the lossy conducting medium, where a traveling wave phase delay (Φ) of the receiving structure is matched to a wave tilt angle (Ψ) associated with the guided surface wave, the wave tilt angle (Ψ) based at least in part upon characteristics of the lossy conducting medium in a vicinity of the receiving structure. A charge terminal of the receiving structure can be suspended at a defined height over a surface of the lossy conducting medium. Electrical power can be extracted from the receiving structure via a coil.

In one or more aspects of these embodiments, the receiving structure can comprise a receiver network coupled between the charge terminal and the lossy conducting medium. The receiver network can comprise a coil coupled to the lossy conducting medium and a supply line conductor coupled between the coil and the charge terminal, where the traveling wave phase delay (Φ) is based upon a phase delay (0_C) of the coil and a phase delay (d_y) of the supply line conductor. Adjusting the traveling wave phase delay (Φ) can comprise adjusting a position of a tap on the coil to vary the phase delay (0_C) of the coil. The supply line conductor can be coupled to the coil via the tap. The charge terminal can hâve an effective spherical diameter, and the defined height of the charge terminal is at least four times the effective spherical diameter to reduce bound capacitance.

In one or more aspects of these embodiments, the receiving structure can be resonated relative to an image plane at a complex depth below the surface of the lossy conducting medium. Resonating the receiving structure can comprise adjusting a load impédance (Z_L) of the charge terminal based upon an image ground plane impédance (Z_/n) associated with the lossy conducting medium. Resonating the receiving structure can establish a standing wave on the receiving structure by exploiting phase delays from transmission line sections of the receiving structure plus phase jumps arising from discontinuities in characteristic impédances of the transmission line sections, the standing wave superposed with a traveling wave on the receiving structure.

In another embodiment, a receiving structure for mode-matching with a guided surface wave established on a lossy conducting medium comprises a charge terminal elevated over the lossy conducting medium; and a receiver network coupled between the charge terminal and the lossy conducting medium, where the receiver network has a phase delay (Φ) that matches a wave tilt angle (Ψ) associated with the guided surface wave, the wave tilt angle (Ψ) based at least in part upon characteristics of the lossy conducting medium in a vicinity of the receiving structure.

In one or more aspects of these embodiments, the charge terminal can hâve a variable load impédance (Z_L). The variable load impédance (Z_L) can be determined based upon an image ground plane impédance (Z_in) associated with the lossy conducting medium in the vicinity of the receiving structure. The load impédance (Z_L) can be adjusted to resonate the receiving structure relative to an image plane at a complex depth below a surface of the lossy conducting medium. Resonating the receiving structure can establishe a standing wave on the receiving structure by exploiting phase delays from transmission line sections of the receiver network plus phase jumps arising from discontinuities in characteristic impédances of the transmission line sections.

In one or more aspects of these embodiments, the receiver network can comprise a coil coupîed to the lossy conducting medium and a supply line conductor coupled between the coil and the charge terminal, where the phase delay (Φ) of the receiver network is based upon a phase delay (0_C) of the coil and a phase delay (9_y) of the supply line conductor. The receiving structure can comprise a variable tap configured to adjust the phase delay (e_c) of the coil. The receiving structure can comprise an impédance matching network coupled to a coil. The impédance matching network can be inductively coupled to the coil.

In another embodiment, a method comprises positioning a receive structure relative to a terrestrial medium; and receiving, via the receive structure, energy conveyed in a form of a guided surface wave on a surface of the terrestrial medium. In one or more aspects of these embodiments, the receive structure can load an excitation source coupled to a guided surface waveguide probe that generates the guided surface wave. The energy can comprise electrical power, and the electrical power can be applied to an electrical load coupled to the receive structure, where the electrical power can be used as a power source for the electrical load. An electrical load can be impédance matched to the receive structure. A maximum power transfer can be established from the receive structure to the electrical load. The receive structure can comprise a magnetic coil, a linear probe and/or a tuned resonator coupled to the terrestrial medium.

In another embodiment, an apparatus comprises a receive structure that receives energy conveyed in a form of a guided surface wave along a surface of a terrestrial medium. In one or more aspects of these embodiments, the receive structure can be configured to load an excitation source coupled to a guided surface waveguide probe that generates the guided surface wave. The energy can comprise electrical power, and the receive structure is coupled to an electrical load, and wherein the electrical power is applied to the electrical load, the electrical power being employed as a power source for the electrical load. The electrical load can be impédance matched with the receive structure. The receive structure can comprise a magnetic coil, a linear probe and/or a tuned resonator. The tuned resonator can comprise a sériés tuned resonator, a parallel tuned resonator and/or a distributed tuned resonator.

in another embodiment, a power transmission System comprises a guided surface waveguide probe that transmits electrical energy in a form of a guided surface wave along a surface of a terrestrial medium; and a receive structure that receives the electrical energy. In one or more aspects of these embodiments, the receive structure can load the guided surface waveguide probe. An electrical load can be coupled to the receive structure and the electrical energy can be used as a power source for the electrical load. The electrical load can be impédance matched to the receive circuit. A maximum power transfer can be established from the receive structure to the electrical load.

In another embodiment, a method comprises transmitting energy conveyed in a form of a guided surface-waveguide mode along a surface of a terrestrial medium by exciting a guided surface waveguide probe. In one or more aspects of these embodiments, transmitting energy conveyed in the form of the guided surface-waveguide mode along the surface of the terrestrial medium by exciting the guided surface waveguide probe can comprise synthesizing a plurality offields that substantially match the guided surface-waveguide mode ofthe terrestrial medium, wherein the fields substantially synthesize a wave front incident at a complex Brewsterangle ofthe terrestrial medium, resulting in a negligible reflection.

In another embodiment, an apparatus comprises a guided surface waveguide probe configured to create a plurality of résultant fields that are substantially mode-matched to a guided surface wave mode on a surface of a lossy conducting medium. In one or more aspects of these embodiments, the lossy conducting medium can comprise a terrestrial medium. The résultant fields can substantially synthesize a wave incident at a complex Brewster angle ofthe lossy conducting medium, resulting in substantially zéro reflection. In another embodiment, a method can comprise positioning a receive circuit relative to a terrestrial medium and receiving, via the receive circuit, energy conveyed in a form of a guided surface wave on a surface of the terrestrial medium. In one or more aspects of these embodiments, an electrical load coupled to the receive circuit can load an excitation source coupled to a guided surface waveguide probe that generates the guided surface wave. The energy can comprise electrical power. The electrical power can be applied to an electrical load coupled to the receive circuit, wherein the electrical power is used as a power source for the electrical load. The electrical load can be impedance-matched to the receive circuit. A maximum power transfer from the receive circuit to the electrical load can be established.

In another embodiment, an apparatus comprises a receive circuit that receives energy conveyed in a form of a guided surface wave along a surface of a lossy conducting medium. In one or more aspects of these embodiments, the lossy conducting medium further comprises a terrestrial medium. An electrical load coupled to the receive circuit can load an excitation source coupled to a guided surface waveguide probe that generates the guided surface wave. The receive circuit can comprise one of a magnetic coil, a linear probe, or a tuned resonator.

In another embodiment, a power transmission system comprises a guided surface waveguide probe that transmits electrical energy in a form of a guided surface wave along a surface of a terrestrial medium and a receive circuit that receives the electrical energy. In one or more aspects of these embodiments, an electrical load coupled to the receive circuit can load the guided surface waveguide probe. The electrical energy can be used as a power source for an electrical load is coupled to the receive circuit. A maximum power transfer can be established from the receive circuit to the electrical load.

In another embodiment, a guided surface waveguide probe, comprises a charge terminal elevated over a lossy conducting medium and a feed network configured to couple an excitation source to the charge terminal. The feed network can be configured to provide a voltage to the charge terminal that estabiishes an electric field having a wave tilt (VP) that intersects the lossy conducting medium at a tangent of a complex Brewster angle (ψ_1ιΒ) at a Hankel crossover distance (R_x) from the guided surface waveguide probe. The lossy conducting medium can be a terrestrial medium.

In one or more aspects of these embodiments, the feed network can comprises a coil coupled between the excitation source and the charge terminal. The coil can be a helical coil. The excitation source can be coupled to the coil via a tap connection. The tap connection can be at an impédance matchîng point on the coil. An impédance matching network can be coupled between the excitation source and the tap connection on the coil. The excitation source can be magnetically coupled to the coil. The charge terminal can be coupled to the coil via a tap connection.

In one or more aspects of these embodiments, the charge terminal can be positioned at a physical height (h_p) corresponding to a magnitude of an effective height of the guided surface waveguide probe, where the effective height is given by h_eff = R_xtanip_iB = h_pe^J<î), with Ψί,Β = (”·/²) - ^θι,β ^and Φ is a phase of the effective height. The phase Φ can be approximately equal to an angle Ψ of the wave tilt of illumination that corresponds to the complex Brewster angle. The charge terminal can hâve an effective spherical diameter, and the charge terminal can be positioned at a height that is at least four times the effective spherical diameter. The height of the charge terminal can be greater than a physical height (/ip) corresponding to a magnitude of an effective height of the guided surface waveguide probe, where the effective height is given by he^ = Rx tanipiB - ΙΐρβΙ^φ, with = (π/2) — Θι,βIn another embodiment, a system comprises a guided surface waveguide probe, including: a charge terminal elevated over a lossy conducting medium, and a feed network configured to provide a voltage to the charge terminal that establishes an electric field having a wave tilt (M/) that intersects the lossy conducting medium at a tangent of a complex Brewster angle at a Hankel crossover distance (Rx) from the guided surface waveguide probe; and an excitation source coupled to the charge terminal via the feed network. The lossy conducting medium can be a terrestrial medium.

In one or more aspects of these embodiments, a probe control System can be configured to adjustthe guided surfacewaveguide probe based at least in part upon characteristics ofthe lossy conducting medium. The feed network can comprise a coil coupled between the excitation source and the charge terminal, where the charge terminal can be coupled to the coil via a variable tap. The coil can be a helical coil. The probe control System can adjust a position of the variable tap in response to a change in the characteristics of the lossy conducting medium. The adjustment of the position of the variable tap can adjust the wave tilt of the electric field to correspond to a wave illumination that intersects the lossy conducting medium at the complex Brewster angle (ψ_£β) at the Hankel crossover distance (R_x).

Other Systems, methods, features, and advantages of the present disclosure will be or become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that ail such additional Systems, methods, features, and advantages be included within this description, be within the scope of the present disclosure, and be protected by the accompanying daims.

In addition, ail optional and preferred features and modifications of the described embodiments are usable in ail aspects of the disclosure taught herein. Furthermore, the individual features of the dépendent daims, as well as ail optional and preferred features and modifications of the described embodiments are combinable and interchangeable with one another.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the disclosure. Moreover, in the drawings, like reference numerals 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 field.

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

FIG. 3 is a drawing that illustrâtes a guided surface waveguide probe disposed with respect to a propagation interface of FIG. 2 according to an embodiment of the present disclosure.

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

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

FIG. 6 is a graphical représentation illustrating the effect of élévation of a charge terminal on the location where the electric field of FIG. 5A intersects with the lossy conducting medium at a Brewster angle according to various embodiments of the présent disclosure.

FIG. 7 is a graphical représentation of an example of a guided surface waveguide probe according to an embodiment of the présent disclosure.

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

FIGS. 9A and 9B are graphical représentations illustrating examples of single-wire transmission line and classic transmission line models of the équivalent image plane models of FIGS. 8B and 8C 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 7 to launch a guided surface wave along the surface of a lossy conducting medium according to various embodiments of the present disclosure.

FIG. 11 is a plot illustrating an example of the relationship between a wave tilt angle and the phase delay of a guided surface waveguide probe of FIGS. 3 and 7 according to various embodiments of the present disclosure.

FIG. 12 is a Smith chart illustrating an example of adjusting the load impédance of the guided surface waveguide probe of FIGS. 3 and 7 according to various embodiments of the present disclosure.

FIG. 13 is a plot comparing measured and theoretical field strength of the guided surface waveguide probe of FIGS. 3 and 7 according to an embodiment of the present disclosure. FIGS. 14A through 140 depict examples of receiving structures that can be employed to receive energy transmitted in the form of a guided surface wave launched by a guided surface waveguide probe according to the various embodiments of the present disclosure.

FIG. 14D is a flow chart illustrating an example of adjusting a receiving structure according to various embodiments of the present disclosure.

FIG. 15 depicts an example of an additional receiving structure that can be employed to receive energy transmitted in the form of a guided surface wave launched by a guided surface waveguide probe according to the various embodiments of the present disclosure.

FIG. 16A depicts a schematic diagram representing the Thevenin-equivalent of the receivers depicted in FIGS. 14A and 14B according to an embodiment of the present disclosure.

FIG. 16B depicts a schematic diagram representing the Norton-equivalent of the receiver depicted in FIG. 15 according to an embodiment of the present disclosure.

FIGS. 17Aand 17B are schematic diagrams representing examples of a conductivity measurement probe and an open wire line probe, respectively, according to an embodiment of the présent disclosure.

