CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims priority to and the benefit of, U.S. Application No. 14/848,565, filed on 09 September 2015, herein incorporated by reference in its entirety.

[0002] This application is related to co-pending U.S. Non-provisional Patent Application entitled "Excitation and Use of Guided Surface Wave Modes on Lossy Media," which was filed on March 7, 2013 and assigned Application Number 13/789,538, and was published on September 1 1 , 2014 as Publication Number US2014/0252886 A1 , and which is incorporated herein by reference in its entirety. This application is also related to co-pending U.S. Non- provisional Patent Application entitled "Excitation and Use of Guided Surface Wave Modes on Lossy Media," which was filed on March 7, 2013 and assigned Application Number 13/789,525, and was published on September 1 1 , 2014 as Publication Number

US2014/0252865 A1 , and which is incorporated herein by reference in its entirety. This application is further related to co-pending U.S. Non-provisional Patent Application entitled "Excitation and Use of Guided Surface Wave Modes on Lossy Media," which was filed on September 10, 2014 and assigned Application Number 14/483,089, and which is

incorporated herein by reference in its entirety. This application is further related to copending U.S. Non-provisional Patent Application entitled "Excitation and Use of Guided Surface Waves," which was filed on June 2, 2015 and assigned Application Number

14/728,507, and which is incorporated herein by reference in its entirety. This application is further related to co-pending U.S. Non-provisional Patent Application entitled "Excitation and Use of Guided Surface Waves," which was filed on June 2, 2015 and assigned Application Number 14/728,492, and which is incorporated herein by reference in its entirety.

SUMMARY

[0003] According to one embodiment, power is distributed within a hierarchical power distribution system from a plurality of guided surface waveguide probes to one or more guided surface wave receive structure. Each of the plurality of guided surface waveguide probes are associated with a respective one of a plurality of frequency sets. Each of the plurality of frequency sets correspond to a transmission class among a plurality of transmission classes. Each of the plurality of guided surface waveguide probes are designated to transmit within a respective transmission class among the plurality of transmission classes. Each of the plurality of guided surface waveguide probes transmit guided surface waves along a terrestrial medium to a guided surface wave receive structure. The guided wave receive structure supplies electrical energy to at least one electrical load coupled to the at least one guided surface wave receive structure.

[0004] In another embodiment, a system includes one or more guided surface waveguide probe associated with a frequency set among multiple frequency sets. Each of the frequency sets are associated with a respective transmission class. The guided surface waveguide probes are configured to transmit guided surface waves along a terrestrial medium to the guided surface wave receive structure within a respective transmission class. The guided surface wave receive structure is configured to supply electrical energy to an electrical load coupled to the guided surface wave receive structure.

[0005] In yet another embodiment, a system includes a guided surface wave receive structure. The guided surface wave receive structure is configured to receive electrical energy from a guided surface wave transmitted along a terrestrial medium. The guided surface wave is transmitted from guided surface waveguide probe. The guided surface waveguide probe transmits in a respective transmission class among a multiple transmission classes. The guided surface wave receive structure is assigned the transmission class. The guided surface wave receive structure is configured to supply the electrical energy to an electrical load coupled to a guided surface wave receive structure. A control system is configured to assign the guided surface waveguide probe to the guided surface wave receive structure in response to a request for power.

[0006] Other systems, methods, features, and advantages of the present disclosure will be or become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, features, and advantages be included within this description, be within the scope of the present disclosure, and be protected by the accompanying claims.

[0007] In addition, all optional and preferred features and modifications of the described embodiments are usable in all aspects of the entire disclosure taught herein. Furthermore, the individual features of the dependent claims, as well as all optional and preferred features and modifications of the described embodiments are combinable and interchangeable with one another. BACKGROUND

[0008] For over a century, signals transmitted by radio waves involved radiation fields launched using conventional antenna structures. In contrast to radio science, electrical power distribution systems in the last century involved the transmission of energy guided along electrical conductors. This understanding of the distinction between radio frequency (RF) and power transmission has existed since the early 1900's.

BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views.

[0010] FIG. 1 is a chart that depicts field strength as a function of distance for a guided electromagnetic field and a radiated electromagnetic field.

[0011] FIG. 2 is a drawing that illustrates a propagation interface with two regions employed for transmission of a guided surface wave according to various embodiments of the present disclosure.

[0012] FIG. 3 is a drawing that illustrates a guided surface waveguide probe disposed with respect to a propagation interface of FIG. 2 according to various embodiments of the present disclosure.

[0013] FIG. 4 is a plot of an example of the magnitudes of close-in and far-out asymptotes of first order Hankel functions according to various embodiments of the present disclosure.

[0014] FIGS. 5A and 5B are drawings that illustrate a complex angle of incidence of an electric field synthesized by a guided surface waveguide probe according to various embodiments of the present disclosure.

[0015] FIG. 6 is a graphical representation illustrating the effect of elevation of a charge terminal on the location where the electric field of FIG. 5A intersects with the lossy conducting medium at a Brewster angle according to various embodiments of the present disclosure.

[0016] FIG. 7 is a graphical representation of an example of a guided surface waveguide probe according to various embodiments of the present disclosure.

[0017] FIGS. 8A through 8C are graphical representations illustrating examples of equivalent image plane models of the guided surface waveguide probe of FIGS. 3 and 7 according to various embodiments of the present disclosure. [0018] FIGS. 9A and 9B are graphical representations illustrating examples of single- wire transmission line and classic transmission line models of the equivalent image plane models of FIGS. 8B and 8C according to various embodiments of the present disclosure.

[0019] FIG. 10 is a flow chart illustrating an example of adjusting a guided surface waveguide probe of FIGS. 3 and 7 to launch a guided surface wave along the surface of a lossy conducting medium according to various embodiments of the present disclosure.

[0020] FIG. 11 is a plot illustrating an example of the relationship between a wave tilt angle and the phase delay of a guided surface waveguide probe of FIGS. 3 and 7 according to various embodiments of the present disclosure.

[0021] FIG. 12 is a drawing that illustrates an example of a guided surface waveguide probe according to various embodiments of the present disclosure.

[0022] FIG. 13 is a graphical representation 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.

[0023] FIG. 14 is a graphical representation of an example of a guided surface waveguide probe of FIG. 12 according to various embodiments of the present disclosure.

[0024] FIG. 15A includes plots of an example of the imaginary and real parts of a phase delay (Φ^) of a charge terminal ΤΊ of a guided surface waveguide probe according to various embodiments of the present disclosure.

[0025] FIG. 15B is a schematic diagram of the guided surface waveguide probe of FIG. 14 according to various embodiments of the present disclosure.

[0026] FIG. 16 is a drawing that illustrates an example of a guided surface waveguide probe according to various embodiments of the present disclosure.

[0027] FIG. 17 is a graphical representation of an example of a guided surface waveguide probe of FIG. 16 according to various embodiments of the present disclosure.

[0028] FIGS. 18A through 18C depict examples of receiving structures that can be employed to receive energy transmitted in the form of a guided surface wave launched by a guided surface waveguide probe according to the various embodiments of the present disclosure.

[0029] FIG. 18D is a flow chart illustrating an example of adjusting a receiving structure according to various embodiments of the present disclosure.

[0030] FIG. 19 depicts an example of an additional receiving structure that can be employed to receive energy transmitted in the form of a guided surface wave launched by a guided surface waveguide probe according to the various embodiments of the present disclosure. [0031] FIGS. 20-20E depict examples of various schematic symbols that are used according to various embodiments of the present disclosure.

[0032] FIG. 21 depicts a representation of a hierarchical power distribution system including transmission classes and frequency sets according to various embodiments of the present disclosure.

[0033] FIG. 22 depicts a representation of a transmission spectrum that the guided surface waveguide probes may be tuned into when transmitting electrical energy in the form of a guided surface wave according to various embodiments of the present disclosure.

[0034] FIG. 23A depicts another representation of a transmission spectrum according to various embodiments of the present disclosure.

[0035] FIG. 23B depicts an example of geographic areas that may be served by respective transmission classes according to various embodiments of the present disclosure.

[0036] FIG. 24 is a drawing of a networked hierarchical power distribution system according to various embodiments of the present disclosure.

[0037] FIG. 25 is a flowchart illustrating one example of the transmission control system according to various embodiments of the present disclosure.

[0038] FIG. 26 is a flowchart illustrating one example of assigning at least one discrete frequency within a frequency set to a guided surface wave receive structure according to various embodiments of the present disclosure.

[0039] FIG. 27 is a schematic block diagram that provides one example illustration of a transmission control system and a receive controller employed in the networked hierarchical power distribution system of FIG. 24 according to various embodiments of the present disclosure.

DETAILED DESCRIPTION

[0040] To begin, some terminology shall be established to provide clarity in the discussion of concepts to follow. First, as contemplated herein, a formal distinction is drawn between radiated electromagnetic fields and guided electromagnetic fields.

[0041] 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 atmosphere 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 inherent in the radiated electromagnetic waves is lost forever. Electrical structures such as antennas are designed to radiate electromagnetic fields by maximizing the ratio of the radiation resistance to the structure loss resistance. Radiated energy spreads out in space and is lost regardless of whether a receiver is present. The energy density of the radiated fields is a function of distance due to geometric spreading. Accordingly, the term "radiate" in all its forms as used herein refers to this form of electromagnetic propagation.

[0042] A guided electromagnetic field is a propagating electromagnetic wave whose energy is concentrated within or near boundaries between media having different electromagnetic properties. In this sense, a guided electromagnetic field is one that is bound to a waveguide and may be characterized as being conveyed by the current flowing in the waveguide. If there is no load to receive and/or dissipate the energy conveyed in a guided electromagnetic wave, then no energy is lost except for that dissipated in the conductivity of the guiding medium. Stated another way, if there is no load for a guided electromagnetic wave, then no energy is consumed. Thus, a generator or other source generating a guided electromagnetic field does not deliver real power unless a resistive load is present. To this end, such a generator or other source essentially runs idle until a load is presented. This is akin to running a generator to generate a 60 Hertz electromagnetic wave that is transmitted over power lines where there is no electrical load. It should be noted that a guided electromagnetic field or wave is the equivalent to what is termed a "transmission line mode." This contrasts with radiated electromagnetic waves in which real power is supplied at all 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 all its forms as used herein refers to this transmission mode of electromagnetic propagation.

[0043] Referring now to FIG. 1 , shown is a graph 100 of field strength in decibels (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.

[0044] 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 / fd 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 112, which occurs at a crossing distance. At distances less than the crossing distance at intersection point 112, the field strength of a guided electromagnetic field is significantly greater at most locations than the field strength of a radiated electromagnetic field. At distances greater than the crossing distance, the opposite is true. Thus, the guided and radiated field strength curves 103 and 106 further illustrate the fundamental propagation difference between guided and radiated electromagnetic fields. For an informal discussion of the difference between guided and radiated electromagnetic fields, reference is made to Milligan, T., Modern Antenna Design, McGraw-Hill, 1 ^st Edition, 1985, pp.8-9, which is incorporated herein by reference in its entirety.

[0045] 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 equation, the wave equation, analytically follows from the boundary conditions imposed on the problem. The Green function for the wave equation, itself, contains the distinction between the nature of radiation and guided waves.

[0046] In empty space, the wave equation is a differential operator whose

eigenfunctions possess a continuous spectrum of eigenvalues on the complex wave-number plane. This transverse electro-magnetic (TEM) field is called the radiation field, and those propagating fields are called "Hertzian waves." However, in the presence of a conducting boundary, the wave equation plus boundary conditions mathematically lead to a spectral representation of wave-numbers composed of a continuous spectrum plus a sum of discrete spectra. To this end, reference is made to Sommerfeld, A., "Uber die Ausbreitung der Wellen in der Drahtlosen Telegraphie," 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, Academic 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 Techniques, Van Nostrand, 1953, pp. 291- 293, each of these references being incorporated herein by reference in its entirety.

[0047] The terms "ground wave" and "surface wave" identify two distinctly different physical propagation phenomena. A surface wave arises analytically from a distinct pole yielding a discrete component in the plane wave spectrum. See, e.g., "The Excitation of Plane Surface Waves" by Cullen, A.L., (Proceedings of the I EE (British), Vol. 101 , Part IV, August 1954, pp. 225-235). In this context, a surface wave is considered to be a guided surface wave. The surface wave (in the Zenneck-Sommerfeld guided wave sense) is, physically and mathematically, not the same as the ground wave (in the Weyl-Norton-FCC sense) that is now so familiar from radio broadcasting. These two propagation mechanisms arise from the excitation of different types of eigenvalue spectra (continuum or discrete) on the complex plane. The field strength of the guided surface wave decays exponentially with distance as illustrated by curve 103 of FIG. 1 (much like propagation in a lossy waveguide) and resembles propagation in a radial transmission line, as opposed to the classical Hertzian radiation of the ground wave, which propagates spherically, possesses a continuum of eigenvalues, falls off geometrically as illustrated by curve 106 of FIG. 1 , and results from branch-cut integrals. As experimentally demonstrated by C.R. Burrows in "The Surface Wave in Radio Propagation over Plane Earth" (Proceedings of the IRE, Vol. 25, No. 2, February, 1937, pp. 219-229) and "The Surface Wave in Radio Transmission" (Bell

Laboratories Record, Vol. 15, June 1937, pp. 321-324), vertical antennas radiate ground waves but do not launch guided surface waves.

[0048] To summarize the above, first, the continuous part of the wave-number eigenvalue spectrum, corresponding to branch-cut integrals, produces the radiation field, and second, the discrete spectra, and corresponding residue sum arising from the poles enclosed by the contour of integration, result 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, Wley, 1956, pp. pp. 214, 283-286, 290, 298-300.

[0049] In free space, antennas excite the continuum eigenvalues of the wave equation, 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 discrete 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 have 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 have 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 have been no known structures for practically accomplishing the launching of open surface guided waves over planar or spherical surfaces of lossy, homogeneous media. [0050] According to the various embodiments of the present disclosure, various guided surface waveguide probes are described that are configured to excite electric fields that couple into a guided surface waveguide mode along the surface of a lossy conducting medium. Such guided electromagnetic fields are substantially mode-matched in magnitude and phase to a guided surface wave mode on the surface of the lossy conducting medium. Such a guided surface wave mode can also be termed a Zenneck waveguide mode. By virtue of the fact that the resultant fields excited by the guided surface waveguide probes described herein are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium, a guided electromagnetic field in the form of a guided surface wave is launched along the surface of the lossy conducting medium.

According to one embodiment, the lossy conducting medium comprises a terrestrial medium such as the Earth.

[0051] Referring to FIG. 2, shown is a propagation interface that provides for an examination of the boundary value solutions to Maxwell's equations derived in 1907 by Jonathan Zenneck as set forth in his paper Zenneck, J., "On the Propagation of Plane Electromagnetic Waves Along a Flat Conducting Surface and their Relation to Wireless Telegraphy," Annalen der Physik, Serial 4, Vol. 23, September 20, 1907, pp. 846-866. FIG. 2 depicts cylindrical coordinates for radially propagating waves along the interface between a lossy conducting medium specified as Region 1 and an insulator specified as Region 2. Region 1 can comprise, for example, any lossy conducting medium. In one example, such a lossy conducting medium can comprise a terrestrial medium such as the Earth or other medium. Region 2 is a second medium that shares a boundary interface with Region 1 and has different constitutive parameters relative to Region 1. Region 2 can comprise, for example, any insulator such as the atmosphere 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.

[0052] According to various embodiments, the present disclosure sets forth various guided surface waveguide probes that generate electromagnetic fields that are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium comprising Region 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 result in zero reflection.

[0053] To explain further, in Region 2, where an β ^]ωΐ field variation is assumed and where p≠ 0 and z≥ 0 (with z being the vertical coordinate normal to the surface of Region 1 , and p being the radial dimension in cylindrical coordinates), Zenneck's closed-form exact solution of Maxwell's equations satisfying the boundary conditions along the interface are expressed by the following electric field and magnetic field components:

E _2P = A ≠? {-jyp and (2)

Ε _2ζ = Α (^) β-^ H ₀ ⁽²⁾ (- ₇ yp). (3)

[0054] In Region 1 , where the β ^]ωΐ field variation is assumed and where p≠ 0 and z≤ 0, Zenneck's closed-form exact solution of Maxwell's equations satisfying the boundary conditions along the interface is expressed by the following electric field and magnetic field components:

[0055] In these expressions, z is the vertical coordinate normal to the surface of Region 1 and p is the radial coordinate, is a complex argument Hankel function of the second kind and order n, u _t is the propagation constant in the positive vertical (z) direction in Region 1 , u ₂ is the propagation constant in the vertical (z) direction in Region 2, σ _χ is the conductivity of Region 1 , ω is equal to 2nf, where is a frequency of excitation, ε ₀ is the permittivity of free space, s _t is the permittivity of Region 1 , A is a source constant imposed by the source, and γ is a surface wave radial propagation constant.

[0056] The propagation constants in the +z directions are determined by separating the wave equation above and below the interface between Regions 1 and 2, and imposing the boundary conditions. This exercise gives, in Region 2,

u ₂ = . ^fc ° _. (7) and gives, in Region 1 ,

The radial propagation constant is given by

which is a complex expression where n is the complex index of refraction given by

In all of the above Equations,

x =— . and where ε _τ comprises the relative permittivity of Region 1 , ^ is the conductivity of Region 1 , ε ₀ is the permittivity of free space, and μ ₀ comprises the permeability of free space. Thus, the generated surface wave propagates parallel to the interface and exponentially decays vertical to it. This is known as evanescence.

[0057] Thus, Equations (1)-(3) can be considered to be a cylindrically-symmetric, radially-propagating waveguide mode. See Barlow, H. M., and Brown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 10-12, 29-33. The present 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 Region 2 and Region 1. This may be better understood with reference to FIG. 3, which shows an example of a guided surface waveguide probe 200a that includes a charge terminal Ti elevated above a lossy conducting medium 203 (e.g., the Earth) along a vertical axis z that is normal to a plane presented by the lossy conducting medium 203. The lossy conducting medium 203 makes up Region 1 , and a second medium 206 makes up Region 2 and shares a boundary interface with the lossy conducting medium 203.

[0058] According to one embodiment, the lossy conducting medium 203 can comprise a terrestrial medium such as the planet Earth. To this end, such a terrestrial medium comprises all structures or formations included thereon whether natural or man-made. For example, such a terrestrial medium can comprise natural elements such as rock, soil, sand, fresh water, sea water, trees, vegetation, and all other natural elements that make up our planet. In addition, such a terrestrial medium can comprise man-made elements such as concrete, asphalt, building materials, and other man-made materials. In other embodiments, the lossy conducting medium 203 can comprise some medium other than the Earth, whether naturally occurring or man-made. In other embodiments, the lossy conducting medium 203 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.

[0059] In the case where the lossy conducting medium 203 comprises a terrestrial medium or Earth, the second medium 206 can comprise the atmosphere above the ground. As such, the atmosphere can be termed an "atmospheric medium" that comprises air and other elements that make up the atmosphere of the Earth. In addition, it is possible that the second medium 206 can comprise other media relative to the lossy conducting medium 203.

[0060] The guided surface waveguide probe 200a includes a feed network 209 that couples an excitation source 212 to the charge terminal Ti via, e.g., a vertical feed line conductor. According to various embodiments, a charge is imposed on the charge terminal ΤΊ to synthesize an electric field based upon the voltage applied to terminal ΤΊ at any given instant. Depending on the angle of incidence of the electric field (E), it is possible to substantially mode-match the electric field to a guided surface waveguide mode on the surface of the lossy conducting medium 203 comprising Region 1.

