Excitation and use of guided surface wave modes on lossy media

240240/2

EXCITATION AND USE OF GUIDED SURFACE WAVE MODES ON LOSSY MEDIA

BACKGROUND

[0001] 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

[0002] 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.

[0003] FIG. 1 is a chart that depicts a field strength as a function of distance for a guided electromagnetic field and a radiated electromagnetic field.

[0004] 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.

[0005] FIG. 3 is a drawing that illustrates a polyphase waveguide probe disposed with respect to a propagation interface of FIG. 2 according to an embodiment of the present disclosure.

[0006] FIG. 4 is a drawing that provides one example illustration of a phase shift in a ground current that facilitates the launching of a guided surface-waveguide mode on a lossy conducting medium in the propagation interface of FIG. 3 according to an embodiment of the present disclosure.

[0007] FIG. 5 is a drawing that illustrates a complex angle of insertion of an electric field synthesized by the polyphase waveguide probes according to the various embodiments of the present disclosure.

240240/2

[0008] FIG. 6 is a schematic of a polyphase waveguide probe according to an embodiment of the present disclosure.

[0009] FIGS. 7A-J are schematics of specific examples of the polyphase waveguide probe of FIG. 6 according to various embodiments of the present disclosure.

[0010] FIGS. 8A-C are graphs that illustrate field strengths of guided surface waves at select transmission frequencies generated by the various embodiments of polyphase waveguide probes according to the various embodiments of the present disclosure.

[0011] FIG. 9 shows one example of a graph of experimental measurements of field strength of a guided surface wave at 59 Megahertz as a function of distance generated by a polyphase waveguide probe according to an embodiment of the present disclosure.

[0012] FIG. 10 shows a graph of experimental measurements of the phase as a function of distance of the guided surface wave of FIG. 9 according to an embodiment of the present disclosure.

[0013] FIG. 11 shows another example of a graph of experimental measurements of field strength as a function of distance of a guided surface wave generated by a polyphase waveguide probe at 1.85 Megahertz according to an embodiment of the present disclosure.

[0014] FIGS. 12A-B depict examples of receivers that can be employed to receive energy transmitted in the form of a guided surface wave launched by a polyphase waveguide probe according to the various embodiments of the present disclosure.

[0015] FIG. 13 depicts an example of an additional receiver that can be employed to receive energy transmitted in the form of a guided surface wave launched by a polyphase waveguide probe according to the various embodiments of the present disclosure.

[0016] FIG. 14A depicts a schematic diagram representing the Thevenin-equivalent of the receivers depicted in FIGS. 12A-B according to an embodiment of the present disclosure.

[0017] FIG. 14B depicts a schematic diagram representing the Norton-equivalent of the receiver depicted in FIG. 13 according to an embodiment of the present disclosure.

DETAILED DESCRIPTION

[0018] Referring to FIG. 1, 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.

[0019] 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

240240/2 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 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 radiated fields is a function of distance due to geometrical spreading. Accordingly, the term “radiate” in all its forms as used herein refers to this form of electromagnetic propagation.

[0020] 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 can 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.

[0021] To further illustrate the distinction between radiated and guided electromagnetic fields, reference is made to FIG. 1 that depicts 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. 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

240240/2 100 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.

[0022] Of interest are the shapes of the curves 103/106 for radiation and for guided wave propagation. The radiated field strength curve 106 falls off geometrically (1/d, where d is distance) and is a straight line on the log-log scale. The guided field strength curve 103, on the other hand, has the characteristic exponential decay of e־^ad/Vd and exhibits a distinctive knee 109. Thus, as shown, the field strength of a guided electromagnetic field falls off at a rate of e~^ad/-Jd, whereas the field strength of a radiated electromagnetic field falls off at a rate of 1/_d, where d is the distance. Due to the fact that the guided field strength curve 103 falls off exponentially, the guided field strength curve 103 features the knee 109 as mentioned above. The guided field strength curve 103 and the radiated field strength curve 106 intersect at a crossover point 113 which occurs at a crossover distance. At distances less than the crossover distance, 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 crossover 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.

[0023] 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.

[0024] 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, 4

240240/2 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. 291293, each of these references being incorporated herein by reference in their entirety.

[0025] 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 Techniques of Applied Mathematics, Wiley, 1956, pp. pp. 214, 283-286, 290, 298-300.

[0026] In free space, antennas excite the continuum eigenvalues of the wave equation, which is a radiation field, where the outwardly propagating RF energy with Ez and H inphase 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.

[0027] According to the various embodiments of the present disclosure, various polyphase waveguide probes are described that are configured to excite radial surface currents having resultant fields that synthesize the form of surface-waveguide modes 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 surface wave mode. By virtue of the fact that the resultant fields excited by the polyphase waveguide probes described herein are substantially mode-matched to a Zenneck surface wave mode on the surface of the lossy conducting medium, a guided electromagnetic field in the form of a Zenneck surface wave is launched along the surface of

240240/2 the lossy conducting medium. According to one embodiment, the lossy conducting medium comprises a terrestrial medium such as the Earth.

[0028] Referring to FIG. 2, shown is a propagation interface that provides for an examination of the boundary value solution 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.

[0029] According to various embodiments, the present disclosure sets forth various polyphase waveguide probes that generate electromagnetic fields that are substantially mode-matched to a Zenneck surface wave 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 results in zero reflection.

[0030] To explain further, in Region 2, where β⁷'^ωί field variation is assumed and where p ψ0 and z>0 (z is a vertical coordinate normal to the surface of Region 1, p is 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:

Η_2φ = Αβ-·^^ζ H1²\-jYp),(1) ^E2p = ^A GS־) ^e-“^{2Z H}1²⁾(-jyP), and(2)

Jwcq/

E2_z = A(--^)e--^2Z H0²K-j_Yp)(3)

[0031] In Region 1, where e^J<ui field variation is assumed and where p ψ 0 and z < 0, Zenneck’s closed-form exact solution of Maxwell’s equations satisfying the boundary

240240/2 conditions along the interface are expressed by the following electric field and magnetic field components:

II, ^1 H1²\-jYp),(4) ^E!p = ^A fc-U ^e“^{1Z H}1^2)(-jyp⁾, ^and(5)

- '-( ' ^'^(6)

In these expressions, W^²⁾(-jyp) is a complex argument Hankel function of the second kind and order n, u1 is the propagation constant in the positive vertical direction in Region 1, u₂ is the propagation constant in the vertical direction in Region 2, 01־ is the conductivity of Region 1, ω is equal to 2nf, where f is a frequency of excitation, ε₀ is the permittivity of free space, ε1 is the permittivity of Region 1, A is a source constant imposed by the source, z is a vertical coordinate normal to the surface of Region 1, γ is a surface wave radial propagation constant, and p is the radial coordinate.

[0032] 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 = _ ^-jk° ² J1+(s_r-jx)' and gives, in Region 1,

U1 = -U₂(8_r —jx).

The radial propagation constant γ is given by γ = j^k0 + u2, which is a complex expression. In all of the above Equations, x = ־01־, and ^0 = ω^μ080, (7) (8) (9) (10) (11) where μ₀ comprises the permeability of free space, 8_r comprises relative permittivity of Region 1. Thus, the surface wave generated propagates parallel to the interface and exponentially decays vertical to it. This is known as evanescence.

[0033] 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 polyphase waveguide probe is provided with charge terminals of 7

240240/2 appropriate size that are positioned relative to each other and are fed with voltages and/or currents so as to excite the relative phasing of the fields of the surface waveguide mode that is to be launched along the boundary interface between Region 2 and Region 1.

[0034] To continue further, the Leontovich impedance boundary condition between Region 1 and Region 2 is stated as ———— n x H2GM,0) = J_s ,(12) ^—^ where n is a unit normal in the positive vertical (+z) direction and H₂ is the magnetic field strength in Region 2 expressed by Equation (1) above. Equation (12) implies that the fields specified in Equations (1)-(3) can be obtained by driving a radial surface current density along the boundary interface, such radial surface current density being specified by

J_p (p') = - AH(²⁾(-J_Zp')| (13) where A is a constant yet to be determined. Further, it should be noted that close-in to the polyphase waveguide probe (for ρ << λ), Equation (13) above has the behavior

J (p) = ---= ' = - h. =--°— ^clos^ π (- p) ^φ 2 πρ (14)

One may wish to note the negative sign. This means that when source current flows vertically upward, the required “close-in” ground current flows radially inward. By field matching on Ηφ “close-in” we find that ^- Io Y (15) in Equations (1)-(6) and (13). Therefore, Equation (13) can be restated as Jp (p) = -Y H<²⁾(-JYP) . (16)

[0035] With reference then to FIG. 3, shown is an example of a polyphase waveguide probe 200 that includes a charge terminal T1 and a charge terminal T2 that are arranged along a vertical axis z. The polyphase waveguide probe 200 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 (FIG. 2) according to one embodiment. In addition, a second medium 206 shares a boundary interface with the lossy conducting medium 203 and makes up Region 2 (FIG. 2). The polyphase waveguide probe 200 includes a probe coupling circuit 209 that couples an excitation source 213 to the charge terminals T1 and T2 as is discussed in greater detail with reference to later figures.

[0036] The charge terminals T1 and T2 are positioned over the lossy conducting medium 203. The charge terminal T1 can be considered a capacitor, and the charge terminal T2 can 8

240240/2 comprise a counterpoise or lower capacitor as is described herein. According to one embodiment, the charge terminal T1 is positioned at height H1, and the charge terminal T2 is positioned directly below T1 along the vertical axis z at height H2, where H2 is less than H1. The height h of the transmission structure presented by the polyphase waveguide probe 200 is h = H1 - H2. Given the foregoing discussion, one can determine asymptotes of the radial Zenneck surface current on the surface of the lossy conducting medium Jp(p) to be J1(p) close-in and J2(p) far-out, where

Close-in (ρ < λ/8): J_p(p) ~ J = 1^{1 +} I² + E^{p (Q1) +} E^{p Q2}^ , and ^{1 1} 2πρ Zp (17) ivmQ 2/ e ־⁽“ ^{+ jβ})^p

Far-out (ρ >> λ/8): J (p) ~ J₂ = ——-1 _x — x ---=— (18)

4 V π pp where I1 is the conduction current feeding the charge Q1 on the first charge terminal T1, and I2 is the conduction current feeding the charge Q2 on the second charge terminal T2. The charge Q1 on the upper charge terminal T1 is determined by Q1 = C1V1, where C1 is the isolated capacitance of the charge terminal T1. Note that there is a third component to J1 set forth above given by (^-), which follows from the Leontovich boundary condition and is the ' ^ZP 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 Q1. The quantity Z_p = ¹ך-is the radial impedance of the lossy conducting medium, where γβ = (]ωμ1σ1 - ω²μ1ε₁)^1/2.

[0037] The asymptotes representing the radial current close-in and far-out as set forth by Equations (17) and (18) are complex quantities. According to various embodiments, a physical surface current, J(r), is synthesized to match as close as possible the current asymptotes in magnitude and phase. That is to say close-in, |J(r)| is to be tangent to |J1|, and far-out |J(r)| is to be tangent to |J2|. Also, according to the various embodiments, the phase of J(r) should transition from the phase of J1 close-in to the phase of J2 far-out.

[0038] According to one embodiment, if any of the various embodiments of a polyphase waveguide probe described herein are adjusted properly, this configuration will give at least an approximate magnitude and phase match to the Zenneck mode and launch Zenneck surface waves. It should be noted that the phase far-out, Φ2, is proportional to the propagation phase corresponding to e* plus a fixed “phase boost” due to the phase of ργ which is arg (Ργ), ]Φ2 (P) = -]βρ + ατρ^Ργ) (19)

240240/2 where γ is expressed in Equation (9) above, and depending on the values for ερ and σ at the site of transmission on the lossy conducting medium and the operating frequency f, arg(Vy), which has two complex roots, is typically on the order of approximately 45° or 225°. Stated another way, in order to match the Zenneck surface wave mode at the site of transmission to launch a Zenneck surface wave, the phase of the surface current |J2| far-out should differ from the phase of the surface current |J1| close-in by the propagation phase corresponding to e^-7'^^(P2-P1) plus a constant of approximately 45 degrees or 225 degrees. This is because there are two roots for Vy, one near π/4 and one near 5π/4. The properly adjusted synthetic radial surface current is ^Jp ⁽Ρ^,φ,0) = ¹°£- ^H1⁽²⁾ (- J2P)(20) .

By Maxwell’s equations, such a J(p) surface current automatically creates fields that conform to

Ηφ = \I e²״־z H(²⁾(-J2P) ,(21) ^ep = ²־^ ^f־Γ]^eu“²z ^h(2)(-Jr P⁾, and(22) י \ ^{J 2} & o ) ^Ez = f 2־) e־’ z^H»²⁾⁽-J 2P⁾ (23)

Thus, the difference in phase between the surface current |J2| far-out and the surface current |J1| close-in for the Zenneck surface wave mode that is to be matched is due to the inherent characteristics of the Hankel functions in Equations (20)-(23) set forth above. It is of significance to recognize that the fields expressed by Equations (1)-(6) and (20) have the nature of a transmission line mode bound to a lossy interface, not radiation fields such as are associated with groundwave propagation. See Barlow, H. M. and Brown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 1-5. These fields automatically satisfy the complex Brewster angle requirement for zero reflection, which means that radiation is negligible, while surface guided wave propagation is dramatically enhanced, as is verified and supported in the experimental results provided below.

[0039] At this point, a review of the nature of the Hankel functions used in Equations (20)-(23) is provided with emphasis on a special property of 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

H^{(} (}x⁾ = J_n (x) + J N_n (x) and (24)

240240/2 (25)

Hi²⁾ (x ) = ^J xx) - j Nn ⁽x) .

