OA18762A - Excitation and use of guided surface wave modes on lossy media. - Google Patents

Abstract

Disclosed are various embodiments for transmitting 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 by exciting a guided surface waveguide probe.

Description

EXCITATION AND USE OF GUIDED SURFACE WAVE MODES ON LOSSY MEDIA

CROSS REFERENCE TO RELATED APPLICATIONS

This Patent Coopération Treaty Application daims priority to, and the benefit of, copending U.S. Patent Application Number 14/483,089 entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media filed on September 10, 2014.

BACKGROUND

For over a century, signais transmitted by radio waves involved radiation fields launched using conventional antenna structures. In contrastto 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 since the early 1900's.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the présent disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead beîng placed upon clearly îllustrating 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 présent disclosure.

FIGS. 3A and 3B are drawings that illustrate a complex angle of insertion of an electric field synthesized by guided surface waveguide probes according to the various embodiments of the présent disclosure.

FIG. 4 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 présent disclosure.

FIG. 5 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 présent disclosure.

FIGS. 6A and 6B are plots îllustrating bound charge on a sphere and the effect on capacitance according to various embodiments of the présent disclosure.

FIG. 7 is a graphical représentation illustrating the effect of élévation of a charge terminal on the location where a Brewster angle intersects with the lossy conductive medium according to various embodiments ofthe present disclosure.

FIGS. 8A and 8B are graphical représentations illustrating the incidence of a synthesized electric field at a complex Brewster angle to match the guided surface waveguide mode at the Hankel crossover distance according to various embodiments of the present disclosure.

FIGS. 9A and 9B are graphical représentations of examples of a guided surface waveguide probe according to an embodiment of the present disclosure.

FIG. 10 is a schematic diagram of the guided surface waveguide probe or FIG. 9A according to an embodiment of the present disclosure.

FIG. 11 includes plots of an example of the imaginary and real parts of a phase delay (Φ_ο) of a charge terminal Τ of a guided surface waveguide probe of FIG. 9A according to an embodiment of the present disclosure.

FIG. 12 is an image of an example of an implemented guided surface waveguide probe of FIG. 9A according to an embodiment of the present disclosure.

FIG. 13 is a plot comparing measured and theoretical field strength ofthe guided surface waveguide probe of FIG. 12 according to an embodiment of the present disclosure.

FIGS. 14A and 14B are an image and graphical représentation of a guided surface waveguide probe according to an embodiment of the present disclosure.

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

FIG. 16 is a plot comparing measured and theoretical field strength of the guided surface waveguide probe of FIGS. 14A and 14B according to an embodiment ofthe present disclosure

FIGS. 17 and 18 are graphical représentations of examples of guided surface waveguide probes according to embodiments of the present disclosure.

FIGS. 19A and 19B depict examples of receivers 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. 20 depicts an example of an additional receiver 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. 21A depicts a schematic diagram representing the Thevenin-equivalent ofthe receivers depicted in FIGS. 19A and 19B according to an embodiment ofthe present disclosure.

FIG. 21B depicts a schematic diagram representing the Norton-equivalent ofthe receiver depicted in FIG. 17 according to an embodiment ofthe present disclosure.

FIGS. 22A and 22B are schematic dîagrams representing examples of a conductîvity measurement probe and an open wire line probe, respectively, according to an embodiment of the présent disclosure.

FIGS. 23A through 23C are schematic drawings of examples of an adaptive control system employed by the probe control system of FIG. 4 according to embodiments of the présent disclosure.

FIGS. 24A and 24B are drawings of an example of a variable terminal for use as a charging terminal according to an embodiment 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 gu/cfed 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 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 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 exceptforthat dissipated in the conductîvity ofthe 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 3 not deliver real power unless a résistive load is présent. 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 (TM) 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/4d 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 fondamental propagation différence between guided and radiated electromagnetic fields. For an informai discussion of the différence between guided and radiated electromagnetic fields, reference is made to Milligan, T., Modem Antenna Design, McGraw-Hill, 1^st Edition, 1985, pp.89.

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 4 for the wave équation, îtself, 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 continuons 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 called “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 Drahtiosen 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 Physics: 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 Theory and Technigues, Van Nostrand, 1953, pp. 291-293.

To summarize the above, first, the continuous part of the 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 Technigues 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 présent disclosure, various guided surface waveguide probes are described that are configured to excite electric fields that couple into a 5 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 terrestrïal medium such as the Earth.

Referring to FIG. 2, shown is a propagation interface that provides for an examination of the boundary value solution 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 cylîndrical 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 terrestrïal 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 zero only for incidence at a complex Brewster angle. See Stratton, J.A., Electromagnetic Theory, McGraw-Hill, 1941, p. 516.

According to various embodiments, the présent disclosure sets forth various guided surface waveguide probes that generate electromagnetic fields that are substantially modematched 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 zero reflection.

To explain further, in Région 2, where an field variation is assumed and where p φ o and z > 0 (with z being the vertical coordinate normal to the surface of Région 1, and p being the radial dimension in cylîndrical 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:

wPt-yyp), (1)

E_2p - A ^{e u}*² and (2)

E₂z = 4-=r)e-^^z W$²⁾(-yyp). (3) \u)Cq/

In Région 1, where the e^iiüt 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 interface are expressed by the following electric field and magnetic field components:

Η_1φ=Αε^ Hβ⁾(-jyp),(4) ^E« = 4ty3;)^e-^zi^2,<-'>'”'^and<

=

In these expressions, z is the vertical coordinate normal to the surface of Région 1 and p is the radial coordinate, H„^2ï(-/yp) is a complex argument Hankel function of the second kind and order n, Ui 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, is the conductivity of Région 1, ω is equal to 2π[, where f is a frequency of excitation, ε₀ is the permittivity of free space, ε^ is the permittivity of Région 1, λ 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, and gives, in Région 1, ui =-u₂(f_r-;x).(8)

The radial propagation constant y is given by y = /7^ + ^u2 = ί ^=2’( which is a complex expression where n is the complex index of refraction given by n = 7^£r ^_ J^x(

In ail of the above Equations, x-Aand(11) ωε₀ k₀ = ω/μ_οε_ο -(12) where μ₀ comprises the permeability of free space, e_r comprises relative permittivity of Région 1. Thus, the generated surface wave propagates parallel to the interface and exponentially decays vertical to it. This is known as evanescence.

Thus, Equations (1)-(3) can be considérée! to be a cylindrically -symmetric, radiallypropagating 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 to produce the complex Brewster angle at the boundary interface to excite the surface waveguide mode with no or minimal reflection. A compensation terminal of appropriate size can be positioned relative to the charge terminal, and fed with voltage and/or current, to refine the Brewster angle at the boundary interface.

To continue, the Leontovich impédance boundary condition between Région 1 and Région 2 is stated as nx H_z(p,<p,0) =J_S, (13) where ή is a unit normal in the positive vertical (+z) direction and W₂ 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, such radial surface current density being specified by

J_P(p') = -ΑΗ[²\-)γρ'} (14) where A is a constant. Further, it should be noted that close-in to the guided surface waveguide probe (for p « 2), Equation (14) above has the behavior

Jclose^P ) = πί-ίγρ’) ~^ΗΦ ~ ~2np>'

The négative sign means that when source current (/₀) flows vertically upward, the required “close-in” ground current flows radially inward. By field matching on Ηψ “close-in” we find that

A = -^ (16) in Equations (1)-(6) and (14). Therefore, the radial surface current density of Equation (14) can be restated as = (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 such as 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 fonctions of the first and second kind and order n are defined as complex combinations of the standard Bessel fonctions of the first and second kinds

Hf¹⁾(x)=/_n(x)+/7V„(x),and (18) /fJ²⁾(x)=/„(x)-;W_n(x), (19)

These fonctions represent cyiindrica! waves propagating radially inward (H„^1J) and ootward (H„²\ respectively. The définition is analogous to the relationship e^±JX = cos x ± j sinx. See, for example, Harrington, R.F., Time-Harmonic Fields, McGraw-Hill, 1961, pp. 460463.

That is an outgoing wave can be recognized from its large argument asymptotic behavior that is obtained directly from the sériés définitions of /_n(x) and /V_n(x). Farout from the guided surface waveguide probe:

=£ J’e-'W, <^20a>

which, when multiplied by e^jCüt, is an outward propagating cylindrical wave of the form _wj_th _a ι/y^ spatial variation. The first order (n = 1) solution can be determined from Equation (20a) to be <20b)

Close-in to the guided surface waveguide probe (for p « 2), the Hankel function of first order and the second kind behaves as:

(21) ¹ x—0 Ttx

Note that these asymptotic expressions are complex quantities. When x is a real quantity, Equations (20b) and (21) differ in phase by yjj, which corresponds to an extra phase advance or “phase boost” of 45° or, equivalently, λ/8. The close-in and far-out asymptotes of the first order Hankel function of the second kind hâve a Hankel “crossover” or transition point where they are of equal magnitude at a distance of p = R_x. The distance to the Hankel crossover point can be found by equating Equations (20b) and (21), and solving for R_x. With x — σ!ωε₀, 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.

Guided surface waveguide probes 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 at a complex angle, thereby excîting radial surface currents by substantially mode-matching to a guided surface wave mode at the Hankel crossover point at R_x.

Referring now to FIG. 3A, shown is a ray optic interprétation of an incident field (E) polarized parallel to a plane of incidence. The electric field vector E is to be synthesized as an incoming non-uniform plane wave, polarized parallel to the plane of incidence. The electric field vector E can be created from independent horizontal and vertical components as:

F(0_O) = E_pp + E_zz.(22)

Geometrically, the illustration in FIG. 3A suggests that the electric field vector E can be given by:

EpÇp.z) = E(p,z) cos0_o, and(23a)

E_z(p,z) = E(p,z)cos(|-G_o) = E(p, z)sin0_o,(23b) which means that the field ratio is ^=tanip₀.(24)

Using the electric field and magnetic field components from the electric field and magnetic field component solutions, the surface waveguide impédances can be expressed.

The radial surface waveguide impédance can be written as

Z == ^p Μ_φ )ωε_ο' and the surface-normal impédance can be written as

Z - ~^Ep - ~^{U; z} Ηφ ;ωε₀ ' (25) (26)

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

W =^= |Μφ'^ψ, (27) ^Ez which is complex and has both magnitude and phase.

For a TEM wave in Région 2, the wave tilt angle 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. 3B, which illustrâtes equi-phase surfaces of a TEM 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 ofthe boundary interface at z = 0.

This may be better understood with reference to FIG. 4, which shows an example of a guided surface waveguide probe 400a that includes an elevated charge terminal T, and a lower compensation terminal T₂ that are arranged along a vertical axis z that is normal to a plane presented by the lossy conducting medium 403. In this respect, the charge terminal T, is placed directly above the compensation terminal T₂ although it is possible that some other arrangement of two or more charge and/or compensation terminais T_N can be used. The guided surface waveguide probe 400a is disposed above a lossy conducting medium 403 according to an embodiment of the présent disclosure. The lossy conducting medium 403 makes up Région 1 (FIGS. 2, 3A and 3B) and a second medium 406 shares a boundary interface with the lossy conducting medium 403 and makes up Région 2 (FIGS. 2, 3A and 3B).

The guided surface waveguide probe 400a includes a coupling circuit 409 that couples an excitation source 412 to the charge and compensation terminais T, and T₂. According to various embodiments, charges Q-ι and Q₂ can be imposed on the respective charge and compensation terminais and T₂, depending on the voltages applied to terminais T, and T₂ at any given instant, h is the conduction current feeding the charge Qî on the charge terminal T-,, and l₂ is the conduction current feeding the charge Q₂ on the compensation terminal T₂.