[0061] By considering the Zenneck closed-form solutions of Equations (1)-(6), the Leontovich impedance boundary condition between Region 1 and Region 2 can be stated as z H ₂ (p, (p, 0) = J _s , (13) where z is a unit normal in the positive vertical (+z) direction and H ₂ is the magnetic field strength in Region 2 expressed by Equation (1) above. Equation (13) implies that the electric and magnetic fields specified in Equations (1)-(3) may result in a radial surface current density along the boundary interface, where the radial surface current density can be specified by

where A is a constant. Further, it should be noted that close-in to the guided surface waveguide probe 200 (for p « λ), Equation (14) above has the behavior

Jciose(p') = _{π( γ} ² ρ = ^~Η Φ = - ^ - ( ¹⁵ ) The negative sign means that when source current (I ₀ ) flows vertically upward as illustrated in FIG. 3, the "close-in" ground current flows radially inward. By field matching on Η _φ "close- in," it can be determined that

_A = - i^ = - ^ _{( 1 6)}

4 4 ' where C1V1 , in Equations (1)-(6) and (14). Therefore, the radial surface current density of Equation (14) can be restated as

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

[0062] 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 equation. One might observe that the Hankel functions of the first and second kind and order n are defined as complex combinations of the standard Bessel functions of the first and second kinds These functions represent cylindrical waves propagating radially inward {H^) and outward

(H®), respectively. The definition is analogous to the relationship e ^±jx = cos x ± j sin x. See, for example, Harrington, R.F., Time-Harmonic Fields, McGraw-Hill, 1961 , pp. 460-463.

[0063] That is an outgoing wave can be recognized from its large argument asymptotic behavior that is obtained directly from the series definitions of J _n (x) and N _n (x). Far-out from the guided surface waveguide probe:

# _n " ²⁾ 0) x—→∞ * j πχ j ^{J n} e ^~]x = β πχ- ^J 7 ^'n e ^"7' (* ^" ¾, (20a) which, when multiplied by β ^]ωΐ , is an outward propagating cylindrical wave of the form _e j(a>t-kp) _w j _t (-, _a spatial variation. The first order (n = 1) solution can be determined from Equation (20a)

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

1 ⁽²⁾ (x)→— . (21)

¹ x→0 πχ ^v '

Note that these asymptotic expressions are complex quantities. When x is a real quantity, Equations (20b) and (21) differ in phase by ], 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 have a Hankel "crossover" or transition point where they are of equal magnitude at a distance of p = R _x .

[0064] Thus, beyond the Hankel crossover point the "far out" representation predominates over the "close-in" representation of the Hankel function. The distance to the Hankel crossover point (or Hankel crossover distance) can be found by equating Equations (20b) and (21) for -y ^' yp, and solving for R _x . With x = σ/ωε ₀ , it can be seen that the far-out and close-in Hankel function asymptotes are frequency dependent, 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.

[0065] Referring to FIG. 4, shown is an example of a plot of the magnitudes of the first order Hankel functions of Equations (20b) and (21) for a Region 1 conductivity of σ = 0.010 mhos/m and relative permittivity e _r = 15, at an operating frequency of 1850 kHz. Curve 115 is the magnitude of the far-out asymptote of Equation (20b) and curve 1 18 is the magnitude of the close-in asymptote of Equation (21), with the Hankel crossover point 121 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 121. It can also be seen that the Hankel crossover distance is much less than a wavelength of the operation frequency.

[0066] Considering the electric field components given by Equations (2) and (3) of the Zenneck closed-form solution in Region 2, it can be seen that the ratio of E _z and E _p asymptotically passes to

where n is the complex index of refraction of Equation (10) and B _t is the angle of incidence of the electric field. In addition, the vertical component of the mode-matched electric field of Equation (3) asymptoti

which is linearly proportional to free charge on the isolated component of the elevated charge terminal's capacitance at the terminal voltage, q _free = C _free x V _T .

[0067] For example, the height Hi of the elevated charge terminal ΤΊ in FIG. 3 affects the amount of free charge on the charge terminal T When the charge terminal ΤΊ is near the ground plane of Region 1 , most of the charge on the terminal is "bound." As the charge terminal ΤΊ is elevated, the bound charge is lessened until the charge terminal ΤΊ reaches a height at which substantially all of the isolated charge is free.

[0068] The advantage of an increased capacitive elevation for the charge terminal ΤΊ is that the charge on the elevated charge terminal ΤΊ is further removed from the ground plane, resulting in an increased amount of free charge q _free to couple energy into the guided surface waveguide mode. As the charge terminal ΤΊ is moved away from the ground plane, the charge distribution becomes more uniformly distributed about the surface of the terminal. The amount of free charge is related to the self-capacitance of the charge terminal TV

[0069] For example, the capacitance of a spherical terminal can be expressed as a function of physical height above the ground plane. The capacitance of a sphere at a physical height of h above a perfect ground is given by

Qlevated sphere = 4ττε ₀ α(1 + M + M ² + M ³ + 2M ⁴ + 3M ⁵ + ··· ), (24) where the diameter of the sphere is 2a, and where M = a/2h with h being the height of the spherical terminal. As can be seen, an increase in the terminal height h reduces the capacitance C of the charge terminal. It can be shown that for elevations of the charge terminal ΤΊ that are at a height of about four times the diameter (W = 8a) or greater, the charge distribution is approximately uniform about the spherical terminal, which can improve the coupling into the guided surface waveguide mode.

[0070] In the case of a sufficiently isolated terminal, the self-capacitance of a

conductive sphere can be approximated by C = 4πε ₀ α, where a is the radius of the sphere in meters, and the self-capacitance of a disk can be approximated by C = 8ε ₀ α, where a is the radius of the disk in meters. The charge terminal ΤΊ can include any shape such as a sphere, a disk, a cylinder, a cone, a torus, a hood, one or more rings, or any other randomized shape or combination of shapes. An equivalent spherical diameter can be determined and used for positioning of the charge terminal J^ .

[0071] This may be further understood with reference to the example of FIG. 3, where the charge terminal Ti is elevated at a physical height of h _p = U _t above the lossy conducting medium 203. To reduce the effects of the "bound" charge, the charge terminal Ti can be positioned at a physical height that is at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal Ti to reduce the bounded charge effects.

[0072] Referring next to FIG. 5A, shown is a ray optics interpretation of the electric field produced by the elevated charge Qi on charge terminal Ti of FIG. 3. As in optics, minimizing the reflection of the incident electric field can improve and/or maximize the energy coupled into the guided surface waveguide mode of the lossy conducting medium 203. For an electric field (£ ^" u) that is polarized parallel to the plane of incidence (not the boundary interface), the amount of reflection of the incident electric field may be determined using the Fresnel reflection coefficient, which can be expressed as

where B _t is the conventional angle of incidence measured with respect to the surface normal.

[0073] In the example of FIG. 5A, the ray optic interpretation shows the incident field polarized parallel to the plane of incidence having an angle of incidence of 6 which is measured with respect to the surface normal (z). There will be no reflection of the incident electric field when Γ^θ;) = 0 and thus the incident electric field will be completely coupled into a guided surface waveguide mode along the surface of the lossy conducting medium 203. It can be seen that the numerator of Equation (25) goes to zero when the angle of incidence is

9 = arctan(J£ _r — jx) = 9 (26) where x = σ/ωε ₀ . This complex angle of incidence (9 _{i B} ) is referred to as the Brewster angle. Referring back to Equation (22), it can be seen that the same complex Brewster angle (θ _{ί Β} ) relationship is present in both Equations (22) and (26). [0074] As illustrated in FIG. 5A, the electric field vector E can be depicted as an incoming non-uniform plane wave, polarized parallel to the plane of incidence. The electric field vector E can be created from independent horizontal and vertical components as

Ε(Θ = E _p p + E _z z. (27) Geometrically, the illustration in FIG. 5A suggests that the electric field vector E can be given by

Ep (p, z) = E(p, z) cos θι, and (28a) E _z (p, z) = E(p, z) cos ( - 0 _£ ) = E(p, z) sin 9 _{i t} (28b) which means that the field ratio is

[0075] A generalized parameter W, called "wave tilt," is noted herein as the ratio of the horizontal electric field component to the vertical electric field component given by

W = ^ = \W\e^, or (30a)

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

[0076] Applying Equation (30b) to a guided surface wave gives

t _an e _{i B} = ^ = ^ = s _r - ix = n = - = ^- e-' ⁱ '. (31) ι· ^ΰ Ep Y ^{r J} W \W\ '

With the angle of incidence equal to the complex Brewster angle (θ _{ί Β} ), the Fresnel reflection coefficient of Equation (25) vanishes, as shown by

Y ( _{Q .} Λ— y(£r-7*)-sin ² gj-(H _r - /¾) cos gj

= 0. (32)

(E _r -jx)-sm ² θι+ (ε _τ -]χ) cos 0;

θί -θί,Β

By adjusting the complex field ratio of Equation (22), an incident field can be synthesized to be incident at a complex angle at which the reflection is reduced or eliminated. Establishing this ratio as n = Js _r - jx results in the synthesized electric field being incident at the complex Brewster angle, making the reflections vanish. [0077] The concept of an electrical effective height can provide further insight into synthesizing an electric field with a complex angle of incidence with a guided surface waveguide probe 200. The electrical effective height (h _eff ) has been defined as

h _eff = ^ z dz (33) for a monopole with a physical height (or length) of h _p . Since the expression depends upon the magnitude and phase of the source distribution along the structure, the effective height (or length) is complex in general. The integration of the distributed current I(z) of the structure is performed over the physical height of the structure (h _p ), and normalized to the ground current (I ₀ ) flowing upward through the base (or input) of the structure. The distributed current along the structure can be expressed by

/(z) = / _c cos(/? ₀ z), (34) where β ₀ is the propagation factor for current propagating on the structure. In the example of FIG. 3, I _c is the current that is distributed along the vertical structure of the guided surface waveguide probe 200a.

[0078] For example, consider a feed network 209 that includes a low loss coil (e.g., a helical coil) at the bottom of the structure and a vertical feed line conductor connected between the coil and the charge terminal Ti . The phase delay due to the coil (or helical delay line) is 0 _C = β _ρ 1 _£ , with a physical length of l _c and a propagation factor of

^ = T _p = ik ^{> (35)} where V _f is the velocity factor on the structure, λ ₀ is the wavelength at the supplied frequency, and λ _ρ is the propagation wavelength resulting from the velocity factor V _f . The phase delay is measured relative to the ground (stake) current I ₀ .

[0079] In addition, the spatial phase delay along the length l _w of the vertical feed line conductor can be given by 0 _y = /5 _W Z _W where ? _w is the propagation phase constant for the vertical feed line conductor. In some implementations, the spatial phase delay may be approximated by θ _ν = β„Η _ρ , since the difference between the physical height h _p of the guided surface waveguide probe 200a and the vertical feed line conductor length l _w is much less than a wavelength at the supplied frequency (A ₀ ). As a result, the total phase delay through the coil and vertical feed line conductor is Φ = 0 _C + 0 _y , and the current fed to the top of the coil from the bottom of the physical structure is

with the total phase delay Φ measured relative to the ground (stake) current I ₀ .

Consequently, the electrical effective height of a guided surface waveguide probe 200 can be approximated by ff = γ _ο C ^P h ** cos(/? ₀ z) dz = h _p e^, (37) for the case where the physical height h _p « λ ₀ . The complex effective height of a monopole, Kff = h _p at an angle (or phase shift) of Φ, may be adjusted to cause the source fields to match a guided surface waveguide mode and cause a guided surface wave to be launched on the lossy conducting medium 203.

[0080] In the example of FIG. 5A, ray optics are used to illustrate the complex angle trigonometry of the incident electric field (E) having a complex Brewster angle of incidence (θ _{ί Β} ) at the Hankel crossover distance (R _x ) 121 . Recall from Equation (26) that, for a lossy conducting medium, the Brewster angle is complex and specified by

Electrically, the geometric parameters are related by the electrical effective height {h _eff ) of the charge terminal by

R _x tan ψ _1ιΒ = R _x W = h _eff = ^7' * (39) where ψ _{ί Β} = (π/2) - θ _{ί Β} is the Brewster angle measured from the surface of the lossy conducting medium. To couple into the guided surface waveguide mode, the wave tilt 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

¾^ = tan ^, _s = l ¾,. (40)

Since both the physical height (h _p ) and the Hankel crossover distance (R _x ) are real quantities, the angle (Ψ) of the desired guided surface wave tilt at the Hankel crossover distance (R _x ) is equal to the phase (Φ) of the complex effective height (h _efr ). This implies that by varying the phase at the supply point of the coil, and thus the phase shift in Equation (37), the phase, Φ, of the complex effective height can be manipulated to match the angle of the wave tilt, Ψ, of the guided surface waveguide mode at the Hankel crossover point 121 : Φ = Ψ.

[0081] In FIG. 5A, a right triangle is depicted having an adjacent side of length R _x along the lossy conducting medium surface and a complex Brewster angle measured between a ray 124 extending between the Hankel crossover point 121 at R _x and the center of the charge terminal Ti , and the lossy conducting medium surface 127 between the Hankel crossover point 121 and the charge terminal Ti . With the charge terminal Ti positioned at physical height h _p and excited with a charge having the appropriate phase delay Φ, the resulting electric field is incident with the lossy conducting medium boundary interface at the Hankel crossover distance R _x , and at the Brewster angle. Under these conditions, the guided surface waveguide mode can be excited without reflection or substantially negligible reflection.

[0082] If the physical height of the charge terminal ΤΊ is decreased without changing the phase shift Φ of the effective height (h _eff ), the resulting electric field intersects the lossy conducting medium 203 at the Brewster angle at a reduced distance from the guided surface waveguide probe 200. FIG. 6 graphically illustrates the effect of decreasing the physical height of the charge terminal ΤΊ on the distance where the electric field is incident at the Brewster angle. As the height is decreased from h ₃ through h ₂ to hi , the point where the electric field intersects with the lossy conducting medium (e.g. , the Earth) at the Brewster angle moves closer to the charge terminal position. However, as Equation (39) indicates, the height Hi (FIG. 3) of the charge terminal ΤΊ should be at or higher than the physical height (h _p ) in order to excite the far-out component of the Hankel function. With the charge terminal ΤΊ positioned at or above the effective height (h _eff ), the lossy conducting medium 203 can be illuminated at the Brewster angle of incidence (ψ _{ί Β} = (π/2) - θ _{ί Β} ) at or beyond the Hankel crossover distance (R _x ) 121 as illustrated in FIG. 5A. To reduce or minimize the bound charge on the charge terminal the height should be at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal ΤΊ as mentioned above.

[0083] A guided surface waveguide probe 200 can be configured to establish an electric field having a wave tilt that corresponds to a wave illuminating the surface of the lossy conducting medium 203 at a complex Brewster angle, thereby exciting radial surface currents by substantially mode-matching to a guided surface wave mode at (or beyond) the Hankel crossover point 121 at R _x .

[0084] Referring to FIG. 7, shown is a graphical representation of an example of a guided surface waveguide probe 200b that includes a charge terminal ΤΊ . An AC source 212 acts as the excitation source for the charge terminal ΤΊ , which is coupled to the guided surface waveguide probe 200b through a feed network 209 (FIG. 3) comprising a coil 215 such as, e.g. , a helical coil. In other implementations, the AC source 212 can be inductively coupled to the coil 215 through a primary coil. In some embodiments, an impedance matching network may be included to improve and/or maximize coupling of the AC source 212 to the coil 215.

[0085] As shown in FIG. 7, the guided surface waveguide probe 200b can include the upper charge terminal ΤΊ (e.g. , a sphere at height h _p ) that is positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. A second medium 206 is located above the lossy conducting medium 203. The charge terminal ΤΊ has a self-capacitance C _T . During operation, charge d is imposed on the terminal ΤΊ depending on the voltage applied to the terminal ΤΊ at any given instant.

[0086] In the example of FIG. 7, the coil 215 is coupled to a ground stake 218 at a first end and to the charge terminal ΤΊ via a vertical feed line conductor 221. In some

implementations, the coil connection to the charge terminal ΤΊ can be adjusted using a tap 224 of the coil 215 as shown in FIG. 7. The coil 215 can be energized at an operating frequency by the AC source 212 through a tap 227 at a lower portion of the coil 215. In other implementations, the AC source 212 can be inductively coupled to the coil 215 through a primary coil.

[0087] The construction and adjustment of the guided surface waveguide probe 200 is based upon various operating conditions, such as the transmission frequency, conditions of the lossy conducting medium (e.g. , soil conductivity σ and relative permittivity e _r ), and size of the charge terminal T^ The index of refraction can be calculated from Equations (10) and (1 1) as

n = J£ _r - jx, (41) where x = σ/ωε ₀ with ω = 2nf. The conductivity σ and relative permittivity e _r can be determined through test measurements of the lossy conducting medium 203. The complex Brewster angle (θ _{ί Β} ) measured from the surface normal can also be determined from Equation (26) as

6 _B = arctan(J£ _r - jx) , (42) or measured from the surface as shown in FIG. 5A as

^i,B = - ₂ - 9i, _B . (43) The wave tilt at the Hankel crossover distance (W _Rx ) can also be found using Equation (40).

[0088] The Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21) for -jyp, and solving for R _x as illustrated by FIG. 4. The electrical effective height can then be determined from Equation (39) using the Hankel crossover distance and the complex Brewster angle as

h _eff = h _p e' ^(p = R _x tan xp _i:B . (44) As can be seen from Equation (44), the complex effective height (h _eff ) includes a magnitude that is associated with the physical height (h _p ) of the charge terminal Ti and a phase delay (Φ) that is to be associated with the angle (Ψ) of the wave tilt at the Hankel crossover distance (R _x ). With these variables and the selected charge terminal configuration, it is possible to determine the configuration of a guided surface waveguide probe 200.

[0089] With the charge terminal positioned at or above the physical height (h _p ), the feed network 209 (FIG. 3) and/or the vertical feed line connecting the feed network to the charge terminal ΤΊ can be adjusted to match the phase (Φ) of the charge Qi on the charge terminal ΤΊ to the angle (Ψ) of the wave tilt (W). The size of the charge terminal Ti can be chosen to provide a sufficiently large surface for the charge Qi imposed on the terminals. In general, it is desirable to make the charge terminal Ti as large as practical. The size of the charge terminal Ti should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal.

