Antenna using Galerkin-Bubnov Indirect

Transcription

1 Frequency Domain Analysis of GPR Dipole B E = t H = J + D = ρ B = 0 Antenna using Galerkin-Bubnov Indirect D t Boundary Element Method To be presented by Dragan Poljak Croatia

2 CONTENTS Introduction to Computational Electromagnetics (CEM) and Electromagnetic Compatibility (EMC) Frequency Domain Analysis of Wire Antennas Stochastic Modeling Computational Examples

3 Introduction to Computational Electromagnetics (CEM) and Electromagnetic Compatibility (EMC) Historical note on modeling in electromagnetics Electromagnetics as a rigorous theory started when James Clerk Maxwell derived his celebrated four equations and published this work in the famous treatise in In addition to Maxwell s equations themselves, relating the behaviour of EM fields and sources we need: the constitutive relations of the medium the imposed boundary conditions of the physical problem of interest.

4 Introduction to Computational Electromagnetics (CEM) and Electromagnetic Compatibility (EMC) Historical note on modeling in electromagnetics One of the first digital computer solution of the Pocklington s equation was reported in This was followed by the one of the first implementations of the Finite Difference Method (FDM) to the solution of partial differential equations in 1966 and time domain integral equation formulations in 1968 and Through 1970s the Finite Element Method (FEM) became widely used in almost all areas of applied EM applications. The Boundary Element Method (BEM) developed in the late seventies for the purposes of civil and mechanical engineering started to be used in electromagnetics in 1980s.

5 Introduction to Computational Electromagnetics (CEM) and Electromagnetic Compatibility (EMC) EMC computational models and solution methods A basic EMC model, includes EMI source (any kind of undesired EMP), coupling path which is related to EM fields propagating in free space, material medium or conductors, and, finally, EMI victim - any kind of electrical equipment, medical electronic equipment (e.g. pacemaker), or even the human body itself. EMI source Coupling path EMI victim A basic EMC model

6 Introduction to Computational Electromagnetics (CEM) and Electromagnetic Compatibility (EMC) EMC computational models and solution methods In principle, all EMC models arise from the rigorous EM theory concepts and foundations based on Maxwell equations. EMC models are analysed using either analytical or numerical methods. Analytical models are not useful for accurate simulation of electric systems, or their use is restricted to the solution of rather simplified geometries. More accurate simulation of various practical engineering problems is possible by the use of numerical methods.

7 Classification of EMC models Introduction to Computational Electromagnetics (CEM) and Electromagnetic Compatibility (EMC) Regarding underlying theoretical background EMC models can be classified as: circuit theory models featuring the concentrated electrical parameters transmission line models using distributed parameters in which low frequency electromagnetic field coupling are taken into account models based on the full-wave approach taking into account radiation effects for the treatment of electromagnetic wave propagation problems

8 Introduction to Computational Electromagnetics (CEM) and Electromagnetic Compatibility (EMC) Summary remarks on EMC modeling The main limits to EMC modeling arise from the physical complexity of the considered electric system. Sometimes even the electrical properties of the system are too difficult to determine, or the number of independent parameters necessary for building a valid EMC model is too large for a practical computer code to handle.

9 Introduction to Computational Electromagnetics (CEM) and Electromagnetic Compatibility (EMC) Summary remarks on EMC modeling The advanced EMC modeling approach is based on integral equation formulations in the FD and TD and related BEM solution featuring the direct and indirect approach, respectively. This approach is preferred over a partial differential equation formulations and related numerical methods of solution, as the integral equation approach is based on the corresponding fundamental solution of the linear operator and, therefore, provides more accurate results. This higher accuracy level is paid with more complex formulation, than it is required within the framework of the partial differential equation approach, and related computational cost.

10 Frequency domain analysis of wire antennas In addition to antenna design the model of horizontal wires above lossy half-space has numerous applications in (EMC) in the analysis of aboveground lines and cables. The current distribution along the multiple wire structure is governed by the set of Pocklington equation for half-space problems. The influence of lossy half-space can be taken into account via the reflection coefficient (RC) approximation.