The concept of an electrical effective height can be used to provide insight into the construction and operation of the guided surface waveguide probe 400a. The electrical effective height (h_efi) has been defined as h_eff = ^Î^I(z)dz (28a) for a monopole with a physical height (or length) of h_p, and as h_en = ity» (28b) for a doublet or dipole. These expressions differ by a factor of 2 since the physical length of a dipole, 2h_p, is twice the physical height of the monopole, h_p. Since the expressions dépend upon the magnitude and phase of the source distribution, effective height (or length) is complex in general. The intégration of the distributed current /(z) of the monopole antenna structure is performed over the physical height of the structure (h_p), and normalized to the ground current (/₀) flowing upward through the base (or input) of the structure. The distributed current along the structure can be expressed by l{z) = /_ccos(/3₀z), (29) where β_ϋ is the propagation factor for free space. In the case of the guided surface waveguide probe 400a of FIG. 4, !_c is the current distributed along the vertical structure.

This may be understood using a coupling circuit 409 that includes a low loss coil (e.g., a helical coil) at the bottom of the structure and a supply conductor connected to the charge terminal T_v With a coil or a helical delay line of physical length l_c and a propagation factor of ^ = Γ ⁼ 7Χ’ ^{(30) r} 4p V fAç where is the velocity factor on the structure, A_o is the wavelength at the supplied frequency, and A_p is the propagation wavelength resulting from any velocity factor ty, the phase delay on the structure is Φ = β_ρΙς, and the current fed to the top of the coil from the bottom of the physical structure is lc(PpQ = (31) with the phase Φ measured relative to the ground (stake) current l₀. Consequently, the electrical effective height of the guided surface waveguide probe 400a in FIG. 4 can be approximated by h_eff = ξÙ cos(/ï₀z) dz - h_pei*, (32) for the case where the physical height h_p « λ₀, the wavelength at the supplied frequency. A dipole antenna structure may be evaluated in a similar fashion. The complex effective height of a monopole, h_e^f = h_p at an angle Φ (or the complex effective length for a dipole h_eff = 2Λ_ρε^7φ), 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 403.

According to the embodiment of FIG. 4, the charge terminal T, is positioned over the lossy conducting medium 403 at a physical height H_b and the compensation terminal T₂ is positioned directly below T, along the vertical axis z at a physical height H₂, where H₂ is less than H-f. The height h of the transmission structure may be calculated as h = H, - H₂ The charge terminal has an isolated capacitance C_b and the compensation terminal T₂ has an isolated capacitance C₂. A mutual capacitance C_M can also exist between the terminais Ti and T₂ depending on the distance therebetween. During operation, charges Ch and Q₂ are imposed on the charge terminal L and compensation terminal T₂, respectively, depending on the voltages applied to the charge terminal Τ_Ί and and compensation terminal T₂ at any given instant

According to one embodiment, the lossy conducting medium 403 comprises 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 concrète, asphalt, building materials, and other man-made materials. In other embodiments, the lossy conducting medium 403 can comprise some medium other than the Earth, whether naturally occurring or manmade. In other embodiments, the lossy conducting medium 403 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 that the lossy conducting medium 403 comprises a terrestrial medium or Earth, the second medium 406 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 406 can comprise other media relative to the lossy conducting medium 403.

Referring back to FIG. 4, the effect of the lossy conducting medium 403 in Région 1 can be examined using image theory analysis. This analysis with respect to the lossy conducting medium assumes the presence of induced effective image charges Qi' and Q₂' beneath the guided surface waveguide probes coinciding with the charges Q-ι and Q₂ on the charge and compensation terminais and T₂ as illustrated in FIG. 4. Such image charges Q,' and Q₂' are not merely 180° out of phase with the primary source charges CL and Q₂ on the charge and compensation terminais Ti and T₂, as they would be in the case of a perfect conductor. A lossy conducting medium such as, for example, a terrestrial medium présents phase shifted images. That is to say, the image charges Q/ and Q₂' are at complex depths. For a discussion of complex images, 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.

Instead of the image charges Q/ and Q₂' being at a depth that is equal to the physical height (H_n) of the charges Q, and Q₂, a conducting image ground plane 415 (representing a perfect conductor) is placed at a complex depth of z = - d/2 and the image charges appear at complex depths (i.e., the “depth” has both magnitude and phase), given by -D_n = (d/2 + d/2 + H_n) #-H_n, where n = 1, 2,..., and for vertically polarized sources, d = ~^=d_r +jd_t = |d|zÇ ,(33) re re where

Ye = - ω²η₁ε_ί, and(34) k_o = (ûfu_oE_Q.(35) as indicated in Equation (12). In the lossy conducting medium, the wave front normal is parallel to the tangent of the conducting image ground plane 415 at z = - d/2, and not at the boundary interface between Régions 1 and 2.

The complex spacing of image charges Q/ and Q₂’, in turn, implies that the external fields will expérience extra phase shifts not encountered when the interface is either a lossless dielectric or a perfect conductor. The essence of the lossy dielectric image-theory technique is to replace the finitely conducting Earth (or lossy dielectric) by a perfect conductor located at the complex depth, z = - d/2 with source images located at complex depths of D_n = d + H_nThereafter, the fields above ground (z > 0) can be calculated using a superposition of the physical charge Q_n (at z = +H_n) plus its image Q_n' (at z' = -D_n).

Given the foregoing discussion, the asymptotes of the radial surface waveguide current at the surface of the lossy conducting medium J_p(p') can be determined to behlp) when closein and /₂(p) when far-out, where

Close-in (ρ < λ/8): J_p(p) = + ^{Ep ÎQ}^^P and (36)

Far-out (p » λ/8): J_p(p) ~J₂ = x x , (37) where a and β are constants related to the decay and propagation phase of the far-out radial surface current density, respectively. As shown in FIG. 4, l_i is the conduction current feeding the charge Ch on the elevated charge terminal ty, and l₂ is the conduction current feeding the charge Q₂ on the lower compensation terminal T₂.

According to one embodiment, the shape ofthe charge terminal is specified to hold as much charge as practically possible. Ultimately, the field strength of a guided surface wave launched by a guided surface waveguide probe 400a is directly proportional to the quantity of charge on the terminal ty. In addition, bound capacitances may exist between the respective charge terminal T, and compensation terminal T₂ and the lossy conducting medium 403 depending on the heights ofthe respective charge terminal Ti and compensation terminal T₂ with respect to the lossy conducting medium 403.

The charge Ch on the upper charge terminal ty may be determined by Ch = where C, is the isolated capacitance of the charge terminal ty and is the voltage applied to the charge terminal ty. In the example of FIG. 4, the spherical charge terminal T, can be considered a capacitor, and the compensation terminal T₂ can comprise a disk or lower capacitor. However, in other embodiments the terminais T, and/or T₂ can comprise any conductive mass that can hold the electrical charge. For example, the terminais ty and/or T₂ can include any shape such as a sphere, a disk, a cylinder, a cône, a torus, a hood, one or more rings, or any other randomized shape or combination of shapes. If the terminais ty and/or T₂ are spheres or disks, the respective self-capacitance C, and C₂ can be calculated. The capacitance of a sphere at a physical height of h above a perfect ground is given by f-elevated sphere ~ 4?Γ£_οίϊ(1 + Μ + + Μ³ + 2M^ + 3M³ + ··· ), (38) where the diameter of the sphere is 2a and M = a/2h.

In the case of a sufficiently isolated terminal, the self-capacitance of a conductive sphere can be approximated by C = 4πε_οα, where a comprises the radius of the sphere in meters, and 14 (39) (40) (41) (42) the self-capacitance of a disk can be approximated by C = 8ε_οα, where a comprises the radius of the disk in meters. Also note that the charge terminal T, and compensation terminal T2 need not be identical as illustrated in FIG. 4. Each terminal can hâve a separate size and shape, and include different conducting materials. A probe control system 418 is configured to control the operation of the guided surface waveguide probe 400a.

Consider the geometry at the interface with the lossy conducting medium 403, with respect to the charge Q1 on the elevated charge terminal Ti. As illustrated in FIG. 3A, the relationship between the field ratio and the wave tilt is — = ^Esin1^ = tani/j = VF — |ΐν|έ?^;ψ, and E_z E cos ψ

E_z EsinO _ „ 1 1 _;ψ ί ⁼ M = ^{tane =} w = m^c

For the spécifie case of a guided surface wave launched in a transmission mode (TM), the wave tilt field ratio is given by ₌ ^u» 1

Ez -J'Y H^(-jyp) when ίύ²\χ) —> /ⁿH®(x). Applying Equation (40) to a guided surface wave gives

With the angle of incidence equal to the complex Brewster angle (θ_ίΒ), the reflection coefficient vanishes, as shown by \ ₌ V(_Cr-Jx)-sin^(_Cr-jx)cosgJ _{= 0 (43)}

Vfrr-jxl-sin² β,+(ε_Γ-/χ)cos0, |_e ,₌₀

By adjusting the complex field ratio, an incident field can be synthesized to be incident at a complex angle at which the reflection is reduced or eliminated. 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 403. A larger reflection can hinder and/or prevent a guided surface wave from being launched. Establishing this ratio as n = y/s_r - jx gives an incidence at the complex Brewster angle, making the reflections vanish.

Referring to FIG. 5, shown is an example of a plot of the magnitudes of the first order Hankel fonctions of Equations (20b) and (21) for a Région 1 conductivity of σ = 0.010 mhos/m and relative permittivity ε_Γ = 15, at an operating frequency of 1850 kHz. Curve 503 is the magnitude of the far-out asymptote of Equation (20b) and curve 506 is the magnitude of the close-in asymptote of Equation (21), with the Hankel crossover point 509 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 509. According to various embodiments, a guided electromagnetic field can be launched in the form of a guided surface wave along the surface of 15 the lossy conducting medium with little or no reflection by matching the complex Brewster angle (θ_ίΒ) at the Hankel crossover point 509.

Out beyond the Hankel crossover point 509, the large argument asymptote prédominâtes over the “close-in représentation of the Hankel function, and the vertical component of the mode-matched electric field of Equation (3) asymptotically passes to

which is linearly proportional to free charge on the isolated component of the elevated charge terminal’s capacitance at the terminal voltage, r?y_ree = Q_ree x V_T. The height Hi of the elevated charge terminal (FIG. 4) affects the amount of free charge on the charge terminal T_r When the charge terminal is nearthe image ground plane 415 (FIG. 4), most of the charge Qi on the terminal is bound to its image charge. As the charge terminal T, is elevated, the bound charge is lessened until the charge terminal T, reaches a height at which substantially ail of the isolated charge is free.

The advantage of an increased capacitive élévation for the charge terminal Ti is that the charge on the elevated charge terminal T, is further removed from the image ground plane 415, resulting in an increased amount of free charge q^_ree to couple energy into the guided surface waveguide mode.

FIGS. 6A and 6B are plots illustrating the effect of élévation (Λ) on the free charge distribution on a spherical charge terminal with a diameter of D = 32 inches. FIG. 6A shows the angular distribution of the charge around the spherical terminal for physical heights of 6 feet (curve 603), 10 feet (curve 606) and 34 feet (curve 609) above a perfect ground plane. As the charge terminal is moved away from the ground plane, the charge distribution becomes more uniformly distributed about the spherical terminal. In FIG. 6B, curve 612 is a plot of the capacitance of the spherical terminal as a function of physical height (h) in feet based upon Equation (38). For a sphere with a diameter of 32 inches, the isolated capacitance (C_iso) is 45.2 pF, which îs illustrated in FIG. 6B as line 615. From FIGS. 6A and 6B, it can be seen that for élévations of the charge terminal Ti that are about four diameters (4D) or greater, the charge distribution is approximately uniform about the spherical terminal, which can improve the coupling into the guided surface waveguide mode. The amount of coupling may be expressed as the efficiency at which a guided surface wave is launched (or “launching efficiency) in the guided surface waveguide mode. A launching efficiency of close to 100% is possible. For example, launching efficiencies of greater than 99%, greater than 98%, greater than 95%, greater than 90%, greater than 85%, greater than 80%, and greater than 75% can be achieved.