[0090] The phase delay 0 _C of a helically-wound coil can be determined from Maxwell's equations as has been discussed by Corum, K.L. and J.F. Corum, "RF Coils, Helical Resonators and Voltage Magnification by Coherent Spatial Modes," Microwave Review, Vol. 7, No. 2, September 2001 , pp. 36-45., which is incorporated herein by reference in its entirety. For a helical coil with H/D > 1, the ratio of the velocity of propagation (υ) of a wave along the coil's longitudinal axis to the speed of light (c), or the "velocity factor," is given by

Vf = - = (45) c

— (!) ² ¾f

where H is the axial length of the solenoidal helix, D is the coil diameter, N is the number of turns of the coil, s = H/N is the turn-to-turn spacing (or helix pitch) of the coil, and λ ₀ is the free-space wavelength. Based upon this relationship, the electrical length, or phase delay, of the helical coil is given by

The principle is the same if the helix is wound spirally or is short and fat, but V _f and 0 _C are easier to obtain by experimental measurement. The expression for the characteristic (wave) impedance of a helical transmission line has also been derived as

[0091] The spatial phase delay 0 _y of the structure can be determined using the traveling wave phase delay of the vertical feed line conductor 221 (FIG. 7). The capacitance of a cylindrical vertical conductor above a prefect ground plane can be expressed as

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

9 _y = βν = ^L h _w = -p- h _w , (49) where ? _w is the propagation phase constant for the vertical feed line conductor, h _w is the vertical length (or height) of the vertical feed line conductor, V _w is the velocity factor on the wire, λ ₀ is the wavelength at the supplied frequency, and A _w is the propagation wavelength resulting from the velocity factor V _w . For a uniform cylindrical conductor, the velocity factor is a constant with V _w ~ 0.94, or in a range from about 0.93 to about 0.98. If the mast is considered to be a uniform transmission line, its average characteristic impedance can be approximated by

where V _w « 0.94 for a uniform cylindrical conductor and a is the radius of the conductor. An alternative expression that has been employed in amateur radio literature for the

characteristic impedance of a single-wire feed line can be given by

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

[0092] With a charge terminal Ti positioned over the lossy conducting medium 203 as shown in FIG. 3, the feed network 209 can be adjusted to excite the charge terminal Ti with the phase shift (Φ) of the complex effective height (h _efr ) equal to the angle (Ψ) of the wave tilt at the Hankel crossover distance, or Φ = Ψ. When this condition is met, the electric field produced by the charge oscillating Qi on the charge terminal Ti is coupled into a guided surface waveguide mode traveling along the surface of a lossy conducting medium 203. For example, if the Brewster angle (θ _{ί Β} ), the phase delay (0 _y ) associated with the vertical feed line conductor 221 (FIG. 7), and the configuration of the coil 215 (FIG. 7) are known, then the position of the tap 224 (FIG. 7) can be determined and adjusted to impose an oscillating charge Qi on the charge terminal Ti with phase Φ = Ψ. The position of the tap 224 may be adjusted to maximize coupling the traveling surface waves into the guided surface waveguide mode. Excess coil length beyond the position of the tap 224 can be removed to reduce the capacitive effects. The vertical wire height and/or the geometrical parameters of the helical coil may also be varied.

[0093] The coupling to the guided surface waveguide mode on the surface of the lossy conducting medium 203 can be improved and/or optimized by tuning the guided surface waveguide probe 200 for standing wave resonance with respect to a complex image plane associated with the charge Qi on the charge terminal Ti . By doing this, the performance of the guided surface waveguide probe 200 can be adjusted for increased and/or maximum voltage (and thus charge Qi) on the charge terminal Ti . Referring back to FIG. 3, the effect of the lossy conducting medium 203 in Region 1 can be examined using image theory analysis.

[0094] Physically, an elevated charge Qi placed over a perfectly conducting plane attracts the free charge on the perfectly conducting plane, which then "piles up" in the region under the elevated charge The resulting distribution of "bound" electricity on the perfectly conducting plane is similar to a bell-shaped curve. The superposition of the potential of the elevated charge plus the potential of the induced "piled up" charge beneath it, forces a zero equipotential surface for the perfectly conducting plane. The boundary value problem solution that describes the fields in the region above the perfectly conducting plane may be obtained using the classical notion of image charges, where the field from the elevated charge is superimposed with the field from a corresponding "image" charge below the perfectly conducting plane.

[0095] This analysis may also be used with respect to a lossy conducting medium 203 by assuming the presence of an effective image charge Q beneath the guided surface waveguide probe 200. The effective image charge QV coincides with the charge Qi on the charge terminal Ti about a conducting image ground plane 130, as illustrated in FIG. 3. However, the image charge Q is not merely located at some real depth and 180° out of phase with the primary source charge Qi on the charge terminal Ti, as they would be in the case of a perfect conductor. Rather, the lossy conducting medium 203 (e.g., a terrestrial medium) presents a phase shifted image. That is to say, the image charge QV is at a complex depth below the surface (or physical boundary) of the lossy conducting medium 203. For a discussion of complex image depth, reference is made to Wait, J. R., "Complex Image Theory— Revisited," IEEE Antennas and Propagation Magazine, Vol. 33, No. 4, August 1991 , pp. 27-29, which is incorporated herein by reference in its entirety.

[0096] Instead of the image charge QV being at a depth that is equal to the physical height (HVj of the charge Q^ the conducting image ground plane 130 (representing a perfect conductor) is located at a complex depth of z = - d/2 and the image charge QV appears at a complex depth (i.e., the "depth" has both magnitude and phase), given by -O _t = -(d/2 + d/2 + Η _χ )≠ Η _χ . For vertically polarized sources over the Earth,

where

Ye ⁼ ϊ ^ω ι ^σ ι ^{~ ω2} μι ^ε ι , ^an d (53) k ₀ = ω^μ ₀ ε ₀ , (54) as indicated in Equation (12). The complex spacing of the image charge, in turn, implies that the external field will experience extra phase shifts not encountered when the interface is either a dielectric or a perfect conductor. In the lossy conducting medium, the wave front normal is parallel to the tangent of the conducting image ground plane 130 at z = - d/2, and not at the boundary interface between Regions 1 and 2. [0097] Consider the case illustrated in FIG. 8A where the lossy conducting medium 203 is a finitely conducting Earth 133 with a physical boundary 136. The finitely conducting Earth 133 may be replaced by a perfectly conducting image ground plane 139 as shown in FIG.8B, which is located at a complex depth z ₁ below the physical boundary 136. This equivalent representation exhibits the same impedance when looking down into the interface at the physical boundary 136. The equivalent representation of FIG. 8B can be modeled as an equivalent transmission line, as shown in FIG. 8C. The cross-section of the equivalent structure is represented as a (z-directed) end-loaded transmission line, with the impedance of the perfectly conducting image plane being a short circuit (z _s = 0). The depth z ₁ can be determined by equating the TEM wave impedance looking down at the Earth to an image ground plane impedance z _in seen looking into the transmission line of FIG. 8C.

[0098] In the case of FIG. 8A, the propagation constant and wave intrinsic impedance in the upper region (air) 142 are

Yo = ju o = 0 + }β ₀ , and (55)

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

Ye = V (° _I + 7 ^' ωε ₁ ) , and (57)

Z _e = ^i. (58) e Ye

For normal incidence, the equivalent representation of FIG. 8B is equivalent to a TEM transmission line whose characteristic impedance is that of air (z ₀ ), with propagation constant of γ ₀ , and whose length is z _x . As such, the image ground plane impedance Z _in seen at the interface for the shorted transmission line of FIG. 8C is given by

Z _in = Z ₀ tanh ( oZi) . (59) Equating the image ground plane impedance Z _in associated with the equivalent model of FIG. 8C to the normal incidence wave impedance of FIG. 8A and solving for z ₁ gives the distance to a short circuit (the perfectly conducting image ground plane 139) as

_Zl = -tanh ^"1 (^) = -tanh ^"1 « - , (60)

Yo \Z ₀ ) Yo _Ye J _Ye '

where only the first term of the series expansion for the inverse hyperbolic tangent is considered for this approximation. Note that in the air region 142, the propagation constant is Yo = ]β ₀ , so Z _in = jZ ₀ tan β ₀ ζ ₁ (which is a purely imaginary quantity for a real z _t ), but z _e is a complex value if σ ≠ 0. Therefore, Z _in = Z _e only when z ₁ is a complex distance.

[0099] Since the equivalent representation of FIG. 8B includes a perfectly conducting image ground plane 139, the image depth for a charge or current lying at the surface of the Earth (physical boundary 136) is equal to distance z _t on the other side of the image ground plane 139, or d = 2 x z _x beneath the Earth's surface (which is located at z = 0). Thus, the distance to the perfectly conducting image ground plane 139 can be approximated by

d = 2z _{1 ¾} (61)

Additionally, the "image charge" will be "equal and opposite" to the real charge, so the potential of the perfectly conducting image ground plane 139 at depth z ₁ = - d/2 will be zero.

[0100] If a charge Qi is elevated a distance Hi above the surface of the Earth as illustrated in FIG. 3, then the image charge Qi' resides at a complex distance of O ₁ = d + U ₁ below the surface, or a complex distance of d/2 + U _t below the image ground plane 130. The guided surface waveguide probe 200b of FIG. 7 can be modeled as an equivalent single-wire transmission line image plane model that can be based upon the perfectly conducting image ground plane 139 of FIG. 8B. FIG. 9A shows an example of the equivalent single-wire transmission line image plane model, and FIG. 9B illustrates an example of the equivalent classic transmission line model, including the shorted

transmission line of FIG. 8C.

[0101] In the equivalent image plane models of FIGS. 9A and 9B, Φ = 0 _y + θ ₀ is the traveling wave phase delay of the guided surface waveguide probe 200 referenced to Earth 133 (or the lossy conducting medium 203), 0 _C = β _ρ Η is the electrical length of the coil 215 (FIG. 7), of physical length H, expressed in degrees, 0 _y = ? _w / _w is the electrical length of the vertical feed line conductor 221 (FIG. 7), of physical length h _w , expressed in degrees, and θ _α = β ₀ d/2 is the phase shift between the image ground plane 139 and the physical boundary 136 of the Earth 133 (or lossy conducting medium 203). In the example of FIGS. 9A and 9B, Z _w is the characteristic impedance of the elevated vertical feed line conductor 221 in ohms, Z _c is the characteristic impedance of the coil 215 in ohms, and Z ₀ is the characteristic impedance of free space.

[0102] At the base of the guided surface waveguide probe 200, the impedance seen "looking up" into the structure is Z _T = Z _base . With a load impedance of:

where C _T is the self-capacitance of the charge terminal Ti, the impedance seen "looking up" into the vertical feed line conductor 221 (FIG. 7) is given by:

z _ _z Z _L +Z _W tanhg/? _w ft _w ) _ Z _L +Z _W tanh(jg _y )

2 ^W Z _W +Z _L tanhO ? _w ft _w ) ^W Z _W +Z _L tanh(y<9 _y ) ' ^ ' and the impedance seen "looking up" into the coil 215 (FIG. 7) is given by:

Z ₂ +Z _c tanhQ-ff _p H) _ Z ₂ +Z _c tanh(j9 _c ) , _R ^base _Zc+ z _{2 tanh} (j _PpH ) ^A _{Zc+¾ tanh0} e _c) ^■ At the base of the guided surface waveguide probe 200, the impedance seen "looking down" into the lossy conducting medium 203 is Z _t = Z _in , which is given by:

_{= z} Z _S+ Z ₀ tanh^ ₀ (d/2)] ₌

l ^{n 0} Z ₀ +Z _s tanh[ _j ? ₀ (d/2)] o ^j a , ^\ ι where Z _s = 0.

[0103] Neglecting losses, the equivalent image plane model can be tuned to resonance when Zi + Z _T = 0 at the physical boundary 136. Or, in the low loss case, ¾ + X _T = 0 at the physical boundary 136, where X is the corresponding reactive component. Thus, the impedance at the physical boundary 136 "looking up" into the guided surface waveguide probe 200 is the conjugate of the impedance at the physical boundary 136 "looking down" into the lossy conducting medium 203. By adjusting the load impedance Z _L of the charge terminal Ti while maintaining the traveling wave phase delay Φ equal to the angle of the media's wave tilt Ψ, so that Φ = Ψ, which improves and/or maximizes coupling of the probe's electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., Earth), the equivalent image plane models of FIGS. 9A and 9B can be tuned to resonance with respect to the image ground plane 139. In this way, the impedance of the equivalent complex image plane model is purely resistive, which maintains a superposed standing wave on the probe structure that maximizes the voltage and elevated charge on terminal Ti , and by equations (1)-(3) and (16) maximizes the propagating surface wave.

[0104] It follows from the Hankel solutions, that the guided surface wave excited by the guided surface waveguide probe 200 is an outward propagating traveling wave. The source distribution along the feed network 209 between the charge terminal Ti and the ground stake 218 of the guided surface waveguide probe 200 (FIGS. 3 and 7) is actually composed of a superposition of a traveling wave plus a standing wave on the structure. With the charge terminal Ti positioned at or above the physical height h _p , the phase delay of the traveling wave moving through the feed network 209 is matched to the angle of the wave tilt associated with the lossy conducting medium 203. This mode-matching allows the traveling wave to be launched along the lossy conducting medium 203. Once the phase delay has been established for the traveling wave, the load impedance Z _L of the charge terminal Ti is adjusted to bring the probe structure into standing wave resonance with respect to the image ground plane (130 of FIG. 3 or 139 of FIG. 8), which is at a complex depth of - d/2. In that case, the impedance seen from the image ground plane has zero reactance and the charge on the charge terminal is maximized.

[0105] The distinction between the traveling wave phenomenon and standing wave phenomena is that (1) the phase delay of traveling waves (θ = βά) on a section of transmission line of length d (sometimes called a "delay line") is due to propagation time delays; whereas (2) the position-dependent phase of standing waves (which are composed of forward and backward propagating waves) depends on both the line length propagation time delay and impedance transitions at interfaces between line sections of different characteristic impedances. In addition to the phase delay that arises due to the physical length of a section of transmission line operating in sinusoidal steady-state, there is an extra reflection coefficient phase at impedance discontinuities that is due to the ratio of Z _oa /Z _ob , where Z _oa and Z _ob are the characteristic impedances of two sections of a transmission line such as, e.g., a helical coil section of characteristic impedance Z _oa = Z _c (FIG. 9B) and a straight section of vertical feed line conductor of characteristic impedance Z _ob = Z _w (FIG. 9B).

[0106] As a result of this phenomenon, two relatively short transmission line sections of widely differing characteristic impedance may be used to provide a very large phase shift. For example, a probe structure composed of two sections of transmission line, one of low impedance and one of high impedance, together totaling a physical length of, say, 0.05 λ, may be fabricated to provide a phase shift of 90° which is equivalent to a 0.25 λ resonance. This is due to the large jump in characteristic impedances. In this way, a physically short probe structure can be electrically longer than the two physical lengths combined. This is illustrated in FIGS. 9A and 9B, where the discontinuities in the impedance ratios provide large jumps in phase. The impedance discontinuity provides a substantial phase shift where the sections are joined together.

[0107] Referring to FIG. 10, shown is a flow chart 150 illustrating an example of adjusting a guided surface waveguide probe 200 (FIGS. 3 and 7) to substantially mode- match to a guided surface waveguide mode on the surface of the lossy conducting medium, which launches a guided surface traveling wave along the surface of a lossy conducting medium 203 (FIG. 3). Beginning with 153, the charge terminal Ti of the guided surface waveguide probe 200 is positioned at a defined height above a lossy conducting medium 203. Utilizing the characteristics of the lossy conducting medium 203 and the operating frequency of the guided surface waveguide probe 200, the Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21) for -y ^' yp, and solving for R _x as illustrated by FIG. 4. The complex index of refraction (n) can be determined using Equation (41), and the complex Brewster angle (θ _{ί Β} ) can then be determined from Equation (42). The physical height (h _p ) of the charge terminal Ti can then be determined from Equation (44). The charge terminal should be at or higher than the physical height (h _p ) in order to excite the far-out component of the Hankel function. This height relationship is initially considered when launching surface waves. To reduce or minimize the bound charge on the charge terminal the height should be at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal TV

[0108] At 156, the electrical phase delay Φ of the elevated charge on the charge terminal ΤΊ is matched to the complex wave tilt angle Ψ. The phase delay (0 _C ) of the helical coil and/or the phase delay (0 _y ) of the vertical feed line conductor can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (W). Based on Equation (31), the angle (Ψ) of the wave tilt can be determined from:

W = ^ = ^— = - = \W\e^. (66)

The electrical phase Φ can then be matched to the angle of the wave tilt. This angular (or phase) relationship is next considered when launching surface waves. For example, the electrical phase delay Φ = 0 _C + 0 _y can be adjusted by varying the geometrical parameters of the coil 215 (FIG. 7) and/or the length (or height) of the vertical feed line conductor 221 (FIG. 7). By matching Φ = Ψ, an electric field can be established at or beyond the Hankel crossover distance (R _x ) with a complex Brewster angle at the boundary interface to excite the surface waveguide mode and launch a traveling wave along the lossy conducting medium 203.

[0109] Next at 159, the load impedance of the charge terminal Ti is tuned to resonate the equivalent image plane model of the guided surface waveguide probe 200. The depth (d/2) of the conducting image ground plane 139 of FIG. 9A and 9B (or 130 of FIG. 3) can be determined using Equations (52), (53) and (54) and the values of the lossy conducting medium 203 (e.g., the Earth), which can be measured. Using that depth, the phase shift (θ _ά ) between the image ground plane 139 and the physical boundary 136 of the lossy conducting medium 203 can be determined using θ _α = β ₀ d/2. The impedance (Z _in ) as seen "looking down" into the lossy conducting medium 203 can then be determined using Equation (65). This resonance relationship can be considered to maximize the launched surface waves.

[0110] Based upon the adjusted parameters of the coil 215 and the length of the vertical feed line conductor 221 , the velocity factor, phase delay, and impedance of the coil 215 and vertical feed line conductor 221 can be determined using Equations (45) through (51). In addition, the self-capacitance (C _T ) of the charge terminal Ti can be determined using, e.g., Equation (24). The propagation factor (β _ρ ) of the coil 215 can be determined using Equation (35) and the propagation phase constant ( ? _w ) for the vertical feed line conductor 221 can be determined using Equation (49). Using the self-capacitance and the determined values of the coil 215 and vertical feed line conductor 221 , the impedance (Z _base ) of the guided surface waveguide probe 200 as seen "looking up" into the coil 215 can be determined using Equations (62), (63) and (64). [0111] The equivalent image plane model of the guided surface waveguide probe 200 can be tuned to resonance by adjusting the load impedance Z _L such that the reactance component X _base of Z _base cancels out the reactance component X _in of Z _in , or X _base + X _in = 0. Thus, the impedance at the physical boundary 136 "looking up" into the guided surface waveguide probe 200 is the conjugate of the impedance at the physical boundary 136 "looking down" into the lossy conducting medium 203. The load impedance Z _L can be adjusted by varying the capacitance (C _T ) of the charge terminal Ti without changing the electrical phase delay Φ = 0 _C + 0 _y of the charge terminal Ti . An iterative approach may be taken to tune the load impedance Z _L for resonance of the equivalent image plane model with respect to the conducting image ground plane 139 (or 130). In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., Earth) can be improved and/or maximized.

[0112] This may be better understood by illustrating the situation with a numerical example. Consider a guided surface waveguide probe 200 comprising a top-loaded vertical stub of physical height h _p with a charge terminal Ti at the top, where the charge terminal Ti is excited through a helical coil and vertical feed line conductor at an operational frequency (/o) of 1.85 MHz. With a height (H- of 16 feet and the lossy conducting medium 203 {e.g., Earth) having a relative permittivity of e _r = 15 and a conductivity of σ _χ = 0.010 mhos/m, several surface wave propagation parameters can be calculated for ₀ = 1.850 MHz. Under these conditions, the Hankel crossover distance can be found to be R _x = 54.5 feet with a physical height of h _p = 5.5 feet, which is well below the actual height of the charge terminal Ti. While a charge terminal height of Hi = 5.5 feet could have been used, the taller probe structure reduced the bound capacitance, permitting a greater percentage of free charge on the charge terminal Ti providing greater field strength and excitation of the traveling wave.

[0113] The wave length can be determined as:

λ ₀ = = 162.162 meters, (67)

where c is the speed of light. The complex index of refraction is:

n = ^s _r - j _x = 7.529 - j 6.546, (68) from Equation (41), where x = σ ₁ /ωε ₀ with ω = 2nf ₀ , and the complex Brewster angle is:

Q _{i B} = arctan(J£ _r - jx) = 85.6 - j 3.744°. (69) from Equation (42). Using Equation (66), the wave tilt values can be determined to be:

W

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

[0114] The velocity factor of the vertical feed line conductor (approximated as a uniform cylindrical conductor with a diameter of 0.27 inches) can be given as ¾, ¾ 0.93. Since h _p « λ ₀ , the propagation phase constant for the vertical feed line conductor can be approximated as:

^ = r = VT ^{= o 2 m~1} - ( ⁷¹ )

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

y = β _ν ,Κ ~ β _ν , = 11.640°. (72) By adjusting the phase delay of the helical coil so that Q _c = 28.974° = 40.614° - 11.640°, Φ will equal Ψ to match the guided surface waveguide mode. To illustrate the relationship between Φ and Ψ, FIG. 1 1 shows a plot of both over a range of frequencies. As both Φ and Ψ are frequency dependent, it can be seen that their respective curves cross over each other at approximately 1.85 MHz.