These functions represent cylindrical waves propagating radially inward (superscript (1)) and outward (superscript (2)), 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.

[0040] That Hn⁽²⁾(kpP) is an outgoing wave is easily recognized from its large argument asymptotic behavior that is obtained directly from the series definitions of Jn(x) and Nn(x), (x) x - - > j e^j (26) /✓ x which, when multiplied by e^j“t , is an outward propagating cylindrical wave of the form e^j(“t־^kp)with a 1/^p spatial variation. The phase of the exponential component is ψ = (ωΐ - kp). It is also evident that

H²’⁽x) ->- > j H02)(x), (27) and, a further useful property of Hankel functions is expressed by ^S-H^ = - Hi²)(x). (28) x x which is described by Jahnke, E., and F. Emde, Tables of Functions, Dover, 1945, p. 145.

[0041] In addition, the small argument and large argument asymptotes of the outward propagating Hankel function are as follows:

^H 1⁽²⁾⁽x) _x-0 ^- ✓x(29)

7(2)/ \ · I 2 j _- jx I 2 ^- j⁽x ^-y^- 4־) ^H}’(x) x >- > j A— ^{e j} = J — ^{e 2 4} .(30)

V x XU x x

[0042] Note that these asymptotic expressions are complex quantities. Also, unlike ordinary sinusoidal functions, the behavior of complex Hankel functions differs in-close and far-out from the origin. When x is a real quantity, Equations (29) and (30) differ in phase by ·JT, which corresponds to an extra phase advance or “phase boost” of 45° or, equivalently λ/8.

[0043] With reference to FIG. 4, to further illustrate the phase transition between J1 (FIG. 3) and J2 (FIG. 3) shown is an illustration of the phases of the surface currents J1 close-in and J2 far-out relative to a position of a polyphase waveguide probe 200 (FIG. 3).

240240/2 As shown in FIG. 4, there are three different observation points P0, P1, and P2. A transition region is located between the observation point P1 and observation point P2. The observation point P0 is located at the position of the polyphase waveguide probe 200. The observation point P1 is positioned “close-in” at a distance R1 from the observation point P0 that places the observation point P1 between the transition region 216 and the observation point P0. The observation point P2 is positioned “far-out” at a distance R2 from the observation point P0 beyond the transition region 216 as shown.

[0044] At observation point P0, the magnitude and phase of the radial current J is expressed as |/_Ρθ|ζφ₀. At observation point P1, the magnitude and phase of the radial current J is expressed as |/_PJz^0 - βΚ1, where the phase shift of βΚ1 is attributable to the distance R1 between the observation points P0 and P1. At observation point P2, the magnitude and phase of the radial current J is expressed as |/_P2 |ζφ₀ - ββ1 + φ_Δ where the phase shift of βΚ1 + φ_Δ is attributable to the distance R2 between the observation points P0 and P2 AND an additional phase shift that occurs in the transition region 216. The additional phase shift φ_Δ occurs as a property of the Hankel function as mentioned above.

[0045] The foregoing reflects the fact that the polyphase waveguide probe 200 generates the surface current J1 close-in and then transitions to the J2 current far-out. In the transition region 216, the phase of the Zenneck surface-waveguide mode transitions by approximately 45 degrees or 1a. This transition or phase shift can be considered a “phase boost” as the phase of the Zenneck surface-waveguide mode appears to advance by 45 degrees in the transition region 216. The transition region 216 appears to occur somewhere less than 1/10 of a wavelength of the operating frequency.

[0046] Referring back to FIG. 3, according to one embodiment, a polyphase waveguide probe can be created that will launch the appropriate radial surface current distribution. According to one embodiment, a Zenneck waveguide mode is created in a radial direction. If the J(r) given by Equation (20) can be created, it will automatically launch Zenneck surface waves.

[0047] In addition, further discussion is provided regarding the charge images Q1' and Q2' of the charges Q1 and Q2 on the charge terminals T1 and T2 of one example polyphase waveguide probe shown in FIG. 3. Analysis with respect to the lossy conducting medium assumes the presence of induced effective image charges Q1' and Q2' beneath the polyphase waveguide probes coinciding with the charges Q1 and Q2 on the charge reservoirs T1 and T2 as described herein. Such image charges Q1' and Q2' must also be considered in the analysis. These image charges Q1' and Q2' are not merely 180° out of phase with the primary source charges Q1 and Q2 on the charge reservoirs T1 and T2, as

240240/2 they would be in the case of a perfect conductor. A lossy conducting medium such as, for example, a terrestrial medium presents phase shifted images. That is to say, the image charges Q1' and Q2' are at complex depths. For a discussion of complex images, reference is made to Wait, J. R., “Complex Image Theory—Revisited,” IEEE Antennas and Propagation Magazine, Vol. 33, No. 4, August 1991, pp. 27-29, which is incorporated herein by reference in its entirety.

[0048] Instead of the image charges Q1' and Q2' being at a depth that is equal to the height of the charges Q1 and Q2 (i.e. z״' = -hn), a conducting mirror 215 is placed at depth z = -d/2 and the image itself appears at a “complex distance” (i.e., the “distance” has both magnitude and phase), given by Zn' = - Dn = - (d + hn) # -hn, where n = 1, 2, and for vertically polarized sources, λ I r 2 + k 2 211 d = e ₂ o * — = d_r + jd_t = d Z ζ,(31) re where re = ΐ&μσ_χ - rf-μ ε1, and(32) ^ko = ^ω^μα ^εα .

[0049] The complex spacing of image charges Q1' and Q2', in turn, implies that the external fields will experience extra phase shifts not encountered when the interface is either a lossless dielectric or a perfect conductor. The essence of the lossy dielectric image-theory technique is to replace the finitely conducting Earth (or lossy dielectric) by a perfect conductor located at a complex depth, z = - d/2. Next, a source image is then located at a complex depth Dn = d/2 + d/2 + hn = d + hn, where n = 1, 2. Thereafter, one can calculate the fields above ground (z > 0) using a superposition of the physical charge (at z = +h) plus its image (at z' = - D). The charge images Q1' and Q2' at complex depths actually assist in obtaining the desired current phases specified in Equations (20) and (21) above.

[0050] From Equations (2) and (3) above, it is noted that the ratio of E2_Z to E2_p in

Region 2 is given by e_{2z =}⁴(z^־)^e2“־^Z Z2 ^(-n^,p)₌ /-jy\ »²״ (-/vp) ^E2p A(-^—)e ‘‘²'II(' ^-)VIP ^p 2״ Λ// (-/pp)

Also, it should be noted that asymptotically, //,/m ---------> j²)₎//״^l(.v).

X^rn ^w

Consequently, it follows directly from Equations (2) and (3) that

E^ — = ^8,- - jx = n = tan^i'B, ^L2p (34) (35) (36)

240240/2 where ^_i>B is the complex Brewster angle. By adjusting source distributions and synthesizing complex Brewster angle illumination at the surface of a lossy conducting medium 203, Zenneck surface waves can be excited.

[0051] With reference to FIG. 5, shown is an incident field E polarized parallel to a plane of incidence. The electric field vector E is to be synthesized as an incoming nonuniform plane wave, polarized parallel to the plane of incidence. The electric field vector E can be created from independent horizontal and vertical components as: ^^ ^E(θ_ο) = E_pp + E_zz.(37)

Geometrically, the illustration in FIG. 5 suggests:

E_p(p,z) = E(p,z)cos ψ_ο, and(38a) (πΊ ^EZ⁽p,^z) = ^E(p,^z) ^cos I ־ ץ Ψο I = ^E(p,^z) ^sm ψο , which means that the field ratio is

Ez — = tan ψο.

^Ep

However, recall that from Equation (36), ^{tan f)}.,b = Js_r - jx (39) (40) so that, for a Zenneck surface wave, we desire Ψο = Θϊ,β, which results in

E ^Ep tan ψ_ο (41)

[0052] The Equations mean that if one controls the magnitude of the complex field ratio and the relative phase between the incident vertical and horizontal components Ez and Ep in a plane parallel to the plane of incidence, then the synthesized E-field vector will effectively be made to be incident at a complex Brewster angle. Such a circumstance will synthetically excite a Zenneck surface wave over the interface between Region 1 and Region 2.

[0053] With reference to FIG. 6, shown is another view of the polyphase waveguide probe 200 disposed above a lossy conducting medium 203 according to an embodiment of the present disclosure. The lossy conducting medium 203 makes up Region 1 (FIG. 2) according to one embodiment. In addition, a second medium 206 shares a boundary interface with the lossy conducting medium 203 and makes up Region 2 (FIG. 2).

240240/2

[0054] According to one embodiment, the lossy conducting medium 203 comprises 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.

[0055] In the case that 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.

[0056] The polyphase waveguide probe 200 comprises a pair of charge terminals T1 and T2. Although two charge terminals T1 and T2are shown, it is understood that there can be more than two charge terminals T1 and T2. According to one embodiment, the charge terminals T1 and T2are positioned above the lossy conducting medium 203 along a vertical axis z that is normal to a plane presented by the lossy conducting medium 203. In this respect, the charge terminal T1 is placed directly above the charge terminal T2 although it is possible that some other arrangement of two or more charge terminals Tn can be used. According to various embodiments, charges Q1 and Q2 can be imposed on the respective charge terminals T1 and T2.

[0057] The charge terminals T1 and/or T2 can comprise any conductive mass that can hold an electrical charge. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. The charge terminals T1 and/or T2 can comprise any shape such as a sphere, a disk, a cylinder, a cone, a torus, a randomized shape, or any other shape. Also note that the charge terminals T1 and T2 need not be identical, but each can have a separate size and shape, and be comprises of different conducting materials. According to one embodiment, the shape of the charge terminal T1 is specified to hold as much charge as practically possible. Ultimately, the field strength of a Zenneck surface wave launched by a polyphase waveguide probe 200 is directly proportional to the quantity of charge on the terminal T1.

240240/2

[0058] If the charge terminals T1 and/or T2 are spheres or disks, the respective selfcapacitance C1 and C2 can be calculated. For example, the self-capacitance of an isolated conductive sphere is C = 4πε₀Γ, where r comprises the radius of the sphere in meters. The self-capacitance of an isolated disk is C = 88₀r, where r comprises the radius of the disk in meters.

[0059] Thus, the charge Q1 stored on the charge terminal T1 can be calculated as Q1 = C1V, given the self-capacitance C1 of the charge reservoir T1 and voltage V that is applied to the charge terminal T1.

[0060] With further reference to FIG. 6, according to one embodiment, the polyphase waveguide probe 200 comprises a probe coupling circuit 209 that is coupled to the charge terminals T1 and T2. The probe coupling circuit 209 facilitates coupling the excitation source 213 to the charge terminals T1 and T2, and facilitates generating respective voltage magnitudes and phases on the charge terminals T1 and T2 for a given frequency of operation. If more than two charge terminals Tn are employed, then the probe coupling circuit 209 would be configured to facilitate the generation of various voltage magnitudes and phases on the respective charge terminals Tn relative to each other. In the embodiment of the polyphase waveguide probe 200, the probe coupling circuit 209 comprises various circuit configurations as will be described.

[0061] In one embodiment, the probe coupling circuit 209 is specified so as to make the polyphase waveguide probe 200 electrically half-wave resonant. This imposes a voltage +V on a first one of the terminals T1 or T2, and a -V on the second one of the charge terminals T1 or T2 at any given time. In such case, the voltages on the respective charge terminals T1 and T2 are 180 degrees out of phase as can be appreciated. In the case that the voltages on the respective charge terminals T1 and T2 are 180 degrees out of phase, the largest voltage magnitude differential is experienced on the charge terminals T1 and T2. Alternatively, the probe coupling circuit 209 can be configured so that the phase differential between the charge terminals T1 and T2 is other than 180 degrees. To this end, the probe coupling circuit 209 can be adjusted to alter the voltage magnitudes and phases during adjustment of the polyphase waveguide probe 200.

[0062] By virtue of the placement of the charge terminal T1 directly above the charge terminal T2, a mutual capacitance CM is created between the charge terminals T1 and T2. Also, the charge terminal T1 has self-capacitance C1, and the charge terminal T2 has a selfcapacitance C2 as mentioned above. There can also be a bound capacitance between the charge terminal T1 and the lossy conducting medium 203, and a bound capacitance between the charge terminal T2 and the lossy conducting medium 203, depending on the respective

240240/2 heights of the charge terminals T1 and T2. The mutual capacitance Cm depends on the distance between the charge terminals T1 and T2.

[0063] Ultimately, the field strength generated by the polyphase waveguide probe 200 will be directly proportional to the magnitude of the charge Q1 that is imposed on the upper terminal T1. The charge Q1 is, in turn, proportional to the self-capacitance C1 associated with the charge terminal T1 since Q1 = C1V, where V is the voltage imposed on the charge terminal T1.

[0064] According to one embodiment, an excitation source 213 is coupled to the probe coupling circuit 209 in order to apply a signal to the polyphase waveguide probe 200. The excitation source 213 can be any suitable electrical source such as a voltage or current source capable of generating the voltage or current at the operating frequency that is applied to the polyphase waveguide probe 200. To this end, the excitation source 213 can comprise, for example, a generator, a function generator, transmitter, or other electrical source.

[0065] In one embodiment, the excitation source 213 can be coupled to the polyphase waveguide probe 200 by way of magnetic coupling, capacitive coupling, or conductive (direct tap) coupling as will be described. In some embodiments, the probe coupling circuit 209 can be coupled to the lossy conducting medium 203. Also, in various embodiments, the excitation source 213 can be coupled to the lossy conducting medium 203 as will be described.