However, with the ray optic interprétation of the incident field (£), at greater charge terminal heights, the rays intersecting the lossy conducting medium at the Brewster angle do so at substantially greater distances from the respective guided surface waveguide probe. FIG. 7 graphically illustrâtes the effect of increasing the physical height of the sphere on the distance where the electric field is incident at the Brewster angle. As the height is increased from h_T through h₂ to h₃, the point where the electric field intersects with the lossy conducting medium (e.g., the earth) at the Brewster angle moves further away from the charge. The weaker electric field strength resulting from géométrie spreading at these greater distances reduces the effectiveness of coupling into the guided surface waveguide mode. Stated another way, the efficiency at which a guided surface wave is launched (or the “launching efficiency”) is reduced. However, compensation can be provided that reduces the distance at which the electric field is incident with the lossy conducting medium at the Brewster angle as will be described.

Referring now to FIG. 8A, an example of the complex angle trigonometry is shown for the ray optic interprétation of the incident electric field (E) of the charge terminal Τί with a complex Brewster angle (0_{i H}) at the Hankel crossover distance (R_x). Recall from Equation (42) that, for a lossy conducting medium, the Brewster angle is complex and specified by tan 0_i>B = !ε_Γ -j-f- = « · (⁴⁵>

Electrically, the géométrie parameters are related by the electrical effective height ofthe charge terminal T, by

R_xtanip_iB = R_xxW = h_e^ = h_pe^j<t>, (46) where ψ_ί;Β = (ττ/2) - 0_iB is the Brewster angle measured from the surface of the lossy conducting medium. To couple into the guided surface waveguide mode, the wave tilt of the electric field at the Hankel crossover distance can be expressed as the ratio of the electrical effective height and the Hankel crossover distance = tan ψΐΒ = W„_x. (47)

Since both the physical height (/i_p) and the Hankel crossover distance (R_x) are real quantities, the angle of the desired guided surface wave tilt at the Hankel crossover distance (lV_Rje) is equal to the phase (Φ) ofthe complex effective height This implies that by varying the phase at the supply point ofthe coil, and thus the phase shift in Equation (32), the complex effective height can be manipulated and the wave tilt adjusted to synthetically match the guided surface waveguide mode at the Hankel crossover point 509.

In FIG. 8A, 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_ΙιΒ measured between a ray extending between the Hankel crossover point at R_x and the center of the charge terminal T|, and the lossy conducting medium surface between the Hankel crossover point and the charge terminal T,. With the charge terminal T, positioned at physical height h_p and excited with a charge having the appropriate phase Φ, the resulting electric field is incident with the lossy conducting medium boundary interface at the Hankel crossover distance R_x, and at the Brewster angle. Under these conditions, the guided surface waveguide mode can be excited without reflection or substantially negligible reflection.

However, Equation (46) means that the physical height of the guided surface waveguide probe 400a (FIG. 4) can be relatively small. While this will excite the guided surface waveguide mode, the proximity ofthe elevated charge CL to its mirror image Q/ (see FIG. 4) can resuit in an unduly large bound charge with little free charge. To compensate, the charge terminal Ti can be raised to an appropriate élévation to increase the amount of free charge. As one example rule of thumb, the charge terminal T can be positioned at an élévation of about 4-5 fîmes (or more) the effective diameter of the charge terminal T_v The challenge is that as the charge terminal height increases, the rays intersecting the lossy conductive medium at the Brewster angle do so at greater distances as shown in FIG. 7, where the electric field is weaker by a factor of jR_x/R_xn.

FIG. 8B illustrâtes the effect of raising the charge terminal T-, above the height of FIG. 8A. The increased élévation causes the distance at which the wave tilt is incident with the lossy conductive medium to move beyond the Hankel crossover point 509. To improve coupling in the guide surface waveguide mode, and thus provide for a greater launching efficiency of the guided surface wave, a lower compensation terminal T₂ can be used to adjust the total effective height (h_TE) of the charge terminal T, such that the wave tilt at the Hankel crossover distance is at the Brewster angle. For example, if the charge terminal T has been elevated to a height where the electric field intersects with the lossy conductive medium at the Brewster angle at a distance greater than the Hankel crossover point 509, as illustrated by line 803, then the compensation terminal T₂ can be used to adjust h_TE by compensating for the increased height. The effect of the compensation terminal T₂ is to reduce the electrical effective height of the guided surface waveguide probe (or effectively raise the lossy medium interface) such that the wave tilt at the Hankel crossover distance is at the Brewster angle, as illustrated by line 806.

The total effective height can be written as the superposition of an upper effective height (h._UE) associated with the charge terminal T, and a lower effective height (h._LE) associated with the compensation terminal T₂ such that h_TE = h_UE + h_LE = = R_xxW, (48) where Φ_σ is the phase delay applied to the upper charge terminal T,, Φ_£ is the phase delay applied to the lower compensation terminal T₂₁ and β = 2jt/à_p is the propagation factor from Equation (30). If extra lead lengths are taken into considération, they can be accounted for by adding the charge terminal lead length z to the physical height h_p ofthe charge terminal T_t and the compensation terminal lead length y to the physical height h_d of the compensation terminal T₂ as shown in h_TE = (,b_p + ζ)βΧ^(^ΛΡ^+ζ)^+φυ) + (h_d + y)e>Whd+y)+4h.) _ _{Rx x w} (₄₉)

The lower effective height can be used to adjust the total effective height (h_rE) to equal the complex effective height (h_eff·) of FIG. 8A.

Equations (48) or (49) can be used to détermine the physical height of the lower disk of the compensation terminal T₂ and the phase angles to feed the terminais in order to obtain the desired wave tilt at the Hankel crossover distance. For example, Equation (49) can be rewritten as the phase shift applied to the charge terminal T, as a function of the compensation terminal height (h_d) to give . ,, _λ oîl , } -1 (R_xxW-(h_d+y)eiï^phd⁺Py+*L)\ ®_u(h_d) = + z) “ J ln } (⁵⁰)

To détermine the positioning ofthe compensation terminal T₂, the relationships discussed above can be utilized. First, the total effective height (h_TE) is the superposition ofthe complex effective height (h_UE) ofthe upper charge terminal T, and the complex effective height (h_L£) of the lower compensation terminal T₂ as expressed in Equation (49). Next, the tangent of the angle of incidence can be expressed geometrically as tan^_E = ^, (51) «X which is the définition of the wave tilt, W. Finally, given the desired Hankel crossover distance R_x, the h_rE can be adjusted to make the wave tilt of the incident electric field match the complex Brewster angle at the Hankel crossover point 509. This can be accomplished by adjusting h_p, Φ_υ, and/or h_d.

These concepts may be better understood when discussed in the context of an example of a guided surface waveguide probe. Referring to FIGS. 9A and 9B, shown are graphical représentations of examples of guided surface waveguide probes 400b and 400c that include a charge terminal T,. An AC source 912 acts as the excitation source (412 of FIG. 4) for the charge terminal T₁₍ which is coupled to the guided surface waveguide probe 400b through a coupling circuit (409 of FIG. 4) comprising a coil 909 such as, e.g., a helical coil. As shown in FIG. 9A, the guided surface waveguide probe 400b can include the upper charge terminal T, (e.g., a sphere at height h_T) and a lower compensation terminal T₂ (e.g., a disk at height h_d) that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 403. A second medium 406 is located above the lossy conducting medium 403. The charge terminal T has a self-capacitance C_p, and the compensation terminal T₂ has a self-capacitance C_d. During operation, charges Ch and Q₂ are imposed on the terminais T, and T_2> respectively, depending on the voltages applied to the terminais T_t and T₂ at any given instant

In the example of FIG. 9A, the coil 909 is coupled to a ground stake 915 at a first end and the compensation terminal T₂ at a second end. In some implémentations, the connection to the compensation terminal T₂ can be adjusted using a tap 921 at the second end of the coil 909 as shown in FIG. 9A. The coil 909 can be energized at an operating frequency by the AC source 912 through a tap 924 at a lower portion of the coil 909. In other implémentations, the AC source 912 can be inductively coupled to the coil 909 through a primary coil. The charge terminal T, is energized through a tap 918 coupled to the coil 909. An ammeter 927 located between the coil 909 and ground stake 915 can be used to provide an indication of the magnitude of the current flow at the base of the guided surface waveguide probe. Alternatively, a current clamp may be used around the conductor coupled to the ground stake 915 to obtain an indication of the magnitude of the current flow. The compensation terminal T₂ is positioned above and substantially parallel with the lossy conducting medium 403 (e.g., the ground).

The construction and adjustment of the guided surface waveguide probe 400 is based upon various operating conditions, such as the transmission frequency, conditions of the lossy conductive medium (e.g., soil conductivity σ and relative permittivity f_r), and size of the charge terminal T_v The index of refraction can be calculated from Equations (10) and (11) as n = j£_r~jx, (52) where x = σ/ωε₀ with ω = 2π/, and complex Brewster angle (θ_{ί Β}) measured from the surface normal can be determined from Equation (42) as ^î,b ⁼ arctan(^/£_r - jx), (53) or measured from the surface as shown in FIG. 8A as

Ψί,Β — j ^— d_iB. (54)

The wave tilt at the Hankel crossover distance can also be found using Equation (47).

The Hankel crossover distance can also be found by equating Equations (20b) and (21), and solving for R_x. The electrical effective height can then be determined from Equation (46) using the Hankel crossover distance and the complex Brewster angle as h-eff ⁼ Κχ^ίΆηΨϊ,β — hp^el<ti· (55)

As can be seen from Equation (55), the complex effective height (h._eff) includes a magnitude that is associated with the physical height (h_p) of charge terminal Tî and a phase (Φ) that is to be associated with the angle of the wave tilt at the Hankel crossover distance (Ψ). With these variables and the selected charge terminal Τί configuration, it is possible to détermine the configuration of a guided surface waveguide probe 400.

With the selected charge terminal T, configuration, a spherical diameter (or the effective spherical diameter) can be determined. For example, if the charge terminal G is not configured as a sphere, then the terminal configuration may be modeled as a spherical capacitance having an effective spherical diameter. The size of the charge terminal T, can be chosen to provide a sufficiently large surface for the charge Q-, imposed on the terminais. In general, it is désirable to make the charge terminal Τ_Ί as large as practical. The size of the charge terminal T, should be large enough to avoid ionization of the surrounding air, which can resuit in electrical discharge or sparking around the charge terminal. As previously discussed with respect to FIGS. 6A and 6B, to reduce the amount of bound charge on the charge terminal T₁₍ the desired élévation of the charge terminal Ti should be 4-5 times the effective spherical diameter (or more). If the élévation of the charge terminal U is less than the physical height (A_p) indicated by the complex effective height (h_eff) determined using Equation (55), then the charge terminal T-, should be positioned at a physical height of h_T — h_p above the lossy conductive medium (e.g., the earth). If the charge terminal T, is located at h_p, then a guided surface wave tilt can be produced at the Hankel crossover distance (R_x) without the use of a compensation terminal T₂. FIG. 9B illustrâtes an example of the guided surface waveguide probe 400c without a compensation terminal T₂.