[0115] For a helical coil having a conductor diameter of 0.0881 inches, a coil diameter (D) of 30 inches and a turn-to-turn spacing (s) of 4 inches, the velocity factor for the coil can be determined using Equation (45) as:

V _f = j ¹ — = 0.069 , (73)

n _\ / n \

1+20 >(?) (έ)

and the propagation factor from Equation (35) is:

2π

= 0.564 m- ¹ . (74)

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

H = ^- = 35.2732 inches . (75)

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

[0116] With the traveling wave phase delay of the coil and vertical feed line conductor adjusted to match the wave tilt angle (Φ = 0 _C + 0 _y = Ψ), the load impedance (Z _L ) of the charge terminal Ti can be adjusted for standing wave resonance of the equivalent image plane model of the guided surface wave probe 200. From the measured permittivity, conductivity and permeability of the Earth, the radial propagation constant can be determined using Equation (57)

y _e = ^]ωη ₁ (σ ₁ + ίωε-^ = 0.25 + j 0.292 m ^_1 , (76) And the complex depth of the conducting image ground plane can be approximated from Equation (52) as:

d * - = 3.364 + j 3.963 meters , (77)

Ye

with a corresponding phase shift between the conducting image ground plane and the physical boundary of the Earth given by: θ _α = β ₀ (ά/2) = 4.015 - / 4.73°. (78) Using Equation (65), the impedance seen "looking down" into the lossy conducting medium 203 (i.e., Earth) can be determined as:

Z _in = Z ₀ tanh(/6> _d ) = R _in + jX _in = 31.191 + j 26.27 ohms. (79)

[0117] By matching the reactive component (X _in ) seen "looking down" into the lossy conducting medium 203 with the reactive component (X _baS e) ^seen "looking up" into the guided surface wave probe 200, the coupling into the guided surface waveguide mode may be maximized. This can be accomplished by adjusting the capacitance of the charge terminal without changing the traveling wave phase delays of the coil and vertical feed line conductor. For example, by adjusting the charge terminal capacitance (C _T ) to 61.8126 pF, the load impedance from Equation (62) is:

Z _L =— = -j 1392 ohms, (80) jojCf

and the reactive components at the boundary are matched.

[0118] Using Equation (51), the impedance of the vertical feed line conductor (having a diameter (2a) of 0.27 inches) is given as

Z _w = 138 log ( ^{1 12} ₂ ³ ^ ^wA °) = 537.534 ohms, (81) and the impedance seen "looking up" into the vertical feed line conductor is given by Equation (63) as:

Using Equation (47), the characteristic impedance of the helical coil is given as

~ ¹ - ⁰²⁷ ] ^{= 1446 ohms} ' ⁽⁸³⁾ and the impedance seen "looking up" into the coil at the base is given by Equation (64)

Zb ^b aassee = Zcc = -j ^J 26.271 ohms. (84) '

When compared to the solution of Equation (79), it can be seen that the reactive

components are opposite and approximately equal, and thus are conjugates of each other. Thus, the impedance (Z _ip ) seen "looking up" into the equivalent image plane model of FIGS. 9A and 9B from the perfectly conducting image ground plane is only resistive or Z _ip = R + jO.

[0119] When the electric fields produced by a guided surface waveguide probe 200 (FIG. 3) are established by matching the traveling wave phase delay of the feed network to the wave tilt angle and the probe structure is resonated with respect to the perfectly conducting image ground plane at complex depth z = -d/2 , the fields are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium, a guided surface traveling wave is launched along the surface of the lossy conducting medium. As illustrated in FIG. 1 , the guided field strength curve 103 of the guided electromagnetic field has a characteristic exponential decay of e ^~ad / fd and exhibits a distinctive knee 109 on the log-log scale.

[0120] In summary, both analytically and experimentally, the traveling wave component on the structure of the guided surface waveguide probe 200 has a phase delay (Φ) at its upper terminal that matches the angle (Ψ) of the wave tilt of the surface traveling wave (Φ = Ψ). Under this condition, the surface waveguide may be considered to be "mode- matched". Furthermore, the resonant standing wave component on the structure of the guided surface waveguide probe 200 has a V _M AX at the charge terminal ΤΊ and a V _M IN down at the image plane 139 (FIG. 8B) where Z _ip = R _ip + j 0 at a complex depth of z = - d/2 , not at the connection at the physical boundary 136 of the lossy conducting medium 203 (FIG. 8B). Lastly, the charge terminal ΤΊ is of sufficient height Hi of FIG. 3 (h≥ R _x an if _{i B} ) so that electromagnetic waves incident onto the lossy conducting medium 203 at the complex Brewster angle do so out at a distance (> R _x ) where the 1/Vr term is predominant. Receive circuits can be utilized with one or more guided surface waveguide probes to facilitate wireless transmission and/or power delivery systems.

[0121] Referring back to FIG. 3, operation of a guided surface waveguide probe 200 may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, an adaptive probe control system 230 can be used to control the feed network 209 and/or the charge terminal ΤΊ to control the operation of the guided surface waveguide probe 200. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 203 (e.g., conductivity σ and relative permittivity e _r ), variations in field strength and/or variations in loading of the guided surface waveguide probe 200. As can be seen from Equations (31), (41) and (42), the index of refraction (n), the complex Brewster angle (θ _{ί Β} ), and the wave tilt (|ν |β ^;ψ ) can be affected by changes in soil conductivity and permittivity resulting from, e.g., weather conditions.

[0122] Equipment such as, e.g., conductivity measurement probes, permittivity sensors, ground parameter meters, field meters, current monitors and/or load receivers can be used to monitor for changes in the operational conditions and provide information about current operational conditions to the adaptive probe control system 230. The probe control system 230 can then make one or more adjustments to the guided surface waveguide probe 200 to maintain specified operational conditions for the guided surface waveguide probe 200. For instance, as the moisture and temperature vary, the conductivity of the soil will also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe 200. Generally, it would be desirable to monitor the conductivity and/or permittivity at or about the Hankel crossover distance R _x for the operational frequency. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations (e.g. , in each quadrant) around the guided surface waveguide probe 200.

[0123] The conductivity measurement probes and/or permittivity sensors can be configured to evaluate the conductivity and/or permittivity on a periodic basis and

communicate the information to the probe control system 230. The information may be communicated to the probe control system 230 through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate wired or wireless communication network. Based upon the monitored conductivity and/or permittivity, the probe control system 230 may evaluate the variation in the index of refraction (n), the complex Brewster angle (θ _{ί Β} ), and/or the wave tilt (|ν |β ^;ψ ) and adjust the guided surface waveguide probe 200 to maintain the phase delay (Φ) of the feed network 209 equal to the wave tilt angle (Ψ) and/or maintain resonance of the equivalent image plane model of the guided surface waveguide probe 200. This can be accomplished by adjusting, e.g. , Q _y , Q _c and/or C _T . For instance, the probe control system 230 can adjust the self-capacitance of the charge terminal Ti and/or the phase delay (Q _y , Q _c ) applied to the charge terminal Ti to maintain the electrical launching efficiency of the guided surface wave at or near its maximum. For example, the self-capacitance of the charge terminal can be varied by changing the size of the terminal. The charge distribution can also be improved by increasing the size of the charge terminal T^ which can reduce the chance of an electrical discharge from the charge terminal Ti. In other embodiments, the charge terminal Ti can include a variable inductance that can be adjusted to change the load impedance Z _L . The phase applied to the charge terminal Ti can be adjusted by varying the tap position on the coil 215 (FIG. 7), and/or by including a plurality of predefined taps along the coil 215 and switching between the different predefined tap locations to maximize the launching efficiency.

[0124] Field or field strength (FS) meters may also be distributed about the guided surface waveguide probe 200 to measure field strength of fields associated with the guided surface wave. The field or FS meters can be configured to detect the field strength and/or changes in the field strength (e.g. , electric field strength) and communicate that information to the probe control system 230. The information may be communicated to the probe control system 230 through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. As the load and/or environmental conditions change or vary during operation, the guided surface waveguide probe 200 may be adjusted to maintain specified field strength(s) at the FS meter locations to ensure appropriate power transmission to the receivers and the loads they supply. [0125] For example, the phase delay (Φ = 6 _y + 6 _C ) applied to the charge terminal ΤΊ can be adjusted to match the wave tilt angle (Ψ). By adjusting one or both phase delays, the guided surface waveguide probe 200 can be adjusted to ensure the wave tilt corresponds to the complex Brewster angle. This can be accomplished by adjusting a tap position on the coil 215 (FIG. 7) to change the phase delay supplied to the charge terminal Ti . The voltage level supplied to the charge terminal Ti can also be increased or decreased to adjust the electric field strength. This may be accomplished by adjusting the output voltage of the excitation source 212 or by adjusting or reconfiguring the feed network 209. For instance, the position of the tap 227 (FIG. 7) for the AC source 212 can be adjusted to increase the voltage seen by the charge terminal Ti . Maintaining field strength levels within predefined ranges can improve coupling by the receivers, reduce ground current losses, and avoid interference with transmissions from other guided surface waveguide probes 200.

[0126] The probe control system 230 can be implemented with hardware, firmware, software executed by hardware, or a combination thereof. For example, the probe control system 230 can include processing circuitry including a processor and a memory, both of which can be coupled to a local interface such as, for example, a data bus with an accompanying control/address bus as can be appreciated by those with ordinary skill in the art. A probe control application may be executed by the processor to adjust the operation of the guided surface waveguide probe 200 based upon monitored conditions. The probe control system 230 can also include one or more network interfaces for communicating with the various monitoring devices. Communications can be through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. The probe control system 230 may comprise, for example, a computer system such as a server, desktop computer, laptop, or other system with like capability.

[0127] Referring back to the example of FIG. 5A, the complex angle trigonometry is shown for the ray optic interpretation of the incident electric field (E) of the charge terminal Ti with a complex Brewster angle (θ _{ί Β} ) at the Hankel crossover distance (R _x ). Recall that, for a lossy conducting medium, the Brewster angle is complex and specified by equation (38). Electrically, the geometric parameters are related by the electrical effective height (h _efr ) of the charge terminal Ti by equation (39). Since both the physical height (h _p ) and the Hankel crossover distance (R _x ) are real quantities, the angle of the desired guided surface wave tilt at the Hankel crossover distance (W _Rx ) is equal to the phase (Φ) of the complex effective height {h _eff ). With the charge terminal Ti positioned at the 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.

[0128] However, Equation (39) means that the physical height of the guided surface waveguide probe 200 can be relatively small. While this will excite the guided surface waveguide mode, this can result in an unduly large bound charge with little free charge. To compensate, the charge terminal ΤΊ can be raised to an appropriate elevation to increase the amount of free charge. As one example rule of thumb, the charge terminal ΤΊ can be positioned at an elevation of about 4-5 times (or more) the effective diameter of the charge terminal T FIG. 6 illustrates the effect of raising the charge terminal Ti above the physical height (h _p ) shown in FIG. 5A. The increased elevation causes the distance at which the wave tilt is incident with the lossy conductive medium to move beyond the Hankel crossover point 121 (FIG. 5A). To improve coupling in the guided surface waveguide mode, and thus provide for a greater launching efficiency of the guided surface wave, a lower compensation terminal T ₂ can be used to adjust the total effective height (h _TE ) of the charge terminal Ti such that the wave tilt at the Hankel crossover distance is at the Brewster angle.

[0129] Referring to FIG. 12, shown is an example of a guided surface waveguide probe 200c that includes an elevated charge terminal Ti and a lower compensation terminal T ₂ that are arranged along a vertical axis z that is normal to a plane presented by the lossy conducting medium 203. In this respect, the charge terminal Ti is placed directly above the compensation terminal T ₂ although it is possible that some other arrangement of two or more charge and/or compensation terminals T _N can be used. The guided surface waveguide probe 200c is disposed above a lossy conducting medium 203 according to an embodiment of the present disclosure. The lossy conducting medium 203 makes up Region 1 with a second medium 206 that makes up Region 2 sharing a boundary interface with the lossy conducting medium 203.

[0130] The guided surface waveguide probe 200c includes a feed network 209 that couples an excitation source 212 to the charge terminal Ti and the compensation terminal T ₂ . According to various embodiments, charges Qi and Q ₂ can be imposed on the respective charge and compensation terminals Ti and T ₂ , depending on the voltages applied to terminals and T ₂ at any given instant, is the conduction current feeding the charge Q- _! on the charge terminal via the terminal lead, and l ₂ is the conduction current feeding the charge Q ₂ on the compensation terminal T ₂ via the terminal lead.

[0131] According to the embodiment of FIG. 12, the charge terminal Ti is positioned over the lossy conducting medium 203 at a physical height H^ and the compensation terminal T ₂ is positioned directly below along the vertical axis z at a physical height H ₂ , where H ₂ is less than Hi. The height h of the transmission structure may be calculated as h = H _! - H _2. The charge terminal ΤΊ has an isolated (or self) capacitance Ci, and the compensation terminal T ₂ has an isolated (or self) capacitance C ₂ . A mutual capacitance C _M can also exist between the terminals ΤΊ and T ₂ depending on the distance therebetween. During operation, charges Qi and Q ₂ are imposed on the charge terminal ΤΊ and the compensation terminal T ₂ , respectively, depending on the voltages applied to the charge terminal ΤΊ and the compensation terminal T ₂ at any given instant.

[0132] Referring next to FIG. 13, shown is a ray optics interpretation of the effects produced by the elevated charge Qi on charge terminal ΤΊ and compensation terminal T ₂ of FIG. 12. With the charge terminal ΤΊ elevated to a height where the ray intersects with the lossy conductive medium at the Brewster angle at a distance greater than the Hankel crossover point 121 as illustrated by line 163, the compensation terminal T ₂ can be used to adjust h _TE by compensating for the increased height. The effect of the compensation terminal T ₂ is to reduce the electrical effective height of the guided surface waveguide probe (or effectively raise the lossy medium interface) such that the wave tilt at the Hankel crossover distance is at the Brewster angle as illustrated by line 166.

[0133] The total effective height can be written as the superposition of an upper effective height (h _UE ) associated with the charge terminal and a lower effective height (h _LE ) associated with the compensation terminal T ₂ such that

h _TE = h _UE + h _LE = h _p e^ ^h v^u) + Η _ά6 ^βη α+Φύ = R _x x W, (85) where is the phase delay applied to the upper charge terminal T O _l is the phase delay applied to the lower compensation terminal T ₂ , β = 2π/λ _ρ is the propagation factor from Equation (35), h _p is the physical height of the charge terminal Ti and h _d is the physical height of the compensation terminal T ₂ . If extra lead lengths are taken into consideration, they can be accounted for by adding the charge terminal lead length z to the physical height h _p of the charge terminal Ti and the compensation terminal lead length y to the physical height h _d of the compensation terminal T ₂ as shown in

h _TE = (h _p + _Z y = R _X x w. (86)

The lower effective height can be used to adjust the total effective height (h _TE ) to equal the complex effective height (h _efr ) of FIG. 5A.

[0134] Equations (85) or (86) can be used to determine the physical height of the lower disk of the compensation terminal T ₂ and the phase angles to feed the terminals in order to obtain the desired wave tilt at the Hankel crossover distance. For example, Equation (86) can be rewritten as the phase shift applied to the charge terminal Ti as a function of the compensation terminal height (h _d ) to give [0135] To determine the positioning of the compensation terminal T ₂ , the relationships discussed above can be utilized. First, the total effective height (h _TE ) is the superposition of the complex effective height (h _UE ) of the upper charge terminal Ti and the complex effective height (h _LE ) of the lower compensation terminal T ₂ as expressed in Equation (86). Next, the tangent of the angle of incidence can be expressed geometrically as

tan^ =^ , (88) which is equal to the definition of the wave tilt, W. Finally, given the desired Hankel crossover distance R _x , the h _TE can be adjusted to make the wave tilt of the incident ray match the complex Brewster angle at the Hankel crossover point 121. This can be accomplished by adjusting h _p , O _y , and/or h _d .

[0136] These concepts may be better understood when discussed in the context of an example of a guided surface waveguide probe. Referring to FIG. 14, shown is a graphical representation of an example of a guided surface waveguide probe 200d including an upper charge terminal Ti (e.g., a sphere at height h _T ) and a lower compensation terminal T ₂ (e.g. , a disk at height h _d ) that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. During operation, charges Qi and Q ₂ are imposed on the charge and compensation terminals Ti and T ₂ , respectively, depending on the voltages applied to the terminals Ti and T ₂ at any given instant.

[0137] An AC source 212 acts as the excitation source for the charge terminal Ti , which is coupled to the guided surface waveguide probe 200d through a feed network 209 comprising a coil 215 such as, e.g. , a helical coil. The AC source 212 can be connected across a lower portion of the coil 215 through a tap 227, as shown in FIG. 14, or can be inductively coupled to the coil 215 by way of a primary coil. The coil 215 can be coupled to a ground stake 218 at a first end and the charge terminal Ti at a second end. In some implementations, the connection to the charge terminal Ti can be adjusted using a tap 224 at the second end of the coil 215. The compensation terminal T ₂ is positioned above and substantially parallel with the lossy conducting medium 203 (e.g., the ground or Earth), and energized through a tap 233 coupled to the coil 215. An ammeter 236 located between the coil 215 and ground stake 218 can be used to provide an indication of the magnitude of the current flow (I 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 218 to obtain an indication of the magnitude of the current flow (I ₀ ).

[0138] In the example of FIG. 14, the coil 215 is coupled to a ground stake 218 at a first end and the charge terminal Ti at a second end via a vertical feed line conductor 221. In some implementations, the connection to the charge terminal Ti can be adjusted using a tap 224 at the second end of the coil 215 as shown in FIG. 14. The coil 215 can be energized at an operating frequency by the AC source 212 through a tap 227 at a lower portion of the coil 215. In other implementations, the AC source 212 can be inductively coupled to the coil 215 through a primary coil. The compensation terminal T ₂ is energized through a tap 233 coupled to the coil 215. An ammeter 236 located between the coil 215 and ground stake 218 can be used to provide an indication of the magnitude of the current flow at the base of the guided surface waveguide probe 200d. Alternatively, a current clamp may be used around the conductor coupled to the ground stake 218 to obtain an indication of the magnitude of the current flow. The compensation terminal T ₂ is positioned above and substantially parallel with the lossy conducting medium 203 (e.g., the ground).

[0139] In the example of FIG. 14, the connection to the charge terminal Ti located on the coil 215 above the connection point of tap 233 for the compensation terminal T ₂ . Such an adjustment allows an increased voltage (and thus a higher charge d) to be applied to the upper charge terminal Ti . In other embodiments, the connection points for the charge terminal Ti and the compensation terminal T ₂ can be reversed. It is possible to adjust the total effective height (h _TE ) of the guided surface waveguide probe 200d to excite an electric field having a guided surface wave tilt at the Hankel crossover distance R _x . The Hankel crossover distance can also be found by equating the magnitudes of equations (20b) and (21) for -jyp, and solving for R _x as illustrated by FIG. 4. The index of refraction (n), the complex Brewster angle (θ _{ί Β} and the wave tilt (|ν |β ^;ψ ) and the complex effective height (h _eff = h _p e ^i<p ) can be determined as described with respect to Equations (41) - (44) above.