[0066] In addition, it should be noted that, according to one embodiment, the polyphase waveguide probe 200 described herein has the property that its radiation resistance Rr is very small or even negligible. One should recall that radiation resistance Rr is the equivalent resistance that would dissipate the same amount of power that is ultimately radiated from an antenna. According to the various embodiments, the polyphase waveguide probe 200 launches a Zenneck surface wave that is a guided electromagnetic wave. According to the various embodiments, the polyphase waveguide probes described herein have little radiation resistance Rr because the height of such polyphase waveguide probes is usually small relative to their operating wavelengths. Stated another way, according to one embodiment, the polyphase waveguide probes described herein are “electrically small.” As contemplated herein, the phrase “electrically small” is defined as a structure such as the various embodiments of polyphase waveguide probes described herein that can be physically bounded by a sphere having a radius equal to λ/2π, where λ is the free-space wavelength. See Fujimoto, K., A. Henderson, K. Hirasawa, and J.R. James, Small Antennas, Wiley, 1987, p. 4.

240240/2

[0067] To discuss further, the radiation resistance Rr for a short monopole antenna is expressed by

R_r = 160π² (j)2 (42) where the short monopole antenna has a height h with a uniform current distribution, and where λ is the wavelength at the frequency of operation. See Stutzman, W.L. et al., “Antenna Theory and Design,” Wiley & Sons, 1981, p. 93.

[0068] Given that the value of the radiation resistance Rr is determined as a function of

G)² it follows that if the height h of the structure is small relative to the wavelength of the operating signal at the operating frequency, then the radiation resistance Rr will also be small. As one example, if the height h of the transmission structure is 10% of the wavelength of the operating signal at the operating frequency, then the resulting value of (J) would be (. 1)² = .01. It would follow that the radiation resistance Rr is correspondingly small.

[0069] Thus, according to various embodiments, if the effective height h of the transmission structure is less than or equal to , where λ is the wavelength at the operating frequency, then the radiation resistance Rr will be relatively small. For the various embodiments of polyphase waveguide probes 200 described below, the height h of the transmission structure can be calculated as h = H1 - H2, where H1 is the height of the charge terminal T1, and H2 is the height of the charge terminal T2. It should be appreciated that the height h of the transmission structure for each embodiment of the polyphase waveguide probes 200 described herein can be determined in a similar manner.

[0070] While is provided as one benchmark, it is understood that the ratio of the height h of the transmission structure over the wavelength of the operating signal at the operating frequency can be any value. However, it is understood that, at a given frequency of operation, as the height of a given transmission structure increases, the radiation resistance Rr will increase accordingly.

[0071] Depending on the actual values for the height h and the wavelength of the operating signal at the operating frequency, it is possible that the radiation resistance Rr can be of a value such that some amount of radiation can occur along with the launching of a Zenneck surface wave. To this end, the polyphase waveguide probe 200 can be constructed to have a short height h relative to the wavelength at the frequency of operation so as to ensure that little or substantially zero energy is lost in the form of radiation.

240240/2

[0072] In addition, the placement of the charge reservoirs T1 and T2 along the vertical axis z provides for symmetry in the Zenneck surface wave that is launched by the polyphase waveguide probe 200 as described by the Hankel functions in Equations (20)-(23) set forth above. Although the polyphase waveguide probe 200 is shown with two charge reservoirs T1 and T2 along the vertical axis z that is normal to a plane making up the surface of the lossy conducting medium 203, it is understood that other configurations can be employed that will also provide for the desired symmetry. For example, additional charge reservoirs Tn can be positioned along the vertical axis z, or some other arrangement can be employed. In some embodiments, symmetry of transmission may not be desired. In such case, the charge reservoirs Tn can be arranged in a configuration other than along a vertical axis z to provide for an alternative transmission distribution pattern.

[0073] When properly adjusted to operate at a predefined operating frequency, the polyphase waveguide probe 200 generates a Zenneck surface wave along the surface of the lossy conducting medium 203. To this end, an excitation source 213 can be employed to generate electrical energy at a predefined frequency that is applied to the polyphase waveguide probe 200 to excite the structure. The energy from the excitation source 213 is transmitted in the form of a Zenneck surface wave by the polyphase waveguide probe 200 to one or more receivers that are also coupled to the lossy conducting medium 203 or that are located within an effective transmission range of the polyphase waveguide probe 200. The energy is thus conveyed in the form of a Zenneck surface wave which is a surfacewaveguide mode or a guided electromagnetic field. In the context of modern power grids using high voltage lines, a Zenneck surface wave comprises a transmission line mode.

[0074] Thus, the Zenneck surface wave generated by the polyphase waveguide probe 200 is not a radiated wave, but a guided wave in the sense that these terms are described above. The Zenneck surface wave is launched by virtue of the fact that the polyphase waveguide probe 200 creates electromagnetic fields that are substantially mode-matched to a Zenneck surface wave mode on the surface of the lossy conducting medium 203. When the electromagnetic fields generated by the polyphase waveguide probe 200 are substantially mode-matched as such, the electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium 203 that results in little or no reflection. Note that if the polyphase waveguide probe 200 is not substantially mode-matched to the Zenneck surface wave mode, then a Zenneck surface wave will not be launched since the complex Brewster angle of the lossy conducting medium 203 would not have been attained.

[0075] In the case that the lossy conducting medium 203 comprises a terrestrial medium such as the Earth, the Zenneck surface wave mode will depend upon the dielectric

240240/2 permittivity ερ and conductivity σ of the site at which the polyphase waveguide probe 200 is located as indicated above in Equations (1)-(11). Thus, the phase of the Hankel functions in Equations (20)-(23) above depends on these constitutive parameters at the launch site and on the frequency of operation.

[0076] In order to excite the fields associated with the Zenneck surface wave mode, according to one embodiment, the polyphase waveguide probe 200 substantially synthesizes the radial surface current density on the lossy conducting medium of the Zenneck surface wave mode as is expressed by Equation (20) set forth above. When this occurs, the electromagnetic fields are then substantially or approximately mode-matched to a Zenneck surface wave mode on the surface of the lossy conducting medium 203. To this end, the match should be as close as is practicable. According to one embodiment, this Zenneck surface wave mode to which the electromagnetic fields are substantially matched is expressed in Equations (21)-(23) set forth above.

[0077] In order to synthesize the radial surface current density in the lossy conducting medium of the Zenneck surface wave mode, the electrical characteristics of the polyphase waveguide probe 200 should be adjusted to impose appropriate voltage magnitudes and phases on the charge terminals T1 and T2 for a given frequency of operation and given the electrical properties of the site of transmission. If more than two charge terminals T n are employed, then appropriate voltage magnitudes and phases would need to be imposed on the respective charge terminals Tn, where N can even be a very large number effectively comprising a continuum of charge terminals.

[0078] In order to obtain the appropriate voltage magnitudes and phases for a given design of a polyphase waveguide probe 200 at a given location, an iterative approach can be used. Specifically, analysis can be performed of a given excitation and configuration of a polyphase waveguide probe 200 taking into account the feed currents to the terminals T1 and T2, the charges on the charge terminals T1 and T2, and their images in the lossy conducting medium 203 in order to determine the radial surface current density generated. This process can be performed iteratively until an optimal configuration and excitation for a given polyphase waveguide probe 200 is determined based on desired parameters. To aid in determining whether a given polyphase waveguide probe 200 is operating at an optimal level, a guided field strength curve 103 (FIG. 1) can be generated using Equations (1)-(11) above based on values for the conductivity of Region 1 (01־) and the permittivity of Region 1 (ε1) at the location of the polyphase waveguide probe 200. Such a guided field strength curve 103 will 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.

240240/2

[0079] In order to arrive at an optimized polyphase waveguide probe 200, various parameters associated with the polyphase waveguide probe 200 can be adjusted. Stated another way, the various parameters associated with the polyphase waveguide probe 200 can be varied to adjust the polyphase waveguide probe 200 to a desired operating configuration.

[0080] One parameter that can be varied to adjust the polyphase waveguide probe 200 is the height of one or both of the charge terminals T1 and/or T2 relative to the surface of the lossy conducting medium 203. In addition, the distance or spacing between the charge terminals T1 and T2 can also be adjusted. In doing so, one can minimize or otherwise alter the mutual capacitance CM or any bound capacitances between the charge terminals T1 and T2 and the lossy conducting medium 203 as can be appreciated.

[0081] Alternatively, another parameter that can be adjusted is the size of the respective charge terminals T1 and/or T2. By changing the size of the charge terminals T1 and/or T2, one will alter the respective self-capacitances C1 and/or C2, and the mutual capacitance CM as can be appreciated. Also, any bound capacitances that exist between the charge terminals T1 and T2 and the lossy conducting medium 203 will be altered. In doing so, the voltage magnitudes and phases on the charge terminals T1 and T2 are altered.

[0082] Still further, another parameter that can be adjusted is the probe coupling circuit 209 associated with the polyphase waveguide probe 200. This can be accomplished by adjusting the size of the inductive and/or capacitive reactances that make up the probe coupling circuit 209. For example, where such inductive reactances comprise coils, the number of turns on such coils can be adjusted. Ultimately, the adjustments to the probe coupling circuit 209 can be made to alter the electrical length of the probe coupling circuit 209, thereby affecting the voltage magnitudes and phases on the charge terminals T1 and T2.

[0083] It is also the case that one can adjust the frequency of an excitation source 213 applied to the polyphase waveguide probe 200 to optimize the transmission of a Zenneck surface wave. However, if one wishes to transmit at a given frequency, other parameters would need to be adjusted to optimize transmission.

[0084] Note that the iterations of transmission performed by making the various adjustments can be implemented by using computer models or by adjusting physical structures as can be appreciated. In one approach, a field meter tuned to the transmission frequency can be placed an appropriate distance from the polyphase waveguide probe 200 and adjustments can be made as set forth above until a maximum or any other desired field strength of a resulting Zenneck surface wave is detected. To this end, the field strength can be compared with a guided field strength curve 103 (FIG. 1) generated at the desired

240240/2 operating frequency and voltages on the terminals T1 and T2. According to one approach, the appropriate distance for placement of such a field meter can be specified as greater than the transition region 216 (FIG. 4) in the “far-out” region described above where the surface current J2 dominates.

[0085] By making the above adjustments, one can create corresponding “close-in” surface current J1 and “far-out” surface current J2 that approximate the same currents J(r) of the Zenneck surface wave mode specified in Equations (17) and (18) set forth above. In doing so, the resulting electromagnetic fields would be substantially or approximately modematched to a Zenneck surface wave mode on the surface of the lossy conducting medium 203.

[0086] Referring next to FIGS. 7A through 7J, shown are additional examples of polyphase waveguide probes 200, denoted herein as polyphase waveguide probes 200a-j, according to various embodiments of the present disclosure. The polyphase waveguide probes 200a-j each include a different probe coupling circuit 209, denoted herein as probe coupling circuits 209a-j, according to various embodiments. While several examples of probe coupling circuits 209a-j are described, it is understood that these embodiments are merely examples and that there can be many other probe coupling circuits 209 not set forth herein that can be employed to provide for the desired voltage magnitudes and phases on the charge terminals T1 and T2 according to the principles set forth herein in order to facilitate the launching of Zenneck surface waves.

[0087] In addition, each of the probe coupling circuits 209a-j can employ, but are not limited to, inductive impedances comprising coils. Even though coils are used, it is understood that other circuit elements, both lumped and distributed, can be employed as reactances. Also, other circuit elements can be included in the probe coupling circuits 209a-j beyond those illustrated herein. In addition, it is noted that the various polyphase waveguide probes 200a-j with their respective probe coupling circuits 209a-j are merely described herein to provide examples. To this end, there can be many other polyphase waveguide probes 200 that employ various probe coupling circuits 209 and other circuitry that can be used to launch Zenneck surface waves according to the various principles set forth herein.

[0088] Referring now to FIG. 7A, shown is a first example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200a, according to one embodiment. The polyphase waveguide probe 200a includes the charge terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on

240240/2 the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance Cm can exist between the charge terminals T1 and T2 depending on the distance there between. In addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

[0089] The polyphase waveguide probe 200a includes a probe coupling circuit 209a that comprises an inductive impedance comprising a coil L1a having a pair of leads that are coupled to respective ones of the charge terminals T1 and T2. In one embodiment, the coil L1a is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the polyphase waveguide probe 200a.

[0090] While the electrical length of the coil L1a is specified as approximately one-half (1/2) the wavelength at the operating frequency, it is understood that the coil L1a can be specified with an electrical length at other values. According to one embodiment, the fact that the coil L1a 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 T1 and T2. Nonetheless, the length or diameter of the coil L1a can be increased or decreased when adjusting the polyphase waveguide probe 200a to obtain optimal excitation of a Zenneck surface wave mode. Alternatively, it can 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 polyphase waveguide probe 200a.

[0091] According to one embodiment, the excitation source 213 is coupled to the probe coupling circuit 209 by way of magnetic coupling. Specifically, the excitation source 213 is coupled to a coil Lp that is inductively coupled to the coil L1a. This can be done by link coupling, a tapped coil, a variable reactance, or other coupling approach as can be appreciated. To this end, the coil Lp acts as a primary, and the coil L1a acts as a secondary as can be appreciated.

[0092] In order to adjust the polyphase waveguide probe 200a for the transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of the coil L1a can be altered by adding or eliminating turns or by changing some other dimension of the coil L1a.

240240/2

[0093] Based on experimentation with the polyphase waveguide probe 200a, this appears to be the easiest of the polyphase waveguide probes 200a-j to adjust and to operate to achieve a desired efficiency.