Referring back to FIG. 9A, a compensation terminal T₂ can be included when the élévation of the charge terminal T-, is greater than the physical height (h_p) indicated by the determined complex effective height As discussed with respect to FIG. 8B, the compensation terminal T₂ can be used to adjust the total effective height (h_TE) of the guided surface waveguide probe 400 to excite an electric field having a guided surface wave tilt at R_x. The compensation terminal T₂ can be positioned below the charge terminal Ti at a physical height of h_d = h_T - h_p, where h_T is the total physical height of the charge terminal T,. With the position of the compensation terminal T₂ fixed and the phase delay 4>_L applied to the lower compensation terminal T₂, the phase delay 4>u applied to the upper charge termina) U can be determined using Equation (50).

When installing a guided surface waveguide probe 400, the phase delays 4>_y and 4>_L of Equations (48)-(50) may be adjusted as follows. Initially, the complex effective height (h_eyy) and the Hankel crossover distance (R_x) are determined for the operational frequency To minimize bound capacitance and corresponding bound charge, the upper charge terminal is positioned at a total physical height (h_r) that is at least four times the spherical diameter (or équivalent spherical diameter) of the charge terminal T_v Note that, at the same time, the upper charge terminal T, should also be positioned at a height that is at least the magnitude (/i_p) of the complex effective height (h_eff)· > ti_p, then the lower compensation terminal T₂ can be 21 positioned at a physical height of h_d = h_T- h_p as shown in FIG. 9A. The compensation terminal T₂ can then be coupled to the coil 909, where the upper charge terminal T, is not yet coupled to the coil 909. The AC source 912 is coupled to the coil 909 in such a manner so as to minimize reflection and maximize coupling into the coil 909. To this end, the AC source 912 may be coupled to the coil 909 at an appropriate point such as at the 50Ω point to maximize coupling. In some embodiments, the AC source 912 may be coupled to the coil 909 via an impédance matching network. For example, a simple L-network comprising capacitors (e.g., tapped or variable) and/or a capacitor/inductor combination (e.g., tapped or variable) can be matched to the operational frequency so that the AC source 912 sees a 50Ω load when coupled to the coil 909. The compensation terminal T₂ can then be adjusted for parallel résonance with at least a portion of the coil at the frequency of operation. For example, the tap 921 at the second end of the coil 909 may be repositioned. While adjusting the compensation terminal circuit for résonance aids the subséquent adjustment of the charge terminal connection, it is not necessary to establish the guided surface wave tilt (W_ffx) at the Hankel crossover distance (R_x). The upper charge terminal U may then be coupled to the coil 909.

In this context, FIG. 10 shows a schematic diagram of the general electrical hookup of FIG. 9A in which V, is the voltage applied to the lower portion of the coil 909 from the AC source 912 through tap 924, V₂ is the voltage at tap 918 that is supplied to the upper charge terminal U, and V₃ is the voltage applied to the lower compensation terminal T₂ through tap 921. The résistances R_p and R_d represent the ground return résistances of the charge terminal Τ_Ί and compensation terminal T₂, respectively. The charge and compensation terminais and T₂ may be configured as spheres, cylinders, toroids, rings, hoods, or any other combination of capacitive structures. The size of the charge and compensation terminais T, and T₂ can be chosen to provide a sufficiently large surface for the charges Qî and Q₂ imposed on the terminais. In general, it is désirable to make the charge terminal as large as practical. The size of the charge terminal T, should be large enough to avoid ionîzation of the surrounding air, which can resuit in electrical discharge or sparking around the charge terminal. The selfcapacitance C_p and C_d can be determîned for the sphère and disk as disclosed, for example, with respect to Equation (38).

As can be seen in FIG. 10, a résonant circuit is formed by at least a portion of the inductance of the coil 909, the self-capacitance C_d of the compensation terminal T₂, and the ground return résistance R_d associated with the compensation terminal T₂. The parallel résonance can be established by adjusting the voltage V₃ applied to the compensation terminal T₂ (e.g, by adjusting a tap 921 position on the coil 909) or by adjusting the height and/or size of the compensation terminal T₂ to adjust C_d. The position of the coil tap 921 can be adjusted for parallel résonance, which will resuit in the ground current through the ground stake 915 and 22

18762 * to improve coupling by iterativeîy adjusting the position ofthe tap 924 iorthe AC source 912 to be at the 50Ω point on the coil 909 and adjusting the position of tap 91S to maximize the ground current through the ammeter 927.

In one experimental example, a guided surface waveguide probe 400h was constructed to verify the operation of the proposed structure at 1.879MHz. The soil cocductîvity at the site cf the guided surface waveguide probe 400b was determined to be σ = 0.0053 mhos/m and the relative permittivity was s_r - 28. Using these values, the index cf re.Tactic:· cjiven by Equation (52) was determined to be n = 6.555 -)3.869. Based upon Equations (53) and (54), the complex Brewster angle was found to be θ_{ί Β} = 83.517 -)3.783 degrees, er _ΰ - 6.483 + )3.783 degrees.

Using Equation (47), the guided surface wave tilt was calculated as = û.113 + )0.067 = 0.131 e'f³⁰·⁵⁵¹ \ A Hankel crossover distance of Rx = 54 feet was found by equating Equations (20b) and (21), and solving for Rx. Using Equation (55), the complex effective height (heff = hpe^J,p) was determined to be hp = 7.094 feet (relative to the lossy conducting medium) and Φ = 30.551 degrees (relative to the ground current). Note that the phase Φ is equal to the argument of the guided surface wave tilt Ψ. However, the physical height of h_p = 7.094 feet is relatively small. While this will excite a guided surface waveguide mode, the proximity of the elevated charge terminal T, to the earth (and its mirror image) will resuit in a large amount of bound charge and very little free charge. Since the guided surface wave field strength is proportional to the free charge on the charge terminal, an increased élévation was désirable.

To increase the amount of free charge, the physical height of the charge terminal T, was set to be /ip = 17 feet, with the compensation terminal T₂ positioned below the charge terminal T_v The extra lead lengths for connections were approximately y = 2.7 feet and z = 1 foot. Using these values, the height ofthe compensation terminal T₂ (h_d) was determined using Equation (50). This is graphically illustrated in FIG. 11, which shows plots 130 and 160 ofthe imaginary and real parts of Φ_υ, respectively. The compensation terminal T₂ is positioned at a height h_d where ΙιπίΦ^,} = 0, as graphically illustrated in plot 130. In this case, setting the imaginary part to zero gives a height of h_d = 8.25 feet At this fixed height, the coil phase Φ^ can be determined from Rc{Tp} as +22.84 degrees, as graphically illustrated in plot 160.

As previously discussed, the total effective height is the superposition of the upper effective height (h_UE) associated with the charge terminal T, and the lower effective height (h_LE) associated with the compensation terminal T₂ as expressed in Equation (49) With the coil tap adjusted to 22.84 degrees, the complex upper effective height is given as h_UE = (Λρ + z)e-'(^/?(^ftP^+z)^+tI’^y) = 14.711 + )10.832 (56) (or 18.006 at 35.21°) and the complex lower effective height is given as h_LE = (Λα + γ)ε>^⁽^⁺Χ)^+φι.) = -8.602 - /6.776 (57) (or 10.950 at-141.773°). The total effective height (h_TE) is the superposition of these two values, which gives h_TE = h_UE + h_LE = 6.109 -/3.606 = 7.094^^{30 551}^. (58)

As can be seen, the coil phase matches the calculated angle of the guided surface wave tilt, lV_Rx. The guided surface waveguide probe can then be adjusted to maximize the ground current. As previously discüssed with respect to FIG. 9A, the guided surface waveguide mode coupling can be improved by iteratively adjusting the position of the tap 924 for the AC source 912 to be at the 50Ω point on the coil 909 and adjusting the position of tap 918 to maximize the ground current through the ammeter 927.

Field strength measurements were carried out to verify the ability of the guided surface waveguide probe 400b (FIG. 9A) to couple into a guided surface wave or a transmission line mode. Referring to FIG. 12, shown is an image of the guided surface waveguide probe used for the field strength measurements. FIG. 12 shows the guided surface waveguide probe 400b including an upper charge terminal T, and a lower compensation terminal T_2l which were both fabricated as rings. An insulating structure supports the charge terminal above the compensation terminal T₂. For example, an RF insulating fiberglass mast can be used to support the charge and compensation terminais T, and T₂. The insulating support structure can be configured to adjust the position of the charge and compensation terminais T, and T₂ using,

e.g., insulated guy wires and pulleys, screw gears, or other appropriate mechanism as can be understood. A coil was used in the coupling circuit with one end of the coil grounded to an 8 foot ground rod near the base of the RF insulating fiberglass mast. The AC source was coupled to the right side of the coil by a tap connection (V,), and taps for the charge terminal T, and compensation terminal T₂ were located at the center (V₂) and the left of the coil (V₃). FIG. 9A graphically illustrâtes the tap locations on the coil 909.

The guided surface waveguide probe 400b was supplied with power at a frequency of 1879 kHz. The voltage on the upper charge terminal L was 15.6Vpeak-peak (5.515V_RMS) with a capacitance of 64pF. Field strength (FS) measurements were taken at predetermined distances along a radial extending from the guided surface waveguide probe 400b using a FIM-41 FS meter (Potomac Instruments, Inc., Silver Spring, MD). The measured data and predicted values for a guided surface wave transmission mode with an electrical launching efficiency of 35% are indicated in TABLE 1 beîow. Beyond the Hankel crossover distance (R_x), the large argument asymptote prédominâtes over the close-in” représentation of the Hankel function, and the vertical component of the mode-matched electric asymptotically passes to Equation (44), which is linearly proportional to free charge on the charge terminal. TABLE 1 shows the measured values and predicted data. When plotted using an accurate plotting application (Mathcad), the 25 measured values were found to fit an electrical launching efficiency curve corresponding to 38%, as illustrated in FIG. 13. For 15.6V_PP on the charge terminal T,, the field strength curve (Zenneck @ 38%) passes through 363pV/m at 1 mile (and 553pV/m at 1 km) and scales linearly with the capacitance (C_p) and applied terminal voltage.

Range (miles)	Measured FS w/ FIM- 41 (pV/m)	Predicted FS (pV/m)
0.64	550	546
1.25	265	263
3.15	67	74
4.48	30	35
6.19	14	13

TABLE 1

The lower electrical launching efficiency may be attributed to the height of the upper charge terminal Τ_υ Even with the charge terminal T, elevated to a physical height of 17 feet, the bound charge reduces the efficiency of the guided surface waveguide probe 400b. While increasing the height of the charge terminal T| would improve the launching efficiency of the guided surface waveguide probe 400b, even at such a low height (/i_d/A = 0.032) the coupled wave was found to match a 38% electric launching efficiency curve. In addition, it can be seen in FIG. 13 that the modest 17 foot guided surface waveguide probe 400b of FIG. 9A (with no ground system other than an 8 foot ground rod) exhibits better field strength than a full quarterwave tower (Λ/4 Norton = 131 feet tall) with an extensive ground system by more than 10dB in the range of 1-6 miles at 1879 kHz. Increasing the élévation of the charge terminal T_n and adjusting the height of the compensation terminal T₂ and the coil phase Φ_α, can improve the guided surface waveguide mode coupling, and thus the resulting electric field strength.