[0140] With the selected charge terminal Ti configuration, a spherical diameter (or the effective spherical diameter) can be determined. For example, if the charge terminal 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 can be chosen to provide a sufficiently large surface for the charge d imposed on the terminals. In general, it is desirable to make the charge terminal Ti as large as practical. The size of the charge terminal Ti should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal. To reduce the amount of bound charge on the charge terminal Ti , the desired elevation to provide free charge on the charge terminal Ti 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 T ₂ can be used to adjust the total effective height (h _TE ) of the guided surface waveguide probe 200d to excite an electric field having a guided surface wave tilt at R _x . The compensation terminal T ₂ 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 Ti . With the position of the compensation terminal T ₂ fixed and the phase delay applied to the upper charge terminal ΤΊ , the phase delay O _l applied to the lower compensation terminal T ₂ can be determine

In alternative embodiments, the compensation terminal T ₂ can be positioned at a height h _d where Im{0 _L } = 0. This is graphically illustrated in FIG. 15A, which shows plots 172 and 175 of the imaginary and real parts of Φ^, respectively. The compensation terminal T ₂ is positioned at a height h _d where Im{Oy} = 0, as graphically illustrated in plot 172. At this fixed height, the coil phase can be determined from Re{Oy}, as graphically illustrated in plot 175.

[0141] With the AC source 212 coupled to the coil 215 (e.g., at the 50Ω point to maximize coupling), the position of tap 233 may be adjusted for parallel resonance of the compensation terminal T ₂ with at least a portion of the coil at the frequency of operation. FIG. 15B shows a schematic diagram of the general electrical hookup of FIG. 14 in which Vi is the voltage applied to the lower portion of the coil 215 from the AC source 212 through tap 227, V ₂ is the voltage at tap 224 that is supplied to the upper charge terminal T^ and V ₃ is the voltage applied to the lower compensation terminal T ₂ through tap 233. The resistances R _p and R _d represent the ground return resistances of the charge terminal and

compensation terminal T ₂ , respectively. The charge and compensation terminals and T ₂ may be configured as spheres, cylinders, toroids, rings, hoods, or any other combination of capacitive structures. The size of the charge and compensation terminals Ti and T ₂ can be chosen to provide a sufficiently large surface for the charges d and Q ₂ imposed on the terminals. In general, it is desirable to make the charge terminal Ti as large as practical. The size of the charge terminal Ti should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal. The self-capacitance C _p and C _d of the charge and compensation terminals Ti and T ₂ respectively, can be determined using, for example, equation (24).

[0142] As can be seen in FIG. 15B, a resonant circuit is formed by at least a portion of the inductance of the coil 215, the self-capacitance C _d of the compensation terminal T ₂ , and the ground return resistance R _d associated with the compensation terminal T ₂ . The parallel resonance can be established by adjusting the voltage V ₃ applied to the compensation terminal T ₂ (e.g., by adjusting a tap 233 position on the coil 215) or by adjusting the height and/or size of the compensation terminal T ₂ to adjust C _d . The position of the coil tap 233 can be adjusted for parallel resonance, which will result in the ground current through the ground stake 218 and through the ammeter 236 reaching a maximum point. After parallel resonance of the compensation terminal T ₂ has been established, the position of the tap 227 for the AC source 212 can be adjusted to the 50Ω point on the coil 215.

[0143] Voltage V ₂ from the coil 215 can be applied to the charge terminal ΤΊ , and the position of tap 224 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 (W _Rx ) at the Hankel crossover distance (R _x ). The position of the coil tap 224 can be adjusted until this operating point is reached, which results in the ground current through the ammeter 236 increasing to a maximum. At this point, the resultant fields excited by the guided surface waveguide probe 200d are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium 203, resulting in the launching of a guided surface wave along the surface of the lossy conducting medium 203. This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe 200.

[0144] Resonance of the circuit including the compensation terminal T ₂ may change with the attachment of the charge terminal Ti and/or with adjustment of the voltage applied to the charge terminal Ti through tap 224. While adjusting the compensation terminal circuit for resonance aids the subsequent adjustment of the charge terminal connection, it is not necessary to establish the guided surface wave tilt (W _Rx ) at the Hankel crossover distance (R _x ). The system may be further adjusted to improve coupling by iteratively adjusting the position of the tap 227 for the AC source 212 to be at the 50Ω point on the coil 215 and adjusting the position of tap 233 to maximize the ground current through the ammeter 236. Resonance of the circuit including the compensation terminal T ₂ may drift as the positions of taps 227 and 233 are adjusted, or when other components are attached to the coil 215.

[0145] In other implementations, the voltage V ₂ from the coil 215 can be applied to the charge terminal Ti , and the position of tap 233 can be adjusted such that the phase (Φ) of the total effective height (h _TE ) approximately equals the angle (Ψ) of the guided surface wave tilt at R _x . The position of the coil tap 224 can be adjusted until the operating point is reached, resulting in the ground current through the ammeter 236 substantially reaching a maximum. The resultant fields are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium 203, and a guided surface wave is launched along the surface of the lossy conducting medium 203. This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe 200. The system may be further adjusted to improve coupling by iteratively adjusting the position of the tap 227 for the AC source 212 to be at the 50Ω point on the coil 215 and adjusting the position of tap 224 and/or 233 to maximize the ground current through the ammeter 236. [0146] Referring back to FIG. 12, operation of a guided surface waveguide probe 200 may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, a probe control system 230 can be used to control the feed network 209 and/or positioning of the charge terminal ΤΊ and/or compensation terminal T ₂ to control the operation of the guided surface waveguide probe 200. Operational conditions can include, but are not limited to, variations in the

characteristics of the lossy conducting medium 203 (e.g. , conductivity σ and relative permittivity e _r ), variations in field strength and/or variations in loading of the guided surface waveguide probe 200. As can be seen from Equations (41) - (44), the index of refraction (n), the complex Brewster angle (θ _{ί Β} and , the wave tilt (|ν |β ^;ψ ) and the complex effective height (h _eff = h _p e ^i<p ) can be affected by changes in soil conductivity and permittivity resulting from, e.g. , weather conditions.

[0147] Equipment such as, e.g. , conductivity measurement probes, permittivity sensors, ground parameter meters, field meters, current monitors and/or load receivers can be used to monitor for changes in the operational conditions and provide information about current operational conditions to the probe control system 230. The probe control system 230 can then make one or more adjustments to the guided surface waveguide probe 200 to maintain specified operational conditions for the guided surface waveguide probe 200. For instance, as the moisture and temperature vary, the conductivity of the soil will also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe 200. Generally, it would be desirable to monitor the conductivity and/or permittivity at or about the Hankel crossover distance R _x for the operational frequency. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations (e.g., in each quadrant) around the guided surface waveguide probe 200.

[0148] With reference then to FIG. 16, shown is an example of a guided surface waveguide probe 200e that includes a charge terminal Ti and a charge terminal T ₂ that are arranged along a vertical axis z. The guided surface waveguide probe 200e is disposed above a lossy conducting medium 203, which makes up Region 1. In addition, a second medium 206 shares a boundary interface with the lossy conducting medium 203 and makes up Region 2. The charge terminals and T ₂ are positioned over the lossy conducting medium 203. The charge terminal is positioned at height and the charge terminal T ₂ is positioned directly below along the vertical axis z at height H ₂ , where H ₂ is less than The height h of the transmission structure presented by the guided surface waveguide probe 200e is h = Hi— H ₂ . The guided surface waveguide probe 200e includes a probe feed network 209 that couples an excitation source 212 to the charge terminals Ti and T ₂ . [0149] The charge terminals ΤΊ and/or T ₂ include a conductive mass that can hold an electrical charge, which may be sized to hold as much charge as practically possible. The charge terminal has a self-capacitance Ci, and the charge terminal T ₂ has a self- capacitance C ₂ , which can be determined using, for example, equation (24). By virtue of the placement of the charge terminal Ti directly above the charge terminal T ₂ , a mutual capacitance C _M is created between the charge terminals Ti and T ₂ . Note that the charge terminals Ti and T ₂ need not be identical, but each can have a separate size and shape, and can include different conducting materials. Ultimately, the field strength of a guided surface wave launched by a guided surface waveguide probe 200e is directly proportional to the quantity of charge on the terminal Ti . The charge Qi is, in turn, proportional to the self- capacitance Ci associated with the charge terminal Ti since Qi = CiV, where V is the voltage imposed on the charge terminal Ti .

[0150] When properly adjusted to operate at a predefined operating frequency, the guided surface waveguide probe 200e generates a guided surface wave along the surface of the lossy conducting medium 203. The excitation source 212 can generate electrical energy at the predefined frequency that is applied to the guided surface waveguide probe 200e to excite the structure. When the electromagnetic fields generated by the guided surface waveguide probe 200e are substantially mode-matched with the lossy conducting medium 203, the electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle that results in little or no reflection. Thus, the surface waveguide probe 200e does not produce a radiated wave, but launches a guided surface traveling wave along the surface of a lossy conducting medium 203. The energy from the excitation source 212 can be transmitted as Zenneck surface currents to one or more receivers that are located within an effective transmission range of the guided surface waveguide probe 200e.

[0151] One can determine asymptotes of the radial Zenneck surface current / _p (p) on the surface of the lossy conducting medium 203 to be Λ(ρ) close-in and / ₂ (p) far-out, where

Close-in (p < λ/8): ;» ~ Λ = ¾ + ^ ^(¾) ^ ^(¾) , and (90)

Far-out (p » λ/8): ; _p (p) ~ _h = ^ x ^ x (91) where l _x is the conduction current feeding the charge Qi on the first charge terminal Ti, and I ₂ is the conduction current feeding the charge Q ₂ on the second charge terminal T ₂ . The charge Qi on the upper charge terminal Ti is determined by Qi = C ₁ V ₁ , where Ci is the isolated capacitance of the charge terminal T^ Note that there is a third component to J ₁ set forth above given by which follows from the Leontovich boundary condition and is the radial current contribution in the lossy conducting medium 203 pumped by the quasi- static field of the elevated oscillating charge on the first charge terminal Qi. The quantity Z _p = )ωμ ₀ /γ _β is the radial impedance of the lossy conducting medium, where y _e =

Ο ^' ωμ^ _! - ω ² μ _ί ε _ί ) ¹ ' ² .

[0152] The asymptotes representing the radial current close-in and far-out as set forth by equations (90) and (91) are complex quantities. According to various embodiments, a physical surface current J(p), is synthesized to match as close as possible the current asymptotes in magnitude and phase. That is to say close-in, |J(p) | is to be tangent to l/ , and far-out |J(p) | is to be tangent to \J ₂ \ . Also, according to the various embodiments, the phase of J(p) should transition from the phase of J _t close-in to the phase of ] ₂ far-out.

[0153] In order to match the guided surface wave mode at the site of transmission to launch a guided surface wave, the phase of the surface current \] ₂ |far-out should differ from the phase of the surface current \J _t \ close-in by the propagation phase corresponding to ρ-ίβ(ρ ^ζ -ρι p| _{us a} constant of approximately 45 degrees or 225 degrees. This is because there are two roots for , one near ττ/4 and one near 5ττ/4. The properly adjusted synthetic radial surface current is

; _p (p, 0, o) = ^ H ₁ ⁽²⁾ (- ₇ y _P ). (92)

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

¾ = ^Z ? ^{e "U2Z} H?HYP), (93) EP ^{= J} {S) ^e~U2Z tfi ⁽²⁾ (- P). ^and ( ⁹⁴ )

^ ^{= 1} ? (≤) ^" ^{2Ζ /} ο ⁽²⁾ (-7> )· 05) Thus, the difference in phase between the surface current \] ₂ \ far-out and the surface current \J _t \ close-in for the guided surface wave mode that is to be matched is due to the

characteristics of the Hankel functions in equations (93)-(95), which are consistent with equations (1)-(3). It is of significance to recognize that the fields expressed by equations (1)- (6) and (17) and equations (92)-(95) have the nature of a transmission line mode bound to a lossy interface, not radiation fields that are associated with groundwave propagation.

[0154] In order to obtain the appropriate voltage magnitudes and phases for a given design of a guided surface waveguide probe 200e at a given location, an iterative approach may be used. Specifically, analysis may be performed of a given excitation and

configuration of a guided surface waveguide probe 200e taking into account the feed currents to the terminals Ti and T ₂ , the charges on the charge terminals Ti and T ₂ , and their images in the lossy conducting medium 203 in order to determine the radial surface current density generated. This process may be performed iteratively until an optimal configuration and excitation for a given guided surface waveguide probe 200e is determined based on desired parameters. To aid in determining whether a given guided surface waveguide probe 200e is operating at an optimal level, a guided field strength curve 103 (FIG. 1) may be generated using equations (1)-(12) based on values for the conductivity of Region 1 (σ- and the permittivity of Region 1 (ε _χ ) at the location of the guided surface waveguide probe 200e. Such a guided field strength curve 103 can provide a benchmark for operation such that measured field strengths can be compared with the magnitudes indicated by the guided field strength curve 103 to determine if optimal transmission has been achieved.

[0155] In order to arrive at an optimized condition, various parameters associated with the guided surface waveguide probe 200e may be adjusted. One parameter that may be varied to adjust the guided surface waveguide probe 200e is the height of one or both of the charge terminals Ti and/or T ₂ relative to the surface of the lossy conducting medium 203. In addition, the distance or spacing between the charge terminals Ti and T ₂ may also be adjusted. In doing so, one may minimize or otherwise alter the mutual capacitance C _M or any bound capacitances between the charge terminals Ti and T ₂ and the lossy conducting medium 203 as can be appreciated. The size of the respective charge terminals Ti and/or T ₂ can also be adjusted. By changing the size of the charge terminals Ti and/or T ₂ , one will alter the respective self-capacitances Ci and/or C ₂ , and the mutual capacitance C _M as can be appreciated.

[0156] Still further, another parameter that can be adjusted is the probe feed network 209 associated with the guided surface waveguide probe 200e. This may be accomplished by adjusting the size of the inductive and/or capacitive reactances that make up the probe feed network 209. For example, where such inductive reactances comprise coils, the number of turns on such coils may be adjusted. Ultimately, the adjustments to the probe feed network 209 can be made to alter the electrical length of the probe feed network 209, thereby affecting the voltage magnitudes and phases on the charge terminals Ti and T ₂ .

[0157] Note that the iterations of transmission performed by making the various adjustments may be implemented by using computer models or by adjusting physical structures as can be appreciated. By making the above adjustments, one can create corresponding "close-in" surface current J _t and "far-out" surface current J ₂ that approximate the same currents J(p) of the guided surface wave mode specified in Equations (90) and (91) set forth above. In doing so, the resulting electromagnetic fields would be substantially or approximately mode-matched to a guided surface wave mode on the surface of the lossy conducting medium 203.

[0158] While not shown in the example of FIG. 16, operation of the guided surface waveguide probe 200e may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe 200. For example, a probe control system 230 shown in FIG. 12 can be used to control the feed network 209 and/or positioning and/or size of the charge terminals ΤΊ and/or T ₂ to control the operation of the guided surface waveguide probe 200e. Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium 203 (e.g. , conductivity σ and relative permittivity e _r ), variations in field strength and/or variations in loading of the guided surface waveguide probe 200e.

[0159] Referring now to FIG. 17, shown is an example of the guided surface waveguide probe 200e of FIG. 16, denoted herein as guided surface waveguide probe 200f. The guided surface waveguide probe 200f includes the charge terminals ΤΊ and T ₂ that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203 (e.g. , the Earth). The second medium 206 is above the lossy conducting medium 203. The charge terminal Ti has a self-capacitance Ci, and the charge terminal T ₂ has a self-capacitance C ₂ . During operation, charges Qi and Q ₂ are imposed on the charge terminals Ti and T ₂ , respectively, depending on the voltages applied to the charge terminals Ti and T ₂ at any given instant. A mutual capacitance C _M may exist between the charge terminals Ti and T ₂ depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals Ti and T ₂ and the lossy conducting medium 203 depending on the heights of the respective charge terminals Ti and T ₂ with respect to the lossy conducting medium 203.

[0160] The guided surface waveguide probe 200f includes a probe feed network 209 that comprises an inductive impedance comprising a coil l_i _a having a pair of leads that are coupled to respective ones of the charge terminals Ti and T ₂ . In one embodiment, the coil l_i _a is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the guided surface waveguide probe 200f.

[0161] While the electrical length of the coil l_i _a is specified as approximately one-half (1/2) the wavelength at the operating frequency, it is understood that the coil l_i _a may be specified with an electrical length at other values. According to one embodiment, the fact that the coil l_i _a has an electrical length of approximately one-half the wavelength at the operating frequency provides for an advantage in that a maximum voltage differential is created on the charge terminals and T ₂ . Nonetheless, the length or diameter of the coil L _a may be increased or decreased when adjusting the guided surface waveguide probe 200f to obtain optimal excitation of a guided surface wave mode. Adjustment of the coil length may be provided by taps located at one or both ends of the coil. In other

embodiments, it may be the case that the inductive impedance is specified to have an electrical length that is significantly less than or greater than ½ the wavelength at the operating frequency of the guided surface waveguide probe 200f. [0162] The excitation source 212 can be coupled to the probe feed network 209 by way of magnetic coupling. Specifically, the excitation source 212 is coupled to a coil L _P that is inductively coupled to the coil L _a . This may be done by link coupling, a tapped coil, a variable reactance, or other coupling approach as can be appreciated. To this end, the coil L _P acts as a primary, and the coil l_i _a acts as a secondary as can be appreciated.

[0163] In order to adjust the guided surface waveguide probe 200f for the transmission of a desired guided surface wave, the heights of the respective charge terminals Ti and T ₂ may be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals Ti and T ₂ may be altered. In addition, the size of the coil L may be altered by adding or eliminating turns or by changing some other dimension of the coil L . The coil l_i _a can also include one or more taps for adjusting the electrical length as shown in FIG. 17. The position of a tap connected to either charge terminal Ti or T ₂ can also be adjusted.

[0164] Referring next to FIGS. 18A, 18B, 18C and 19, shown are examples of generalized receive circuits for using the surface-guided waves in wireless power delivery systems. FIGS. 18A and 18B-18C include a linear probe 303 and a tuned resonator 306, respectively. FIG. 19 is a magnetic coil 309 according to various embodiments of the present disclosure. According to various embodiments, each one of the linear probe 303, the tuned resonator 306, and the magnetic coil 309 may be employed to receive power transmitted in the form of a guided surface wave on the surface of a lossy conducting medium 203 according to various embodiments. As mentioned above, in one embodiment the lossy conducting medium 203 comprises a terrestrial medium (or Earth).

[0165] With specific reference to FIG. 18A, the open-circuit terminal voltage at the output terminals 312 of the linear probe 303 depends upon the effective height of the linear probe 303. To this end, the terminal point voltage may be calculated as

V _T = ^e Einc - dl, (96) where E _inc is the strength of the incident electric field induced on the linear probe 303 in Volts per meter, dl is an element of integration along the direction of the linear probe 303, and h _e is the effective height of the linear probe 303. An electrical load 315 is coupled to the output terminals 312 through an impedance matching network 318.

