TWINS II ANNA ŠUŠNJARA, VICKO DORIĆ & DRAGAN POLJAK Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture University of Split, Croatia Training School on Ground Penetrating Radar for civil engineering and cultural Roma, Italy,
TWINS I = Thin WIre Numerical Solver Suzana = Sustav za analizu antena (Croatian) 2 - Used for educational purpose and scientific research What is solved? - Governing equations for a current distribution along the wire radiating above a half space Frequency domain solver (FD) Time domain solver (TD) Numerical method? - Galerkin-Bubnov scheme of Indirect Boundary Element Method (GB-IBEM) Programming language: - C++
TD vs. FD techniques 3 Time domain (TD) techniques: Frequency domain (FD) techniques: Applied for a large frequency spectrum, but for single source More complex formulation requiring more computational effort Deeper physical insight Applied for more sources, but at a single frequency Formulation is substantially less demanding In general, regulatory standards are specified in FD form; convenient for analyzing EMC properties
TWINS I Frequency Domain (FD) 4
TWINS I Time Domain (TD) 5
TWINS II - Frequency Domain (FD) - Presented on final GPR conference in Warsaw What is solved? Integral formulas for the electric field transmitted into the subsurface. NEW! The subsurface is non-homogenous. (0,h) h z ε r1 σ 1 ε r2 σ 2 ε r3 σ 3 d 1 d 2 L ϑ T (x,z) (L,h) x Transmitted wave 6
Talk Layout THEORETICAL BACKGROUND Model geometry INSTALLATION PROCEDURE USER GUIDE: Antenna parameters Ground parameters Frequency domain (FD) Time domain (TD) Run the simulation EXAMPLES REFERENCES 7
Model geometry 8 (0,h) (L,h) L Ground penetrating radar dipole antenna above a lossy half space. h ϑ The half space is modeled as a three layered medium. x d 1 ε r1 σ 1 ε r2 σ 2 d 2 T (x,z) Transmitted wave z ε r3 σ 3
Pocklington s integro-differential equation: E exc x = j ω μ L 4π න 2 I x g x, x dx L 2 L 1 j4πωε 0 x න 2 I x L x g x, x dx 2 z 9 Wire above a lossy half-space: I(x )... the unknown current g(x,x )... Green s function g x, x = g 0 x, x R TM g i x, x h x ϑ R 0 x x ε 0, μ 0 R i g 0 (x,x )... for free space g 0 x, x = e jk 0R 0 R 0 x σ, ε r μ 0 g i (x,x )... from image theory g i x, x = e jk 0R i R i -h x
Electric field integral relations: 10 1 E x = j4πωε 0 L I x න 0 x g x, y, z, x x L dx γ 2 න I x g x, x dx 0 E y = 1 L I x න j4πωε 0 0 x g x, y, z, x y dx Numerical procedure: E z = 1 L I x න j4πωε 0 0 x g x, y, z, x z dx Galerkin Bubnov scheme of Indirect Boundary Element Method (GB-IBEM) g x, x = T TM g 0 x, x
Reflection/transmission coefficients for three layered media configuration: z Air + Layer1 + Layer2 11 h ϑ 0 L ~ ε 0, σ = 0 μ 0, 01 x The plane wave approximation with oblique incidence at interfaces. The continuity conditions for the tangential components of electric and magnetic fields at two interfaces (01 and 12) and the Snell s law for wave propagation yield the following equations d 1 ϑ 1 ϑ 2 ε 0 ε r1 σ 1 μ 0, ε 0 ε r2 σ 2 μ 0 12 E 0 + = 1 τ 01 E 1 + + ρ 01 τ 01 E 1 E 0 = ρ 01 τ 01 E 1 + + 1 τ 01 E 1 E 1 + = 1 τ 12 e γ 1 cos θ 1 d 1 E 2 + E 1 = ρ 12 τ 12 e γ 1 cos θ 1 d 1 E 2 + E 0 +, E 1 +, E 0 & E 1...the magnitude of electric field at the layer interfaces for the wave propagating in the negative and positive direction of z axis, respectively
Fresnel s reflection/transmission coefficient approximation at interfaces z 12 L ~ h d 1 ϑ 0 ϑ 1 ε 0, σ = 0 μ 0, ε 0 ε r1 σ 1 μ 0, x mn = 01 mn = 12 ρ mn = Z n cos θ n Z m cos θ m Z n cos θ n + Z m cos θ m τ mn = 2Z n cos θ m Z n cos θ n + Z m cos θ m m = 0,1 n = 1,2 ϑ 2 ε 0 ε r2 σ 2 μ 0 Impedance of medium complex propagation constant Z k = jωμ 0 σ k + jωε 0 ε rk γ k = jωμ 0 σ k + jωε 0 ε rk k = 0,1,2
13 E 0 + E 0 = M 01 E 1 + E 1 E 1 + E 1 = P 12M 12 E 2 + E 2... Matrix form for equations that solve for field values at layer s interfaces M mn = 1 1 ρ mn τ mn ρ mn 1 m = 0,1, n = 1,2... matching matrix P 12 = eγ 1 cos θ 1 d 1 0 0 e γ 1 cos θ 1 d 1... propagation matrix The layers k = 3,... are readily included by adding the propagation and matching matrices :
By setting E 0 + = 1 V/m, all the other magnitudes E m + and E m are easily determined. 14 Electric field can be calculated using the following expressions: E 0 = E 1 + cos θ 0 Ԧe x + sin θ 0 Ԧe z e γ 0 x sin θ 0 z cos θ 0 + E 0 cos θ 0 Ԧe x sin θ 0 Ԧe z e γ 0 x sin θ 0 +z cos θ 0 E 1 = E 1 + cos θ 1 Ԧe x + sin θ 1 Ԧe z e γ 1 x sin θ 1 z cos θ 1 + E 1 cos θ 1 Ԧe x sin θ 1 Ԧe z e γ 1 x sin θ 1 +z cos θ 1 E 2 = E 2 + cos θ 2 Ԧe x + sin θ 2 Ԧe z e γ 2 x sin θ 2 z cos θ 2 The reflection and transmission coefficients R TM and T TM appearing in the Green s function are calculated as the ratio between the field value at the ground surface and the field value obtained from the equations above at the given observation point.