In another experimental example, a guided surface waveguide probe 400 was constructed to verify the operation of the proposed structure at 52MHz (corresponding to ω = 2nf = 3.267 χ 10⁸ radians/sec). FIG. 14A shows an image of the guided surface waveguide probe 400. Fig. 14B is a schematic diagram of the guided surface waveguide probe 400 of FIG 14A. The complex effective height between the charge and compensation terminais Ti and T₂ of the doublet probe was adjusted to match R_x times the guided surface wave tilt, at the Hankel crossover distance to launch a guided surface wave. This can be accomplished by changing the physical spacing between terminais, the magnetic link coupling and its position between the AC source 912 and the coil 909, the relative phase of the voltage between the terminais T, and T₂, the height of the charge and compensation terminal T, and T₂ relative to ground or the lossy conducting medium, or a combination thereof. The conductivity of the lossy conducting medium at the site of the guided surface waveguide probe 400 was determined to be σ = 0.067 mhos/m and the relative permittivity was £_r = 82.5. Using these values, the index of refraction was determined to be n = 9.170 —y'1.263. The complex Brewster angle was found to be ψ_{ί Β} = 6.110 + y0.8835 degrees.

A Hankel crossover distance of R_x = 2 feet was found by equating Equations (20b) and (21), and solving for R_x. FIG. 15 shows a graphical représentation of the crossover distance R_x at 52 Hz. Curve 533 is a plot of the ‘far-out asymptote. Curve 536 is a plot of the “close-in asymptote. The magnitudes of the two sets of mathematical asymptotes in this example are equal at a Hankel crossover point 539 of two feet. The graph was calculated for water with a conductivity of 0.067 mhos/m and a relative dielectric constant (permittivity) of ε_Γ = 82.5, at an operating frequency of 52 MHz. At lower frequencies, the Hankel crossover point 539 moves farther out. The guided surface wave tilt was calculated as lV_Rx = 0.108 e^/(7851°⁾. For the doublet configuration with a total height of 6 feet, the complex effective height (he^ = 2/ΐρε^;φ = Rx tan ψ; _fl) was determined to be 2/t_p = 6 înches with Φ = - 172 degrees. When adjusting the phase delay of the compensation terminal T₂ to the actual conditions, it was found that Φ = 174 degrees maximized the mode matching of the guided surface wave, which was within experimental error.

Field strength measurements were carried out to verify the ability of the guided surface waveguide probe 400 of FIGS. 14A and 14B to couple into a guided surface wave or a transmission line mode. With 10V peak-to-peak applied to the 3.5 pF terminais and T₂, the electric fields excited by the guided surface waveguide probe 400 were measured and plotted in FIG. 16. As can be seen, the measured field strengths fell between the Zenneck curves for 90% and 100%. The measured values for a Norton half wave dipole antenna were significantly less.

Referring next to FIG. 17, shown is a graphical représentation of another example of a guided surface waveguide probe 400d including an upper charge terminal T, (e.g., a sphère at height h_T) and a lower compensation terminal T₂ (e.g., a disk at height h_d) that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 403. During operation, charges Q, and Q₂ are imposed on the charge and compensation terminais Tî and T₂, respectively, depending on the voltages applied to the terminais Tî and T₂ at any given instant.

As in FIGS. 9A and 9B, an AC source 912 acts as the excitation source (412 of FIG. 4) for the charge terminal D. The AC source 912 is coupled to the guided surface waveguide probe 400d through a coupling circuit (409 of FIG. 4) comprising a coil 909. The AC source 912 can be connected across a lower portion of the coil 909 through a tap 924, as shown in FIG. 17, or can be inductively coupled to the coil 909 by way of a primary coil. The coil 909 can be coupled to a ground stake 915 at a first end and the charge terminal T, at a second end. In some implémentations, the connection to the charge terminal T, can be adjusted using a tap 930 at the second end of the coil 909. The compensation terminal T₂ is positicned abovo and substantially parallel with the lossy conducting medium 403 (e.g., the ground or earth), and energized through a tap 933 coupled to the coil 909. An ammeter 927 located between the coil 909 and ground stake 915 can be used to provide an indication of the magnitude or the current flow (/₀) at the base of the guided surface waveguide probe. Alternatively, a current clamp may be used around the conductor coupled to the ground stake 915 to obtain an indication of the magnitude of the current flow (7₀).

In the embodiment of FIG. 17, the connection to the charge terminal T, (tap 930) has been moved up above the connection point of tap 933 for the compensation terminal T₂ as compared to the configuration of FIG. 9A. Such an adjustment allows an increased voltage (and thus a higher charge Q,) to be applied to the upper charge terminal T_v As with the guided surface waveguide probe 400b of FIG. 9A, it is possible to adjust the total effective height (h_TE) of the guided surface waveguide probe 400d to excite an electric field having a guided surface wave tilt at the Hankei crossover distance R_x. The Hankel crossover distance can also be found by equating Equations (20b) and (21), and solving for R_x. The index of refraction (n), the complex Brewster angle (0_{i B} and ψ_ίβ), the wave tilt (|W |ε^7ψ) and the complex effective height (h_eff = h_pe^;tt>) can be determined as described with respect to Equations (52) - (55) above.

With the selected charge terminal T, configuration, a spherical diameter (or the effective spherical diameter) can be determined. For example, if the charge terminal T) is not configured as a sphere, then the terminal configuration may be modeled as a spherical capacitance having an effective spherical diameter. The size ofthe charge terminal Ti can be chosen to provide a sufficiently large surface for the charge Q·, imposed on the terminais. In general, it is désirable to make the charge terminal as large as practical. The size of the charge terminal Τ_Ί should be large enough to avoid ionization of the surrounding air, which can resuit in electrical discharge or sparking around the charge terminal. To reduce the amount of bound charge on the charge terminal T,, the desired élévation to provide free charge on the charge terminal T! for launching a guided surface wave should be at least 4-5 times the effective spherical diameter above the lossy conductive medium (e.g., the earth). The compensation terminal T₂ can be used to adjust the total effective height (h_TE) of the guided surface waveguide probe 400d to excite an electric field having a guided surface wave tilt at R_K. The compensation terminal T₂ can be positioned below the charge terminal Ti at h_d = h_T - h_p, where h_T is the total physical height of the charge terminal T_b With the position of the compensation terminal T₂ fixed and the phase delay Φ_;; applied to the upper charge terminal T_b the phase delay Φ_Λ applied to the lower compensation terminal T₂ can be determined using the relationships of Equation (49).

*M = -M, +y) (59)

In alternative embodiments, the compensation terminal T₂ can be positioned at a height h_d where Ιγπ{Φ_£} = 0.

. With the AC source 912 coupled to the coil 909 (e.g., at the 50Ω point to maximize coupling), the position of tap 933 may be adjusted for parallel résonance of the compensation terminal T₂ with at least a portion of the coil at the frequency of operation. Voltage V₂ from the coil 909 can be applied to the charge terminal T,, and the position of tap 930 can be adjusted such that the phase (Ψ) of the total effective height (h_TE) approximately equals the angle of the guided surface wave tilt (VV_Hx) at the Hankel crossover distance (R_x). The position of the coil tap 930 can be adjusted until this operating point is reached, which results in the ground current through the ammeter 927 increasing to a maximum. At this point, the résultant fields excited by the guided surface waveguide probe 400d are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium 403, resulting in the launching of a guided surface wave along the surface of the lossy conducting medium 403. This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe 400.

In other implémentations, the voltage V₂ from the coil 909 can be applied to the charge terminal Τ_Ί, and the position of tap 933 can be adjusted such that the phase (Φ) of the total effective height (h_TE) approximately equals the angle of the guided surface wave tilt (Ψ) at R_x. The position of the coil tap 930 can be adjusted until the operating point is reached, resulting in the ground current through the ammeter 927 substantially reaching a maximum. The résultant fields are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium 403, and a guided surface wave is launched along the surface of the lossy conducting medium 403. This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe 400. The System may be further adjusted to improve coupling by îteratively adjusting the position of the tap 924 for the AC source 912 to be at the 50Ω point on the coil 909 and adjusting the position of tap 930 and/or 933 to maximize the ground current through the ammeter 927.

FIG. 18 is a graphical représentation illustrating another example of a guided surface waveguide probe 400e including an upper charge terminal T-, (e.g., a sphere at height h_T) and a lower compensation terminal T₂ (e.g., a disk at height h_d) that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 403. In the example of FIG. 18, the charge terminal T) (e.g., a sphere at height h_T) and compensation terminal T₂(e.g., a disk at height h_d) are coupled to opposite ends of the coil 909. For example, charge terminal ty can be connected via tap 936 at a first end of coil 909 and compensation terminal T₂can be connected via tap 939 at a second end of coil 909 as shown in FIG. 18. The compensation terminal T₂ is positioned above and substantially paralle! with the lossy conducting medium 403 (e.g., the ground or earth). During operation, charges Q, and Q₂ are imposed on the charge and compensation terminais and T₂, respectively, depending on the voltages applied to the terminais ty and T₂ at any given instant.

An AC source 912 acts as the excitation source (412 of FIG. 4) for the charge termina! ty. The AC source 912 is coupled to the guided surface waveguide probe 400e through a coupling circuit (409 of FIG. 4) comprising a coil 909. In the example of FIG. 18, th i AC source 912 is connected across a middle portion of the coil 909 through tapped connections 942 and 943. In other embodiments, the AC source 912 can be inductively coupled to the coil 909 through a primary coil. One side of the AC source 912 is also coupled to a ground stake 915, which provides a ground point on the coil 909. An ammeter 927 located between the coil 909 and ground stake 915 can be used to provide an indication of the magnitude of the current flow at the base of the guided surface waveguide probe 400e. Alternatively, a current clamp may be used around the conductor coupled to the ground stake 915 to obtain an indication of the magnitude of the current flow.

It is possible to adjust the total effective height (h_TE) of the guided surface waveguide probe 400e to excite an electric field having a guided surface wave tilt at the Hankel crossover distance R_x, as has been previously discussed. The Hankel crossover distance can also be found by equating Equations (20b) and (21), and solving for R_x. The index of refraction (n), the complex Brewster angle (d_iB and ψ_{ί β}) and the complex effective height (h_eff = h_pe^J,p) can be determined as described with respect to Equations (52) - (55) above.

A spherical diameter (or the effective spherical diameter) can be determined for the selected charge terminal ty configuration. For example, if the charge terminal ty is not configured as a sphere, then the terminal configuration may be modeled as a spherical capacitance having an effective spherical diameter. To reduce the amount of bound charge on the charge terminal T,, the desired élévation to provide free charge on the charge terminal T, for launching a guided surface wave should be at least 4-5 times the effective spherical diameter above the lossy conductive medium (e.g., the earth). The compensation terminal T₂ can be positioned below the charge terminal ty at h_d = h_T - h_p, where h_T is the total physical height of the charge terminal ty. With the positions of the charge terminal ty and the compensation terminal T₂ fixed and the AC source 912 coupled to the coil 909 (e.g., at the 50Ω point to maximize coupling), the position of tap 939 may be adjusted for parallel résonance of the compensation terminal T₂ with at least a portion of the coil at the frequency of operation. While adjusting the compensation terminal circuit for résonance aids the subséquent adjustment of the charge terminal connection, it is not necessary to establish the guided surface wave tilt J at the Hankel crossover distance (R_x). One or both cf the phase delays 4>_L and Φ_μ applied to the upper charge terminal and lower compensation terminal T₂ can be adjusted by repositioning one or both of the taps 936 and/or 939 on the coil 909. In addition, the phase delays Φ_ί and Φ_π may be adjusted by repositioning one or bcth of the taps 942 of the AC source 912. The ‘ position of the coil tap(s) 936, 939 and/or 942 can be adjusted until this operating point is reached, which résulte in the ground current through the ammeter 927 increasing to a maximum. This can be verifîed by measuring field strength along a radial extending from the guided surface waveguide probe 400. The phase delays may then be adjusted by repositicning these tap(s) to increase (or maximize) the ground current.

