A Near-to-Far Field Transformation Using Spherical Expansions Phase 1: Verification on Simulated Antennas

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1 TECHNICAL REPORT 258 September 214 A Near-to-Far Field Transformation Using Spherical Expansions Phase 1: Verification on Simulated Antennas J. H. Meloling J. C. Allen Approved for public release. SSC Pacific San Diego, CA

2 SSC Pacific San Diego, California K. J. Rothenhaus, CAPT, USN Commanding Officer C. A. Keeney Executive Director ADMINISTRATIVE INFORMATION The work described in this report was performed for the Intelligence Advanced Research Projects Activity (IARPA) HF Geolocation Program (HFGeo) by the Applied Electromagnetics Branch (Code 5225) of the Systems of Systems and Platform Design Division (Code 552), Space and Naval Warfare Systems Center Pacific (SSC Pacific), San Diego, CA. The program manager of this project is Chris W. Reed. The HFGeo Program s research thrust falls under IARPA s Office of Smart Collection.. Released by D. E. Hurdsman, Head Applied Electromagnetics Branch Under authority of J. McGee, Head SoS and Platform Design Division This is a work of the United States Government and therefore is not copyrighted. This work may be copied and disseminated without restriction. SB

3 Executive Summary High-Frequency (HF) antenna design is challenged by advanced HF communication systems, such as HF-MIMO and HF-NVIS for vehicular units, and the new vectorsensing HF direction-finding systems. These systems require small, wideband, and efficient HF antennas with specified radiation patterns. Regardless of the tradeoffs that are made to get to a specific HF antenna, its far-field pattern is the final governance of system performance. Measuring far-field HF patterns is challenging HF wavelengths range from 1 m at 3 MHz to 1 m at 3 MHz. Typical pattern ranges measure the electric field on a hemisphere with a radius 1 wavelengths from the antenna. Consequently, an HF pattern range requires a 1-km radius and necessitates other approaches. One approach measures the near field on a standard pattern range and extends the resulting nearfield model to a far-field estimate. Tapan Sarkar wrote a substantial technical paper on near-to-far transformation using spherical expansions [12]. An implementation of Sarkar s near-to-far transformation is assessed in this report. Phase 1 of this report assesses this implementation on the following antenna models: Analytic antenna models where the near field, far field, and the spherical expansions are computable in closed form. Mini-NEC antenna simulations where the near and far fields are computable and numerical quadratures obtain the spherical expansions. This implementation does recover the far field from the near field provided the spherical expansion includes a sufficient number of terms. Phase 2 applies this expansion to pattern-range measurements and extends the implementation to estimate the number of terms and account for measurement noise. iii

4 Contents Executive Summary iii 1 HF Antenna Measurements 1 2 Near-to-Far Field Transformations 4 3 Sarkar s Near-to-Far Transform dpn m (cos(θ))/ ( ) 3.2 / r (kr) rh (2) n 4 Hertzian Electric Dipole Hertzian Dipole Expansions Hertzian Dipole Quadratures Hertzian Dipole Synthesis Hertzian Dipole in the Far Field The Small Loop Antenna Expansions Quadratures Synthesis Far-Field Pattern A Synthetic α Field 33 7 A Synthetic δ Field 38 8 Horizontal Electric Dipole Derivation Analysis Synthesis Far Field... 5 iv

5 9 Tilted Dipole Synthesis Far Field Shifted Tilted Dipole Synthesis Far Field V Dipole Synthesis Far Field Far-Field Gains Phase 2 Tasking 75 v

6 List of Figures 1 Compact HF-NVIS antenna for vehicle and ground forces (Reproduced by permission of Military Hi-Q [9]) Compact HF direction-finding antenna (Reproduced by permission of the Institution of Engineering & Technology [11]) Antenna Pattern Range at SSC Pacific Computational chart for the near-to-far transform Flow chart of the near-to-far comparison using an antenna model Spherical coordinates for the electric field showing the tangent vectors (E r is omitted) Spherical Bessel functions of the second kind Legendre polynomials Hertzian dipole in spherical coordinates: r = r r(θ, φ) E r (r, θ, φ) : i A = 1 (A), λ = 1 (m), l = λ/1 (m) E θ (r, θ, φ) : i A = 1 (A), λ = 1 (m), l = λ/1 (m) Hertzian dipole E θ (a,θ,φ) ; i A = 1 amp, λ = 1 meter, l = λ/1 meter; a =3λ, θ =18, φ = Hertzian dipole: error in R(α 1 ) as a function of the sampling sphere Synthesis of E θ (a,θ,φ) with m n Synthesis of E φ (a,θ,φ) with m n Exact E θ (r, θ, φ) Synthetic E θ (r, θ, φ) Far-field pattern comparison: i A = 1 A(A), λ = 1 (m), l = λ/1 (m) E φ (a,θ,φ) of the small loop antenna; i A = 1 amp, λ = 1 meter, r l = λ/1 meter; a =3λ, θ =18, φ = Small Loop Antenna: error in I(γ 1 ) as a function of the sampling sphere Small Loop Synthesis E φ (a,θ,φ) ; i A = 1 amp, λ = 1 meter, r l = λ/1 meter; a =3λ, θ =18, φ = Small Loop Antenna E φ (r, θ, φ) Synthetic E φ (r, θ, φ) vi

7 24 Far-field pattern comparison: i A = 1 (A), λ = 1 (m), r l = λ/1 (m) Synthesis of the α 11 = 1 field Synthesis of the δ 12 =1field Horizontal Dipole Horizontal dipole at 3 MHz in 3-D; i A = 1 amp, l = λ/1; upper panels are the simulated fields, lower panels are the synthesized fields Horizontal dipole images at 3 MHz; i A = 1 amp, l = λ/1; upper panels are the simulated fields, lower panels are the synthesized fields Pattern function for the horizontal dipole: i A = 1 amp, l = λ/1 m Pattern function approximation at r = 1m Synthetic pattern function approximation at r = 1 m Dipole tilted in the x z-plane and lifted over a perfect ground MiniNEC simulation of the tilted dipole at 29 MHz; electric fields are measuredat8ft MiniNEC simulation of the tilted dipole at 29 MHz; electric fields are measuredat8ft MiniNEC simulation of the tilted dipole at 29 MHz; electric fields are measuredat8ft Tilted dipole synthesis at 29 MHz and the 8-ft radius. The upper panels are the sampled fields computed from MiniNEC. The lower panels are the synthesized fields Comparing far-field patterns of the tilted dipole at 29 MHz. (upper panels) sampled fields computed from MiniNEC at the 5-ft radius. (lower panels) synthesized fields computed from the sampled fields at the 8-ft radius and extended to the 5-ft radius Dipole shifted along the x-axis, tilted in the x z-plane and lifted over aperfectground MiniNEC simulation of the shifted, tilted dipole at 29 MHz; electric fields are measured at 8 ft Comparing far-field patterns of the shifted, tilted dipole at 29 MHz. (upper panels) sampled fields computed from MiniNEC at the 5-ft radius. (lower panels) synthesized fields computed at the 8-ft radius and extended to the 5-ft radius V dipole in the y z-plane and lifted over a perfect ground vii

8 43 MiniNEC simulation of the V dipole at 3 MHz; electric fields are measuredat8ft Sarkar s coefficients estimated from E θ and E φ sampled on the sphere at 8 ft for the V dipole Near-field synthesis using N = 11, M = Near-field synthesis using N = 15, M = Near-field synthesis using N = 19, M = V dipole synthesis at r = 5 ft from E θ and E φ sampled on the upper hemisphere at 8 ft V dipole radiation intensity at r = 5 ft from E θ and E φ sampled on the upper hemisphere at 8 ft Reference antenna in the SSC Pacific Antenna Pattern Range List of Tables 1 Notation Legendre polynomials [3, Table 12.1] R(α mn ) estimated from E θ and E φ sampled on the sphere a =3λ I(α mn ) estimated from E θ and E φ sampled on the sphere a =3λ R(γ mn ) estimated from E θ and E φ sampled on the sphere a =3λ I(γ mn ) estimated from E θ and E φ sampled on the sphere a =3λ β mn estimated from E θ and E φ sampled on the sphere a =3λ δ mn estimated from E θ and E φ sampled on the sphere a =3λ α mn estimated from E θ and E φ sampled on the sphere a =3λ γ mn estimated from E θ and E φ sampled on the sphere a =3λ α mn estimated from E θ and E φ sampled on the sphere a =3λ for the δ 12 =1antenna β mn estimated from E θ and E φ sampled on the sphere r =3λ for the δ 12 =1antenna γ mn estimated from E θ and E φ sampled on the sphere r =3λ for the δ 12 =1antenna... 4 viii

9 14 δ mn estimated from E θ and E φ sampled on the sphere r =3λ for the δ 12 =1antenna α mn estimated from E θ and E φ sampled on the sphere a =3λ for the Horizontal Dipole β mn estimated from E θ and E φ sampled on the sphere r =3λ for the Horizontal Dipole γ mn estimated from E θ and E φ sampled on the sphere r =3λ for the Horizontal Dipole δ mn estimated from E θ and E φ sampled on the sphere r =3λ for the Horizontal Dipole MiniNEC description α mn estimated from E θ and E φ sampled on the sphere at 8 ft for the tilteddipole β mn estimated from E θ and E φ sampled on the sphere at 8 ft for the tilteddipole γ mn estimated from E θ and E φ sampled on the sphere at 8 ft for the tilteddipole δ mn estimated from E θ and E φ sampled on the sphere at 8 ft for the tilteddipole MiniNEC description of the shifted, tilted dipole α mn estimated from E θ and E φ sampled on the sphere at 8 ft for the shifted, tilted dipole β mn estimated from E θ and E φ sampled on the sphere at 8 ft for the shifted, tilted dipole γ mn estimated from E θ and E φ sampled on the sphere at 8 ft for the shifted, tilted dipole δ mn estimated from E θ and E φ sampled on the sphere at 8 ft for the shifted, tilted dipole MiniNEC description of the V dipole ix