FIG. 18 is a schematic drawing of an example of an adaptive control system employed by the probe control system of FIG. 3 according to various embodiments of the présent disclosure. FIGS. 19A-19B and 20 are drawings of examples of variable terminais for use as a charging terminal according to various embodiments of the présent disclosure.

DETAILED DESCRIPTION

To begin, some terminology shall be established to provide clarity in the discussion of concepts to follow. 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 energy that is emitted from a source structure in the form of waves that are not bound to a waveguide. For example, a radiated electromagnetic field is generally a field 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 medium of propagation (such as air) independent of their source until they dissipate regardless of whether the source continues to operate. Once electromagnetic waves are radiated, they are not recoverable unless intercepted, and, if not intercepted, the energy inhérent in the radiated electromagnetic waves is lost forever. Electrical structures such as antennas are designed to radiate electromagnetic fields by maximizing the ratio of the radiation résistance to the structure loss résistance. Radiated energy spreads out in space and is lost regardless of whether a receiver is présent. The energy density of the radiated fields is a function of distance due to géométrie spreading. Accordingly, the term “radiate” in ail its forms as used herein refers to this form of electromagnetic propagation.

A guided electromagnetic field is a propagating electromagnetic wave whose energy is concentrated within or near boundaries between media having different electromagnetic properties. In this sense, a guided electromagnetic 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 lost except for that dissipated in the conductivity of the guiding medium. Stated another way, if there is no load for a guided electromagnetic wave, then no energy is consumed. Thus, a generator or other source generating a guided electromagnetic field does not deliver real power unless a résistive load is present. To this end, such a generator or other source essentially runs idle until a load is presented. This is akin to running a generator to generate a 60 Hertz electromagnetic wave that is transmitted over power lines where there is no electrical load. It should be noted that a guided electromagnetic field or wave is the équivalent to what is termed a “transmission line mode.” This contrasts with radiated electromagnetic waves in which real power is supplied at ail times in order to generate radiated waves. Unlike radiated electromagnetic waves, guided electromagnetic energy does not continue to propagate along a finite length waveguide after the 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 field strength in décibels (dB) above an arbitrary reference in volts per meter as a function of distance in kilometers on a log-dB plot to further illustrate the distinction between radiated and guided electromagnetic fields. The graph 100 of FIG. 1 depicts a guided field strength curve 103 that shows the field strength of a guided electromagnetic field as a function of distance. This guided field strength curve 103 is essentially the same as a transmission line mode. Also, 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 radiation propagation, respectively. The radiated field strength curve 106 falls off geometrically (1/d, where d is distance), which is depicted as a straight line on the log-log scale. The guided field strength curve 103, on the other hand, has a characteristic exponential decay of e~^ad/^[d and exhibits a distinctive knee 109 on the log-log scale. The guided field strength curve 103 and the radiated field strength curve 106 intersect at point 113, which occurs at a Crossing distance. At distances less than the Crossing distance at intersection point 113, 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 fundamental 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 Milligan, T., Modem Antenna Design, McGraw-Hill, 1^st 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 formally and placed on a rigorous basis. That two such diverse solutions could emerge 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, itself, 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 propagating fields are calied “Hertzian waves.” However, in the presence of a conducting boundary, the wave équation plus boundary conditions mathematically lead to a spectral représentation of wavenumbers composed of a continuous spectrum plus a sum of discrète spectra. To this end, reference is made to Sommerfeld, A., “Liber die Ausbreitung der Wellen in der Drahtlosen Télégraphié,” Annalen der Physik, Vol. 28, 1909, pp. 665-736. Also see Sommerfeld, A., “Problems of Radio,” published as Chapter 6 in Partial Differential Equations in Physics Lectures on Theoretical Phvsics: Volume VI, 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^th Century Controversies,” IEEE Antennas and Propagation Magazine, Vol. 46, No. 2, April 2004, pp. 64-79; and Reich, H. J., Ordnung, P.F, Krauss, H.L., and Skalnik, J.G., Microwave Theorv and Techniques, Van Nostrand, 1953, pp. 291-293, each of these référencés being incorporated herein by reference in their entirety.

The terms “ground wave” and “surface wave” identify two distinctly different physical propagation phenomena. A surface wave arises analytically from a distinct pôle yielding a discrète component in the plane wave spectrum. See, e.g., “The Excitation of Plane Surface Waves” by Cullen, A.L., (Proceedings of the IEE (British), Vol. 101, 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 familiar from radio broadeasting. These two propagation mechanisms arise from the excitation of different types of eigenvalue spectra (continuum or discrète) on the complex plane. The field strength of the guided surface wave decays exponentially with distance as illustrated by curve 103 of FIG. 1 (much like propagation in a lossy waveguide) and resembles propagation in a radial transmission line, as opposed to the classical Hertzian radiation of the ground wave, which propagates spherically, possesses a continuum of eigenvalues, falls off geometrically as illustrated by curve 106 of FIG. 1, and results from branch-cut intégrais. As experimentally demonstrated by C.R. Burrows in “The Surface Wave in Radio Propagation over Plane Earth” (Proceedings ofthe IRE, Vol. 25, No. 2, February, 1937, pp. 219-229) and “The Surface Wave in Radio Transmission” (Bell Laboratories Record, Vol. 15, June 1937, pp. 321-324), vertical antennas radiate ground waves but do not launch guided surface waves. To summarize the above, first, the continuous part ofthe wave-number eigenvalue spectrum, corresponding to branch-cut intégrais, produces the radiation field, and second, the discrète spectra, and corresponding residue sum arising from the pôles enclosed by the contour of intégration, resuit in non-TEM traveling surface waves that are exponentially damped in the direction transverse to the propagation. Such surface waves are guided transmission line modes. For further explanation, reference is made to Friedman, B., Principles and Techniques of Applied Mathematics, Wiley, 1956, pp. pp. 214, 283-286, 290, 298-300. In free space, antennas excite the continuum eigenvalues of the wave équation, which is a radiation field, where the outwardly propagating RF energy with E_z and Η_φ in-phase is lost forever. On the other hand, waveguide probes excite discrète eigenvalues, which results in transmission line propagation. See Collin, R. E., Field Theory of Guided Waves, McGraw-Hill, 1960, pp. 453, 474-477. While such theoretical analyses hâve held out the hypothetical possibility of launching open surface guided waves over planar or spherical surfaces of lossy, homogeneous media, for more than a century no known structures in the engineering arts hâve existed for accomplishing this with any practical efficiency. Unfortunately, since it emerged in the early 1900’s, the theoretical analysis set forth above has essentially remained a theory and there hâve been no known structures for practically accomplishing the launching 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 electric fields that couple into a guided surface waveguide mode along the surface of a lossy conducting medium. Such guided electromagnetic fields are substantially mode-matched in magnitude and phase to a guided surface wave mode on the surface of the lossy conducting medium. Such a guided surface wave mode can also be termed a Zenneck waveguide mode. By virtue of the fact that the résultant fields excited by the guided surface waveguide probes described herein are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium, a guided electromagnetic field 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 terrestrial medium such as the Earth.

Referring to FIG. 2, shown is a propagation interface that provides for an examination of the boundary value solutions to Maxwell’s équations derived in 1907 by Jonathan Zenneck as set forth in his paper Zenneck, J., “On the Propagation of Plane Electromagnetic Waves Along a Fiat Conducting Surface and their Relation to Wireless Telegraphy,” Annalen der Physik, Serial 4, Vol. 23, September 20, 1907, pp. 846-866. FIG. 2 depicts cylindrical coordinates for radially propagating waves along the interface between a lossy conducting medium specified as Région 1 and an insulator specified as Région 2. Région 1 can comprise, for example, any lossy conducting medium. In one example, such a lossy conducting medium can comprise a terrestrial medium such as the Earth or other medium. 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 example, 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 angle. See Stratton, J.A., Electromagnetic Theorv, McGrawHill, 1941, p. 516.

According to various embodiments, the présent disclosure sets forth various guided surface waveguide probes that generate electromagnetic fields that are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium comprising Région 1. According to various embodiments, such electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium that can resuit in zéro reflection.

To explain further, in Région 2, where an e⁷'^wt field variation is assumed and where ρ * 0 and z > 0 (with z being the vertical coordinate normal to the surface of Région 1, and p being the radial dimension in cylindrical coordinates), Zenneck’s closed-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;

Η_2φ = Ae~^^z H^X-jyp),(1)

E_2p = A e^-U2Z H^i-jyp), and(2) ^E^ = ^A(Ër)^e~^{U2Z h}β⁾(-Jyp)·&

\UJCq/

In Région 1, where the field variation is assumed and where p Ψ 0 and z < 0, Zenneck’s closed-form exact solution of Maxwell’s équations satisfying the boundary conditions along the (6) interface is expressed-by the following electric field and magnetic field components: //^=^^//^(-^),(4) ^£1P = 4 (^fe) ^e“^1Z «F’C-J/P).<

^=4(^)e^//®(-;_Kp).

In these expressions, z is the vertical coordinate normal to the surface of Région 1 and p is the radial coordinate, (-;yp) is a complex argument Hankel function of the second kind and ordern, u_r is the propagation constant in the positive vertical (z) direction in Région 1, u₂ is the propagation constant in the vertical (z) direction in Région 2, σ_Ύ is the conductivity of Région 1, ω is equal to 2π/, where / is à frequency of excitation, ε₀ is the permittivity of free space, ε_χ is the permittivity of Région 1, A is a source constant imposed by the source, and y is a surface wave radial propagation constant.

The propagation constants in the ±z directions are determined by separating the wave équation above and below the interface between Régions 1 and 2, and imposing the boundary conditions. This exercise gives, in Région 2, „ - , ~jk°_ ^U2 - _ (7) and gives, in Région 1,

Ui = -u₂(p_r — jx).(8)

The radial propagation constant y is given by y = ; VfcF+üï = j(9) which is a complex expression where n is the complex index of refraction given by n — /_r — jx.(10)

In ail of the above Equations, x = —, and(11) ωε₀ '' k₀ = 0)///=(12) where comprises the relative permittivity of Région 1, σ_± is the conductivity of Région 1, ε_ο is the permittivity of free space, and μ₀ comprises the permeability of free space. Thus, the generated surface wave propagates para llel to the interface and exponentially decays vertical to it. This is known as evanescence.

Thus, Equations (1)-(3) can be considered to be a cylindrrcally-symmetric, radially-propagating waveguide mode. See Barlow, H. M., and Brown, J., Radio Surface Waves, Oxford University Press, 1962, pp, 10-12, 29-33. The présent disclosure details structures that excite this “open boundary” waveguide mode. Specifically, according to various embodiments, a guided surface waveguide probe is provided with a charge terminal 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 référencé to FIG. 3, which shows an example of a guided surface waveguide probe 300a that includes a charge terminal B elevated above a lossy conducting medium 303 (e.g., the earth) along a vertical axis zthat is normal to a plane presented by the lossy conducting medium 303. The lossy conducting medium 303 makes up Région 1, and a second medium 306 makes up Région 2 and shares a boundary interface with the lossy conducting medium 303.

According to one embodiment, the lossy conducting medium 303 can comprise a terrestrial medium such as the planet Earth. To this end, such a terrestrial medium comprises ail structures or formations included thereon whether natural 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 natural éléments that make up our planet. In addition, such a terrestrial medium can comprise man-made éléments such as concrete, asphalt, building materials, and other man-made materials. In other embodiments, the lossy conducting medium 303 can comprise some medium other than the Earth, whether naturally occurring or man-made. In other embodiments, the lossy conducting medium 303 can comprise other media such as man-made surfaces and structures such as automobiles, aircraft, man-made materials (such as plywood, plastic sheeting, or other materials) or other media.

In the case where the lossy conducting medium 303 comprises a terrestrial medium or Earth, the second medium 306 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 306 can comprise other media relative to the lossy conducting medium 303.

The guided surface waveguide probe 300a includes a feed network 309 that couples an excitation source 312 to the charge terminal Τ_Ί via, e.g., a vertical feed line conductor. According to various embodiments, a charge Cb is imposed on the charge terminal Tj to synthesize an electric field based upon the voltage applied to terminal Ti at any given instant. Depending on the angle of incidence (0_É) of the electric field (F), it is possible to substantîally mode-match the electric field to a guided surface waveguide mode on the surface of the lossy conducting medium 303 comprising Région 1.

