Patent Publication Number: US-10320200-B2

Title: Chemically enhanced isolated capacitance

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. utility application entitled “CHEMICALLY ENHANCED ISOLATED CAPACITANCE” having application Ser. No. 14/847,390 filed on Sep. 8, 2015, which claims the benefit of, and priority to, U.S. Provisional Patent Application No. 62/049,046 entitled “CHEMICALLY ENHANCED ISOLATED CAPACITANCE” filed on Sep. 11, 2014, which are incorporated herein by reference in their entireties. 
     This application is related to U.S. Non-provisional patent application entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on Mar. 7, 2013 and assigned application Ser. No. 13/789,538, and was published on Sep. 11, 2014 as Publication Number US2014/0252886 A1, and which is incorporated herein by reference in its entirety. This application is also related to U.S. Non-provisional patent application entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on Mar. 7, 2013 and assigned application Ser. No. 13/789,525, and was published on Sep. 11, 2014 as Publication Number US2014/0252865 A1, and which is incorporated herein by reference in its entirety. This application is further related to co-pending U.S. Non-provisional patent application entitled “Excitation and Use of Guided Surface Wave Modes on Lossy Media,” which was filed on Sep. 10, 2014 and assigned application Ser. No. 14/483,089, and which is incorporated herein by reference in its entirety. This application is further related to co-pending U.S. Non-provisional patent application entitled “Excitation and Use of Guided Surface Waves,” which was filed on Jun. 2, 2015 and assigned application Ser. No. 14/728,507, and which is incorporated herein by reference in its entirety. This application is further related to U.S. Non-provisional patent application entitled “Excitation and Use of Guided Surface Waves,” which was filed on Jun. 2, 2015 and assigned application Ser. No. 14/728,492, and which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND 
     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&#39;s. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Many aspects of the present disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views. 
         FIG. 1  is a chart that depicts field strength as a function of distance for a guided electromagnetic field and a radiated electromagnetic field. 
         FIG. 2  is a drawing that illustrates a propagation interface with two regions employed for transmission of a guided surface wave according to various embodiments of the present disclosure. 
         FIG. 3  is a drawing that illustrates a guided surface waveguide probe disposed with respect to a propagation interface of  FIG. 2  according to various embodiments of the present disclosure. 
         FIG. 4  is a plot of an example of the magnitudes of close-in and far-out asymptotes of first order Hankel functions according to various embodiments of the present disclosure. 
         FIGS. 5A and 5B  are drawings that illustrate a complex angle of incidence of an electric field synthesized by a guided surface waveguide probe according to various embodiments of the present disclosure. 
         FIG. 6  is a graphical representation illustrating the effect of elevation of a charge terminal on the location where the electric field of  FIG. 5A  intersects with the lossy conducting medium at a Brewster angle according to various embodiments of the present disclosure. 
         FIG. 7  is a graphical representation of an example of a guided surface waveguide probe according to various embodiments of the present disclosure. 
         FIGS. 8A through 8C  are graphical representations illustrating examples of equivalent image plane models of the guided surface waveguide probe of  FIGS. 3 and 7  according to various embodiments of the present disclosure. 
         FIGS. 9A and 9B  are graphical representations illustrating examples of single-wire transmission line and classic transmission line models of the equivalent image plane models of  FIGS. 8B and 8C  according to various embodiments of the present disclosure. 
         FIG. 10  is a flow chart illustrating an example of adjusting a guided surface waveguide probe of  FIGS. 3 and 7  to launch a guided surface wave along the surface of a lossy conducting medium according to various embodiments of the present disclosure. 
         FIG. 11  is a plot illustrating an example of the relationship between a wave tilt angle and the phase delay of a guided surface waveguide probe of  FIGS. 3 and 7  according to various embodiments of the present disclosure. 
         FIG. 12  is a drawing that illustrates an example of a guided surface waveguide probe according to various embodiments of the present disclosure. 
         FIG. 13  is a graphical representation illustrating the incidence of a synthesized electric field at a complex Brewster angle to match the guided surface waveguide mode at the Hankel crossover distance according to various embodiments of the present disclosure. 
         FIG. 14  is a graphical representation of an example of a guided surface waveguide probe of  FIG. 12  according to various embodiments of the present disclosure. 
         FIG. 15A  includes plots of an example of the imaginary and real parts of a phase delay (Φ U ) of a charge terminal T 1  of a guided surface waveguide probe according to various embodiments of the present disclosure. 
         FIG. 15B  is a schematic diagram of the guided surface waveguide probe of  FIG. 14  according to various embodiments of the present disclosure. 
         FIG. 16  is a drawing that illustrates an example of a guided surface waveguide probe according to various embodiments of the present disclosure. 
         FIG. 17  is a graphical representation of an example of a guided surface waveguide probe of  FIG. 16  according to various embodiments of the present disclosure. 
         FIGS. 18A through 18C  depict examples of receiving structures that can be employed to receive energy transmitted in the form of a guided surface wave launched by a guided surface waveguide probe according to the various embodiments of the present disclosure. 
         FIG. 18D  is a flow chart illustrating an example of adjusting a receiving structure according to various embodiments of the present disclosure. 
         FIG. 19  depicts an example of an additional receiving structure that can be employed to receive energy transmitted in the form of a guided surface wave launched by a guided surface waveguide probe according to the various embodiments of the present disclosure. 
         FIGS. 20A-E  depict examples of various schematic symbols that are used with reference to embodiments of the present disclosure. 
         FIGS. 21 and 22  are graphical representations of various embodiments of a charge device for enhancing the isolated capacitance of a charge terminal in a guided surface waveguide probe in accordance with embodiments of the present disclosure. 
         FIG. 23  is a graphical representation of an example of a guided surface waveguide probe utilizing charge devices according to an embodiment of the present disclosure. 
         FIG. 24  is a flow chart diagram describing an example of the operation of a portion of a guided surface waveguide probe according to an embodiment of the present disclosure. 
         FIG. 25  is a graphical representation of an example of a guided surface waveguide probe having a chemically enhanced charge terminal according to various embodiments of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     To begin, some terminology shall be established to provide clarity in the discussion of concepts to follow. First, as contemplated herein, a formal distinction is drawn between radiated electromagnetic fields and guided electromagnetic fields. 
     As contemplated herein, a radiated electromagnetic field comprises electromagnetic energy that is emitted from a source structure in the form of waves that are not bound to a waveguide. For example, a radiated electromagnetic field is generally a field that leaves an electric structure such as an antenna and propagates through the atmosphere or other medium and is not bound to any waveguide structure. Once radiated electromagnetic waves leave an electric structure such as an antenna, they continue to propagate in the medium of propagation (such as air) independent of their source until they dissipate regardless of whether the source continues to operate. Once electromagnetic waves are radiated, they are not recoverable unless intercepted, and, if not intercepted, the energy inherent in the radiated electromagnetic waves is lost forever. Electrical structures such as antennas are designed to radiate electromagnetic fields by maximizing the ratio of the radiation resistance to the structure loss resistance. Radiated energy spreads out in space and is lost regardless of whether a receiver is present. The energy density of the radiated fields is a function of distance due to geometric spreading. Accordingly, the term “radiate” in all its forms as used herein refers to this form of electromagnetic propagation. 
     A guided electromagnetic field is a propagating electromagnetic wave whose energy is concentrated within or near boundaries between media having different electromagnetic properties. In this sense, a guided electromagnetic field is one that is bound to a waveguide and may be characterized as being conveyed by the current flowing in the waveguide. If there is no load to receive and/or dissipate the energy conveyed in a guided electromagnetic wave, then no energy is lost except for that dissipated in the conductivity of the guiding medium. Stated another way, if there is no load for a guided electromagnetic wave, then no energy is consumed. Thus, a generator or other source generating a guided electromagnetic field does not deliver real power unless a resistive load is present. To this end, such a generator or other source essentially runs idle until a load is presented. This is akin to running a generator to generate a 60 Hertz electromagnetic wave that is transmitted over power lines where there is no electrical load. It should be noted that a guided electromagnetic field or wave is the equivalent to what is termed a “transmission line mode.” This contrasts with radiated electromagnetic waves in which real power is supplied at all times in order to generate radiated waves. Unlike radiated electromagnetic waves, guided electromagnetic energy does not continue to propagate along a finite length waveguide after the energy source is turned off. Accordingly, the term “guide” in all its forms as used herein refers to this transmission mode of electromagnetic propagation. 
     Referring now to  FIG. 1 , shown is a graph  100  of field strength in decibels (dB) above an arbitrary reference in volts per meter as a function of distance in kilometers on a log-dB plot to further illustrate the distinction between radiated and guided electromagnetic fields. The graph  100  of  FIG. 1  depicts a guided field strength curve  103  that shows the field strength of a guided electromagnetic field as a function of distance. This guided field strength curve  103  is essentially the same as a transmission line mode. Also, the graph  100  of  FIG. 1  depicts a radiated field strength curve  106  that shows the field strength of a radiated electromagnetic field as a function of distance. 
     Of interest are the shapes of the curves  103  and  106  for guided wave and for radiation propagation, respectively. The radiated field strength curve  106  falls off geometrically (1/d, where d is distance), which is depicted as a straight line on the log-log scale. The guided field strength curve  103 , on the other hand, has a characteristic exponential decay of e −αd /√{square root over (d)} and exhibits a distinctive knee  109  on the log-log scale. The guided field strength curve  103  and the radiated field strength curve  106  intersect at point  112 , which occurs at a crossing distance. At distances less than the crossing distance at intersection point  112 , the field strength of a guided electromagnetic field is significantly greater at most locations than the field strength of a radiated electromagnetic field. At distances greater than the crossing distance, the opposite is true. Thus, the guided and radiated field strength curves  103  and  106  further illustrate the fundamental propagation difference between guided and radiated electromagnetic fields. For an informal discussion of the difference between guided and radiated electromagnetic fields, reference is made to Milligan, T.,  Modern Antenna Desiqn , McGraw-Hill, 1 st  Edition, 1985, pp. 8-9, which is incorporated herein by reference in its entirety. 
     The distinction between radiated and guided electromagnetic waves, made above, is readily expressed formally and placed on a rigorous basis. That two such diverse solutions could emerge from one and the same linear partial differential 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. 
     In empty space, the wave equation is a differential operator whose eigenfunctions possess a continuous spectrum of eigenvalues on the complex wave-number plane. This transverse electro-magnetic (TEM) field is called the radiation field, and those propagating fields are called “Hertzian waves.” However, in the presence of a conducting boundary, the wave equation plus boundary conditions mathematically lead to a spectral representation of wave-numbers composed of a continuous spectrum plus a sum of discrete spectra. To this end, reference is made to Sommerfeld, A., “Uber die Ausbreitung der Wellen in der Drahtlosen Telegraphie,” Annalen der Physik, Vol. 28, 1909, pp. 665-736. Also see Sommerfeld, A., “Problems of Radio,” published as Chapter 6 in  Partial Differential Equations in Physics—Lectures on Theoretical Physics: Volume VI , Academic Press, 1949, pp. 236-289, 295-296; Collin, R. E., “Hertzian Dipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20 th  Century Controversies,”  IEEE Antennas and Propagation Maqazine , Vol. 46, No. 2, April 2004, pp. 64-79; and Reich, H. J., Ordnung, P. F, Krauss, H. L., and Skalnik, J. G.,  Microwave Theory and Techniques , Van Nostrand, 1953, pp. 291-293, each of these references being incorporated herein by reference in its entirety. 
     The terms “ground wave” and “surface wave” identify two distinctly different physical propagation phenomena. A surface wave arises analytically from a distinct pole yielding a discrete component in the plane wave spectrum. See, e.g., “The Excitation of Plane Surface Waves” by Cullen, A. L., ( Proceedings of the IEE  (British), Vol. 101, Part IV, August 1954, pp. 225-235). In this context, a surface wave is considered to be a guided surface wave. The surface wave (in the Zenneck-Sommerfeld guided wave sense) is, physically and mathematically, not the same as the ground wave (in the Weyl-Norton-FCC sense) that is now so familiar from radio broadcasting. These two propagation mechanisms arise from the excitation of different types of eigenvalue spectra (continuum or discrete) on the complex plane. The field strength of the guided surface wave decays exponentially with distance as illustrated by curve  103  of  FIG. 1  (much like propagation in a lossy waveguide) and resembles propagation in a radial transmission line, as opposed to the classical Hertzian radiation of the ground wave, which propagates spherically, possesses a continuum of eigenvalues, falls off geometrically as illustrated by curve  106  of  FIG. 1 , and results from branch-cut integrals. As experimentally demonstrated by C. R. Burrows in “The Surface Wave in Radio Propagation over Plane Earth” ( Proceedings of the IRE , Vol. 25, No. 2, February, 1937, pp. 219-229) and “The Surface Wave in Radio Transmission” ( Bell Laboratories Record , Vol. 15, June 1937, pp. 321-324), vertical antennas radiate ground waves but do not launch guided surface waves. 
     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. 
     In free space, antennas excite the continuum eigenvalues of the wave equation, which is a radiation field, where the outwardly propagating RF energy with E z  and H φ  in-phase is lost forever. On the other hand, waveguide probes excite discrete eigenvalues, which results in transmission line propagation. See Collin, R. E.,  Field Theory of Guided Waves , McGraw-Hill, 1960, pp. 453, 474-477. While such theoretical analyses have held out the hypothetical possibility of launching open surface guided waves over planar or spherical surfaces of lossy, homogeneous media, for more than a century no known structures in the engineering arts have existed for accomplishing this with any practical efficiency. Unfortunately, since it emerged in the early 1900&#39;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. 
     According to the various embodiments of the present disclosure, various guided surface waveguide probes are described that are configured to excite electric fields that couple into a guided surface waveguide mode along the surface of a lossy conducting medium. Such guided electromagnetic fields are substantially mode-matched in magnitude and phase to a guided surface wave mode on the surface of the lossy conducting medium. Such a guided surface wave mode can also be termed a Zenneck waveguide mode. By virtue of the fact that the resultant fields excited by the guided surface waveguide probes described herein are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium, a guided electromagnetic field in the form of a guided surface wave is launched along the surface of the lossy conducting medium. According to one embodiment, the lossy conducting medium comprises a terrestrial medium such as the Earth. 
     Referring to  FIG. 2 , shown is a propagation interface that provides for an examination of the boundary value solutions to Maxwell&#39;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, Sep. 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. 
     According to various embodiments, the present disclosure sets forth various guided surface waveguide probes that generate electromagnetic fields that are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium comprising Region  1 . According to various embodiments, such electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle of the lossy conducting medium that can result in zero reflection. 
     To explain further, in Region  2 , where an e jωt  field variation is assumed and where ρ≠0 and z≥0 (with z being the vertical coordinate normal to the surface of Region  1 , and ρ being the radial dimension in cylindrical coordinates), Zenneck&#39;s closed-form exact solution of Maxwell&#39;s equations satisfying the boundary conditions along the interface are expressed by the following electric field and magnetic field components: 
     
       
         
           
             
               
                 
                   
                     
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     In these expressions, z is the vertical coordinate normal to the surface of Region  1  and ρ is the radial coordinate, H n   (2) (−jγφ )is a complex argument Hankel function of the second kind and order n, u 1  is the propagation constant in the positive vertical (z) direction in Region  1 , u 2  is the propagation constant in the vertical (z) direction in Region  2 , σ 1  is the conductivity of Region  1 , ω is equal to 2πf, where f is a frequency of excitation, ε o  is the permittivity of free space, ε 1  is the permittivity of Region  1 , A is a source constant imposed by the source, and γ is a surface wave radial propagation constant. 
     The propagation constants in the ±z directions are determined by separating the wave equation above and below the interface between Regions  1  and  2 , and imposing the boundary conditions. This exercise gives, in Region  2 , 
                     u   2     =       -     jk   o           1   +     (       ɛ   r     -   jx     )                   (   7   )               
and gives, in Region  1 ,
 
 u   1   =−u   2 ( ε r   −jx ).  (8)
 
The radial propagation constant γ is given by
 
                     γ   =       j   ⁢         k   o   2     +     u   2   2           =     j   ⁢         k   o     ⁢   n         1   +     n   2                 ,           (   9   )               
which is a complex expression where n is the complex index of refraction given by
 
 n =√{square root over (ε r   −jx )}.  (10)
 
In all of the above Equations,
 
                     x   =       σ   1       ωɛ   o         ,   and           (   11   )                   k   o     =       ω   ⁢         μ   o     ⁢     ɛ   o           =       λ   o       2   ⁢   π           ,           (   12   )               
where ε r  comprises the relative permittivity of Region  1 , σ 1  is the conductivity of Region  1 , ε o  is the permittivity of free space, and μ o  comprises the permeability of free space. Thus, the generated surface wave propagates parallel to the interface and exponentially decays vertical to it. This is known as evanescence.
 