10 1 HF Antenna Measurements HF antenna design is challenged by advanced HF communication systems and developments in HF direction finding. Figure 1 shows a recent HF-NVIS antenna developed for ground-mobile applications. Challenges for any compact HF-NVIS antenna are the trade-offs between bandwidth, pattern (upward radiation), and efficiency with the necessary adaptive tuning. This antenna and its inevitable tradeoffs should be verified by measurements. Figure 1: Compact HF-NVIS antenna for vehicle and ground forces (Reproduced by permission of Military Hi-Q [9]). Figure 2 shows co-located HF antennas for direction finding. Rather than an HF array with the half-wavelength spacing, this co-located array offers direction finding using polarization rather than spatial diversity. One challenge for HF vector sensors is array calibration. Array calibration for a co-located HF array is equivalent to measuring the array s response to an electric field for each azimuth, elevation, and polarization. Figure 2: Compact HF direction-finding antenna (Reproduced by permission of the Institution of Engineering & Technology [11]). 1

11 Consequently, both the HF-NVIS antennas and HF vector sensors require farfield measurements to establish antenna performance. Figure 3 shows the Antenna Pattern Range located at the Model Range, Space and Naval Warefare Systems Center Pacific (SSC Pacific). The electric field produced by an antenna placed in the Antenna Pattern Range is measured over the upper hemisphere at a radius of approximately 8 ft (24 m). For 3-MHz HF antennas, the Antenna Pattern Range measures their near fields at 24 m or 1/4 wavelength. For 3-MHz HF antennas, their near fields are measured at 2.4 wavelengths. This variation in the near-field measurements raises the following question: Is the near-to-far transform stable with respect to Antenna Pattern Range measurements ranging over.25 to 2.4 wavelengths? Figure 3: Antenna Pattern Range at SSC Pacific. This report assesses a near-to-far transformation with the goal of mapping the near-field measurements taken on a grid of azimuth and elevation angles on the Antenna Pattern Range into far-field patterns. Because the far-field pattern is not directly measurable, this Phase 1 discussion relies on bounding the error in analytic models before undertaking the leap to the Antenna Pattern Range measurements. Accordingly, Section 2 reviews the near-to-far transformation and its discrete implementation with the errors arising from truncation of the spherical sums and the discrete sampling of the electric field. 2

12 Section 3 reviews Tapan Sarkar s near-to-far transformation [12]. The spherical expansions and the special functions are made explicit with correct normalizations, end points of the expansions, and three-term recurrence relations are supported for computational processing. Sections 4 11 apply the near-to-far transform to selected antenna models. Each section undertakes the near-field and far-field models of an antenna, an analysis of the near field in spherical modes using only discrete samples of the electric field, and a comparison the resulting synthetic spherical expansion against the modeled antenna in both the near and far fields. These examples show that controlling the truncation error is more critical than the discretization errors. That is, sampling the electric field at 5 to 18 increments (discretization error) contributes less error than omitting higher-order terms in the spherical expansion (truncation error). The final section outlines the tasking for Phase 2 where the near-to-far transformation will be applied to measurements collected on the Antenna Pattern Range. 3

13 2 Near-to-Far Field Transformations A canonical problem in antenna engineering is estimating the inaccessible far field from near-field measurements. This report assesses one implementation of Tapan Sarkar s near-to-far transformation [12]. Figure 4 illustrates this near-to-far implementation. Figure 4: Computational chart for the near-to-far transform. Discrete near-field samples of the electric field are collected on a sphere of radius r = a. The analysis box uses these samples to compute electric-field coefficients {s m,n } by numerical quadratures. The main source of error in these coefficients {s m,n } arises from the discrete samples because the numerical quadratures operate on twodimensional splines of the electric field samples. That is, the coefficients {s m,n } are accurate representations of an electric field recovered from the sampled electric field [6]. The synthesis box computes the electric field at any radius r a as a sum of Sarkar s orthonormal basis functions {e m,n }: n E(r, θ, φ) = s m,n e m,n (r, θ, φ) =:S(a; r, θ, φ). (1) n= m= The key observation is that the expansion holds regardless of the measurement sphere: S(a 1 ; r, θ, φ) =S(a 2 ; r, θ, φ) for r a 1,a 2. Assessing the quality of this expansion focuses not only on computing the far-field pattern using a truncated sum N n E(r, θ, φ) s m,n e m,n (r, θ, φ) =:S(a; N; r, θ, φ), n= m= 4

14 but a truncated sum where the coefficients of the expansion have accuracy limited by the discrete sampling: E(r, θ, φ) N n s m,n ( θ, φ)e m,n (r, θ, φ) n= m= =: S(a; θ, φ, N; r, θ, φ) (2) Thus, the expansion s accuracy is degraded by truncation error and limited by the discretization errors. Phase 1 of this report assesses Sarkar s expansion on several antenna models. The Hertzian dipole is the baseline computational vehicle for Phase 1. The goal is to verify the numerical computation and the sensitivity of the approximation E(r, θ, φ) S(a; θ, φ, N; r, θ, φ) on this analytic dipole model. Once this ideal dipole is analyzed, other antennas models are treated in the same format: analysis, synthesis, and the far-field pattern. Figure 5 summarizes this process where the ground truth of the antenna model s electric field is compared against the synthetic field. Figure 5: Flow chart of the near-to-far comparison using an antenna model. 5

15 Table 1: Notation. Variable Description k Wavenumber λ Wavelength (m) f Frequency (Hz) ω Frequency (rad/sec) η Free space impedance 12π (ohms) ɛ Permittivity (F/m) r Radius (m) θ Zenith (radians) φ Azimuth(radians) E Electric field vector (V/m) := Definition 6

16 3 Sarkar s Near-to-Far Transform Figure 6 illustrates spherical coordinates and the decomposition of the electric field in the θ φ plane. Sarkar expands the electric field using spherical Bessel functions of the second kind h (2) n (kr), associated Legendre functions of the first kind Pn m (cos(θ)) with argument cos(θ), and coefficients α mn, β mn, γ mn, and δ mn determined by the near-field boundary [12, Equation 5]: E θ (r, θ, φ) =jωµ + n n= m= 1 r and [12, Equation 6] E φ (r, θ, φ) =jωµ n n= m= 1 r n mh (2) n n= m= r r ( rh (2) n n n= m= ( rh (2) n (kr) P n m(cos(θ)) {γ mn sin(mφ) δ mn cos(mφ)} sin(θ) (kr)) dp m n (cos(θ)) h (2) n {α mn cos(mφ)+β mn sin(mφ)}. (kr) dp n m(cos(θ)) {γ mn cos(mφ)+δ mn sin(mφ)} (kr)) mp m n (cos(θ)) sin(θ) {α mn sin(mφ) β mn cos(mφ)}. Figure 6: Spherical coordinates for the electric field showing the tangent vectors (E r is omitted). 7

17 The coefficients are computed from the near field at radius r = a [12, Equations 14 and 27]: α mn = a (2n + 1)(n m)! 2n(n + 1)(n + m)! ɛ 2π π m dφ 2π kah (2) n (ka)+h (2) n (ka) { E θ (a,θ,φ) dp n m (cos(θ)) sin(θ) cos(mφ) me φ (a,θ,φ)pn m (cos(θ)) sin(mφ) [12, Equations 14 and Eq. 31]: β mn = a (2n + 1)(n m)! 2n(n + 1)(n + m)! ɛ 2π π m dφ 2π kah (2) n (ka)+h (2) n (ka) { E θ (a,θ,φ) dp n m(cos(θ)) sin(θ) sin(mφ)+me φ (a,θ,φ)pn m (cos(θ)) cos(mφ) [12, Equations 13 and 32]: γ mn = 1 jωµh (2) n (ka) (2n + 1)(n m)! 2n(n + 1)(n + m)! ɛ 2π π m dφ 2π { E φ (a,θ,φ) dp } n m (cos(θ)) sin(θ) cos(mφ)+me θ (a,θ,φ)pn m (cos(θ)) sin(mφ), [12, Equations 13 and 33]: δ mn = 1 jωµh (2) n (ka) (2n + 1)(n m)! 2n(n + 1)(n + m)! ɛ 2π π m dφ 2π { E φ (a,θ,φ) dp n m (cos(θ)) sin(θ) sin(mφ) me θ (a,θ,φ)pn m (cos(θ)) cos(mφ) The twiddle factor is the standard adjustment for working with the sinusoidal basis: { 1 m = ɛ m := 2 m>. The spherical Bessel functions of the second kind are defined in Arfken as [3, Equation 11.15]: π h (2) n (r) := 2r H(2) n+1/2 (r), (3) where H n (2) (r) is the Hankel function of the second kind. Figure 7 plots the first few }, }, }. 8

18 spherical Bessel functions [3, Equation 11.15]: h (2) (r) = exp( jr) j r 1 (r) = exp( jr) { 1r + jr } 2 h (2) h (2) 2 (r) = exp( jr) { j r 3 r + j3 }. 2 r 3 Figure 7: Spherical Bessel functions of the second kind. The associated Legendre functions of the first kind Pn m (x) are [3, Equation and footnote]: Pn m (x) :=( 1) m (1 x 2 m/2 dm ) dx P n(x), m where P n (x) s are the Legendre polynomials. There is the handy observation that P n (x) =P n(x). A stand-alone form adapted from [2, Equation 4.12] Pn m (x) := ( 1)m 2 n n! (1 x2 m/2 dm+n ( ) (x 2 1) n) (4) dx m+n 9

19 that computes P m n (x) = for m>n (5) Table 2 lists the initial Legendre polynomials and their derivatives. Figure 8 plots the first few Legendre polynomials. Table 2: Legendre polynomials [3, Table 12.1]. n P n (x) P n(x) 1 1 x (3x2 1) 3x (5x3 1 3x) 2 (15x2 3) (35x4 3x ) 2 (35x3 15x) Figure 8: Legendre polynomials. An ambiguity at n = occurs because α mn, β mn, γ mn and δ mn contain a 1/ term for n =. If these coefficients are finite, the contribution of n = to the 1