[0166] When the linear probe 303 is subjected to a guided surface wave as described above, a voltage is developed across the output terminals 312 that may be applied to the electrical load 315 through a conjugate impedance matching network 318 as the case may be. In order to facilitate the flow of power to the electrical load 315, the electrical load 315 should be substantially impedance matched to the linear probe 303 as will be described below. [0167] Referring to FIG. 18B, a ground current excited coil 306a possessing a phase shift equal to the wave tilt of the guided surface wave includes a charge terminal T _R that is elevated (or suspended) above the lossy conducting medium 203. The charge terminal T _R has a self-capacitance C _R . In addition, there may also be a bound capacitance (not shown) between the charge terminal T _R and the lossy conducting medium 203 depending on the height of the charge terminal T _R above the lossy conducting medium 203. The bound capacitance should preferably be minimized as much as is practicable, although this may not be entirely necessary in every instance.

[0168] The tuned resonator 306a also includes a receiver network comprising a coil L _R having a phase shift Φ. One end of the coil L _R is coupled to the charge terminal T _R , and the other end of the coil L _R is coupled to the lossy conducting medium 203. The receiver network can include a vertical supply line conductor that couples the coil L _R to the charge terminal T _R . To this end, the coil L _R (which may also be referred to as tuned resonator L _R - C _R ) comprises a series-adjusted resonator as the charge terminal C _R and the coil L _R are situated in series. The phase delay of the coil L _R can be adjusted by changing the size and/or height of the charge terminal T _R , and/or adjusting the size of the coil L _R so that the phase Φ of the structure is made substantially equal to the angle of the wave tilt Ψ. The phase delay of the vertical supply line can also be adjusted by, e.g., changing length of the conductor.

[0169] For example, the reactance presented by the self-capacitance C _R is calculated as l/ wC _s . Note that the total capacitance of the structure 306a may also include capacitance between the charge terminal T _R and the lossy conducting medium 203, where the total capacitance of the structure 306a 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 203 as previously discussed.

[0170] The inductive reactance presented by a discrete-element coil L _R may be calculated as y ^' coL, where L is the lumped-element inductance of the coil L _R . If the coil L _R is a distributed element, its equivalent terminal-point inductive reactance may be determined by conventional approaches. To tune the structure 306a, one would make adjustments so that the phase delay is equal to the wave tilt for the purpose of mode-matching to the surface waveguide at the frequency of operation. Under this condition, the receiving structure may be considered to be "mode-matched" with the surface waveguide. A transformer link around the structure and/or an impedance matching network 324 may be inserted between the probe and the electrical load 327 in order to couple power to the load. Inserting the impedance matching network 324 between the probe terminals 321 and the electrical load 327 can effect a conjugate-match condition for maximum power transfer to the electrical load 327.

[0171] When placed in the presence of surface currents at the operating frequencies power will be delivered from the surface guided wave to the electrical load 327. To this end, an electrical load 327 may be coupled to the structure 306a by way of magnetic coupling, capacitive coupling, or conductive (direct tap) coupling. The elements of the coupling network may be lumped components or distributed elements as can be appreciated.

[0172] In the embodiment shown in FIG. 18B, magnetic coupling is employed where a coil l_s is positioned as a secondary relative to the coil L _R that acts as a transformer primary. The coil L _s may be link-coupled to the coil L _R by geometrically winding it around the same core structure and adjusting the coupled magnetic flux as can be appreciated. In addition, while the receiving structure 306a comprises a series-tuned resonator, a parallel-tuned resonator or even a distributed-element resonator of the appropriate phase delay may also be used.

[0173] While a receiving structure immersed in an electromagnetic field may couple energy from the field, it can be appreciated that polarization-matched structures work best by maximizing the coupling, and conventional rules for probe-coupling to waveguide modes should be observed. For example, a TE ₂₀ (transverse electric mode) waveguide probe may be optimal for extracting energy from a conventional waveguide excited in the TE ₂₀ mode. Similarly, in these cases, a mode-matched and phase-matched receiving structure can be optimized for coupling power from a surface-guided wave. The guided surface wave excited by a guided surface waveguide probe 200 on the surface of the lossy conducting medium 203 can be considered a waveguide mode of an open waveguide. Excluding waveguide losses, the source energy can be completely recovered. Useful receiving structures may be E-field coupled, H-field coupled, or surface-current excited.

[0174] The receiving structure can be adjusted to increase or maximize coupling with the guided surface wave based upon the local characteristics of the lossy conducting medium 203 in the vicinity of the receiving structure. To accomplish this, the phase delay (Φ) of the receiving structure can be adjusted to match the angle (Ψ) of the wave tilt of the surface traveling wave at the receiving structure. If configured appropriately, the receiving structure may then be tuned for resonance with respect to the perfectly conducting image ground plane at complex depth z = -d/2.

[0175] For example, consider a receiving structure comprising the tuned resonator 306a of FIG. 18B, including a coil L _R and a vertical supply line connected between the coil L _R and a charge terminal T _R . With the charge terminal T _R positioned at a defined height above the lossy conducting medium 203, the total phase shift Φ of the coil L _R and vertical supply line can be matched with the angle (Ψ) of the wave tilt at the location of the tuned resonator 306a. From Equation (22), it can be seen that the wave tilt asymptotically passes to

where e _r comprises the relative permittivity and σ ₁ is the conductivity of the lossy conducting medium 203 at the location of the receiving structure, ε ₀ is the permittivity of free space, and ω = 2nf, where is the frequency of excitation. Thus, the wave tilt angle (Ψ) can be determined from Equation (97).

[0176] The total phase shift (Φ = 0 _C + 0 _y ) of the tuned resonator 306a includes both the phase delay (0 _C ) through the coil L _R and the phase delay of the vertical supply line (0 _y ). The spatial phase delay along the conductor length l _w of the vertical supply line can be given by e _y = _w l _w , where ? _w is the propagation phase constant for the vertical supply line

conductor. The phase delay due to the coil (or helical delay line) is 0 _C = β _ρ 1 _£ , with a physical length of l _c and a propagation factor of

where V _r is the velocity factor on the structure, λ ₀ is the wavelength at the supplied frequency, and λ _ρ is the propagation wavelength resulting from the velocity factor V _f . One or both of the phase delays (0 _C + 0 _y ) can be adjusted to match the phase shift Φ to the angle (Ψ) of the wave tilt. For example, a tap position may be adjusted on the coil L _R of FIG. 18B to adjust the coil phase delay (0 _C ) to match the total phase shift to the wave tilt angle (Φ = Ψ). For example, a portion of the coil can be bypassed by the tap connection as illustrated in FIG. 18B. The vertical supply line conductor can also be connected to the coil L _R via a tap, whose position on the coil may be adjusted to match the total phase shift to the angle of the wave tilt.

[0177] Once the phase delay (Φ) of the tuned resonator 306a has been adjusted, the impedance of the charge terminal T _R can then be adjusted to tune to resonance with respect to the perfectly conducting image ground plane at complex depth z = -d/2. This can be accomplished by adjusting the capacitance of the charge terminal Ti without changing the traveling wave phase delays of the coil L _R and vertical supply line. The adjustments are similar to those described with respect to FIGS. 9A and 9B.

[0178] The impedance seen "looking down" into the lossy conducting medium 203 to the complex image plane is given by:

Ztn = ^R in + J ^' Xin = ?o tanh(/jff ₀ (d/2)) , (99) where β ₀ = ω^μ ₀ ε ₀ . For vertically polarized sources over the Earth, the depth of the complex image plane can be given by:

d/2 « 1/ _Λ /7 ^' ωμ ₁ σ ₁ — ω ² μ ₁ ε ₁ , (100) where μ ₁ is the permeability of the lossy conducting medium 203 and ε ₁ = ε _Γ ε ₀ .

[0179] At the base of the tuned resonator 306a, the impedance seen "looking up" into the receiving structure is Z _T = Z _BASE as illustrated in FIG. 9A. With a terminal impedance of:

ZR =— , (101 ) where C _R is the self-capacitance of the charge terminal T _R , the impedance seen "looking up" into the vertical supply line conductor of the tuned resonator 306a is given by:

z _ _z Z _R +Z _w tanhg _w h _w ) _ Z _R +Z _W tanh(jg _y )

2 ^W Z _W +Z _R tanh(7/S _w ft _w ) ^w Z _W +Z _R tanh(y<9 _y ) ' ^ ' and the impedance seen "looking up" into the coil L _R of the tuned resonator 306a is given by:

_ Z _Z +Z _R tanhQe _c )

~ ε _{Ζβ+¾ tanh06c)} ^{■ l} ' ^{U J}

By matching the reactive component (X _IN ) seen "looking down" into the lossy conducting medium 203 with the reactive component (X _BASE ) seen "looking up" into the tuned resonator 306a, the coupling into the guided surface waveguide mode may be maximized.

[0180] Referring next to FIG. 18C, shown is an example of a tuned resonator 306b that does not include a charge terminal T _R at the top of the receiving structure. In this

embodiment, the tuned resonator 306b does not include a vertical supply line coupled between the coil L _R and the charge terminal T _R . Thus, the total phase shift (Φ) of the tuned resonator 306b includes only the phase delay (0 _C ) through the coil L _R . As with the tuned resonator 306a of FIG. 18B, the coil phase delay 0 _c can be adjusted to match the angle (Ψ) of the wave tilt determined from Equation (97), which results in Φ = Ψ. While power extraction is possible with the receiving structure coupled into the surface waveguide mode, it is difficult to adjust the receiving structure to maximize coupling with the guided surface wave without the variable reactive load provided by the charge terminal T _R .

[0181] Referring to FIG. 18D, shown is a flow chart 180 illustrating an example of adjusting a receiving structure to substantially mode-match to a guided surface waveguide mode on the surface of the lossy conducting medium 203. Beginning with 181 , if the receiving structure includes a charge terminal T _R (e.g. , of the tuned resonator 306a of FIG. 18B), then the charge terminal T _R is positioned at a defined height above a lossy conducting medium 203 at 184. As the surface guided wave has been established by a guided surface waveguide probe 200, the physical height (h _p ) of the charge terminal T _R may be below that of the effective height. The physical height may be selected to reduce or minimize the bound charge on the charge terminal T _R (e.g. , four times the spherical diameter of the charge terminal). If the receiving structure does not include a charge terminal T _R (e.g., of the tuned resonator 306b of FIG. 18C), then the flow proceeds to 187.

[0182] At 187, the electrical phase delay Φ of the receiving structure is matched to the complex wave tilt angle Ψ defined by the local characteristics of the lossy conducting medium 203. The phase delay (0 _C ) of the helical coil and/or the phase delay (0 _y ) of the vertical supply line can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (W). The angle (Ψ) of the wave tilt can be determined from Equation (86). The electrical phase Φ can then be matched to the angle of the wave tilt. For example, the electrical phase delay Φ = 0 _C + e _y can be adjusted by varying the geometrical parameters of the coil L _R and/or the length (or height) of the vertical supply line conductor.

[0183] Next at 190, the load impedance of the charge terminal T _R can be tuned to resonate the equivalent image plane model of the tuned resonator 306a. The depth (d/2) of the conducting image ground plane 139 (FIG. 9A) below the receiving structure can be determined using Equation (100) and the values of the lossy conducting medium 203 (e.g., the Earth) at the receiving structure, which can be locally measured. Using that complex depth, the phase shift (θ _α ) between the image ground plane 139 and the physical boundary 136 (FIG. 9A) of the lossy conducting medium 203 can be determined using θ _α = β ₀ d/2. The impedance (Z _in ) as seen "looking down" into the lossy conducting medium 203 can then be determined using Equation (99). This resonance relationship can be considered to maximize coupling with the guided surface waves.

[0184] Based upon the adjusted parameters of the coil L _R and the length of the vertical supply line conductor, the velocity factor, phase delay, and impedance of the coil L _R and vertical supply line can be determined. In addition, the self-capacitance (C _R ) of the charge terminal T _R can be determined using, e.g., Equation (24). The propagation factor (β _ρ ) of the coil L _R can be determined using Equation (98), and the propagation phase constant ( ? _w ) for the vertical supply line can be determined using Equation (49). Using the self-capacitance and the determined values of the coil L _R and vertical supply line, the impedance (Z _base ) of the tuned resonator 306a as seen "looking up" into the coil L _R can be determined using Equations (101), (102), and (103).

[0185] The equivalent image plane model of FIG. 9A also applies to the tuned resonator 306a of FIG. 18B. The tuned resonator 306a can be tuned to resonance with respect to the complex image plane by adjusting the load impedance Z _R of the charge terminal T _R such that the reactance component X _base of Z _base cancels out the reactance component of X _in of Z _in , or X _base + X _in = 0. Thus, the impedance at the physical boundary 136 (FIG. 9A) "looking up" into the coil of the tuned resonator 306a is the conjugate of the impedance at the physical boundary 136 "looking down" into the lossy conducting medium 203. The load impedance Z _R can be adjusted by varying the capacitance (C _R ) of the charge terminal T _R without changing the electrical phase delay Φ = 0 _c + 0 _y seen by the charge terminal T _R . An iterative approach may be taken to tune the load impedance Z _R for resonance of the equivalent image plane model with respect to the conducting image ground plane 139. In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium 203 (e.g., Earth) can be improved and/or maximized.

[0186] Referring to FIG. 19, the magnetic coil 309 comprises a receive circuit that is coupled through an impedance matching network 333 to an electrical load 336. In order to facilitate reception and/or extraction of electrical power from a guided surface wave, the magnetic coil 309 may be positioned so that the magnetic flux of the guided surface wave, Η _φ , passes through the magnetic coil 309, thereby inducing a current in the magnetic coil 309 and producing a terminal point voltage at its output terminals 330. The magnetic flux of the guided surface wave coupled to a single turn coil is expressed by

T = ii _Acs r oH - MA (104) where T is the coupled magnetic flux, μ _ν is the effective relative permeability of the core of the magnetic coil 309, μ ₀ is the permeability of free space, H is the incident magnetic field strength vector, n is a unit vector normal to the cross-sectional area of the turns, and A _cs is the area enclosed by each loop. For an N-turn magnetic coil 309 oriented for maximum coupling to an incident magnetic field that is uniform over the cross-sectional area of the magnetic coil 309, the open-circuit induced voltage appearing at the output terminals 330 of the magnetic coil 309 is

^V = ^~N _t ^¾ -j«>WoNHA _cs , (105) where the variables are defined above. The magnetic coil 309 may be tuned to the guided surface wave frequency either as a distributed resonator or with an external capacitor across its output terminals 330, as the case may be, and then impedance-matched to an external electrical load 336 through a conjugate impedance matching network 333.

[0187] Assuming that the resulting circuit presented by the magnetic coil 309 and the electrical load 336 are properly adjusted and conjugate impedance matched, via impedance matching network 333, then the current induced in the magnetic coil 309 may be employed to optimally power the electrical load 336. The receive circuit presented by the magnetic coil 309 provides an advantage in that it does not have to be physically connected to the ground.

[0188] With reference to FIGS. 18A, 18B, 18C and 19, the receive circuits presented by the linear probe 303, the mode-matched structure 306, and the magnetic coil 309 each facilitate receiving electrical power transmitted from any one of the embodiments of guided surface waveguide probes 200 described above. To this end, the energy received may be used to supply power to an electrical load 315/327/336 via a conjugate matching network as can be appreciated. This contrasts with the signals that may be received in a receiver that were transmitted in the form of a radiated electromagnetic field. Such signals have very low available power, and receivers of such signals do not load the transmitters.

[0189] It is also characteristic of the present guided surface waves generated using the guided surface waveguide probes 200 described above that the receive circuits presented by the linear probe 303, the mode-matched structure 306, and the magnetic coil 309 will load the excitation source 212 (e.g., FIGS. 3, 12 and 16) that is applied to the guided surface waveguide probe 200, thereby generating the guided surface wave to which such receive circuits are subjected. This reflects the fact that the guided surface wave generated by a given guided surface waveguide probe 200 described above comprises a transmission line mode. By way of contrast, a power source that drives a radiating antenna that generates a radiated electromagnetic wave is not loaded by the receivers, regardless of the number of receivers employed.

[0190] Thus, together one or more guided surface waveguide probes 200 and one or more receive circuits in the form of the linear probe 303, the tuned mode-matched structure 306, and/or the magnetic coil 309 can make up a wireless distribution system. Given that the distance of transmission of a guided surface wave using a guided surface waveguide probe 200 as set forth above depends upon the frequency, it is possible that wireless power distribution can be achieved across wide areas and even globally.

[0191] The conventional wireless-power transmission/distribution systems extensively investigated today include "energy harvesting" from radiation fields and also sensor coupling to inductive or reactive near-fields. In contrast, the present wireless-power system does not waste power in the form of radiation which, if not intercepted, is lost forever. Nor is the presently disclosed wireless-power system limited to extremely short ranges as with conventional mutual-reactance coupled near-field systems. The wireless-power system disclosed herein probe-couples to the novel surface-guided transmission line mode, which is equivalent to delivering power to a load by a wave-guide or a load directly wired to the distant power generator. Not counting the power required to maintain transmission field strength plus that dissipated in the surface waveguide, which at extremely low frequencies is insignificant relative to the transmission losses in conventional high-tension power lines at 60 Hz, all of the generator power goes only to the desired electrical load. When the electrical load demand is terminated, the source power generation is relatively idle.

[0192] Referring next to FIGS. 20A-E, shown are examples of various schematic symbols that are used with reference to the discussion that follows. With specific reference to FIG. 20A, shown is a symbol that represents any one of the guided surface waveguide probes 200a, 200b, 200c, 200e, 200d, or 200f; or any variations thereof. In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface waveguide probe P. For the sake of simplicity in the following discussion, any reference to the guided surface waveguide probe P is a reference to any one of the guided surface waveguide probes 200a, 200b, 200c, 200e, 200d, or 200f; or variations thereof.

[0193] Similarly, with reference to FIG. 20B, shown is a symbol that represents a guided surface wave receive structure that may comprise any one of the linear probe 303 (FIG. 18A), the tuned resonator 306 (FIGS. 18B-18C), or the magnetic coil 309 (FIG. 19). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R. For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R is a reference to any one of the linear probe 303, the tuned resonator 306, or the magnetic coil 309; or variations thereof.

[0194] Further, with reference to FIG. 20C, shown is a symbol that specifically represents the linear probe 303 (FIG. 18A). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R _P . For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R _P is a reference to the linear probe 303 or variations thereof.

[0195] Further, with reference to FIG. 20D, shown is a symbol that specifically represents the tuned resonator 306 (FIGS. 18B-18C). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R _R . For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R _R is a reference to the tuned resonator 306 or variations thereof.

[0196] Further, with reference to FIG. 20E, shown is a symbol that specifically represents the magnetic coil 309 (FIG. 19). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R _M . For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R _M is a reference to the magnetic coil 309 or variations thereof.

[0197] Referring next to FIG. 21 , shown is an example of a hierarchical power distribution system 2100 according to various embodiments. The hierarchical power distribution system 2100 includes guided surface waveguide probes Pa, Pb... PN. According to various embodiments, each of the guided surface waveguide probes Pa... N may be employed to transmit electrical energy in the form of a guided surface wave on the surface of a lossy conducting medium 203 (FIG. 3). Any N number of guided surface waveguide probes Pa... N may be employed to transmit electrical energy in the form of a guided surface wave to one or more guided surface wave receive structures R (FIG. 20B), which supply the electrical energy to one or more electrical loads.