By considering the Zenneck closed-form solutions of Equations (1)-(6), the Leontovich impédance boundary condition between Région 1 and Région 2 can be stated as zxW₂(p,<p,0) =/_s, (13) where z is a unit normal in the positive vertical (+z) direction and H₂ is the magnetic field strength in Région 2 expressed by Equation (1) above. Equation (13) implies that the electric and magnetic fields specified in Equations (1)-(3) may resuit in a radial surface current density along the boundary interface, where the radial surface current density can be specified by J_p(fil =-A H⁽²\-jYP’) ' (14) where A is a constant. Further, it should be noted that close-in to the guided surface waveguide probe 300 (forp « Λ), Equation (14) above has the behavior /_rto„(p-)=^= =(15)

The négative sign means that when source current (f₀) flows vertically upward as iliustrated in FIG. 3, the “close-in” ground current flows radially inward. By field matching on Η_φ “close-in,” it can be determined that _{a = (16)} 44 where qï= CM, in Equations (1)-(6) and (14). Therefore, the radial surface current density of Equation (14) can be restated as

J_P(pf = ~Hβ\-j_Yp^f).(17)

The fields expressed by Equations (1)-(6) and (17) hâve the nature of a transmission line mode bound to a lossy interface, not radiation fields that are associated with groundwave propagation. See Barlow, H. M. and Brown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 1-5.

At this point, a review of the nature of the Hankel functions used in Equations (1)-(6) and (17) is provided for these solutions of the wave équation. One might observe that the Hankel functions of the first and second kind and ordern are defined as complex combinations ofthe standard Bessel functions of the first and second kinds

G) = Jn G) + JN_n (x), and (18)

Hβ\x)=J_n(x)~jN_n(x), (19)

These functions represent cylindrical waves propagating radially inward and outward (H®), respectively. The définition is analogous to the relationship e^±jx = cos x± j sin x.

See, for example, Harrington, R.F., Time-Harmonic Fields, McGraw-Hill, 1961, pp. 460-463.

That H^(k_pp) is an outgoing wave can be recognized from its large argument asymptotic behavior that is obtained directly from the serres définitions of /_n(x) and JV_n(x). Far-out from the guided surface waveguide probe:

H®(x)—> jⁿe~^JX =(20a) ^Tt x—>00 \ πχ ^J ·\]πχ ^Jv which, when multiplied by ε^;ωί, is an outward propagating cylindrical wave ofthe form _ei(_ù)t-kp) _w|_th _a i/^ spatial variation. The first order (n = 1) solution can be determined from Equation (20a) to be η¥\χ)—>₇· ËZ = fE _e-X^x-Î-ï).(20b) ¹ x->œ ·\]πχ ·χ]πχ'

Close-in to the guided surface waveguide probe (for p « Λ), the Hankel function of first order and the second kind behaves as h[²Xx)^^-.(21)

Note that these asymptotic expressions are complex quantifies. When x is a real quantity, Equations (20b) and (21) differ in phase by J], which corresponds to an extra phase advance or “phase boost” of 45° or, equivalently, λ/8. The close-in and far-out asymptotes ofthe 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 out” représentation prédominâtes over the “close-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 -σ/ωε₍₁, it can be seen that the far-out and close-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 also vary as the conductivity (σ) of the lossy conducting medium changes. For example, the conductivity of the soil 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 σ = 0.010 mhos/m and relative permittivity ε_τ = 15, at an operating frequency of 1850 kHz. Curve 403 is the magnitude of the far-out asymptote of Equation (20b) and curve 406 is the magnitude of the close-in asymptote of Equation (21), with the Hankel crossover point 409 occurring at a distance of R_x = 54 feet. While the magnitudes are equal, a phase offset exists between the two asymptotes at the Hankel crossover point 409. It can also be seen that the Hankel crossover distance is much less than a wavelength of the operation frequency.

Considering the electric field components given by Equations (2) and (3) of the Zenneck closed-form solution in Région 2, it can be seen that the ratio of E_z and E_p asymptotically passes to ^EP ^£r~J— = n = tan (22) where n is the complex index of refraction of Equation (10) and θι is the angle of incidence of the electric field. In addition, the vertical component of the mode-matched electric field of Equation (3) asymptotically passes to

E_2z —, pi-) E _e-u₂z p-tos \ ε₀ / y 8π fp ' ⁷ which is linearly proportional to free charge on the isolated component of the elevated charge terminal’s capacitance at the terminal voltage, q_free = C_free x V_T.

For example, the height H-ι of the elevated charge terminal Tj in FIG. 3 affects the amount of free charge on the charge terminal T-). When the charge terminal Tî is near the ground plane of Région 1, most of the charge Ch on the terminal is “bound.” As the charge terminal T| is elevated, the bound charge is lessened until the charge terminal Tj 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-ι is that the charge on the elevated charge terminal Tj is further removed from the ground plane, resulting in an increased amount of free charge qf_ree to couple energy into the guided surface waveguide mode. As the charge terminal Tj 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 Τ_υ 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

Ceievated sphere = 4πε_οα(1 + Μ + Μ² + Μ³ + 2M⁴ + 3M⁵ + -), (24) where the diameter of the sphere is 2a, and where M = a/2h with h being the height of the 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 Tj that are at a height of about four times the diameter (4Ό — 8a) or greater, the charge distribution is approximately uniform about the spherical terminal, which can improve the coupling into the guided surface waveguide mode.

In the case of a sufficiently isolated terminal, the self-capacitance of a conductive sphere can be approximated by C = 4πε_οα, where a is the radius of the sphere in meters, and the selfcapacitance of a disk can be approximated by C — 8ε_οα, where a is the radius of the disk in _meters. The charge terminal Tt can include any shape such as a sphere, a disk, a cyiinder, 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 Τμ

This may be further understood with référencé to the example of FIG. 3, where the charge terminal Τ_Ί is elevated at a physical height of h_p = Hj above the lossy conducting medium 303. To reduce the effects of the “bound” charge, the charge terminal Ti can be positioned at a physical height that is at least four times the spherical diameter (or équivalent spherical diameter) of the charge terminal 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 Qi on charge terminal Ti 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 303. For an electric field (£j|) that is polarized parallel to the plane of incidence (not the boundary interface), the amount of reflection of the incident electric field may be determined using the Fresnel reflection (25) coefficient, which can be expressed as r (a\ _ _ d(.Sr-N)-sin^Yë~i-C_r-jx') cos 8_t ¹¹ En,; 7(^£r-^)-^sin2 9i+(£_r-jx) cos θί where is the conventional 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 0^, which is measured with respect to the surface normal (z). There will be no reflection of the incident electric field when Γ||(0ί) — 0 and thus the incident electric field will be completely coupled into a guided surface waveguide mode along the surface of the lossy conducting medium 303. It can be seen that the numerator of Equation (25) goes to zéro when the angle of incidence is

0i - arctan(_A/e_r -jx) = 9_irB, (26) where x = σ/ωε₀. This complex angle of incidence (θ_1β) is referred to as the Brewster angle. Referring back to Equation (22), it can be seen that the same complex Brewster angle (0_ijB) relationship is présent in both Equations (22) and (26).

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

Ε(θί) = E_pp + E_zz.(27)

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

Ep(p,z) = E(p,z) cos0j, and(28a)

E_z(p,z) = E(p,z)cos(j- = E(p,z)sin0_É,(28b) which means that the field ratio is — = ¹ _n = tan ψ;.(29)

E_z tan0j ¹¹'

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 = = [W|β'^ψ, or(30a) ^Ez

A = = tan θι = Α^ε“^7Ψ·( which is complex and has both magnitude and phase. For an electromagnetic wave in Région 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 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 normals for a radial cylindrical guided surface wave. At the boundary interface (z = 0) with a perfect 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 tanéu = = -;x = n = i = (31)

With the angle of incidence equal to the complex Brewster angle (0_i>B), the Fresnel reflection coefficient of Equation (25) vanishes, as shown by

r. (a > _ y (fi_r-jx)-sïn? 0j-(s_r-jx) cos Θ, _ _ ®i+(^£r^_A)COSθί g.₌g._g

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 = — jx results in the synthesized electric field being incident at the complex

Brewster angle, making the reflections vanish.

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 300. The electrical effective height (h_eff) has been defined as ^hMf = r_oiï^Pi<^^dz (33) for a monopole with a physical height (or length) of h_p. Since the expression dépends upon 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 Z(z) of the structure is performed over the physical height of the structure (Zi_p), and normalized to the ground current (Z_o) flowing upward through the base (or input) of the structure. The distributed current along the structure can be expressed by

I(z) = Z_ccos(/Ï₀z), (34) where β₀ is the propagation factor for current propagating on the structure. In the example of FIG. 3, Z_c is the current that is distributed along the vertical structure of the guided surface waveguide probe 300a.

For example, consider a feed network 309 that includes a low loss coil (e.g., a helical 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 delay line) is d_c = β_ρ1_€, with a physical length of l_c and a propagation factor of « 2ΤΓ 2JT zoc\ ⁽³⁵⁾ where is the velocity factor on the structure, λ₀ is the wavelength at the supplied frequency, and λ_ρ is the propagation wavelength resulting from the velocity factor v_f. The phase delay is measured relative to the ground (stake) current Z_o.

In addition, the spatial phase delay along the length l_w of the vertical feed line conductor can be given by θ_γ = p_wl_w where /?_w is the propagation phase constant for the vertical feed line conductor. In some implémentations, the spatial phase delay may be approximated by 6_y — Bwhp, since the différence between the physical height Zip of the guided surface waveguide probe 300a and the vertical feed line conductor length l_w is much less than a wavelength at the supplied frequency (λ₀). As a resuit, the total phase delay through the coil and vertical feed line conductor is Φ = Q_c + Q_y, and the current fed to the top of the coil from the bottom of the physical structure is l_c^_c + 0_y) = Ζ_οβ>^φ, (36) with the total phase delay Φ measured relative to the ground (stake) current Z_o. Consequently, the electrical effective height of a guided surface waveguide probe 300 can be approximated by

M = T_o Û cos(&z) dz = h_pei*, (37) for the case where the physical height hp « λ₀. The complex effective height of a monopole, h_eff = at an angle (or phase shift) of Φ, may be adjusted to cause the source fields to match a guided surface waveguide mode and cause a guided surface wave to be launched on the lossy conducting medium 303.

In the example of FIG. 5A, ray optics are used to illustrate the complex angle trigonometry of the incident electric field (E) having a complex Brewster angle of incidence (0_{ί β}) at the Hankel crossover distance (ÆJ 315. Recall from Equation (26) that, for a lossy conducting medium, the Brewster angle is complex and specified by tan θ_ίιΒ = ε_Γ -j-^- = n. (38)

Electrically, the géométrie parameters are related by the electrical effective height (h_eff) of the charge terminal T| by = R_xx.W = h_eff — Ερβ^^φ, (39) where ψ_{ί Β} — (π/2) - θ_{ί Β} is the Brewster angle measured from the surface of the lossy conducting medium. To couple into the guided surface waveguide mode, the wave tilt ofthe electric field at the Hankel crossover distance can be expressed as the ratio of the electrical effective height and the Hankel crossover distance = tanÿ_iJ} = W_Rx. (40)

Since both the physical height (hp) and the Hankel crossover distance (R_x) are real quantifies, the angle (Ψ) of the desired guided surface wave tilt at the Hankel crossover distance (R_x) is equal to the phase (Φ) of the complex effective height (h_eff). This implies that by varying the phase at the supply point of the coil, and thus the phase shift in Equation (37), the phase, Φ, of the complex effective height can be manipulated to match the angle ofthe wave tilt, Ψ, of the guided surface waveguide mode at the Hankel crossover point 315: Φ = Ψ.

In FIG. 5A, a right triangle is depicted having an adjacent side of length R_x along the lossy conducting medium surface and a complex Brewster angle ψ_ίιΒ measured between a ray 316 extending between the Hankel crossover point 315 at R_x and the center ofthe charge terminal Ti, and the lossy conducting medium surface 317 between the Hankel crossover point 315 and the charge terminal Τ_ή. With the charge terminal Τ_Ί positioned at 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.

If the physical height of the charge terminal is decreased without changing the phase shift Φ of the effective height the resulting electric field intersects the lossy conducting medium 303 at the Brewster angle at a reduced distance from the guided surface waveguide probe 300. 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 the Brewster angle. As the height is decreased from h₃ through h₂ to h_1r 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 he physical height (/t_p) in order to excite the far-out component of the Hankel function. With the charge terminal T_q positioned at or above the effective height (/i_eyy), the lossy conducting medium 303 can be illuminated at the Brewster angle of incidence (ψ_£Β = (jt/2) - θ_{ί Β}) at or beyond the Hankel crossover distance (R_x) 315 as illustrated in FIG. 5A. 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 as mentioned above.

A guided surface waveguide probe 300 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 303 at a complex Brewster angle, thereby exciting radial surface currents by substantially mode-matching to a guided surface wave mode at (or beyond) the Hankel crossover point 315 at R_x. Referring to FIG. 7, shown is a graphical représentation of an example of a guided surface waveguide probe 300b that includes a charge terminal T_b An AC source 712 acts as the excitation source (312 of FIG. 3) for the charge terminal Tn which is coupled to the guided surface waveguide probe 300b through a feed network (309 of FIG. 3) comprising a coil 709 such as, e.g., a helical coil. In other implémentations, the AC source 712 can be inductively coupled to the coil 709 through a primary coil. In some embodiments, an impédance matching network may be included to improve and/or maximize coupling of the AC source 712 to the coil 709.

