Rančić, Milica

Rančić, Predrag D.

Digitale Bibliothek Thüringen

  PROCEEDINGS 11-15 September 2006      FACULTY OF ELECTRICAL ENGINEERING AND INFORMATION SCIENCE    INFORMATION TECHNOLOGY AND ELECTRICAL ENGINEERING - DEVICES AND SYSTEMS, MATERIALS AND TECHNOLOGIES FOR THE FUTURE      Startseite / Index: http://www.db-thueringen.de/servlets/DocumentServlet?id=12391 51. IWK Internationales Wissenschaftliches Kolloquium International Scientific Colloquium Impressum  Herausgeber: Der Rektor der Technischen Universität llmenau  Univ.-Prof. Dr. rer. nat. habil. Peter Scharff  Redaktion: Referat Marketing und Studentische Angelegenheiten  Andrea Schneider   Fakultät für Elektrotechnik und Informationstechnik  Susanne Jakob  Dipl.-Ing. Helge Drumm  Redaktionsschluss: 07. Juli 2006  Technische Realisierung (CD-Rom-Ausgabe):  Institut für Medientechnik an der TU Ilmenau  Dipl.-Ing. Christian Weigel  Dipl.-Ing. Marco Albrecht  Dipl.-Ing. Helge Drumm  Technische Realisierung (Online-Ausgabe):  Universitätsbibliothek Ilmenau    Postfach 10 05 65  98684 Ilmenau  Verlag:   Verlag ISLE, Betriebsstätte des ISLE e.V.  Werner-von-Siemens-Str. 16  98693 llrnenau   © Technische Universität llmenau (Thür.) 2006  Diese Publikationen und alle in ihr enthaltenen Beiträge und Abbildungen sind urheberrechtlich geschützt. Mit Ausnahme der gesetzlich zugelassenen Fälle ist eine Verwertung ohne Einwilligung der Redaktion strafbar.   ISBN (Druckausgabe): 3-938843-15-2 ISBN (CD-Rom-Ausgabe): 3-938843-16-0  Startseite / Index: http://www.db-thueringen.de/servlets/DocumentServlet?id=12391  51st Internationales Wissenschaftliches Kolloquium Technische Universität Ilmenau  September 11 – 15, 2006  M. Rančić, P. Rančić   EM Field of Vertical Hertz's Dipole (VHD) Placed above a Lossy Half-space: Approximate Expression for the Sommerfeld's Integral   INTRODUCTION   The electromagnetic field in the air in the surroundings of the vertical Hertz's dipole (VHD) placed above linear, isotropic and homogenous lossy half-space is determined in this paper. The influence of the electrical parameters of the lossy half-space on the EM field components is described by the Sommerfeld's integral kernel that, in numerical sense, presents a very complex problem. A very simple and accurate mathematical model, which is not limited by the values of electrical parameters of the lossy half-space or the VHD location, is proposed for solution of this numerically complex problem. General theoretical solution for the EM field, which was proposed by Sommerfeld in 1909, [1], was formulated by a class of semi-infinite integrals, in literature often referred to as integrals of Sommerfeld's type, or Sommerfeld's integral kernel (SIK). Accurate numerical evaluation of integrals of this type demands very complex numerical procedure that depends on the values of the electrical parameters of the lossy half-space 1σ , r1 0 1ε = ε ε  and 1 0µ = µ  ( 1 [0, )σ ∈ ∞  - conductivity, r1 [1,81]ε ∈  - relative permittivity), and the location of the VHD above a lossy half-space, kz [0, )′ ∈ ∞ . Thus, a great number of models for approximate evaluation of the SIK were proposed in order to solve this complex problem in a simpler way. However, the majority of these approximations were developed under certain assumptions, such as limited values of electrical parameters for which it is valid. Authors of this paper have also proposed a number of simple models that are, besides accurate, general in a sense that they are not limited in any way by the values of the electrical parameters of the lossy half-space.  The SIK model that is presented in this paper is similar to the one proposed by the same authors in [3, 4] and the one used in [5] for evaluation of the input impedance of the vertical asymmetrical wire dipole antenna placed above a lossy half-space. In this paper, a SIK model, which represents arithmetical mean of the models from [3], [4] and [5] is proposed.   BRIEF THEORETICAL DESCRIPTION   Problem layout  The VHD is placed in the air ( 0 0σ = , 0ε , 0µ ) at height kz′  above a homogenous, isotropic lossy half-space of known electrical parameters ( 1σ , 1 1 0rε = ε ε , 1 0µ = µ ). Following labels are also used: iσ = i ijσ + ωε , 0,1i =  - complex conductivity, iγ =  1/ 2( )i i i ij j= α + β = ωµ σ , 0,1i =  - complex propagation constant, 1rε = 1 1r ijε − ε =  1 1 060r j= ε − σ λ  - complex relative permittivity, rn 1/ 210 1( )= ε  - complex refraction coefficient, 0λ - wave length in the air, and 2 fω = π - angular frequency.   