When the electric fields produced by a guided surface waveguide probe 400 has a guided surface wave tilt at the Hankel crossover distance R_x, they are substantially modematched to a guided surface waveguide mode on the surface of the lossy conducting medium, and a guided electromagnetic field in the form of a guided surface wave is launched along the surface of the lossy conducting medium. As illustrated in FIG. 1, the guided field strength curve 103 of the guided electromagnetic field has a characteristic exponential decay of e~^aà/y[d and exhibits a distinctive knee 109 on the log-îog scale. Receive circuits can be utilized with one or more guided surface waveguide probe to facilitate wireless transmission and/or power delivery Systems.

Referring next to FIGS. 19A, 19B, and 20, shown are examples of generalized receive circuits for using the surface-guided waves in wireless power delivery Systems. FIGS. 19A and 19B include a linear probe 703 and a tuned resonator 706, respectively. FIG. 20 is a magnetic coil 709 according to various embodiments of the présent disclosure. According to various embodiments, each one of the linear probe 703, the tuned resonator 706, and the magnetiG coil 709 may be employed to receive power transmitted in the form of a guided surface wave on the surface of a lossy conducting medium 403 (FIG. 4) according to various embodiments. As mçntioned above, in one embodiment the lossy conducting medium 403 comprises a terrestrial medium (or earth).

With spécifie reference to FIG. 19A, the open-circuit terminal voltage at the output terminais 713 of the linear probe 703 dépends upon the effective height of the linear probe 703. To this end, the terminal point voltage may be calculated as

Vt= (60) where E_inc is the strength of the electric field on the linear probe 703 in Volts per nieter, dl is an element of intégration along the direction of the linear probe 703, and is the effective height of the linear probe 703. An electrical load 716 is coupled to the output terminais 713 through an impédance matching network 719.

When the linear probe 703 is subjected to a guided surface wave as described above, a voltage is developed across the output terminais 713 that may be applied to the electrical load 716 through a conjugale impédance matching network 719 as the case may be. In order tu facilitate the flow of power to the electrical load 716, the electrical load 716 should be substantially impédance matched to the linear probe 703 as will be described below.

Referring to FIG. 19B, the tuned resonator 706 includes a charge terminal T_R that is elevated abcve the lossy conducting medium 403. The charge terminal Tr has a selfcapacitance C_R. In addition, there may also be a bound capacitance (not shown) between the charge terminal Tr and the lossy conducting medium 403 depending on the height of the charge terminal T_R above the lossy conducting medium 403. 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 400.

The tuned resonator 706 also includes a coil L_R. 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 403. To this end, the tuned resonator 706 (which may also be referred to as tuned resonator Lr-Cr) comprises a series-tuned resonator as the charge terminal C_R and the coil L_R are situated in sériés. The tuned resonator 706 is tuned by adjustîng the size and/or height of the charge terminal T_R, and/or adjusting the size of the coil L_R so that the reactive impédance of the structure is substantially eliminated.

For example, the reactance presented by the self-capacitance C_R is calculated as Note that the total capacitance of the tuned resonator 706 may also include capacitance between the charge terminal T_R and the lossy conducting medium 403, where the total capacitance of the tuned resonator 706 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 403.

The inductive reactance presented by a discrete-element coil L_R may be calculated as jœL, where L is the lumped-element inductance of the coil L_R. If the coil L_R is a distributed element, its équivalent terminal-point inductive reactance may be determined by conventional approaches. To tune the tuned resonator 706, one would make adjustments so that the 32 inductive reactance presented by the coil L_R equals the capacitive reactance presented by the tuned resonator 706 so that the resulting net reactance of the tuned resonator 706 is substantially zéro at the frequency of operation. An impédance matching network 723 may be inserted between the probe terminais 721 and ihe electrical load 726 in order to effect a conjugate-match condition for maxim power transfer to the electrical load 726.

When placed in the presence of a guided surface wave, generated at the frequency of the tuned resonator 706 and the conjugale matching network 723, as described sIjovc* maximum power will be delivered from the surface guided wave to the electrical load 726. That is, once conjugale impédance matching is established between the tuned resonator 706 and the electrical load 726, power will be delivered from the structure to the electrical load 726. To this end, an electrical load 726 may be coupled to the tuned resonator 706 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. 19B, magnetic coupling is employed where a coil L_s is positioned as a secondary relative to the coil L_R that acts as a transformer primary. The coil L_s may be link coupled to the coil L_R by geometrically winding it around the same core structure and adjusting the coupled magnetic flux as can be appreciated. In addition, while the tuned resonator 706 comprises a series-tuned resonator, a parallel-tuned resonator or even a distributed-element resonator may also be used.

Referring to FIG. 20, the magnetic coil 709 comprises a receive circuit that is coupled through an impédance matching network 733 to an electrical load 736. In order to facilitate réception and/or extraction of electrical power from a guided surface wave, the magnetic coil 709 may be positioned so that the magnetic flux of the guided surface wave, Η_ψι passes through the magnetic coil 709, thereby inducing a current in the magnetic coil 709 and producing a terminal point voltage at its output terminais 729. The magnetic flux of the guided surface wave coupled to a single turn coil is expressed by ^ψ = ÎÎ_Acs8_r^oîî -ndA (61) where Ψ is the coupled magnetic flux, μ_Γ is the effective relative permeability of the core of the magnetic coil 709, μ_σ 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 709 oriented for maximum coupling to an incident magnetic field that is uniform over the cross-secticnal area of the magnetic coil 709, the open-circuit induced voltage appearing at the output terminais 729 of the magnetic coil 709 is

F = « -jüw_rp₀HA_cs, (62) where the variables are defined above. The magnetic coil 709 may be tuned to the guided surface wave frequency either as a distributed resonator or with an external capacitor across its output terminais 729, as the case may be, and then impedance-matched to an external electrical load 736 through a conjugale impédance matching network 733.

Assuming that the resulting circuit presented by the magnetic coil 709 and the electrical load 736 are properly adjusted and conjugale impédance matched, via impédance matching network 733, then the current induced in the magnetic oeil 709 may be employai to sptimalîy power the electrical load 736. The receive circuit presented by the magnetic coil 709 provides an advantage in that it does not hâve to be physically connected to the ground.

With reference to FIGS. 19A, 19B, and 20, the receive circuits presented by the linear probe 703, the tuned resonator 706, and the magnetic coil 709 each facilitate receiving electrical power transmitted from any one of the embodiments of guided surface waveguide probes 400 described abeve. To this end, the energy received may be used to supply power to an electrical load 716/726/736 via a conjugale matching network as can be appreciated. This contrasls 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 présent guided surface waves generated using the guided surface waveguide probes 400 described above that the receive circuits presented by the linear probe 703, the tuned resonator 706, and the magnetic coil 709 will load the excitation source 413 (FIG. 4) that is applied to the guided surface waveguide probe 400, 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 400 described above comprises a transmission line mode. By way of contrast, a power source that drives a radiating antenna that generales 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 400 and one or more receive circuits in the form of the linear probe 703, the tuned resonator 706, and/or the magnetic coil 709 can together make up a wireless distribution system. Given that the distance of transmission of a guided surface wave using a guided surface waveguide prebe 400 as set forth above dépends upon the frequency, it is possible that wireless power distribution can be achieved across wide areas and even globally.

The conventional wireless-power transmission/distribution Systems extensively investigated today include energy harvesting from radiation fields and also sensor coupling to inductive or reactive near-fields. In contrast, the présent wireless-power system does not waste power in the form of radiation which, if not intercepted, is lost forever. Nor is the presently disclosed wireless-power system limited to extremeiy short ranges as with conventionai mutuaireactance coupled near-field Systems. The wireless-power system disclosed herein probecouples to the novel surface-guided transmission line mode, which is équivalent to deüvering 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 extremeiy low frequencies is insignificant relative to the transmission losses in conventionai high-tensicn power lines at 60 Hz, ail 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. 21A shown is a schematic that represents the linear probe 703 and the tuned resonator 706. FIG 21B shows a schematic that represents the magnetic coil 709. The linear probe 703 and the tuned resonator 706 may each be considerad 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 709 may be viewed as a Norton équivalent represented by a short-circuit terminal current source Is and a dead network terminal point impédance Zs. Each electrical load 716/726/736 (FIGS. 19A, 19B and 20) may be represented by a load impédance Z_L. The source impédance Z_s comprises both real and imaginary components and takes the form Z₃ = R_s + jXsAccording to one embodiment, the electrical load 716/726/736 is impédance matched to each receive circuit, respectively. Specifically, each electrical load 716/726/736 présents through a respective impédance matching network 719/723/733 a load on the probe network specified as Z_L' expressed as Z_L’ = R_L' + j Xl', which will be equal to Z_L‘ = 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 716/726/736 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 716/726/736 is impédance matched to the respective receive circuit and that maximum power transfer is established to the respectiva electrical load 716/726/736.

Operation of a guided surface waveguide probe 400 may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 400. For example, a probe control System 418 (FIG. 4) can be used to control the coupling circuit 409 and/or positioning of the charge terminal T-ι and/or compensation terminal T₂ to control the operation of the guided surface waveguide probe 400. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 403 (e.g., 35 conductivity σ and relative permittivity ε_Γ), variations in field strength and/or variations in loading of the guided surface waveguide probe 400. As can be seen from Equations (52) - (55). the index of refraction (n), the complex Brewster angle (θ_ίΒ and \p_{i B}), the wave tilt (|IV|β'^ψ) and the complex effective height (h_eff = h_pe^J<t>) can be affected by changes in soil conductivity and permittivity resulting from, e.g., weather conditions.

Equipment such as, e.g., conductivity measurernent probes, permittivity senscrs, 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 operationa! conditions to the probe control system 418. The probe control system 418 can then make one or more adjustments to the guided surface waveguide probe 400 to maintain specified 'operational conditions for the guided surface waveguide probe 400. For instance, as the ' moisture and température vary, the conductivity of the soil will also vary. Conductivity ' measurernent probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe 400. 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 measurernent probes and/or permittivity sensors may be located at multiple locations (e.g., in each quadrant) around the guided surface waveguide probe 400.

FIG. 22A shows an example of a conductivity measurernent probe that can be installed for monitoring changes in soil conductivity. As shown in FIG. 22A, a sériés of measurernent 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 applyîng 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 scpply 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 ^rsoil. As illustrated in FIG. 22B, impédance is measured between the tops of two rods inrerted 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 s.84 r « 1 io⁶ r r 1 ^σ=^{and =}s^Iï^J· ^<63) 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 418 (FIG. 4). The information may be communicated to the probe control System 413 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 monîtored conductivity and/or permittivity, the probe control System 418 may evaluate the variation in the index of refraction (n), the complex Brewster angle {9_ilt and . -ke wave tilt (|ΐν|β^;ψ) and/or the complex effective height (heyz- — hpe^j<p) and adjust the guided surface waveguide probe 400 to maintain the wave tilt at the Hankel crossover distance so that the illumination remains at the complex Brewster angle. This can be accomplished by adjusting, e.g., /ip=, Φυ, Φ_Δ and/or h_d. For instance, the probe control System 418 can adjust the height (/i_d) of the compensation terminal T₂ or the phase delay (Φ^, ΦΗ applied to the charge terminal Ti and/or compensation terminal T₂, respectively, to maintain the electrical launching efficiency of the guided surface wave at or near its maximum. The phase applied to the charge terminal T, and/or compensation terminal T₂ can be adjusted by varying the tap position on the coil 909, and/or by including a plurality of predefined taps along the coil 909 and switching between the different predefined tap locations to maximize the launching efficiency.