20 near-to-far transformation is zero: n n= m= + jωµm h(2) n (kr) sin(θ) P n m (cos(θ)){γ mn sin(mφ) δ mn cos(mφ)} = 1 r r =. n n= m= ( rh (2) n 1 r r ( rh (2) n (kr) ) dp (cos(θ)) (kr) ) dp m n (cos(θ)) α {α mn cos(mφ)+β mn sin(mφ)} The last equality follows because the derivative of P (x) =P (x) = 1 is zero. Therefore, the near-to-far transformation starts the double sum at n = dp m n (cos(θ))/ Differentiating the generating function [3, Equation 12.84] computes this derivative in terms of these functions. Recall that Pn(x) =P n (x) denotes the Legendre polynomials. If m<n, dp m n (x) dx = d dx {( 1) m (1 x 2 m/2 dm ) dx P n(x) m = ( 1) m m(1 x 2 ) m/2 1 x dm dx P n(x)+( 1) m (1 x 2 m/2 dm+1 ) m dx P n(x) m+1 = m x 1 x 2 ( 1)m (1 x 2 m/2 dm ) dx P n(x) m ( 1) m+1 (1 x 2 ) 1/2 (1 x 2 (m+1)/2 dm+1 ) dx P n(x) m+1 = m x 1 x P m 2 n (x) (1 x 2 ) 1/2 Pn m+1 (x). In terms of the elevation angle, } dp m n (cos(θ)) = dp n m (x) dx { dx = m x } 1 x 2P n m (x) (1 x 2 ) 1/2 Pn m+1 (x) ( sin(θ)) { = m cos(θ) sin 2 (θ) P n m (cos(θ)) + 1 } sin(θ) P n m+1 (cos(θ)) sin(θ) = m cot(θ)pn m (cos(θ)) + Pn m+1 (cos(θ)). 11

21 If m = n, the (n + 1) derivative of P n (x) is zero using P n+1 n (x) = from Equation 4. Consequently, dp m n (cos(θ)) = m cot(θ)p m n (cos(θ)) + P m+1 n (cos(θ)) (6) holds for all m n. When this derivative is weighted by the sine function, 3.2 / r dp m n (cos(θ)) ( ) rh (2) n (kr) sin(θ) =m cos(θ)p m n (cos(θ)) + sin(θ)p m+1 n (cos(θ)), (7) The radial term is computed from the spherical Hankel function of the second kind using the three-term recurrence relation [1, Equation ]: h (2) n (r) =h (2) n 1(r) n +1 r h (2) n (r). Consequently, the derivative of the spherical Hankel functions can be computed using these functions themselves: ( rh (2) n (kr) ) = h (2) r n (kr)+krh n (2) (kr) { h (2) = h (2) n (kr)+kr n 1(kr) n +1 kr } h(2) n (kr) = krh (2) n 1 (kr) nh(2) n (kr). (8) The interpretation of [12, Equation 14] is computed as follows: N = r (rh(2) n (kr)) r=a = kah (2) n 1(ka) nh (2) n (ka). 12

22 4 Hertzian Electric Dipole The Hertzian dipole is the baseline to assess Sarkar s spherical expansion. Figure 9 illustrates spherical coordinates around a short antenna carrying a ẑ-directed current i A in free space on a wire of length l. Figure 9: Hertzian dipole in spherical coordinates: r = r r(θ, φ). The next two figures present the nonzero magnetic and electric fields of the Hertzian dipole [7, Section 17.2]: H φ (r, θ, φ) = i { A l j 4π k2 sin(θ) exp( jkr) kr + 1 }, (kr) 2 E r (r, θ, φ) = η i { A l 1 2π k2 cos(θ) exp( jkr) (kr) j }, 2 (kr) 3 E θ (r, θ, φ) = η i { A l j 4π k2 sin(θ) exp( jkr) kr + 1 (kr) j }. (9) 2 (kr) 3 13

23 Figure 1: E r (r, θ, φ) : i A = 1 (A), λ = 1 (m), l = λ/1 (m). Figure 11: E θ (r, θ, φ) : i A = 1 (A), λ = 1 (m), l = λ/1 (m). 14

24 The analysis and synthesis of this Hertzian dipole consists of sampling the fields near the antenna (r = a) and recovering the far-field patterns. With [ ] 1 E r (r, θ, φ) = O (kr) 2 E θ (r, θ, φ) = η i A l 4π k2 sin(θ) exp( jkr) j kr + O [ 1 ] (kr) 2 as r a, the pattern function for the electrical field is [7, Section 17.4] F (θ, φ) := lim r E(r, θ, φ) = η i A l k sin(θ). (1) r 4π 4.1 Hertzian Dipole Expansions This section uses the electric field of the Hertzian dipole to assess Sarkar s expansion in two stages. For the first stage, Sarkar s electric-field expansion is computed from the physical model of the dipole. The Hertzian dipole has electric fields (Equation 9): E (r) := η i A l 4π k2 exp( jkr) E θ (r, θ, φ) = E (r) sin(θ) E φ (r, θ, φ) =. { j kr + 1 (kr) 2 j (kr) 3 Second, the coefficients determined by this physical model must match the coefficients computed by quadratures of the electric field measured on any near-field sphere. The first stage will verify that the Hertzian dipole s coefficients are all zero, with the exception of α 1 = ηk i A l 4π. (11) Comparing the physical model with Sarkar s expansion forces the expansion in function of φ to vanish or that the sums collapse to m =: E θ (r, θ, φ) =E (r) sin(θ) = jωµ = = + n n= m= n n= m= n= n= α n 1 r α n 1 r 1 r r r mh (2) n r m (kr)p n (cos(θ)) {γ mn sin(mφ) δ mn cos(mφ)} sin(θ) ( rh (2) n (kr) ) dp m n (cos(θ)) ( rh (2) n (kr)) dpn(cos(θ)) ( rh (2) n (kr) ) dp n (cos(θ)) 15, } {α mn cos(mφ)+β mn sin(mφ)}

25 where the P n s are the Legendre polynomials. More precisely, α mn =,β mn =, γ mn =,δ mn = for m >. Table 2 provides the derivatives of the Legendre polynomials to substitute into E θ : 1 ( ) E θ (r, θ, φ) = α 1 rh (2) 1 (kr) P r r 1(cos(θ))( sin(θ)) 1 ( ) +α 2 rh (2) 2 (kr) P 2 (cos(θ))( sin(θ)) +... r r 1 ( ) = α 1 rh (2) 1 (kr) sin(θ) r r 1 ( ) α 2 rh (2) 2 r r (kr) 3 cos(θ) sin(θ)+... Matching the sinusoidal basis in θ gives that this electric field component expands as 1 ( ) E θ (r, θ, φ) = α 1 rh (2) 1 (kr) sin(θ). r r The spherical Bessel term expands as 1 r { } rh (2) 1 (kr) = k exp( jkr) r { j kr + 1 (kr) 2 j }. (12) (kr) 3 Substituting into the electric field component gives { j E θ (r, θ, φ) = α 1 sin(θ)k exp( jkr) kr + 1 (kr) j }. 2 (kr) 3 Comparison with the analytic formula for E θ in Equation 9 forces the closed-form solution given in Equation 11. That is, starting with the physical model of E θ, the Hertzian dipole s coefficients are all zero, excepting α 1. To complete this first stage, it is necessary to verify that this expansion explains E φ using Sarkar s expansion (α mn =,β mn =,γ mn =,δ mn = for m>): [12, Equation 6] E φ (r, θ, φ) =jωµ = jωµ =, n n= m= n= 1 r h (2) n n h (2) n n= m= r ( rh (2) n (kr) dp n(cos(θ)) provided the γ mn s vanish entirely. (kr) dp n m(cos(θ)) {γ mn cos(mφ)+δ mn sin(mφ)} (kr)) mp m n (cos(θ)) sin(θ) γ n 16 {α mn sin(mφ) β mn cos(mφ)}

26 The second stage verifies that the coefficients in the expansion can be computed using only the electric fields measured on a given sphere. Sarkar computes α 1 using E θ and E φ measured on a sphere of radius a whereas the expansion of the electric fields are independent of any spherical measurements. Consequently, Sarkar s formulas must cancel the measurements on the sphere. Equivalently, Sarkar s formulas must be independent of the radius: α 1 = = = a kah (2) n (ka)+h (2) n (ka) { E θ (a,θ,φ) dp n m (cos(θ)) a kah (2) 1 (ka)+h (2) 1 (ka) a kah (2) 1 (ka)+h (2) 1 (ka) a (2n + 1)(n m)! 2n(n + 1)(n + m)! ɛ 2π π m dφ 2π sin(θ) cos(mφ) me φ (a,θ,φ)p m n (cos(θ)) sin(mφ) 1 2π 2π π π = 3 4 kah (2) 1 (ka)+h (2) 1 (ka) Equation 12 computes the denominator: π E θ (a,θ,φ) dp 1(cos(θ)) E θ (a,θ,φ)( sin(θ)) sin(θ) E θ (a, θ) sin(θ) 2. kah (2) 1 (ka)+h (2) 1 (ka) = r (rh(2) On the sphere, the electric field integrates as π n = exp( jka) E θ (a, θ) =η i A l k sin(θ) exp( jka) 4π a E θ (a, θ) sin(θ) 2 = η i A l k 4π = η i A l 4π (kr)) r=a sin(θ) { j + 1 ka j }. (ka) 2 { j + 1 ka j } (ka) 2 { a exp( jka) j + 1 ka j } π sin(θ) 3 (ka) 2 { k a exp( jka) j + 1 ka j } 4 (ka) 2 3. } m= n=1 17

27 Combining the numerator and denominator α 1 = 3 4 = 3 4 a kah (2) 1 (ka)+h (2) 1 (ka) a { exp( jka) exp( jka) j + 1 { j + 1 ka π }η j ka (ka) 2 j (ka) 2 E θ (a, θ) sin(θ) 2 i A l k 4π a } 4 3 = ηk i A l 4π. Consequently, Sarkar s denominator term cancels the effect of the radius a in the measured electric field to produce an α 1 that is independent of the measuring sphere. 18