[0094] Referring now to FIG. 7B, shown is an example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200b, according to one embodiment. The polyphase waveguide probe 200b includes the charge terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminals T1 and T2 are positioned along a vertical axis z to provide for cylindrical symmetry in the resulting Zenneck surface wave as was described above. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance Cm can exist between the charge terminals T1 and T2 depending on the distance there between. In addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

[0095] The polyphase waveguide probe 200b also includes a probe coupling circuit 209b comprising a first coil L1b and a second coil L2b. The first coil L1b is coupled to each of the charge terminals T1 and T2 as shown. The second coil L2b is coupled to the charge terminal T2 and to the lossy conducting medium 203.

[0096] The excitation source 213 is magnetically coupled to the probe coupling circuit 209b in a manner similar as was mentioned with respect to the polyphase waveguide probe 200a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil Lp that acts as a primary and the coil L1b acts as a secondary. Alternatively, the coil L2b can act as a secondary as well.

[0097] In order to adjust the polyphase waveguide probe 200b for the transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of each of the coils L1b and L2b can be altered by adding or eliminating turns or by changing some other dimension of the respective coils L1b or L2b.

[0098] Referring now to FIG. 7C, shown is another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200c, according to one embodiment. The polyphase waveguide probe 200c includes the charge

240240/2 terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance Cm can exist between the charge terminals T1 and T2 depending on the distance there between. In addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

[0099] The polyphase waveguide probe 200c also includes a probe coupling circuit 209c comprising a coil L1c. One end of the coil L1c is coupled to the charge terminal T1 as shown. The second end of the coil L1c is coupled to the lossy conducting medium 203. A tap that is coupled to the charge terminal T2 is positioned along the coil L1c.

[0100] The excitation source 213 is magnetically coupled to the probe coupling circuit 209c in a manner similar as was mentioned with respect to the polyphase waveguide probe 200a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil Lp that acts as a primary and the coil L1c acts as a secondary. The coil LP can be positioned at any location along the coil L1c.

[0101] In order to adjust the polyphase waveguide probe 200c for the excitation and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of the coil L1c can be altered by adding or eliminating turns, or by changing some other dimension of the coil L1c. In addition, the inductance presented by the portions of the coil L1c above and below the tap can be adjusted by moving the position of the tap.

[0102] Referring now to FIG. 7D, shown is yet another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200d, according to one embodiment. The polyphase waveguide probe 200d includes the charge terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance Cm can exist between the charge terminals T1 and T2 depending on the distance there between. In

240240/2 addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

[0103] The polyphase waveguide probe 200d also includes a probe coupling circuit 209d comprising a first coil L1d and a second coil L2d. A first lead of the first coil L1d is coupled to the charge terminal T1, and the second lead of the first coil L1d is coupled to the lossy conducting medium 203. A first lead of the second coil L2d is coupled to the charge terminal T2, and the second lead of the second coil L2d is coupled to the lossy conducting medium 203.

[0104] The excitation source 213 is magnetically coupled to the probe coupling circuit 209d in a manner similar as was mentioned with respect to the polyphase waveguide probe 200a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil Lp that acts as a primary and the coil L2d acts as a secondary. Alternatively, the coil L1d can act as a secondary as well.

[0105] In order to adjust the polyphase waveguide probe 200d for the excitation and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of each of the coils L1d and L2d can be altered by adding or eliminating turns or by changing some other dimension of the respective coils L1d or L2d.

[0106] Referring now to FIG. 7E, shown is still another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200e, according to one embodiment. The polyphase waveguide probe 200e includes the charge terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminals T1 and T2 are positioned along a vertical axis z to provide for cylindrical symmetry in the resulting Zenneck surface wave as was described above. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance CM can exist between the charge terminals T1 and T2 depending on the distance there between. In addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

240240/2

[0107] The polyphase waveguide probe 200e also includes a probe coupling circuit 209e comprising a first coil L1e and a resistor R2. A first lead of the first coil L1e is coupled to the charge terminal T1, and the second lead of the first coil L1e is coupled to the lossy conducting medium 203. A first lead of the resistor R2 is coupled to the charge terminal T2, and the second lead of the resistor R2 is coupled to the lossy conducting medium 203.

[0108] The excitation source 213 is magnetically coupled to the probe coupling circuit 209e in a manner similar as was mentioned with respect to the polyphase waveguide probe 200a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil Lp that acts as a primary and the coil L1e acts as a secondary.

[0109] In order to adjust the polyphase waveguide probe 200e for the transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of the coil L1e can be altered by adding or eliminating turns or by changing some other dimension of the respective coils L1e. In addition, the magnitude of the resistance R2 can be adjusted as well.

[0110] Referring now to FIG. 7F, shown is a further example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200f, according to one embodiment. The polyphase waveguide probe 200f includes a charge terminal T1 and a ground screen G that acts as a second charge terminal. The charge terminal T1 and the ground screen G are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. Note that to calculate the height h of the transmission structure, the height H2 of the ground screen G is subtracted from the height H1 of the charge terminal T1.

[0111] The charge terminal T1 has a self-capacitance C1, and the ground screen G has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on the charge terminal T1 and the ground screen G, respectively, depending on the voltages applied to the charge terminal T1 and the ground screen G at any given instant. A mutual capacitance CM can exist between the charge terminal T1 and the ground screen G depending on the distance there between. In addition, bound capacitances can exist between the charge terminal T1 and/or the ground screen G and the lossy conducting medium 203 depending on the heights of the charge terminal T1 and the ground screen G with respect to the lossy conducting medium 203. Generally, a bound capacitance will exist between the ground screen G and the lossy conducting medium 203 due to its proximity to the lossy conducting medium 203.

240240/2

[0112] The polyphase waveguide probe 200f includes a probe coupling circuit 209f that is made up of an inductive impedance comprising a coil L1f having a pair of leads that are coupled to the charge terminal T1 and ground screen G. In one embodiment, the coil L1f is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the polyphase waveguide probe 200f.

[0113] While the electrical length of the coil L1f is specified as approximately one-half (1/2) the wavelength at the operating frequency, it is understood that the coil L1f can be specified with an electrical length at other values. According to one embodiment, the fact that the coil L1f 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 terminal T1 and the ground screen G. Nonetheless, the length or diameter of the coil L1f can be increased or decreased when adjusting the polyphase waveguide probe 200f to obtain optimal transmission of a Zenneck surface wave. Alternatively, it can 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 polyphase waveguide probe 200f.

[0114] According to one embodiment, the excitation source 213 is coupled to the probe coupling circuit 209f by way of magnetic coupling. Specifically, the excitation source 213 is coupled to a coil Lp that is inductively coupled to the coil L1f. This can be done by link coupling, a phasor/coupling network, or other approach as can be appreciated. To this end, the coil Lp acts as a primary, and the coil L1f acts as a secondary as can be appreciated.

[0115] In order to adjust the polyphase waveguide probe 200a for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of the coil L1f can be altered by adding or eliminating turns or by changing some other dimension of the coil L1f.

[0116] Referring now to FIG. 7G, shown is another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200g, according to one embodiment. The polyphase waveguide probe 200g includes the charge terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminals T1 and T2 are positioned along a vertical axis z to provide for cylindrical symmetry in the resulting Zenneck surface wave as was described above. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are

240240/2 imposed on the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance Cm can exist between the charge terminals T1 and T2 depending on the distance there between. In addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

[0117] The polyphase waveguide probe 200g also includes a probe coupling circuit 209g comprising a first coil L1g, a second coil L2g, and a variable capacitor Cv. The first coil L1g is coupled to each of the charge terminals T1 and T2 as shown. The second coil L2g has a first lead that is coupled to a variable capacitor CV and a second lead that is coupled to the lossy conducting medium 203. The variable capacitor CV, in turn, is coupled to the charge terminal T2 and the first coil L1g.

[0118] The excitation source 213 is magnetically coupled to the probe coupling circuit 209g in a manner similar as was mentioned with respect to the polyphase waveguide probe 200a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil Lp that acts as a primary and either the coil L1g or the coil L2g can act as a secondary.

[0119] In order to adjust the polyphase waveguide probe 200g for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of each of the coils L1g and L2g can be altered by adding or eliminating turns or by changing some other dimension of the respective coils L1g or L2g. In addition, the variable capacitance CV can be adjusted.

[0120] Referring now to FIG. 7H, shown is yet another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200h, according to one embodiment. The polyphase waveguide probe 200h includes the charge terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance CM can exist between the charge terminals T1 and T2 depending on the distance there between. In addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

240240/2

[0121] The polyphase waveguide probe 200h also includes a probe coupling circuit 209h comprising a first coil L1h and a second coil L2h. The first lead of the first coil L1h is coupled to the charge terminal T1, and the second lead of the first coil L1h is coupled to the second charge terminal T2. A first lead of the second coil L2h is coupled to a terminal TT, and the second lead of the second coil L2h is coupled to the lossy conducting medium 203. The terminal Tt is positioned relative to the charge terminal T2 such that a coupling capacitance CC exists between the charge terminal T2 and the terminal TT.

[0122] The excitation source 213 is magnetically coupled to the probe coupling circuit 209h in a manner similar as was mentioned with respect to the polyphase waveguide probe 200a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil Lp that acts as a primary and the coil L2h acts as a secondary. Alternatively, the coil L1h can act as a secondary as well.

[0123] In order to adjust the polyphase waveguide probe 200h for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of each of the coils L1h and L2h can be altered by adding or eliminating turns or by changing some other dimension of the respective coils L1h or L2h. Also the spacing between the charge terminal T2 and the terminal TT can be altered, thereby modifying the coupling capacitance CC as can be appreciated.

[0124] Referring now to FIG. 7I, shown is yet another example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200i, according to one embodiment. The polyphase waveguide probe 200i is very similar to the polyphase waveguide probe 200h (FIG. 7H) except for the fact that the excitation source 213 is series-coupled to the probe coupling circuit 209i as will be described.

[0125] To this end, the polyphase waveguide probe 200i includes the charge terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance CM can exist between the charge terminals T1 and T2 depending on the distance there between. In addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

240240/2

[0126] The polyphase waveguide probe 200i also includes a probe coupling circuit 209i comprising a first coil L1i and a second coil L2i. The first lead of the first coil L1i is coupled to the charge terminal T1, and the second lead of the first coil L·״ is coupled to the second charge terminal T2. A first lead of the second coil L2i is coupled to a terminal Tt, and the second lead of the second coil L2i is coupled to an output of the excitation source 213. Also, a ground lead of the excitation source 213 is coupled to the lossy conducting medium 203. The terminal Tt is positioned relative to the charge terminal T2 such that a coupling capacitance CC exists between the charge terminal T2 and the terminal TT.

[0127] The polyphase waveguide probe 200i provides one example where the excitation source 213 is series-coupled to the probe coupling circuit 209i as mentioned above. Specifically, the excitation source 213 is coupled between the coil L2i and the lossy conducting medium 203.

[0128] In order to adjust the polyphase waveguide probe 200i for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of each of the coils L1i and L2i can be altered by adding or eliminating turns or by changing some other dimension of the respective coils L1i or L2i. Also the spacing between the charge terminal T2 and the terminal TT can be altered, thereby modifying the coupling capacitance CC as can be appreciated.

[0129] Referring now to FIG. 7J, shown is an example of the polyphase waveguide probe 200 (FIG. 6), denoted herein as polyphase waveguide probe 200j, according to one embodiment. The polyphase waveguide probe 200j includes the charge terminals T1 and T2 that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium 203. The second medium 206 is above the lossy conducting medium 203. In this embodiment, the charge terminal T1 comprises a sphere and the charge terminal T2 comprises a disk. In this respect, the polyphase waveguide probe 200j provides an illustration that the charge terminals TN can comprise any shape.

[0130] The charge terminal T1 has a self-capacitance C1, and the charge terminal T2 has a self-capacitance C2. During operation, charges Q1 and Q2 are imposed on the charge terminals T1 and T2, respectively, depending on the voltages applied to the charge terminals T1 and T2 at any given instant. A mutual capacitance CM can exist between the charge terminals T1 and T2 depending on the distance there between. In addition, bound capacitances can exist between the respective charge terminals T1 and T2 and the lossy conducting medium 203 depending on the heights of the respective charge terminals T1 and T2 with respect to the lossy conducting medium 203.

240240/2

[0131] The polyphase waveguide probe 200j includes a probe coupling circuit 209j comprising an inductive impedance comprising a coil L1j having a pair of leads that are coupled to respective ones of the charge terminals T1 and T2. In one embodiment, the coil L1j is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the polyphase waveguide probe 200j. While the electrical length of the coil L1j is specified as approximately one-half (1/2) the wavelength at the operating frequency, it is understood that the coil L1j can be specified with an electrical length at other values as was discussed with reference to the polyphase waveguide probe 200a (FIG. 7A) described above. In addition, the probe coupling circuit 209j includes a tap 223 on the coil L1j that is coupled to the lossy conducting medium 203.

[0132] The excitation source 213 is magnetically coupled to the probe coupling circuit 209j in a manner similar as was mentioned with respect to the polyphase waveguide probe 200a (FIG. 7A) set forth above. To this end, the excitation source 213 is coupled to a coil Lp that acts as a primary and the coil L1j acts as a secondary. The coil LP can be positioned at any location along the coil L1j. Also, the coil Lp can be positioned above or below the tap 223.

[0133] In order to adjust the polyphase waveguide probe 200j for the launching and transmission of a desired Zenneck surface wave, the heights of the respective charge terminals T1 and T2 can be altered with respect to the lossy conducting medium 203 and with respect to each other. Also, the sizes of the charge terminals T1 and T2 can be altered. In addition, the size of the coil L1j can be altered by adding or eliminating turns or by changing some other dimension of the coil L1j. Further, the position of the tap 223 on the coil L1j can be adjusted.