     Thus, Equations (1)-(3) can be considered to be a cylindrically-symmetric, radially-propagating waveguide mode. See Barlow, H. M., and Brown, J.,  Radio Surface Waves , Oxford University Press, 1962, pp. 10-12, 29-33. The present disclosure details structures that excite this “open boundary” waveguide mode. Specifically, according to various embodiments, a guided surface waveguide probe is provided with a charge terminal of appropriate size that is fed with voltage and/or current and is positioned relative to the boundary interface between Region  2  and Region  1 . This may be better understood with reference to  FIG. 3 , which shows an example of a guided surface waveguide probe  200   a  that includes a charge terminal T 1  elevated above a lossy conducting medium  203  (e.g., the Earth) along a vertical axis z that is normal to a plane presented by the lossy conducting medium  203 . The lossy conducting medium  203  makes up Region  1 , and a second medium  206  makes up Region  2  and shares a boundary interface with the lossy conducting medium  203 . 
     According to one embodiment, the lossy conducting medium  203  can comprise a terrestrial medium such as the planet Earth. To this end, such a terrestrial medium comprises all structures or formations included thereon whether natural or man-made. For example, such a terrestrial medium can comprise natural elements such as rock, soil, sand, fresh water, sea water, trees, vegetation, and all other natural elements that make up our planet. In addition, such a terrestrial medium can comprise man-made elements such as concrete, asphalt, building materials, and other man-made materials. In other embodiments, the lossy conducting medium  203  can comprise some medium other than the Earth, whether naturally occurring or man-made. In other embodiments, the lossy conducting medium  203  can comprise other media such as man-made surfaces and structures such as automobiles, aircraft, man-made materials (such as plywood, plastic sheeting, or other materials) or other media. 
     In the case where the lossy conducting medium  203  comprises a terrestrial medium or Earth, the second medium  206  can comprise the atmosphere above the ground. As such, the atmosphere can be termed an “atmospheric medium” that comprises air and other elements that make up the atmosphere of the Earth. In addition, it is possible that the second medium  206  can comprise other media relative to the lossy conducting medium  203 . 
     The guided surface waveguide probe  200   a  includes a feed network  209  that couples an excitation source  212  to the charge terminal T 1  via, e.g., a vertical feed line conductor. According to various embodiments, a charge Q 1  is imposed on the charge terminal T 1  to synthesize an electric field based upon the voltage applied to terminal T 1  at any given instant. Depending on the angle of incidence (θ i ) of the electric field (E), it is possible to substantially mode-match the electric field to a guided surface waveguide mode on the surface of the lossy conducting medium  203  comprising Region  1 . 
     By considering the Zenneck closed-form solutions of Equations (1)-(6), the Leontovich impedance boundary condition between Region  1  and Region  2  can be stated as
 
 {circumflex over (z)}×{right arrow over (H)}   2 (ρ,φ,0)= {right arrow over (J)}   S ,  (13)
 
where {circumflex over (z)} is a unit normal in the positive vertical (+z) direction and {right arrow over (H)} 2  is the magnetic field strength in Region  2  expressed by Equation (1) above. Equation (13) implies that the electric and magnetic fields specified in Equations (1)-(3) may result in a radial surface current density along the boundary interface, where the radial surface current density can be specified by
 
 J   ρ (ρ′)=− AH   1   (2) (− j γρ′)  (14)
 
where A is a constant. Further, it should be noted that close-in to the guided surface waveguide probe  200  (for ρ&lt;&lt;λ), Equation (14) above has the behavior
 
                       J   close     ⁡     (     ρ   ′     )       =         -     A   ⁡     (     j   ⁢           ⁢   2     )           π   ⁡     (       -   j     ⁢           ⁢     γρ   ′       )         =       -     H   ϕ       =     -         I   o       2   ⁢     πρ   ′         .                   (   15   )               
The negative sign means that when source current (I o ) flows vertically upward as illustrated in  FIG. 3 , the “close-in” ground current flows radially inward. By field matching on H φ  “close-in,” it can be determined that
 
                   A   =       -         I   o     ⁢   γ     4       =     -       ω   ⁢           ⁢     q   1     ⁢   γ     4                 (   16   )               
where q 1 =C 1 V 1 , in Equations (1)-(6) and (14). Therefore, the radial surface current density of Equation (14) can be restated as
 
                       J   ρ     ⁡     (     ρ   ′     )       =           I   o     ⁢   γ     4     ⁢         H   1     (   2   )       ⁡     (       -   j     ⁢           ⁢   γ   ⁢           ⁢     ρ   ′       )       .               (   17   )               
The fields expressed by Equations (1)-(6) and (17) have the nature of a transmission line mode bound to a lossy interface, not radiation fields that are associated with groundwave propagation. See Barlow, H. M. and Brown, J.,  Radio Surface Waves , Oxford University Press, 1962, pp. 1-5.
 
     At this point, a review of the nature of the Hankel functions used in Equations (1)-(6) and (17) is provided for these solutions of the wave equation. One might observe that the Hankel functions of the first and second kind and order n are defined as complex combinations of the standard Bessel functions of the first and second kinds
 
 H   n   (1) ( x )= J   n ( x )+ jN   n ( x ), and  (18)
 
 H   n   (2) ( x )= J   n ( x )− jN   n ( x ),  (19)
 
These functions represent cylindrical waves propagating radially inward (H n   (1) ) and outward (H n   (2) ), respectively. The definition is analogous to the relationship e ±/x =cos x±j sin x. See, for example, Harrington, R. F.,  Time - Harmonic Fields , McGraw-Hill, 1961, pp. 460-463.
 
     That H n   (2) (k ρ ρ) is an outgoing wave can be recognized from its large argument asymptotic behavior that is obtained directly from the series definitions of J n (x) and N n (x). Far-out from the guided surface waveguide probe: 
                             H   n     (   2   )       ⁡     (   x   )       ⁢     ⟶     x   →   ∞       ⁢         2   ⁢   j       π   ⁢           ⁢   x           ⁢     j   n     ⁢     e     -   jx         =         2     π   ⁢           ⁢   x         ⁢     j   n     ⁢     e     -     j   ⁡     (     x   -     π   4       )               ,           (     20   ⁢   a     )               
which, when multiplied by e jωt , is an outward propagating cylindrical wave of the form e j(ωt-kρ)  with a 1/√{square root over (ρ)} spatial variation. The first order (n=1) solution can be determined from Equation (20a) to be
 
                           H   1     (   2   )       ⁡     (   x   )       ⁢     ⟶     x   →   ∞       ⁢   j     ⁢         2   ⁢   j       π   ⁢           ⁢   x         ⁢     e     -   jx         =         2     π   ⁢           ⁢   x         ⁢       e     -     j   ⁡     (     x   -     π   2     -     π   4       )           .               (     20   ⁢   b     )               
Close-in to the guided surface waveguide probe (for ρ&lt;&lt;λ), the Hankel function of first order and the second kind behaves as
 
                         H   1     (   2   )       ⁡     (   x   )       ⁢     ⟶     x   →   0       ⁢       2   ⁢   j       π   ⁢           ⁢   x         .           (   21   )               
Note that these asymptotic expressions are complex quantities. When x is a real quantity, Equations (20b) and (21) differ in phase by √{square root over (j)}, which corresponds to an extra phase advance or “phase boost” of 45° or, equivalently, λ/8. The close-in and far-out asymptotes of the first order Hankel function of the second kind have a Hankel “crossover” or transition point where they are of equal magnitude at a distance of ρ=R x .
 
     Thus, beyond the Hankel crossover point the “far out” representation predominates over the “close-in” representation of the Hankel function. The distance to the Hankel crossover point (or Hankel crossover distance) can be found by equating Equations (20b) and (21) for −jγρ, and solving for R x . With x=σ/ωε o , it can be seen that the far-out and close-in Hankel function asymptotes are frequency dependent, with the Hankel crossover point moving out as the frequency is lowered. It should also be noted that the Hankel function asymptotes may also vary as the conductivity (σ) of the lossy conducting medium changes. For example, the conductivity of the soil can vary with changes in weather conditions. 
     Referring to  FIG. 4 , shown is an example of a plot of the magnitudes of the first order Hankel functions of Equations (20b) and (21) for a Region  1  conductivity of σ=0.010 mhos/m and relative permittivity ε r =15, at an operating frequency of 1850 kHz. Curve  115  is the magnitude of the far-out asymptote of Equation (20b) and curve  118  is the magnitude of the close-in asymptote of Equation (21), with the Hankel crossover point  121  occurring at a distance of R x =54 feet. While the magnitudes are equal, a phase offset exists between the two asymptotes at the Hankel crossover point  121 . It can also be seen that the Hankel crossover distance is much less than a wavelength of the operation frequency. 
     Considering the electric field components given by Equations (2) and (3) of the Zenneck closed-form solution in Region  2 , it can be seen that the ratio of E z  and E ρ  asymptotically passes to 
                         E   z       E   ρ       =         (         -   j     ⁢           ⁢   γ       u   2       )     ⁢           H   0     (   2   )       ⁡     (       -   j     ⁢           ⁢   γ   ⁢           ⁢   ρ     )           H   1     (   2   )       ⁡     (       -   j     ⁢           ⁢   γ   ⁢           ⁢   ρ     )         ⁢     ⟶     ρ   ⁢           →   ∞       ⁢         ɛ   r     -     j   ⁢     σ     ω   ⁢           ⁢     ɛ   o                   =     n   =     tan   ⁢           ⁢     θ   i             ,           (   22   )               
where n is the complex index of refraction of Equation (10) and θ i  is the angle of incidence of the electric field. In addition, the vertical component of the mode-matched electric field of Equation (3) asymptotically passes to
 
                         E     2   ⁢   z       ⁢     ⟶     ρ   ⁢           →   ∞       ⁢     (       q   free       ɛ   o       )       ⁢         γ   3       8   ⁢           ⁢   π         ⁢     e       -     u   2       ⁢   z   ⁢       e     -     j   ⁡     (       γ   ⁢           ⁢   ρ     -     π   /   4       )             ρ             ,           (   23   )               
which is linearly proportional to free charge on the isolated component of the elevated charge terminal&#39;s capacitance at the terminal voltage, q free =C free ×V T .
 
     For example, the height H 1  of the elevated charge terminal T 1  in  FIG. 3  affects the amount of free charge on the charge terminal T 1 . When the charge terminal T 1  is near the ground plane of Region  1 , most of the charge Q 1  on the terminal is “bound.” As the charge terminal T 1  is elevated, the bound charge is lessened until the charge terminal T 1  reaches a height at which substantially all of the isolated charge is free. 
     The advantage of an increased capacitive elevation for the charge terminal T 1  is that the charge on the elevated charge terminal T 1  is further removed from the ground plane, resulting in an increased amount of free charge q free  to couple energy into the guided surface waveguide mode. As the charge terminal T 1  is moved away from the ground plane, the charge distribution becomes more uniformly distributed about the surface of the terminal. The amount of free charge is related to the self-capacitance of the charge terminal T 1 . 
     For example, the capacitance of a spherical terminal can be expressed as a function of physical height above the ground plane. The capacitance of a sphere at a physical height of h above a perfect ground is given by
 
 C   elevated sphere =4πε o   a (1+ M+M   2   +M   3 +2 M   4 +3 M   5 + . . . ),  (24)
 
where the diameter of the sphere is 2a, and where M=a/2h with h being the height of the spherical terminal. As can be seen, an increase in the terminal height h reduces the capacitance C of the charge terminal. It can be shown that for elevations of the charge terminal T 1  that are at a height of about four times the diameter (4D=8a) or greater, the charge distribution is approximately uniform about the spherical terminal, which can improve the coupling into the guided surface waveguide mode.
 
     In the case of a sufficiently isolated terminal, the self-capacitance of a conductive sphere can be approximated by C=47 ε o a, where a is the radius of the sphere in meters, and the self-capacitance of a disk can be approximated by C=8 ε o a, where a is the radius of the disk in meters. The charge terminal T 1  can include any shape such as a sphere, a disk, a cylinder, a cone, a torus, a hood, one or more rings, or any other randomized shape or combination of shapes. An equivalent spherical diameter can be determined and used for positioning of the charge terminal T 1 . 
     This may be further understood with reference to the example of  FIG. 3 , where the charge terminal T 1  is elevated at a physical height of h p =H 1  above the lossy conducting medium  203 . To reduce the effects of the “bound” charge, the charge terminal T 1  can be positioned at a physical height that is at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal T 1  to reduce the bounded charge effects. 
     Referring next to  FIG. 5A , shown is a ray optics interpretation of the electric field produced by the elevated charge Q 1  on charge terminal T 1  of  FIG. 3 . As in optics, minimizing the reflection of the incident electric field can improve and/or maximize the energy coupled into the guided surface waveguide mode of the lossy conducting medium  203 . For an electric field (E ∥ ) that is polarized parallel to the plane of incidence (not the boundary interface), the amount of reflection of the incident electric field may be determined using the Fresnel reflection coefficient, which can be expressed as 
                         Γ   ∥     ⁡     (     θ   i     )       =         E     ∥     ,   R           E     ∥     ,   i           =             (       ɛ   r     -   jx     )     -       sin   2     ⁢     θ   i           -       (       ɛ   r     -   jx     )     ⁢   cos   ⁢           ⁢     θ   i                 (       ɛ   r     -   jx     )     -       sin   2     ⁢     θ   i           +       (       ɛ   r     -   jx     )     ⁢   cos   ⁢           ⁢     θ   i               ,           (   25   )               
where θ i  is the conventional angle of incidence measured with respect to the surface normal.
 
     In the example of  FIG. 5A , the ray optic interpretation shows the incident field polarized parallel to the plane of incidence having an angle of incidence of θ i , which is measured with respect to the surface normal ({circumflex over (z)}). There will be no reflection of the incident electric field when Γ ∥ (θ i )=0 and thus the incident electric field will be completely coupled into a guided surface waveguide mode along the surface of the lossy conducting medium  203 . It can be seen that the numerator of Equation (25) goes to zero when the angle of incidence is
 
θ i =arctan(√{square root over (ε r   −jx )})=θ i,B ,  (26)
 
where x=σ/ωε o . This complex angle of incidence (θ i,B ) is referred to as the Brewster angle. Referring back to Equation (22), it can be seen that the same complex Brewster angle (θ i,B ) relationship is present in both Equations (22) and (26).
 
     As illustrated in  FIG. 5A , the electric field vector E can be depicted as an incoming non-uniform plane wave, polarized parallel to the plane of incidence. The electric field vector E can be created from independent horizontal and vertical components as
 
 {right arrow over (E)} (θ i )= E   ρ   {circumflex over (ρ)}+E   z   {circumflex over (z)}.   (27)
 
Geometrically, the illustration in  FIG. 5A  suggests that the electric field vector E can be given by
 
                         E   ρ     ⁡     (     ρ   ,   z     )       =       E   ⁡     (     ρ   ,   z     )       ⁢   cos   ⁢           ⁢     θ   i         ,   and           (     28   ⁢   a     )                     E   z     ⁡     (     ρ   ,   z     )       =         E   ⁡     (     ρ   ,   z     )       ⁢     cos   ⁡     (       π   2     -     θ   i       )         =       E   ⁡     (     ρ   ,   z     )       ⁢           ⁢   sin   ⁢           ⁢     θ   i           ,           (     28   ⁢   b     )               
which means that the field ratio is
 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       ρ 
                     
                     
                       E 
                       z 
                     
                   
                   = 
                   
                     
                       1 
                       
                         tan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           i 
                         
                       
                     
                     = 
                     
                       tan 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ψ 
                           i 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     A generalized parameter W, called “wave tilt,” is noted herein as the ratio of the horizontal electric field component to the vertical electric field component given by 
                     W   =         E   ρ       E   z       =          W        ⁢     e     j   ⁢           ⁢   Ψ             ,   or           (     30   ⁢   a     )                   1   W     =         E   z       E   ρ       =       tan   ⁢           ⁢     θ   i       =       1        W          ⁢     e       -   j     ⁢           ⁢   Ψ               ,           (     30   ⁢   b     )               
which is complex and has both magnitude and phase. For an electromagnetic wave in Region  2 , the wave tilt angle (Ψ) is equal to the angle between the normal of the wave-front at the boundary interface with Region  1  and the tangent to the boundary interface. This may be easier to see in  FIG. 5B , which illustrates equi-phase surfaces of an electromagnetic wave and their normals for a radial cylindrical guided surface wave. At the boundary interface (z=0) with a perfect conductor, the wave-front normal is parallel to the tangent of the boundary interface, resulting in W=0. However, in the case of a lossy dielectric, a wave tilt W exists because the wave-front normal is not parallel with the tangent of the boundary interface at z=0.
 
     Applying Equation (30b) to a guided surface wave gives 
                     tan   ⁢           ⁢     θ     i   ,   B         =         E   z       E   ρ       =         u   2     γ     =           ɛ   r     -   jx       =     n   =       1   W     =       1        W          ⁢       e       -   j     ⁢           ⁢   Ψ       .                         (   31   )               
With the angle of incidence equal to the complex Brewster angle (θ i,B ), the Fresnel reflection coefficient of Equation (25) vanishes, as shown by
 
                             Γ   ∥     ⁡     (     θ     i   ,   B       )       =             (       ɛ   r     -   jx     )     -       sin   2     ⁢     θ   i           -       (       ɛ   r     -   jx     )     ⁢   cos   ⁢           ⁢     θ   i                 (       ɛ   r     -   jx     )     -       sin   2     ⁢     θ   i           +       (       ɛ   r     -   jx     )     ⁢   cos   ⁢           ⁢     θ   i                      θ   i     =     θ     i   ,   B           =   0.           (   32   )               
By adjusting the complex field ratio of Equation (22), an incident field can be synthesized to be incident at a complex angle at which the reflection is reduced or eliminated. Establishing this ratio as n=√{square root over (ε r −jx)} results in the synthesized electric field being incident at the complex Brewster angle, making the reflections vanish.
 
     The concept of an electrical effective height can provide further insight into synthesizing an electric field with a complex angle of incidence with a guided surface waveguide probe  200 . The electrical effective height (h eff ) has been defined as 
                     h   eff     =       1     I   0       ⁢       ∫   0     h   p       ⁢       I   ⁡     (   z   )       ⁢   dz                 (   33   )               
for a monopole with a physical height (or length) of h p . Since the expression depends upon the magnitude and phase of the source distribution along the structure, the effective height (or length) is complex in general. The integration of the distributed current I(z) of the structure is performed over the physical height of the structure (h p ), and normalized to the ground current (I 0 ) flowing upward through the base (or input) of the structure. The distributed current along the structure can be expressed by
 
 I ( z )= I   C  cos(β 0   z ),  (34)
 
where β 0  is the propagation factor for current propagating on the structure. In the example of  FIG. 3 , I C  is the current that is distributed along the vertical structure of the guided surface waveguide probe  200   a.  
 
     For example, consider a feed network  209  that includes a low loss coil (e.g., a helical coil) at the bottom of the structure and a vertical feed line conductor connected between the coil and the charge terminal T 1 . The phase delay due to the coil (or helical delay line) is θ c =β p l C , with a physical length of l C  and a propagation factor of 
                       β   p     =         2   ⁢           ⁢   π       λ   p       =       2   ⁢           ⁢   π         V   f     ⁢     λ   0             ,           (   35   )               
where V f  is the velocity factor on the structure, λ 0  is the wavelength at the supplied frequency, and λ p  is the propagation wavelength resulting from the velocity factor V f . The phase delay is measured relative to the ground (stake) current I 0 .
 