28 4.2 Hertzian Dipole Quadratures The Hertzian dipole has electric fields (Equation 9): E (r) := η i A l 4π k2 exp( jkr) E θ (r, θ, φ) = E (r) sin(θ), E φ (r, θ, φ) =. { j kr + 1 (kr) 2 j }, (kr) 3 Figure 12 plots the magnitude of E θ sampled at 18 increments. on the sphere a =3λ over the angular grid Figure 12: Hertzian dipole E θ (a,θ,φ) ; i A = 1 amp, λ = 1 meter, l = λ/1 meter; a =3λ, θ =18, φ =18. The Hertzian dipole has only the single non-zero coefficient given by Equation 11: α 1 = ηk i A l 4π = , where the numerical value is computed using the parameters in Figure 12. Tables 3 and 4 report on the coefficients α mn estimated from the sampled values on the sphere a =3λ. The real part of these α mn s is accurate to 5 places whereas the imaginary part is accurate to 6 places. Tables 3 and 4 compute the α mn from the electric fields sampled on a sphere of fixed radius a = 3λ. Figure 13 displays the effect of varying the sampling radius. The accuracy remains at 5 places for λ a 5λ. This stability of α 1 as a function of the sampling radius provides a partial answer to the question of Section 1. The nearto-far transformation is stable for near-fields of the ideal dipole where the sampling sphere ranges as λ a 5λ. 19

29 Table 3: R(α mn ) estimated from E θ and E φ sampled on the sphere a =3λ. n =1 n =2 n =3 m = m =1... m =2.. m =3. Table 4: I(α mn ) estimated from E θ and E φ sampled on the sphere a =3λ. n =1 n =2 n =3 m = m = m = m = Figure 13: Hertzian dipole: error in R(α 1 ) as a function of the sampling sphere. 2

30 4.3 Hertzian Dipole Synthesis Once the spherical coefficients are computed from measurement on a sphere of radius r = a, Sarkar s expansions compute the electric field at distances r a. This section verifies that the sampled electric field can be recovered on the measurement sphere. Figure 14 reports on the expansion of E θ (r, θ, φ) atr = a using the numerical coefficients in Tables 3 and 4. Comparison with the analytic E θ (a,θ,φ) in Figure 12 shows credible agreement. In the notation of Section 2, the synthetic E θ (a,θ,φ) computed using Sarkar s expansion is a credible approximation despite the truncation (N = 3) and discretization ( θ =18, φ =18 ): E θ (a,θ,φ) S(a =3; θ =18, φ =18,N =3;a,θ,φ). Figure 14: Synthesis of E θ (a,θ,φ) with m n 3. 21

31 Likewise, Figure 15 reports on the expansion at E φ (r, θ, φ) atr = a using the numerical coefficients in Tables 3 and 4. Because E φ (r, θ, φ) =, the absolute error is approximately Figure 15: Synthesis of E φ (a,θ,φ) with m n Hertzian Dipole in the Far Field The last step in Sarkar s near-to-far transformation is to take the coefficients computed in the near field and synthesize the far-field pattern. Figure 16 shows E θ (r, θ, φ) in the r-θ plane out to r = 1λ. Figure 17 shows the corresponding image of the synthesized E θ (r, θ, φ) determined by the synthesis code. The synthesis code implements Sarkar s expansion given the coefficients α, β, γ, and δ. This code was tested by inserting the exact value of the coefficients. That is, all coefficients are zeros excepting α 1 = set by Equation 11. Comparison of both figures shows that the magnitude of the synthesized electric field is a credible approximation to the analytic field. 22

32 Figure 16: Exact E θ (r, θ, φ). Figure 17: Synthetic E θ (r, θ, φ). 23

33 The far-field pattern function for the electric field is given in Equation 1 and repeated here for convenience: F (θ, φ) := lim r E(r, θ, φ) = η i A l r 4π k sin(θ). Figure 18 compares this pattern function against two approximations. The approximation labeled Exact is r E θ (r, θ, φ) evaluated at r = 1 m. The approximation labeled Synthetic is the corresponding computation using the synthesis code. The approximations are relatively close to the true pattern. Figure 18: Far-field pattern comparison: i A = 1 A(A), λ = 1 (m), l = λ/1 (m). 24

34 5 The Small Loop Antenna The Hertzian dipole s computations required only E θ because E φ is zero. A loop antenna tests the near-to-far transformation when E φ is non-zero and E θ is zero. The electric field for a small loop antenna is [4, Equation 5-19] E r (r, θ, φ) = E θ (r, θ, φ) = E φ (r, θ, φ) = η i A S 4π k3 sin(θ) exp( jkr) { 1 kr j }, (13) (kr) 2 where S = πrl 2 is the area of the loop. This electric field expanded and synthesized to assess Sarkar s expansion. 5.1 Expansions This analytic form of the electric field determines the coefficients of Sarkar s expansion. Sarkar s expansion for E φ is repeated here for convenience: [12, Equation 6] E φ (r, θ, φ) =jωµ n n= m= 1 r r n n= m= ( rh (2) n h (2) n (kr) dp n m (cos(θ)) {γ mn cos(mφ)+δ mn sin(mφ)} (kr) ) mp m n (cos(θ)) sin(θ) {α mn sin(mφ) β mn cos(mφ)}. The symmetry of E φ with respect to φ forces the sums to collapse to m =: E φ (r, θ, φ) =jωµ h (2) n (kr)dp n(cos(θ)) γ n. n= More precisely, this expansion forces α mn =,β mn =,γ mn =, and δ mn = for m>. Consequently, the expansion for E θ must take the form [12, Equation 5]: E θ (r, θ, φ) =jωµ = + n= m= n= 1 r n r 1 r n mh (2) n n= m= r ( rh (2) n ( rh (2) n m (kr)p n (cos(θ)) {γ mn sin(mφ) δ mn cos(mφ)} sin(θ) (kr) ) dp m n (cos(θ)) (kr)) dp n (cos(θ)) α n. {α mn cos(mφ)+β mn sin(mφ)} 25

35 The vanishing E θ = forces α mn = for all m, n. Returning to the expansion of E φ, the product of the radial term and elevation term for n =1is h (2) 1 (kr) dp 1(cos(θ)) = exp( jr) { 1r + jr } ( sin(θ)) 2 = sin(θ) exp( jr) { 1 r j r 2 Comparison with the physical model of E φ of Equation 13 forces all γ n s to zero with the exception E φ (r, θ, φ) = η i { } A S 4π k3 h (2) 1 (kr)dp 1(cos(θ)) = jωµh (2) 1 (kr)dp 1(cos(θ)) γ 1. That is, the physical model of the small loop antenna forces jωµγ 1 = η i A S 4π k3. For convenience, recall that [5, Equation 9-14], [5, Equation 9-28], [5, Equation 1-33]: Solving for the coefficient c = 1 ɛµ, η = µ ɛ, k = ω c. γ 1 = 1 η i A S k 3 j µ 4π ω 1 i A S k 2 = j µɛ 4π c = jc i A S 4π k 2 c = j i A S 4π k2. That is, the small loop s coefficients are all zero, excepting }. γ 1 = j i A S 4π k2. (14) The second task is to verify that the coefficients in the expansion can be computed using only the electric fields measured on a given sphere. Sarkar computes γ 1 using E θ and E φ measured on a sphere of radius a whereas the expansion of the electric fields 26

36 are independent of any spherical measurements. Consequently, Sarkar s formulas must cancel the measurements on the sphere. Equivalently, Sarkar s formulas must be independent of the radius: γ 1 = 1 (2n + 1)(n m)! 2n(n + 1)(n + m)! jωµh (2) n (ka) { E φ (a,θ,φ) dp n m(cos(θ)) 2π ɛ 2π π m dφ 2π sin(θ) cos(mφ)+me θ (a,θ,φ)p m n (cos(θ)) sin(mφ) π = dφ E jωµh (2) φ (a,θ,φ) dp 1 (cos(θ)) sin(θ) 1 (ka) 4 2π 1 3 π { } d = E jωµh (2) φ (a,θ,φ) 1 (ka) 4 cos(θ) sin(θ) 1 3 π = E jωµh (2) φ (a,θ,φ) sin(θ) 2 1 (ka) 4 1 = jωµh (2) 1 (ka) η i { A S 1 4π k3 exp( jka) ka 1 = jωµh (2) 1 (ka) η i A S 4π k3 ( h (2) 1 (ka)) = ηk3 jωµ i A S 4π j } 3 π sin(θ) 3 (ka) 2 4 } m= n=1 = jk 2i A S 4π. That is, γ 1 determined by the small loop s physical model (Equation 14) may also be computed by a quadrature that integrates the electric fields measured over a sphere. 27

37 5.2 Quadratures The small loop antenna has electric fields E θ (r, θ, φ) = and E φ (r, θ, φ) =η i { A S 1 4π k3 sin(θ) exp( jkr) kr j }, (kr) 2 where S = πr 2 l is the area of the loop. Figure 19 plots the magnitude of E φ on the sphere a =3λ over the angular grid sampled at 18 increments. Figure 19: E φ (a,θ,φ) of the small loop antenna; i A = 1 amp, λ = 1 meter, r l = λ/1 meter; a =3λ, θ =18, φ =18. The small loop antenna has only the single non-zero coefficient given by Equation 14: γ 1 = j i A S 4π k2 = j.98696, where the numerical value is computed using the parameters in Figure 19. Tables 5 and 6 report the coefficients γ mn estimated from the sampled values on the sphere a =3λ. The real part of these γ mn s is accurate to 9 places whereas the imaginary part is accurate to 5 places. Figure 2 displays the effect of varying the sampling radius. The accuracy remains at five decimal places for λ a 5λ. This stability 28

38 Table 5: R(γ mn ) estimated from E θ and E φ sampled on the sphere a =3λ n =1 n =2 n =3 m = m = m = m = Table 6: I(γ mn ) estimated from E θ and E φ sampled on the sphere a =3λ n =1 n =2 n =3 m = m =1... m =2.. m =3. of γ 1 as a function of the sampling radius provides a partial answer to the question of Section 1 the near-to-far transformation is stable for the near-fields of the small loop antenna where the sampling sphere ranges as λ a 5λ. Figure 2: Small Loop Antenna: error in I(γ 1 ) as a function of the sampling sphere. 29

39 5.3 Synthesis Synthesis uses the coefficients estimated by the quadratures to reconstruct the fields of the small loop antenna. Figure 21 plots E φ synthesized from the estimated coefficients using only the relatively coarse sampling of the E φ and E θ = fields. The shape and peak values are in agreement with Figure 19. Figure 21: Small Loop Synthesis E φ (a,θ,φ) ; i A = 1 amp, λ = 1 meter, r l = λ/1 meter; a =3λ, θ =18, φ =18. 3