[0198] According to various embodiments, each of the guided surface waveguide probes Pa... N may transmit electrical energy at frequencies associated with one or more respective frequency sets 2103a, 2103b...2103N on a transmission spectrum 2109. In some embodiments, each of the guided surface waveguide probes Pa... N transmits electrical energy at frequencies within one of the respective frequency sets 2103a... N on the transmission spectrum 2109. The frequency sets 2103a... N may comprise frequencies that fall within respective frequency bands or may correspond to sets of discrete frequencies that do not fall within a single frequency band. Accordingly, each of the guided surface waveguide probes Pa... N may transmit electrical energy in the form of a guided surface wave at frequencies within a unique frequency set relative to other frequency sets

2103a... N. In alternative embodiments, each of the guided surface waveguide probes Pa... N may transmit electrical energy at frequencies within more than one of the respective frequency sets 2103a... N on the transmission spectrum 2109. In some embodiments, each of the guided surface waveguide probes Pa... N may transmit electrical energy at one or more discrete frequencies within one of the respective frequency sets 2103a... N. In other words, each of the guided surface waveguide probes Pa... N may transmit within a narrow frequency set within one of the respective frequency sets 2103a... N. Accordingly, each of the guided surface waveguide probes Pa... N may be configured to transmit electrical energy in the form of a guided surface wave at a unique frequency relative to all other frequencies in the transmission spectrum 2109.

[0199] Alternatively, multiple the guided surface waveguide probes Pa... N may be configured to transmit electrical energy at different discrete frequencies (narrow frequency sets) relative to each other within a same frequency set 2103a... N. Each of the guided surface waveguide probes Pa... N may be configured to transmit electrical energy in one or more transmission classes 2106a, 2106b...2106N. Each of the guided surface waveguide probes Pa... N may be assigned to transmit in one or more of the transmission classes 2106a... N based upon one or more loads to which the electrical energy may be supplied. For example, guided surface waveguide probe Pa may transmit electrical energy that is received by a load designated to receive energy transmitted via the first transmission class 2106a, while guided surface waveguide probe Pb may transmit electrical energy that is received by a load designated to receive energy transmitted via the second transmission class 2106b.

[0200] Each transmission class 2106a, 2106b...2106N corresponds to at least one of the respective frequency sets 2103a... N comprising at least one discrete frequency by which each of the guided surface waveguide probes Pa... N may transmit electrical energy.

According to various embodiments, each of the transmission classes 2106a... N is associated with a priority relative to one another for transmitting electrical energy in the hierarchical power distribution system 2100. The priority may be based upon the one or more loads that are being supplied with electrical energy. For example, a load that is designated to receive energy transmitted by the first transmission class 2106a may be classified as being more critical to receive electrical energy than a load designated to receive electrical energy transmitted by the second transmission class 2106b. Thus, in a situation where the hierarchical power distribution system 2100 must cease supplying energy to one of the aforementioned loads due to a loss of power generation capacity, the hierarchical power distribution system 2100 may cease supplying energy to the load designated to receive electrical energy transmitted by the second transmission class because it is classified as being less critical. Alternatively or additionally, the hierarchical power distribution system 2100 may reassign one or more guided surface wave receive structures R that are coupled to the load to a different one of the frequency sets 2103a... N to prevent overloading one or more power sources.

[0201] The hierarchical power distribution system 2100 may assign one or more specific frequencies and/or one or more of the frequency sets 2103a... N to one or more guided surface wave receive structures R to prevent overloading one or more power sources coupled to respective guided surface waveguide probes Pa... N. The specific frequencies and/or one or more of the frequency sets 2103a... N may be based upon one or more loads that need to receive electrical energy. The hierarchical power distribution system 2100 may identify a frequency set 2103a... N associated with a load, and accordingly determine one or more available discrete frequencies within the frequency set 2103a... N to assign to the one or more guided surface wave receive structures R.

[0202] The hierarchical power distribution system 2100 may determine that a power source coupled to one of the guided surface waveguide probes Pa... N is at risk of being overloaded. Therefore, the hierarchical power distribution system 2100 may prevent electrical energy from being supplied to one or more of the loads to prevent overloading one or more power sources. Alternatively or additionally, the hierarchical power distribution system 2100 may reassign one or more guided surface wave receive structures R to a different one of the frequency sets 2103a... N to prevent overloading the one or more power sources. For example, the hierarchical power distribution system 200 may communicate with a receive controller to instruct the one or more guided surface wave receive structures R to adjust to a frequency other than the one on which it is receiving electrical energy. In other embodiments, the hierarchical power distribution system 200 may communicate with a computer associated with one or more of the guided surface waveguide probes Pa... N and shut down transmission upon detection that a power source is being overloaded. In embodiments where one guided surface waveguide probe P is transmitting electrical energy being supplied to more than one load, the hierarchical power distribution system 200 may cease transmission of electrical energy to specific loads based upon one or more

characteristics associated with the load. A load may include for example, medical devices, transportation machines, buildings, or any other structure that may be supplied with electrical energy. In some embodiments, a range of sizes of the loads determines which of the transmission classes 2106a... N the loads belong to. The type of load may also determine which one or more frequencies within one or more of the respective frequency sets

2103a... N a guided surface waveguide probe Pa... N will be adjusted to when transmitting electrical energy.

[0203] Referring next to FIG. 22, shown is an example of a transmission spectrum 2109 which comprises one or more frequencies by which the guided surface waveguide probes Pa... N (FIG. 21) may transmit electrical energy in the form of a guided surface wave according to various embodiments. In this example, transmission spectrum 2109 includes critical load frequency set 2206, non-critical load frequency set 2209, and unused frequency sets 2203a, 2203b, and 2203c. According to various embodiments, loads may be classified as either critical or non-critical. Thus, to supply electrical energy to one or more loads that are classified as critical, a guided surface waveguide probe Pa... N may transmit electrical energy at one or more frequencies within the critical load frequency set 2206. For example, a guided surface waveguide probe Pa... N may transmit at a discrete frequency 2212 within the critical load frequency set 2206. As another example, a guided surface waveguide probe Pa... N may transmit at multiple discrete frequencies 2212 within the critical load frequency set 2206. As yet another example, multiple guided surface waveguide probes Pa... N may transmit at different discrete frequencies 2212 within the critical load frequency set 2206. Alternatively, to supply electrical energy to one or more loads that are classified as non-critical, a guided surface waveguide probe Pa... N may transmit at one or more discrete frequencies 2212 within the non-critical load frequency set 2209 to transmit electrical energy that will be supplied to a non-critical load.

[0204] In alternative embodiments, loads may be classified in varying degrees of criticality. For example, loads may be classified as high criticality loads, medium criticality loads, low criticality loads, or non-critical loads. Thus, the critical load frequency set 2206 may comprise several frequency sets 2103a... N (FIG. 21), such as a high critical load frequency set, a medium critical load frequency set, and a low critical load frequency set. Accordingly, each of the frequency sets 2103a... N may be reserved for a predefined degree of criticality of one or more loads. For example, a load that is classified as a high criticality load may comprise an emergency medical services critical load such as a load associated with an ambulance, a hospital, or a medical device. A load that is classified as a medium criticality load may be for example, an air conditioning load. A load that is classified as a low criticality load may be for example, a lighting fixture load.

[0205] The guided surface waveguide probes Pa... N may transmit electrical energy to one or more load at discrete frequencies 2212 within each of the frequency sets 2103a... N. In some embodiments, the discrete frequencies 2212 within one or more frequency sets 2103a... N are reserved for specific types of loads, such as loads within a specific size range or loads within a specific degree of criticality. In various embodiments, unused frequency sets 2203a 2203b, and 2203c comprise frequencies between frequency sets 2103a... N, such as the critical load frequency set 2206 and the non-critical load frequency set 2209. In some embodiments, the unused frequency sets 2203a, 2203b, 2203c, may become used if they are reserved. Each of the frequency sets 2103a... N (FIG. 21) may comprise one or more reserved frequency sets 2215. The reserved frequency sets 2215 may be dedicated to specific types of loads and/or specific transmission classes 2106a... N (FIG. 21).

[0206] As previously mentioned, the classification of loads may be determined by whether the loads are critical loads or non-critical loads. Criticality may be determined by a type of load that is being supplied with electrical energy. For example, a medically critical load, such as a pacemaker or a life support machine, may be classified as a critical load because if it is not supplied with electrical energy, a person's life or health may be put at risk. Therefore, it may be more important that the medically critical load be supplied with electrical energy compared with the importance of supplying electrical energy to another type of load. As another example, a critical transportation load, such as an emergency services vehicle such as ambulance or a fire truck, may be classified as a critical load because if it is not supplied with electrical energy, a person's life or health may be put at risk. As a contrary example, a load such as a smart watch, may be classified as non-critical because it is not likely that a person's life or health is at risk if the smart watch does not receive electrical energy.

[0207] The classification of loads may additionally or alternatively be determined by a range of sizes of the loads. For example, loads may be classified as small loads, medium loads, or large loads. As one example, a portable personal device may be classified as a small load and supplied with electrical energy transmitted from a guided surface waveguide probe Pa... N operating in one or more of the transmission classes 2106a... N (FIG. 21) corresponding to one or more frequencies within one or more of the frequency sets

2103a... N that is necessary for the portable personal device to receive electrical energy. Examples of portable personal devices include pacemakers, watches, mobile phones, mobile computing devices, etc. As another example, an air conditioning unit may be classified as a medium load and supplied with electrical energy transmitted from a guided surface waveguide probe Pa... N operating in one or more of the transmission classes 2106a... N (FIG. 21) corresponding to one or more frequencies within one or more of the frequency sets 2103a... N that is necessary for the air conditioning unit to receive electrical energy. Other examples of medium loads include household appliances such as refrigerators, water pumps, etc. As yet another example, an automobile may classified as a large load and supplied with electrical energy transmitted from a guided surface waveguide probe Pa... N operating in one or more of the transmission classes 2106a... N (FIG. 21) corresponding to one or more frequencies within one or more of the frequency sets

2103a... N that is necessary for the automobile to receive electrical energy.

[0208] The classification of loads may additionally or alternatively be determined by whether a load being supplied with electrical energy is mobile or non-mobile. For example, a personal portable device, such as a watch or a cell phone may be classified as a mobile load. As another example, a household appliance, such as a refrigerator, may be classified as a non-mobile load. The classification of loads may also be determined by the nature of the load. For example, loads may be classified as resistive, capacitive, inductive, or a combination thereof. As one example, a light bulb may be classified as a resistive load. As another example, an electric motor may classified as a capacitive load. As yet another example, a transformer may be classified as an inductive load. The nature of the load may also correspond to a level or degree of criticality. The loads may also be classified by a function of the load. For example, loads for lighting, loads for receptacles, and loads for power may all be classified differently based upon the function that each carries out. As one example, a light bulb may classified as a lighting load. As another example, an item that couples to typical power outlets may be classified as a receptacle load. In various embodiments, the loads may be classified by which consumer category they are associated. Consumer categories may include residential, commercial, industrial, and/or governmental or municipal categories. As one example, a household appliance may be classified as a residential load. As another example, an air conditioning unit for an office building may be classified as a commercial load. As another example, a transformer may be classified as an industrial load. As yet another example, loads for street lights may be classified as a municipal load. Additionally, loads may vary in criticality depending on what consumer category that they are associated with. In some embodiments, each consumer category corresponds to a different transmission class. For example, loads for street lights, loads for drainage purposes, and loads for water supply, which may all be classified as governmental or municipal loads may be in a transmission class with a higher priority than a load to power a residential air conditioning unit. [0209] Referring next to FIG. 23A, shown is an example of a transmission spectrum 2109 which comprises one or more frequencies by which the guided surface waveguide probes Pa... N (FIG. 21) may be transmitting electrical energy in the form of guided surface waves according to various embodiments. In this example, transmission spectrum 2109 includes class A load high frequency set 2303a, class B load medium frequency set 2303b, and class C load low frequency set 2303c. Additionally, transmission spectrum 2109 includes reserved frequency set 2306a, reserved frequency set 2306b, and discrete frequencies 2212 within the various frequency sets.

[0210] In the example, class A load high frequency set 2303a includes the highest frequency set within the transmission spectrum 2109. The class A load high frequency set 2303a may correspond to local frequency set 2315 depicted in FIG. 23B. Class B load medium frequency set 2303b includes frequencies within the frequency set that are higher than the frequencies in the class C load low frequency set and lower than the frequencies in the class A load high frequency set. The class B load medium frequency set 2303b may correspond to regional frequency set 2312 depicted in FIG. 23B. The class C load low frequency set 2303c may include the lowest frequency set within the transmission spectrum 2109. The class C load low frequency set 2303c may correspond to nationwide (or global) frequency set 2309 depicted in FIG. 23B.

[0211] According to various embodiments, guided surface wave receive structures R may be assigned one or more discrete frequencies 2212 (FIG. 22) within one or more frequency sets 2103a... N (FIG. 21) based upon one or more loads that are coupled to them. In this example, loads may be classified according to a range of frequencies by which electrical energy is transmitted. For example, Class A load high frequency set 2303a may be used to transmit electrical energy, for example, to loads for personal portable devices that receive electrical energy transmitted at high frequencies. Thus, a guided surface wave receive structure R (FIG. 20B) coupled to a load that receives electrical energy transmitted at a high frequency, may be assigned to one or more discrete frequencies 2212 within the class A load high frequency set 2303a. The class B load medium frequency set 2303b may be used to transmit electrical energy to, for example, loads that service predefined geographic regions, such as a utility plant. Thus, a guided surface wave receive structure R coupled to a load that receives electrical energy transmitted at medium frequency, may be assigned to one or more discrete frequencies 2212 within the class B load medium frequency set 2303b. The class C load low frequency set 2303c may be used to transmit electrical energy to types of loads that receive electrical energy via low frequencies. Thus, a guided surface wave receive structure R coupled to a load that receives electrical energy transmitted at low frequency, may be assigned to one or more discrete frequencies 2212 within the class C load low frequency set 2303c.

[0212] Referring next to FIG. 23B, shown is a representation of the United States and a local frequency set 2315, regional frequency set 2312, and nationwide frequency set 2309. In this example, each of the transmission classes 2106a... N (FIG. 21) may only transmit in a predefined range. Generally, the area of transmission is dictated by frequency and potentially other factors. For example, a guided surface waveguide probe Pa... N (FIG. 21) transmitting electrical energy in a transmission class 2106a... N corresponding to the local frequency set 2315 for example, may only be configured to transmit in a predefined local geographic area. As another example, a guided surface waveguide probe Pa... N (FIG. 21) transmitting electrical energy in a transmission class 2106a... N corresponding to the regional frequency set 2315 for example, may only be configured to transmit in a predefined regional geographic area. As yet another example, a guided surface waveguide probe Pa... N (FIG. 21) transmitting electrical energy in a transmission class 2106a... N

corresponding to the national frequency set 2315 for example, may only be configured to transmit in a predefined national geographic area.

[0213] Referring next to FIG. 24, shown is a networked hierarchical power distribution system 2400 according to various embodiments. The networked hierarchical power distribution system 2400 includes a transmission control system 2403 in data communication with one or more guided surface waveguide probes P and one or more receive controllers 2412 by way of network 2415. The network 2415 includes, for example, the Internet, intranets, extranets, wide area networks (WANs), local area networks (LANs), wired networks, wireless networks, or other suitable networks, etc., or any combination of two or more such networks.

[0214] The transmission control system 2403 may include, for example, one or more server computers or one or more server banks, located in a single installation or distributed among many different geographical locations. The transmission control system 2403 may also include one or more sensors and one or more data stores that store various data. The data stored in the data store, for example, is associated with the operation of the various functional entities described below.

[0215] The one or more of the guided surface waveguide probes P may operate in one or more of the transmission classes 2106a... N (FIG. 21) at one or more frequencies within one or more of the frequency sets 2103a... N (FIG. 21) to transmit electrical energy in the form of guided surface waves to one or more guided surface wave receive structures R (FIG. 20B), which supply the electrical energy to one or more loads 2409. Additionally or alternatively, multiple guided surface waveguide probes P may transmit electrical energy to one guided surface wave receive structure R. The one or more guided surface waveguide probes P may be coupled to one or more power sources 2418.

[0216] Each of the one or more guided surface wave receiver structures R may supply electrical energy to one or more loads 2409. The one or more guided surface wave receive structures R may be coupled to one or more receive controllers 2412, which are in data communication with the transmission control system 2403 by way of network 2415.

[0217] Next, a general description of the operation of the various components of the networked hierarchical power distribution system 2400 is provided. To begin, the transmission control system 2403 may receive a request for electrical energy from one or more receive controllers 2412 coupled to one or more guided surface waveguide receive structures R (FIG. 20B) by way of network 2415. The transmission control system 2403 may send a communication back to the one or more receive controllers 2412 requesting identification of one or more loads to which the electrical energy will be supplied.

Alternatively, the one or more receive controllers 2412 may identify one or more loads 2409 that the electrical energy will be supplied to in the request. Additionally, the one or more receive controllers 2412 may state the class of the one or more loads 2409 in the request. In some embodiments, the transmission control system 2403 accesses a record of each of the guided surface wave receive structures R and each of the one or more loads 2409 in a data store. For example, each of the guided surface wave receive structures R and each of the loads 2409 may already be registered in a record contained in the data store. Thus, in some embodiments, the transmission control system 2406 may already have an

identification of the one or more loads 2409 that will be supplied with electrical energy.

[0218] Next, the transmission control system 2403 may assign each of the guided surface wave receive structures R to one or more discrete frequencies 2212 (FIG. 22) within one or more frequency sets 2103a... N (FIG. 21). Additionally or alternatively, the transmission control system 2403 may assign each of the guided surface wave receive structures R to one or more discrete frequencies 2212 (FIG. 22) within one or more reserved frequency sets 2215 (FIG. 21). As previously discussed, one or more frequencies within the one or more frequency sets 2103a... N (FIG. 21) may correspond to one or more transmission classes 2106a... N (FIG. 21), which are determined based at least in part on the one or more loads 2409 that are being supplied with electrical energy. Thus, each of the guided surface wave receive structures R may be assigned to one or more discrete frequencies 2212 (FIG. 22) and/or one or more frequency sets 2103a... N (FIG. 21) based upon the one or more loads 2409 to which each of the guided surface wave receive structures R is coupled. The transmission control system 2403 may communicate the assignment to one or more receive controllers 2412 via a message over the network 2415. During assignment of the guided surface wave receive structures R, the transmission control system 2403 may evaluate whether any of the power sources 2418 coupled to any of the guided surface waveguide probes P is overloaded or in danger of being overloaded. In some embodiments, one or more of the guided surface waveguide probes P may already be transmitting electrical energy when a request is received by the transmission control system 2403. Thus, the transmission control system 2403 may assign one or more guided surface wave receive structures R to one or more guided surface waveguide probes P after determining that none of the power sources 2418 coupled to the one or more guided surface waveguide probes P is at risk of being overloaded or at overcapacity.

[0219] In other embodiments, the one or more guided surface wave receive structures R may not be coupled to one or more receive controllers 2412, and may not be in data communication with any other components of the networked hierarchical power distribution system 2400 over the network 2415. Thus, the transmission control system 2403 may assign one or more discrete frequencies 2212 (FIG. 22) and/or one or more respective frequency sets 2103a... N (FIG. 21) to the one or more guided surface wave receive structures R that are not in data communication with any other components of the networked hierarchical power distribution system 2400 over the network 2415. Alternatively or additionally, one or more guided surface waveguide probes P may be assigned to transmit guided surface waves to the one or more guided surface wave receive structures R that are not in data communication with any other components of the networked hierarchical power distribution system 2400 over the network 2415. There is a possibility that the one or more guided surface waveguide probes P may become overloaded. In response to a detection that one or more power sources 2418 is at risk of being overloaded or in response to detecting that the one or more guided surface waveguide probes P does not have requisite generation capacity, one or more guided surface waveguide probes P may be prevented from transmitting guided surface waves to the one or more guided surface wave receive structures R that are not in data communication with any other components of the networked hierarchical power distribution system 2400 over the network 2415.