     In addition, the spatial phase delay along the length l w  of the vertical feed line conductor can be given by θ y =β w l w  where β w  is the propagation phase constant for the vertical feed line conductor. In some implementations, the spatial phase delay may be approximated by θ y =β w h p , since the difference between the physical height h p  of the guided surface waveguide probe  200   a  and the vertical feed line conductor length l w  is much less than a wavelength at the supplied frequency (λ 0 ). As a result, the total phase delay through the coil and vertical feed line conductor is Φ=θ c +θ y , and the current fed to the top of the coil from the bottom of the physical structure is
 
 I   C (θ c +θ y )= I   0   e   jΦ ,  (36)
 
with the total phase delay Φ measured relative to the ground (stake) current I 0 . Consequently, the electrical effective height of a guided surface waveguide probe  200  can be approximated by
 
                       h   eff     =         1     I   0       ⁢       ∫   0     h   p       ⁢       I   0     ⁢           ⁢     e     j   ⁢           ⁢   Φ       ⁢     cos   ⁡     (       β   0     ⁢   z     )       ⁢   dz         ≅       h   p     ⁢     e     j   ⁢           ⁢   Φ             ,           (   37   )               
for the case where the physical height h p &lt;&lt;λ 0 . The complex effective height of a monopole, h eff =h p  at an angle (or phase shift) of Φ, may be adjusted to cause the source fields to match a guided surface waveguide mode and cause a guided surface wave to be launched on the lossy conducting medium  203 .
 
     In the example of  FIG. 5A , ray optics are used to illustrate the complex angle trigonometry of the incident electric field (E) having a complex Brewster angle of incidence (θ i,B ) at the Hankel crossover distance (R x )  121 . Recall from Equation (26) that, for a lossy conducting medium, the Brewster angle is complex and specified by 
                     tan   ⁢           ⁢     θ     i   ,   B         =           ɛ   r     -     j   ⁢     σ     ω   ⁢           ⁢     ɛ   o               =     n   .               (   38   )               
Electrically, the geometric parameters are related by the electrical effective height (h eff ) of the charge terminal T 1  by
 
 R   x  tan ψ i,B   =R   x   ×W=h   eff   =h   p   e   jΦ ,  (39)
 
where ψ i,B =(π/2)−θ i,B  is the Brewster angle measured from the surface of the lossy conducting medium. To couple into the guided surface waveguide mode, the wave tilt of the electric field at the Hankel crossover distance can be expressed as the ratio of the electrical effective height and the Hankel crossover distance
 
                       h   eff       R   x       =       tan   ⁢           ⁢     ψ     i   ,   B         =       W   Rx     .               (   40   )               
Since both the physical height (h p ) and the Hankel crossover distance (R x ) are real quantities, the angle (Ψ) of the desired guided surface wave tilt at the Hankel crossover distance (R x ) is equal to the phase (Φ) of the complex effective height (h eff ). This implies that by varying the phase at the supply point of the coil, and thus the phase shift in Equation (37), the phase, Φ, of the complex effective height can be manipulated to match the angle of the wave tilt, Ψ, of the guided surface waveguide mode at the Hankel crossover point  121 : Φ=Ψ.
 
     In  FIG. 5A , a right triangle is depicted having an adjacent side of length R x  along the lossy conducting medium surface and a complex Brewster angle ψ i,B  measured between a ray  124  extending between the Hankel crossover point  121  at R x  and the center of the charge terminal T 1 , and the lossy conducting medium surface  127  between the Hankel crossover point  121  and the charge terminal T 1 . With the charge terminal T 1  positioned at physical height h p  and excited with a charge having the appropriate phase delay Φ, the resulting electric field is incident with the lossy conducting medium boundary interface at the Hankel crossover distance R x , and at the Brewster angle. Under these conditions, the guided surface waveguide mode can be excited without reflection or substantially negligible reflection. 
     If the physical height of the charge terminal T 1  is decreased without changing the phase shift Φ of the effective height (h eff ), the resulting electric field intersects the lossy conducting medium  203  at the Brewster angle at a reduced distance from the guided surface waveguide probe  200 .  FIG. 6  graphically illustrates the effect of decreasing the physical height of the charge terminal T 1  on the distance where the electric field is incident at the Brewster angle. As the height is decreased from h 3  through h 2  to h 1 , the point where the electric field intersects with the lossy conducting medium (e.g., the Earth) at the Brewster angle moves closer to the charge terminal position. However, as Equation (39) indicates, the height H 1  ( FIG. 3 ) of the charge terminal T 1  should be at or higher than the physical height (h p ) in order to excite the far-out component of the Hankel function. With the charge terminal T 1  positioned at or above the effective height (h eff ), the lossy conducting medium  203  can be illuminated at the Brewster angle of incidence (ψ i,B =(π/2)−θ i,B ) at or beyond the Hankel crossover distance (R x )  121  as illustrated in  FIG. 5A . To reduce or minimize the bound charge on the charge terminal T 1 , the height should be at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal T 1  as mentioned above. 
     A guided surface waveguide probe  200  can be configured to establish an electric field having a wave tilt that corresponds to a wave illuminating the surface of the lossy conducting medium  203  at a complex Brewster angle, thereby exciting radial surface currents by substantially mode-matching to a guided surface wave mode at (or beyond) the Hankel crossover point  121  at R x . 
     Referring to  FIG. 7 , shown is a graphical representation of an example of a guided surface waveguide probe  200   b  that includes a charge terminal T 1 . An AC source  212  acts as the excitation source for the charge terminal T 1 , which is coupled to the guided surface waveguide probe  200   b  through a feed network  209  ( FIG. 3 ) comprising a coil  215  such as, e.g., a helical coil. In other implementations, the AC source  212  can be inductively coupled to the coil  215  through a primary coil. In some embodiments, an impedance matching network may be included to improve and/or maximize coupling of the AC source  212  to the coil  215 . 
     As shown in  FIG. 7 , the guided surface waveguide probe  200   b  can include the upper charge terminal T 1  (e.g., a sphere at height h p ) that is positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium  203 . A second medium  206  is located above the lossy conducting medium  203 . The charge terminal T 1  has a self-capacitance C T . During operation, charge Q 1  is imposed on the terminal T 1  depending on the voltage applied to the terminal T 1  at any given instant. 
     In the example of  FIG. 7 , the coil  215  is coupled to a ground stake  218  at a first end and to the charge terminal T 1  via a vertical feed line conductor  221 . In some implementations, the coil connection to the charge terminal T 1  can be adjusted using a tap  224  of the coil  215  as shown in  FIG. 7 . The coil  215  can be energized at an operating frequency by the AC source  212  through a tap  227  at a lower portion of the coil  215 . In other implementations, the AC source  212  can be inductively coupled to the coil  215  through a primary coil. 
     The construction and adjustment of the guided surface waveguide probe  200  is based upon various operating conditions, such as the transmission frequency, conditions of the lossy conducting medium (e.g., soil conductivity σ and relative permittivity ε r ), and size of the charge terminal T 1 . The index of refraction can be calculated from Equations (10) and (11) as
 
 n =√{square root over (ε r   −jx )},  (41)
 
where x=σ/ωε o  with ω=2πf. The conductivity σ and relative permittivity ε r  can be determined through test measurements of the lossy conducting medium  203 . The complex Brewster angle (θ i,B ) measured from the surface normal can also be determined from Equation (26) as
 
θ i,B =arctan(√{square root over (ε r   −jx )}),  (42)
 
or measured from the surface as shown in  FIG. 5A  as
 
                     ψ     i   ,   B       =       π   2     -       θ     i   ,   B       .               (   43   )               
The wave tilt at the Hankel crossover distance (W Rx ) can also be found using Equation (40).
 
     The Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21) for −jγρ, and solving for R x  as illustrated by  FIG. 4 . The electrical effective height can then be determined from Equation (39) using the Hankel crossover distance and the complex Brewster angle as
 
 h   eff   =h   p   e   jΦ   R   x  tan ψ i,B .  (44)
 
As can be seen from Equation (44), the complex effective height (h eff ) includes a magnitude that is associated with the physical height (h p ) of the charge terminal T 1  and a phase delay (Φ) that is to be associated with the angle (Ψ) of the wave tilt at the Hankel crossover distance (R x ). With these variables and the selected charge terminal T 1  configuration, it is possible to determine the configuration of a guided surface waveguide probe  200 .
 
     With the charge terminal T 1  positioned at or above the physical height (h p ), the feed network  209  ( FIG. 3 ) and/or the vertical feed line connecting the feed network to the charge terminal T 1  can be adjusted to match the phase (Φ) of the charge Q 1  on the charge terminal T 1  to the angle (Ψ) of the wave tilt (W). The size of the charge terminal T 1  can be chosen to provide a sufficiently large surface for the charge Q 1  imposed on the terminals. In general, it is desirable to make the charge terminal T 1  as large as practical. The size of the charge terminal T 1  should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal. 
     The phase delay θ c  of a helically-wound coil can be determined from Maxwell&#39;s equations as has been discussed by Corum, K. L. and J. F. Corum, “RF Coils, Helical Resonators and Voltage Magnification by Coherent Spatial Modes,”  Microwave Review , Vol. 7, No. 2, September 2001, pp. 36-45, which is incorporated herein by reference in its entirety. For a helical coil with H/D&gt;1, the ratio of the velocity of propagation (v) of a wave along the coil&#39;s longitudinal axis to the speed of light (c), or the “velocity factor,” is given by 
                       V   f     =       υ   c     =     1       1   +     20   ⁢       (     D   s     )     2.5     ⁢       (     D     λ   o       )     0.5                 ,           (   45   )               
where H is the axial length of the solenoidal helix, D is the coil diameter, N is the number of turns of the coil, s=H/N is the turn-to-turn spacing (or helix pitch) of the coil, and λ o  is the free-space wavelength. Based upon this relationship, the electrical length, or phase delay, of the helical coil is given by
 
                     θ   c     =         β   p     ⁢   H     =           2   ⁢   π       λ   p       ⁢   H     =         2   ⁢   π         V   f     ⁢     λ   0         ⁢     H   .                   (   46   )               
The principle is the same if the helix is wound spirally or is short and fat, but V f  and θ c  are easier to obtain by experimental measurement. The expression for the characteristic (wave) impedance of a helical transmission line has also been derived as
 
     
       
         
           
             
               
                 
                   
                     Z 
                     c 
                   
                   = 
                   
                     
                       
                         60 
                         
                           V 
                           f 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             ln 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     V 
                                     f 
                                   
                                   ⁢ 
                                   
                                     λ 
                                     0 
                                   
                                 
                                 D 
                               
                               ) 
                             
                           
                           - 
                           1.027 
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     The spatial phase delay θ y  of the structure can be determined using the traveling wave phase delay of the vertical feed line conductor  221  ( FIG. 7 ). The capacitance of a cylindrical vertical conductor above a prefect ground plane can be expressed as 
                       C   A     =         2   ⁢   π   ⁢           ⁢     ɛ   o     ⁢     h   w           ln   ⁡     (     h   a     )       -   1       ⁢           ⁢   Farads       ,           (   48   )               
where h w  is the vertical length (or height) of the conductor and a is the radius (in mks units). As with the helical coil, the traveling wave phase delay of the vertical feed line conductor can be given by
 
                       θ   y     =         β   w     ⁢     h   w       =           2   ⁢   π       λ   w       ⁢     h   w       =         2   ⁢   π         V   w     ⁢     λ   0         ⁢     h   w             ,           (   49   )               
where β w  is the propagation phase constant for the vertical feed line conductor, h w  is the vertical length (or height) of the vertical feed line conductor, V w  is the velocity factor on the wire, λ 0  is the wavelength at the supplied frequency, and λ w  is the propagation wavelength resulting from the velocity factor V w . For a uniform cylindrical conductor, the velocity factor is a constant with V w ≈0.94, or in a range from about 0.93 to about 0.98. If the mast is considered to be a uniform transmission line, its average characteristic impedance can be approximated by
 
                       Z   w     =       60     V   w       ⁡     [       ln   ⁡     (       h   w     a     )       -   1     ]         ,           (   50   )               
where V w ≈0.94 for a uniform cylindrical conductor and a is the radius of the conductor. An alternative expression that has been employed in amateur radio literature for the characteristic impedance of a single-wire feed line can be given by
 
                     Z   w     =     138   ⁢           ⁢       log   ⁡     (       1.123   ⁢           ⁢     V   w     ⁢     λ   0         2   ⁢   π   ⁢           ⁢   a       )       .               (   51   )               
Equation (51) implies that Z w  for a single-wire feeder varies with frequency. The phase delay can be determined based upon the capacitance and characteristic impedance.
 
     With a charge terminal T 1  positioned over the lossy conducting medium  203  as shown in  FIG. 3 , the feed network  209  can be adjusted to excite the charge terminal T 1  with the phase shift (Φ) of the complex effective height (h eff ) equal to the angle (Ψ) of the wave tilt at the Hankel crossover distance, or Φ=Ψ. When this condition is met, the electric field produced by the charge oscillating Q 1  on the charge terminal T 1  is coupled into a guided surface waveguide mode traveling along the surface of a lossy conducting medium  203 . For example, if the Brewster angle (θ i,B ), the phase delay (θ y ) associated with the vertical feed line conductor  221  ( FIG. 7 ), and the configuration of the coil  215  ( FIG. 7 ) are known, then the position of the tap  224  ( FIG. 7 ) can be determined and adjusted to impose an oscillating charge Q 1  on the charge terminal T 1  with phase Φ=Ψ. The position of the tap  224  may be adjusted to maximize coupling the traveling surface waves into the guided surface waveguide mode. Excess coil length beyond the position of the tap  224  can be removed to reduce the capacitive effects. The vertical wire height and/or the geometrical parameters of the helical coil may also be varied. 
     The coupling to the guided surface waveguide mode on the surface of the lossy conducting medium  203  can be improved and/or optimized by tuning the guided surface waveguide probe  200  for standing wave resonance with respect to a complex image plane associated with the charge Q 1  on the charge terminal T 1 . By doing this, the performance of the guided surface waveguide probe  200  can be adjusted for increased and/or maximum voltage (and thus charge Q 1 ) on the charge terminal T 1 . Referring back to  FIG. 3 , the effect of the lossy conducting medium  203  in Region  1  can be examined using image theory analysis. 
     Physically, an elevated charge Q 1  placed over a perfectly conducting plane attracts the free charge on the perfectly conducting plane, which then “piles up” in the region under the elevated charge Q 1 . The resulting distribution of “bound” electricity on the perfectly conducting plane is similar to a bell-shaped curve. The superposition of the potential of the elevated charge Q 1 , plus the potential of the induced “piled up” charge beneath it, forces a zero equipotential surface for the perfectly conducting plane. The boundary value problem solution that describes the fields in the region above the perfectly conducting plane may be obtained using the classical notion of image charges, where the field from the elevated charge is superimposed with the field from a corresponding “image” charge below the perfectly conducting plane. 
     This analysis may also be used with respect to a lossy conducting medium  203  by assuming the presence of an effective image charge Q 1 ′ beneath the guided surface waveguide probe  200 . The effective image charge Q 1 ′ coincides with the charge Q 1  on the charge terminal T 1  about a conducting image ground plane  130 , as illustrated in  FIG. 3 . However, the image charge Q 1 ′ is not merely located at some real depth and 180° out of phase with the primary source charge Q 1  on the charge terminal T 1 , as they would be in the case of a perfect conductor. Rather, the lossy conducting medium  203  (e.g., a terrestrial medium) presents a phase shifted image. That is to say, the image charge Q 1 ′ is at a complex depth below the surface (or physical boundary) of the lossy conducting medium  203 . For a discussion of complex image depth, reference is made to Wait, J. R., “Complex Image Theory—Revisited,”  IEEE Antennas and Propagation Magazine , Vol. 33, No. 4, August 1991, pp. 27-29, which is incorporated herein by reference in its entirety. 
     Instead of the image charge Q 1 ′ being at a depth that is equal to the physical height (H 1 ) of the charge Q 1 , the conducting image ground plane  130  (representing a perfect conductor) is located at a complex depth of z=−d/2 and the image charge Q 1 ′ appears at a complex depth (i.e., the “depth” has both magnitude and phase), given by −D 1 =−(d/2+d/2+H 1 )≠H 1 . For vertically polarized sources over the Earth, 
                     d   =           2   ⁢         γ   e   2     +     k   0   2             γ   e   2       ≈     2     γ   e         =         d   r     +     jd   i       =          d        ⁢   ∠ζ           ,           (   52   )               
where
 
γ e   2   =jωμ   1 σ 1 −ω 2 μ 1 ε 1 , and  (53)
 
 k   o =ω√{square root over (μ o ε o )},  (54)
 
as indicated in Equation (12). The complex spacing of the image charge, in turn, implies that the external field will experience extra phase shifts not encountered when the interface is either a dielectric or a perfect conductor. In the lossy conducting medium, the wave front normal is parallel to the tangent of the conducting image ground plane  130  at z=−d/2, and not at the boundary interface between Regions  1  and  2 .
 