As shown in FIG. 7, the guided surface waveguide probe 300b can include the upper charge terminal Ti (e.g., a sphere at height hp) that is positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 303. A second medium 306 is located above the lossy conducting medium 303. The charge terminal T| has a self-capacitance C_T. During operation, charge Q-i is imposed on the terminal T| depending on the voltage applied to the terminal at any given instant.

In the example of FIG. 7, the coil 709 is coupled to a ground stake 715 at a first end and to the charge terminal Ti via a vertical feed line conductor 718. In some implémentations, the coil connection to the charge terminal Tî can be adjusted using a tap 721 of the coil 709 as shown in FIG. 7. The coil 709 can be energized at an operating frequency by the AC source 712 through a tap 724 at a lower portion of the coil 709. In other implémentations, the AC source 712 can be inductively coupled to the coil 709 through a primary coil.

The construction and adjustment of the guided surface waveguide probe 300 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 ε_Γ), and size of the charge terminal Tî. The index of refraction can be calculated from Equations (10) and (11) as n = —jx, (41 ) where x = σ/ωε_ο with ω = 2π/. The conductivity σ and relative permittivity s_r can be determined through test measurements of the lossy conducting medium 303. The complex Brewster angle (θ_ίΒ) measured from the surface normal can also be determined from Equation (26) as

0_itB = arctan^f,. —jx), (42) or measured from the surface as shown in FIG. 5A as (43)

The wave tilt at the Hankel crossover distance (ΙΤ_Λζ) 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 -jyp, and solving for R_x as illustrated by FIG. 4. The electrical effective height can then be determined from Equation (39) using the Hankel crossover distance and the complex Brewster angle as h_eff = hpe^ = R_x tanip_iiB. (44)

As can be seen from Equation (44), the complex effective height includes a magnitude that is associated with the physical height (/ip) of the charge terminal Ti and a phase delay (Φ) that is to be associated with the angle (Ψ) of the wave tilt at the Hankel crossover distance (β_χ). With these variables and the selected charge terminal Ti configuration, it is possible to détermine the configuration of a guided surface waveguide probe 300.

With the charge terminal T| positioned at or above the physical height (Λ_ρ), the feed network (309 of FIG. 3) and/or the vertical feed line connecting the feed network to the charge terminal Ti can be adjusted to match the phase (Φ) of the charge Ch on the charge terminal T-i to the angle (Ψ) of the wave tilt (VF). The size of the charge terminal 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 D as large as practical. The size of the charge terminal Tj 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 6_C 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, Helical 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 (υ) of a wave along the coil’s longitudinal axis to the speed of light (c), or the “velocity factor,” is given by

Vf=-_C= j ¹ ---(45) 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

The principle is the same if the hélix is wound spirally or is short and fat, but Vf and Q_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 ^^h-Cv)-¹·⁰²⁷]·<

The spatial phase delay 6_y of the structure can be determined using the traveling wave phase delay of the vertical feed line conductor 718 (FIG. 7). The capacitance of a cylindrical vertical conductor above a prefect ground plane can be expressed as

Farads,(48) where h_w is the vertical length (or height) of the conductor and a is the radius (in mks units). As with the helical coil, the traveling wave phase delay of the vertical feed line conductor can be given by

0_y^f_wh_w = ^h_w = ^j-h_w,(49) where f_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, λ₀ is the wavelength at the supplied frequency, and A_w is the propagation wavelength resulting from the velocity factor V_w. For a uniform cylindrical conductor, the velocity factor is a constant with V_w « 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

Z_w = ^[/n(^)-l], (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 = 138to_a(Ü^A). (51)

Equation (51 ) implies that Z_o 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 Tq positioned over the lossy conducting medium 303 as shown in FIG. 3, the feed network 309 can be adjusted to excite the charge terminal Tq with the phase shift (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 Q, on the charge terminal Tq is coupled into a guided surface waveguide mode traveling along the surface of a lossy conducting medium 303. For example, if the Brewster angle (θ_ίΒ), the phase delay (0_y) associated with the vertical feed line conductor 718 (FIG- 7), and the configuration of the coil 709 (FIG. 7) are known, then the position of the tap 721 (FIG. 7) can be determined and adjusted to impose an oscillating charge Q) on the charge terminal Τ_Ί with phase Φ = Ψ. The position of the tap 721 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 721 can be removed to reduce the capacitive effects. The vertical wire height and/or the geometrical parameters of the helical coil may also be varied.

The coupling to the guided surface waveguide mode on the surface of the lossy conducting medium 303 can be improved and/or optimized by tuning the guided surface waveguide probe 300 for standing wave résonance with respect to a complex image plane associated with the charge Qq on the charge terminal Tq. By doing this, the performance ofthe guided surface waveguide probe 300 can be adjusted for increased and/or maximum voltage (and thus charge Q,) on the charge terminal T_b Referring back to FIG. 3, the effect of the lossy conducting medium 303 in Région 1 can be examined using image theory analysis.

Physically, an elevated charge Q₁ placed over a perfectly conducting plane attracts the free charge on the perfectly conducting plane, which then “piles up” in the région under the elevated charge Qq. The resulting distribution of “bound” electricity on the perfectly conducting plane is similarto a bell-shaped curve. The superposition ofthe potential ofthe elevated charge Qq, plus the potential ofthe 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 303 by assuming the presence of an effective image charge Qi' beneath the guided surface waveguide probe 300. The effective image charge Qq' coïncides with the charge Q) on the charge terminal Tq about a conducting image ground plane 318, as illustrated in FIG. 3. However, the image ,

I 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 T_b as they would be in the case of a perfect conductor. Rather, the lossy conducting medium 303 (e.g., a terrestrial medium) présents a phase shifted image. That is to say, the image charge Q/ is at a complex depth below the surface (or physical boundary) of the lossy conducting medium 303. 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 (H-ι) of the charge Q-ι, the conducting image ground plane 318 (representing a perfect conductor) is located at a complex depth of z - - d/2 and the image charge Q/ appears at a complex depth (/.e., the “depth” has both magnitude and phase), given by -Di = -(d/2 + d/2 + H!) ψ H_r For vertically polarized sources over the earth, ²Υε+*ο 2 d = — * γ = d_r + jdt = |d|zÇ ,(52) where

Ye = ίωμ^σ^ — ω²μ₁ε₁, and(53) k₀ = ^ω#ο^εο-(54) 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 parallel to the tangent of the conducting image ground plane 318 atz - - 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 303 is a finitely conducting earth 803 with a physical boundary 806. The finitely conducting earth 803 may be replaced by a perfectly conducting image ground plane 809 as shown in FIG.8B, which is located at a complex depth z_± below the physical boundary 806. This équivalent représentation exhibits the same impédance when looking down into the interface at the physical boundary 806. The équivalent représentation of FIG. 8B can be modeled as an équivalent transmission line, as shown in FIG. 8C. 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 z_x 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) 812 are

Yo = Ϊω^μ_οε_ο = 0 + Jβ_ΰ . and (55)

(56)

In the lossy earth 803, the propagation constant and wave intrinsic impédance are

Ye = + ί^ωει) > ^and (57)

Z_e = (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 γ₀, and whose length is ζ_χ. As such, the image ground plane impédance Z_in seen at the interface for the shorted transmission line of FIG. 80 is given by ^zin = ^zo 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_± gives the distance to a short circuit (the perfectly conducting image ground plane 809) as z_r = —tanh^-1 (—) = —tanh^-1(60) ^Λ γ₀ \Z₀J γ_ο \y_eJ y_e '' 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 812, the propagation constant is Yo = ϊβο> ^s° ^zin = î^zo tan^Zi (which is a purely imaginary quantity for a real z_r), but z_e is a complex value if σ Φ 0. Therefore, Z_in = Z_e only when z_± is a complex distance.

Since the équivalent représentation of FIG. 8B includes a perfectly conducting image ground plane 809, the image depth for a charge or current Iving at the surface of the earth (physical boundary 806) is equal to distance z_± on the other side of the image ground plane 809, 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 809 can be approximated by d = 2zi«-. (61)

Additionally, the “image charge” will be “equal and opposite to the real charge, so the potential ofthe perfectly conducting image ground plane 809 at depth ζ_γ = - d/2 will be zéro. If a charge Q-ι is elevated a distance H-ι above the surface ofthe earth as illustrated in FIG. 3, then the image charge Q/ résides at a complex distance of Di = d + Hi below the surface, or a complex distance of d/2 + Hj below the image ground plane 318. The guided surface waveguide probe 300b of FIG. 7 can be modeled as an équivalent single-wire transmission line image plane model that can be based upon the perfectly conducting image ground plane 809 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. 80.

In the équivalent image plane models of FIGS. 9A and 9B, ¢ = + the traveling wave phase delay of the guided surface waveguide probe 300 referenced to earth (or the lossy conducting medium 303), = β_ρΗ is the electrical length of the coil 709 (FIG. 7), of physical length H, expressed in degrees, 6_y = p_wh_w is the electrical length of the vertical feed line conductor 718 (FIG. 7), of physical length h_w, expressed in degrees, and θ_α - β₀ d/2 is the phase shift between the image ground plane 809 and the physical boundary 806 of the earth (or lossy conducting medium 303). In the example of FIGS. 9A and 9B, Z_w is the characteristic impédance of the elevated vertical feed line conductor 718 in ohms, Z_c is the characteristic impédance of the coil 709 in ohms, and Z_o is the characteristic impédance of free space.

At the base of the guided surface waveguide probe 300, the impédance seen “looking up” into the structure is = Z_base. With a load impédance of:

= Ά-.

(62) where C_T is the self-capacitance of the charge terminal Ti, the impédance seen “looking up into the vertical feed line conductor 718 (FIG. 7) is given by:

y y tanhCj/Aykw) y tanh(jfly)/63) ^{2 lV} Zu,+ZLtanh(jfiwhw) ^w Zw+Z_L tanh(j6_y) ’' ' and the impédance seen “looking up” into the coil 709 (FIG. 7) is given by:

_ Z₂+Z_ctanh(jp_pff) _ Z₂+Z_ctanh(je_c) ^{base c} Zc+Z2 tanh(y/îpH) ^c Zc+Z₂ tanh(j0_c) ’'

At the base of the guided surface waveguide probe 300, the impédance seen “looking down” into the lossy conducting medium 303 is Z± = Z_in, which is given by:

Z_in = 4 ^Zj+Z°^t3n^⁷î°Îl/²Î! = 4 tanh(/0_d) ,(65) ^{m 0} Z₀+Z_s tanin |j j3₀ (d/²)] ^{0 V aV} where Z_s — 0.

Neglecting losses, the équivalent image plane model can be tuned to résonance when

Z_t + Z_T = 0 at the physical boundary 806. Or, in the low loss case, Xj. + X_T — 0 at the physical boundary 806, where X is the corresponding reactive component. Thus, the impédance at the physical boundary 806 “looking up” into the guided surface waveguide probe 300 is the conjugate of the impédance at the physical boundary 806 “looking down” into the lossy conducting medium 303. By adjusting 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 of the lossy conducting medium 303 (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 809. 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 Ti , and by équations (1)-(3) and (16) maximizes the propagating surface wave.

It follows from the Hankel solutions, that the guided surface wave excited by the guided surface waveguide probe 300 is an outward propagating travelinq wave. The source distribution along the feed network 309 between the charge terminal Τ_Ί and the ground stake 715 ofthe guided surface waveguide probe 300 (FIGS. 3 and 7) is actually composed of a superposition of a travelinq wave plus a standing wave on the structure. With the charge terminal T, positioned at or above the physical height hp, the phase delay of the traveling wave moving through the feed network 309 is matched to the angle of the wave tilt associated with the lossy conducting medium 303. This mode-matching allows the traveling wave to be launched along the lossy conducting medium 303. Once the phase delay has been established for the traveling wave, the load impédance Z_L ofthe charge terminal Τ_Ί is adjusted to bring the probe structure into standing wave résonance with respect to the image ground plane (318 of FIG. 3 or 809 of FIG. 8), which is at a complex depth of - d/2. In that case, the impédance seen from the image ground plane has zéro 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). The effect of such discontinuous phase jumps can be seen in the Smith chart plots in FIG. 12.

As a resuit ofthis 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, but is especially ciear in FIG. 12 where the discontinuities in the impédance ratios provide large jumps in phase between the different plotted sections on the

Smith chart. The impédance discontinuity provides a substantial phase shift where the sections are joined together.

Referring to FIG. 10, shown is a flow chart illustrating an example of adjusting a guided surface waveguide probe 300 (FIG. 3) 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 303 (FIG. 3). Beginning with 1003, the charge terminal T-, of the guided surface waveguide probe 300 is positioned at a defined height above a lossy conducting medium 303. Utilizing the characteristics of the lossy conducting medium 303 and the operating frequency of the guided surface waveguide probe 300, 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 (0_iB) can then be determined from Equation (42). The physical height (h_p) of the charge terminal Ti can then be determined from Equation (44). The charge terminal T| should be at or higher than the physical height (ftp) 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_1r the height should be at least four times the spherical drameter (or équivalent spherical diameter) of the charge terminal N.

