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4B. Input ImpedancesThe antenna is excited by means of two generators, for thisreason, monopole 1 input voltage V 1 will be taken as the phasereference and monopole 2 input voltage V 2 will be given byV2 K V1 ej φ2(15)Thus, the parameter K is the amplitude ratio between V 1 andV2 , while φ2 is the phase difference between them. Voltagesand currents are taken as effective values.The port 1 input impedance isV11 I1Y11 Y12 K ej φ2The port 2 input impedance isZ1 R1 j X1 (16)V2K (17) jφI2Y12 e 2 K Y22It can be seen that the input impedances depend on a stronginteraction between both generators.Z2 R2 j X2 Fig. 6. a) CFA operating in the first regime (I) G12 0. b) CFA equivalentcircuit.C. Input PowerThe active or real power W 1 I1 2 R1 , produced bygenerator 1, is given byW1 V1 2 (G11 K G12 cos φ2 K B12 sin φ2 ) (18)The active or real power W 2 I2 2 R2 , produced bygenerator 2, is given byW2 V1 2 (K2 G22 K G12 cos φ2 K B12 sin φ2 ) (19)Then, the total input power will be the sum of the inputpowers of both generators, W in W1 W2 , that isWin V1 2 (G11 2 K G12 cos φ2 K2 G22 )(20)Equation (20) shows that the input power depends on thecosine of the voltage phase difference φ 2 . For this reason,there are three cases:Fig. 7. a) CFA operating in the second regime (II) G12 0. b) CFAequivalent circuit.(I) When G12 0, the input power is maximum forφ2 0 (360 ) and minimum for φ 2 180 .(II) When G12 0, the input power is maximum forφ2 180 and minimum for φ 2 0 (360 ).(III) When G12 0, the input power is constant for anyvalue of φ2 .III. E LECTROMAGNETIC F IELDThese three cases or regimes depend on the antenna physicaldimensions and frequency, and once these are given, theantenna will operate in only one of those regimes.In Fig. 6, a CFA operating in the first regime is shown(G12 0, φ2 0) as well as the antenna equivalent circuit,where the currents of both generators are in phase. In Fig. 7a CFA operating in the second regime is shown (G 12 0,φ2 180 ), where the generator currents are out of phase.When G12 0, the CFA will operate either as in Figs. 6 or 7according to the highest voltage generator, however, the totalinput power will be independent of φ 2 .The electromagnetic field radiated from the antenna is thesum of the radiation produced by the array of the two CFAmonopoles. Monopole 1 is a top-loaded short vertical antennaand monopole 2 is practically a Hertz monopole makes up bythe disk and its feeding vertical lead.Because of the very short distance between both radiators,the radiation center is located on the CFA vertical geometricaxis at a zero height. The feeding currents of each radiatingsource have different amplitudes and phases depending on thevoltages of both generators.A. Near FieldThe near field of a top-loaded monopole has been determined in [1] using Image Theory, according to Fig. 8geometry.Then, the magnetic and electric near fields of bothmonopoles on the ground plane, at z 0, are given by [1]

5Hφ Hφ1 Hφ2(23)Ez Ez1 Ez2(24)On the surface of the earth, a thin metallic layer of radius R 0is laid down, which is called the artificial ground plane. Fromthis metallic layer radius R 0 , the bare soil is considered up toa distance of half-wavelength. This is the surface of the earthto be taken into account in the soil power loss calculations.Over the metallic layer or artificial ground plane the electric field is practically perpendicular, having only the E zcomponent, due to the boundary conditions of a very highconductivity medium, where the surface impedance Z g is givenby [26], [27]Fig. 8. Monopole geometry used to calculate the near field in cylindricalcoordinates.Hφi ImiHijψi jβ(ri Hi ) jβρe e1 e(21)4πρri Hi e jψi 1 e jβ(ri Hi ) e jβρriEzi j Imi4 π 0 ω ρ ejψi(22)ρ e jβ(ri Hi )ri HiHi jβ 1 r2iri e jψiρ e jβ(ri Hi )ri HiHi jβ 1 r2iri jβ e