     Consider the case illustrated in  FIG. 8A  where the lossy conducting medium  203  is a finitely conducting Earth  133  with a physical boundary  136 . The finitely conducting Earth  133  may be replaced by a perfectly conducting image ground plane  139  as shown in  FIG. 8B , which is located at a complex depth z 1  below the physical boundary  136 . This equivalent representation exhibits the same impedance when looking down into the interface at the physical boundary  136 . The equivalent representation of  FIG. 8B  can be modeled as an equivalent transmission line, as shown in  FIG. 8C . The cross-section of the equivalent structure is represented as a (z-directed) end-loaded transmission line, with the impedance of the perfectly conducting image plane being a short circuit (z s =0). The depth z 1  can be determined by equating the TEM wave impedance looking down at the Earth to an image ground plane impedance z in  seen looking into the transmission line of  FIG. 8C . 
     In the case of  FIG. 8A , the propagation constant and wave intrinsic impedance in the upper region (air)  142  are 
                       γ   o     =       j   ⁢           ⁢   ω   ⁢         μ   o     ⁢     ɛ   o           =     0   +     j   ⁢           ⁢     β   o             ,   and           (   55   )                 z   o     =         j   ⁢           ⁢   ω   ⁢           ⁢     μ   o         γ   o       =           μ   o       ɛ   o         .               (   56   )               
In the lossy Earth  133 , the propagation constant and wave intrinsic impedance are
 
                       γ   e     =       j   ⁢           ⁢       ωμ   1     ⁡     (       σ   1     +     j   ⁢           ⁢   ω   ⁢           ⁢     ɛ   1         )             ,   and           (   57   )                 Z   e     =         j   ⁢           ⁢   ω   ⁢           ⁢     μ   1         γ   e       .             (   58   )               
For normal incidence, the equivalent representation of  FIG. 8B  is equivalent to a TEM transmission line whose characteristic impedance is that of air (z o ), with propagation constant of γ o , and whose length is z 1 . As such, the image ground plane impedance Z in  seen at the interface for the shorted transmission line of  FIG. 8C  is given by
 
 Z   in   =Z   o  tan  h (γ o   z   1 ).  (59)
 
Equating the image ground plane impedance Z in  associated with the equivalent model of  FIG. 8C  to the normal incidence wave impedance of  FIG. 8A  and solving for z 1  gives the distance to a short circuit (the perfectly conducting image ground plane  139 ) as
 
                       z   1     =         1     γ   o       ⁢       tanh     -   1       ⁡     (       z   e       z   o       )         =         1     γ   o       ⁢       tanh     -   1       ⁡     (       γ   o       γ   e       )         ≈     1     γ   e             ,           (   60   )               
where only the first term of the series expansion for the inverse hyperbolic tangent is considered for this approximation. Note that in the air region  142 , the propagation constant is γ o =jβ o , so Z in =jZ o  tan β o z 1  (which is a purely imaginary quantity for a real z 1 ), but z e  is a complex value if σ≠0. Therefore, Z in =Z e  only when z 1  is a complex distance.
 
     Since the equivalent representation of  FIG. 8B  includes a perfectly conducting image ground plane  139 , the image depth for a charge or current  lying at the surface of the Earth (physical boundary  136 ) is equal to distance z 1  on the other side of the image ground plane  139 , or d=2 z 1  beneath the Earth&#39;s surface (which is located at z=0). Thus, the distance to the perfectly conducting image ground plane  139  can be approximated by 
                   d   =       2   ⁢     z   1       ≈       2     γ   e       .               (   61   )               
Additionally, the “image charge” will be “equal and opposite” to the real charge, so the potential of the perfectly conducting image ground plane  139  at depth z 1 =−d/2 will be zero.
 
     If a charge Q 1  is elevated a distance H 1  above the surface of the Earth as illustrated in  FIG. 3 , then the image charge Q 1 ′ resides at a complex distance of D 1 =d+H 1  below the surface, or a complex distance of d/2+H 1  below the image ground plane  130 . The guided surface waveguide probe  200   b  of  FIG. 7  can be modeled as an equivalent single-wire transmission line image plane model that can be based upon the perfectly conducting image ground plane  139  of  FIG. 8B .  FIG. 9A  shows an example of the equivalent single-wire transmission line image plane model, and  FIG. 9B  illustrates an example of the equivalent classic transmission line model, including the shorted transmission line of  FIG. 8C . 
     In the equivalent image plane models of  FIGS. 9A and 9B , Φ=θ y +θ c  is the traveling wave phase delay of the guided surface waveguide probe  200  referenced to Earth  133  (or the lossy conducting medium  203 ), θ c =β p H is the electrical length of the coil  215  ( FIG. 7 ), of physical length H, expressed in degrees, θ y =β w h w  is the electrical length of the vertical feed line conductor  221  ( FIG. 7 ), of physical length h w , expressed in degrees, and θ d =β o  d/2 is the phase shift between the image ground plane  139  and the physical boundary  136  of the Earth  133  (or lossy conducting medium  203 ). In the example of  FIGS. 9A and 9B , Z w  is the characteristic impedance of the elevated vertical feed line conductor  221  in ohms, Z c  is the characteristic impedance of the coil  215  in ohms, and Z O  is the characteristic impedance of free space. 
     At the base of the guided surface waveguide probe  200 , the impedance seen “looking up” into the structure is Z ↑ =Z base . With a load impedance of: 
                       Z   L     =     1     j   ⁢           ⁢   ω   ⁢           ⁢     C   T           ,           (   62   )               
where C T  is the self-capacitance of the charge terminal T 1 , the impedance seen “looking up” into the vertical feed line conductor  221  ( FIG. 7 ) is given by:
 
                       Z   2     =         Z   W     ⁢         Z   L     +       Z   w     ⁢     tanh   ⁡     (     j   ⁢           ⁢     β   w     ⁢     h   w       )               Z   w     +       Z   L     ⁢     tanh   ⁡     (     j   ⁢           ⁢     β   w     ⁢     h   w       )               =       Z   W     ⁢         Z   L     +       Z   w     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   y       )               Z   w     +       Z   L     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   y       )                   ,           (   63   )               
and the impedance seen “looking up” into the coil  215  ( FIG. 7 ) is given by:
 
                     Z   base     =         Z   c     ⁢         Z   2     +       Z   c     ⁢     tanh   ⁡     (     j   ⁢           ⁢     β   p     ⁢   H     )               Z   c     +       Z   2     ⁢     tanh   ⁡     (     j   ⁢           ⁢     β   p     ⁢   H     )               =       Z   c     ⁢           Z   2     +       Z   c     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   c       )               Z   c     +       Z   2     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   c       )             .                 (   64   )               
At the base of the guided surface waveguide probe  200 , the impedance seen “looking down” into the lossy conducting medium  203  is Z ↓ =Z in , which is given by:
 
                       Z     i   ⁢           ⁢   n       =         Z   o     ⁢         Z   s     +       Z   o     ⁢     tanh   ⁡     [     j   ⁢           ⁢       β   o     ⁡     (     d   /   2     )         ]               Z   o     +       Z   s     ⁢     tanh   ⁡     [     j   ⁢           ⁢       β   o     ⁡     (     d   /   2     )         ]               =       Z   o     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   d       )             ,           (   65   )               
where Z s =0.
 
     Neglecting losses, the equivalent image plane model can be tuned to resonance when Z ↓ +Z ↑ =0 at the physical boundary  136 . Or, in the low loss case, X ↓ +X ↑ =0 at the physical boundary  136 , where X is the corresponding reactive component. Thus, the impedance at the physical boundary  136  “looking up” into the guided surface waveguide probe  200  is the conjugate of the impedance at the physical boundary  136  “looking down” into the lossy conducting medium  203 . By adjusting the load impedance Z L  of the charge terminal T 1  while maintaining the traveling wave phase delay Φ equal to the angle of the media&#39;s wave tilt Ψ, so that Φ=Ψ, which improves and/or maximizes coupling of the probe&#39;s electric field to a guided surface waveguide mode along the surface of the lossy conducting medium  203  (e.g., Earth), the equivalent image plane models of  FIGS. 9A and 9B  can be tuned to resonance with respect to the image ground plane  139 . In this way, the impedance of the equivalent complex image plane model is purely resistive, which maintains a superposed standing wave on the probe structure that maximizes the voltage and elevated charge on terminal T 1 , and by equations (1)-(3) and (16) maximizes the propagating surface wave. 
     It follows from the Hankel solutions, that the guided surface wave excited by the guided surface waveguide probe  200  is an outward propagating traveling wave. The source distribution along the feed network  209  between the charge terminal T 1  and the ground stake  218  of the guided surface waveguide probe  200  ( FIGS. 3 and 7 ) is actually composed of a superposition of a traveling wave plus a standing wave on the structure. With the charge terminal T 1  positioned at or above the physical height h p , the phase delay of the traveling wave moving through the feed network  209  is matched to the angle of the wave tilt associated with the lossy conducting medium  203 . This mode-matching allows the traveling wave to be launched along the lossy conducting medium  203 . Once the phase delay has been established for the traveling wave, the load impedance Z L  of the charge terminal T 1  is adjusted to bring the probe structure into standing wave resonance with respect to the image ground plane ( 130  of  FIG. 3 or 139  of  FIG. 8 ), which is at a complex depth of −d/2. In that case, the impedance seen from the image ground plane has zero reactance and the charge on the charge terminal T 1  is maximized. 
     The distinction between the traveling wave phenomenon and standing wave phenomena is that (1) the phase delay of traveling waves (θ=βd) on a section of transmission line of length d (sometimes called a “delay line”) is due to propagation time delays; whereas (2) the position-dependent phase of standing waves (which are composed of forward and backward propagating waves) depends on both the line length propagation time delay and impedance transitions at interfaces between line sections of different characteristic impedances. In addition to the phase delay that arises due to the physical length of a section of transmission line operating in sinusoidal steady-state, there is an extra reflection coefficient phase at impedance discontinuities that is due to the ratio of Z oa /Z ob , where Z oa  and Z ob  are the characteristic impedances of two sections of a transmission line such as, e.g., a helical coil section of characteristic impedance Z oa =Z c  ( FIG. 9B ) and a straight section of vertical feed line conductor of characteristic impedance Z ob =Z w  ( FIG. 9B ). 
     As a result of this phenomenon, two relatively short transmission line sections of widely differing characteristic impedance may be used to provide a very large phase shift. For example, a probe structure composed of two sections of transmission line, one of low impedance and one of high impedance, together totaling a physical length of, say, 0.05λ, may be fabricated to provide a phase shift of 90° which is equivalent to a 0.25λ resonance. This is due to the large jump in characteristic impedances. In this way, a physically short probe structure can be electrically longer than the two physical lengths combined. This is illustrated in  FIGS. 9A and 9B , where the discontinuities in the impedance ratios provide large jumps in phase. The impedance discontinuity provides a substantial phase shift where the sections are joined together. 
     Referring to  FIG. 10 , shown is a flow chart  150  illustrating an example of adjusting a guided surface waveguide probe  200  ( FIGS. 3 and 7 ) to substantially mode-match to a guided surface waveguide mode on the surface of the lossy conducting medium, which launches a guided surface traveling wave along the surface of a lossy conducting medium  203  ( FIG. 3 ). Beginning with  153 , the charge terminal T 1  of the guided surface waveguide probe  200  is positioned at a defined height above a lossy conducting medium  203 . Utilizing the characteristics of the lossy conducting medium  203  and the operating frequency of the guided surface waveguide probe  200 , the Hankel crossover distance can also be found by equating the magnitudes of Equations (20b) and (21) for −jγρ, and solving for R x  as illustrated by  FIG. 4 . The complex index of refraction (n) can be determined using Equation (41), and the complex Brewster angle (θ i,B ) can then be determined from Equation (42). The physical height (h p ) of the charge terminal T 1  can then be determined from Equation (44). The charge terminal T 1  should be at or higher than the physical height (h p ) in order to excite the far-out component of the Hankel function. This height relationship is initially considered when launching surface waves. To reduce or minimize the bound charge on the charge terminal T 1 , the height should be at least four times the spherical diameter (or equivalent spherical diameter) of the charge terminal T 1 . 
     At  156 , the electrical phase delay Φ of the elevated charge Q 1  on the charge terminal T 1  is matched to the complex wave tilt angle Ψ. The phase delay (θ c ) of the helical coil and/or the phase delay (θ y ) of the vertical feed line conductor can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (W). Based on Equation (31), the angle (Ψ) of the wave tilt can be determined from: 
                   W   =         E   ρ       E   z       =       1     tan   ⁢           ⁢     θ     i   ,   B           =       1   n     =          W        ⁢       e     j   ⁢           ⁢   Ψ       .                     (   66   )               
The electrical phase Φ can then be matched to the angle of the wave tilt. This angular (or phase) relationship is next considered when launching surface waves. For example, the electrical phase delay Φ=θ c +θ y  can be adjusted by varying the geometrical parameters of the coil  215  ( FIG. 7 ) and/or the length (or height) of the vertical feed line conductor  221  ( FIG. 7 ). By matching Φ=Ψ, an electric field can be established at or beyond the Hankel crossover distance (R x ) with a complex Brewster angle at the boundary interface to excite the surface waveguide mode and launch a traveling wave along the lossy conducting medium  203 .
 
     Next at  159 , the load impedance of the charge terminal T 1  is tuned to resonate the equivalent image plane model of the guided surface waveguide probe  200 . The depth (d/2) of the conducting image ground plane  139  of  FIGS. 9A and 9B  (or  130  of  FIG. 3 ) can be determined using Equations (52), (53) and (54) and the values of the lossy conducting medium  203  (e.g., the Earth), which can be measured. Using that depth, the phase shift (θ d ) between the image ground plane  139  and the physical boundary  136  of the lossy conducting medium  203  can be determined using θ d =β o  d/2. The impedance (Z in ) as seen “looking down” into the lossy conducting medium  203  can then be determined using Equation (65). This resonance relationship can be considered to maximize the launched surface waves. 
     Based upon the adjusted parameters of the coil  215  and the length of the vertical feed line conductor  221 , the velocity factor, phase delay, and impedance of the coil  215  and vertical feed line conductor  221  can be determined using Equations (45) through (51). In addition, the self-capacitance (C T ) of the charge terminal T 1  can be determined using, e.g., Equation (24). The propagation factor (β p ) of the coil  215  can be determined using Equation (35) and the propagation phase constant (β w ) for the vertical feed line conductor  221  can be determined using Equation (49). Using the self-capacitance and the determined values of the coil  215  and vertical feed line conductor  221 , the impedance (Z base ) of the guided surface waveguide probe  200  as seen “looking up” into the coil  215  can be determined using Equations (62), (63) and (64). 
     The equivalent image plane model of the guided surface waveguide probe  200  can be tuned to resonance by adjusting the load impedance Z L  such that the reactance component X base  of Z base  cancels out the reactance component X in  of Z in , or X base +X in =0. Thus, the impedance at the physical boundary  136  “looking up” into the guided surface waveguide probe  200  is the conjugate of the impedance at the physical boundary  136  “looking down” into the lossy conducting medium  203 . The load impedance Z L  can be adjusted by varying the capacitance (C T ) of the charge terminal T 1  without changing the electrical phase delay Φ=θ c + y , of the charge terminal T 1 . An iterative approach may be taken to tune the load impedance Z L  for resonance of the equivalent image plane model with respect to the conducting image ground plane  139  (or  130 ). In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium  203  (e.g., Earth) can be improved and/or maximized. 
     This may be better understood by illustrating the situation with a numerical example. Consider a guided surface waveguide probe  200  comprising a top-loaded vertical stub of physical height h p  with a charge terminal T 1  at the top, where the charge terminal T 1  is excited through a helical coil and vertical feed line conductor at an operational frequency (f o ) of 1.85 MHz. With a height (H 1 ) of 16 feet and the lossy conducting medium  203  (e.g., Earth) having a relative permittivity of ε r =15 and a conductivity of σ 1 =0.010 mhos/m, several surface wave propagation parameters can be calculated for f o =1.850 MHz. Under these conditions, the Hankel crossover distance can be found to be R x =54.5 feet with a physical height of h p =5.5 feet, which is well below the actual height of the charge terminal T 1 . While a charge terminal height of H 1 =5.5 feet could have been used, the taller probe structure reduced the bound capacitance, permitting a greater percentage of free charge on the charge terminal T 1  providing greater field strength and excitation of the traveling wave. 
     The wave length can be determined as: 
                       λ   o     =       c     f   o       =     162.162   ⁢           ⁢   meters         ,           (   67   )               
where c is the speed of light. The complex index of refraction is:
 
 n =√{square root over (ε r   −jx )}=7.529− j 6.546,  (68)
 
from Equation (41), where x=σ 1 /ωε o  with ω=2πf o , and the complex Brewster angle is:
 
θ i,B =arctan(√{square root over (ε r   −jx )})=85.6− j 3.7440°.  (69)
 
from Equation (42). Using Equation (66), the wave tilt values can be determined to be:
 
                   W   =       1     tan   ⁢           ⁢     θ     i   ,   B           =       1   n     =            W        ⁢     e     j   ⁢           ⁢   Ψ         =     0.101   ⁢           ⁢       e     j   ⁢           ⁢   40.614   ⁢   °       .                     (   70   )               
Thus, the helical coil can be adjusted to match Φ=Ψ=40.614°.
 
     The velocity factor of the vertical feed line conductor (approximated as a uniform cylindrical conductor with a diameter of 0.27 inches) can be given as V w ≈0.93. Since h p &lt;&lt;λ o , the propagation phase constant for the vertical feed line conductor can be approximated as: 
                     β   w     =         2   ⁢           ⁢   π       λ   w       =         2   ⁢           ⁢   π         V   w     ⁢     λ   0         =     0.042   ⁢           ⁢       m     -   1       .                   (   71   )               
From Equation (49) the phase delay of the vertical feed line conductor is:
 
θ y =β w   h   w ≈β w   h   p =11.640°.  (72)
 
By adjusting the phase delay of the helical coil so that θ c =28.974°=40.614°−11.6400, Φ will equal Ψ to match the guided surface waveguide mode. To illustrate the relationship between Φ and Ψ,  FIG. 11  shows a plot of both over a range of frequencies. As both Φ and Ψ are frequency dependent, it can be seen that their respective curves cross over each other at approximately 1.85 MHz.
 
     For a helical coil having a conductor diameter of 0.0881 inches, a coil diameter (D) of 30 inches and a turn-to-turn spacing (s) of 4 inches, the velocity factor for the coil can be determined using Equation (45) as: 
                       V   f     =       1       1   +     20   ⁢       (     D   s     )     2.5     ⁢       (     D     λ   0       )     0.5             =   0.069       ,           (   73   )               
and the propagation factor from Equation (35) is:
 
     
       
         
           
             
               
                 
                   
                     β 
                     p 
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                       
                         
                           V 
                           f 
                         
                         ⁢ 
                         
                           λ 
                           0 
                         
                       
                     
                     = 
                     
                       0.564 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           m 
                           
                             - 
                             1 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
           
         
       
     
     With θ c =28.974°, the axial length of the solenoidal helix (H) can be determined using Equation (46) such that: 
                   H   =         θ   c       β   p       =     35.2732   ⁢           ⁢     inches   .                 (   75   )               
This height determines the location on the helical coil where the vertical feed line conductor is connected, resulting in a coil with 8.818 turns (N=H/s).
 