40 5.4 Far-Field Pattern This section compares the exact far-field patterns of the small loop antenna to the synthesized patterns. Figure 22 plots Eφ as a function of range r and angle θ from Equation 13. Figure 23 plots Eφ generated using the synthesis code with the exact values for Sarkar s coefficients: α, β, γ, and δ are all zero excepting γ1 = j These slices of the exact and synthetic electric field verify that the synthesis is accurate for this antenna. Figure 22: Small Loop Antenna Eφ (r, θ, φ). Figure 23: Synthetic Eφ (r, θ, φ). 31

41 The pattern function for the small loop antenna is obtained from Equation 13: F (θ, φ) = lim r E(r, θ, φ) = η i A S r 4π k2 sin(θ). Figure 24 compares this pattern function against two approximations. The approximation labeled Exact is r E θ (r, θ, φ) evaluated at r = 1 m. The approximation labeled Synthetic is the corresponding computation using the synthesis code. The approximations are relatively close to the small loop pattern. Figure 24: Far-field pattern comparison: i A = 1 (A), λ = 1 (m), r l = λ/1 (m). 32

42 6 A Synthetic α Field Define a synthetic electric field with α 11 = 1 as the only non-zero coefficient. The angular components of the electric field are E θ (r, θ, φ) = + 1 ( ) rh (2) dp 1 1 (kr) 1 (cos(θ)) cos(φ) r r E φ (r, θ, φ) = 1 ( ) rh (2) P 1 1 (kr) 1 (cos(θ)) sin(φ). r r sin(θ) Use [3, Table 12.3 with footnote] Use [3, Equation 11.15]: P 1 1 (cos(θ)) = sin(θ) = dp 1 1 (cos(θ)) h (2) 1 (r) =exp( jr) { 1r + jr } 2 = cos(θ). to compute 1 ( ) rh (2) 1 (kr) r r = 1 ( ) rh (2) 1 (kr) r r { j = exp( jkr) r + 1 kr 2 = exp( jkr)k =: E (r). j } k 2 r 3 { j kr + 1 (kr) 2 j (kr) 3 } The electric field forced by α 11 = 1 has the form: E θ (r, θ, φ) = E (r) cos(θ) cos(φ) E φ (r, θ, φ) = +E (r) sin(φ). 33

43 Figure 25 reports the electric field and the resulting synthesis. The upper panels are sampled using N φ = 73 and N θ = 43 sample points for a angular grid size of φ =5 and θ = 4.3. Only these sample values are input to the analysis code to estimate Sarkar s coefficients. The lower panels show the resulting synthesis. Comparison of the upper panels and lower panels shows that the magnitude of the synthesized electric is a credible approximation to the sampled field. Figure 25: Synthesis of the α 11 = 1 field. 34

44 The electric field was synthesized in the preceding plots. However, there is the question if the electric field determined by the α 11 is represented by this single term. Question: Is α 11 the only non-zero coefficient of this electric field? The following analysis answers this question positively and leads to the second question regarding the numerical quadrature. Question: How well does the numerical quadrature estimate α 11 compare to the vanishing coefficients of this electric field? The β mn s vanish because of the orthogonality between cos(φ) and sin(φ) and the basis functions sin(mφ) and cos(mφ), respectively. β mn = kah (2) a n (ka)+h (2) n (ka) (2n + 1)(n m)! 2n(n + 1)(n + m)! ɛ m 2π {b φ;mn + b θ;mn }. Integrating each term separately, b φ;mn = = 2π 2π π dφ dφ = me (r) = π 2π me φ (a,θ,φ)p m n (cos(θ)) cos(mφ) m{e (r) sin(φ)}p m n (cos(θ)) cos(mφ) π sin(φ) cos(mφ)dφ Pn m (cos(θ)) b θ;mn = 2π = E (r) = E (r) =. π dφ 2π 2π E θ (a,θ,φ) dp n m (cos(θ)) sin(θ) sin(mφ) π dφ {cos(θ) cos(φ)} dp n m (cos(θ)) cos(φ) sin(mφ)dφ π dp m n (cos(θ)) sin(θ) sin(mφ) sin(θ) cos(θ) Thus, the β mn s must vanish in a credible quadrature. Table 7 reports the quadrature computations β mn from Figure 25. These coefficients vanish nearly to the machine ɛ. The δ mn s vanish mainly because the orthogonality of the sinusoids, excepting for m =1. 1 (2n + 1)(n m)! ɛ m δ mn = jωµh (2) n (ka) 2n(n + 1)(n + m)! 2π {d φ,mn d θ,mn }. 35

45 Table 7: β mn estimated from E θ and E φ sampled on the sphere a =3λ. n =1 n =2 n =3 m = m = m = m = d φ;mn = 2π 2π π dφ π E φ (a,θ,φ) dp m n (cos(θ)) sin(θ) sin(mφ) = dφ {E (r) sin(φ)} dp n m(cos(θ)) sin(θ) sin(mφ) 2π π dpn m (cos(θ)) = E (r) sin(φ) sin(mφ)dφ sin(θ) π dpn 1 (cos(θ)) = E (r)πχ(m =1) sin(θ) by parts = E (r)πχ(m =1) {P 1m(cos(θ)) sin(θ) π π = E (r)πχ(m =1) π cos(θ)p 1 n(cos(θ)). For the contribution of the elevation components, d θ;mn = = m 2π dφ 2π = me (r) π dφ π 2π = me (r)πχ(m =1) = E (r)πχ(m =1) me θ (a,θ,φ)p m n (cos(θ)) cos(mφ) } cos(θ)pn(cos(θ)) 1 { E (r) cos(θ) cos(φ)}pn m (cos(θ)) cos(mφ) π cos(φ) cos(mφ)dφ cos(θ)pn m (cos(θ)) π π cos(θ)p 1 n(cos(θ)) cos(θ)p 1 n (cos(θ)). Therefore, d φ;mn d θ;mn = and the δ mn s must also vanish. Table 8 reports the quadrature computations δ mn from Figure 25. These coefficients vanish nearly to the machine ɛ, with the exception of δ

46 Table 8: δ mn estimated from E θ and E φ sampled on the sphere a =3λ. n =1 n =2 n =3 m = m = m = m = Tables 9 and 1 report the magnitude of α mn and γ mn. The questions raised in this section are both answered affirmatively with the numerical quadrature and the analysis. Table 9: α mn estimated from E θ and E φ sampled on the sphere a =3λ n =1 n =2 n =3 m = m = m = m = Table 1: γ mn estimated from E θ and E φ sampled on the sphere a =3λ n =1 n =2 n =3 m = m = m = m =

47 7 A Synthetic δ Field Define a synthetic electric field with δ 12 = 1 as the only non-zero coefficient. The angular components of this electric field are E θ (r, θ, φ) = jωµh (2) 2 (kr) P 2 1 (cos(θ)) cos(φ) sin(θ) E φ (r, θ, φ) = jωµh (2) 2 (kr) dp 2 1 (cos(θ)) sin(φ). For brevity, introduce E (r) :=jωµh (2) 2 (kr) =jωµexp( jkr) { j kr 3 (kr) 2 + using [3, Equation 11.15]. Use [3, Table 12.3 with footnote] P 1 2 (cos(θ)) = 3 cos(θ) sin(θ) j3 }, (kr) 3 to compute The electric fields are dp 1 2 (cos(θ)) =3 6 cos(θ) 2. E θ (r, θ, φ) = 3E (r) cos(θ) cos(φ) E φ (r, θ, φ) = 3E (r){1 2 cos(θ) 2 } sin(φ). 38

48 Figure 26 reports the analysis and synthesis of this electric field at 3 MHz. The upper panels show E θ (r, θ, φ) and E φ (r, θ, φ) sampled on an angular grid at r =3m. The angular grid uses N φ = 73 and N θ = 43 sample points for an angular grid size of φ =5 and θ =4.3. Figure 26: Synthesis of the δ 12 = 1 field. The following tables report the magnitude of Sarkar s coefficients estimated from the sampled electric fields. All coefficients are smaller than 1 3 excepting δ

49 Table 11: α mn estimated from E θ and E φ sampled on the sphere a =3λ for the δ 12 = 1 antenna. n =1 n =2 n =3 m = m = m = m = Table 12: β mn estimated from E θ and E φ sampled on the sphere r =3λ for the δ 12 = 1 antenna. n =1 n =2 n =3 m = m = m = m = Table 13: γ mn estimated from E θ δ 12 = 1 antenna. and E φ sampled on the sphere r =3λ for the n =1 n =2 n =3 m = m = m = m = Table 14: δ mn estimated from E θ and E φ sampled on the sphere r =3λ for the δ 12 = 1 antenna. n =1 n =2 n =3 m = m = m = m =

50 8 Horizontal Electric Dipole Figure 27 shows an electric dipole of length l centered at the origin and oriented along the x-axis. Figure 27: Horizontal Dipole. If the dipole is carrying current i A amps, the near field is { } 1 E θ (r, θ, φ) = E 1 (r) cos(θ) 2 sin(2θ) sin(φ)2 + cos(φ) +E (r) sin(θ) cos(θ) 2 sin(φ) 2 (15) E φ (r, θ, φ) = E 1 (r) cos(θ) sin(φ)(cos(θ) cos(φ) sin(θ)) +E (r) sin(θ) sin(φ)(sin(θ) + cos(φ) cos(θ)), (16) where E (r) := η i A l { j 4π k2 exp( jkr) E 1 (r) := η i A l 2π k2 exp( jkr) kr + 1 (kr) j } 2 (kr) { 3 1 (kr) j 2 (kr) 3 (17) }. (18) 41