At 1006, the electrical phase delay Φ of the elevated charge Ch on the charge terminal Τ_ή is matched to the complex wave tilt angle Ψ. The phase delay (0_C) of the helical coil and/or the phase delay (θ_γ) of the vertical feed line conductor can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (I¥). Based on Equation (31 ), the angle (Ψ) of the wave tilt can be determined from:

W = ÿ = —^- = -= |ΜΖ|β7^ψ.

E_z tan0i_/B n (66)

The electrical phase 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 — R_c + 0_y can be adjusted by varying the geometrical parameters of the coil 709 (FIG. 7) and/or the length (or height) of the vertical feed line conductor 718 (FIG. 7). 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 303.

Next at 1009, the load impédance of the charge terminal Tj is tuned to resonate the équivalent image plane model of the guided surface waveguide probe 300. The depth (d/2) of the conducting image ground plane 809 (or 318 of FIG. 3) can be determined using Equations (52), (53) and (54) and the values of the lossy conducting medium 303 (e.g., the earth), which can be measured. Using that depth, the phase shift (0_d) between the image ground plane 809 and the physical boundary 806 of the lossy conducting medium 303 can be determined using 6_d = β₀ d/2. The impédance (Z_in) as seen “looking down” into the lossy conducting medium 303 can then be determined using Equation (65). This résonance relationship can be considered to maximize the launched surface waves.

Based upon the adjusted parameters of the coil 709 and the length of the vertical feed line conductor 718, the velocity factor, phase delay, and impédance of the coil 709 and vertical feed line conductor 718 can be determined using Equations (45) through (51). In addition, the self-capacitance (C_T) of the charge terminal T| can be determined using, e.g., Equation (24). The propagation factor (β_ρ) of the coil 709 can be determined using Equation (35) and the propagation phase constant (f_w) for the vertical feed line conductor 718 can be determined using Equation (49). Using the self-capacitance and the determined values of the coil 709 and vertical feed line conductor 718, the impédance (Z_base) of the guided surface waveguide probe 300 as seen “looking up into the coil 709 can be determined using Equations (62), (63) and (64)The équivalent image plane model of the guided surface waveguide probe 300 can be tuned to résonance by adjusting the load impédance Z_L such that the reactance component X_ftase of cancels out the reactance component X_in of Z_in, orX_base +X_ln = 0. Thus, the impédance at the physical boundary 806 “looking up” into the guided surface waveguide probe 300 is the conjugale of the impédance at the physical boundary 806 “looking down” into the lossy conducting medium 303. The load impédance Z_L can be adjusted by varying the capacitance (C_T) of the charge terminal Tî without changing the electrical phase delay Φ = 8_C + 0_y of the charge terminal Tp 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 809 (or 318). In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 303 (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 300 comprising a top-loaded vertical stub of physical height ft._p with a charge terminal Ti at the top, where the charge terminal Tj is excited through a helical coil and vertical feed line conductor at an operational frequency (f_o) of 1.85 MHz. With a height (H-ι) of 16 feet and the lossy conducting medium 303 (i.e., earth) having a relative permittivity of s_r = 15 and a conductivity of = 0.010 mhos/m, several surface wave propagation parameters can be calculated for/_D = 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 hp = 5.5 feet, which is well below the actual height of the charge terminal T-|. While a charge terminal height of Hi = 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 Τ_Ί providing greater field strength and excitation of the traveling wave.

The wave length can be determined as:

λ_ο = = 162.162 meters,(67) where c is the speed of light. The complex index of refraction is:

n - — jx = 7.529 -j 6.546,(68) from Equation (41), where x = σ_ί/ωε₀ with ω = 2zr/_o, and the complex Brewster angle is: = arctanQs_r - jx) = 85.6 - j 3.744°.(69) from Equation (42). Using Equation (66), the wave tilt values can be determined to be: W = —1— = - = |Ρ7|β>^ψ = 0.101e^j40-⁶¹⁴°.(70) tandis n ¹S'/

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:

ftv = F = FT = °·^{042 m-1}· (71)

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

By = R_wh_w « R_wh_p = 11.640°. (72)

By adjusting the phase delay of the helical coil so that θ_(: - 28.974° = 40.614° - 11.640°, Φ will equal Ψ to match the guided surface waveguide mode. To illustrate the relationship between Φ and Ψ, 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:

i

V_f = | = 0.069 ,(73) and the propagation factor from Equation (35) is:

^=^ = °.⁵⁶⁴^C

With = 28.974°, the axial length of the solenoidal hélix (H) can be determined using Equation (46) such that: a

H = / = 35.2732 inches .(75)

Pp

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 = = Ψ), the load impédance (Z_L) ofthe charge terminal U can be adjusted for standing wave résonance ofthe équivalent image plane model ofthe guided surface wave probe 300. From the measured permittivity, conductivity and permeabiiity of the earth, the radial propagation constant can be determined using Equation (57) γ_ε = + 1^ωει) ~ θ-25 + j 0.292 m^-1,(76)

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

0_d = β₀(ά/2) = 4.015 -j 4.73°.(78)

Using Equation (65), the impédance seen “looking down” into the lossy conducting medium 303 (i.e., earth) can be determined as:

Z_in = Z_o tanh(/0_d) = R_in +jX_in = 31.191 + j 26.27 ohms. (79)

By matching the reactive component (X_in) seen “looking down” into the lossy conducting medium 303 with the reactive component (Xbase)^seen “looking up” into the guided surface wave probe 300, the coupling into the guided surface waveguide mode may be maximized. This can be accomplished by adjusting the capacitance of the charge terminal Τ_Ί 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_L = -H— = -j 1392 ohms, (80) juC_T ^{J K z} and the reactive components at the boundary are matched.

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

Z_w = 138 log (^1-12^²°) = 537.534 ohms,(81 ) and the impédance seen “looking up” into the vertical feed line conductor is given by Equation (63) as:

Z₂ = Z_w = —j 835.438 ohms.(82) ^{4 w} Zw+ZLtanhQftj,) ^Jv

Using Equation (47), the characteristic impédance ofthe helical coil is given as z_c = pTi - 1.027] = 1446 ohms,(83) and the impédance seen “looking up” into the coil at the base is given by Equation (64) as:

Z_base = Z_c ^{Zz+Zc tan}Î/.^ = ~j 26.271 ohms. (84) oase c z_c+z₂tanh(je_c) ^{J v} '

When compared to the solution of Equation (79), it can be seen that the reactive components are opposite and approximately equal, and thus are conjugates of each other. Thus, the impédance (Z_ip) seen “iooking 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 + JO.

Referring to FIG. 12, shown is a Smith chart 1200 that graphically illustrâtes an example of the effect of the discontinuous phase jumps on the impédance (Z_ip) seen “Iooking up” into the équivalent image plane model of FIG. 9B. First, because of the transition between the charge terminal and the vertical feed line conductor, the actual load impédance Z_L is normalized with respect to the characteristic impédance (Z_w) of the vertical feed line conductor and is entered on Smith chart 1200 at point 1203 (Z_z/Z_w). The normalized impédance is then transferred along the vertical feed line section by an electrical distance of Q_y - f_wh_w « f_wh_p (which is clockwise through an angle of 20_y on the Smith chart 1200) to point 1206 (Z₂/Z_w). The impédance at point 1206 is now converted to the actual impédance (Z₂) seen “Iooking up” into the vertical feed line conductor using Z_w.

Second, because of the transition between the vertical feed line conductor and the helical coil, the impédance Z₂ is then normalized with respect to the characteristic impédance (Z_c) of the helical coil. This normalized impédance can now be entered on the Smith chart 1200 at point 1209 (Z₂/Z_c) and transferred along the helical coil transmission line section by an electrical distance 0_C = β_ρΗ, (which is clockwise through an angle equal to 26_c on the Smith chart 1200) to point 1212 (Z_base/Z_c). The jump between point 1206 and point 1209 is a resuit of the discontinuity in the impédance ratios. The impédance Iooking into the base of the coil at point 1212 is then converted to the actual impédance (Z_base) seen “Iooking up” into the base of the coil (or the guided surface wave probe 300) using Z_c.

Third, because of the transition between the helical coil and the lossy conducting medium, the impédance at Z_base is then normalized with respect to the characteristic impédance (Z_o) of the modeled image space below the physical boundary of the lossy conducting medium (e.g., the ground surface). This normalized impédance can now be entered on the Smith chart 1200 at point 1215 (Z_base/Z₀), and transferred along the subsurface image transmission line section by an electrical distance θ_ά = β₀ d/2 (which is clockwise through an angle equal to 20_d on the Smith chart 1200) to point 1218 (Z_ip/Z₀). The jump between point 1212 and point 1215 is a resuit of the discontinuity in the impédance ratios. The impédance Iooking into the subsurface image transmission line at point 1218 is now converted to an actual impédance (Z_£p) using Z_o. When this System is resonated, the impédance at point 1218 is Z_ip = R_ip + j 0. On the Smith chart 1200, Z_base/Z₀ is a larger reactance than Z_base/Z_c. This is because the characteristic impédance (Z_c) of a helical coil is considerably larger than the characteristic impédance z_o of free space.

When properly adjusted and tuned, the oscillations on a structure of sufficient physical height are actually composed of a traveling wave, which is phase delayed to match the angle of the wave tilt associated with the lossy conducting medium (Φ - Ψ), plus a standing wave which is electrically brought into résonance (Z_ip = R + JO) by a combination of the phase delays of the transmission line sections of the guided surface waveguide probe 300 plus the phase discontinurties due to jumps in the ratios ofthe characteristic impédances, as illustrated on the Smith chart 1200 of FIG. 12. The above example illustrâtes how the three considérations discussed above can be satisfied for launching guided surface traveling waves on the lossy conducting medium.

Field strength measurements were carried out to verify the ability of the guided surface waveguide probe 300b (FIG. 7) to couple into a guided surface wave or a transmission line mode. A 70 pF circular plate capacitor was elevated to a height of 16 feet (4.88 meters) and charged to 30 volts (peak-to-peak) at a frequency of f₀ — 1.85 MHz (λ₀ = 162.162 meters) over soil with constitutive parameters, measured at/₀, with a relative permittivity of s_r = 15 and a conductivity of oy - 0.010 mhos/m. The measured data (documented with a NIST-traceable field strength meter) is tabulated below in TABLE 1.

Range (miles)	Measured FS w/ FIM- 41 (pV/m)	Predicted FS (pV/m)	Percent Différence
0.6	3400	3415	-0.44%
2	1300	1296	+0.31%
3	840	814	+3.19%
4	560	542	+3.32%
5	380	373	+1.88%
6	270	262	+3.05%
7	190	187	+ 1.60%
8	140	134	+4.48%
9	100	97	+3.09%
10	70	71	-1.41%

TABLE 1

Referring to FIG. 13, shown is the measured field strength in mV/m (circles) vs. range (in miles) with respect to the theoretical Zenneck surface wave field strengths for 100% and 85 % electric charge, as well as the conventional Norton radiated ground-wave predicted for a 16 foot top-loaded vertical mast (monopole at 2.5% radiation efficiency). The quantity h corresponds to the height of the vertical conducting mast for Norton ground wave radiation with a 55 ohm ground stake. The predicted Zenneck fields were calculated from Equation (3), and the standard Norton ground wave was calculated by conventional means. A statistical analysis gave a minimized RMS déviation between the measured and theoretical fields for an electrical efficiency of 97.4%.

When the electric fields produced by a guided surface waveguide probe 300 (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 ofthe guided electromagnetic field has a characteristic exponential decay of e^_<ld/Vd and exhibits a distinctive knee 109 on the loglog scale.

In summary, both analytically and experimentally, the traveling wave comportent on the structure ofthe guided surface waveguide probe 300 has a phase delay (Φ) at its upper terminal that matches the angle (Ψ) of the wave tilt ofthe 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 ofthe guided surface waveguide probe 300 has a Vma_X at the charge terminal Ti and a V_M,_N down at the image plane 809 (FIG. 8) where Z_lp = R_lp + j 0 at a complex depth of z = - d/2, not at the connection at the physical boundary 806 ofthe lossy conducting medium 303 (FIG. 8). Lastly, the charge terminal is of sufficient height Ht of FIG. 3 (h > tani/>_i;B) so that electromagnetic waves incident onto the lossy conducting medium 303 at the complex Brewster angle do so out at a distance (> R_x) where the 1/Vr 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 nextto FIGS. 14A, 14B, 14C and 15, shown are examples of generaiized receive circuits for using the surface-guided waves in wireless power delivery Systems. FIGS. 14A and 14B-14C include a iinear probe 1403 and a tuned resonator 1406, respectively. FIG. 15 is a magnetic coil 1409 according to various embodiments ofthe present disclosure.

According to various embodiments, each one ofthe Iinear probe 1403, the tuned resonator 1406, and the magnetic coil 1409 may be employed to receive power transmitted in the form of a guided surface wave on the surface of a lossy conducting medium 303 (FIG. 3) according to various embodiments. As mentioned above, in one embodiment the lossy conducting medium 303 comprises a terrestrial medium (or earth).