In the described case of the VHD, the Sommerfeld's integral kernel (SIK), which describes the field at point ≥0( , , 0)M x y z  scattered from the lossy half-space, is given by the following expression:    d00 2 10 0 20( ) ( ) ( , )v k z kS r R K r∞α== α ⋅ α α∫   , z n u uR n u u210 0 110 210 0 1( )( )( )−α = + , (1) where: z zR R u10 10 0( ) ( )α =   - spectral reflection coefficient (SRC), 2 2 , i=0, 1i iu = α + γ , and 0 2( , )kK rα  - spectral form of the standard potential kernel from the image in the flat mirror, i.e.   d dk kr u z zk kkK r K r Jr u20 0( )0 2 0 2 02 00 0e e( ) ( , ) ( )−γ ∞ ∞ ′− +α= α== = α α = α αρ α∫ ∫ , (2) Fig.1. Schematic illustration of the VHD  placed above a lossy half-space.  where J0( )αρ  is the zero-th Bessel function of the first kind, k kr z z2 22 ( )′= ρ + +  - distance from the image in the flat mirror to the observed field point, and ρ  - radial distance, ρ = +2 2x y .   Approximate solution for the SIK  In the papers [3] and [5], authors have proposed two similar models for the approximation of the SIK. Firstly, the SRC is assumed in a form of a rational function with two unknown constants, which are determined by matching the SCR at characteristic points. Based on these models, the approximate form for the SRC proposed in this paper has the following form:  0000010 //)''(5.0)(~ uABuAABuRz γ+=γ+′+≅ , (3) where B , A′  and A ′′  are unknown constants, obtained by matching 10 0( )zR u  at points 0u → ∞ , 0 0u = γ  and 2 1/ 20 100 /( 1)u n= γ + , respectively. Their values are as follows:  B n n2 210 10( 1) /( 1)= − + , BnnA −+−=′ )1(/)1( 1010 , 1/ 210 +−=′′ nBA . (4) Substituting (4) into (1), the following approximate SIK model is obtained:  )()()()''(5.0)()( 20202020200 kkkkkv rLArBKrLAArBKrS γ+=γ+′+≅ , (5) where 2( )kL r  is the new integral kernel,  ( ) d dkkz zk kv kvv=z+z vL r K r v K r v N j J2 0 2 0 2 0 0 0 00( ) ( ) [ ( ) ( )]2′+∞′ =π= = − − β ρ + β ρ∫ ∫ . (6a) In some practical cases when ′+ >> ρkz z  (e.g. [9]), the new integral kernel kL r2( )  can be very accurately numerically solved using the Li  integrals, = −Li X Ci X jSi X( ) ( ) ( )  ([10]). Approximate expression for these calculations is  ( ) − ≅ − + −   kjRk k kk kR R eL r j Li R jRR R22 22 2 22 2( ) 1 ( ) 14 4, (6b) where: = β ρR 0 , = βk kR r2 0 2  and when the condition << kR R2  is satisfied.   The Hertz’s vector and electrical scalar potential  The Hertz’s vector potential, z z0 0 ˆΠ = ΠJJG, in the case of the VHD placed above a lossy medium is given, considering (5), by the following expression:  ( ) ( ) ( )vz zz k 2k k k kp pK r S r K r BK r A L r0 00 0 1 00 0 1 0 2 200 0( ) ( )4 4   Π = + ≅ + + γ  πσ πσ , (7) where: zp 0  - moment of the Hertz's dipole, and k kr z z2 2 1/ 21 [ ( ) ]′= ρ + −  - distance from the VHD to the observed field point.  The electric scalar potential in the air in the vicinity of the VHD is as follows:  ( )z z k k kp K r BK r A K rz z0 000 0 1 0 2 0 200div ( ) ( )4∂Π ∂  ϕ = − Π = − = − + − γ  ∂ πσ ∂ JJG. (8)   EM field structure  The electric, E r 20 00 0( ) grad= − ϕ − γ ΠJG G JJG, and magnetic field, H r0 00( ) rot= σ ΠJG G JJG, in the vicinity of the VHD can be determined using the following definition expressions:  ∂ Π= ∂ ∂zxE x z200 , ∂ Π= ∂ ∂zyE y z200 , ∂ Π= − γ Π∂zz zEz2200 002 , (9)  ∂Π= σ ∂zxH y00 0 , ∂Π= −σ ∂zyH x00 0 , =zH 0 0 . (10) Complete form for the EM field components is obtained substituting the approximate solution for the Hertz's vector potential given by (7) into (9) and (10).   NUMERICAL RESULTS   Two groups of numerical results will be presented in this section: • The first one corresponds to numerical experiments that illustrate the accuracy of the proposed model for SIK calculation; and • The second one illustrates changes of EM field components on the ground surface in the air, versus y-, or ρ - coordinate for =x 0 . The refraction coefficient n10  and position of the VHD ′kz  are taken as parameters in these calculations. Sommerfeld’s integral kernel. The validity of the proposed SIK model was investigated through a number of numerical experiments, and the obtained results are compared to the accurate ones from [6].  