     With the traveling wave phase delay of the coil and vertical feed line conductor adjusted to match the wave tilt angle (Φ=θ c +θ y =Ψ), the load impedance (Z L ) of the charge terminal T 1  can be adjusted for standing wave resonance of the equivalent image plane model of the guided surface wave probe  200 . From the measured permittivity, conductivity and permeability of the Earth, the radial propagation constant can be determined using Equation (57)
 
γ e =√{square root over ( jωu   1 (σ 1   +jωε   1 ))}=0.25+ j 0.292 m −1 ,  (76)
 
And the complex depth of the conducting image ground plane can be approximated from Equation (52) as:
 
                       d   ≈     2     γ   e         =     3.364   +     j   ⁢           ⁢   3.963   ⁢           ⁢   meters         ,           (   77   )               
with a corresponding phase shift between the conducting image ground plane and the physical boundary of the Earth given by:
 
θ d =β o ( d/ 2)=4.015− j 4.73°.  (78)
 
Using Equation (65), the impedance seen “looking down” into the lossy conducting medium  203  (i.e., Earth) can be determined as:
 
 Z   in   =Z   o  tan  h ( jθ   d )= R   in   +jX   in =31.191+ j 26.27 ohms.  (79)
 
     By matching the reactive component (X in ) seen “looking down” into the lossy conducting medium  203  with the reactive component (X base ) seen “looking up” into the guided surface wave probe  200 , the coupling into the guided surface waveguide mode may be maximized. This can be accomplished by adjusting the capacitance of the charge terminal T 1  without changing the traveling wave phase delays of the coil and vertical feed line conductor. For example, by adjusting the charge terminal capacitance (C T ) to 61.8126 pF, the load impedance from Equation (62) is: 
                       Z   L     =       1     j   ⁢           ⁢   ω   ⁢           ⁢     C   T         =       -   j     ⁢           ⁢   1392   ⁢           ⁢   ohms         ,           (   80   )               
and the reactive components at the boundary are matched.
 
     Using Equation (51), the impedance of the vertical feed line conductor (having a diameter (2a) of 0.27 inches) is given as 
                       Z   w     =       138   ⁢           ⁢     log   ⁡     (       1.123   ⁢           ⁢     V   w     ⁢     λ   0         2   ⁢           ⁢   π   ⁢           ⁢   a       )         =     537.534   ⁢           ⁢   ohms         ,           (   81   )               
and the impedance seen “looking up” into the vertical feed line conductor is given by Equation (63) as:
 
                     Z   2     =         Z   W     ⁢         Z   L     +       Z   w     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   y       )               Z   w     +       Z   L     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   y       )               =       -   j     ⁢           ⁢   835.438   ⁢           ⁢     ohms   .                 (   82   )               
Using Equation (47), the characteristic impedance of the helical coil is given as
 
                       Z   c     =         60     V   f       ⁡     [       ln   ⁡     (         V   f     ⁢     λ   0       D     )       -   1.027     ]       =     1446   ⁢           ⁢   ohms         ,           (   83   )               
and the impedance seen “looking up” into the coil at the base is given by Equation (64) as:
 
                     Z   base     =         Z   c     ⁢         Z   2     +       Z   c     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   c       )               Z   c     +       Z   2     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   c       )               =       -   j     ⁢           ⁢   26.27   ⁢             ⁢             ⁢     ohms   .                 (   84   )               
When compared to the solution of Equation (79), it can be seen that the reactive components are opposite and approximately equal, and thus are conjugates of each other. Thus, the impedance (Z ip ) seen “looking up” into the equivalent image plane model of  FIGS. 9A and 9B  from the perfectly conducting image ground plane is only resistive or Z ip =R+j0.
 
     When the electric fields produced by a guided surface waveguide probe  200  ( FIG. 3 ) are established by matching the traveling wave phase delay of the feed network to the wave tilt angle and the probe structure is resonated with respect to the perfectly conducting image ground plane at complex depth z=−d/2, the fields are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium, a guided surface traveling wave is launched along the surface of the lossy conducting medium. As illustrated in  FIG. 1 , the guided field strength curve  103  of the guided electromagnetic field has a characteristic exponential decay of e −αd /√{square root over (d)} and exhibits a distinctive knee  109  on the log-log scale. 
     In summary, both analytically and experimentally, the traveling wave component on the structure of the guided surface waveguide probe  200  has a phase delay (Φ) at its upper terminal that matches the angle (Ψ) of the wave tilt of the surface traveling wave (Φ=Ψ). Under this condition, the surface waveguide may be considered to be “mode-matched”. Furthermore, the resonant standing wave component on the structure of the guided surface waveguide probe  200  has a V MAX  at the charge terminal T 1  and a V MIN  down at the image plane  139  ( FIG. 8B ) where Z ip =R ip +j0 at a complex depth of z=−d/2, not at the connection at the physical boundary  136  of the lossy conducting medium  203  ( FIG. 8B ). Lastly, the charge terminal T 1  is of sufficient height H 1  of  FIG. 3  (h≥R x  tan ψ i,B ) so that electromagnetic waves incident onto the lossy conducting medium  203  at the complex Brewster angle do so out at a distance (≥R x ) where the 1/√{square root over (r)} term is predominant. Receive circuits can be utilized with one or more guided surface waveguide probes to facilitate wireless transmission and/or power delivery systems. 
     Referring back to  FIG. 3 , operation of a guided surface waveguide probe  200  may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe  200 . For example, an adaptive probe control system  230  can be used to control the feed network  209  and/or the charge terminal T 1  to control the operation of the guided surface waveguide probe  200 . Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium  203  (e.g., conductivity σ and relative permittivity ε r ), variations in field strength and/or variations in loading of the guided surface waveguide probe  200 . As can be seen from Equations (31), (41) and (42), the index of refraction (n), the complex Brewster angle (θ i,B ), and the wave tilt (|W|e jΨ ) can be affected by changes in soil conductivity and permittivity resulting from, e.g., weather conditions. 
     Equipment such as, e.g., conductivity measurement probes, permittivity sensors, ground parameter meters, field meters, current monitors and/or load receivers can be used to monitor for changes in the operational conditions and provide information about current operational conditions to the adaptive probe control system  230 . The probe control system  230  can then make one or more adjustments to the guided surface waveguide probe  200  to maintain specified operational conditions for the guided surface waveguide probe  200 . For instance, as the moisture and temperature vary, the conductivity of the soil will also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe  200 . Generally, it would be desirable to monitor the conductivity and/or permittivity at or about the Hankel crossover distance R x  for the operational frequency. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations (e.g., in each quadrant) around the guided surface waveguide probe  200 . 
     The conductivity measurement probes and/or permittivity sensors can be configured to evaluate the conductivity and/or permittivity on a periodic basis and communicate the information to the probe control system  230 . The information may be communicated to the probe control system  230  through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate wired or wireless communication network. Based upon the monitored conductivity and/or permittivity, the probe control system  230  may evaluate the variation in the index of refraction (n), the complex Brewster angle (θ i,B ), and/or the wave tilt (|W|e jΨ ) and adjust the guided surface waveguide probe  200  to maintain the phase delay (Φ) of the feed network  209  equal to the wave tilt angle (Ψ) and/or maintain resonance of the equivalent image plane model of the guided surface waveguide probe  200 . This can be accomplished by adjusting, e.g., θ y , θ c  and/or C T . For instance, the probe control system  230  can adjust the self-capacitance of the charge terminal T 1  and/or the phase delay (θ y , θ c ) applied to the charge terminal T 1  to maintain the electrical launching efficiency of the guided surface wave at or near its maximum. For example, the self-capacitance of the charge terminal T 1  can be varied by changing the size of the terminal. The charge distribution can also be improved by increasing the size of the charge terminal T 1 , which can reduce the chance of an electrical discharge from the charge terminal T 1 . In other embodiments, the charge terminal T 1  can include a variable inductance that can be adjusted to change the load impedance Z L . The phase applied to the charge terminal T 1  can be adjusted by varying the tap position on the coil  215  ( FIG. 7 ), and/or by including a plurality of predefined taps along the coil  215  and switching between the different predefined tap locations to maximize the launching efficiency. 
     Field or field strength (FS) meters may also be distributed about the guided surface waveguide probe  200  to measure field strength of fields associated with the guided surface wave. The field or FS meters can be configured to detect the field strength and/or changes in the field strength (e.g., electric field strength) and communicate that information to the probe control system  230 . The information may be communicated to the probe control system  230  through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. As the load and/or environmental conditions change or vary during operation, the guided surface waveguide probe  200  may be adjusted to maintain specified field strength(s) at the FS meter locations to ensure appropriate power transmission to the receivers and the loads they supply. 
     For example, the phase delay (Φ=θ y +θ c ) applied to the charge terminal T 1  can be adjusted to match the wave tilt angle (Ψ). By adjusting one or both phase delays, the guided surface waveguide probe  200  can be adjusted to ensure the wave tilt corresponds to the complex Brewster angle. This can be accomplished by adjusting a tap position on the coil  215  ( FIG. 7 ) to change the phase delay supplied to the charge terminal T 1 . The voltage level supplied to the charge terminal T 1  can also be increased or decreased to adjust the electric field strength. This may be accomplished by adjusting the output voltage of the excitation source  212  or by adjusting or reconfiguring the feed network  209 . For instance, the position of the tap  227  ( FIG. 7 ) for the AC source  212  can be adjusted to increase the voltage seen by the charge terminal T 1 . Maintaining field strength levels within predefined ranges can improve coupling by the receivers, reduce ground current losses, and avoid interference with transmissions from other guided surface waveguide probes  200 . 
     The probe control system  230  can be implemented with hardware, firmware, software executed by hardware, or a combination thereof. For example, the probe control system  230  can include processing circuitry including a processor and a memory, both of which can be coupled to a local interface such as, for example, a data bus with an accompanying control/address bus as can be appreciated by those with ordinary skill in the art. A probe control application may be executed by the processor to adjust the operation of the guided surface waveguide probe  200  based upon monitored conditions. The probe control system  230  can also include one or more network interfaces for communicating with the various monitoring devices. Communications can be through a network such as, but not limited to, a LAN, WLAN, cellular network, or other appropriate communication network. The probe control system  230  may comprise, for example, a computer system such as a server, desktop computer, laptop, or other system with like capability. 
     Referring back to the example of  FIG. 5A , the complex angle trigonometry is shown for the ray optic interpretation of the incident electric field (E) of the charge terminal T 1  with a complex Brewster angle (θ i,B ) at the Hankel crossover distance (R x ). Recall that, for a lossy conducting medium, the Brewster angle is complex and specified by equation (38). Electrically, the geometric parameters are related by the electrical effective height (h eff ) of the charge terminal T 1  by equation (39). Since both the physical height (h p ) and the Hankel crossover distance (R x ) are real quantities, the angle of the desired guided surface wave tilt at the Hankel crossover distance (W Rx ) is equal to the phase (Φ) of the complex effective height (h eff ). With the charge terminal T 1  positioned at the physical height h p  and excited with a charge having the appropriate phase Φ, the resulting electric field is incident with the lossy conducting medium boundary interface at the Hankel crossover distance R x , and at the Brewster angle. Under these conditions, the guided surface waveguide mode can be excited without reflection or substantially negligible reflection. 
     However, Equation (39) means that the physical height of the guided surface waveguide probe  200  can be relatively small. While this will excite the guided surface waveguide mode, this can result in an unduly large bound charge with little free charge. To compensate, the charge terminal T 1  can be raised to an appropriate elevation to increase the amount of free charge. As one example rule of thumb, the charge terminal T 1  can be positioned at an elevation of about 4-5 times (or more) the effective diameter of the charge terminal T 1 .  FIG. 6  illustrates the effect of raising the charge terminal T 1  above the physical height (h p ) shown in  FIG. 5A . The increased elevation causes the distance at which the wave tilt is incident with the lossy conductive medium to move beyond the Hankel crossover point  121  ( FIG. 5A ). To improve coupling in the guided surface waveguide mode, and thus provide for a greater launching efficiency of the guided surface wave, a lower compensation terminal T 2  can be used to adjust the total effective height (h TE ) of the charge terminal T 1  such that the wave tilt at the Hankel crossover distance is at the Brewster angle. 
     Referring to  FIG. 12 , shown is an example of a guided surface waveguide probe  200   c  that includes an elevated charge terminal T 1  and a lower compensation terminal T 2  that are arranged along a vertical axis z that is normal to a plane presented by the lossy conducting medium  203 . In this respect, the charge terminal T 1  is placed directly above the compensation terminal T 2  although it is possible that some other arrangement of two or more charge and/or compensation terminals T N  can be used. The guided surface waveguide probe  200   c  is disposed above a lossy conducting medium  203  according to an embodiment of the present disclosure. The lossy conducting medium  203  makes up Region  1  with a second medium  206  that makes up Region  2  sharing a boundary interface with the lossy conducting medium  203 . 
     The guided surface waveguide probe  200   c  includes a feed network  209  that couples an excitation source  212  to the charge terminal T 1  and the compensation terminal T 2 . According to various embodiments, charges Q 1  and Q 2  can be imposed on the respective charge and compensation terminals T 1  and T 2 , depending on the voltages applied to terminals T 1  and T 2  at any given instant.  11  is the conduction current feeding the charge Q 1  on the charge terminal T 1  via the terminal lead, and I 2  is the conduction current feeding the charge Q 2  on the compensation terminal T 2  via the terminal lead. 
     According to the embodiment of  FIG. 12 , the charge terminal T 1  is positioned over the lossy conducting medium  203  at a physical height H 1 , and the compensation terminal T 2  is positioned directly below T 1  along the vertical axis z at a physical height H 2 , where H 2  is less than H 1 . The height h of the transmission structure may be calculated as h=H 1 −H 2 . The charge terminal T 1  has an isolated (or self) capacitance C 1 , and the compensation terminal T 2  has an isolated (or self) capacitance C 2 . A mutual capacitance C M  can also exist between the terminals T 1  and T 2  depending on the distance therebetween. During operation, charges Q 1  and Q 2  are imposed on the charge terminal T 1  and the compensation terminal T 2 , respectively, depending on the voltages applied to the charge terminal T 1  and the compensation terminal T 2  at any given instant. 
     Referring next to  FIG. 13 , shown is a ray optics interpretation of the effects produced by the elevated charge Q 1  on charge terminal T 1  and compensation terminal T 2  of  FIG. 12 . With the charge terminal T 1  elevated to a height where the ray intersects with the lossy conductive medium at the Brewster angle at a distance greater than the Hankel crossover point  121  as illustrated by line  163 , the compensation terminal T 2  can be used to adjust h TE  by compensating for the increased height. The effect of the compensation terminal T 2  is to reduce the electrical effective height of the guided surface waveguide probe (or effectively raise the lossy medium interface) such that the wave tilt at the Hankel crossover distance is at the Brewster angle as illustrated by line  166 . 
     The total effective height can be written as the superposition of an upper effective height (h UE ) associated with the charge terminal T 1  and a lower effective height (h LE ) associated with the compensation terminal T 2  such that
 
 h   TE   h   UE   +h   LE   =h   p   e   j(βh     p     +Φ     U     )   +h   d   e   j(βh     d     +Φ     L     )   =R   x   ×W,   (85)
 
where Φ U  is the phase delay applied to the upper charge terminal T 1 , Φ L  is the phase delay applied to the lower compensation terminal T 2 , β=2π/λ p  is the propagation factor from Equation (35), h p  is the physical height of the charge terminal T 1  and h d  is the physical height of the compensation terminal T 2 . If extra lead lengths are taken into consideration, they can be accounted for by adding the charge terminal lead length z to the physical height h p  of the charge terminal T 1  and the compensation terminal lead length y to the physical height h d  of the compensation terminal T 2  as shown in
 
 h   TE =( h   p   +z ) e   j(β(h     p     +z)+Φ     U     ) +( h   d   +y ) e   j(β(h     d     y)+Φ     L     )   =R   x   ×W.   (86)
 
The lower effective height can be used to adjust the total effective height (h TE ) to equal the complex effective height (h eff ) of  FIG. 5A .
 
     Equations (85) or (86) can be used to determine the physical height of the lower disk of the compensation terminal T 2  and the phase angles to feed the terminals in order to obtain the desired wave tilt at the Hankel crossover distance. For example, Equation (86) can be rewritten as the phase shift applied to the charge terminal T 1  as a function of the compensation terminal height (h d ) to give 
     
       
         
           
             
               
                 
                   
                     
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     To determine the positioning of the compensation terminal T 2 , the relationships discussed above can be utilized. First, the total effective height (h TE ) is the superposition of the complex effective height (h UE ) of the upper charge terminal T 1  and the complex effective height (h LE ) of the lower compensation terminal T 2  as expressed in Equation (86). Next, the tangent of the angle of incidence can be expressed geometrically as 
                       tan   ⁢           ⁢     ψ   E       =       h   TE       R   x         ,           (   88   )               
which is equal to the definition of the wave tilt, W. Finally, given the desired Hankel crossover distance R x , the h TE  can be adjusted to make the wave tilt of the incident ray match the complex Brewster angle at the Hankel crossover point  121 . This can be accomplished by adjusting h p , Φ U , and/or h d .
 