With spécifie reference to FIG. 14A, the open-circuit terminal voltage at the output terminais 1413 of the linear probe 1403 dépends upon the effective height of the linear probe 1403. To this end, the terminal point voltage may be calculated as

Vt= (85) where E_inc is the strength of the incident electric field induced on the linear probe 1403 in Volts per meter, dl is an element of intégration along the direction of the linear probe 1403, and h_e is the effective height of the linear probe 1403. An electrical load 1416 is coupled to the output terminais 1413 through an impédance matching network 1419.

When the linear probe 1403 is subjected to a guided surface wave as described above, a voltage is developed across the output terminais 1413 that may be applied to the electrical load 1416 through a conjugate impédance matching network 1419 as the case may be. In order to facilitate the flow of power to the electrical load 1416, the electrical load 1416 should be substantially impédance matched to the linear probe 1403 as will be described below. Referring to FIG. 14B, a ground current excited coil 1406a possessing a phase shift 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 303. The charge terminal T_R has a selfcapacitance C_R. In addition, there may also be a bound capacitance (not shown) between the charge terminal T_R and the lossy conducting medium 303 depending on the height of the charge terminal T_R above the lossy conducting medium 303. The bound capacitance should preferably be minimized as much as is practicable, although this may not be entirely necessary in every instance of a guided surface waveguide probe 300.

The tuned resonator 1406a also includes a receiver network comprising a coil L_R having a phase shift Φ. One end of the coil L_R is coupled to the charge terminal T_R, and the other end of the coil L_R is coupled to the lossy conducting medium 303. 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 1406a (which may also be referred to as tuned resonator L_R-C_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 1406a 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 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//ωϋ_Λ. Note that the total capacitance of the structure 1406a may also include capacitance between the charge terminal T_R and the lossy conducting medium 303, where the total capacitance of the structure 1406a may be calculated from both the self-capacitance 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 303 as previously discussed.

The inductive reactance presented by a discrete-element coil L_R may be calculated as jtoL, where L is the lumped-element inductance of the coil L_R. If the coil 1__R is a distributed element, its équivalent terminal-point inductive reactance may be determined by conventional approaches. To tune the structure 1406a, 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 1423 may be inserted between the probe and the electrical load 1426 in order to couple power to the load. Inserting the impédance matching network 1423 between the probe terminais 1421 and the electrical load 1426 can effect a conjugate-match condition for maximum power transfer to the eiectrical load 1426.

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 1426. To this end, an electrical load 1426 may be coupled to the structure 1406a 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. 14B, 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 receiving structure 1406a comprises a series-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 optrmized for coupling power from a surface-guided wave. The guided surface wave excited by a guided surface waveguide probe 300 on the surface of the lossy conducting medium 303 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 303 in the vicinity of the receiving structure. To accomplish this, the phase delay (Φ) of the 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 1406a of FIG. 14B, 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 303, the total phase shift 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 1406a. From Equation (22), it can be seen that the wave tilt asymptotically passes to

W - \W\e^ - ---> . ¹ , (86) ' ' E_z p->co ^V ’ y ^r ωε₀ where e_r comprises the relative permittivity and is the conductivity of the lossy conducting medium 303 at the location of the receiving structure, ε_ο is the permittivity of free space, and ω = 2nf, where f is the frequency of excitation. Thus, the wave tilt angle (Ψ) can be determined from Equation (86).

The total phase shift (Φ = 0_C + 0_y) of the tuned resonator 1406a includes both the phase delay (0_C) through the coil L_R and the phase delay of the vertical supply line (0_y). The spatial phase delay along the conductor length l_w of the vertical supply line can be given by 9_y = B_wl_w, where 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 = f_pl_c, with a physical length of l_c and a propagation factor of ⁼ Γ ⁼ (⁸⁷>

where V_f is the velocity factor on the structure, A_o is the wavelength at the supplied frequency, and λ_ρ is the propagation wavelength resulting from the velocity factor V_f. One or both of the phase delays (0_C + 9_y) can be adjusted to match the phase shift Φ to the angle (Ψ) of the wave tilt. For example, a tap position may be adjusted on the coil L_R of FIG. 14B to adjust the coil phase delay (0_C) to match the total phase shift to the wave tilt angle (Φ — Ψ). For example, a portion of the coil can be bypassed by the tap connection as illustrated in FIG.

14B. The vertical supply line conductor can also be connected to the coil L_R via a tap, whose position on the coil may be adjusted to match the total phase shift to the angle of the wave tilt.

Once the phase delay (Φ) of the tuned resonator 1406a 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 ofthe charge terminal η without changing the traveling wave phase delays of the coil L_R and vertical supply line. The adjustments are similar to those described with respect to FIGS. 9A and 9B and the Smith chart of FIG. 12.

The impédance seen “looking down” into the lossy conducting medium 303 to the complex image plane is given by:

2in - Rin +jXtn = tanh(j/î₀ (d/2)),(88) where β₀ = ω/μ₀ε_ο. For verticallypolarized sources overthe earth, the depth ofthe complex image plane can be given by:

d/2 = l/T/â/M - ω²/ζ₁ε₁ ,(89) where μ₄ is the permeability of the lossy conducting medium 303 and = ε_Γε₀. At the base ofthe tuned resonator 1406a, the impédance seen “looking up” into the receiving structure is Z₄ = Z_base as illustrated in FIG. 9A. With a terminal impédance of:

^Zr=7^’(90) where C_R is the self-capacitance ofthe charge terminal T_R, the impédance seen “looking up” into the vertical supply line conductor of the tuned resonator 1406a is given by:

g - g tanhÇ/'/Î^h^) tanh (;&>,) ² “ ^Z^^z_fit_anh(;e_y) ’ '( and the impédance seen “looking up” into the coil L_R of the tuned resonator 1406a is given by:

Z base ~ Phase T jXbase

- z Zz+Zr tanh(jj?_pH) _ Z₂+Z_R tanhÇ/gç) ^R ZR +z2 tanh(;£ptf) ^c zR+z₂ tanh(/0_c) ' (92)

By matching the reactive component (X_in) seen “looking down” into the lossy conducting medium 303 with the reactive component (X_base) seen “looking up” into the tuned resonator 1406a, the coupling into the guided surface waveguide mode may be maxrmized. Referring next to FIG. 14C, shown is an example of a tuned resonator 1406b that does not include a charge terminal T_R at the top of the receiving structure. In this embodiment, the tuned resonator 1406b does not include a vertical supply line coupled between the coil L_R and the charge terminal T_R. Thus, the total phase shift (Φ) of the tuned resonator 1406b includes only the phase delay (0_C) through the coil L_R, As with the tuned resonator 1406a of FIG. 14B, the coil phase delay 0_ccan be adjusted to match the angle (Ψ) ofthe wave tilt determined from Equation (86), 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.

Referring to FIG. 14D, shown is a flow chart illustratrng 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 303. Beginning with 1453, if the receiving structure includes a charge terminal T_R (e.g., of the tuned resonator 1406a of FIG. 14B), then the charge terminal T_R is positioned at a defined height above a lossy conducting medium 303 at 1456. As the surface guided wave has been established by a guided surface waveguide probe 300, the physical height (h_p) of the charge terminal T_R height 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 1406b of FIG. 14C), then the flow proceeds to 1459.

At 1459, 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 medium 303. The phase delay (0_C) of the helical coil and/or the phase delay (0_y) of the vertical supply line can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (PF). The angle (Ψ) of the wave tilt can be determined from Equation (86). The electrical phase Φ can then be matched to the angle of the wave tilt. For example, the electrical phase delay Φ - 9_C + 6_y 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 1462, the load impédance of the charge terminal T_R can be tuned to resonate the équivalent image plane model of the tuned resonator 1406a. The depth (d/2) of the conducting image ground plane 809 (FIG. 9A) below the receiving structure can be determined using Equation (89) and the values of the lossy conducting medium 303 (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 809 and the physical boundary 806 (FIG. 9A) of the lossy conducting medium 303 can be determined using θ_ά = β_ο d/2. The impédance (Z_in) as seen “looking down” into the lossy conducting medium 303 can then be determined using Equation (88). 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 vélocity 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 (£_p) of the coil L_R can be determined using Equation (87) and the propagation phase constant 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_base) of the tuned resonator 1406a as seen “looking up” into the coil L_R can be determined using Equations (90), (91), and (92).

The équivalent image plane model of FIG. 9A also applies to the tuned resonator 1406a of FIG. 14B. The tuned resonator 1406a 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 Xbase + Xin - θ· Thus, the impédance at the physical boundary 806 (FIG. 9A) “looking up” into the coil of the tuned resonator 1406a is the conjugate of the impédance at the physical boundary 806 “looking down” into the lossy conducting medium 303. The load impédance Z_R can be adjusted by varying the capacitance (C_R) of the charge terminal T_R without changing the electrical phase delay Φ = d_c + θ_γ seen by the charge terminal T_R. An itérative approach may be taken to tune the load impédance Z_R for résonance of the équivalent image plane model with respect to the conducting image ground plane 809. In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 303 (e.g., earth) can be improved and/or maximized.

Referring to FIG. 15, the magnetic coil 1409 comprises a receive circuit that is coupled through an impédance matching network 1433 to an electrical load 1436. In order to facilitate réception and/or extraction of electrical power from a guided surface wave, the magnetic coil 1409 may be positioned so that the magnetic flux of the guided surface wave, Η_φ, passes through the magnetic coil 1409, thereby inducing a current in the magnetic coil 1409 and producing a terminal point voltage at its output terminais 1429. The magnetic flux of the guided surface wave coupled to a single turn coil is expressed by

F = fi μ_Γμ_οΗ ndA (93) where T is the coupled magnetic flux, is the effective relative permeability of the core of the magnetic coil 1409, 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 is the area enclosed by each loop. For an N-turn magnetic coil 1409 oriented for maximum coupling to an incident magnetic field that is uniform over the cross-sectional area of the magnetic coil 1409, the open-circuit induced voltage appearing at the output terminais 1429 of the magnetic coil 1409 is v = -N Μ - -/ωμ_Γμ₀/νHA_CS, (94) where the variables are defined above. The magnetic coil 1409 may be tuned to the guided surface wave frequency either as a distributed resonator or with an external capacitor across its output terminais 1429, as the case may be, and then impedance-matched to an external electrical load 1436 through a conjugate impédance matching network 1433.

Assuming that the resulting circuit presented by the magnetic coil 1409 and the electrical load 1436 are properly adjusted and conjugate impédance matched, via impédance matching network 1433, then the current induced in the magnetic coil 1409 may be employed to optimally power the electrical load 1436. The receive circuit presented by the magnetic coil 1409 provides an advantage in that it does not hâve to be physically connected to the ground. With référencé to FIGS. 14A, 14B, 14C and 15, the receive circuits presented by the linear probe 1403, the mode-matched structure 1406, and the magnetic coil 1409 each facilitate receiving electrical power transmitted from any one of the embodiments of guided surface waveguide probes 300 described above. To this end, the energy received may be used to supply power to an electrical load 1416/1426/1436 via a conjugate matching network as can be appreciated. This contrasts with the signais that may be received in a receiver that were transmitted in 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 300 described above that the receive circuits presented by the linear probe 1403, the mode-matched structure 1406, and the magnetic coil 1409 will load the excitation source 312 (FIG. 3) that is applied to the guided surface waveguide probe 300, 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 300 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 300 and one or more receive circuits in the form of the linear probe 1403, the tuned mode-matched structure 1406, and/or the magnetic coil 1409 can together make up a wireless distribution system. Given that the distance of transmission of a guided surface wave using a guided surface waveguide probe 300 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 present 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 mutualreactance coupled near-field Systems. The wireless-power system disclosed herein probecouples to the novel surface-guided transmission line mode, which is équivalent to deiivering 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 Unes at 60 Hz, ali 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.

Referring next to FIG. 16A shown is a schematic that représente the linear probe 1403 and the mode-matched structure 1406. FIG 16B shows a schematic that represents the magnetic coil 1409. The linear probe 1403 and the mode-matched structure 1406 may each be considered a Thevenin équivalent represented by an open-circuit terminal voltage source V_s and a dead network terminal point impédance Z_s. The magnetic coil 1409 may be viewed as a Norton équivalent represented by a short-circuit terminal current source l_s and a dead network terminal point impédance Z_s. Each electrical ioad 1416/1426/1436 (FIGS. 14A, 14B and 15) may be represented by a load impédance Z_L. The source impédance Z_s comprises both real and imaginary components and takes the form Z_s = Rs + jX_s.