jβρri ρ2 H2i 2XiiImi Ii 1 Z0mi Xiiψi arctanZ0miIt is assumed, as in [1], that only the vertical part of a toploaded monopole produces a net electromagnetic field. Thedistance ρ along the surface of the earth is measured from thecenter of the cylindrical coordinates or antenna vertical axis.The CFA total near electromagnetic fields on the groundplane (z 0) will beρ R0(25)(26)Where Zs is the bare soil impedance, which can be calculated from the soil conductivity, σ, and permittivity, , asfollows [26], [27]j ω µ0σ jω forρ R0(27)B. Wave ImpedanceThe wave impedance in space, just above the earth surfacein the air, is the ratio between the near electric and magneticfields Ez and Hφ on the ground plane, and is a function ofdistance ρ/λ, thereforeZ0 Where i 1, 2 andforEρ Zs HφZs Rs j Xs ω µ0(1 j)2 σmWhere σm is the metallic layer conductivity.On the bare soil, the electric field develops a radial component Eρ due to the low conductivity compared to that of themetallic layer, and is given by j β e jβρ Zg Rg j Xg EzHφ(28)In the far field zone, ρ λ, the wave impedance ispractically equal to the free space intrinsic impedanceZ00 120π Ω. The wave impedance Z 0 is almost pureimaginary or reactive very close to the antenna and almost purereal or resistive at a distance greater than half-wavelength.C. Far FieldIn the far field zone, when spherical coordinates are used,the CFA total far electric field E θ will be the sum of themonopoles 1 and 2 far electric fields, E θ1 and Eθ2 , which aregiven by [22], [23]Eθi j 60 Ii βHeiWheree j β rsin θri 1, 2(29)

6r is the distance from the antenna to any point in space [m].θ is the zenith angle.The total far electric field, E θ Eθ1 Eθ2 , will beEθ j 60 ( I1 βHe1 I2 βHe2 )e j β rsin θr(30)This equation is exactly the same as for any standard shortmonopole. This field is the non attenuated radiated electricfield, because it depends only on the inverse distance law.The actual field intensity along the earth is affected by thesoil physical constants and the diffraction due to the sphericalearth [28], [29].Thensin θrWhere the CFA effective height H e is defined as I2 He2 He He1 1 I1 He1 Eθ 60 I1 βHeIV. A NTENNA T UNING AND L OSSES(31)(32)Therefore, the CFA radiation resistance referred to monopole 1 can be defined as2Rrad 40 (βHe )(33)orRrad 2 I2 He2 40 (βHe1 ) 1 I1 He1 2(34)It is very important to understand that this radiation resistance is not the input resistance in any of the antenna portsand it depends on the current and effective height relationshipsof both monopoles.The antenna radiated power can be calculated asWrad I1 2 RradQLi XLiRLii 1, 2(41)Where RLi and XLi are the i-th monopole tuning coilresistance and reactance, respectively.Network parameters can be written in the following waywhen losses and tuning coils are present:Zii Z ii Rci Rgpi RLi j XLii 1, 2(42)(35)Then, monopole 1 effective input current I 1 , for a givenradiated power W rad , becomes Wrad I1 (36)RradEquations (31), (33) and (36) give the following far effectiveelectric field: 30 Wrad D Eθ sin θ(37)rThus, the CFA antenna directivity is D 3 due to the sin θfar field radiation pattern, as given by any short monopole ofheight less than 0.1λ, as was confirmed by antenna patternmeasurements using a model [6].Taking into account the antenna efficiency η, the radiatedpower Wrad and gain G are given by [26]Wrad η Win(38)G ηD(39)Then, the CFA far effective electric field becomes 30 Win G Eθ sin θrIn the antenna circuit there are losses in conductors, insulators, tuning system and in the earth surface within a circlehalf-wavelength in radius. In general, insulator losses are verylow compared to the other and can be neglected.In order to tune the antenna, a coil is connected in series toeach port, obtaining real input impedances. These coils havemerit factors given by(40)Z12 Z21 R12 j X12(43)WhereZ ii Rradi