Characteristic Modes Part I: Introduction Miloslav ˇ Capek Department of Electromagnetic Field Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz Seminar May 2, 2018 ˇ Capek, M. Characteristic Modes – Part I: Introduction 1 / 39
Characteristic Modes Conventionally, characteristic modes I n are defined as XI n = λ n RI n , in which Z = R + j X is the impedance matrix. Dominant characteristic mode of helicopter model discretized into 18989 basis functions, ka = 1 / 2, decomposed into characteristic modes in AToM in 47 s. ˇ Capek, M. Characteristic Modes – Part I: Introduction 2 / 39
Characteristic Modes Conventionally, characteristic modes I n are defined as XI n = λ n RI n , in which Z = R + j X is the impedance matrix. However, who knows: ◮ What is impedance matrix and how to get it? ◮ What the hell are the characteristic modes? Dominant characteristic mode of helicopter model discretized into 18989 basis functions, ◮ Why are they of our interest? ka = 1 / 2, decomposed into characteristic modes in AToM in 47 s. ˇ Capek, M. Characteristic Modes – Part I: Introduction 2 / 39
Characteristic Modes Conventionally, characteristic modes I n are defined as XI n = λ n RI n , in which Z = R + j X is the impedance matrix. However, who knows: ◮ What is impedance matrix and how to get it? ◮ What the hell are the characteristic modes? Dominant characteristic mode of helicopter model discretized into 18989 basis functions, ◮ Why are they of our interest? ka = 1 / 2, decomposed into characteristic modes in AToM in 47 s. Therefore, . . . . . . the characteristic mode theory is to be systematically derived. ˇ Capek, M. Characteristic Modes – Part I: Introduction 2 / 39
Characteristic Modes Conventionally, characteristic modes I n are defined as XI n = λ n RI n , in which Z = R + j X is the impedance matrix. However, who knows: ◮ What is impedance matrix and how to get it? ◮ What the hell are the characteristic modes? Dominant characteristic mode of helicopter model discretized into 18989 basis functions, ◮ Why are they of our interest? ka = 1 / 2, decomposed into characteristic modes in AToM in 47 s. Therefore, . . . . . . the characteristic mode theory is to be systematically derived. Disclaimer: There will be equations! Brace yourself and be prepared. . . ˇ Capek, M. Characteristic Modes – Part I: Introduction 2 / 39
Outline 1 Necessary Background 2 Discretization and Method of Moments 3 Definition of Characteristic Modes 4 Properties of Characteristic Modes 5 Activities at the Department J 1 ( r , t ) 6 Concluding Remarks J 2 ( r , t ) This talk concerns: ◮ electric currents in vacuum (generalization is, however, straightforward), ◮ time-harmonic quantities, i.e. , A ( r , t ) = Re { A ( r ) exp (j ωt ) } . ˇ Capek, M. Characteristic Modes – Part I: Introduction 3 / 39
Necessary Background Electric Field Integral Equation 1 σ → ∞ (PEC) Ω Original problem. 1 R. F. Harrington, Time-Harmonic Electromagnetic Fields , 2nd ed. Wiley – IEEE Press, 2001 ˇ Capek, M. Characteristic Modes – Part I: Introduction 4 / 39
Necessary Background Electric Field Integral Equation 1 E s ( r ) k σ → ∞ (PEC) Ω E i ( r ) k Original problem. � � r ′ � � r ′ � � r ′ ∈ Ω n × + E i = 0 , ˆ E s 1 R. F. Harrington, Time-Harmonic Electromagnetic Fields , 2nd ed. Wiley – IEEE Press, 2001 ˇ Capek, M. Characteristic Modes – Part I: Introduction 4 / 39
Necessary Background Electric Field Integral Equation 1 E s ( r ) k ǫ 0 , µ 0 σ → ∞ (PEC) Ω Ω E i ( r ) k Original problem. Equivalent problem. � � r ′ � � r ′ � � r ′ ∈ Ω n × + E i = 0 , ˆ E s 1 R. F. Harrington, Time-Harmonic Electromagnetic Fields , 2nd ed. Wiley – IEEE Press, 2001 ˇ Capek, M. Characteristic Modes – Part I: Introduction 4 / 39
Necessary Background Electric Field Integral Equation 1 E s ( r ) E s ( r ) k k ǫ 0 , µ 0 σ → ∞ (PEC) J ( r ′ ) Ω Ω E i ( r ) k Original problem. Equivalent problem. � � r ′ � � r ′ � � � r ′ � � r ′ � r ′ ∈ Ω n × + E i = 0 , − ˆ n × ˆ n × E i = Z ( J ) , J = J ˆ E s 1 R. F. Harrington, Time-Harmonic Electromagnetic Fields , 2nd ed. Wiley – IEEE Press, 2001 ˇ Capek, M. Characteristic Modes – Part I: Introduction 4 / 39