Field orfield 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 400 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 418. The information may be communicated to the probe control System 418 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 400 may be adjusted to maintain specified field strength(s) at the FS meter locations to ensure appropriate power transmission to the receivers and the loads they supply.

< For example, the phase delay (Φ^, ΦΗ applied to the charge terminal and/or compensation terminal T₂, respectively, can be adjusted to improve and/or maximize the electrical launching efficiency of the guided surface waveguide probe 400. By adjusting one or both phase delays, the guided surface waveguide probe 400 can be adjusted to ensure the wave tilt at the Hankel crossover distance remains at the complex Brewster angle. This can be accomplished by adjusting a tap position on the coil 909 to change the phase delay supplied to the charge terminal T, and/or compensation terminal T₂. The voltage level supplied to the charge terminal h 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 412 (FIG. 4) or by adjusting or reconfiguring the coupling circuit 409 (FIG. 4). For instance, the position of the tap 924 (FIG. 4) for the AC source 912 (FIG. 4) can be adjusted to increase the voltage seen by the charge terminal T,. Maintaining field strength levels within predefined ranges can împrove coupling by the receivers, reduce ground current losses, and avoid interférence with transmissions from other guided surface waveguide probes 400.

Referring to FIG. 23A, shown is an example of an adaptive control system 430 including the probe control system 418 of FIG. 4, which is configured to adjust the operation cf a guided surface waveguide probe 400, based upon monitored conditions. The probe control system 418 can be implemented with hardware, firmware, software executed by hardware, or a combination thereof. For example, the probe control system 418 can include processing 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 apprecîated by those with ordinary skill in the art. A probe control application may be executed by the processor to adjust the operation ofthe guided surface waveguide probe 400 based upon monitored conditions. The probe control system 418 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 418 may comprise, for example, a computer system such as a server, desktop computer, laptop, or other system with like capability.

The adaptive control system 430 can include one or more ground parameter meter(s) 433 such as, but not limited to, a conductivity measurement probe of FIG. 22A and/or an open wire probe of FIG. 22B. The ground parameter meter(s) 433 can be distributed about the guided surface waveguide probe 400 at about the Hankel crossover distance (R_x) associated with the probe operating frequency. For example, an open wire probe of FIG. 22B may be located in each quadrant around the guided surface waveguide probe 400 to monitor the conductivity and permittivity of the lossy conducting medium as previously described. The ground parameter meter(s) 433 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 418 for potential adjustment ofthe guided surface waveguide probe 400. In some cases, the ground parameter meter(s) 433 may communicate the information to the probe control System 418 only when a change in the monitored conditions is detected.

The adaptive control System 430 can also include one or more field meter(s) 436 such as, but not limited to, an electric field strength (FS) meter. The field meter(s) 436 can be distributed about the guided surface waveguide probe 400 beyond the Hankel crossover 38 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 436 may be located along one or more radiais extending outward from the guided surface waveguide probe 400 to monitor the electric field strength as previousiy described. The field meter(s) 436 can be configured to détermine the field strength on a penodic basis and communicate the information to the probe control system 418 for potential adjustment of the guided surface waveguide probe 400. In some cases, the field meter(s) 436 may communicate the infcnnaticn to the probe ucntrô' · system 418 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 400. For instance, the ground current flowing through the ground stake 915 (FIG. 9A-9B, 17 and 18) can be used to monitor the operation ofthe guided surface waveguide probe 400. For example, the ground current can provîde^: an indication of changes in the loading of the guided surface waveguide probe 400 and/or the coupling of the electric field into the guided surface wave mode on the surface of the lossy conducting medium 403. Real power delivery may be determined by monitoring of the AC source 912 (or excitation source 412 of FIG. 4). In some implémentations, the guided surface waveguide probe 400 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 supplied to the charge terminal Τη and/or compensation terminal T₂, the wave tilt at the Hankel crossover distance can be maintained for illumination at the complex Brewster angle for guided surface wave transmissions in the lossy conducting medium 403 (e.g., the earth). This can be accomplished by adjusting the tap position on the coil 909. 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 for loading of the guided surface waveguide probe 400 is taking place.

The excitation source 412 (or AC source 912) can also be monitored to ensure that overloading does not occur. As real load on the guided surface waveguide probe 400 increases, the output voltage of the excitation source 412, or the voltage supplied to the charge terminal Τη 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 418. The information may be communicated to the probe control system 418 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 418 can then adjust the guided surface waveguide probe 400 for continued operation. For example, the phase delay (Φ_σ, Φ_ύ) 39 applied to the charge terminal T, and/or compensation termina! T₂, respectively, can be adjusted to improve and/or maximize the electrical launching efficiency of the guided surface waveguide probe 400, to supply the load demands of the receivers. In some cases, the probe control System 418 may adjust the guided surface waveguide probe 400 to reduce loading on the excitation source 412 and/or guided surface waveguide probe 400. For examp:e, 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 400 can be adjusted by the probe control System 418 using, e.g., one or more tap controllers 439. In FIG. 23A, the connection from the coil 909 to the upper charge terminal ty is controlled by a tap controller 439. In respcnse to a change m 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 439 to initiale a change in the tap position. The tap controller 439 can be configured to vary the tap position continuously along the coil 909 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 of the charge terminal ty can be adjusted to improve the launching efficiency of the guided surface waveguide mode.

While FIG. 23A illustrâtes a tap controller 439 coupled between the coil 909 and the charge terminal ty, in other embodiments the connection 442 from the coil 909 to the lower compensation terminal T₂ can also include a tap controller 439. FIG. 23B shows another embodiment of the guided surface waveguide probe 400 with a tap contrôliez 439 for adjusting the phase delay of the compensation terminal T₂. FIG. 23C shows an embodiment of the guided surface waveguide probe 400 where the phase delay of both terminal ty and 1₂ can be controlled using tap controllers 439. The tap controllers 439 may be controlled independently or concurrently by the probe control system 418. In both embodiments, an impédance matching network 445 is included for coupling the AC source 912 to the coil 909. In some implémentations, the AC source 912 may be coupled to the coil 909 through a tap controller 439, which may be controlled by the probe control system 418 to maintain a-matched condition for maximum power transfer from the AC source.

Referring back to FIG. 23A, the guided surface waveguide probe 400 can also be adjusted by the probe control system 418 using, e.g., a charge terminal positioning system 448 and/or a compensation terminal positioning system 451. By adjusting the height of the charge terminal ty and/or the compensation terminal T₂, and thus the distance between the two, it is possible to adjust the coupling into the guided surface waveguide mode. The terminal positioning Systems 448 and 451 can be configured to change the height of the terminais T, and 40

T₂ by linearly raising or lowering the terminal along the z-axis normal to the lossy conducting medium 403. For example, linear motors may be used to translate the charge and compensation terminais T and T₂ upward or downward using însulated shafts coupled to the terminais. Other embodiments can include insulated gearing and/or guy wires and pulleys, screw gears, or other apprcpriate mechanism that can control the positioning of the charge and compensation terminais Τ_ή and T₂. Insulation of the terminal positioning Systems 443 and 451 prevents discharge ofthe charge that is présent on the charge and eompensaîiun terminais Ti and T₂. For instance, an insulating structure can support the charge terminal T, above the compensation terminal T₂. For example, an RF insulating fiberglass mast can be used to support the charge and compensation terminais T-ι and T₂. The charge and compensation terminais T and T₂ can be individually positioned using the charge terminal positioning system 448 and/or compensation terminal positioning system 451 to improve and/or maximize the electrical launching efficiency of the guided surface waveguide probe 400.

As has been discussed, the probe control system 418 ofthe adaptive control system 430 can monitor the operating conditions of the guided surface waveguide probe 400 by communicating with one or more remotely located monitoring devices such as, but not limited to, a ground parameter meter 433 and/or a field meter 436. The probe control system 418 can also monitor other conditions by accessing information from, e.g., the ground curren. ammeter 927 (FIGS. 23B and 23C) and/or the AC source 912 (or excitation source 412). Based upon the monitored information, the probe control system 418 can détermine if adjustment of the guided surface waveguide probe 400 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 418 can initiate an adjustment of one or more of the phase delay (Φ^, Φ_Δ) applied tG the charge terminal Tî and/or compensation terminal T_2l respectively, and/or the physical height (h_p, h_d) of the charge termina! T and/or compensation terminal T₂, respectively. in some implantations, the probe control system 418 can evaluate the monitored conditions to identify the source ofthe change. If the monitored condition(s) was caused by a change in receiver load, then adjustment of the guided surface waveguide probe 400 may be avoided. If the monitored condition(s) affect the launching efficiency of the guided surface waveguide probe 400, then the probe control system 418 can initiate adjustments oftheguided surface waveguide probe 400 to improve and/or maximize the launching efficiency.

In some embodiments, the size of the charge terminal T, may also be adjusted te· control the coupling into the guided surface waveguide mode. 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 ofthe charge terminal T, which can reduce the chance of an electrical discharge from the charge terminal T_v Control of the charge terminal T 41 size can be provided by the probe control system 418 through the charge terminal pcsitioning system 448 or through a separate control system.

FIGS. 24A and 24B illustrate an example of a variable terminal 203 that can be used as a charge terminal Ti of the guided surface waveguide probe 400. 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. 24A, the cylindrically shaped variable tsi minai 202 t+ 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 exlended to increase the surface area as shown in FIG. 24B. This may be accomplished 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 addition to the forgoing, the various embodiments of the présent disclosure include, but are not limited to, the embodiments set forth in the following clauses.

Clause 1. A guided surface waveguide probe, comprising: a charge terminal elevated over a lossy conducting medium; and a coupling circuit configured to couple an excitation source to the charge terminal, the coupling circuit configured to provide a voltage to the charge terminal that establishes an electric field having a wave tilt (IV) that intersects the lossy conducting medium at a tangent of a complex Brewster angle (ψ_ίιΒ) at a Hankel crossover distance (R_x) from the guided surface waveguide probe.

Clause 2. The guided surface waveguide probe of clause 1, wherein the coupling circuit comprises a coil coupled between the excitation source and the charge terminal.

Clause 3. The guided surface waveguide probe of clause 2, wherein the coil is a helical coil.

Clause 4. The guided surface waveguide probe of any one of clauses 2 and 3, wherein the excitation source is coupled to the coil via a tap connection.

Clause 5. The guided surface waveguide probe of any one of clauses 2-4, wherein the tap .connection is at an impédance matching point on the coil.

. Clause 6. The guided surface waveguide probe of any one of clauses 2-5, wherein an impédance matching network is coupled between the excitation source and the tap connection on the coil.

Clause 7. The guided surface waveguide probe of any one of clauses 2-6, wherein the excitation source is magnetically coupled to the coil.

Clause 8. The guided surface waveguide probe of any one of clauses 2-7, wherein the charge terminal is coupled to the coil via a tap connection.

Clause 9. The guided surface waveguide probe of any one cf clauses 1-8, wherein the charge terminai is positioned at a physical height (/i_p) corresponding to a magnitude cf an effective height of the guided surface waveguide probe, where the effective height is given by h_eff = ^χ^Ψϊ,β = hpe^, with ψ_ίιΒ = (π/2) - θ_1ιΒ and Φ is a phase cf the effective height

Clause 10. The guided surface waveguide probe of clause 9, wherein the phase Φ is approximately equal to an angle Ψ of the wave tilt of illumination that corresponds to the complex Brewster angle.

Clause 11. The guided surface waveguide probe of any one of clauses 1-10, wherein Hie charge terminal has an effective spherical diameter, and the charge terminal is positioned at a height that is at least four times the effective spherical diameter.