[0134] With reference to the various embodiments of the polyphase waveguide probes 200a-j in FIGS. 7A-J, each of the polyphase waveguide probes 200a-j can be excited to transmit energy conveyed in the form of a guided wave, or waveguide mode along the surface of the lossy conducting medium 203. To facilitate such transmission, the elements of each of the polyphase waveguide probes 200a-j can be adjusted to impose a desired voltage magnitude and phase on the respective charge terminals T1 and T2 when the respective polyphase waveguide probe 200a-j is excited. Such excitation can occur by applying energy from an excitation source 213 to the respective polyphase waveguide probe 200a-j as was described above.

[0135] The voltage magnitudes and phases imposed on the charge terminals T1 and T2 can be adjusted in order to substantially synthesize the fields that are substantially modematched to the guided or Zenneck surface-waveguide mode of the lossy conducting medium 203 at the site of transmission given the local permittivity ε,, conductivity σ, and potentially

240240/2 other parameters of the lossy conducting medium 203. The waveguide mode of the surfaceguided wave is expressed in Equations (21), (22), and (23) set forth above. This surfacewaveguide mode has a radial surface current density expressed in Equation (20) in Amperes per meter.

[0136] It is understood that it can be difficult to synthesize fields that exactly match the surface-waveguide mode expressed in Equations (21), (22), and (23) set forth above. However, a guided surface wave can be launched if such fields at least approximate the surface-waveguide mode. According to various embodiments, the fields are synthesized to match the surface-waveguide mode within an acceptable engineering tolerance so as to launch a guided surface wave.

[0137] Likewise, it can be difficult to synthesize a radial surface current density that exactly matches the radial surface current density of the Zenneck surface-waveguide mode, where the synthesized radial surface current density results from the synthesized fields described above. According to various embodiments, the polyphase waveguide probes 200 can be adjusted to match the radial surface current density of the guided surface-waveguide mode within an acceptable engineering tolerance so as to launch a Zenneck surface wave mode. By creating specific charge distributions plus their images at complex distances, the various polyphase waveguide probes 200a-j set forth above excite surface currents, the fields of which are designed to approximately match a propagating Zenneck surface wave mode and a Zenneck surface wave is launched. By virtue of this complex image technique inherent in the various polyphase waveguide probes 200a-j described above, one is able to substantially mode-match to the surface waveguide modes that the guiding interface wants to support at the location of transmission. The guiding interface is the interface between Region 1 (FIG. 2) and Region 2 (FIG. 2) as described above. According to one embodiment, the guiding interface is the interface between the lossy conducting medium 203 presented by the Earth and the atmospheric medium as described above.

[0138] When the voltage magnitudes and phases imposed on the charge terminals T1 and T2 are adjusted so that they, plus their effective images at complex depths, excite complex surface currents whose fields synthesize the fields that substantially match the Zenneck surface-waveguide mode of the lossy conducting medium 203 at the site of transmission, by virtue of the Leontovich boundary condition, such fields will automatically substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium 203, resulting in zero reflection. This is the condition of wave matching at the boundary.

[0139] Referring next to FIGS. 8A, 8B, and 8C, shown are examples of graphs 300a, 300b, and 300c that depict field strength in Volts per meter as a function of distance in

240240/2 kilometers for purposes of comparison between a Zenneck surface wave and conventionally radiated fields. In addition, the various graphs 300a, 300b, and 300c illustrate how the distance of transmission of a Zenneck surface wave varies with the frequency of transmission.

[0140] Each graph 300a, 300b, and 300c depicts a corresponding guided field strength curve 303a, 303b, and 303c and corresponding radiated field strength curves 306a, 306b, and 306c. The guided field strength curves 303a, 303b, and 303c were generated assuming various parameters. Specifically, the graphs 300a, 300b, and 300c were calculated with a constant charge Q1 (FIG. 3) applied to the upper terminal T1 (FIG. 3) at frequencies of 10 MHz, 1 MHz, and 0.1 MHz, respectively. Constitutive parameters of εr = 15 and σ = 0.008 mhos/m, which are taken from the R-3 map for central Ohio set forth by the Federal Communications Commission (FCC), were assumed for purposes of calculation. The following table provides the assumed polyphase waveguide probe operating parameters for the generation of each of the guided field strength curve 303a, 303b, and 303c.

<td> Curve</td><td> Frequency (MHz)</td><td> λ</td><td> Height of Terminal T1 (m)</td><td> H/λ</td><td> Self-Cap of Terminal T1 (pF)</td><td> Voltage on Term T1 (V)</td><td> Charge Q1 (Coulombs)</td>

<td> 303a</td><td> 10</td><td> 30</td><td> 0.8</td><td> 0.027</td><td> 50</td><td> 10,000</td><td> 5x107־</td>

<td> 303b</td><td> 1.0</td><td> 300</td><td> 8</td><td> 0.027</td><td> 100</td><td> 5,000</td><td> 5x107־</td>

<td> 303c</td><td> 0.1</td><td> 3000</td><td> 8</td><td> 0.0027</td><td> 100</td><td> 5,000</td><td> 5x107־</td>

[0141] In order to have physically realizable operation, the height of terminal T1 was specified at Ht1 = 8 meters for f = 0.1 MHz and 1.0 MHz, but shortened to 0.8 meters at 10 MHz in order to keep the current distribution uniform. Also, the self-capacitance C1 of the terminal T1 was set to 100 pF for operation at f = 0.1 MHz and 1.0 MHz. This capacitance is unreasonably large for use at 10 MHz, so the self-capacitance C1 was reduced for that case. However, the resulting terminal charge Qt1 which is the controlling parameter for field strength was kept at the same value for all three guided field strength curve 303a, 303b, and 303c.

[0142] From the graphs it can be seen that the lower the frequency, the less the propagation attenuation and the fields reach out over greater distances. However, 34

240240/2 consistent with conservation of energy, the energy density decreases with distance. Another way to state that is that the higher the frequency, the smaller the region over which the energy is spread, so the greater the energy density. Thus, the “knee” of the Zenneck surface wave shrinks in range as the frequency is increased. Alternatively, the lower the frequency, the less the propagation attenuation and the greater the field strength of the Zenneck surface wave at very large distances from the site of transmission using a polyphase waveguide probe 200 (FIG. 6).

[0143] The Zenneck surface wave for each case is identified as guided field strength curves 303a, 303b, and 303c, respectively. The Norton ground wave field strength in Volts per meter for a short vertical monopole antenna, of the same height as the respective polyphase waveguide probe 200, with an assumed ground loss of 10 ohms, is represented by the radiated field strength curves 306a, 306b, and 306c, respectively. It is asserted that this is a reasonably realistic assumption for monopole antenna structures operating at these frequencies. The critical point is that a properly mode-matched polyphase waveguide probe launches a guided surface wave which dramatically outperforms the radiation field of any monopole at distances out to just beyond the “knee” in the guided field strength curves 303ac of the respective Zenneck surface waves.

[0144] Given the foregoing, according to one embodiment, the propagation distance of a guided surface wave varies as a function of the frequency of transmission. Specifically, the lower the transmission frequency, the less the exponential attenuation of the guided surface wave and, therefore, the farther the guided surface wave will propagate. As e-“^dmentioned above, the field strength of a guided surface wave falls off at a rate of -^=-, whereas the field strength of a radiated electromagnetic field falls off geometrically, proportional to 1/_d, where d is the distance in kilometers. Thus, each of the guided field strength curves 303a, 303b, and 303c feature a knee as was described above. As the frequency of transmission of a polyphase waveguide probe described herein decreases, the knee of the corresponding guided field strength curve 303a, 303b, and 303c will push to the right in the graph.

[0145] FIG. 8A shows a guided field strength curve 303a and a radiated field strength curve 306a generated at a frequency of 10 Megahertz. As shown, the guided surface wave falls off below 10 kilometers. In FIG. 8B, the guided field strength curve 303b and the radiated field strength curve 306b are generated at a frequency of 1 Megahertz. The guided field strength curve 303b falls off at approximately 100 Kilometers. Finally, in FIG. 8C, the guided field strength curve 303c and the radiated field strength curve 306c are generated at

240240/2 a frequency of 100 Kilohertz (which is .1 Megahertz). The guided field strength curve 303c falls off at between 4000-7000 Kilometers.

[0146] Note that if the frequency is low enough, it can be possible to transmit a guided surface wave around the entire Earth. It is believed that such frequencies can be at or below approximately 20-25 kilohertz. It should be noted that at such low frequencies, the lossy conducting medium 203 (FIG. 6) ceases to be a plane and becomes a sphere. Thus, when a lossy conducting medium 203 comprises a terrestrial medium, the calculation of guided field strength curves will be altered to take into account the spherical shape at low frequencies where the propagation distances approach size of the terrestrial medium.

[0147] Given the foregoing, next some general guidance is provided in constructing a polyphase waveguide probe 200 (FIG. 6) using the terrestrial medium of Earth as the lossy conducting medium 203 according to the various embodiments. As a practical approach, one might specify the frequency of operation and identify the desired field strength of the guided surface wave at a distance of interest from the respective polyphase waveguide probe 200 to be constructed.

[0148] Given these parameters, next one can determine the charge Q1 (FIG. 6) that is to be imposed on the upper charge terminal T1 (FIG. 6) in order to produce the desired field strength at the specified distance. To determine the charge Q1 needed, one would need to obtain the permittivity εr and the conductivity σ of the Earth at the transmission site. These values can be obtained by measurement or by reference to conductivity charts published, for example, by the Federal Communications Commission or the Committee Consultif International Radio (CCIR). When the permittivity ε,, conductivity σ, and the desired field strength at the specified distance are known, the needed charge Q1 can be determined by direct calculation of the field strength from Zenneck’s exact expressions set forth in Equations (21)-(23) above.

[0149] Once the needed charge Q1 is determined, next one would need to identify what self-capacitance C1 of the charge terminal T1 at what voltage V would produce the needed charge Q1 on the charge terminal T1. A charge Q on any charge terminal T is calculated as Q=CV. In one approach, one can choose what is deemed to be an acceptable voltage V that can be placed on the charge terminal T1, and then construct the charge terminal T1 so as to have the required self-capacitance C1 to achieve the needed charge Q1. Alternatively, in another approach, one could determine what is an achievable self-capacitance C1 by virtue of the specific construction of the charge terminal T1, and then raise the resulting charge terminal T1 to the required voltage V to achieve the needed charge Q1.

[0150] In addition, there is an issue of operational bandwidth that should be considered when determining the needed self-capacitance C1 of the charge terminal T1 and voltage V to

240240/2 be imposed on the charge terminal T1. Specifically, the bandwidth of the polyphase waveguide probes 200 described herein is relatively large. This results in a significant degree of flexibility in specifying the self-capacitance C1 or the voltage V as described above. However, it should be understood that as the self-capacitance C1 is reduced and the voltage V increased, the bandwidth of the resulting polyphase waveguide probe 200 will diminish.

[0151] Experimentally, it should be noted that a smaller self-capacitance C1 can make a given polyphase waveguide probe 200 more sensitive to small variations in the permittivity ε! or conductivity σ of the Earth at or near the transmission site. Such variation in the permittivity ε! or conductivity σ might occur due to variation in the climate given the transition between the seasons or due to changes in local weather conditions such as the onset of rain, drought, and/or other changes in local weather. Consequently, according to one embodiment, the charge terminal T1 can be specified so as to have a relatively large selfcapacitance C1 as is practicable.

[0152] Once the self-capacitance C1 of the charge terminal T1 and the voltage to be imposed thereon are determined, next the self-capacitance C2 and physical location of the second charge terminal T2 are to be determined. As a practical matter, it has been found easiest to specify the self-capacitance C2 of the charge terminal T2 to be the same as the self-capacitance C1 of the charge terminal T1. This can be accomplished by making the size and shape of the charge terminal T2 the same as the size and shape of the charge terminal T1. This would ensure that symmetry is maintained and will avoid the possibility of unusual phase shifts between the two charge terminals T1 and T2 that might negatively affect achieving a match with the complex Brewster angle as described above. The fact that the self-capacitances C1 and C2 are the same for both charge terminals T1 and T2 will result in the same voltage magnitudes on the charge terminals T1 and T2. However, it is understood that the self-capacitances C1 and C2 can differ, and the shape and size of the charge terminals T1 and T2 can differ.

[0153] To promote symmetry, the charge terminal T2 can be positioned directly under the charge terminal T1 along the vertical axis z (FIG. 6) as described above. Alternatively, it can be possible to position the charge terminal T2 at some other location with some resulting effect.

[0154] The distance between the charge terminals T1 and T2 should be specified so as to provide for the best match between the fields generated by the polyphase waveguide probe 200 and the guided surface-waveguide mode at the site of transmission. As a suggested starting point, this distance can be set so that the mutual capacitance CM (FIG. 6) between the charge terminals T1 and T2 is the same or less than the isolated capacitance C1

240240/2 on the charge terminal T1. Ultimately, one should specify the distance between the charge terminals T1 and T2 to make the mutual capacitance Cm as small as is practicable. The mutual capacitance Cm can be determined by measurement, and the charge terminals T1 and T2 can be positioned accordingly.

[0155] Next, the proper height h = H1-H2 (FIGS. 7A-J) of the polyphase waveguide probe 200 is determined. The so-called “image complex-depth” phenomenon comes into bearing here. This would entail consideration of the superposed fields on the surface of the Earth from the charge reservoirs T1 and T2 that have charges Q1 and Q2, and from the subsurface images of the charges Q1 and Q2 as height h is varied. Due to the significant number of variables to consider to ensure that a given polyphase waveguide probe 200 is mode matched with the guided surface-waveguide mode of the Earth at the site of transmission, a practical starting point is a height h at which the bound capacitance of each of the charge reservoirs T1 and T2 with respect to the ground is negligible such that the capacitance associated with the charge terminals T1 and T2 is essentially their isolated selfcapacitance C1 and C2, respectively.