According to one embodiment, the electrical load 1416/1426/1436 is impédance matched to each receive circuit, respectively. Specifically, each electrical load 1416/1426/1436 présents through a respective impédance matching network 1419/1423/1433 a load on the probe network specified as Z_L' expressed as Z_L' = R_L' + j XJ, which will be equal to ZJ = Z_s * = R_s - j Xs, where the presented load impédance Z_L' is the complex conjugate of the actual source impédance Z_s. The conjugate match theorem, which States that if, in a cascaded network, a conjugate match occurs at any terminal pair then it will occur at ail terminal pairs, then asserts that the actual electrical load 1416/1426/1436 will also see a conjugate match to its impédance, Z_L'. See Everitt, W.L. and G.E. Anner, Communication Engineering, McGraw-Hill, 3^rd édition, 1956, p. 407. This ensures that the respective electrical load 1416/1426/1436 is impédance matched to the respective receive circuit and that maximum power transfer is established to the respective electrical load 1416/1426/1436.

Operation of a guided surface waveguide probe 300 may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 300. For example, an adaptive probe control System 321 (FIG. 3) can be used to control the feed network 309 and/or the charge terminal T-ι to control the operation of the guided surface waveguide probe 300. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 303 (e.g., conductivity σ and relative permittivity ε_Γ), variations in field strength and/or variations in loading of the guided surface waveguide probe 300. As can be seen from Equations (31), (41) and (42), the index of refraction (n), the complex Brewster angle (d_iiB), and the wave tilt (|W\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 321. The probe control system 321 can then make one or more adjustments to the guided surface waveguide probe 300 to maintain specified operational conditions for the guided surface waveguide probe 300. For instance, as the moisture and température vary, the conductivity of the soi! wili also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe 300. 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 300.

FIG. 17A shows an example of a conductivity measurement probe that can be installed for monitoring changes in soil conductivity. As shown in FIG. 17A, a sériés of measurement probes are inserted along a straight line in the soil. For example, the probes may be 9/16-inch diameter rods with a pénétration depth of 12 inches or more, and spaced apart by d = 18 inches. DS1 is a 100 Watt light bulb and R1 is a 5 Watt, 14.6 Ohm résistance. By applying an AC voltage to the circuit and measuring V1 across the résistance and V2 across the center probes, the conductivity can be determined by the weighted ratio of σ = 21(V1/V2). The measurements can be filtered to obtain measurements related only to the AC voltage supply frequency. Different configurations using other voltages, frequencies, probe sizes, depths and/or spacing may also be utilized.

Open wire line probes can also be used to measure conductivity and permittivity of the soil.

As illustrated in FIG. 17B, impédance is measured between the tops of two rods inserted into the soil (lossy medium) using, e.g., an impédance analyzer. If an impédance analyzer is utilized, measurements (R + JX) can be made over a range of frequencies and the conductivity and permittivity determined from the frequency dépendent measurements using ^σ=^and =wïl^J· ⁽⁹⁵>

where C_o is the capacitance in pF of the probe in air.

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 321 (FIG. 3). The information may be communicated to the probe control System 321 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 321 may evaluate the variation in the index of refraction (n), the complex Brewster angle (d_iB), and/or the wave tilt (|W|e^) and adjust the guided surface waveguide probe 300 to maintain the phase delay (Φ) of the feed network 309 equal to the wave tilt angle (Ψ) and/or maintain résonance of the 46 équivalent image plane model of the guided surface waveguide probe 300. This can be accomplished by adjusting, e.g., 6_y, 6_C and/or C_T. For instance, the probe control system 321 can adjust the self-capacitance ofthe charge terminal X orthe phase delay (6_y, 6_C) applied to the charge terminal X to maintain the electrical launching efficiency ofthe guided surface wave at or near its maximum. The phase applied to the charge terminal T-, can be adjusted by varying the tap position on the coil 709, and/or by including a plurality of predefined taps along the coil 709 and switching between the different predefined tap locations to maximize the launching efficiency.

Field or field strength (FS) meters (e.g., a FIM-41 FS meter, Potomac Instruments, Inc., Silver Spring, MD) may also be distributed about the guided surface waveguide probe 300 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 in the field strength (e.g., electric field strength) and communicate that information to the probe control system 321. The information may be communicated to the probe control system 321 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 300 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 (¢ = 65, + 6_C) applied to the charge terminal X can be adjusted to match the wave tilt angle (Ψ). By adjusting one or both phase delays, the guided surface waveguide probe 300 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 709 (FIG. 7) to change the phase delay supplied to the charge terminal T1. The voltage level supplied to the charge terminal Ti 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 312 (FIG. 3) or by adjusting or reconfiguring the feed network 309 (FIG. 3). For instance, the position of the tap 724 (FIG. 7) for the AC source 712 (FIG. 7) can be adjusted to increase the voltage seen by the charge terminal Ti- 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 300.

Referring to FIG. 18, shown is an example of an adaptive control system 330 including the probe control system 321 of FIG. 3, which is configured to adjust the operation of a guided surface waveguide probe 300, based upon monitored conditions. As in FIGS. 3 and 7, an AC source 712 acts as the excitation source (312 of FIG. 3) for the charge terminal T,. The AC source 712 is coupled to the guided surface waveguide probe 400d through a feed network (309 of FIG. 3) comprising a coil 709. The AC source 712 can be connected across a lower portion of the coil 709 through a tap 724, as shown in FIG. 7, or can be inductively coupled to the coil 709 by way of a primary coil. The coil 709 can be coupled to a ground stake 715 (FIG.

7) at a first end and the charge terminal Ti at a second end. In some implémentations, the connection to the charge terminal Τ_Ί can be adjusted using a tap 721 (FIG. 7) at the second end of the coil 709. An ammeter located between the coil 709 and ground stake 715 can be used to provide an indication of the magnitude of the current flow (/₀) at the base of the guided surface waveguide probe 300. Alternatively, a current clamp may be used around the conductor coupled to the ground stake 715 to obtain an indication of the magnitude of the current flow (/₀).

The probe control System 321 can be implemented with hardware, firmware, software executed by hardware, or a combination thereof. For example, the probe control system 321 can include processrng circuitry 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 400 based upon monitored conditions. The probe control system 321 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 321 may comprise, for example, a computer system such as a server, desktop computer, laptop, or other System with like capability.

The adaptive control system 330 can include one or more ground parameter meter(s) 333 such as, but not limited to, a conductivity measurement probe of FIG. 17A and/or an open wire probe of FIG. 17B. The ground parameter meter(s) 333 can be distributed about the guided surface waveguide probe 300 at, e.g., about the Hankel crossover distance (R_x) associated with the probe operating frequency. For example, an open wire probe of FIG. 17B may be located in each quadrant around the guided surface waveguide probe 300 to monitor the conductivity and permittivity of the lossy conducting medium as previousiy described. The ground parameter meter(s) 333 can be configured to détermine the conductivity and permittivity of the lossy conducting medium on a periodic basis and communicate the information to the probe control system 321 for potential adjustment of the guided surface waveguide probe 300. In some cases, the ground parameter meter(s) 333 may communicate the information to the probe control system 321 only when a change in the monitored conditions is detected.

The adaptive control system 330 can also include one or more field meterfs) 336 such as, but not limited to, an electric field strength (FS) meter. The field meter(s) 336 can be distributed about the guided surface waveguide probe 300 beyond the Hankel crossover distance (R_x) where the guided field strength curve 103 (FIG. 1) dominâtes the radiated field strength curve 106 (FIG. 1). For example, a plurality of filed meters 336 may be located along one or more radiais extending outward from the guided surface waveguide probe 300 to monitor the electric field strength as previousiy described. The field meter(s) 336 can be configured to détermine the field strength on a periodic basis and communicate the information to the probe control System 321 for potential adjustment of the guided surface waveguide probe 300. In some cases, the field meter(s) 336 may communicate the information to the probe control System 321 only when a change in the monitored conditions is detected.

Other variables can also be monitored and used to adjust the operation of the guided surface waveguide probe 300. For instance, the ground current flowing through the ground stake 715 (FIG. 7) can be used to monitor the operation of the guided surface waveguide probe 300. For example, the ground current can provide an indication of changes in the loading of the guided surface waveguide probe 300 and/or the coupling of the electric field into the guided surface wave mode on the surface of the lossy conducting medium 303. Real power delivery may be determined by monitoring the AC source 712 (or excitation source 312 of FIG. 3). In some implémentations, the guided surface waveguide probe 300 may be adjusted to maximize coupling into the guided surface waveguide mode based at least in part upon the current indication. By adjusting the phase delay (Φ = 8_y + 0_C) supplied to the charge terminal T_b the match with the wave tilt angle (Ψ) can be maintained for illumination at the complex Brewster angle for guided surface wave transmissions in the lossy conducting medium 303 (e.g., the earth). This can be accomplished by adjusting the tap position on the coil 709.

However, the ground current can also be affected by receiver loading. If the ground current is above the expected current level, then this may indicate that unaccounted loading of the guided surface waveguide probe 400 is taking place.

The excitation source 312 (or AC source 712) can also be monitored to ensure that overloading does not occur. As real load on the guided surface waveguide probe 300 increases, the output voltage of the excitation source 312, or the voltage supplied to the charge terminal T) from the coil, can be increased to increase field strength levels, thereby avoiding additional load currents. In some cases, the receivers themselves can be used as sensors monitoring the condition of the guided surface waveguide mode. For example, the receivers can monitor field strength and/or load demand at the receiver. The receivers can be configured to communicate information about current operational conditions to the probe control system 321. The information may be communicated to the probe control System 321 through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. Based upon the information, the probe control system 321 can then adjust the guided surface waveguide probe 300 for continued operation. For example, the phase delay (Φ = 0_y + 0_C) applied to the charge terminal Ti can be adjusted to maintain the electrical launching efficiency of the guided surface waveguide probe 300, to supply the load demande of the receivers. In some cases, the probe control system 321 may adjust the guided surface waveguide probe 300 to reduce loading on the excitation source 312 and/or guided surface waveguide probe 300. For example, the voltage supplied to the charge terminal T, may be reduced to lower field strength and prevent coupling to a portion of the most distant load devices.

The guided surface waveguide probe 300 can be adjusted by the probe control system 321 using, e.g., one or more tap controllers 339. In FIG. 18, the connection from the coil 709 to the upper charge terminal T-ι is controlled by a tap controller 339. In response to a change in the monitored conditions (e.g., a change in conductivity, permittivity, and/or electric field strength), the probe control system can communicate a control signal to the tap controller 339 to initiate a change in the tap position. The tap controller 339 can be configured to vary the tap position continuously along the coil 709 or incrementally based upon predefined tap connections. The control signal can include a specified tap position or indicate a change by a defined number of tap connections. By adjusting the tap position, the phase delay (Φ) of the charge terminal can be adjusted to maintain and/or improve coupling of the guided surface waveguide mode.

The guided surface waveguide probe 300 can also be adjusted by the probe control system 321 using, e.g., a charge terminal control system 348. By adjusting the impédance of the charge terminal Ti, it is possible to adjust the coupling into the guided surface waveguide mode. The charge terminal control system 348 can be configured to change the capacitance of the charge terminal T·,. By adjusting the load impédance Z_L of the charge terminal Ti while maintaining Φ = Ψ, résonance with respect to the conductive image ground plane can be maintarned. In this way, coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 303 (e.g., earth) can be improved and/or maximized.

As has been discussed, the probe control system 321 of the adaptive control system 330 can monitor the operating conditions of the guided surface waveguide probe 300 by communicating with one or more remotely located monitoring devices such as, but not limited to, a ground parameter meter 333 and/or a field meter 336. The probe control system 321 can also monitor other conditions by accessing information from, e.g., the AC source 712 (or excitation source 312). Based upon the monitored information, the probe control system 321 can détermine if adjustment of the guided surface waveguide probe 300 is needed to improve and/or maximize the launching efficiency. In response to a change in one or more of the monitored conditions, the probe control system 321 can initiate an adjustment of one or more of the phase delay (Q_y, 0_C) applied to the charge terminal T-ι and/or the load impédance Z_L of the charge terminal In some implantations, the probe control system 321 can evaluate the monitored conditions to identify the source of the change. If the monitored condition(s) was caused by a change in receiver load, then adjustment of the guided surface waveguide probe 300 may be avoided. If the monitored condition(s) affect the launching efficiency of the guided surface waveguide probe 400, then the probe control System 321 can initiate adjustments of the guided surface waveguide probe 300 to improve and/or maximize the launching efficiency. In some embodiments, the size of the charge terminal T-, can be adjusted to control the load impédance Z_L of the guided surface waveguide probe 300. For example, the self-capacitance of the charge terminal T| 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 T-i, which can reduce the chance of an electrical discharge from the charge terminal T_v In other embodiments, the charge terminal Ti can include a variable inductance that can be adjusted to change the load impédance Z_L. Control of the charge terminal T, size can be provided by the probe control system 321 through the charge terminal control System 348 or through a separate control System.