[0220] The transmission control system 2403 may also generate a record of assignment of each of the guided surface wave receive structures R and each load 2409 being supplied with electrical energy. Each of the guided surface wave receive structures R and each load 2409 may be identified with a unique identification number that is stored in the data store. The record may also include the one or more discrete frequencies 2212 (FIG. 22), the frequency sets 2103a... N (FIG. 21), and the transmission classes 2106a... N (FIG. 21) that have been previously assigned and/or associated with each of the guided surface wave receive structures R and each load 2409. [0221] The transmission control system 2403 may also identify a priority of power distribution for each one of the guided surface wave receive structures R and each load 2409. As previously discussed, loads 2409 may be classified by a criticality or non-critical ity of the load 2409. Accordingly, priority may be identified by a classification of the load 2409 as a critical load or a non-critical load. For example, a load classified as a critical load may be assigned to one or more discrete frequencies 2212 (FIG. 22) with a higher priority for receiving electrical energy over one or more discrete frequencies 2212 (FIG. 22) than a non- critical load assigned to a different one or more discrete frequencies 2212. In some embodiments, loads 2409 are classified according to a degree of criticality. There may be numerous degrees of criticality classifications for loads 2409. Accordingly, there may be many levels of priority for receiving electrical energy among one or more discrete

frequencies 2212 (FIG. 22) and/or one or more frequency sets 2103a... N (FIG. 21). In one embodiment, each degree of criticality corresponds to a level of priority. For example, a highest criticality classification may correspond to a first priority. A medium criticality classification may correspond to a second priority. A low criticality classification may correspond to a third priority.

[0222] Additionally or alternatively, loads 2409 may be classified by a range of sizes. Thus, in various embodiments, priority for receiving electrical energy may be determined by a range of sizes of the loads 2409. In some embodiments the priority for receiving electrical energy may increase as the sizes of the loads 2409 increase. In other embodiments, the priority for receiving electrical energy may increase as the sizes of the loads 2409 decrease. Loads 2409 may also be classified as mobile or non-mobile. In some embodiments, loads 2409 that are mobile may have priority for receiving electrical energy over loads 2409 that are non-mobile, while in other embodiments loads 2409 that are non-mobile may have priority for receiving electrical energy over loads 2409 that are mobile.

[0223] The transmission control system 2403 may reassign one or more loads 2409 to another one or more discrete frequencies 2212 in response to detecting that one or more power sources 2418 is at risk of being overloaded or in response to detecting that one or more guided surface waveguide probes P does not have requisite generation capacity. Additionally or alternatively, the transmission control system 2403 may determine to cease transmitting electrical energy to one or more of the guided surface wave receive structures R and/or one or more of the loads 2409 in response to detecting that one or more power sources 2418 coupled to the one or more guided surface waveguide probes P is at risk of being overloaded. The transmission control system 2403 may select which of the one or more of the guided surface wave receive structures R and/or one or more of the loads 2409 will cease receiving electrical energy based upon one or more classifications. [0224] Referring next to FIG. 25, shown is a flowchart that provides one example of the operation of a portion of the transmission control system 2403 (FIG. 24) according to various embodiments. It is understood that the flowchart of FIG. 25 provides merely an example of the many different types of arrangements that may be employed to implement the operation of the portion of the transmission control system 2403 as described herein. As an alternative, the flowchart of FIG. 25 may be viewed as depicting an example of steps of a method implemented according to one or more embodiments.

[0225] Beginning with box 2503, the transmission control system 2403 associates each of the guided surface waveguide probes P with at least one frequency within a respective frequency set 2103a... N (FIG. 21) corresponding to one of the transmission classes

2106a... N (FIG. 21). Each of the guided surface waveguide probes P may transmit within multiple frequencies within one of the respective frequency sets 2103a... N (FIG. 21) or at a single frequency within one of the respective frequency sets 2103a... N (FIG. 21).

[0226] Next, at box 2506, the transmission control system 2403 receives at least one request for electrical energy over network 2415 (FIG. 24). The request may be received, for example, from one or more receive controllers 2412 (FIG. 24) coupled to one or more guided surface wave receive structures R (FIG. 20B). The request may identify, for example, one or more loads 2409 (FIG. 24).

[0227] Next, at box 2509, electrical energy in the form of at least one guided surface wave is transmitted along a terrestrial medium to at least one guided surface wave receive structure R. One or more loads 2409 may be coupled to the at least one guided surface wave receive structure R to receive the electrical energy.

[0228] Moving on to FIG. 26, shown is a flowchart that provides on example of assigning a frequency to a guided surface wave receive structure R (FIG. 20B) according to various embodiments. Beginning at box 2603, the transmission control system 2403 (FIG. 24) receives a request from a receive controller 2412 (FIG. 24) coupled to a guided surface wave receive structure R. The request may identify a load 2409 (FIG. 24) that is coupled to the guided surface wave receive structure R. Additionally, the request may identify a classification of the load 2409 and a magnitude of the load 2409. Alternatively, the request may identify multiple loads 2409 that are coupled to the guided surface wave receive structure R.

[0229] Next, at box 2606, the transmission control system 2403 determines all of the available frequencies for the guided surface wave receive structure R to be assigned. In making the determination, factors to consider may include for example, identifying a classification for the load 2409 coupled to the guided surface wave receive structure R, the size of the load 2409, or other factors. By identifying a classification of the load 2409, the transmission control system 2403 may determine a frequency set 2103a... N (FIG. 21) and a transmission class 2106a... N (FIG. 21) associated with the load 2409. Accordingly, the transmission control system 2403 can determine one or more discrete frequencies 2212 (FIG. 22) within the frequency set 2103a... N that are available for use (i.e. that have capacity).

[0230] Next, at box 2609, the transmission control system 2403 may assign one or more discrete frequencies 2212 to the guided surface wave receive structure R. The transmission control system 2403 may instruct the guided surface wave receive structure R to tune into the one or more discrete frequencies 2212 by communicating a message over a network 2415 (FIG. 24). For example, the guided surface wave receive structures R may be coupled to variable capacitors and/or inductors that may be electronically changed by varying the tap positions on a coil or by making other adjustments as can be appreciated.

[0231] Finally, at box 2612, the transmission control system 2403 instructs one or more guided surface waveguide probes P to transmit electrical energy in the form of at least one guided surface wave along a terrestrial medium to the guided surface wave receive structure R.

[0232] Referring next to FIG. 27, shown is a schematic block diagram of the

transmission control system 2403 (FIG. 24) and the receive controller 2412 (FIG. 24) according to an embodiment of the present disclosure. The transmission control system 2403 and the receive controller 2412 each include one or more computing devices 2703. Each computing device 2703 includes at least one processor circuit, for example, having a processor 2706 and a memory 2709, both of which are coupled to a local interface 2715. To this end, each computing device 2703 may comprise, for example, at least one server computer or like device. The local interface 2715 may comprise, for example, a data bus with an accompanying address/control bus or other bus structure as can be appreciated.

[0233] In the transmission control system 2403, stored in the memory 2709 are both data and several components that are executable by the processor 2706. In particular, stored in the memory 2709 and executable by the processor 2706 are the power distribution application 2715, and potentially other applications. Also stored in the memory 2709 may be a data store 2712 and other data. In addition, an operating system may be stored in the memory 2709 and executable by the processor 2706.

[0234] In the receive controller 2412, stored in the memory 2709 are both data and several components that are executable by the processor 2706. In particular, stored in the memory 2709 and executable by the processor 2706 are the receive structure control application 2718, and potentially other applications. Also stored in the memory 2709 may be a data store 2712 and other data. In addition, an operating system may be stored in the memory 2709 and executable by the processor 2706.

[0235] It is understood that there may be other applications that are stored in the memory 2709 and are executable by the processor 2706 as can be appreciated. Where any component discussed herein is implemented in the form of software, any one of a number of programming languages may be employed such as, for example, C, C++, C#, Objective C, Java ^® , JavaScript ^® , Perl, PHP, Visual Basic ^® , Python ^® , Ruby, Flash ^® , or other programming languages.

[0236] A number of software components are stored in the memory 2709 and are executable by the processor 2706. In this respect, the term "executable" means a program file that is in a form that can ultimately be run by the processor 2706. Examples of executable programs may be, for example, a compiled program that can be translated into machine code in a format that can be loaded into a random access portion of the memory 2709 and run by the processor 2706, source code that may be expressed in proper format such as object code that is capable of being loaded into a random access portion of the memory 2709 and executed by the processor 2706, or source code that may be interpreted by another executable program to generate instructions in a random access portion of the memory 2709 to be executed by the processor 2706, etc. An executable program may be stored in any portion or component of the memory 2709 including, for example, random access memory (RAM), read-only memory (ROM), hard drive, solid-state drive, USB flash drive, memory card, optical disc such as compact disc (CD) or digital versatile disc (DVD), floppy disk, magnetic tape, or other memory components.

[0237] The memory 2709 is defined herein as including both volatile and nonvolatile memory and data storage components. Volatile components are those that do not retain data values upon loss of power. Nonvolatile components are those that retain data upon a loss of power. Thus, the memory 2709 may comprise, for example, random access memory (RAM), read-only memory (ROM), hard disk drives, solid-state drives, USB flash drives, memory cards accessed via a memory card reader, floppy disks accessed via an associated floppy disk drive, optical discs accessed via an optical disc drive, magnetic tapes accessed via an appropriate tape drive, and/or other memory components, or a combination of any two or more of these memory components. In addition, the RAM may comprise, for example, static random access memory (SRAM), dynamic random access memory (DRAM), or magnetic random access memory (MRAM) and other such devices. The ROM may comprise, for example, a programmable read-only memory (PROM), an erasable

programmable read-only memory (EPROM), an electrically erasable programmable readonly memory (EEPROM), or other like memory device. [0238] Also, the processor 2706 may represent multiple processors 2706 and/or multiple processor cores and the memory 2709 may represent multiple memories 2709 that operate in parallel processing circuits, respectively. In such a case, the local interface 2715 may be an appropriate network that facilitates communication between any two of the multiple processors 2706, between any processor 2706 and any of the memories 2709, or between any two of the memories 2709, etc. The local interface 2715 may comprise additional systems designed to coordinate this communication, including, for example, performing load balancing. The processor 2706 may be of electrical or of some other available construction.

[0239] Although power distribution application 2715, receive structure control application 2718, and other various systems described herein may be embodied in software or code executed by general purpose hardware as discussed above, as an alternative the same may also be embodied in dedicated hardware or a combination of software/general purpose hardware and dedicated hardware. If embodied in dedicated hardware, each can be implemented as a circuit or state machine that employs any one of or a combination of a number of technologies. These technologies may include, but are not limited to, discrete logic circuits having logic gates for implementing various logic functions upon an application of one or more data signals, application specific integrated circuits (ASICs) having appropriate logic gates, field-programmable gate arrays (FPGAs), or other components, etc. Such technologies are generally well known by those skilled in the art and, consequently, are not described in detail herein.

[0240] The flowcharts of FIGS. 25 and 26 show the functionality and operation of an implementation of portions of the transmission control system 2403 and the receive controller 2412. If embodied in software, each block may represent a module, segment, or portion of code that comprises program instructions to implement the specified logical function(s). The program instructions may be embodied in the form of source code that comprises human-readable statements written in a programming language or machine code that comprises numerical instructions recognizable by a suitable execution system such as a processor 2706 in a computer system or other system. The machine code may be converted from the source code, etc. If embodied in hardware, each block may represent a circuit or a number of interconnected circuits to implement the specified logical function(s).

[0241] Although the flowcharts of FIGS. 25 and 26 show a specific order of execution, it is understood that the order of execution may differ from that which is depicted. For example, the order of execution of two or more blocks may be scrambled relative to the order shown. Also, two or more blocks shown in succession in FIGS. 25 and 26 may be executed concurrently or with partial concurrence. Further, in some embodiments, one or more of the blocks shown in FIGS. 25 and 26 may be skipped or omitted. In addition, any number of counters, state variables, warning semaphores, or messages might be added to the logical flow described herein, for purposes of enhanced utility, accounting, performance measurement, or providing troubleshooting aids, etc. It is understood that all such variations are within the scope of the present disclosure.

[0242] Also, any logic or application described herein, including the power distribution application 2715 and the receive structure control application 2718, that comprises software or code can be embodied in any non-transitory computer-readable medium for use by or in connection with an instruction execution system such as, for example, a processor 2706 in a computer system or other system. In this sense, the logic may comprise, for example, statements including instructions and declarations that can be fetched from the computer- readable medium and executed by the instruction execution system. In the context of the present disclosure, a "computer-readable medium" can be any medium that can contain, store, or maintain the logic or application described herein for use by or in connection with the instruction execution system.

[0243] The computer-readable medium can comprise any one of many physical media such as, for example, magnetic, optical, or semiconductor media. More specific examples of a suitable computer-readable medium would include, but are not limited to, magnetic tapes, magnetic floppy diskettes, magnetic hard drives, memory cards, solid-state drives, USB flash drives, or optical discs. Also, the computer-readable medium may be a random access memory (RAM) including, for example, static random access memory (SRAM) and dynamic random access memory (DRAM), or magnetic random access memory (MRAM). In addition, the computer-readable medium may be a read-only memory (ROM), a programmable readonly memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or other type of memory device.

[0244] Further, any logic or application described herein, including the power distribution application 2715 and the receive structure control application 2718, may be implemented and structured in a variety of ways. For example, one or more applications described may be implemented as modules or components of a single application. Further, one or more applications described herein may be executed in shared or separate computing devices or a combination thereof. For example, a plurality of the applications described herein may execute in the same computing device 2703, or in multiple computing devices in the same computing environment. Additionally, it is understood that terms such as "application," "service," "system," "engine," "module," and so on may be interchangeable and are not intended to be limiting. [0245] In addition to the foregoing, the various embodiments of the present disclosure include, but are not limited to, the embodiments set forth in the following clauses:

[0246] Clause 1. A method of distributing power within a hierarchical power distribution system from a plurality of guided surface waveguide probes to at least one guided surface wave receive structure comprising: associating each of the plurality of guided surface waveguide probes with a respective one of a plurality of frequency sets, each of the plurality of frequency sets corresponding to a transmission class among a plurality of transmission classes, wherein each of the plurality of guided surface waveguide probes is designated to transmit within a respective transmission class among the plurality of transmission classes; transmitting, by each of the plurality of guided surface waveguide probes, at least one guided surface wave along a terrestrial medium to the at least one guided surface wave receive structure; and supplying, by the at least one guided surface wave receive structure, electrical energy to at least one electrical load coupled to the at least one guided surface wave receive structure.

[0247] Clause 2. The method of clause 1 , wherein at least one characteristic is associated with each of the plurality of transmission classes.

[0248] Clause 3. The method of clause 2, wherein the at least one characteristic specifies a range of sizes of the at least one electrical load.

[0249] Clause 4. The method of clause 3, wherein the at least one electrical load encompasses at least one portable personal device.

[0250] Clause 5. The method of any of clauses 2-4, wherein the at least one characteristic comprises a criticality of the at least one electrical load.

[0251] Clause 6. The method of clause 5, wherein each of the plurality of frequency sets comprises a plurality of discrete transmission frequencies reserved for a predefined criticality type of the at least one electrical load.

[0252] Clause 7. The method of clause 6, wherein the criticality comprises a medical criticality associated with at least one medically critical load, wherein the at least one medically critical load is vital to a life of at least one entity associated with the at least one medically critical load.

[0253] Clause 8. The method of clause 7, wherein at least one of the plurality of frequency sets includes a subset of frequencies reserved for the medically critical loads.

[0254] Clause 9. The method of clause 6 or 7, wherein the criticality comprises a transportation criticality.

[0255] Clause 10. The method of any of clauses 1-9, wherein each of the plurality of frequency sets services electrical loads of a predefined type. [0256] Clause 1 1. The method of any of clauses 1-10, further comprising: receiving a request for power over a network from the at least one guided surface wave receive structure; and assigning the at least one guided surface wave receive structure to a discrete transmission frequency within a respective one of the plurality of frequency sets.

[0257] Clause 12. The method of clause 11 , further comprising: generating a record including the assigning of the at least one guided surface wave receive structure to the discrete transmission frequency within the respective one of the plurality of frequency sets; and determining a priority of power distribution associated with the at least one guided surface wave receive structure based at least in part upon the at least one electrical load.

[0258] Clause 13. The method of clause 12, wherein the respective one of the plurality of frequency sets includes a reserved frequency set, and the priority of power distribution is further based at least in part on whether the at least one guided surface wave receive structure is assigned to the reserved frequency set within the respective one of the plurality of frequency sets.

[0259] Clause 14. A system comprising: at least one guided surface waveguide probe associated with at least one respective frequency set among a plurality of frequency sets, each of a plurality of transmission classes being associated with each of the plurality of frequency sets; wherein the at least one guided surface waveguide probe is configured to transmit at least one guided surface wave along a terrestrial medium to at least one guided surface wave receive structure within a respective transmission class among the plurality of transmission classes; and wherein the at least one guided surface wave receive structure is configured to supply electrical energy to at least one electrical load coupled to the at least one guided surface wave receive structure.

[0260] Clause 15. The system of clause 14, wherein the at least one electrical load is experienced as at least one load at an excitation source coupled to the at least one guided surface waveguide probe transmitting the at least one guided surface wave.

[0261] Clause 16. The system of clause 14 or 15, wherein the at least one guided surface waveguide probe comprises a charge terminal elevated over a lossy conducting medium configured to generate at least one resultant field that synthesizes a wave front incident at a complex Brewster angel of incidence (θ _{ί Β} ) of the lossy conducting medium.

[0262] Clause 17. The system of clause 16, wherein one of the transmission classes specifies a geographical transmission class.

[0263] Clause 18. The system of clause 17, wherein the geographical transmission class is at least one of a local class, a regional class, or a nationwide class.

[0264] Clause 19. The system of any of clauses 14-18, further comprising at least one control system in communication with the at least one guided surface waveguide probe, the control system configured to assign the at least one guided surface waveguide probe to the at least one guided surface wave receive structure.

[0265] Clause 20. The system of any of clauses 14-18, further comprising at least one control system in communication with the at least one guided surface waveguide probe, the control system configured to assign the at least one guided surface wave receive structure to a respective frequency set associated with the at least one guided surface waveguide probe.

[0266] Clause 21. The system of clause 20, wherein the respective frequency set is assigned based at least in part upon power data associated with a power source that is coupled to the at least one guided surface waveguide probe.

[0267] Clause 22. The system of clause any of claims 14-21 , wherein at least one of the plurality of frequency sets comprises a reserved frequency set that is selected for use in connection with a specific category of electrical loads.

[0268] Clause 23. A system comprising: at least one guided surface wave receive structure configured to: receive electrical energy from at least one guided surface wave transmitted along a terrestrial medium from at least one guided surface waveguide probe transmitting in a respective transmission class among a plurality of transmission classes, the at least one guided surface wave receive structure being assigned the transmission class; and supply the electrical energy to at least one electrical load coupled to the at least one guided surface wave receive structure; and at least one control system configured to assign the at least one guided surface waveguide probe to the at least one guided surface wave receive structure in response to a request for power.

[0269] Disjunctive language such as the phrase "at least one of X, Y, or Z," unless specifically stated otherwise, is otherwise understood with the context as used in general to present that an item, term, etc., may be either X, Y, or Z, or any combination thereof (e.g., X, Y, and/or Z). Thus, such disjunctive language is not generally intended to, and should not, imply that certain embodiments require at least one of X, at least one of Y, or at least one of Z to each be present.

[0270] It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear

understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims.

[0271] It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear

understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims. In addition, all optional and preferred features and modifications of the described embodiments and dependent claims are usable in all aspects of the disclosure taught

herein. Furthermore, the individual features of the dependent claims, as well as all optional and preferred features and modifications of the described embodiments are combinable and interchangeable with one another.