[0156] Another consideration to take into account when determining the height h associated with a polyphase waveguide probe 200 is whether radiation is to be avoided. Specifically, as the height h of the polyphase waveguide probe 200 approaches an appreciable portion of a wavelength at the frequency of operation, the radiation resistance Rr will grow quadratically with height h and radiation will begin to dominate over the generation of a guided surface wave as described above. One benchmark set forth above that ensures the Zenneck surface wave will dominate over any radiation is to make sure the height h is less than 10% of the wavelength at the frequency of operation, although other benchmarks can be specified. In some cases, it can be desired to allow some degree of radiation to occur in addition to launching a guided surface wave, where the height h can be specified accordingly.

[0157] Next, the probe coupling circuit 209 (FIG. 6) is specified to provide for the voltage phase between the charge terminals T1 and T2. The voltage phase appears to have a significant effect on creating fields that mode-match the guided surface-waveguide mode at the site of transmission. Assuming that the placement of the charge terminals T1 and T2 is along the vertical z axis to promote symmetry, the probe coupling circuit 209 can be specified to provide for a voltage phase differential of 180 degrees on the charge terminals T1 and T2. That is to say, the probe coupling circuit 209 is specified so that voltage V on the charge terminal T1 is 180 degrees out of phase with respect to the voltage on the charge terminal T2.

240240/2

[0158] As was described above, one example approach is to place a coil L1a (FIG. 7A) between the charge terminals T1 and T2 as described above with reference to the polyphase waveguide probe 200a and adjust the coil L1a until the resulting system is electrically halfwave resonant. This would place a voltage V on the charge terminal T1 and voltage -V on the charge terminal T2 such that the largest voltages are placed on the charge terminals T1 and T2 180 degrees out of phase.

[0159] The excitation source 213 (FIG. 6) can then be coupled to the probe coupling circuit 209 and the output voltage adjusted to achieve the required voltage V to provide for the needed charge Q1 as described above. The excitation source 213 can be coupled via magnetic coupling, capacitive coupling, or conductive coupling (direct) to the probe coupling circuit 209. Note that the output of the excitation source 213 can be stepped up using a transformer or via some other approach if necessary. The location of the coil L1a can be at any location such as down on the ground by the excitation source 213. Alternatively, as per best RF practice, the coil L1a can be positioned directly between the charge reservoirs T1 and T2. Principles of impedance matching can be applied when coupling the excitation source 213 to the probe coupling circuit 209.

[0160] Note that the phase differential does not necessarily have to be 180 degrees. To this end, one has the option of raising and lowering one or both of the charge terminals T1 and/or T2, adjusting the voltages V on the charge terminals T1 and/or T2, or adjusting the probe coupling circuit 209 to adjust the voltage magnitudes and phases to create fields that most closely match the guided surface-waveguide mode in order to generate a guided surface wave.

Experimental Results

[0161] The above disclosures are supported by experimental measurements and documentation. With reference to FIG. 9, shown is a graph that presents the measured field strength of an electromagnetic field transmitted by one embodiment of an experimental polyphase waveguide probe measured on October 14, 2012 in Plymouth, New Hampshire. The frequency of transmission was 59 MHz with a voltage of 60 mV imposed on the charge terminal T1 of the experimental polyphase waveguide probe. The self-capacitance C1 of the experimental polyphase waveguide probe was 8.5 pF. The conductivity σ of the ground at the test site is 0.0002 mhos/m, and the permittivity εr of the ground at the test site was 5. These values were measured in situ at the frequency in use.

[0162] The graph includes a guided field strength curve 400 that is labeled a “Zenneck” curve at 80% efficiency and a radiated field strength curve 403 that is labeled a “Norton” curve at 100% radiation efficiency, which is the best possible. To this end, the radiated field

240240/2 strength curve 403 represents the radiated electromagnetic fields that would be generated by a % wavelength monopole antenna operating at a frequency of 59 MHz. The circles 406 on the graph represent measured field strengths produced by the experimental polyphase waveguide probe. The field strength measurements were performed with a NIST-traceable Potomac Instruments FIM-71 commercial VHF field strength meter. As can be seen, the measured field strengths fall along the theoretical guided field strength curve 400. These measured field strengths are consistent with the propagation of a guided or Zenneck surface wave.

[0163] Referring next to FIG. 10, shown is a graph that presents the measured phase of the transmitted electromagnetic wave from the experimental polyphase waveguide probe. The curve J(r) indicates the phase of the fields incident to the currents J1 and J2 with a transition between the currents J1 and J2 as shown. The curve 503 indicates the asymptote depicting the phase of the current J1, and the curve 506 indicates the asymptote depicting the phase of the current J2. A difference of approximately 45 degrees generally exists between the phases of the respective currents J1 and J2. The circles 509 indicate measurements of the phase of the current J(r) generated by the experimental polyphase waveguide probe operating at 59 MHz as with FIG. 9. As shown, the circles 509 fall along the curve J(r) indicating that there is a transition of the phase of the current J(r) from the curve 503 to the curve 506. This indicates that the phase of the current J(r) generated by the experimental polyphase waveguide probe transitions from the phase generated by the close-in current J1 to the far-out current J2. Thus, these phase measurements are consistent with the phase with the presence of a guided or Zenneck surface wave.

[0164] With reference to FIG. 11 shown is a graph of a second set of measured data that depicts the field strength of an electromagnetic field transmitted by a second embodiment of an experimental polyphase waveguide probe measured on November 1, 2003 in the vicinity of Ashland, New Hampshire and across the region north of Lake Winnipesaukee. The frequency of transmission was 1850 kHz with a voltage of 1250 V imposed on the charge terminal T1 of the experimental polyphase waveguide probe. The experimental polyphase waveguide probe had a physical height of H1 = 2 meters. The selfcapacitance C1 of the experimental polyphase waveguide probe in this experiment, which was a flat conducting disk of 1 meter radius, was measured to be 70 pF. The polyphase waveguide probe was arranged as illustrated in FIG. 7J, with spacing h = 1 meter and the height of the charge terminal T2 above the ground (the lossy conducting medium 203) being H2 = 1 meter. The average conductivity σ of the ground in the vicinity of experimentation was 0.006 mhos/m, and the relative permittivity εr of the ground was on the order of 15. These were determined at the frequency in use.

240240/2

[0165] The graph includes a guided field strength curve 600 that is launched by the experimental polyphase waveguide probe, labeled as “Zenneck” curve at 85% efficiency, and a radiated field strength curve 603 that is labeled a “Norton” curve as radiated from a resonated monopole of the same height, H2 = 2 meters, over a ground screen composed of 20 radial wires equally spaced and of length 200 feet each. To this end, the radiated field strength curve 603 represents the conventional Norton ground wave field radiated from a conventional stub monopole antenna operating at a frequency of 1850 kHz over the lossy Earth. The circles 606 on the graph represent measured field strengths produced by the experimental polyphase waveguide probe.

[0166] As can be seen, the measured field strengths fall closely along the theoretical Zenneck guided field strength curve 600. Special mention of the field strength measured at the r = 7 mile point may be made. This field strength data point was measured adjacent to the shore of a lake, and this may account for the data departing slightly above the theoretical Zenneck guided field strength curve 600, i.e. the constitutive parameters, εr and/or σ, at that location are likely to have departed significantly from the path-average constitutive parameters.

[0167] The field strength measurements were performed with a NIST-traceable Potomac Instruments FIM-41 MF/HF field strength meter. The measured field strength data are consistent with the presence of a guided or Zenneck surface wave. It is apparent from the experimental data that the measured field strengths observed at distances less than 15 miles could not possibly be due to conventional Norton ground wave propagation, and can only be due to guided surface wave propagation launched by the polyphase probe operating as disclosed above. Under the given 1.85 MHz experimental conditions, out at 20 miles it appears that a Norton ground wave component has finally overtaken the Zenneck surface wave component.

[0168] A comparison of the measured Zenneck surface wave data shown in FIG. 9 at 59 MHz with the measured data in FIG. 11 at 1.85 MHz illustrates the great advantage of employing a polyphase waveguide probe according to the various embodiments at lower frequencies.

[0169] These experimental data confirm that the present polyphase waveguide probes, comprising a plurality of appropriately phased and adjusted charge terminals, as taught herein, induce a phase-advanced surface current with a unique phase boost of arg(//), and whose fields synthesize surface illumination at the complex Brewster angle for the lossy boundary as disclosed herein. The consequence is the efficient launching of cylindrical Zenneck-like wave propagation, guided by the boundary surface as an evanescent, single41

240240/2 _e 'd conductor radial transmission-line mode, which attenuates as -n=, not as a radiation field, Jd which would decrease as 1/d due to geometrical spreading.

[0170] Referring next to FIGS. 12A, 12B, and 13, shown are examples of generalized receive circuits for using the surface-guided waves in wireless power delivery systems. FIGS. 12A and 12B include a linear probe 703 and a tuned resonator 706. FIG. 13 is a magnetic coil 709 according to various embodiments of the present disclosure. According to various embodiments, each one of the linear probe 703, the tuned resonator 706, and the magnetic coil 709 can be employed to receive power transmitted in the form of a guided surface wave on the surface of a lossy conducting medium 203 (FIG. 6) according to various embodiments. As mentioned above, in one embodiment the lossy conducting medium 203 comprises a terrestrial medium.

[0171] With specific reference to FIG. 12A, the open-circuit terminal voltage at the output terminals 713 of the linear probe 703 depends upon the effective height of the linear probe 703. To this end, the terminal point voltage can be calculated as

Vt= C^eE_inc^dl, (43) where Einc is the strength of the electric field in vector on the linear probe 703 in Volts per meter, dl is an element of integration along the direction of the linear probe 703, and he is the effective height of the linear probe 703. An electrical load 716 is coupled to the output terminals 713 through an impedance matching network 719.

[0172] When the linear probe 703 is subjected to a guided surface wave as described above, a voltage is developed across the output terminals 713 that can be applied to the electrical load 716 through a conjugate impedance matching network 719 as the case may be. In order to facilitate the flow of power to the electrical load 716, the electrical load 716 should be substantially impedance matched to the linear probe 703 as will be described below.

[0173] Referring to FIG. 12B, the tuned resonator 706 includes a charge terminal Tr that is elevated above the lossy conducting medium 203. The charge terminal TR has a selfcapacitance Cr. In addition, there can also be a bound capacitance (not shown) between the charge terminal Tr and the lossy conducting medium 203 depending on the height of the charge terminal TR 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 of a polyphase waveguide probe 200.

240240/2

[0174] The tuned resonator 706 also includes a coil Lr. One end of the coil Lr is coupled to the charge terminal Tr, and the other end of the coil Lr is coupled to the lossy conducting medium 203. To this end, the tuned resonator 706 (which can also be referred to as tuned resonator Lr-Cr) comprises a series-tuned resonator as the charge terminal Cr and the coil Lr are situated in series. The tuned resonator 706 is tuned by adjusting the size and/or height of the charge terminal Tr, and/or adjusting the size of the coil Lr so that the reactive impedance of the structure is substantially eliminated.

[0175] For example, the reactance presented by the self-capacitance Cr is calculated 1 as ך—. Note that the total capacitance of the tuned resonator 706 can also include JMC_r capacitance between the charge terminal Tr and the lossy conducting medium 203, where the total capacitance of the tuned resonator 706 can be calculated from both the selfcapacitance Cr and any bound capacitance as can be appreciated. According to one embodiment, the charge terminal Tr can be raised to a height so as to substantially reduce or eliminate any bound capacitance. The existence of a bound capacitance can be determined from capacitance measurements between the charge terminal Tr and the lossy conducting medium 203.

[0176] The inductive reactance presented by a discrete-element coil Lr can be calculated as /«A, where L is the lumped-element inductance of the coil Lr. If the coil Lr is a distributed element, its equivalent terminal-point inductive reactance can be determined by conventional approaches. To tune the tuned resonator 706, one would make adjustments so that the inductive reactance presented by the coil Lr equals the capacitive reactance presented by the tuned resonator 706 so that the resulting net reactance of the tuned resonator 706 is substantially zero at the frequency of operation. An impedance matching network 723 can be inserted between the probe terminals 721 and the electrical load 726 in order to effect a conjugate-match condition for maxim power transfer to the electrical load 726.

[0177] When placed in the presence of a guided surface wave, generated at the frequency of the tuned resonator 706 and the conjugate matching network 723, as described above, maximum power will be delivered from the surface guided wave to the electrical load 726. That is, once conjugate impedance matching is established between the tuned resonator 706 and the electrical load 726, power will be delivered from the structure to the electrical load 726. To this end, an electrical load 726 can be coupled to the tuned resonator 706 by way of magnetic coupling, capacitive coupling, or conductive (direct tap) coupling. The elements of the coupling network can be lumped components or distributed elements as can be appreciated. In the embodiment shown in FIG. 12B, magnetic coupling is employed where a coil Ls is positioned as a secondary relative to the coil Lr that acts as a transformer 43

240240/2 primary. The coil Ls can be link coupled to the coil Lr by geometrically winding it around the same core structure and adjusting the coupled magnetic flux as can be appreciated. In addition, while the tuned resonator 706 comprises a series-tuned resonator, a parallel-tuned resonator or even a distributed-element resonator can also be used.