Based on the approximate SIK model given by (5), modulus of the normalized SIK, β00 2 0( ) /v kS r , was determined for different values of the parameters on which it depends. Modulus of the normalized SIK versus normalized radial distance β ρ0( )Log , is shown in Figs. 2a, b and c. Position of the VHD ′kz , is taken as a parameter, relative permittivity is ε =1 2r , and Figs. 2a, b and c correspond to the following values of the normalized conductivity −σ λ = 41 0 10 S , σ λ =1 0 0.041667S  and σ λ =1 0 10S , respectively. The results obtained applying the proposed SIK model (solid line) are compared to the ones obtained by accurate calculations from [6] (solid down triangle). For another group of electrical parameters, i.e. ε =1 10r , and −σ λ = 41 0 10 S , σ λ =1 0 0.175S  and σ λ =1 0 10S , modulus of the normalized SIK is presented in Figs. 3a, b and c. The results obtained applying the proposed model when constant A  takes value = 0A , are also given in the same figures (open up triangle).  As it can be seen from both Figs.2 and 3, the results obtained applying the proposed model for the SIK are in very good accordance with the ones of accurate calculations, regardless of the electrical parameters of the lossy half-space or the VHD position. In that sense, the proposed model can be, besides simple, characterized as general and accurate. EM field components. Based on the expressions for the electric field components (9), and (10) that describe the magnetic ones, a number of numerical calculations were performed, and the obtained results are presented further in this section. The values for the normalized z- component of the electrical field, βzE 20 0/  and Am10 =zp , versus normalized radial distance β ρ0( )Log , are shown in Figs.4a-c. Position of the VHD ′kz  is taken as a parameter, relative permittivity is ε =1 2r , and Figs.4a-c correspond to the values of the normalized conductivity −σ λ = 41 0 10 S , σ λ =1 0 0.041667S  and σ λ =1 0 10S , respectively. Normalized z- component of the electric field versus normalized radial distance are also shown in Figs.5a-c, but for different value of the relative permittivity ε =1 10r , taking the     Fig.2. Modulus of the normalized SIK versus normalized radial distance. Relative permittivity is ε =1 2r , and VHD position ′kz  and ε = σ λ1 1 060i  are parameters.    Fig.3. Modulus of the normalized SIK versus normalized radial distance. Relative permittivity is ε =1 10r , and VHD position ′kz  and ε = σ λ1 1 060i  are parameters.    Fig.4. Modulus of the z- component of the electrical field versus normalized radial distance. Relative permittivity is ε =1 2r , and VHD position ′kz  and ε = σ λ1 1 060i  are parameters.   Fig.5. Modulus of the z- component of the electrical field versus normalized radial distance. Relative permittivity is ε =1 10r , and VHD position ′kz  and ε = σ λ1 1 060i  are parameters.            Fig.6. Modulus of the x- component of the magnetic field versus normalized radial distance. Relative permittivity is ε =1 2r  (6a-c) and ε =1 10r  (6d-f), and VHD position ′kz  and ε = σ λ1 1 060i  are parameters.  position of the VHD as parameter. Figs. 5a-c correspond to the following values of the normalized conductivity −σ λ = 41 0 10 S , σ λ =1 0 0.175S  and σ λ =1 0 10S , respectively.  The results obtained for the x- component of the normalized magnetic field, βxH 20 0/  and Am10 =zp , (for =x 0 , =yH 0 0 ), versus normalized radial distance are presented in Figs.6a-f, taking the position of the of the VHD as parameter. Values shown in Figs.6a-c correspond to relative permittivity ε =1 2r , and normalized conductivity −σ λ = 41 0 10 S , σ λ =1 0 0.041667S  and σ λ =1 0 10S , respectively. The results in  Figs.6d-f correspond to the lossy half-space of relative permittivity ε =1 10r  and normalized conductivity −σ λ = 41 0 10 S , σ λ =1 0 0.175S  and σ λ =1 0 10S , respectively. Based on the presented results, it can be concluded that the greatest changes of the values of the EM field components ( 0zE  and 0xH ) occur for the normalized radial distances up to β ρ ≅0 1, where the proposed SIK model has performed most accurately. It should also be noted that yE 0 - component of the electric field, so-called surface wave, exists and varies for different parameters of the lossy half-space, which is not the case when the ground is treated as an ideally conducting medium. Namely, based on the previous experience, the application of the proposed approximate SIK model as a function of normalized conductivity 01λσ , i.e. 011 60 λσ=ε i , gives numerical results that correspond to the ones of the case of an ideally conducting ground, for all characte-ristics of the EM field already for S1001 >λσ . Numerical values for the surface wave ( 0yE  component) are significant, i.e. it vanishes for much greater values of the norma-lized conductivity S10501 >λσ . Therefore, the analysis of the surface wave in the frame of the proposed SIK model demands wider elaboration as well as much more space. Thus, the validity analysis of the proposed approximate models for the SIK will be the subject of further work.   