FIGS. 19A and 19B illustrate an example of a variable terminal 203 that can be used as a charge terminal of the guided surface waveguide probe 300 or a charge terminal T_R of the tuned resonator 1406 (FIGS. 14B and 14C). For example, the variable terminal 203 can include an inner cylindrical section 206 nested inside of an outer cylindrical section 209. The inner and outer cylindrical sections 206 and 209 can include plates across the bottom and top, respectively. In FIG. 19A, the cylindrically shaped variable terminal 203 is shown in a contracted condition having a first size, which can be associated with a first effective spherical diameter. To change the size of the terminal, and thus the effective spherical diameter, one or both sections of the variable terminal 203 can be extended to increase the surface area as shown in FIG. 19B. This may be accomplished by using a driving mechanism such as an electric motor or hydraulic cylinder that is electrically isolated to prevent discharge of the charge on the terminal. In this way, the capacitance (C, or C_R) of the charge terminal T₁ or T_R, and thus the load impédance (Z_£ or Z_ff) of the charge terminal T| or T_R, can be adjusted. Referring next to FIG. 20, shown is a schematic représentation illustrating a variable terminal 212 including a variable inductance 215 within the outer surface 218 of the terminal 212. By placing the variable inductor within the terminal 212, the load impédance Z_£ of the guided surface waveguide probe 300 of FIG. 3 (or the load impédance Z_R of the tuned resonator 1406 of FIGS. 14B and 14C) can be adjusted by adjusting the inductance 215, without affecting the charge surface of the charge terminal In some embodiments, the variable terminal 203 of FIGS. 19A and 19B can include a variable inductance 215 within the cylindrical sections 206 and 209. Such a combination can provide a wider range of control over the load impédance Z_L of the guided surface waveguide probe 300.

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 abovedescribed embodiment(s) without departing substantially from the sprrit and principles of the disclosure. Ail 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 (19)

1. Therefore, the following is claimed:

   1. A method, comprising:

   positioning a charge terminal at a defined height over a lossy conducting medium; adjusting a phase delay (Φ) of a feed network connected to the charge terminal to match a wave tilt angle (Ψ) corresponding to a complex Brewster angle of incidence (θ_ίΒ) associated with the lossy conducting medium;

   adjusting a load impédance (Z_L) of the charge terminal based upon an image ground plane impédance (Z_in) associated with the lossy conducting medium; and exciting the charge terminal with an excitation voltage via the feed network, where the excitation voltage establishes an electric field that couples into a guided surface waveguide mode along a surface of the lossy conducting medium.
2. 2. The method of claim 1, wherein the feed network comprises a feed line conductor coupled to the charge terminal and a coil coupled between the lossy conducting medium and the feed line conductor, where the phase delay (Φ) of the feed network includes a phase delay (6_y) associated with the feed line conductor and a phase delay (0_C) associated with the coil.
3. 3. The method of claim 2, wherein adjusting the phase delay (Φ) comprises adjusting the phase delay (0_C) associated with the coil.
4. 4. The method of any of claims 2 and 3, wherein a connection of the feed line conductor is repositioned on the coil to adjust the phase delay (0_C) associated with the coil.
5. 5. The method of claim 4, wherein the connection of the feed line conductor is repositioned on the coil via a variable tap.
6. 6. The method of any of claims 1-5, wherein the complex Brewster angle of incidence (0_ÉB) associated with the lossy conducting medium rs based upon an operational frequency of the excitation voltage and characteristics ofthe lossy conducting medium.
7. 7. The method of claim 6, wherein the characteristics of the lossy conducting medium include conductivity and permittivity.

   !
8. 8. The method of any of claims 1-7, wherein the image ground plane impédance (Z_in) is based at least in part upon a phase shift (6_d) between a physical boundary of the lossy conducting medium and a conducting image ground plane.
9. 9. The method of claim 8, wherein the physical boundary of the lossy conducting medium and the conducting image ground plane are separated by a complex depth.
10. 10. The method of any of claims 1-9, wherein the load impédance (Z_L) of the charge terminal is adjusted based upon a reactive component of the image ground plane impédance (Z_itl).
11. 11. The method of claim 10, wherein the load impédance (Z_L) of the charge terminal is adjusted to match the reactive component ofthe image ground plane impédance (Z_in) with a structure impédance (Z_base) associated with the feed network and the charge terminal.
12. 12. The method of any of claims 1-11, wherein the phase delay (Φ) of the feed network is fixed while the load impédance (Z_L) ofthe charge terminal is adjusted.
13. 13. The method of any of claims 1-12, wherein the charge terminal has an effective spherical diameter, and the defined height of the charge terminal is at least four times an effective spherical diameter to reduce the bound capacitance.
14. 14. The method of any of claims 1-13, wherein the charge terminal is coupled to an excitation source via a coil.
15. 15. The method of any of claims 1 -14, comprising:

    sensing a change in a characteristic of the lossy conducting medium; and adjusting the phase delay (Φ) of the feed network connected to the charge terminal to match a modified wave tilt angle in response to the change in the characteristic ofthe lossy conducting medium, the modified wave tilt angle corresponding to a complex Brewster angle of incidence associated with the lossy conducting medium having the changed characteristic.
16. 16. The method of claim 15, comprising:

    adjusting the load impédance (Z_L) ofthe charge terminal based upon a new image ground plane impédance based upon the lossy conducting medium having the changed characteristic.
17. 17. The method of any of daims 1-16, wherein the lossy conducting medium is a terrestrial medium.
18. 18. A guided surface waveguide probe, comprising;

    a charge terminal elevated over a lossy conducting medium; and a feed network configured to couple an excitation source to the charge terminal, the feed network configured to provide a voltage to the charge terminal with a phase delay (Φ) that matches a wave tilt angle (Ψ) associated with a complex Brewster angle of incidence (e_/e) associated with the lossy conducting medium, and the charge terminal having a load impédance (Z_L) that is determined based upon an image ground piane impédance (Z_i?l) associated with the lossy conducting medium.
19. 19. The guided surface waveguide probe of claim 18, wherein the feed network comprises a feed line conductor coupled to the charge terminal and a coil coupled between the lossy conducting medium and the feed line conductor, where the phase delay (Φ) of the feed network includes a phase delay (9_y) associated with the feed line conductor and a phase delay (9_C) associated with the coil.

    !0. The guided surface waveguide probe of claim 19, wherein the coil is a helical coil.

    1. The guided surface waveguide probe of any of daims 19 and 20, wherein the excitation source is coupled to the coil via a tap connection.

    2. The guided surface waveguide probe of any of daims 19-21, wherein an impédance matching network is coupled between the excitation source and the tap connection on the coil.

    3. The guided surface waveguide probe of any of daims 19 and 20, wherein the excitation source is magnetically coupled to the coil.

    L The guided surface waveguide probe of any of daims 19-23, wherein the charge terminal is coupled to the coil via a tap connection.

    i >. The guided surface waveguide probe of any of daims 18-24, wherein the feed network is configured to vary the phase delay (Φ) to match the wave tilt angle (Ψ).

    i

    55 !

    26. The guided surface waveguide probe of any of daims 18-25, comprising a probe control system configured to adjust the feed network based at least in part upon characteristics of the lossy conducting medium.

    27. The guided surface waveguide probe of claim 26, wherein the feed network comprises a coil coupled between the excitation source and the charge terminal, the charge terminal coupled to the coil via a variable tap.

    28. The guided surface waveguide probe of claim 27, wherein the probe control system adjusts a position of the variable tap in response to a change in the characteristics of the lossy conducting medium.

    29. A method, comprising:

    coupling a receiving structure to a lossy conducting medium; and mode-matching with a guided surface wave established on the lossy conducting medium, where a traveling wave phase delay (Φ) of the receiving structure is matched to a wave tilt angle (Ψ) associated with the guided surface wave, the wave tilt angle (Ψ) based at least in part upon characteristics of the lossy conducting medium in a vicinity of the receiving structure.

    30. The method of claim 29, comprising suspending a charge terminal of the receiving structure at a defined height over a surface of the lossy conducting medium.

    31. The method of claim 30, wherein the receiving structure comprises a receiver network coupled between the charge terminal and the lossy conducting medium.

    32. The method of claim 31, wherein the receiver network comprises a coil coupled to the lossy conducting medium and a supply line conductor coupled between the coil and the charge terminal, where the traveling wave phase delay (Φ) is based upon a phase delay (0_C) of the coil and a phase delay (3_y) of the supply line conductor.

    33. The method of claim 32, wherein adjusting the traveling wave phase delay (Φ) comprises adjusting a position of a tap on the coil to vary the phase delay (0_C) of the coil.

    34. The method of claim 33, wherein the supply line conductor is coupled to the coil via the tap.

    35. The method of any of claims 30-34, wherein the charge terminal has an effective spherical diameter, and the defined height of the charge terminal is at least four fîmes the effective spherical diameter to reduce bound capacitance.

    36. The method of any of claims 30-35, comprising resonating the receiving structure relative to an image plane at a complex depth below the surface of the lossy conducting medium.

    37. The method of claim 36, wherein resonating the receiving structure comprises adjusting a load impédance (Z_L) of the charge terminal based upon an image ground plane impédance (Z_in) associated with the lossy conducting medium.

    38. The method of any of claims 36 and 37, wherein resonating the receiving structure establishes a standing wave on the receiving structure by exploiting phase delays from transmission line sections of the receiving structure plus phase jumps arising from discontinuities in characteristic impédances of the transmission line sections, the standing wave superposed with a traveling wave on the receiving structure.

    39. The method of any of claims 29-38, comprising extracting electrical power from the receiving structure via a coil.

    40. A receiving structure for mode-matching with a guided surface wave established on a lossy conducting medium, the receiving structure comprising:

    a charge terminal elevated over the lossy conducting medium; and a receiver network coupled between the charge terminal and the lossy conducting medium, the receiver network having a phase delay (Φ) that matches a wave tilt angle (Ψ) associated with the guided surface wave, the wave tilt angle (Ψ) based at least in part upon characteristics of the lossy conducting medium in a vicinity of the receiving structure.

    41. The receiving structure of claim 40, wherein the charge terminal has a variable load impédance (Z_L).

    42. The receiving structure of claim 41, wherein the variable load impédance (Z_L) is determined based upon an image ground plane impédance (Z_in) associated with the lossy conducting medium in the vicinity of the receiving structure.

    43. The receiving structure of any of daims 41 and 42, wherein the load impédance (Z_L) is adjusted to resonate the receiving structure relative to an image plane at a complex depth below a surface of the lossy conducting medium.

    44. The receiving structure of any of daims 40-43, wherein resonating the receiving structure establishes a standing wave on the receiving structure by exploiting phase delays from transmission line sections of the receiver network plus phase jumps arising from discontinuities in characteristic impédances of the transmission line sections.

    45. The receiving structure of any of daims 40-44, wherein the receiver network comprises a coil coupled to the lossy conducting medium and a supply line conductor coupled between the coil and the charge terminal, where the phase delay (Φ) of the receiver network is based upon a phase delay (0_C) of the coil and a phase delay (θ_γ) of the supply line conductor.

    46. The receiving structure of claim 45, further comprising a variable tap configured to adjust the phase delay (Ô_c) of the coil.

    47. The receiving structure of any of daims 45 and 46, comprising an impédance matching network coupled to the coil.

    48. The receiving structure of claim 47, wherein the impédance matching network is inductively coupled to the coil.

    49. The method of claim 29, further comprising:

    receiving, via the receive structure, energy conveyed by the guided surface wave on the lossy conducting medium.

    50. The method of claim 49, wherein the receive structure loads an excitation source coupled to a guided surface waveguide probe that generates the guided surface wave.

    51. The method of any of daims 49 and 50, wherein the energy further comprises electrical power, and the method further comprises applying the electrical power to an electrical load coupled to the receive structure, where the electrical power is used as a power source for the electrical load.

    52. The method of any of claims 49-51, further comprising impédance matching an electrical load to the receive structure.

    53. The method of claim 52, further comprising establishing a maximum power transfer from the receive structure to the electrical load.

    54. The method of any of claims 49-53, wherein the receive structure further comprises a magnetic coil, a linear probe or a tuned resonator coupled to the lossy conducting medium.

    55. A power transmission system, comprising:

    a guided surface waveguide probe that transmits electrical energy in a form of a guided surface wave along a surface of a terrestrial medium, the guided surface waveguide probe comprising a feed network configured to provide a voltage to a charge terminal with a phase delay (Φ) that matches a wave tilt angle (Ψ) associated with a complex Brewster angle of incidence associated with the terrestrial medium, where the charge terminal has a load impédance (Z_L) that is determined based upon an image ground plane impédance (Z_in) associated with the terrestrial medium; and a receive structure that receives the electrical energy.

    56. The power transmission system of claim 55, wherein the receive structure loads the guided surface waveguide probe.

    57. The power transmission system of any of claims 55 and 56, wherein an electrical load is coupled to the receive structure and the electrical energy is used as a power source for the electrical load.

    58. The power transmission system of claim 57, wherein the electrical load is impédance matched to the receive circuit.

    59. The power transmission system of any of claims 57 and 58, wherein a maximum power transfer is established from the receive structure to the electrical load.

    60. The power transmission system of any of claims 55-59, wherein the receive structure further comprises a magnetic coil, a linear probe, or a tuned resonator.

    i

    The apparatus of claim 60, wherein the tuned resonator comprises a sériés tuned resonator, a parallel tuned resonator, or a distributed tuned resonator.