[0178] Referring to FIG. 13, the magnetic coil 709 comprises a receive circuit that is coupled through an impedance coupling network 733 to an electrical load 736. In order to facilitate reception and/or extraction of electrical power from a guided surface wave, the magnetic coil 709 can be positioned so that the magnetic flux of the guided surface wave, H&lt;p , passes through the magnetic coil 709, thereby inducing a current in the magnetic coil 709 and producing a terminal point voltage at its output terminals 729. The magnetic flux of the guided surface wave coupled to a single turn coil is expressed by

Ψ = ^_Α(:εμ_Γμ_οϊΙ·ηάΑ (44) where Ψ is the coupled magnetic flux, μr is the effective relative permeability of the core of the magnetic coil 709, μo 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 Acs is the area enclosed by each loop. For an N-turn magnetic coil 709 oriented for maximum coupling to an incident magnetic field that is uniform over the cross-sectional area of the magnetic coil 709, the open-circuit induced voltage appearing at the output terminals 729 of the magnetic coil 709 is ^V = -^N~^x -jWrV0HA_cs, (45) where the variables are defined above. The magnetic coil 709 can be tuned to the guided wave frequency either as a distributed resonator or with an external capacitor across its output terminals 729, as the case may be, and then impedance-matched to an external electrical load 736 through a conjugate impedance matching network 733.

[0179] Assuming that the resulting circuit presented by the magnetic coil 709 and the electrical load 736 are properly adjusted and conjugate impedance matched, via impedance matching network 733, then the current induced in the magnetic coil 709 can be employed to

240240/2 optimally power the electrical load 736. The receive circuit presented by the magnetic coil 709 provides an advantage in that it does not have to be physically connected to the ground.

[0180] With reference to FIGS. 12A, 12B, and 13, the receive circuits presented by the linear probe 703, the tuned resonator 706, and the magnetic coil 709 each facilitate receiving electrical power transmitted from any one of the embodiments of polyphase waveguide probes 200 described above. To this end, the energy received can be used to supply power to an electrical load 716/726/736 via a conjugate matching network as can be appreciated. This contrasts with the signals that can 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.

[0181] It is also characteristic of the present guided surface waves generated using the polyphase waveguide probes 200 described above that the receive circuits presented by the linear probe 703, the tuned resonator 706, and the magnetic coil 709 will load the excitation source 213 (FIG. 3) that is applied to the polyphase 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 polyphase 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.

[0182] Thus, together a given polyphase waveguide probe 200 and receive circuits in the form of the linear probe 703, the tuned resonator 706, and/or the magnetic coil 709 can together make up a wireless distribution system. Given that the distance of transmission of a guided surface wave using a polyphase 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.

[0183] 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

240240/2 Hz, all the generator power goes only to the desired electrical load. When the electrical load demand is terminated, the source power generation is relatively idle.

[0184] Referring next to FIG. 14A shown is a schematic that represents the linear probe 703 and the tuned resonator 706. FIG 14B shows a schematic that represents the magnetic coil 709. The linear probe 703 and the tuned resonator 706 can each be considered a Thevenin equivalent represented by an open-circuit terminal voltage source VS and a dead network terminal point impedance ZS. The magnetic coil 709 can be viewed as a Norton equivalent represented by a short-circuit terminal current source Is and a dead network terminal point impedance Zs. Each electrical load 716/726/736 (FIGS. 12A-B and FIG. 13) can be represented by a load impedance ZL. The source impedance ZS comprises both real and imaginary components and takes the form Zs = Rs + jXs.

[0185] According to one embodiment, the electrical load 716/726/736 is impedance matched to each receive circuit, respectively. Specifically, each electrical load 716/726/736 presents through a respective impedance matching network 719/723/733 a load on the probe network specified as Zl' expressed as Zl' = Rl' + j Xl', which will be equal to Zl' = Zs * = Rs - j Xs, where the presented load impedance Zl' is the complex conjugate of the actual source impedance Zs. The conjugate match theorem, which states that if, in a cascaded network, a conjugate match occurs at any terminal pair then it will occur at all terminal pairs, then asserts that the actual electrical load 716/726/736 will also see a conjugate match to its impedance, Zl'. See Everitt, W.L. and G.E. Tanner, Communication Engineering, McGrawHill, 3^rd edition, 1956, p. 407. This ensures that the respective electrical load 716/726/736 is impedance matched to the respective receive circuit and that maximum power transfer is established to the respective electrical load 716/726/736.

[0186] In addition to the forgoing, the various embodiments of the present disclosure include, but are not limited to, the embodiments set forth in the following clauses.

[0187] Clause 1. A method, comprising the step of: transmitting energy conveyed in a form of a guided surface-waveguide mode along a surface of a terrestrial medium by exciting a polyphase waveguide probe.

[0188] Clause 2. The method of clause 1, wherein the step of transmitting energy conveyed in the form of the guided surface-waveguide mode along the surface of the terrestrial medium by exciting the polyphase waveguide probe further comprises the step of synthesizing a plurality of fields that substantially match the guided surface-waveguide mode of the terrestrial medium.

[0189] Clause 3. The method of any one of clauses 1 or 2, wherein a radial surface current density of the guided surface-waveguide mode is substantially expressed by

240240/2

J_p (ρ,φ,ΰ) = Io- H&lt;²&gt; (-j pp), where γ is a surface wave radial propagation constant given by = jyk 0 ^{+ u} 2 and u2 is a vertical propagation constant given by u ₂ , . ^o . , where x '1 + ^{p -} jx) σ is a conductivity of the terrestrial medium, ω is equal to 2nf, where f is a frequency of excitation of the polyphase waveguide probe, εο is a permittivity of free space, εΓ is a relative permittivity of the terrestrial medium, and a free-space wave number ko is equal to 2^π, where

Λ-0 λo is a free-space wavelength of the polyphase waveguide probe, j is equal to V-1, p is a radial coordinate, z is a vertical coordinate normal to the terrestrial medium, φ is an azimuthal coordinate, Io is a net polyphase probe current, and H1⁽²⁾( -j γ p) is a Hankel function of a second kind and a first order with a complex argument -j γ p for an β⁺⁷'^ωί time variation, where t is time.

[0190] Clause 4. The method of any one of clauses 1-3, wherein the guided surfacewaveguide mode is substantially expressed as

Ηφ = ^-= ' eי^u2z H' (-jpp) , ^EP = ^-I (Y::} e ^uu2z^H(2)(־ j-p⁾, and

Ez = -I [i- ] e u u Η 0²⁾ (יj PP&gt;

where Η_φ is an azimuthal magnetic field strength, Ep is a radial electric field strength, Ez is a vertical electric field strength, where γ is a surface wave radial propagation constant given by = = j^k} + u 2 and u2 is a vertical propagation constant given by u₂ = . ° , , where x

5/1 + P j jx) σ

------, σ is a conductivity of the terrestrial ® ^po medium, ω is equal to 2nf, where f is a frequency of excitation of the polyphase waveguide probe, εο is a permittivity of free space, εΓ is a relative permittivity of the terrestrial medium, 2 2π and a free-space wave number ko is equal to — , where λo is a free-space wavelength of the polyphase waveguide probe, j is equal to V-1, p is a radial coordinate, z is a vertical coordinate normal to the terrestrial medium, φ is an azimuthal coordinate, Io is a net polyphase probe current, H1⁽²⁾( -j p p) is a Hankel function of a second kind and first order

240240/2 with complex argument -j γρ, and H0⁽²⁾( -]γ p) is a Hankel function of a second kind and zero order with complex argument - j γρ for an β⁺⁷'^ωί time variation, where t is time.

[0191] Clause 5. The method of any one of clauses 2-4, wherein the fields substantially synthesize a wave front incident at a complex Brewster angle of the terrestrial medium, resulting in a negligible reflection.

[0192] Clause 6. The method of any one of clauses 1-5 wherein the polyphase waveguide probe comprises a plurality of charge terminals, the method further comprising the step of adjusting the polyphase waveguide probe by adjusting a height of at least one of the charge terminals.

[0193] Clause 7. The method of any one of clauses 1-5, wherein the polyphase waveguide probe comprises a plurality of charge terminals, the method further comprising the step of tuning the polyphase waveguide probe by adjusting a distance between the charge terminals.

[0194] Clause 8. The method of any one of clauses 1-5, wherein the polyphase waveguide probe comprises a plurality of charge terminals, the method further comprising the step of tuning the polyphase waveguide probe by adjusting a size of at least one of the charge terminals.

[0195] Clause 9. The method of any one of clauses 1-5, wherein the polyphase waveguide probe comprises a plurality of charge terminals, the method further comprising the step of tuning the polyphase waveguide probe by adjusting a probe coupling circuit coupled to the charge terminals.

[0196] Clause 10. An apparatus, comprising: a polyphase waveguide probe configured to create a plurality of resultant fields that are substantially mode-matched to a Zenneck surface wave mode on a surface of a lossy conducting medium.

[0197] Clause 11. The apparatus of clause 10, wherein the lossy conducting medium further comprises a terrestrial medium.

[0198] Clause 12. The apparatus of any one of clauses 10 or 11, wherein a radiation resistance of the polyphase waveguide probe is substantially zero.

[0199] Clause 13. The apparatus of any one of clauses 10-12, wherein a height of the polyphase waveguide probe is less than at an operating frequency of the polyphase waveguide probe, where λ is a wavelength at the operating frequency.

[0200] Clause 14. The apparatus of any one of clauses 10-13, wherein the resultant fields substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium, resulting in substantially zero reflection.

240240/2

[0201] Clause 15. The apparatus of any one of clauses 10-14, wherein an excitation source is electrically coupled to the polyphase waveguide probe.

[0202] Clause 16. The apparatus of any one of clauses 10-15, wherein a radial surface current density of the Zenneck surface wave mode is substantially expressed by

J_p (ρ,φ,Ο) = IT H&lt;²&gt; (-j γρ), where γ is a surface wave radial propagation constant given by γ = j^k \ + u 2 and u2 is a vertical propagation constant given by u ₂ — 1 FL· -.

. , where ^x v ^{1 + (s}r ^— J^x) is a conductivity of the lossy conducting medium, ω is equal to 2nf, where f is a frequency of excitation of the polyphase waveguide probe, εο is a permittivity of free space, εΓ is a relative permittivity of the lossy conducting medium, and a free-space wave number ko is equal to 2^,

Λ-0 where λο is a free-space wavelength of the polyphase waveguide probe, j is equal to V—1, p is a radial coordinate, z is a vertical coordinate normal to the lossy conducting medium, φ is an azimuthal coordinate, Io is a net polyphase probe current, and Hr²'( — jγρ) is a Hankel function of a second kind and first order with complex argument — j γ p for an β⁺⁷'^ωί time variation, where t is time.

[0203] Clause 17. The apparatus of any one of clauses 10-16, wherein the Zenneck surface wave mode is substantially expressed as

Ηφ = ^γ ' eי^u2z H(²(-j/p) , ^EP = {7ύ2ΓΊ euu²z^H 1⁽²⁾(- j'P), and \^ ^{j UJS} O J

Ez = '\ .^γ 1 e u² zH 02’⁽—j rP) where Ηφ is an azimuthal magnetic field strength, Ep is a radial electric field strength, Ez is a vertical electric field strength, where γ is a surface wave radial propagation constant given by T = j^k²_g + u 2 and u2 is a vertical propagation constant given by u₂ = . ^o , , where x

1/ג + (tr — jx) σ

------, σ is a conductivity of the lossy ® ^So conducting medium, ω is equal to 2nf, where f is a frequency of excitation of the polyphase waveguide probe, εο is a permittivity of free space, ε, is a relative permittivity of the

240240/2 2π conducting lossy medium, and a free-space wave number ko is equal to — , where λο is a free-space wavelength of the polyphase waveguide probe, j is equal to V—1, ρ is a radial coordinate, z is a vertical coordinate normal to the lossy conducting medium, φ is an azimuthal coordinate, Io is a net polyphase probe current, H1⁽²⁾( -j γ p) is a Hankel function of a second kind and first order with complex argument -j γρ, and H0⁽²⁾( -jy p) is a Hankel function of a second kind and zero order with complex argument -jγ p for an β⁺⁷'^ωίtime variation, where t is time.

[0204] Clause 18. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises a plurality of charge terminals, the polyphase waveguide probe being further configured to impose a plurality of voltage magnitudes and a plurality of phases on the charge terminals.

[0205] Clause 19. The apparatus of any one of clauses 10-18, wherein the polyphase waveguide probe further comprises a probe coupling circuit coupled to the charge terminals, the probe coupling circuit being configured to impose the voltage magnitudes and the phases on the charge terminals.

[0206] Clause 20. The apparatus of any one of clauses 10-19, wherein both the voltage magnitudes and the phases vary as a function of a geometrical position of the charge terminals relative to each other.

[0207] Clause 21. The apparatus of any one of clauses 10-20, wherein both the voltage magnitudes and the phases vary as a function of a geometrical position of each of the charge terminals relative to the lossy conducting medium.

[0208] Clause 22. The apparatus of any one of clauses 10-21, wherein both the voltage magnitudes and the phases vary as a function of a physical size of the charge terminals.

[0209] Clause 23. The apparatus of any one of clauses 10-22, wherein both the voltage magnitudes and the phases vary as a function of an electrical circuit.

[0210] Clause 24. The apparatus of any one of clauses 10-23, wherein the charge terminals are positioned along an axis.

[0211] Clause 25. The apparatus of any one of clauses 10-24, wherein an excitation source is series-coupled to the polyphase waveguide probe.

[0212] Clause 26. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises a coil coupled to both a first charge terminal and a second charge terminal.

[0213] Clause 27. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises a first coil and a second coil, wherein the first coil is

240240/2 coupled to both a first charge terminal and a second charge terminal, and the second coil is coupled to the second charge terminal and the lossy conducting medium.

[0214] Clause 28. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises: a coil having a first end coupled to a first charge terminal and a second end coupled to the lossy conducting medium; and a tap is coupled to the second charge terminal and positioned along the coil.

[0215] Clause 29. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises a first coil and a second coil, wherein the first coil is coupled to both a first charge terminal and the lossy conducting medium, and the second coil is coupled to both a second charge terminal and the lossy conducting medium.