     These concepts may be better understood when discussed in the context of an example of a guided surface waveguide probe. Referring to  FIG. 14 , shown is a graphical representation of an example of a guided surface waveguide probe  200   d  including an upper charge terminal T 1  (e.g., a sphere at height h T ) and a lower compensation terminal T 2  (e.g., a disk at height ha) that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium  203 . During operation, charges Q 1  and Q 2  are imposed on the charge and compensation terminals T 1  and T 2 , respectively, depending on the voltages applied to the terminals T 1  and T 2  at any given instant. 
     An AC source  212  acts as the excitation source for the charge terminal T 1 , which is coupled to the guided surface waveguide probe  200   d  through a feed network  209  comprising a coil  215  such as, e.g., a helical coil. The AC source  212  can be connected across a lower portion of the coil  215  through a tap  227 , as shown in  FIG. 14 , or can be inductively coupled to the coil  215  by way of a primary coil. The coil  215  can be coupled to a ground stake  218  at a first end and the charge terminal T 1  at a second end. In some implementations, the connection to the charge terminal T 1  can be adjusted using a tap  224  at the second end of the coil  215 . The compensation terminal T 2  is positioned above and substantially parallel with the lossy conducting medium  203  (e.g., the ground or Earth), and energized through a tap  233  coupled to the coil  215 . An ammeter  236  located between the coil  215  and ground stake  218  can be used to provide an indication of the magnitude of the current flow (I 0 ) at the base of the guided surface waveguide probe. Alternatively, a current clamp may be used around the conductor coupled to the ground stake  218  to obtain an indication of the magnitude of the current flow (I 0 ). 
     In the example of  FIG. 14 , the coil  215  is coupled to a ground stake  218  at a first end and the charge terminal T 1  at a second end via a vertical feed line conductor  221 . In some implementations, the connection to the charge terminal T 1  can be adjusted using a tap  224  at the second end of the coil  215  as shown in  FIG. 14 . The coil  215  can be energized at an operating frequency by the AC source  212  through a tap  227  at a lower portion of the coil  215 . In other implementations, the AC source  212  can be inductively coupled to the coil  215  through a primary coil. The compensation terminal T 2  is energized through a tap  233  coupled to the coil  215 . An ammeter  236  located between the coil  215  and ground stake  218  can be used to provide an indication of the magnitude of the current flow at the base of the guided surface waveguide probe  200   d . Alternatively, a current clamp may be used around the conductor coupled to the ground stake  218  to obtain an indication of the magnitude of the current flow. The compensation terminal T 2  is positioned above and substantially parallel with the lossy conducting medium  203  (e.g., the ground). 
     In the example of  FIG. 14 , the connection to the charge terminal T 1  located on the coil  215  above the connection point of tap  233  for the compensation terminal T 2 . Such an adjustment allows an increased voltage (and thus a higher charge Q 1 ) to be applied to the upper charge terminal T 1 . In other embodiments, the connection points for the charge terminal T 1  and the compensation terminal T 2  can be reversed. It is possible to adjust the total effective height (h TE ) of the guided surface waveguide probe  200   d  to excite an electric field having a guided surface wave tilt at the Hankel crossover distance R x . The Hankel crossover distance can also be found by equating the magnitudes of equations (20b) and (21) for −jγρ, and solving for R x  as illustrated by  FIG. 4 . The index of refraction (n), the complex Brewster angle (θ i,B  and ψ i,B ), the wave tilt (|W|e jΨ ) and the complex effective height (h eff =h p e jΦ ) can be determined as described with respect to Equations (41)-(44) above. 
     With the selected charge terminal T 1  configuration, a spherical diameter (or the effective spherical diameter) can be determined. For example, if the charge terminal T 1  is not configured as a sphere, then the terminal configuration may be modeled as a spherical capacitance having an effective spherical diameter. The size of the charge terminal T 1  can be chosen to provide a sufficiently large surface for the charge Q 1  imposed on the terminals. In general, it is desirable to make the charge terminal T 1  as large as practical. The size of the charge terminal T 1  should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal. To reduce the amount of bound charge on the charge terminal T 1 , the desired elevation to provide free charge on the charge terminal T 1  for launching a guided surface wave should be at least 4-5 times the effective spherical diameter above the lossy conductive medium (e.g., the Earth). The compensation terminal T 2  can be used to adjust the total effective height (h TE ) of the guided surface waveguide probe  200   d  to excite an electric field having a guided surface wave tilt at R x . The compensation terminal T 2  can be positioned below the charge terminal T 1  at h d =h T −h p , where h T  is the total physical height of the charge terminal T 1 . With the position of the compensation terminal T 2  fixed and the phase delay Φ U  applied to the upper charge terminal T 1 , the phase delay Φ L  applied to the lower compensation terminal T 2  can be determined using the relationships of Equation (86), such that: 
                       Φ   U     ⁡     (     h   d     )       =       -     β   ⁡     (       h   d     +   y     )         -     j   ⁢           ⁢       ln   ⁡     (           R   x     ×   W     -       (       h   d     +   z     )     ⁢     e     j   ⁡     (       β   ⁢           ⁢     h   p       +     β   ⁢           ⁢   z     +     Φ   L       )               (       h   d     +   y     )       )       .                 (   89   )               
In alternative embodiments, the compensation terminal T 2  can be positioned at a height h d  where Im{Φ L }=0. This is graphically illustrated in  FIG. 15A , which shows plots  172  and  175  of the imaginary and real parts of Φ U , respectively. The compensation terminal T 2  is positioned at a height h d  where Im{Φ U }=0, as graphically illustrated in plot  172 . At this fixed height, the coil phase Φ U  can be determined from Re{Φ U }, as graphically illustrated in plot  175 .
 
     With the AC source  212  coupled to the coil  215  (e.g., at the 50Ω point to maximize coupling), the position of tap  233  may be adjusted for parallel resonance of the compensation terminal T 2  with at least a portion of the coil at the frequency of operation.  FIG. 15B  shows a schematic diagram of the general electrical hookup of  FIG. 14  in which V 1  is the voltage applied to the lower portion of the coil  215  from the AC source  212  through tap  227 , V 2  is the voltage at tap  224  that is supplied to the upper charge terminal T 1 , and V 3  is the voltage applied to the lower compensation terminal T 2  through tap  233 . The resistances R p  and R d  represent the ground return resistances of the charge terminal T 1  and compensation terminal T 2 , respectively. The charge and compensation terminals T 1  and T 2  may be configured as spheres, cylinders, toroids, rings, hoods, or any other combination of capacitive structures. The size of the charge and compensation terminals T 1  and T 2  can be chosen to provide a sufficiently large surface for the charges Q 1  and Q 2  imposed on the terminals. In general, it is desirable to make the charge terminal T 1  as large as practical. The size of the charge terminal T 1  should be large enough to avoid ionization of the surrounding air, which can result in electrical discharge or sparking around the charge terminal. The self-capacitance C p  and C d  of the charge and compensation terminals T 1  and T 2  respectively, can be determined using, for example, equation (24). 
     As can be seen in  FIG. 15B , a resonant circuit is formed by at least a portion of the inductance of the coil  215 , the self-capacitance C d  of the compensation terminal T 2 , and the ground return resistance R d  associated with the compensation terminal T 2 . The parallel resonance can be established by adjusting the voltage V 3  applied to the compensation terminal T 2  (e.g., by adjusting a tap  233  position on the coil  215 ) or by adjusting the height and/or size of the compensation terminal T 2  to adjust C d . The position of the coil tap  233  can be adjusted for parallel resonance, which will result in the ground current through the ground stake  218  and through the ammeter  236  reaching a maximum point. After parallel resonance of the compensation terminal T 2  has been established, the position of the tap  227  for the AC source  212  can be adjusted to the 50Ω point on the coil  215 . 
     Voltage V 2  from the coil  215  can be applied to the charge terminal T 1 , and the position of tap  224  can be adjusted such that the phase (Φ) of the total effective height (h TE ) approximately equals the angle of the guided surface wave tilt (W Rx ) at the Hankel crossover distance (R x ). The position of the coil tap  224  can be adjusted until this operating point is reached, which results in the ground current through the ammeter  236  increasing to a maximum. At this point, the resultant fields excited by the guided surface waveguide probe  200   d  are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium  203 , resulting in the launching of a guided surface wave along the surface of the lossy conducting medium  203 . This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe  200 . 
     Resonance of the circuit including the compensation terminal T 2  may change with the attachment of the charge terminal T 1  and/or with adjustment of the voltage applied to the charge terminal T 1  through tap  224 . While adjusting the compensation terminal circuit for resonance aids the subsequent adjustment of the charge terminal connection, it is not necessary to establish the guided surface wave tilt (W Rx ) at the Hankel crossover distance (R x ). The system may be further adjusted to improve coupling by iteratively adjusting the position of the tap  227  for the AC source  212  to be at the 50Ω point on the coil  215  and adjusting the position of tap  233  to maximize the ground current through the ammeter  236 . Resonance of the circuit including the compensation terminal T 2  may drift as the positions of taps  227  and  233  are adjusted, or when other components are attached to the coil  215 . 
     In other implementations, the voltage V 2  from the coil  215  can be applied to the charge terminal T 1 , and the position of tap  233  can be adjusted such that the phase (Φ) of the total effective height (h TE ) approximately equals the angle (Ψ) of the guided surface wave tilt at R x . The position of the coil tap  224  can be adjusted until the operating point is reached, resulting in the ground current through the ammeter  236  substantially reaching a maximum. The resultant fields are substantially mode-matched to a guided surface waveguide mode on the surface of the lossy conducting medium  203 , and a guided surface wave is launched along the surface of the lossy conducting medium  203 . This can be verified by measuring field strength along a radial extending from the guided surface waveguide probe  200 . The system may be further adjusted to improve coupling by iteratively adjusting the position of the tap  227  for the AC source  212  to be at the 50Ω point on the coil  215  and adjusting the position of tap  224  and/or  233  to maximize the ground current through the ammeter  236 . 
     Referring back to  FIG. 12 , operation of a guided surface waveguide probe  200  may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe  200 . For example, a probe control system  230  can be used to control the feed network  209  and/or positioning of the charge terminal T 1  and/or compensation terminal T 2  to control the operation of the guided surface waveguide probe  200 . Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium  203  (e.g., conductivity σ and relative permittivity ε r ), variations in field strength and/or variations in loading of the guided surface waveguide probe  200 . As can be seen from Equations (41)-(44), the index of refraction (n), the complex Brewster angle (θ i,B  and ψ i,B ), the wave tilt (|W|e jΨ ) and the complex effective height (h eff =h p e jΦ ) can be affected by changes in soil conductivity and permittivity resulting from, e.g., weather conditions. 
     Equipment such as, e.g., conductivity measurement probes, permittivity sensors, ground parameter meters, field meters, current monitors and/or load receivers can be used to monitor for changes in the operational conditions and provide information about current operational conditions to the probe control system  230 . The probe control system  230  can then make one or more adjustments to the guided surface waveguide probe  200  to maintain specified operational conditions for the guided surface waveguide probe  200 . For instance, as the moisture and temperature vary, the conductivity of the soil will also vary. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations around the guided surface waveguide probe  200 . Generally, it would be desirable to monitor the conductivity and/or permittivity at or about the Hankel crossover distance R x  for the operational frequency. Conductivity measurement probes and/or permittivity sensors may be located at multiple locations (e.g., in each quadrant) around the guided surface waveguide probe  200 . 
     With reference then to  FIG. 16 , shown is an example of a guided surface waveguide probe  200   e  that includes a charge terminal T 1  and a charge terminal T 2  that are arranged along a vertical axis z. The guided surface waveguide probe  200   e  is disposed above a lossy conducting medium  203 , which makes up Region  1 . In addition, a second medium  206  shares a boundary interface with the lossy conducting medium  203  and makes up Region  2 . The charge terminals T 1  and T 2  are positioned over the lossy conducting medium  203 . The charge terminal T 1  is positioned at height H 1 , and the charge terminal T 2  is positioned directly below T 1  along the vertical axis z at height H 2 , where H 2  is less than H 1 . The height h of the transmission structure presented by the guided surface waveguide probe  200   e  is h=H 1 −H 2 . The guided surface waveguide probe  200   e  includes a feed network  209  that couples an excitation source  212  to the charge terminals T 1  and T 2 . 
     The charge terminals T 1  and/or T 2  include a conductive mass that can hold an electrical charge, which may be sized to hold as much charge as practically possible. The charge terminal T 1  has a self-capacitance C 1 , and the charge terminal T 2  has a self-capacitance C 2 , which can be determined using, for example, equation (24). By virtue of the placement of the charge terminal T 1  directly above the charge terminal T 2 , a mutual capacitance C M  is created between the charge terminals T 1  and T 2 . Note that the charge terminals T 1  and T 2  need not be identical, but each can have a separate size and shape, and can include different conducting materials. Ultimately, the field strength of a guided surface wave launched by a guided surface waveguide probe  200   e  is directly proportional to the quantity of charge on the terminal T 1 . The charge Q 1  is, in turn, proportional to the self-capacitance C 1  associated with the charge terminal T 1  since Q 1 =C 1 V, where V is the voltage imposed on the charge terminal T 1 . 
     When properly adjusted to operate at a predefined operating frequency, the guided surface waveguide probe  200   e  generates a guided surface wave along the surface of the lossy conducting medium  203 . The excitation source  212  can generate electrical energy at the predefined frequency that is applied to the guided surface waveguide probe  200   e  to excite the structure. When the electromagnetic fields generated by the guided surface waveguide probe  200   e  are substantially mode-matched with the lossy conducting medium  203 , the electromagnetic fields substantially synthesize a wave front incident at a complex Brewster angle that results in little or no reflection. Thus, the surface waveguide probe  200   e  does not produce a radiated wave, but launches a guided surface traveling wave along the surface of a lossy conducting medium  203 . The energy from the excitation source  212  can be transmitted as Zenneck surface currents to one or more receivers that are located within an effective transmission range of the guided surface waveguide probe  200   e.    
     One can determine asymptotes of the radial Zenneck surface current J ρ (ρ) on the surface of the lossy conducting medium  203  to be J 1 (ρ) close-in and J 2 (ρ) far-out, where 
                       Close   ⁢     -     ⁢   in   ⁢           ⁢     (     ρ   &lt;     λ   /   8       )     ⁢     :     ⁢           ⁢       J   ρ     ⁡     (   ρ   )       ⁢     ~     ⁢     J   1       =           I   1     +     I   2         2   ⁢   πρ       +           E   ρ   QS     ⁡     (     Q   1     )       +       E   ρ   QS     ⁡     (     Q   2     )           Z   ρ           ,   and           (   90   )                       ⁢         Far   ⁢     -     ⁢   out   ⁢           ⁢     (     ρ   ⪢     λ   /   8       )       :         J   ρ     ⁡     (   ρ   )       ⁢     ~     ⁢     J   2         =         j   ⁢           ⁢   γω   ⁢           ⁢     Q   1       4     ×         2   ⁢   γ     π       ×         e       -     (     α   +     j   ⁢           ⁢   β       )       ⁢   ρ         ρ       .                 (   91   )               
where I 1  is the conduction current feeding the charge Q 1  on the first charge terminal T 1 , and I 2  is the conduction current feeding the charge Q 2  on the second charge terminal T 2 . The charge Q 1  on the upper charge terminal T 1  is determined by Q 1 =C 1 V 1 , where C 1  is the isolated capacitance of the charge terminal T 1 . Note that there is a third component to J 1  set forth above given by (E ρ   Q     i   )/Z ρ , which follows from the Leontovich boundary condition and is the radial current contribution in the lossy conducting medium  203  pumped by the quasi-static field of the elevated oscillating charge on the first charge terminal Q 1 . The quantity Z p =jωμ o /γ e  is the radial impedance of the lossy conducting medium, where γ e =(jωμ 1 σ 1 −ω 2 μ 1 ε 1 ) 1/2 .
 
     The asymptotes representing the radial current close-in and far-out as set forth by equations (90) and (91) are complex quantities. According to various embodiments, a physical surface current J(ρ), is synthesized to match as close as possible the current asymptotes in magnitude and phase. That is to say close-in, |J(ρ)| is to be tangent to |J 1 |, and far-out |J(ρ)| is to be tangent to |J 2 |. Also, according to the various embodiments, the phase of J(ρ) should transition from the phase of J 1  close-in to the phase of J 2  far-out. 
     In order to match the guided surface wave mode at the site of transmission to launch a guided surface wave, the phase of the surface current |J 2 | far-out should differ from the phase of the surface current |J 1 | close-in by the propagation phase corresponding to e −jβ(ρ     2     −ρ     1     )  plus a constant of approximately 45 degrees or 225 degrees. This is because there are two roots for √{square root over (γ)}, one near π/4 and one near 5π/4. The properly adjusted synthetic radial surface current is 
                       J   ρ     ⁡     (     ρ   ,   ϕ   ,   0     )       =           I   o     ⁢   γ     4     ⁢           ⁢         H   1     (   2   )       ⁡     (       -   j     ⁢           ⁢   γρ     )       .               (   92   )               
Note that this is consistent with equation (17). By Maxwell&#39;s equations, such a J(ρ) surface current automatically creates fields that conform to
 
                       H   ϕ     =           -   γ     ⁢           ⁢     I   o       4     ⁢     e       -     u   2       ⁢   z       ⁢           ⁢       H   1     (   2   )       ⁡     (       -   j     ⁢           ⁢   γρ     )           ,           (   93   )                   E   ρ     =           -   γ     ⁢           ⁢     I   o       4     ⁢     (       u   2       j   ⁢           ⁢     ωɛ   o         )     ⁢     e       -     u   2       ⁢   z       ⁢           ⁢       H   1     (   2   )       ⁡     (       -   j     ⁢           ⁢   γρ     )           ,   and           (   94   )                 E   z     =           -   γ     ⁢           ⁢     I   o       4     ⁢     (       -   γ       ωɛ   o       )     ⁢     e       -     u   2       ⁢   z       ⁢           ⁢         H   0     (   2   )       ⁡     (       -   j     ⁢           ⁢   γρ     )       .               (   95   )               
Thus, the difference in phase between the surface current |J 2 | far-out and the surface current |J 1 | close-in for the guided surface wave mode that is to be matched is due to the characteristics of the Hankel functions in equations (93)-(95), which are consistent with equations (1)-(3). It is of significance to recognize that the fields expressed by equations (1)-(6) and (17) and equations (92)-(95) have the nature of a transmission line mode bound to a lossy interface, not radiation fields that are associated with groundwave propagation.
 