11 FD analysis of wire antennas The geometry of interest consists of M parallel straight wires horizontally placed above a lossy ground at height h. The geometry of the problem All wires are assumed to have same radius a and the length of the m-th wire is equal L m.

12 FD analysis of wire antennas The analysis starts by considering a single straight wire above a dissipative half-space. Horizontal antenna over imperfect ground

13 FD analysis of wire antennas The integral equation can be derived by enforcing the interface conditions for the E-field at the wire surface: exc sct e E + E = x ( ) 0 The excitation represents the sum of the incident field and field reflected from the lossy ground: E = E + E exc inc ref The scattered field can be written as: = ω ϕ sct E j A where A is the magnetic vector potential and φis the scalar potential. According to the thin wire approximation (TWA) only the axial component of the magnetic potential differs from zero: sct ϕ Ex = jω Ax L L µ 1 x Ax = I ( x ') g( x, x ') dx ' 4π ϕ( x) = q( x ') g( x, x ') dx ' 4πε 0 0 while q(x) is the charge distribution and I(x ) is the induced current along the wire.

14 FD analysis of wire antennas Green function g(x,x ) is given by: g( x, x ') = g ( x, x ') R g ( x, x ') 0 where g 0 (x, x ) is the free space-green function and g i (x, x ) arises from the image theory: TM i g o ( x x' ) jk o o R e, = gi ( x, x' ) R o = e jk R o i R i R o and Ri, respectively, is the distance from the source to the observation point, and the reflection coefficient is R TM = ncos ncos 2 Θ n sin Θ 2 r Θ + n sin Θ ωε0 n = ε j σ Θ = x x' arctg 2 h

15 FD analysis of wire antennas The linear charge density and the current distribution along the line are related through the equation of continuity: q = 1 di jω dx After mathematical manipulation it follows: L 1 I ( x ') ϕ( x) = g( x, x ') dx ' j4 πωε x ' 0 leading to the following integral relationship for the scattered field: L L sct µ 1 I ( x') Ex = jω I( x') g( x, x ') dx ' g( x, x') dx ' 4 π + j4 πωε x x ' 0 0

16 FD analysis of wire antennas Combining previous equations results in the following integral equation for the current distribution induced along the wire: 0 0 ( ') L L exc µ 1 I x Ex = jω I ( x ') g ( x, x ') dx ' g ( x, x ') dx ' 4 π j4 πωε x x ' This equation is well-known in antenna theory representing one of the most commonly used variants of the Pocklington s integro-differential equation for half space problems. This integro-differential equation is particularly attractive for numerical modeling, as there is no second-order differential operator under the integral sign.

17 FD analysis of wire antennas The electric field components are: 1 I ( x ') g( x, x ') Ex = dx + k g x x I x dx j4 x ' x ' L L 2 ' (, ') ( ') ' πωε 0 L L E z = 1 I ( x ') g( x ', z) j4 πωε x ' z 0 L L dx ' E y = L 1 I ( x ') g( x ', y) j4 πωε x ' y 0 L dx '

18 FD analysis of wire antennas Vertical wire above a real ground

19 FD analysis of wire antennas Integro-differential equation for vertical wire

20 FD analysis of wire antennas Vertical wire penetrating the ground

21 FD analysis of wire antennas Integro-differential equation for vertical penetrating the ground

22 FD analysis of wire antennas Integro-differential equation for vertical penetrating the ground

23 FD analysis of wire antennas Integro-differential equation for vertical penetrating the ground

24 FD analysis of wire antennas An extension to the wire array is straightforward and results in the the set of coupled Pocklington integral equations: M Ln / 2 2 exc 1 2 x = 2 1 [ 0mn TM imn ] n j4πωε + 0 n= 1 L / 2 x E k g ( x, x ) R g ( x, x ) I ( x ) dx m = 1,2,... M n where I n (x ) is the unknown current distribution induced on the n- th wire axis, g 0mn (x,x`) is the free space Green function, while g imn (x,x`) arises from the image theory: g 0mn ( x, x ) = e jk1r1 mn R 1mn g 0mn ( x, x ) = e jk1r1 mn R 1mn