[0216] Clause 30. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises a coil coupled to a first charge terminal and the lossy conducting medium, and a resistor coupled to a second charge terminal and the lossy conducting medium.

[0217] Clause 31. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises a coil coupled to both a first charge terminal and a ground screen.

[0218] Clause 32. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises: a first coil coupled to both a first charge terminal and a second charge terminal; a second coil coupled to the lossy conducting medium and a capacitance; and the capacitance being further coupled to the second charge terminal.

[0219] Clause 33. The apparatus of clause 32, wherein the capacitance is a variable capacitance.

[0220] Clause 34. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises: a first coil coupled to both a first charge terminal and a second charge terminal; and a second coil coupled to a terminal and the lossy conducting medium, wherein the terminal is positioned relative to the second charge terminal resulting in a coupling capacitance between the terminal and the second charge terminal.

[0221] Clause 35. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises: a first coil coupled to both a first charge terminal and a second charge terminal; a second coil coupled to a terminal, wherein the terminal is positioned relative to the second charge terminal resulting in a coupling capacitance between the terminal and the second charge terminal; and wherein the excitation source is coupled to the second coil and the lossy conducting medium.

240240/2

[0222] Clause 36. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises a plurality of charge terminals, wherein respective ones of the terminals comprise a sphere or a disk.

[0223] Clause 37. The apparatus of any one of clauses 10-17, wherein the polyphase waveguide probe further comprises: a coil having a first end coupled to a first charge terminal and a second end coupled to a second charge terminal; and a tap coupled to the lossy conducting medium and positioned along the coil.

[0224] Clause 38. The apparatus of any one of clauses 26-34, 36, and 37, further comprising an excitation source coupled to a primary coil, wherein the primary coil is magnetically coupled to the polyphase waveguide probe.

[0225] Clause 39. An apparatus, comprising: a polyphase waveguide probe configured to create a plurality of resultant fields; and wherein the resultant fields are substantially mode-matched to a Zenneck surface wave mode on a surface of a terrestrial medium.

[0226] Clause 40. The apparatus of clause 39, wherein the resultant fields substantially synthesize a wave incident at a complex Brewster angle of the terrestrial medium, resulting in substantially zero reflection.

[0227] Clause 41. The apparatus of any one of clauses 39 or 40, wherein a radial surface current density of the Zenneck surface wave mode is substantially expressed by

J_p (p,£0) = I-ot- H &lt;²⁾ (-j γρ), where γ is a surface wave radial propagation constant given by γ j^k 0 + u 2 and u2 is a vertical propagation constant given by u ₂ = ^o , where ^x = , σ + (s_r - jx) &amp;So is a conductivity of the terrestrial medium, ω is equal to 2nf, where f is a frequency of excitation of the polyphase waveguide probe, εο is a permittivity of free space, εΓ is a relative permittivity of the terrestrial medium, and a free-space wave number ko is equal to 2^, where

Λ-0 λo is a free-space wavelength of the polyphase waveguide probe, j is equal to V-1, ρ is a radial coordinate, z is a vertical coordinate normal to the terrestrial medium, φ is an azimuthal coordinate, Io is a net polyphase probe current, and H1⁽²⁾( -j γ p) is a Hankel function of a second kind and first order with complex argument -j γ p for an β⁺⁷'^ωί time variation, where t is time.

[0228] Clause 42. The apparatus of any one of clauses 39-41, wherein the Zenneck surface wave mode is substantially expressed as

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Η = \' e u u ·z«i^2&gt; (־ j1P) , ^EP = ^{11 (}J.:Ίe2יי־z^H 2 '^]rP)^and t j o j

Ez = -¹IT GuT ] e u u H 02&gt; (-.j 1P) where Ηφ is an azimuthal magnetic field strength, Ep is a radial electric field strength, Ez is a vertical electric field strength, where γ is a surface wave radial propagation constant given by 1 ⁼ j^kl ⁺ u 2 and u2 is a vertical propagation constant given by u2 = . _z^o , where x

1/ג + (tr j jx) σ is a conductivity of the terrestrial medium, ω is equal to 2nf, where f is a frequency of excitation of the polyphase waveguide probe, to is a permittivity of free space, tr is a relative permittivity of the terrestrial medium, 2 2π and a free-space wave number ko is equal to — , where λo is a free-space wavelength of the polyphase waveguide probe, j is equal to V-1, p is a radial coordinate, z is a vertical coordinate normal to the terrestrial medium, φ is an azimuthal coordinate, Io is a net polyphase probe current, H1⁽²⁾( -j γ p) is a Hankel function of a second kind and first order with complex argument -j γρ, and H0⁽²⁾( - jr p) is a Hankel function of a second kind and zero order with complex argument -j γ p for an β⁺⁷'^ωί time variation, where t is time.

[0229] Clause 43. The apparatus of any one of clauses 39-42, wherein the polyphase waveguide probe further comprises a pair of charge terminals, the polyphase waveguide probe being further configured to impose a plurality of voltage magnitudes and a plurality of phases on the charge terminals.

[0230] Clause 44. The apparatus of clause 43, wherein the polyphase waveguide probe further comprises distribution circuitry coupled to the charge terminals.

[0231] Clause 45. The apparatus of clause 44, wherein an electrical source is coupled to the distribution circuitry.

[0232] Clause 46. The apparatus of any one of clauses 44 or 45, wherein the distribution circuitry further comprises a coil.

[0233] Clause 47. The apparatus of clause 43, wherein the polyphase waveguide probe further comprises a coil coupled between the charge terminals.

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[0234] Clause 48. The apparatus of clause 43, wherein both the voltage magnitudes and the phases vary as a function of a geometrical position of the charge terminals relative to each other.

[0235] Clause 49. The apparatus of clause 43, wherein both the voltage magnitudes and the phases vary as a function of a geometrical position of each of the charge terminals relative to the terrestrial medium.

[0236] Clause 50. The apparatus of clause 43, wherein both the voltage magnitudes and the phases vary as a function of a physical size of the charge terminals.

[0237] Clause 51. The apparatus of clause 43, wherein both the voltage magnitudes and the phases vary as a function of an electrical circuit.

[0238] Clause 52. The apparatus of any one of clauses 39-44 and 46-51, wherein an excitation source is electrically coupled to the polyphase waveguide probe.

[0239] Clause 53. A method, comprising the step of: positioning a receive circuit relative to a terrestrial medium; and receiving, via the receive circuit, energy conveyed in a form of a Zenneck surface wave on a surface of the terrestrial medium.

[0240] Clause 54. The method of clause 53, wherein an electrical load coupled to the receive circuit loads an excitation source coupled to a polyphase waveguide probe that generates the Zenneck surface wave.

[0241] Clause 55. The method of any one of clauses 53 or 54, wherein the energy further comprises electrical power, and the method further comprises the step of applying the electrical power to an electrical load coupled to the receive circuit, where the electrical power is used as a power source for the electrical load.

[0242] Clause 56. The method of any one of clauses 53-55, further comprising the step of impedance-matching an electrical load to the receive circuit.

[0243] Clause 57. The method of any one of clauses 53-56, further comprising the step of establishing a maximum power transfer from the receive circuit to the electrical load.

[0244] Clause 58. The method of any one of clauses 53-57, wherein the receive circuit further comprises a magnetic coil.

[0245] Clause 59. The method of any one of clauses 53-57, wherein the receive circuit further comprises a linear probe.

[0246] Clause 60. The method of any one of clauses 53-57, wherein the receive circuit further comprises a tuned resonator coupled to the terrestrial medium.

[0247] Clause 61. An apparatus, comprising: a receive circuit that receives energy conveyed in a form of a Zenneck surface wave along a surface of a lossy conducting medium.

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[0248] Clause 62. The apparatus of clause 61, wherein the lossy conducting medium further comprises a terrestrial medium.

[0249] Clause 63. The apparatus of any one of clauses 61 or 62, wherein an electrical load coupled to the receive circuit loads an excitation source coupled to a polyphase waveguide probe that generates the Zenneck surface wave.

[0250] Clause 64. The apparatus of any one of clauses 61 or 62, wherein the energy further comprises electrical power, and the receive circuit is coupled to an electrical load, and wherein the electrical power is applied to the electrical load, the electrical power being employed as a power source for the electrical load.

[0251] Clause 65. The apparatus of any one of clauses 63 or 64, wherein the electrical load is impedance-matched with the receive circuit.

[0252] Clause 66. The apparatus of any one of clauses 61-65, wherein the receive circuit further comprises a magnetic coil.

[0253] Clause 67. The apparatus of any one of clauses 61-65, wherein the receive circuit further comprises a linear probe.

[0254] Clause 68. The apparatus of any one of clauses 61-65, wherein the receive circuit further comprises a tuned resonator.

[0255] Clause 69. The apparatus of clause 68, wherein the tuned resonator comprises a series tuned resonator.

[0256] Clause 70. The apparatus of clause 68, wherein the tuned resonator comprises a parallel tuned resonator.

[0257] Clause 71. The apparatus of clause 68, wherein the tuned resonator comprises a distributed tuned resonator.

[0258] Clause 72. A power transmission system, comprising: a polyphase waveguide probe that transmits electrical energy in a form of a guided surface wave along a surface of a terrestrial medium; and a receive circuit that receives the electrical energy.

[0259] Clause 73. The power transmission system of clause 72, wherein an electrical load coupled to the receive circuit loads the polyphase waveguide probe.

[0260] Clause 74. The power transmission system of clause 72, wherein an electrical load is coupled to the receive circuit and the electrical energy is used as a power source for the electrical load.

[0261] Clause 75. The power transmission system of any one of clauses 73 or 74, wherein the electrical load is impedance-matched to the receive circuit.

[0262] Clause 76. The power transmission system of any one of clauses 73 or 74, wherein a maximum power transfer is established from the receive circuit to the electrical load.

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[0263] Clause 77. The power transmission system of any one of clauses 72-76, wherein the receive circuit further comprises a magnetic coil.

[0264] Clause 78. The power transmission system of any one of clauses 72-76, wherein the receive circuit further comprises a linear probe.

[0265] Clause 79. The power transmission system of any one of clauses 72-76, wherein the receive circuit further comprises a tuned resonator.

[0266] Clause 80. The power transmission system of any one of clauses 72-79, wherein the polyphase waveguide probe is configured to create a plurality of resultant fields that are substantially mode-matched to a guided surface wave mode on the surface of the terrestrial medium.

[0267] Clause 81. The power transmission system of any one of clauses 72-80, wherein a radiation resistance of the polyphase waveguide probe is substantially zero.

[0268] Clause 82. The power transmission system of any one of clauses 72-81, wherein a height of the polyphase waveguide probe is less than at an operating frequency of the polyphase waveguide probe, where λ is a wavelength at the operating frequency.

[0269] Clause 83. The power transmission system of clause 80, wherein the resultant fields substantially synthesize a wave front incident at a complex Brewster angle of the lossy medium, resulting in substantially zero reflection.

[0270] Clause 84. The power transmission system of any one of clauses 72-83, wherein an excitation source is electrically coupled to the polyphase waveguide probe.

[0271] Clause 85. The power transmission system of clause 80, wherein a radial surface current density of the guided surface wave mode is substantially expressed by J _p (p,£0) = I-^ H&lt;²⁾ (- j γρ), where γ is a surface wave radial propagation constant given by γ j^k 0 + u 2 and u2 is a vertical propagation constant given by u ₂ , _z^o . , where ^x

V ^{1 + (s}r ^{J j X)} is a conductivity of the lossy medium, ω is equal to 2nf, where f is a frequency of excitation of the polyphase waveguide probe, εο is a permittivity of free space, εΓ is a relative permittivity of the lossy medium, and a free-space wave number ko is equal to 2^, where λο is

Λ-0 a free-space wavelength of the polyphase waveguide probe, j is equal to V-1, ρ is a radial coordinate, z is a vertical coordinate normal to the lossy medium, φ is an azimuthal coordinate, Io is a net polyphase probe current, and Hr²'(-j γ p) is a Hankel function of a

240240/2 second kind and first order with complex argument -j γρ for an β⁺⁷'^ωί time variation, where t is time.

[0272] Clause 86. The power transmission system of clause 80, wherein the guided surface wave mode is substantially expressed as

H = !‘p e ^- u² Η (- j&gt;P), ^EP = - (ju:) e ־ u^{2 (2) (-} ׳γρ⁾ and e = -(^)^e-u h»²⁾(-׳γρ) where Ηψ is an azimuthal magnetic field strength, Ep is a radial electric field strength, Ez is a vertical electric field strength, where γ is a surface wave radial propagation constant given by 1 = j^o ⁺ u 2 and u2 is a vertical propagation constant given by u₂ = . ^o , , where x

1/ג + pr - jx) σ

------, σ is a conductivity of the lossy medium, ® ^po ω is equal to 2nf, where f is a frequency of excitation of the polyphase waveguide probe, εο is a permittivity of free space, ε, is a relative permittivity of the lossy medium, and a free2π space wave number ko is equal to — , where λo is a free-space wavelength of the polyphase waveguide probe, j is equal to V-1, p is a radial coordinate, z is a vertical coordinate normal to the lossy medium, φ is an azimuthal coordinate, Io is a net polyphase probe current, H1⁽²⁾( -j γ ρ) is a Hankel function of a second kind and first order with complex argument -j1 ρ , and H0⁽²⁾( -j γ ρ) is a Hankel function of a second kind and zero order with complex argument -γ'γ ρ for an β⁺⁷'^ωί time variation, where t is time.

[0273] 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 can 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

240240/2 interchangeable with one another where applicable. To this end, the various embodiments described above disclose elements that can optionally be combined in a variety of ways depending on the desired implementation.