     In order to obtain the appropriate voltage magnitudes and phases for a given design of a guided surface waveguide probe  200   e  at a given location, an iterative approach may be used. Specifically, analysis may be performed of a given excitation and configuration of a guided surface waveguide probe  200   e  taking into account the feed currents to the terminals T 1  and T 2 , the charges on the charge terminals T 1  and T 2 , and their images in the lossy conducting medium  203  in order to determine the radial surface current density generated. This process may be performed iteratively until an optimal configuration and excitation for a given guided surface waveguide probe  200   e  is determined based on desired parameters. To aid in determining whether a given guided surface waveguide probe  200   e  is operating at an optimal level, a guided field strength curve  103  ( FIG. 1 ) may be generated using equations (1)-(12) based on values for the conductivity of Region  1  (σ 1 ) and the permittivity of Region  1  ( ε 1 ) at the location of the guided surface waveguide probe  200   e . Such a guided field strength curve  103  can provide a benchmark for operation such that measured field strengths can be compared with the magnitudes indicated by the guided field strength curve  103  to determine if optimal transmission has been achieved. 
     In order to arrive at an optimized condition, various parameters associated with the guided surface waveguide probe  200   e  may be adjusted. One parameter that may be varied to adjust the guided surface waveguide probe  200   e  is the height of one or both of the charge terminals T 1  and/or T 2  relative to the surface of the lossy conducting medium  203 . In addition, the distance or spacing between the charge terminals T 1  and T 2  may also be adjusted. In doing so, one may minimize or otherwise alter the mutual capacitance C M  or any bound capacitances between the charge terminals T 1  and T 2  and the lossy conducting medium  203  as can be appreciated. The size of the respective charge terminals T 1  and/or T 2  can also be adjusted. By changing the size of the charge terminals T 1  and/or T 2 , one will alter the respective self-capacitances C 1  and/or C 2 , and the mutual capacitance C M  as can be appreciated. 
     Still further, another parameter that can be adjusted is the feed network  209  associated with the guided surface waveguide probe  200   e . This may be accomplished by adjusting the size of the inductive and/or capacitive reactances that make up the feed network  209 . For example, where such inductive reactances comprise coils, the number of turns on such coils may be adjusted. Ultimately, the adjustments to the feed network  209  can be made to alter the electrical length of the feed network  209 , thereby affecting the voltage magnitudes and phases on the charge terminals T 1  and T 2 . 
     Note that the iterations of transmission performed by making the various adjustments may be implemented by using computer models or by adjusting physical structures as can be appreciated. By making the above adjustments, one can create corresponding “close-in” surface current J 1  and “far-out” surface current J 2  that approximate the same currents J(ρ) of the guided surface wave mode specified in Equations (90) and (91) set forth above. In doing so, the resulting electromagnetic fields would be substantially or approximately mode-matched to a guided surface wave mode on the surface of the lossy conducting medium  203 . 
     While not shown in the example of  FIG. 16 , operation of the guided surface waveguide probe  200   e  may be controlled to adjust for variations in operational conditions associated with the guided surface waveguide probe  200 . For example, a probe control system  230  shown in  FIG. 12  can be used to control the feed network  209  and/or positioning and/or size of the charge terminals T 1  and/or T 2  to control the operation of the guided surface waveguide probe  200   e . Operational conditions can include, but are not limited to, variations in the characteristics of the lossy conducting medium  203  (e.g., conductivity σ and relative permittivity ε r ), variations in field strength and/or variations in loading of the guided surface waveguide probe  200   e.    
     Referring now to  FIG. 17 , shown is an example of the guided surface waveguide probe  200   e  of  FIG. 16 , denoted herein as guided surface waveguide probe  200   f . The guided surface waveguide probe  200   f  includes the charge terminals T 1  and T 2  that are positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium  203  (e.g., the Earth). The second medium  206  is above the lossy conducting medium  203 . The charge terminal T 1  has a self-capacitance C 1 , and the charge terminal T 2  has a self-capacitance C 2 . During operation, charges Q 1  and Q 2  are imposed on the charge terminals T 1  and T 2 , respectively, depending on the voltages applied to the charge terminals T 1  and T 2  at any given instant. A mutual capacitance C M  may exist between the charge terminals T 1  and T 2  depending on the distance there between. In addition, bound capacitances may exist between the respective charge terminals T 1  and T 2  and the lossy conducting medium  203  depending on the heights of the respective charge terminals T 1  and T 2  with respect to the lossy conducting medium  203 . 
     The guided surface waveguide probe  200   f  includes a feed network  209  that comprises an inductive impedance comprising a coil L 1a  having a pair of leads that are coupled to respective ones of the charge terminals T 1  and T 2 . In one embodiment, the coil L 1a  is specified to have an electrical length that is one-half (½) of the wavelength at the operating frequency of the guided surface waveguide probe  200   f.    
     While the electrical length of the coil L 1a  is specified as approximately one-half (½) the wavelength at the operating frequency, it is understood that the coil L 1a  may be specified with an electrical length at other values. According to one embodiment, the fact that the coil L 1a  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 T 1  and T 2 . Nonetheless, the length or diameter of the coil L 1a  may be increased or decreased when adjusting the guided surface waveguide probe  200   f  to obtain optimal excitation of a guided surface wave mode. Adjustment of the coil length may be provided by taps located at one or both ends of the coil. In other embodiments, it may be the case that the inductive impedance is specified to have an electrical length that is significantly less than or greater than ½ the wavelength at the operating frequency of the guided surface waveguide probe  200   f.    
     The excitation source  212  can be coupled to the feed network  209  by way of magnetic coupling. Specifically, the excitation source  212  is coupled to a coil L P  that is inductively coupled to the coil L 1a . This may be done by link coupling, a tapped coil, a variable reactance, or other coupling approach as can be appreciated. To this end, the coil L P  acts as a primary, and the coil L 1a  acts as a secondary as can be appreciated. 
     In order to adjust the guided surface waveguide probe  200   f  for the transmission of a desired guided surface wave, the heights of the respective charge terminals T 1  and T 2  may be altered with respect to the lossy conducting medium  203  and with respect to each other. Also, the sizes of the charge terminals T 1  and T 2  may be altered. In addition, the size of the coil L 1a  may be altered by adding or eliminating turns or by changing some other dimension of the coil L 1a . The coil L 1a  can also include one or more taps for adjusting the electrical length as shown in  FIG. 17 . The position of a tap connected to either charge terminal T 1  or T 2  can also be adjusted. 
     Referring next to  FIGS. 18A, 18B, 18C and 19 , shown are examples of generalized receive circuits for using the surface-guided waves in wireless power delivery systems.  FIGS. 18A and 18B-18C  include a linear probe  303  and a tuned resonator  306 , respectively.  FIG. 19  is a magnetic coil  309  according to various embodiments of the present disclosure. According to various embodiments, each one of the linear probe  303 , the tuned resonator  306 , and the magnetic coil  309  may be employed to receive power transmitted in the form of a guided surface wave on the surface of a lossy conducting medium  203  according to various embodiments. As mentioned above, in one embodiment the lossy conducting medium  203  comprises a terrestrial medium (or Earth). 
     With specific reference to  FIG. 18A , the open-circuit terminal voltage at the output terminals  312  of the linear probe  303  depends upon the effective height of the linear probe  303 . To this end, the terminal point voltage may be calculated as
 
 V   T =∫ 0   h     e     E   inc   ·dl,   (96)
 
where E inc  is the strength of the incident electric field induced on the linear probe  303  in Volts per meter, dl is an element of integration along the direction of the linear probe  303 , and h e  is the effective height of the linear probe  303 . An electrical load  315  is coupled to the output terminals  312  through an impedance matching network  318 .
 
     When the linear probe  303  is subjected to a guided surface wave as described above, a voltage is developed across the output terminals  312  that may be applied to the electrical load  315  through a conjugate impedance matching network  318  as the case may be. In order to facilitate the flow of power to the electrical load  315 , the electrical load  315  should be substantially impedance matched to the linear probe  303  as will be described below. 
     Referring to  FIG. 18B , a ground current excited coil  306   a  possessing a phase shift equal to the wave tilt of the guided surface wave includes a charge terminal T R  that is elevated (or suspended) above the lossy conducting medium  203 . The charge terminal T R  has a self-capacitance C R . In addition, there may also be a bound capacitance (not shown) between the charge terminal T R  and the lossy conducting medium  203  depending on the height of the charge terminal T R  above the lossy conducting medium  203 . The bound capacitance should preferably be minimized as much as is practicable, although this may not be entirely necessary in every instance. 
     The tuned resonator  306   a  also includes a receiver network comprising a coil L R  having a phase shift Φ. One end of the coil L R  is coupled to the charge terminal T R , and the other end of the coil L R  is coupled to the lossy conducting medium  203 . The receiver network can include a vertical supply line conductor that couples the coil L R  to the charge terminal T R . To this end, the coil L R  (which may also be referred to as tuned resonator L R −C R ) comprises a series-adjusted resonator as the charge terminal C R  and the coil L R  are situated in series. The phase delay of the coil L R  can be adjusted by changing the size and/or height of the charge terminal T R , and/or adjusting the size of the coil L R  so that the phase Φ of the structure is made substantially equal to the angle of the wave tilt Ψ. The phase delay of the vertical supply line can also be adjusted by, e.g., changing length of the conductor. 
     For example, the reactance presented by the self-capacitance C R  is calculated as 1/jωC R . Note that the total capacitance of the structure  306   a  may also include capacitance between the charge terminal T R  and the lossy conducting medium  203 , where the total capacitance of the structure  306   a  may be calculated from both the self-capacitance C R  and any bound capacitance as can be appreciated. According to one embodiment, the charge terminal T R  may be raised to a height so as to substantially reduce or eliminate any bound capacitance. The existence of a bound capacitance may be determined from capacitance measurements between the charge terminal T R  and the lossy conducting medium  203  as previously discussed. 
     The inductive reactance presented by a discrete-element coil L R  may be calculated as jωL, where L is the lumped-element inductance of the coil L R . If the coil L R  is a distributed element, its equivalent terminal-point inductive reactance may be determined by conventional approaches. To tune the structure  306   a , one would make adjustments so that the phase delay is equal to the wave tilt for the purpose of mode-matching to the surface waveguide at the frequency of operation. Under this condition, the receiving structure may be considered to be “mode-matched” with the surface waveguide. A transformer link around the structure and/or an impedance matching network  324  may be inserted between the probe and the electrical load  327  in order to couple power to the load. Inserting the impedance matching network  324  between the probe terminals  321  and the electrical load  327  can effect a conjugate-match condition for maximum power transfer to the electrical load  327 . 
     When placed in the presence of surface currents at the operating frequencies power will be delivered from the surface guided wave to the electrical load  327 . To this end, an electrical load  327  may be coupled to the structure  306   a  by way of magnetic coupling, capacitive coupling, or conductive (direct tap) coupling. The elements of the coupling network may be lumped components or distributed elements as can be appreciated. 
     In the embodiment shown in  FIG. 18B , magnetic coupling is employed where a coil Ls is positioned as a secondary relative to the coil L R  that acts as a transformer primary. The coil Ls may be link-coupled to the coil L R  by geometrically winding it around the same core structure and adjusting the coupled magnetic flux as can be appreciated. In addition, while the receiving structure  306   a  comprises a series-tuned resonator, a parallel-tuned resonator or even a distributed-element resonator of the appropriate phase delay may also be used. 
     While a receiving structure immersed in an electromagnetic field may couple energy from the field, it can be appreciated that polarization-matched structures work best by maximizing the coupling, and conventional rules for probe-coupling to waveguide modes should be observed. For example, a TE 20  (transverse electric mode) waveguide probe may be optimal for extracting energy from a conventional waveguide excited in the TE 20  mode. Similarly, in these cases, a mode-matched and phase-matched receiving structure can be optimized for coupling power from a surface-guided wave. The guided surface wave excited by a guided surface waveguide probe  200  on the surface of the lossy conducting medium  203  can be considered a waveguide mode of an open waveguide. Excluding waveguide losses, the source energy can be completely recovered. Useful receiving structures may be E-field coupled, H-field coupled, or surface-current excited. 
     The receiving structure can be adjusted to increase or maximize coupling with the guided surface wave based upon the local characteristics of the lossy conducting medium  203  in the vicinity of the receiving structure. To accomplish this, the phase delay (Φ) of the receiving structure can be adjusted to match the angle (Ψ) of the wave tilt of the surface traveling wave at the receiving structure. If configured appropriately, the receiving structure may then be tuned for resonance with respect to the perfectly conducting image ground plane at complex depth z=−d/2. 
     For example, consider a receiving structure comprising the tuned resonator  306   a  of  FIG. 18B , including a coil L R  and a vertical supply line connected between the coil L R  and a charge terminal T R . With the charge terminal T R  positioned at a defined height above the lossy conducting medium  203 , the total phase shift Φ of the coil L R  and vertical supply line can be matched with the angle (Ψ) of the wave tilt at the location of the tuned resonator  306   a . From Equation (22), it can be seen that the wave tilt asymptotically passes to 
                     W   =            W        ⁢     e       j   ⁢           ⁢   Ψ     ⁢                 =         E   ρ       E   z       ⁢     →     ρ   →   ∞       ⁢     1         ɛ   r     -     j   ⁢       σ   1       ωɛ   o                     ,           (   97   )               
where ε r  comprises the relative permittivity and σ 1  is the conductivity of the lossy conducting medium  203  at the location of the receiving structure, ε o  is the permittivity of free space, and ω=2πf, where f is the frequency of excitation. Thus, the wave tilt angle (Ψ) can be determined from Equation (97).
 
     The total phase shift (Φ=θ c +θ y ) of the tuned resonator  306   a  includes both the phase delay (θ c ) through the coil L R  and the phase delay of the vertical supply line (θ y ). The spatial phase delay along the conductor length l w  of the vertical supply line can be given by θ y =β w l w , where β w  is the propagation phase constant for the vertical supply line conductor. The phase delay due to the coil (or helical delay line) is θ c =β p l C , with a physical length of l C  and a propagation factor of 
                       β   p     =         2   ⁢   π       λ   p       =       2   ⁢   π         V   f     ⁢     λ   o             ,           (   98   )               
where V f  is the velocity factor on the structure, λ 0  is the wavelength at the supplied frequency, and λ p  is the propagation wavelength resulting from the velocity factor V f . One or both of the phase delays (θ c +θ y ) can be adjusted to match the phase shift Φ to the angle (Ψ) of the wave tilt. For example, a tap position may be adjusted on the coil L R  of  FIG. 18B  to adjust the coil phase delay (θ c ) to match the total phase shift to the wave tilt angle (Φ=Ψ). For example, a portion of the coil can be bypassed by the tap connection as illustrated in  FIG. 18B . The vertical supply line conductor can also be connected to the coil L R  via a tap, whose position on the coil may be adjusted to match the total phase shift to the angle of the wave tilt.
 
     Once the phase delay (Φ) of the tuned resonator  306   a  has been adjusted, the impedance of the charge terminal T R  can then be adjusted to tune to resonance with respect to the perfectly conducting image ground plane at complex depth z=−d/2. This can be accomplished by adjusting the capacitance of the charge terminal T 1  without changing the traveling wave phase delays of the coil L R  and vertical supply line. The adjustments are similar to those described with respect to  FIGS. 9A and 9B . 
     The impedance seen “looking down” into the lossy conducting medium  203  to the complex image plane is given by:
 
 Z   in   =R   in   +jX   in   =Z   o  tan  h ( jβ   o ( d/ 2)),  (99)
 
where β o =ω√{square root over (μ o ε o )}. For vertically polarized sources over the Earth, the depth of the complex image plane can be given by:
 
 d/ 2≈1/√{square root over ( jωμ   1 σ 1 −ω 2 μ 1 ε 1 )},  (100)
 
where μ 1  is the permeability of the lossy conducting medium  203  and ε 1 =ε r ε o .
 
     At the base of the tuned resonator  306   a , the impedance seen “looking up” into the receiving structure is Z ↑ =Z base  as illustrated in  FIG. 9A . With a terminal impedance of: 
                       Z   R     =     1     j   ⁢           ⁢   ω   ⁢           ⁢     C   R           ,           (   101   )               
where C R  is the self-capacitance of the charge terminal T R , the impedance seen “looking up” into the vertical supply line conductor of the tuned resonator  306   a  is given by:
 
                       Z   2     =         Z   W     ⁢         Z   R     +       Z   w     ⁢     tanh   ⁡     (     j   ⁢           ⁢     β   w     ⁢     h   w       )               Z   w     +       Z   R     ⁢     tanh   ⁡     (     j   ⁢           ⁢     β   w     ⁢     h   w       )               =       Z   W     ⁢         Z   R     +       Z   w     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   y       )               Z   w     +       Z   R     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   y       )                   ,           (   102   )               
and the impedance seen “looking up” into the coil L R  of the tuned resonator  306   a  is given by:
 
                     Z   base     =         R   base     +     jX   base       =         Z   R     ⁢         Z   2     +       Z   R     ⁢     tanh   ⁡     (     j   ⁢           ⁢     β   p     ⁢   H     )               Z   R     +       Z   2     ⁢     tanh   ⁡     (     j   ⁢           ⁢     β   p     ⁢   H     )               =       Z   c     ⁢           Z   2     +       Z   R     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   c       )               Z   R     +       Z   2     ⁢     tanh   ⁡     (     j   ⁢           ⁢     θ   c       )             .                   (   103   )               
By matching the reactive component (X in ) seen “looking down” into the lossy conducting medium  203  with the reactive component (X base ) seen “looking up” into the tuned resonator  306   a , the coupling into the guided surface waveguide mode may be maximized.
 