Clause 12. The guided surface waveguide probe of clause 11, wherein the charge terminal is a spherical terminal with the effective spherical diameter equal to a diameter of the spherical terminal.

Clause 13. The guided surface waveguide probe of any one of clauses 11 and 12, wherein the height of the charge terminal is greater than 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_x tan ψ_ίΒ = with ψ_{ί Β} = (π/2) Clause 14. The guided surface waveguide probe of any one of clauses 11-13, further comprising a compensation terminal positioned below the charge terminal, the compensation terminal coupled to the coupling circuit.

Clause 15. The guided surface waveguide probe of any one of clauses 11-14, wherein the compensation terminal is positioned below the charge terminal at a distance equal to the physical height (/i_p).

Clause 16. The guided surface waveguide probe of any one of clauses 11-15, wherein Φ is a complex phase différence between the compensation terminal and the charge terminal.

Clause 17. The guided surface waveguide probe of any one of clauses 1-16, wherein the lossy conducting medium is a terrestrial medium.

Clause 18. A System, comprising: a guided surface waveguide probe, including: a charge terminal elevated over a lossy conducting medium; and a coupling circuit configured to provide a voltage to the charge terminal that establishes an electric field having a wave tilt (VF) that intersects the lossy conducting medium at a tangent of a complex Brewster angle (ψ_ίβ) at a Hankel crossover distance (K_x) from the guided surface waveguide probe; and an excitation source coupled to the charge terminal via the coupling circuit.

Clause 19. The system of clause 18, further comprising a probe control system configured to adjust the guided surface waveguide probe based at least in part upon characteristics of the lossy conducting medium.

Clause 20. The System of any one of clauses 18 and 19, wherein the lossy ccnducti.ig medium is a terrestrial medium.

Clause 21. The system of any one of clauses 18-20, wherein the coupling circuit comprises a coil coupled between the excitation source and the charge temiinal, thi charge terminal coupled to the coil via a variable tap.

Clause 22. The system of clause 21, wherein the coil is a helîcal coil.

Clause 23. The system of any one of clauses 21 and 22, 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.

Clause 24. The system of any one of clauses 21-23, wherein the adjustment of the position of the variable tap adjusts 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_ίβ) at the Hankel crossover distance (R_x).

Clause 25. The System of any one of clauses 21-24, wherein the guided surface waveguide probe further comprises a compensation terminal positioned below the charge terminal, the compensation terminal coupled to the coupling circuit.

Clause 26. The System of any one of clauses 21-25, wherein the compensation terminal is positioned below the charge terminal at a distance equal to a physical height (Ji_p) corresponding to a magnitude of an effective height of the guided surface waveguide probe, where the effective height is given by = R_xtanÿ_ig = Ιι_ρβΙ^φ, with ψ_ίβ = (π/2) - 0_ίβ and wherein Φ is a complex phase différence between the compensation terminal and the charge terminal.

Clause 27. The System of any one of clauses 21-26, wherein the probe control system adjusts a position of the compensation terminal in response to a change in the characteristics of the lossy conducting medium.

Clause 28. A method, comprising:

positioning a charge terminal at a defined height over a lossy conducting rnedium;

positioning a compensation terminal below the charge terminal, the compensation terminal separated by a defined distance; and exciting the charge terminal and the compensation terminal with excitation voltages having a complex phase différence, where the excitation voltages establish an electric field having a wave tilt (IV) that corresponds to a wave illuminating the lossy conducting medium at a complex Brewster angle (ψ_ί>Β) at a Hankel crossover distance (R_x) from the charge terminal and the compensation terminal.

Clause 29. The method of clause 28, wherein the charge terminal has an effective spherical diameter, and the charge terminal is positioned at the defined height is at least four times the effective spherical diameter.

Clause 30. The method of any one of clauses 28 and 29, wherein the defined distance is equal to a physical height (k_p) corresponding to a tr.âgrûtude of an effec.ive iïeight of iiie cna_tge^: terminal, where the effective height is given by hejf = flxtani/i1B = hpe^Jli\ with ψίΒ = (tt/2) θ_ίΒ and wherein Φ is the complex phase différence between the compensation terminal and the charge terminal.

Clause 31. The method of any one of clauses 28-30, wherein the charge terminal and the compensation terminal are coupled to an excitation source via a coil, the charge terminal coupled to the coil by a variable tap.

Clause 32. The method of clause 31, further comprising adjusting a position of the variable tap to establish the electric field having the wave tilt that intersects the lossy conducting medium at the complex Brewster angle (0_iB) at the Hankel crossover distance (R_x).

The invention is not limited to the embodiment/s illustrated in the drawings. Accordingly it should be understood that where features mentioned in the appended daims are followed by reference signs, such signs are included solely for the purpose of enhancing the intelligibility of the daims and are in no way limiting on the scope of the daims.

Claims (30)

1. Therefore, the following is claimed:

   1. A guided surface waveguide probe (400, 400a, 400b, 400c), comprising:

   a single charge terminal (T^ elevated over a lossy conducting medium (403);

   an excitation source (412); and a coupling circuit (409) configured to couple the excitation source (412) to the single charge terminal, the coupling circuit configured to provide a voltage ro the charge terminal that establishes an electric field having a wave tilt (IV) 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.
2. 2. The guided surface waveguide probe of claim 1, wherein the coupling circuit comprises a coil coupled between the excitation source and the single charge terminal.
3. 3. The guided surface waveguide probe of claim 2, wherein the coil is a helical coil.
4. 4 The guided surface waveguide probe of any one of daims 2 and 3, wherein the excitation source is coupled to the coil via a tap connection.
5. 5. The guided surface waveguide probe of any one of daims 2-4, wherein the tap connection is at an impédance matching point on the coil.
6. 6. The guided surface waveguide probe of any one of daims 2-5, wherein an impédance matching network is coupled between the excitation source and the tap connection on the coil.
7. 7. The guided surface waveguide probe of any one of daims 2-6, wherein the excitation source is magnetically coupled to the coil.

   3. The guided surface waveguide probe of any one of daims 2-7, wherein the single charge terminal is coupled to the coil via a tap connection.
8. 9. The guided surface waveguide probe of any one of daims 1-3, wherein the single charge terminal is positioned at a physical height (h_p) correspcnding to a magnitude of an effective height of the guided surface waveguide probe, where the effective height is given by h_eff = R_x tan ip_itB = h_pe^j<*\ with ψ_ΛΒ = 0/2) - d_{i B} and Φ is a phase of the effective height.
9. 10. The guided surface waveguide probe of claim 9, wherein the phase Φ is approximately equal to an angle Ψ of the wave tilt of illumination that corresponds to the complex Brewster angle.
10. 11. The guided surface waveguide probe of any one of daims 1-10, wherein the single charge terminal has an effective spherical diameter, and the single charge terminal is positioned at a height that is at least four times the effective spherical diameter.
11. 12. The guided surface waveguide probe of claim 11, wherein the single charge terminal is a spherical terminal with the effective spherical diameter equal to a diameter of the spherical terminal.
12. 13. The guided surface waveguide probe of any one of daims 11 and 12, wherein the height of the single charge terminal is greater than 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_e^ = R_xtan?/i_ÎB = h_pe^i<x>, with ψ_ίΒ = (π/2) - θ< _B.
13. 14. The guided surface waveguide probe of any one of daims 11-13, further comprising a compensation terminal (T₂) positioned below ihe single charge terminai and above the lossy conducting medium, the compensation terminal coupled to the coupling circuit.
14. 15. The guided surface waveguide probe of any one of daims 11-14, wherein the compensation terminal is positioned below the charge terminal at a distance equal to the physical height (h_p).
15. 16. The guided surface waveguide probe of any one of daims 11-15, wherein Φ is a complex phase différence between the compensation terminal and the single charge terminal.
16. 17. The guided surface waveguide probe of any one of daims 1-16, wherein the lossy conducting medium is a terrestrial medium.
17. 18. A system, comprising:

    a probe control system (418); and a guided surface waveguide probe (400, 400a, 400b, or 400c), including:

    a single charge terminal (TO elevated over a lossy conducting medium (403);

    a coupling circuit (409) configured to provide a voltage to the single charge terminal that establishes an electric field having a wave tilt (H') that intersects the lossy conducting medium at a tangent of a complex Brewster angle (ψ, _Β) at a Hankel crossover distance (R_x) from the guided surface waveguide probe; and an excitation source (412) coupled to the single charge terminal via the coupling circuit.
18. 19. The system of claim 18, further comprising a wherein the probe control System is configured to adjust the position of the compensation terminal oased at least in part upon one or more characteristics of the lossy conducVng medium.
19. 20. The System of any one of daims 18 and 19, wherein the lossy conducting medium is a terrestrial medium.
20. 21. The system of any one of daims 18-20, wherein the coupling circuit comprises a coil coupled between the excitation source and the single charge terminal, the single charge terminal coupled to the coil via a variable tap.
21. 22. The system of claim 21, wherein the coil is a helical coil.
22. 23. The system of any one of daims 21 and 22, wherein the probe control system is further configured to adjust a position of the variable tap in response to a change in the characteristics of the lossy conducting medium.
23. 24. The system of any one of daims 21-23, wherein the adjustment of the position of the variable tap includes an adjustment to 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_ίιΒ) at the Hankel crossover distance (R_x).
24. 25. The system of any one of daims 21-24, wherein the guided surface waveguide probe further comprises a compensation terminal (T₂) positioned below the single charge terminal, the compensation terminal coupled to the coupling circuit.
25. 26. The system of any one of daims 21-25, wherein the compensation terminal is positioned below the single charge terminal at a distance equal to 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 = /?_x tan-0_i£, = h_pe^j<i>, with ψ_ίιΒ =

    10 (rr/2) - θ_{ί Β} and wherein Φ is a complex phase différence between the compensation terminal and the single charge terminal.
26. 27. The system of any one of daims 21-26, wherein the probe control system further adjusts “ the position of the compensation terminal in response to a change in the one or more

    15 characteristics of the lossy conducting medium.
27. 28. A method, comprising:

    positioning a single charge terminal (T,) at a defined height (H,, h_T) over a lossy conducting medium (403);

    20 positioning a compensation terminal (T₂) below the single charge terminal and above the lossy conducting medium, the compensation terminal separated from the single charge terminal by a defined distance (h); and exciting the single charge terminal and the compensation terminal with excitation voltages having a complex phase différence, where the excitation voltages establish an

    25 electric field having a wave tilt (IV) that corresponds to a wave illuminating the lossy conducting medium at a complex Brewster angle (i/\_B) at a Hankel crossover distance (R_x) from the single charge terminal and the compensation terminal.
28. 29. The method of claim 28, wherein the single charge terminal has an affective sphericai diameter, and the single charge terminal is positioned at the defined height is at least four times the effective sphericai diameter.

    5
29. 30. The method of any one of daims 28 and 29, wherein the defined distance is equal to a physical height (h_p) correspcnding to a magnitude of an effective height cf the single charge terminal, where the effective height is given by h_eff - ϋ_χ^ηψ_:Β = /ι_ρβ^;φ, with Ψι,Β ~ (λ/2) ^{- θ}ί.β ^and wnerein Φ is the complex phase différence be.ween the compensation terminal and the single charge terminal.
30. 31. The method of any one of daims 28-30, wherein the single charge terminal and the compensation terminal are coupled to an excitation source via a coil, the single charge terminal coupled to the coil by a variable tap.

    15 32. The method of daim 31, further comprising adjusting a position of the variable tap to establish the electric field having the wave tilt that intersects the lossy conducting medium at the complex Brewster angle (ψ_ίιΒ) at the Hankel crossover distance (R_x).