25 FD analysis of wire antennas The wires are excited by a plane wave of arbitrary incidence z u E T E u H T E y u E T M r E T M r E T E r H T M x n r r H T E n r Reflect. wave Incident wave n u u H T M n u θ φ θ t Plane incidence of t H T M t E T M n t t E T E t H T E Transmitt. wave n t Incident, reflected and transmitted wave

26 FD analysis of wire antennas The tangential component of an incident plane wave can be represented in terms of its vertical E V and horizontal E H component: E = E + E = exc i r x x x E 0 (sinα sinφ cosα cosθ cos φ) 0 jk1ni r e + ( sinα sinφ cosα cosθ cosφ ) + E R + R e TE TM jk n r where α is an angle between E-field vector and the plane of incidence. R TM and R TE are the vertical and horizontal Fresnel reflection coefficients at the air-earth interface given by: R TM = n cosθ n sin n cosθ + n sin 2 2 θ θ R TE = cosθ n sin cosθ + n sin 2 2 θ θ ni r = x sinθ cosφ y sinθ sinφ z cosθ n r = x sinθ cosφ y sinθ sinφ + z cosθ r 1 r

27 FD analysis of wire antennas The E-field components are given, as follows: M Ln L 1 n I n( x') G nm( x, x') 2 Ex = dx' k Gnm ( x, x') In( x') dx' j4 πωε 0 n = + 1 x ' x ' Ln L n E y = L M n 1 In ( x ') Gnm ( x, x ') j4 x ' y πωε 0 n= 1 L n dx ' E z = 1 M L πωε 0 n= 1 L n I ( x ') G ( x, x ') j4 x ' z n n nm dx ' where m=1, 2,, M and Green function G is given by: G ( x, x ') = g ( x, x ') R g ( x, x ') nm 0nm TM inm

28 FD analysis of wire antennas BEM solution of Pocklington equation system x x ' x ' x I( x ') = I1 i + I2i x x The BEM procedure starts, as follows: 2i 1 Performing certain mathematical manipulations and BEM discretisation results in the following matrix equation: N e - the total number of elements [ Z ] { I} = { V} [Z] pk - the interaction matrix: 2 [ ] = { } { } ji + { } { } e T T N e k = 1 pk k p=1,2,,m Z D D ' g ( x, x ') dx ' dx k f f ' g ( x, x ') dx ' dx pk p k l k l l l l p k p k Vectors {f}and {f } contain shape functions f n (x) and f n (x ), while {D} and {D } contain their derivatives. The vector {V} p represents the voltage along the segment: inc { } 4 πωε ( ){ } p ji = 0 l V j E x f dx p x i p p

29 FD analysis of wire antennas The BEM field calculation Applying the BEM formalism to field expressions it follows: x M N x j 1, 1, 1 i+ n i+ n Ii+ 1, n Ii, n Gnm ( x, x ') 2 Ex = dx ' + k Gnm ( x, x ') Iin ( x ') dx ' ; j4 πωε 0 n= 1 i= 1 x x x ' i, n x i, n m = 1,2,..., M 0 N x i+ 1, n i, n nm i+ 1, n I I G x x M j 1 (, ') Ey = dx '; m = 1,2,..., M j4πωε x y n= 1 i= 1 x i, n 0 N x i+ 1, n i, n nm i+ 1, n I I G x x M j 1 (, ') Ez = dx '; m = 1,2,..., M j4πωε x z n= 1 i= 1 x i, n - N j is the total number of boundary elements on the j-th wire

30 FD analysis of wire antennas Computational examples Vertical wire: Single wire above a lossy ground Wire penetrating the interfacearray above a lossy ground

31 FD analysis of wire antennas

32 FD analysis of wire antennas

33 FD analysis of wire antennas

34 FD analysis of wire antennas Computational examples Numerical results are obtained via TWiNS code for: Single wire above a lossy ground Wire array above a lossy ground Practical example: Yagi-Uda array for VHF TV applications Practical example: single LPDA for ILS