51 8.1 Derivation One way to obtain the near field of a horizontal dipole is to rotate the electric field of the Hertzian dipole. The Hertzian dipole has electric fields [7, Section 17.3]: E θ (V ; r, θ, φ) = E (r) sin(θ) E φ (V ; r, θ, φ) = E r (V ; r, θ, φ) = E 1 (r) cos(θ), where the V makes explicit that these fields reference the vertical dipole. right-handed matrix R that rotates ẑ into x while leaving ŷ fixed is 1 R = 1. 1 The If E(H; r, θ, φ) denotes the horizontal dipole field, E(H; r, θ, φ) = RE(V ; r, θ, φ) { } = R E r (V ; r, θ, φ) r + E θ (V ; r, θ, φ) θ = E r (V ; r, θ, φ)r r + E θ (V ; r, θ, φ)r θ { } = E r (V ; r, θ, φ) R r, r r + R r, θ θ + R r, φ φ { } +E θ (V ; r, θ, φ) R θ, r r + R θ, θ θ + R θ, φ φ { } = r E r (V ; r, θ, φ) R r, r + E θ (V ; r, θ, φ) R θ, r { } + θ E r (V ; r, θ, φ) R r, θ + E θ (V ; r, θ, φ) R θ, θ { } + φ E r (V ; r, θ, φ) R r, φ + E θ (V ; r, θ, φ) R θ, φ. The action of the rotation matrix R on the radial vector 1 sin(θ) cos(φ) R r = 1 sin(θ) sin(φ) 1 cos(θ) cos(θ) = sin(θ) sin(φ) sin(θ) cos(φ) = R r, r r + R r, θ θ + R r, φ φ 42

52 has coefficients R r, r = sin(θ) 2 sin(φ) 2 R r, θ = 1 2 sin(2θ) sin(φ)2 + cos(φ) R r, φ = sin(φ)(cos(θ) cos(φ) sin(θ)). The action of the rotation matrix R on the elevation vector 1 cos(θ) cos(φ) R θ = 1 cos(θ) sin(φ) 1 sin(θ) sin(θ) = cos(θ) sin(φ) cos(θ) cos(φ) has coefficients = R θ, r r + R θ, θ θ + R θ, φ φ R θ, r = 1 2 sin(2θ) sin(φ)2 cos(φ) R θ, θ = cos(θ) 2 sin(φ) 2 R θ, φ = sin(φ)(sin(θ) + cos(φ) cos(θ)). The elevation component of the horizonal dipole is E θ (H; r, θ, φ) = E r (V ; r, θ, φ) R r, θ + E θ (V ; r, θ, φ) R θ, { } θ 1 = E r (V ; r, θ, φ) 2 sin(2θ) sin(φ)2 + cos(φ) +E θ (V ; r, θ, φ) cos(θ) 2 sin(φ) 2 { } 1 = E 1 (r) cos(θ) 2 sin(2θ) sin(φ)2 + cos(φ) +E (r) sin(θ) cos(θ) 2 sin(φ) 2. The azimuth component of the horizonal dipole is E φ (H; r, θ, φ) = E r (V ; r, θ, φ) R r, φ + E θ (V ; r, θ, φ) R θ, φ = E r (V ; r, θ, φ) sin(φ)(cos(θ) cos(φ) sin(θ)) +E θ (V ; r, θ, φ) sin(φ)(sin(θ) + cos(φ) cos(θ)) = E 1 (r) cos(θ) sin(φ)(cos(θ) cos(φ) sin(θ)) +E (r) sin(θ) sin(φ)(sin(θ) + cos(φ) cos(θ)). 43

53 8.2 Analysis This section verifies that the β mn s and the γ mn s of Sarkar s expansion vanish. The β mn s vanish because of the orthogonality between cos(φ) and sin(φ) and the basis functions sin(mφ) and cos(mφ), respectively, β mn = kah (2) a n (ka)+h (2) n (ka) (2n + 1)(n m)! 2n(n + 1)(n + m)! ɛ m 2π {b φ;mn + b θ;mn }. Integrating each term separately, b φ;mn = = 2π 2π dφ π dφ π me φ (a,θ,φ)p m n (cos(θ)) cos(mφ) m{ E 1 (r) cos(θ) sin(φ)(cos(θ) cos(φ) sin(θ)) +E (r) sin(θ) sin(φ)(sin(θ) + cos(φ) cos(θ))}p m n (cos(θ)) cos(mφ) = me 1 (a) = +me 1 (a) +me (a) +me (a) 2π 2π 2π 2π sin(φ) cos(mφ)dφ π cos(θ) 2 P m n (cos(θ)) π sin(φ) cos(φ) cos(mφ)dφ cos(θ) sin(θ)pn m (cos(θ)) π sin(φ) cos(mφ)dφ sin(θ) 2 Pn m (cos(θ)) sin(φ) cos(φ) cos(mφ)dφ π sin(θ) cos(θ)p m n (cos(θ)) because the cos(mφ) s are even functions while sin(φ) and sin(φ) cos(φ) are odd functions. 2π π b θ;mn = dφ E θ (a,θ,φ) dp n m(cos(θ)) sin(θ) sin(mφ) 2π π = E (a) cos(φ) sin(mφ)dφ cos(θ) sin(θ) dp n m (cos(θ)) 2π π = E (a) cos(φ) sin(mφ)dφ sin(θ) cos(θ) dp n m (cos(θ)) =. Thus, the β mn s must vanish in a credible quadrature (see Table 16). 44

54 The γ mn s vanish because of the orthogonality between cos(φ) and sin(φ) and the basis functions sin(mφ) and cos(mφ), respectively. γ mn = 1 jωµh (2) n (ka) Integrating each term separately, c φ;mn := = 2π 2π dφ dφ π π (2n + 1)(n m)! 2n(n + 1)(n + m)! {c φ;mn + c θ;mn }. E φ (a,θ,φ) dp m n (cos(θ)) sin(θ) cos(mφ) { E 1 (a) cos(θ) sin(φ)(cos(θ) cos(φ) sin(θ)) +E (a) sin(θ) sin(φ)(sin(θ) + cos(φ) cos(θ))} dp n m(cos(θ)) sin(θ) cos(mφ) 2π π = E 1 (a) sin(φ) cos(mφ)dφ sin(θ) cos(θ) 2 dp n m (cos(θ)) = +E 1 (a) +E (a) +E (a) 2π 2π 2π sin(φ) cos(φ) cos(mφ)dφ π sin(φ) cos(mφ)dφ sin(φ) cos(φ) cos(mφ)dφ π cos(θ) sin(θ) 2 dp n m (cos(θ)) sin(θ) 3 dp n m(cos(θ)) π sin(θ) 2 cos(θ) dp n m (cos(θ)) because the cos(mφ) s are even functions while sin(φ) and sin(φ) cos(φ) are odd functions. 2π π c θ;mn := dφ me θ (a,θ,φ)pn m (cos(θ)) sin(mφ) = 2π dφ π m{e 1 (a) cos(θ){sin(2θ) sin(φ) 2 /2 + cos(φ)} +E (a) sin(θ) cos(θ) 2 sin(φ) 2 }P m n = m 2 E 1(a) = +me 1 (a) +me (a) 2π 2π 2π sin(φ) 2 sin(mφ)dφ cos(φ) sin(mφ)dφ sin(φ) 2 sin(mφ)dφ (cos(θ)) sin(mφ) π π π cos(θ) sin(2θ)p m n (cos(θ)) cos(θ)p m n (cos(θ)) sin(θ) cos(θ) 2 P m n (cos(θ)) because the sin(mφ) s are odd functions while cos(φ) are sin(φ) 2 are even functions over [, 2π]. Thus, the γ mn s must vanish in a credible quadrature (see Table 17). 45

55 Table 15: α mn estimated from E θ and E φ sampled on the sphere a =3λ for the Horizontal Dipole. n =1 n =2 n =3 n =4 n =5 m = m = m = m = m = m = Table 16: β mn estimated from E θ and E φ sampled on the sphere r =3λ for the Horizontal Dipole. n =1 n =2 n =3 n =4 n =5 m = m = m = m = m = m = Table 17: γ mn estimated from E θ Horizontal Dipole. and E φ sampled on the sphere r =3λ for the n =1 n =2 n =3 n =4 n =5 m = m = m = m = m = m =

56 Table 18: δ mn estimated from E θ and E φ sampled on the sphere r =3λ for the Horizontal Dipole. n =1 n =2 n =3 n =4 n =5 m = m = m = m = m = m =

57 8.3 Synthesis The following figures show the synthesis for this antenna. The electric fields are sampled on an angular grid at a radial distance of 3 m. This angular grid is determined by sampling N φ = 73 and N θ = 43 for a angular grid size of φ =5 and θ =4.3. Figure 28 is a 3-D view of the magnitude of the electric fields. The upper panels show E θ (r, θ, φ) and E φ (r, θ, φ) sampled on an angular grid. The lower panels show the corresponding synthesis. Figure 29 displays the same information in two-dimensional images. Figure 28: Horizontal dipole at 3 MHz in 3-D; i A = 1 amp, l = λ/1; upper panels are the simulated fields, lower panels are the synthesized fields. 48

58 Figure 29: Horizontal dipole images at 3 MHz; i A = 1 amp, l = λ/1; upper panels are the simulated fields, lower panels are the synthesized fields. 49

59 8.4 Far Field From Equations 17 and 18, the radial terms behave like E (r) = η i A l 4π k exp( jkr)j r + O[r 2 ] E 1 (r) = O[r 2 ]. Substitution into Equations 15 and 16 gives the behavior of the electric field: E θ (r, θ, φ) = E (r) sin(θ) cos(θ) 2 sin(φ) 2 + O[r 2 ] E φ (r, θ, φ) = E (r) sin(θ) sin(φ)(sin(θ) + cos(φ) cos(θ)) + O[r 2 ]. The magnitude squared of the far field has the expansion E(r, θ, φ) 2 = E θ (r, θ, φ) 2 + E φ (r, θ, φ) 2 + E r (r, θ, φ) 2 = E (r) 2 sin(θ) 2 cos(θ) 4 sin(φ) 4 + E (r) 2 sin(θ) 2 sin(φ) 2 (sin(θ) + cos(φ) cos(θ)) 2 + O[(kr) 3 ] = E (r) 2 sin(θ) 2 sin(φ) { 2 cos(θ) 4 sin(φ) 2 + (sin(θ)+cos(φ) cos(θ)) 2} +O[(kr) 3 ]. The pattern function of the electric field is F (θ, φ) = lim r E(r, θ, φ) r = lim r E (r) sin(θ) sin(φ) r cos(θ) 4 sin(φ) 2 + (sin(θ) + cos(φ) cos(θ)) 2 = η i A lk sin(θ) sin(φ) 4π cos(θ) 4 sin(φ) 2 + (sin(θ) + cos(φ) cos(θ)) 2. Figure 3 plots this far-field pattern. Figure 31 is the approximation to the pattern F (θ, φ) r E θ (r, θ, φ) 2 + E φ (r, θ, φ) 2. at r = 1 m. Figure 32 plots the corresponding approximation to the pattern function using the electric field synthesized by Sarkar s coefficients. The pattern functions are comparable in all three figures. 5