j Xii is the i-th monopole self-impedancewith no losses and no tuning coils [Ω].Rradi is the i-th monopole radiation resistance [Ω].Rci is the i-th monopole conductor resistance [Ω].Rgpi is the ground plane loss resistance due to the i-thmonopole current distribution and soil conditions [Ω].The mutual resistance R 12 and all the reactances are practically not affected by the antenna losses, because they dependon the very near field distribution, which is not appreciablyaffected by the finite soil conductivity [30], [31].The CFA equivalent network with losses and tuning coilscan be seen in Fig. 9.A. Conductor Loss ResistanceConductor loss resistance R c1 , due to monopole 1 conductors, can be calculated using the following expression,determined from the current distribution on the monopolestructure and its top-load [1]:

7X11 is the monopole 1 self-reactance [Ω].Z0m1 is the monopole 1 average characteristic impedance[Ω].H1 is the monopole 1 height [m].L1 is the monopole 1 top-load length [m].n is the number of the monopole 1 top-load branches.Fig. 9.The current distribution of the top-loaded monopole hasbeen determined in [1] and can be seen in Fig. 4.Conductor resistance per unit length, taking into account theskin effect, is given by [26] f µ01RcL1 (45)a1 4 π σcCFA equivalent network with losses and tuning coils.Whereσc is the conductor conductivity [S/m].a1 is the wire radius [m].f is the operation frequency [Hz].If the disk radius L 2 is sufficiently smaller than the wavelength, a linear current distribution over the disk can beassumed, as shown in Fig. 5, thereforeI(ρ) I2L2 ρL2 r hr h ρ L2(46)WhereI(ρ) is the current distribution over the disk [A].I2 is the effective current at the disk center [A].L2 is the disk radius [m].rh is the disk hole radius [m].The disk feeding lead has a constant current distribution likea Hertz monopole (I(z) I 2 in Fig. 5). From the disk andfeeding lead current distributions, the monopole 2 conductorresistance, Rc2 , can be calculated (see Appendix A), thereforeFig. 10. CFA near electric field lines (displacement current) and groundplane surface conductive currents.Rc1 RcL12 1 X211Z20m1 (47)The feeding lead conductor resistance per unit length, R cL2 ,is given by (45) with subscript 2 instead of 1.H1 X11 1 cos 2βH1X211 sin 2βH1 1 2Z0m12βZ0m1β 2 1X11sin βH1cos βH1 nZ 0m1 1 L1 1 tan2 βL1 cos 2βL1 11sin 2βL1 1 2ββ tan βL1tan2 βL1 ω µ01Rc2 RcL2 H2 24π (L2 rh )2 σc 2LL r2h222L2 ln 2 L2 (L2 rh ) rh2B. Ground Plane Loss ResistanceThe artificial ground plane [32] is made up by means of acircular metallic layer R 0 in radius with a small surface resistance Rg . Then, the ground plane equivalent loss resistancefor each monopole is given by [1](44)WhereRcL1 is the monopole 1 wire resistance per unit length[Ω/m].Rgpi2π Ii 2 R00 Hφi Rg ρ dρ 2λ/2 Hφi Rs ρ dρ2R0(48)Where i 1, 2 andRgpi is the i-th monopole ground plane loss resistance [Ω].

8Hφi is the i-th monopole near magnetic field on the groundsurface, given by (21) [A/m].Rg is the surface resistance of the artificial ground plane ormetallic layer, given by (25) [Ω].Rs is the surface resistance of the natural ground plane orsoil, given by (27) [Ω].Fig. 10 shows a sketch of the displacement currents and theconductive currents flowing on the ground plane at a specifictime.The ground plane is considered up to a distance ρ λ/2because this is the maximum distance covered by the groundsurface currents under the antenna, closing the antenna electriccircuit. Beyond this distance, the ground currents do not returnto the antenna generators and are taken into account in thesurface wave propagation calculations [1].V. E FFICIENCY AND G AINAntenna efficiency η is defined as the ratio between theradiated power W rad and the input power W in . The powerradiated by the CFA antenna is given by (35), while theantenna input power can