Alternative simplified approach to calculate tranmission / reflection coeffiicients: - Modified image theory based approach - convenient for TD computations - a comparison with the plane wave approximation in FD needed for benchmark 15 T(x,z ) The MIT approach assumes the normal incidence of a propagating EM wave, therefore the incidence angles are zero: ϑ m = 0º. ϑ inc The expressions for the reflection and transmission coefficients for the two adjacent media indexed with m and n are given as: ϑ=0 T(x,z) ρ MIT mn = Z m 2 Z2 n Z 2 2 m + Z n τ MIT mn = 2Z m 2 Z m 2 + Z n 2 m = 0,1 n = 1,2
Installation procedure 16 Run transmitted_e_field_pkg.exe. This action installs MATLAB Compiler Runtime (MCR) version 8.1. and extracts the following files: - transmitted_e_field.exe - MCRInstaller.exe - readme.txt file
17...
18 User guide ANTENNA PARAMETERS GROUND PARAMETERS FREQUENCY DOMAIN (FD) TIME DOMAIN (TD) RUN THE SIMULATION
19 Graphical user interface 1. 3. 2. 5. 4.
20 Antenna parameters Ground parameters (0,h) L (L,h) ε r1 σ 1 d 1 ε r2 σ 2 d 2 ε r3 σ 3
Frequency domain (FD) 21 Figures generated: - E vs. x (z) if only one z (x) point is defined - contours if xz grid is defined Figure generated: - E vs. frequency
Frequency domain (FD) 22 Figure generated: - None - The results are saved in.txt file Freq. (MHz) x(m) z(m) Real(E x ) Imag(E x ) Real(E z ) Imag(E z )
Time domain (TD) 23 The excitation is given as a Gaussian waveform where the parameters t w and t 0 represent the half width and time delay of a pulse Figure generated upon the calculation: - E vs. time for tangential component of electric field
24 Examples 1. E vs. xz 2. E vs. z 3. E vs. freq 4. E vs. time
1. E vs. xz 25
1. E vs. xz f = 10 MHz 26
1. E vs. xz f = 10 MHz 27
1. E vs. xz f = 300 MHz 28
1. E vs. xz f = 300 MHz 29
1. E vs. xz 30
1. E vs. xz Freq. (MHz) x(m) Real(I) Imag(I) 31 Freq. (MHz) x(m) z(m) Real(E x ) Imag(E x ) Real(E z ) Imag(E z )
32 2. E vs. z
33 3. E vs. f
34 3. E vs. f
3. E vs. time 35
REFERENCES [1] D. Poljak, Advanced Modelling in Computational Electromagnetic Compatibility, John Wiley and Sons, New York 2007. 36 [2] D. Poljak, V. Dorić, Transmitted field in the lossy ground from ground penetrating radar (GPR) dipole antenna, Computational Methods and Experimental Measurments XVII 3, WIT Transactions on Modelling and Simulation, Vol 59, pp. 3-11, 2015 [3] A. Šušnjara, D. Poljak, S. Šesnić, V. Dorić Time Domain Integral Equation versus Frequency Domain Boundary Element model for calculation of Transmitted Electrical Field for Ground Penetrating Radar (GPR) Antenna [4] A. Šušnjara, D. Poljak, V. Dorić, Electric Field Radiated By a Dipole Antenna Above a Lossy Half Space: Comparison of Plane Wave Approximation with the Modified Image Theory Approach, SoftCOM conference, Split, 2017 [5] A. Šušnjara, V. Dorić, D. Poljak, Electric Field Radiated By a Dipole Antenna and Transmitted Into a Two-Layered Lossy Half Space, submitted for SpliTECH conference, Split,