Necessary Background Electric Field Integral Equation – Problem Formalization Key role of the impedance operator Z ( J ) � r ′ � n × ˆ ˆ n × E s = Z ( J ) = − ˆ n × ˆ n × (j ω A + ∇ ϕ ) . Substituting for Lorenz gauge-calibrated potentials 2 A and ϕ gives � � r , r ′ � � r ′ � Z ( J ) = j kZ 0 · J d S G Ω 2 J. D. Jackson, Classical Electrodynamics , 3rd ed. Wiley, 1998 ˇ Capek, M. Characteristic Modes – Part I: Introduction 5 / 39
Necessary Background Electric Field Integral Equation – Problem Formalization Key role of the impedance operator Z ( J ) � r ′ � n × ˆ ˆ n × E s = Z ( J ) = − ˆ n × ˆ n × (j ω A + ∇ ϕ ) . Substituting for Lorenz gauge-calibrated potentials 2 A and ϕ gives � � � � � r , r ′ � � r ′ � � r ′ � e − j k | r ′ − r | 1 + 1 Z ( J ) = j kZ 0 · J d S = j kZ 0 k 2 ∇∇ · J 4 π | r ′ − r | d S, G Ω Ω ◮ Impedance operator Z is linear, symmetric (reciprocal, thus non-Hermitian). ◮ Alternative formulation MFIE 3 , common extension towards CFIE 3 . 2 J. D. Jackson, Classical Electrodynamics , 3rd ed. Wiley, 1998 3 W. C. Gibson, The Method of Moments in Electromagnetics , 2nd ed. Chapman and Hall/CRC, 2014 ˇ Capek, M. Characteristic Modes – Part I: Introduction 5 / 39
Discretization and Method of Moments Dicretization of the Problem Only canonical bodies can typically be evaluated analytically. n × E i ( r ′ ) = Z ( J ) has to be solved numerically! Problem − ˆ n × ˆ ǫ 0 , µ 0 Ω Equivalent problem. 4 J. A. De Loera, J. Rambau, and F. Santos, Triangulations – Structures for Algorithms and Applications . Berlin, Germany: Springer, 2010 ˇ Capek, M. Characteristic Modes – Part I: Introduction 6 / 39
Discretization and Method of Moments Dicretization of the Problem Only canonical bodies can typically be evaluated analytically. n × E i ( r ′ ) = Z ( J ) has to be solved numerically! Problem − ˆ n × ˆ ◮ Discretization 4 Ω → Ω T is needed (nontrivial task!) ǫ 0 , µ 0 Ω Ω T Triangularized domain Ω T . Equivalent problem. 4 J. A. De Loera, J. Rambau, and F. Santos, Triangulations – Structures for Algorithms and Applications . Berlin, Germany: Springer, 2010 ˇ Capek, M. Characteristic Modes – Part I: Introduction 6 / 39
Discretization and Method of Moments Representation of the Operator (Recap.) Engineers like linear systems L g f L ( f ) = h . ◮ Typically unsolvable for f in the present state Linear system with input f and output g . (how to invert L ?). ˇ Capek, M. Characteristic Modes – Part I: Introduction 7 / 39
Discretization and Method of Moments Representation of the Operator (Recap.) Engineers like linear systems L g f L ( f ) = h . ◮ Typically unsolvable for f in the present state Linear system with input f and output g . (how to invert L ?). Representation in a basis { ψ n } and linearity of operator L readily gives 5 N � I n L ( ψ n ) = h . n =1 ◮ One equation for N unknowns → still unsolvable. 5 R. F. Harrington, Field Computation by Moment Methods . Wiley – IEEE Press, 1993 ˇ Capek, M. Characteristic Modes – Part I: Introduction 7 / 39
Discretization and Method of Moments Representation of the Operator (Recap.) Using proper inner product �· , ·� and N tests from left, we get N � I n � χ n , L ( ψ n ) � = � χ n , h � , n =1 i.e. , in matrix form the method of moments 5 relation reads LI = H . 5 R. F. Harrington, Field Computation by Moment Methods . Wiley – IEEE Press, 1993 ˇ Capek, M. Characteristic Modes – Part I: Introduction 8 / 39
Discretization and Method of Moments Algebraic Solution – Method of Moments Piecewise basis functions 6 l n ρ ± ( r ) T + l n ρ − P − ψ n ( r ) = n n n ρ + 2 A ± n n P + A − A + n n T − n n r z y x O RWG basis function ψ n . 6 S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape”, IEEE Trans. Antennas Propag. , vol. 30, no. 3, pp. 409–418, 1982. doi : 10.1109/TAP.1982.1142818 ˇ Capek, M. Characteristic Modes – Part I: Introduction 9 / 39
Discretization and Method of Moments Algebraic Solution – Method of Moments Piecewise basis functions 6 l n ρ ± ( r ) T + l n ρ − P − ψ n ( r ) = n n n ρ + 2 A ± n n P + A − A + n n T − n are applied to approximate J ( r ) as n r z � N y x O J ( r ) ≈ I n ψ n ( r ) . n =1 RWG basis function ψ n . 6 S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape”, IEEE Trans. Antennas Propag. , vol. 30, no. 3, pp. 409–418, 1982. doi : 10.1109/TAP.1982.1142818 ˇ Capek, M. Characteristic Modes – Part I: Introduction 9 / 39
Recommend
More recommend