35 FD analysis of wire antennas h=1m h=0.2m Dipole above a PEC ground, f=300mhz, L=λ

36 FD analysis of wire antennas h=1m h=0.2m Dipole above a lossy ground, f=300mhz, L=λ/2, ε r = 30, σ=0.04 S/m

37 FD analysis of wire antennas h=1m h=0.2m XYplane: Currents and far-field pattern for the Yagi-Uda array above a PEC ground (reflector, fed element + director), a=0.0025m, L r =0.479m, L f =0.453m i L d =0.451m, d=0.25m V g =1V

38 FD analysis of wire antennas h=1m h=0.2m XYplane: Currents and far-field pattern for the Yagi-Uda array above a real ground (reflector, fed element + director), a=0.0025m, L r =0.479m, L f =0.453m i L d =0.451m, d=0.25m V g =1V

39 FD analysis of wire antennas Yagi-Uda array for VHF TV applications Geometry of Yagi-Uda array with 15 elements

40 FD analysis of wire antennas Yagi-Uda array: technical parameters Number of wires N=15 Number of directors 13 Operating frequency f=216mhz (frequency of 13th TV channel) Wire radius: a= λ=0.0118m Director lengths l 1 =l 2 =0.424λ=0.589m, l 3 =0.420λ=0.583m, l 4 =0.407λ=0.565m, l 5 =0.403λ=0.56m, l 6 =0.398λ=0.553m, l 7 =0.394λ=0.547m, l 8 - l 13 =0.390λ=0.542m Reflector lengths l 14 =0.475λ=0.66m fed-element length l 15 =0.466λ=0.647m Distance between directors d d =0.308λ=0.427m Distance between reflector and fed-element d r =0.2λ=0.278m Computational aspects l 2a L tot = 5.83m, N tot =225

41 FD analysis of wire antennas free space real ground XYplane: Currents and far-field pattern for the Yagi-Uda array SoftCOM 2016 Split, September 2016

42 FD analysis of wire antennas Log-periodic dipole array LPDA impedance and radiation properties repeat periodically as the logarithm of frequency (VHF and UHF bands; 30MHz to 3GHz). The LPDA antennas are easy to optimize, while the crossing of the feeder between each dipole element leads to a mutual cancellation of backlobe components from the individual elements yielding to a very low level of backlobe radiation (around 25dB below main lobe gain at HF and 35dB at VHF and UHF). The cutoff frequencies of the truncated structure is determined by the electrical lengths of the largest and shortest elements of the structure. The use of logarithmic antenna arrays is very often related with electronic beam steering. An important application of LPDA antennas is in air traffic, as it an essential part of localizer antenna array. A typical localizer antenna system is a part of the electronic systems known as Instrumental Landing System (ILS). Localizer shapes a radiation pattern providing lateral guidance to the aircraft beginning its descent, intercepting the projected runway center line, and then making a final approach.

43 FD analysis of wire antennas The length of actual wire is obtained by multiplying the previous length and factor T: = L n+ 1 τ L n A look at a real localizer antenna element geometry... LPDA geometry

44 FD analysis of wire antennas LPDA in free space LPDA is composed from 12 dipoles insulated in free space. The radius of all wires is a=0.004m while the length of wires are determined by the length of 1st wire L 1 =1.5m, and factor T=0.9. All dipoles are fed by the voltage generator V g =1V with variable phase (each time phase is changed for 180 ). The operating frequency is varied from 100 MHz to 300 MHz.