60 Figure 3: Pattern function for the horizontal dipole: ia = 1 amp, ` = λ/1 m. Figure 31: Pattern function approximation at r = 1 m. 51

61 Figure 32: Synthetic pattern function approximation at r = 1 m. 52

62 9 Tilted Dipole Figure 33 is a schematic of a tilted dipole raised over an infinite ground plane. Figure 33: Dipole tilted in the x z-plane and lifted over a perfect ground. This dipole is simulated in Mini-NEC with the following specifications: Length l =8ft Tilted from vertical in the x z-plane by θ T =45 Lifted over the ground by h G =6ft Table 19 lists the Mini-NEC description for completeness, where the dipole is driven at 29 MHz. Figure 34 reports the resulting electric fields measured on a hemisphere with a radius of 8 ft or 24 m. 53

63 Table 19: MiniNEC description. D:\antenna\NearToFarField\TiltedDipole\TiltedDipole1_ :51:19 Tilted Dipole 45 degrees elevation in the x-z plane 8 long 6 ft above the ground GEOMETRY Dimensions in feet Environment: perfect ground wire caps X Y Z radius segs 1 none Number of wires = 1 current nodes = 1 minimum maximum Individual wires wire value wire value segment length segment/radius ratio radius 1 1.E E-3 ELECTRICAL DESCRIPTION Frequencies (MHz) frequency no. of segment length (wavelengths) no. lowest step steps minimum maximum Sources source node sector magnitude phase type voltage 54

64 Figure 34: MiniNEC simulation of the tilted dipole at 29 MHz; electric fields are measured at 8 ft. 55

65 9.1 Synthesis Synthesis requires extending the electric fields from the upper hemisphere to the entire sphere. If E + θ (θ, φ) and E+ φ (θ, φ) denote the simulated electric fields on the upper hemisphere, make the even extension { E + θ (θ, φ) θ π/2 E θ (θ, φ) = (π θ, φ) π/2 θ π and the odd extension E φ (θ, φ) = E + θ { E + φ E + φ (θ, φ) θ π/2. (π θ, φ) π/2 θ π Figures 35 and 36 report the magnitude of the sampled fields. The mesh plot is used to emphasize that fields are sampled on the upper hemisphere with N φ = 73 samples in φ 36 using the azimuth increment φ =5 and N θ = 19 samples in φ 9 using the zenith increment θ =5. Figure 36 verifies that E φ vanishes at the ground plane. Figure 35: MiniNEC simulation of the tilted dipole at 29 MHz; electric fields are measured at 8 ft. Tables 2, 21, 22, and 23 report the coefficients estimated from the sampled fields. The relatively small size of γ mn suggests that these coefficients are zero. Likewise, the β mn and δ mn are relatively small with respect to α mn. Consequently, when the synthesis undertaken, the α terms will dominate. 56

66 Figure 36: MiniNEC simulation of the tilted dipole at 29 MHz; electric fields are measured at 8 ft. Table 2: α mn estimated from E θ and E φ sampled on the sphere at 8 ft for the tilted dipole. n =1 n =2 n =3 m = m = m = m = Table 21: β mn estimated from E θ and E φ sampled on the sphere at 8 ft for the tilted dipole. n =1 n =2 n =3 m = m = m = m =

67 Table 22: γ mn estimated from E θ and E φ sampled on the sphere at 8 ft for the tilted dipole. n =1 n =2 n =3 m = m = m = m = Table 23: δ mn estimated from E θ and E φ sampled on the sphere at 8 ft for the tilted dipole. n =1 n =2 n =3 m = m = m = m =

68 Figure 37 compares the sampled and synthesized fields at 29 MHz on a upper hemisphere with an 8-ft radius. The upper panels are the sampled fields computed from MiniNEC. The lower panels are the synthesized fields computed from Sarkar s expansion with only N = 3 terms and the coefficients α mn, β mn, γ mn, δ mn, estimated from the sampled fields in Figures 35 and 36. These images use the same linear color scale with the magnitude of the fields plotted in V/m. Comparison between the sampled fields from MiniNEC and the synthesized field of Sarkar shows that differences in the magnitudes are relatively small. Figure 37: Tilted dipole synthesis at 29 MHz and the 8-ft radius. The upper panels are the sampled fields computed from MiniNEC. The lower panels are the synthesized fields. 59

69 9.2 Far Field Figure 38 compares the far field computed by MiniNEC to Sarkar s expansion. The upper panels show the magnitude of the electric fields (V/m) computed by MiniNEC at the 5-ft radius. The sampling still uses the 5 increments. The lower panels show the magnitude of the electric fields computed by Sarkar s expansion. That is, the electric fields were sampled at an 8-ft radius, Sarkar s coefficients were estimated from these samples, and these coefficients compute the field at the 5-ft radius. The magnitude of these plots show good agreement. Figure 38: Comparing far-field patterns of the tilted dipole at 29 MHz. (upper panels) sampled fields computed from MiniNEC at the 5-ft radius. (lower panels) synthesized fields computed from the sampled fields at the 8-ft radius and extended to the 5-ft radius. 6

70 1 Shifted Tilted Dipole Figure 33 is a schematic of the tilted dipole from the preceeding section that is shifted along the x-axis. Figure 39: Dipole shifted along the x-axis, tilted in the x z-plane and lifted over a perfect ground. This dipole is simulated in MiniNEC with the following specifications: Length l = 8 feet Tilted from vertical in the x z-plane by θ T =45 Lifted over the ground by h G =6ft Shifted along the x-axis by 1 ft Table 24 lists the MiniNEC description for completeness, where the dipole is driven at 29 MHz. Figure 4 reports the resulting electric fields measured on a hemisphere with a radius of 8 ft or 24 m. At 29 MHz, the wavelength is approximately 1 m so that electric fields are measured on a hemisphere roughly 2.5 wavelengths from the antenna. 61

71 Table 24: MiniNEC description of the shifted, tilted dipole. D:\antenna\NearToFarField\TiltedDipoleShifted\ShftedTiltedDipole_ :5:23 Tilted Dipole 45 degrees elevation in the x-z plane 8 long 6 ft above the ground Shifted 1 ft along the x-axis GEOMETRY Dimensions in feet Environment: perfect ground wire caps X Y Z radius segs 1 none Number of wires = 1 current nodes = 11 minimum maximum Individual wires wire value wire value segment length segment/radius ratio radius 1 1.E E-3 ELECTRICAL DESCRIPTION Frequencies (MHz) frequency no. of segment length (wavelengths) no. lowest step steps minimum maximum Sources source node sector magnitude phase type voltage 62

72 Figure 4: MiniNEC simulation of the shifted, tilted dipole at 29 MHz; electric fields are measured at 8 ft. 63

73 1.1 Synthesis Synthesis requires extending the electric fields from the upper hemisphere to the entire sphere. If E + θ (θ, φ) and E+ φ (θ, φ) denote the simulated electric fields on the upper hemisphere, make the even extension { E + θ (θ, φ) θ π/2 E θ (θ, φ) = (π θ, φ) π/2 θ π and the odd extension E φ (θ, φ) = E + θ { E + φ E + φ (θ, φ) θ π/2. (π θ, φ) π/2 θ π Tables 25, 26, 27, and 28 report the coefficients estimated from E θ and E φ sampled on the upper hemisphere. The hemisphere is sampled using 5 increments. There are N φ = 73 samples in φ 36 azimuth and N θ = 19 samples in φ 9 zenith. The 1-ft offset of the antenna required that the number of terms increase to N = 5 over N = 3 when there is no offset. Table 25: α mn estimated from E θ and E φ sampled on the sphere at 8 ft for the shifted, tilted dipole. n =1 n =2 n =3 n =4 n =5 m = m = m = m = m = m =

74 Table 26: β mn estimated from E θ and E φ sampled on the sphere at 8 ft for the shifted, tilted dipole. n =1 n =2 n =3 n =4 n =5 m = m = m = m = m = m = Table 27: γ mn estimated from E θ and E φ sampled on the sphere at 8 ft for the shifted, tilted dipole. n =1 n =2 n =3 n =4 n =5 m = m = m = m = m = m = Table 28: δ mn estimated from E θ and E φ sampled on the sphere at 8 ft for the shifted, tilted dipole. n =1 n =2 n =3 n =4 n =5 m = m = m = m = m = m =

75 1.2 Far Field Figure 41 compares the MiniNEC fields to Sarkar s synthesized fields at a 5-ft radius. The upper panels are the sampled fields computed from MiniNEC. The lower panels are the synthesized fields computed from Sarkar s expansion using the coefficients α mn, β mn, γ mn, δ mn, estimated from the sampled fields on the upper hemisphere at the 8-ft radius. These images use the same linear color scale with the magnitude of the fields plotted in V/m. Comparison between the sampled fields from MiniNEC and the synthesized field of Sarkar shows that differences in the magnitudes are relatively small. Figure 41: Comparing far-field patterns of the shifted, tilted dipole at 29 MHz. (upper panels) sampled fields computed from MiniNEC at the 5-ft radius. (lower panels) synthesized fields computed at the 8-ft radius and extended to the 5-ft radius. 66

76 11 V Dipole Figure 42 is a schematic of a half-wavelength, V dipole lifted over the ground plane. Figure 42: V dipole in the y z-plane and lifted over a perfect ground. This dipole is simulated in MiniNEC at 3 MHz: Each leg of the V has length of 1/4 m The angle between the legs is θ V =45 The vertex is lifted over the ground by h G =2m Table 29 lists the MiniNEC description for completeness, where the dipole is driven at 3 MHz. Figure 43 reports the resulting electric fields measured on a hemisphere with a radius of 8 ft or 24 m. At 3 MHz, the wavelength is approximately 1 m so that electric fields are measured on a hemisphere roughly 24 wavelengths from the antenna. 67