CONCLUSION   Simple and satisfyingly accurate model for numerical calculation of the SIK (Sommerfeld's integral kernel, [1]) is proposed in this paper. This model is one of many simple ones proposed by signed authors ([2], [3], [4], [5], [8]). Based on theoretical and numerical results presented in the paper, it can be concluded that the proposed model is reliable, simple, satisfyingly accurate and general, and it can be used for calculating potentials ( Πz0  and ϕ0 ) and EM field components (JGE0  and JGH0 ) in the surroundings of the VHD in the air. It should be noted that these quantities are very accurately modelled at points on the ground surface. This is emphasized because the Hertz's vector in the ground, Π = ΠJJG z z1 1 ˆ , ≤z 0 , can also be approximately, but still very accurately, presented in a following form: Π ≅ Π = ⋅ ≤z z x y z f n z1 0 1 10( , , 0) ( , 0) , where Πz0  is evaluated according to (7), and f1 is new exponential function that depends on electrical parameters of the ground and z- coordinate of the observed point in the ground. Thus, the EM field in the ground can be also modelled in a very simple and accurate manner. The proposed model can be also very successfully used for modelling EM field structure of atmospheric discharge in frequency domain in the range ∈f (0,10MHz] . This presents basis for evaluation of the Lightning EM field in time domain, at points in the surroundings of the Lightning channel using the FFT (Fast Fourier Transform). For this purpose, the model from [3] was successfully used in preliminary researches in [9].  References: [1] Sommerfeld, A.N., Über die Ausbreitung der Wellen in der dratloslen Telegraphie. Ann. der Phzsik 28, pp. 665-736, 1909. [2] Rančić, P.D., Kitanović, M.I., A New Model for Analysis of Vertical Asymmetrical Linear Antenna Above a Lossy Half-Space, Int. J. Electron. Commun. AEÜ, Vol. 51, No. 3, pp. 155-162, May 1997. [3] Rančić, M.P., Rančić, P.D., Vertical Linear Antennas in the Presence of a Lossy Half-Space: An Improved Approximate Model, Int. J. Electron. Comun. AEÜ, Vol. 60, No. 5, pp. 376-386, May 2006. [4] Rančić M.P., Rančić P.D., Vertical Hertz's Dipole above a Lossy Half-Space: Review of Simple Models for the Sommerfeld's Integral, Proc. of the XLIX ETRAN’05, Vol. II, pp. 272-275, Budva, S&M, 2005. (the best paper in the AP session - in Serbian) [5] Rančić, M.P., Rančić, P.D., An Approximate Model for Analyzing Vertical Antennas above a Ground Plane, Proc. of the XII TELFOR 2004, Nov. 23-25, 2004, Belgrade, S&M. http://www.telfor.org.yu/telfor2004/radovi/PEL-9-28.pdf [6] Petrović, V.V., Private communications, ETF Belgrade, 2005. [7] Djordjević, A.R., Baždar, M.B., Petrović, V.V., Olćan, D.I., Sarkar, T.K., Harrington, R.F., AWAS for Windows, Version 2.0, Analysis of Wire Antennas and Scatterers – Software and User's Manual, Artech House, 2002. [8] Rančić, M.P., Rančić, P.D., An approximation of the Sommerfeld’s integral kernel in EM field analysis of the VHD, L ETRAN'06, Belgrade, Serbia, 2006. (in Serbian) [9] Javor V., Rančić, P.D., One Model for Calculating the Lightning Generated Electromagnetic Field above Real Ground, 51th IWK, Ilemnau, Germany, 2006. (accepted for presentation) [10] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, INC., New York, 1970.  Authors:    Dipl.ing Milica Rančić   Prof. dr Predrag Rančić Faculty of Electrical Engineering, P.O.Box 73 18000, Niš, Serbia Phone: +381 18 529 423 Fax: +381 18 588 399 E-mail: milica@elfak.ni.ac.yu, prancic@elfak.ni.ac.yu 

EM field of vertical Hertz's dipole (VHD) placed above a lossy half-space: approximate expression for the Sommerfeld's integral

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EM field of vertical Hertz's dipole (VHD) placed above a lossy half-space: approximate expression for the Sommerfeld's integral

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