be written asWin I1 2 R1 I2 2 R2(49)Where Ri is the i-th port input resistance, i 1, 2.Then, the CFA efficiency will beη RradR1 I2 /I1 2 R2(50)Where 2 2 I2 K 2 Z1 (51) I1 Z2 and the input impedances Z 1 and Z2 are given by (16) and(17), respectively.It can be seen that the CFA efficiency η depends on theantenna geometry, operation frequency, the voltage amplituderatio K and the phase difference φ 2 .Antenna gain, taking into account the directivity D 3, isG η D 3 η.VI. BANDWIDTHThe CFA antenna has two ports, for this reason, two bandwidths can be defined, according to the reflection coefficientsof both ports. In this case, taking into account that both inputpowers, W1 and W2 , must be positive, it is more convenientto define a CFA average reflection coefficient as a weightedgeometric mean of the reflection coefficients, Γ 1 and Γ2 , ofboth ports. Thus, the CFA average reflection coefficient isgiven by Γ W1 Γ1 2 W2 Γ2 2Win(52)Then, the CFA VSWR will beVSWR 1 Γ 1 Γ (53)TABLE ICFA DIMENSIONS IN METERS .Monopole 1 heightH110Barrel diameterD13.0Monopole 1 top-load lengthL10.0Barrel base heightδ11.5Monopole 1 wire radiusa16 · 10 31.0Monopole 2 (disk) heightH2Disk radius (monopole 2 top-load length)L22.5Disk hole radiusrh0.05Monopole 2 wire radiusa26 · 10 3Artificial ground plane radiusR05.0TABLE IICFA EQUIVALENT NETWORK MATRIX .METHODZ11Z12Z22TLM2.18 j 4100.11 j 1240.09 j 856MoM2.13 j 4300.20 j 1170.13 j 844VII. E XAMPLESHaving the available theory expressed previously, theCrossed Field Antenna analysis can be made by means ofnumerical examples.A. CFA Operating WindowsAs a first example, a CFA model is analyzed at the frequency of 1 MHz, with the antenna dimensions indicated inTable I and over average soil (σ 10 2 S/m, r 10).The CFA equivalent network impedance matrix can becalculated using the Transmission Line Method (TLM) andby means of the Method of Moments (MoM) [33] (wire gridmodel [25]). Both methods produce practically the same result,as can be seen in Table II for 1 MHz and average ground.In this example, monopole 1 is a barrel build up by twentyfour copper wires (σ c 5.8 · 107 S/m) of radius a 1 , as shownin Fig. 4. Barrel diameter, D 1 , wire radius, a 1 , and barrel baseheight, δ1 , are taken into account in the calculation of themonopole 1 equivalent transmission line average characteristicimpedance Z 0m1 [1].As a first analysis, input impedances of both antenna portsare calculated as functions of voltage phase difference φ 2 forK 1 and over average ground, and can be seen in Figs. 11and 12.In Fig. 11, negative input resistances R 1 and R2 can be seenfor some ranges of φ 2 , while in Fig. 12 the input reactances X 1and X2 are always negative. There exist two small windows,difficult to appreciate in Fig. 11, where both input resistancesare positive, permitting the antenna operation. One window islocated in the neighborhood of φ 2 180 and the other nearφ2 360 .Input impedances are shown in both windows in Tables IIIand IV. It is interesting to see that the input reactances arepractically constant in both windows, where the antenna canoperate.

9R1RR1R300R1R22[Ω] 20052[Ω]01000 5 100R1R2 200φ2 [ ] 30003060φ [ ]2 100306090 120 150 180 210 240 270 300 330 36090 120 150 180 210 240 270 300 330 360Fig. 11. CFA input resistances R1 and R2 as functions of φ2 for K 1and over average ground.Fig. 13. Tuned CFA input resistances R1 and R2 as functions of φ2 forK 1. The 180 -window can clearly be seen where R1 and R2 are positive.X1 300X2 400X1X2[Ω] 500[Ω]6X1X242 600 7000 800 2 900X1X 1000 1100 12000 4φ [ ]23060290 120 150 180 210 240 270 300 330 360Fig. 12. CFA input reactances X1 and X2 as functions of φ2 for K 1and over average ground.φ2 [ ] 60306090 120 150 180 210 240 270 300 330 360Fig. 14. Tuned CFA input reactances X1 and X2 as functions of φ2 forK 1 and in the 180 -window.TABLE IIICFA INPUT IMPEDANCES IN 180 - WINDOW.