77 Table 29: MiniNEC description of the V dipole. D:\antenna\NearToFarField\vDipole\vDipole1_3MHz_8feet :46:29 Vee dipole: 45 degrees 2 m above perfect ground GEOMETRY Dimensions in meters Environment: perfect ground wire caps X Y Z radius segs 1 none none Number of wires = 2 current nodes = 9 minimum maximum Individual wires wire value wire value segment length segment/radius ratio radius 1 2.5E E-3 ELECTRICAL DESCRIPTION Frequencies (MHz) frequency no. of segment length (wavelengths) no. lowest step steps minimum maximum Sources source node sector magnitude phase type voltage 68

78 Figure 43: MiniNEC simulation of the V dipole at 3 MHz; electric fields are measured at 8 ft. 69

79 11.1 Synthesis Synthesis requires extending the electric fields from the upper hemisphere to the entire sphere. If E + θ (θ, φ) and E+ φ (θ, φ) denote the simulated electric fields on the upper hemisphere, make the even extension { E + θ (θ, φ) θ π/2 E θ (θ, φ) = (π θ, φ) π/2 θ π and the odd extension E φ (θ, φ) = E + θ { E + φ E + φ (θ, φ) θ π/2. (π θ, φ) π/2 θ π Figure 44 reports Sarkar s coefficients estimated from E θ and E φ sampled on upper hemisphere of an 8-ft sphere. This hemisphere is sampled using 5 increments. There are N φ = 73 samples in azimuth ( φ 36 ) and N θ = 19 samples in zenith ( φ 9 ). Figure 44: Sarkar s coefficients estimated from E θ and E φ sampled on the sphere at 8 ft for the V dipole. The coefficients are plotted in db (i.e., 1 log 1 ( α(m, n) )) so that each decade corresponds to a power of 1. The plot shows contributions from β and γ. The slow roll-off of β implies that convergence will be slow. 7

80 For example, Figure 45 reports on using N = 11 coefficients to synthesize the electric fields at 8 ft. The upper panels are the electric fields computed from Mini- NEC at a radius of 8 ft and are the ground truth. The lower panels are the synthesized electric fields at 8 ft. The synthesis starts by sampling both fields on the 8-ft upper hemisphere at 5 increments, extending both fields to the whole 8- ft sphere, estimating Sarkar s coefficients from the samples, and finally, using the estimated coefficients to synthesize the field at r = 8 ft. The lower panels show that the gross structure of the fields are approximated while the magnitudes needs more terms to fill-in. Figure 45: Near-field synthesis using N = 11, M =3. Figures 46 and 47 show that increasing the number of terms produces synthetic fields with increasing accuracy. The final plot is considered good enough to extend to the far field. 71

81 Figure 46: Near-field synthesis using N = 15, M = 3. Figure 47: Near-field synthesis using N = 19, M = 3. 72

82 11.2 Far Field Figure 48 reports on the far-field synthesis at 5 ft. The upper panels are the electric fields computed from Mini-NEC at 5 ft. The lower panels are the corresponding synthetic field at 5 ft. These synthetic fields were produced by sampling the Mini- NEC fields at r = 8 ft on upper hemisphere at 5 increments, extending both fields to the whole 8-ft sphere, estimating Sarkar s coefficients from these samples, and finally, using these estimated coefficients to synthesize the field at r = 5 ft. The lower panels show that the N = 19 terms produces an approximation that can support a gain computations. Figure 48: V dipole synthesis at r = 5 ft from E θ and E φ sampled on the upper hemisphere at 8 ft. 73

83 11.3 Far-Field Gains The radiation intensity is defined for the far field as [4, Equation 2-12a] U(θ, φ) := r2 { Eθ (θ, φ) 2 + E φ (θ, φ) 2} [Watts]. 2η Figure 49 compares the radiation intensity for the V dipole between the Mini- NEC ground truth (upper panel) and the a synthetic approximation constructed from N = 19 terms (lower panel). The range for both fields is 5 ft. At 3 MHz, this corresponds to 152 wavelengths. The synthetic radiation intensity exhibits discrepancies for increasing zenith or decreasing elevation angle. For zenith of, the synthesis algorithms used in this report set the field to zero Figure 49: V dipole radiation intensity at r = 5 ft from E θ and E φ sampled on the upper hemisphere at 8 ft. 74

84 12 Phase 2 Tasking Phase 1 assessed an implementation of Sarkar s near-to-far transformation [12] on analytic antenna models and Mini-NEC antenna simulations. For both classes of antenna models, Phase 1 verified the implementation recovers the far field from the near field provided the spherical expansion includes a sufficient number of terms. That is, the truncation error governs the near-to-far transform rather than the discretization error assuming noise-free antenna models (see the discussion for Equation 2). Phase 2 applies Sarkar s near-to-far transformation to measurements taken at SSC Pacific s Antenna Pattern Range. These electric field measurements are degraded by additive noise, in-band RF interference, and the mechanical and electrical variations in the Antenna Pattern Range. Therefore, an initial task for Phase 2 is to assess the robustness of the near-to-far transformation with respect to the noise-corrupted measurements. Figure 5 shows the candidate reference antenna placed on the Antenna Pattern Range. This antenna consists of a monopole and a loop above the ground plane. Figure 5: Reference antenna in the SSC Pacific Antenna Pattern Range. Because this antenna is amenable to modeling, comparison of the ideal model against the measured antenna is possible and will produce a noise model for the spherical coefficients. Although the near-to-far transformation is not expected to amplify the noise in the far field, the accuracy of the spherical coefficients may degrade the asymptotic estimates of the truncation error. The noise model can set a truncation limit increasing the number terms in the expansion may not improve the accuracy because the higher-order coefficients are lost in the noise. 75

85 Classical statistical smoothing is a standard approach to reduce additive noise but will not mitigate system measurement errors. Assessing the payoff for this smoothing is a necessary Phase 2 task. A more promising approach uses model identification to mitigate noise. In this approach, an antenna model is fitted to the noisy measured near field. For example, measurements of the candidate monopole may be fitted by varying antenna length, radius, orientation, and material. Once a model is determined that best explains the measurements, this model is then used to compute a noise-free far field and extrapolate to other frequencies. 76

86 References [1] Abramowitz, Milton and Irene A. Stegun [1972] Handbook of Mathematical Functions, Dover Publications, Inc., New York, NY. [2] Andrews, Larry C. [1985] Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York, NY. [3] Arfken [197] Mathematical Methods for Physicists, Academic Press, Inc., New York, NY. [4] Balainis, Constantine A. [1997] Antenna Theory, second edition, John Wiley & Sons, Inc., New York, NY. [5] Corson, Dale R. and Paul Lorrain [1962] Introduction to Electromagnetic Waves, W. H. Freeman and Company, San Francisco, CA. [6] Rockway J., J. Meloling, J. C. Allen [211] Interpolating Spherical Harmonics for Computing Antenna Patterns, SPAWAR Technical Report [7] Edminister, Joseph A. [1993] Electromagnetics, second edition, Schaum s Outline Series, McGraw-Hill, New York, NY. [8] Francis, Michael H. ; Ronald C. Wittmann; Jin-Seob Kang [25] Uncertainties in Spherical Near-Field Antenna Measurements, Proceedings of ISAP25, Seoul, Korea. [9] [1] Jackson, J. D. [1975] Classical Electrodynamics, 2nd edition, Wiley. [11] Marie, F.; L. Bertel; Y. Erhel; D. Lemur; M. Oger [25] Beam Forming Techniques Operating on HF Collocated Antennas. [12] Sarkar, T.K., P. Petre, A. Taaghol, R.F. Harrington [1997] An Alternative Spherical Near Field to Far Field Transformation, Progress in Electromagnetics Research, PIER 16, pages [13] Stoer, J. and R. Bulirsch [198] Introduction to Numerical Analysis, Springer- Verlag, New York, NY. 77

87 REPORT DOCUMENTATION PAGE Form Approved OMB No The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden to Department of Defense, Washington Headquarters Services Directorate for Information Operations and Reports (74-188), 1215 Jefferson Davis Highway, Suite 124, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (From - To) September 214 Final 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER A Near-to-Far Transformation using Spherical Expansions Phase 1: Verification on Simulated Antennas 6. AUTHORS J. H. Meloling J.C. Allen 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER SSC Pacific, 5356 Hull Street, San Diego, CA TR SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) Office of the Director of National Intelligence Intelligence Advanced Research Projects Activity Washington, DC SPONSOR/MONITOR S ACRONYM(S) 11. SPONSOR/MONITOR S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release. 13. SUPPLEMENTARY NOTES This is work of the United States Government and therefore is not copyrighted. This work may be copied and disseminated without restriction ABSTRACT Tapan Sarkar wrote a substantial technical paper on a transformation mapping an antenna s near-field into its far field. A numerical implementation of Sarkar s near-to-far transformation is assessed in this report. This report assesses this near-to-far implementation on the following antenna models using only a finite number of near-field samples: Analytic antenna models where the near field, far field, and the spherical expansions are computable in closed form, Mini-NEC antenna simulations where the near field, far fields and spherical expansions are numerically computed. The implementation described in this report demonstrates that an antenna s far field can be extracted from sampled near field measurements provided the spherical expansion includes a sufficient number of terms and that the near-field samples are sufficiently dense. Phase II of this effort extends the implementation to estimate the number of spherical coefficients necessary to deliver user-specified accuracy in the far field and how noisy the near-field measurements affect the far field approximation. 15. SUBJECT TERMS Mission Area: Communications near-to-far field transformations Sarkar s near-to-far transformations Hertzian electric dipole small loop antenna horizontal electric dipole tilted dipole shifted tilted dipole V dipole 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE ABSTRACT OF J. H. Meloling PAGES 19B. TELEPHONE NUMBER (Include area code) U U U U 8 (619) Standard Form 298 (Rev. 8/98) Prescribed by ANSI Std. Z39.18

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