     Referring next to  FIG. 18C , shown is an example of a tuned resonator  306   b  that does not include a charge terminal T R  at the top of the receiving structure. In this embodiment, the tuned resonator  306   b  does not include a vertical supply line coupled between the coil L R  and the charge terminal T R . Thus, the total phase shift (Φ) of the tuned resonator  306   b  includes only the phase delay (θ c ) through the coil L R . As with the tuned resonator  306   a  of  FIG. 18B , the coil phase delay θ c  can be adjusted to match the angle (Ψ) of the wave tilt determined from Equation (97), which results in Φ=Ψ. While power extraction is possible with the receiving structure coupled into the surface waveguide mode, it is difficult to adjust the receiving structure to maximize coupling with the guided surface wave without the variable reactive load provided by the charge terminal T R . 
     Referring to  FIG. 18D , shown is a flow chart  180  illustrating an example of adjusting a receiving structure to substantially mode-match to a guided surface waveguide mode on the surface of the lossy conducting medium  203 . Beginning with  181 , if the receiving structure includes a charge terminal T R  (e.g., of the tuned resonator  306   a  of  FIG. 18B ), then the charge terminal T R  is positioned at a defined height above a lossy conducting medium  203  at  184 . As the surface guided wave has been established by a guided surface waveguide probe  200 , the physical height (h p ) of the charge terminal T R  may be below that of the effective height. The physical height may be selected to reduce or minimize the bound charge on the charge terminal T R  (e.g., four times the spherical diameter of the charge terminal). If the receiving structure does not include a charge terminal T R  (e.g., of the tuned resonator  306   b  of  FIG. 18C ), then the flow proceeds to  187 . 
     At  187 , the electrical phase delay Φ of the receiving structure is matched to the complex wave tilt angle Ψ defined by the local characteristics of the lossy conducting medium  203 . The phase delay (θ c ) of the helical coil and/or the phase delay (θ y ) of the vertical supply line can be adjusted to make Φ equal to the angle (Ψ) of the wave tilt (W). The angle (Ψ) of the wave tilt can be determined from Equation (86). The electrical phase Φ can then be matched to the angle of the wave tilt. For example, the electrical phase delay Φ=θ c +θ y  can be adjusted by varying the geometrical parameters of the coil L R  and/or the length (or height) of the vertical supply line conductor. 
     Next at  190 , the load impedance of the charge terminal T R  can be tuned to resonate the equivalent image plane model of the tuned resonator  306   a . The depth (d/2) of the conducting image ground plane  139  ( FIG. 9A ) below the receiving structure can be determined using Equation (100) and the values of the lossy conducting medium  203  (e.g., the Earth) at the receiving structure, which can be locally measured. Using that complex depth, the phase shift (θ d ) between the image ground plane  139  and the physical boundary  136  ( FIG. 9A ) of the lossy conducting medium  203  can be determined using θ d =β o d/2. The impedance (Z in ) as seen “looking down” into the lossy conducting medium  203  can then be determined using Equation (99). This resonance relationship can be considered to maximize coupling with the guided surface waves. 
     Based upon the adjusted parameters of the coil L R  and the length of the vertical supply line conductor, the velocity factor, phase delay, and impedance of the coil L R  and vertical supply line can be determined. In addition, the self-capacitance (C R ) of the charge terminal T R  can be determined using, e.g., Equation (24). The propagation factor (β p ) of the coil L R  can be determined using Equation (98), and the propagation phase constant (β w ) for the vertical supply line can be determined using Equation (49). Using the self-capacitance and the determined values of the coil L R  and vertical supply line, the impedance (Z base ) of the tuned resonator  306   a  as seen “looking up” into the coil L R  can be determined using Equations (101), (102), and (103). 
     The equivalent image plane model of  FIG. 9A  also applies to the tuned resonator  306   a  of  FIG. 18B . The tuned resonator  306   a  can be tuned to resonance with respect to the complex image plane by adjusting the load impedance Z R  of the charge terminal T R  such that the reactance component X base  of Z base  cancels out the reactance component of X in  of Z in , or X base +X in =0. Thus, the impedance at the physical boundary  136  ( FIG. 9A ) “looking up” into the coil of the tuned resonator  306   a  is the conjugate of the impedance at the physical boundary  136  “looking down” into the lossy conducting medium  203 . The load impedance Z R  can be adjusted by varying the capacitance (C R ) of the charge terminal T R  without changing the electrical phase delay Φ=θ c +θ y  seen by the charge terminal T R . An iterative approach may be taken to tune the load impedance Z R  for resonance of the equivalent image plane model with respect to the conducting image ground plane  139 . In this way, the coupling of the electric field to a guided surface waveguide mode along the surface of the lossy conducting medium  203  (e.g., Earth) can be improved and/or maximized. 
     Referring to  FIG. 19 , the magnetic coil  309  comprises a receive circuit that is coupled through an impedance matching network  333  to an electrical load  336 . In order to facilitate reception and/or extraction of electrical power from a guided surface wave, the magnetic coil  309  may be positioned so that the magnetic flux of the guided surface wave, H φ , passes through the magnetic coil  309 , thereby inducing a current in the magnetic coil  309  and producing a terminal point voltage at its output terminals  330 . The magnetic flux of the guided surface wave coupled to a single turn coil is expressed by
 
 =∫∫ A     cs   μ r μ o   {right arrow over (H)}·{circumflex over (n)}dA   (104)
 
where   is the coupled magnetic flux, μ r  is the effective relative permeability of the core of the magnetic coil  309 , μ o  is the permeability of free space, {right arrow over (H)} is the incident magnetic field strength vector, {circumflex over (n)} is a unit vector normal to the cross-sectional area of the turns, and A cs  is the area enclosed by each loop. For an N-turn magnetic coil  309  oriented for maximum coupling to an incident magnetic field that is uniform over the cross-sectional area of the magnetic coil  309 , the open-circuit induced voltage appearing at the output terminals  330  of the magnetic coil  309  is
 
                     V   =         -   N     ⁢       d   ⁢           ⁢   ℱ     dt       ≈       -   j     ⁢           ⁢     ωμ   r     ⁢     μ   0     ⁢     NHA   CS           ,           (   105   )               
where the variables are defined above. The magnetic coil  309  may be tuned to the guided surface wave frequency either as a distributed resonator or with an external capacitor across its output terminals  330 , as the case may be, and then impedance-matched to an external electrical load  336  through a conjugate impedance matching network  333 .
 
     Assuming that the resulting circuit presented by the magnetic coil  309  and the electrical load  336  are properly adjusted and conjugate impedance matched, via impedance matching network  333 , then the current induced in the magnetic coil  309  may be employed to optimally power the electrical load  336 . The receive circuit presented by the magnetic coil  309  provides an advantage in that it does not have to be physically connected to the ground. 
     With reference to  FIGS. 18A, 18B, 18C and 19 , the receive circuits presented by the linear probe  303 , the mode-matched structure  306 , and the magnetic coil  309  each facilitate receiving electrical power transmitted from any one of the embodiments of guided surface waveguide probes  200  described above. To this end, the energy received may be used to supply power to an electrical load  315 / 327 / 336  via a conjugate matching network as can be appreciated. This contrasts with the signals that may be received in a receiver that were transmitted in the form of a radiated electromagnetic field. Such signals have very low available power, and receivers of such signals do not load the transmitters. 
     It is also characteristic of the present guided surface waves generated using the guided surface waveguide probes  200  described above that the receive circuits presented by the linear probe  303 , the mode-matched structure  306 , and the magnetic coil  309  will load the excitation source  212  (e.g.,  FIGS. 3, 12 and 16 ) that is applied to the guided surface waveguide probe  200 , thereby generating the guided surface wave to which such receive circuits are subjected. This reflects the fact that the guided surface wave generated by a given guided surface waveguide probe  200  described above comprises a transmission line mode. By way of contrast, a power source that drives a radiating antenna that generates a radiated electromagnetic wave is not loaded by the receivers, regardless of the number of receivers employed. 
     Thus, together one or more guided surface waveguide probes  200  and one or more receive circuits in the form of the linear probe  303 , the tuned mode-matched structure  306 , and/or the magnetic coil  309  can make up a wireless distribution system. Given that the distance of transmission of a guided surface wave using a guided surface waveguide probe  200  as set forth above depends upon the frequency, it is possible that wireless power distribution can be achieved across wide areas and even globally. 
     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 waveguide or a load directly wired to the distant power generator. Not counting the power required to maintain transmission field strength plus that dissipated in the surface waveguide, which at extremely low frequencies is insignificant relative to the transmission losses in conventional high-tension power lines at 60 Hz, all of the generator power goes only to the desired electrical load. When the electrical load demand is terminated, the source power generation is relatively idle. 
     Referring next to  FIGS. 20A-E , shown are examples of various schematic symbols that are used with reference to the discussion that follows. With specific reference to  FIG. 20A , shown is a symbol that represents any one of the guided surface waveguide probes  200   a ,  200   b ,  200   c ,  200   e ,  200   d , or  200   f ; or any variations thereof. In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface waveguide probe P. For the sake of simplicity in the following discussion, any reference to the guided surface waveguide probe P is a reference to any one of the guided surface waveguide probes  200   a ,  200   b ,  200   c ,  200   e ,  200   d , or  200   f ; or variations thereof. 
     Similarly, with reference to  FIG. 20B , shown is a symbol that represents a guided surface wave receive structure that may comprise any one of the linear probe  303  ( FIG. 18A ), the tuned resonator  306  ( FIGS. 18B-18C ), or the magnetic coil  309  ( FIG. 19 ). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R. For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R is a reference to any one of the linear probe  303 , the tuned resonator  306 , or the magnetic coil  309 ; or variations thereof. 
     Further, with reference to  FIG. 20C , shown is a symbol that specifically represents the linear probe  303  ( FIG. 18A ). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R P . For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R P  is a reference to the linear probe  303  or variations thereof. 
     Further, with reference to  FIG. 20D , shown is a symbol that specifically represents the tuned resonator  306  ( FIGS. 18B-18C ). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R R . For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R R  is a reference to the tuned resonator  306  or variations thereof. 
     Further, with reference to  FIG. 20E , shown is a symbol that specifically represents the magnetic coil  309  ( FIG. 19 ). In the following drawings and discussion, a depiction of this symbol will be referred to as a guided surface wave receive structure R M . For the sake of simplicity in the following discussion, any reference to the guided surface wave receive structure R M  is a reference to the magnetic coil  309  or variations thereof. 
     Various embodiments of a guided surface waveguide probe have been discussed, where such embodiments of the guided surface waveguide probe include one or more elevated charge terminals T 1 , T 2  (i.e., T N ). As discussed, the charge terminals T 1  and/or T 2  may comprise any conductive mass that can hold an electrical charge. Further, the charge terminal T 1 , T 2  has a self-capacitance and may comprise any shape such as a sphere, a disk, a cylinder, a cone, a torus, a randomized shape, or any shape. Ultimately, the field strength of a guided surface wave launched by a guided surface waveguide probe is proportional to the quantity of charge on the terminal T 1 . In accordance with an embodiment of the present disclosure, compensation can be provided to elevate the charge at terminal T 1  or terminal T 2 . 
     Referring now to  FIG. 21 , the free charge associated with a charge terminal T N  may be elevated by adding or mounting charge devices to the outside of the charge terminal T N . In one embodiment, charge devices  2110  are mounted across the surface of the terminal T N  so as to create a uniform field along the surface. Further, the charge devices  2110   a  may each be a container that is filled with an insulator substance  2120  that can be ionized to produce free charges that enhance the free charges attributed to the charge terminal T N  to which the charge devices are attached or integrated. The insulator substance  2120  may be, but is not limited to, one or more noble gases, such as neon, xenon, argon, krypton, helium, hydrogen, etc. Accordingly, there may be other fluidic materials that can also be used besides gas mixtures or substances. In one embodiment, the portion of the container of the charge device  2110   a  that holds the insulator substance  2120  is a solid insulation material  2130 , such as, but not limited to, glass. Therefore, a container of the charge device  2110   a  can be a glass tube. 
     In one embodiment, the charge device  2120  also contains an electrode  2140   a . In particular, one end of the electrode  2140   a  is in contact with a surface  2150  of the charge terminal T N  and an opposite end is enclosed in the container with the insulator substance  2120  at a desired pressure that is conducive for ionizing the insulator substance  2120  and low enough to not allow for the insulator substance  2120  to spark or become conductive. Further, the charge device  2110   a  may be joined to the charge terminal T N  by a metallic socket  2160  or other fastening means, such as a nut, weld, screw, etc., in some embodiments. The charge device  2110  may take different forms in various embodiments. As an example,  FIG. 22  shows a charge device  2110   b  having a convex or curved shape. In one embodiment, a bottom surface of the charge device  2110   b  contains an electrode  2140   b  that is mountable with a surface of the charge terminal T N . 
     Accordingly, after affixing (or integrating) the charge device  2110  to the charge terminal T N , the voltage inside the charge device  2110  is raised as the voltage applied to the charge terminal T N  is raised. Therefore, as the voltage applied to the charge terminal T N  increases towards a breakdown voltage for the insulator substance  2120 , the insulator substance  2120  will ionize thereby generating free charges of electrons. The generated free charge of the charge device  2110  may then enhance the free charges generated by the charge terminal T N . Although in some embodiments, a charge device  2110  may be small in size, its isolated capacitance can be high on a comparative scale, due in part to a volume of charge being created as opposed to a surface of charge. In particular, the ionized insulator substance  2120 , such as gas, of the charge devices  2110  cause the charge device  2110  to have a high isolated capacitance. 
     To demonstrate its effective, reference is now made back to the discussion for  FIG. 7  and examples of the guided surface waveguide probe P being discussed. It is understood, however, that the guided surface waveguide probe  2300  is shown as an example to illustrate the various embodiments of the present disclosure and that other embodiments of guided surface waveguide probes described herein may be employed. 
     In the present non-limiting example, instead of elevating the physical height h p  to reduce the amount of bounded charge on the terminal T 1 , free charge of the terminal T 1  may be chemically enhanced via charge device(s)  2110  in one embodiment of a guided surface waveguide probe  2300 , as shown in  FIG. 23 . For example, when the charge terminal T 1  is near the image ground plane  130  ( FIG. 3 ), most of the charge Q 1  on the charge terminal T 1  is “bound” to its image charge. However, by adding individual charge devices with the charge terminal T 1 , an increased amount of free charge q free  from each charge device is made available to couple energy into the guided surface waveguide mode. Accordingly, capacitance of the charge terminal T 1  is increased without increasing the size of the charge terminal T 1 . Therefore, such compensation may be used to enhance the capacitance of charge terminal T N  in other embodiments previously discussed and is contemplated by the present disclosure. 
     Referring next to  FIG. 24 , shown is a flowchart that provides one example of the operation of a portion of a guided surface waveguide probe P according to various embodiments. Beginning with box  2403 , a plurality of charge devices are positioned (e.g., mounted) on a surface of a charge terminal, where a respective one of the plurality of charge devices is configured to generate a free charge that is made available to couple energy into a guided surface waveguide mode. Next, in box  2405 , a plurality of resultant fields is generated by a guided surface waveguide probe that is substantially mode-matched to the guided surface waveguide mode on a surface of a terrestrial medium, where the guided surface waveguide probe comprises the charge terminal. Correspondingly, in box  2407 , electrical energy is transmitted in a form of a guided surface wave along a surface of the terrestrial medium using the guided surface waveguide probe. 
     Referring now to  FIG. 25 , in various embodiments, the free charge associated with a charge terminal T N  may be elevated by substituting or replacing the charge terminal T N  with a chemically enhanced charge terminal CET N . 
     In one embodiment, the charge terminal CET N  may be a dielectric container, such as glass, that is filled with an insulator substance  2120  that can be ionized to produce free charges. The insulator substance  2120  may be, but is not limited to, one or more noble gases, such as neon, xenon, argon, krypton, helium, hydrogen, etc. Accordingly, there may be other fluidic materials that can also be used besides gas mixtures or substances. In one embodiment, the portion of the charge terminal CET N  that holds the insulator substance  2120  is a solid insulation material  2130 , such as, but not limited to, glass. Therefore, the charge terminal CET N  can be a glass tube or sphere. 
     In  FIG. 25 , an AC source  212  acts as the excitation source for the charge terminal CET 1 , which is coupled to the guided surface waveguide probe  200   b  through a feed network  209  ( FIG. 3 ) comprising a coil  215  such as, e.g., a helical coil. In other implementations, the AC source  212  can be inductively coupled to the coil  215  through a primary coil. In some embodiments, an impedance matching network may be included to improve and/or maximize coupling of the AC source  212  to the coil  215 . 
     As shown in  FIG. 25 , the guided surface waveguide probe P,  2500  can include the upper charge terminal CET 1  (e.g., a sphere at height h p ) that is positioned along a vertical axis z that is substantially normal to the plane presented by the lossy conducting medium  203 . A second medium  206  is located above the lossy conducting medium  203 . During operation, charge Q 1  is imposed on the terminal CET 1  depending on the voltage applied to the terminal T 1  at any given instant. 
     In this non-limiting example, the coil  215  is coupled to a ground stake  218  at a first end and to the charge terminal CET 1  via a vertical feed line conductor  221 . In some implementations, the coil connection to the charge terminal T 1  can be adjusted using a tap  224  of the coil  215  as shown in  FIG. 25 . The coil  215  can be energized at an operating frequency by the AC source  212  through a tap  227  at a lower portion of the coil  215 . In other implementations, the AC source  212  can be inductively coupled to the coil  215  through a primary coil. 
     In one embodiment, the charge terminal CET 1  contains an electrode  2140 . In particular, one end of the electrode  2140  is in contact with the vertical feed line conductor  221  and an opposite end is enclosed in the container with the insulator substance  2120  at a desired pressure that is conducive for ionizing the insulator substance  2120  and low enough to not allow for the insulator substance  2120  to spark or become conductive. Due to the charge terminal CET 1  comprising a dielectric structure, the charge terminal CET 1  does not contribute a self-capacitance and there is not a bound capacitance between the charge terminal and the lossy conducting medium  203 . Therefore, in various embodiments, the height h p  of the charge terminal CET 1  is not constrained by bound capacitance. 
     It is understood that the guided surface waveguide probe of  FIG. 25  is shown as an example to illustrate the various embodiments of the present disclosure and that other types and embodiments of guided surface waveguide probes described herein may be employed. As an illustrative and non-limiting example, for the guided surface waveguide probe of  FIG. 16 , a top charge terminal T 1  can be replaced with a chemically enhanced charge terminal CET 1 . 
     It should be emphasized that the above-described embodiments of the present disclosure are merely possible examples of implementations set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims. In addition, all optional and preferred features and modifications of the described embodiments and dependent claims are usable in all aspects of the disclosure taught herein. Furthermore, the individual features of the dependent claims, as well as all optional and preferred features and modifications of the described embodiments are combinable and interchangeable with one another.