φ2R1X1R2degΩΩΩΩ179.42.30 343 0.71 629179.82.00 3430.31 629180.21.70 3431.32 629180.61.39 3432.34 629181.01.09 3433.36 629181.40.79 3434.37 629181.80.49 3435.39 629182.20.19 3436.41 629182.6 0.12 3437.42 629X2TABLE IVCFA INPUT IMPEDANCES IN 360 - WINDOW.φ2R1X1R2degΩΩΩΩ358.0 0.12 45915.7 1171358.20.15 45914.0 1171358.40.42 45912.2 1171358.60.69 45910.4 1172358.80.96 4598.67 1172359.01.23 4596.90 1172359.21.50 4595.14 1172359.41.77 4593.37 1172359.62.04 4591.61 1172359.82.31 459 0.16 1172X2In order to tune the antenna, opposite reactances are connected in series to both ports. If the 180 -window is chosen,the reactance to be connected in series with port 1 isXL1 343 Ω and with port 2 is X L2 629 Ω. Under thiscondition, the input resistances and reactances of both portsare shown in Figs. 13 and 14, where they have a very smallvariation as functions of φ 2 , also the input impedances ofboth ports are practically real close to φ 2 180 . In thisexample, only the antenna losses are taken into account andthe tuning coils are supposed to be of infinite merit factorQL QL1 QL2 , permitting to know the antenna operationwithout any other effect.In the 180 -window, when the generators have the samevoltage amplitude (K 1), the antenna is operational becausethe input powers on both ports are positive for φ 2 between 50 and 290 , where both input resistances are positive. In thiscase, the equivalent network mutual conductance is negative,G12 0.253 S, and the antenna operates as in Fig. 7.If the 360 -window is chosen, the reactance to be connectedin series with port 1 is XL1 459 Ω and with port 2 isXL2 1172 Ω. Under this condition, the input resistancesand reactances of both ports are shown in Figs. 15 and 16,where they also have a very small variation as functions ofφ2 . The input impedances of both ports are practically realfor φ2 between 0 and 100 and between 240 and 360 .In this case, the antenna is operational too, because the inputpowers in both ports are positive. Also, because the equivalentnetwork mutual conductance is positive, G 12 0.165 S, the

10R1RTABLE V5CFA PERFORMANCE IN 180 - WINDOW.2[Ω]Kφ2R1X1R2X2 degΩΩΩΩΩ1.6130.61.040.683.47 1.080.591.7155.31.040.423.50 073.540.950.602.0209.61.05 0.103.551.530.602.1230.31.05 0.323.582.280.60KV1 V2 I1 I2 GE VVAAdBimV/m1.627.343.621.912.0 0.691601.724.741.921.912.0 0.691601.823.442.221.911.9 0.691601.922.943.521.911.9 0.6816042.023.045.921.911.9 0.6816022.123.950.221.911.8 0.681600 5R1R2 1003060φ2 [ ]90 120 150 180 210 240 270 300 330 360Fig. 15. Tuned CFA input resistances R1 and R2 as functions of φ2 forK 1. The 360 -window can clearly be seen where R1 and R2 are positive.X1X2[Ω]8X1X26Rrad0TABLE VICFA PERFORMANCE IN 360 - WINDOW. 2 4 6 80φ2 [ ]306090 120 150 180 210 240 270 300 330 360Fig. 16. Tuned CFA input reactances X1 and X2 as functions of φ2 forK 1 and in the 360 -window.antenna operates as in Fig. 6.From this analysis, the range of the voltage phase differenceφ2 was obtained under antenna resonance condition, when theinput impedances of both ports are practically real or resistive.The φ2 value of interest in these ranges or windows is thatwhich permits to have the input powers, W 1 and W2 , undercontrol, with a given relationship K w W2 /W1 .In some cases, it is possible to find values of K, φ 2 and V1 ,which permit to get a given total input power,Win W1 W2 , and a given power ratio K w W2 /W1 ,under the conditions W 1 0 and W2 0. In other

3 Fig. 4. CFA monopole 1 current distribution. Fig. 5. CFA monopole 2 and top-loading disk current distributions. Where Rradi is the i-th monopole radiation resistance [Ω]. Rci is the i-th monopole conductor resistance [Ω]. Rgpi is the i-th monopole ground plane loss resistance [Ω]. Z0mi is the average characteristic impedance of the ith- monopole equivalent transmission line [1] [Ω].