45 FD analysis of wire antennas LPDA in free space Absolute value of Current distribution along 12 dipoles versus BEM nodes at f=100mhz, f=250mhz and f=300mhz

46 FD analysis of wire antennas LPDA in free space Radiation pattern f=100mhz (XY plane) (YZ plane) f=250mhz (XY plane) (YZ plane) f=300mhz

47 FD analysis of wire antennas LPDA above a PEC ground Radiation pattern (XY plane) LPDA above a real ground f=100mhz f=250mhz f=300mhz

48 FD analysis of wire antennas realistic geometries of localizer antenna systems. f=110mhz T=0.983 σ= L 1 =1.27m d 1 =0.4765m n=7 wires per LPDA a=0.002 N seg =11 - segments per wire N LPDA =14 h=1.82m σ=0.005 ε r =13

49 FD analysis of wire antennas realistic geometries of localizer antenna systems.

50 FD analysis of wire antennas Dipole antenna for Ground Penetrating Radar (GPR) applications GPR dipole antenna above a lossy half-space Broadside transmitted field (V/m) into the ground for different frequencies (L=1 m, a=2 mm, h=0.25 m, V T =1 V, ε rg =10, σ=10 ms/m)

51 Frequency domain analysis: Formulation Pocklington integro-differential equation: L/2 0 L/2 ( ') L/2 L/2 exc µ 1 I x Ex = jω I ( x ') g ( x, x ') dx ' g ( x, x ') dx ' 4 π j4 πωε x x ' The transmitted electric field components: L/2 L/2 1 I( x ') G( x, x ', z) 2 Ex = dx ' γ G( x, x ', z) I( x ') dx ' j4 πωε eff x ' x ' L/2 L/2 E z = L/2 1 I ( x ') G( x, x ', z) j4 πωε x ' z eff L/2 dx ' MIT G( x, x ') = Γ g ( x, x ', z) tr E MIT 2n Γ tr = n + 1

52 Frequency domain analysis: Numerical solution The current and its first derivative at the i-th boundary element are given by: x x ' x ' x I( x ') = I + I x x 2i 1i 1i 2i I ( x') x ' Matrix equation: [ Z ] { I} = { V} M j= 1 ji i j = I I 2i 1i x Mutual impedance matrix, voltage vector: 1 Z D D ' g x, x ' dx ' dx f f ' g x, x ' dx ' dx ji j i j i j4ωπε eff l j li l j l i T 2 T [ ] = { } { } ( ) + γ { } { } ( ) inc { } ( ){ } V E x f dx j = l j x j

53 Frequency domain analysis: Numerical solution The field formulas: N x j x i ij I 2i I1 i G( x, x ', z) 2 x2i x ' x ' x1 i Ex = dx ' γ I1 i + I2i G( x, x ', z) I ( x ') dx ' j4 πωε eff i= 1 x j x ' x x x1 ij x 1ij E z = N M j 2ij 1 (, ', ) 2ij 1ij j4πωε = = x z eff j 1 i 1 j x x I I G x x z 1ij dx '

54 Frequency domain analysis: Numerical results The computational example; dipole antenna (L=1m, a=2mm, h=0,25m, ε rg =10, σ=10ms/m). Terminal voltage is V T =1V. The operating frequency: from 1MHz to 100MHz. E x component E z - component Transmitted field (V/m) into the ground at f = 1MHz

55 Frequency domain analysis: Numerical results f = 10MHz f = 100MHz E x component E z - component Transmitted field (V/m) into the ground

56 Frequency domain analysis: Numerical results E x component of the transmitted field versus depth in the broadside direction for different operating frequencies Broadside transmitted field (V/m) into the ground for different frequencies

57 Frequency domain analysis: Concluding remarks The FD analysis of E-field transmitted into the material halfspace due to the GPR dipole antenna radiation is based on the Pocklington IDE and related field formulas. The influence of the earth-air interface is taken into account via the simplified reflection/transmission coefficient arising from the Modified Image Theory (MIT). The Pocklington IDE is solved via the Galerkin-Bubnov variant of the Indirect Boundary Element Method (GB- IBEM) and the corresponding transmitted field is determined by using BEM formalism, as well.

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64 Stochastic Modeling : Numerical results

65 There is scarcely a subject that cannot be mathematically treated and the effect calculated beforehand, or the results determined beforehand from the available theoretical and practical data. Nikola Tesla 65

66 We made models in science, but we also made them in everyday life. STEPHEN HAWKING Thank you for your attention B E = t 66