Nguyễn Công Phương Engineering Electromagnetics Poisson’s & Laplace’s Equations Contents I Introduction II Vector Analysis III Coulomb’s Law & Electric Field Intensity IV Electric Flux Density, Gauss’ Law & Divergence V Energy & Potential VI Current & Conductors VII Dielectrics & Capacitance VIII Poisson’s & Laplace’s Equations IX The Steady Magnetic Field X Magnetic Forces & Inductance XI Time – Varying Fields & Maxwell’s Equations XII Transmission Lines XIII The Uniform Plane Wave XIV Plane Wave Reflection & Dispersion XV Guided Waves & Radiation Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations Poisson’s Equation Laplace’s Equation Uniqueness Theorem Examples of the Solution of Laplace’s Equation Examples of the Solution of Poisson’s Equation Product Solution of Laplace’s Equation Finite Difference Method Finite Element Method Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson's Equation (1) Gauss’s Law: ∇ D = ρ v D = ε 0E Gradient: E = −∇V → ∇ D = ∇.(ε E) = −∇.(ε∇V ) = ρv ρv → ∇ ∇ V = − ε (Poisson's Equation) ∂V ∂V ∂V ∇V = ax + ay + az ∂x ∂y ∂z ∂Ax ∂Ay ∂Az ∇.A = + + ∂x ∂y ∂z ∂ ∂Vx ∂ ∂Vy ∂ ∂Vz ∂ 2V ∂ 2V ∂ 2V → ∇.∇V = + + = + + ∂x ∂x ∂y ∂y ∂z ∂z ∂x ∂y ∂z Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson's Equation (2) ρv ∇ ∇ V = − ε ∂V ∂V ∂V ∇.∇V = + + ∂x ∂y ∂z Define ∇ ∇ = ∇ 2 ∂ ∂V ρ ρ ∂ρ ∂ρ 2 2 ∂ V ∂ V ∂ V ρv ∇ V = + + =− ∂x ∂y ∂z ε (rectangular) ∂ 2V ∂ 2V ρv + =− + 2 ∂z ε ρ ∂ϕ (cylindrical) ∂ ∂V ∂ ∂V ∂ 2V ρv =− r + sin θ + 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂ϕ ε (spherical) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s Equation (3) Ex Find the Laplacian of the following scalar fields: a ) A = xy z b) B = c) C = cos2ϕ ρ 20 sin θ r3 Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations Poisson’s Equation Laplace’s Equation Uniqueness Theorem Examples of the Solution of Laplace’s Equation Examples of the Solution of Poisson’s Equation Product Solution of Laplace’s Equation Finite Difference Method Finite Element Method Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Laplace’s Equation Poisson's Equation: 2 ∂ V ∂ V ∂ V ρv ∇ V = + + =− ∂x ∂y ∂z ε ρv = 2 ∂ V ∂ V ∂ V ∇ V = + + = (Laplace’s equation, rectangular) ∂x ∂y ∂z ∂ ∂V ∂ 2V ∂ 2V ρ + + = (cylindrical) ∂z ρ ∂ρ ∂ρ ρ ∂ϕ ∂ ∂V ∂ ∂V ∂ 2V =0 r + sin θ + 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂ϕ (spherical) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson’s & Laplace’s Equations Poisson’s Equation Laplace’s Equation Uniqueness Theorem Examples of the Solution of Laplace’s Equation Examples of the Solution of Poisson’s Equation Product Solution of Laplace’s Equation Finite Difference Method Finite Element Method Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn Uniqueness Theorem (1) 2 ∂ V ∂ V ∂ V ∇ V = + + =0 ∂x ∂y ∂z Assume two solutions V1 & V2, : ∇ 2V1 = ∇2V2 = → ∇2 (V1 − V2 ) = Assume the boundary condition Vb → V1b = V2b = Vb ∇ (VD) = V (∇ D) + D.(∇V ) V = V1 − V2 D = ∇ (V1 − V2 ) → ∇.[(V1 − V2 )∇(V1 − V2 )] = (V1 − V2 )[∇.∇(V1 − V2 )] + + ∇(V1 − V2 ).∇(V1 − V2 ) Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 10 Finite Element Method (5) Ve = [1 x ( x2 y3 − x3 y2 ) ( x3 y1 − x1 y3 ) ( x1 y2 − x2 y1 ) y] ( y − y ) ( y − y ) ( y − y ) 3 1 x1 y1 ( x3 − x2 ) ( x1 − x3 ) ( x2 − x1 ) x2 y2 x3 −1 Ve1 Ve Ve y3 Ve = ∑αi ( x, y )Vei i =1 [( x2 y3 − x3 y2 ) + ( y2 − y3 ) x + ( x3 − x2 ) y], 2A [( x3 y1 − x1 y3 ) + ( y3 − y1 ) x + ( x1 − x3 ) y ], α2 = 2A α = [( x1 y2 − x2 y1 ) + ( y1 − y2 ) x + ( x2 − x1 ) y ], 2A A = [( x2 − x1 )( y3 − y1 ) − ( x3 − x1 )( y2 − y1 )] α1 = Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 54 Finite Element Method (6) ax + b = ∇ 2V = We = ε E dS e ∫ S E = −∇V → We = d ax + bx + c =0 dx ε ∇ V e dS ∫ S Ve = ∑ αi ( x, y)Vei i =1 → ∇Ve = ∑Vei ∇α i i =1 3 → We = ∑∑ εVei ∫ (∇α i )(∇α j )dS Vej S i =1 j =1 Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 55 Finite Element Method (7) 3 We = ∑∑ εVei ∫ (∇αi )(∇α j )dS Vej S i =1 j =1 Cij( e) = ∫ (∇αi )(∇α j )dS S Ve1 [Ve ] = Ve Ve3 C11(e ) C (e ) = C21(e ) C31(e ) C12(e ) C22(e ) C32(e ) C13(e ) C23(e ) C33(e ) T → We = ε [Ve ] C ( e ) [Ve ] Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 56 Finite Element Method (8) Cij( e) = ∫ (∇αi )(∇α j )dS S C12( e) = ∫ (∇α1 )(∇α )dS S [( x2 y3 − x3 y2 ) + ( y2 − y3 ) x + ( x3 − x2 ) y], 2A [( x3 y1 − x1 y3 ) + ( y3 − y1 ) x + ( x1 − x3 ) y ], α2 = 2A A = [( x2 − x1 )( y3 − y1 ) − ( x3 − x1 )( y2 − y1 )] α1 = → C12( e ) = [( y2 − y3 )( y3 − y1 ) + ( x3 − x2 )( x1 − x3 )] 4A Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 57 Finite Element Method (9) [( y2 − y3 )( y3 − y1 ) + ( x3 − x2 )( x1 − x3 )] 4A = [( y2 − y3 )( y1 − y2 ) + ( x3 − x2 )( x2 − x1 )] 4A = [( y3 − y1 )( y1 − y2 ) + ( x1 − x3 )( x2 − x1 )] 4A = [( y2 − y3 )2 + ( x3 − x2 )2 ] 4A = [( y3 − y1 )2 + ( x1 − x3 ) ] 4A = [( y1 − y2 )2 + ( x2 − x1 )2 ] 4A = C12( e ) , C31( e ) = C13(e ) , C32(e ) = C 23( e) C12( e) = C13( e) C 23( e) C11( e) C 22( e) C33( e) C 21( e) → Cij( e ) = (PP + Qi Q j ) i j 4A P Q − PQ A= 3 2 P1 = y2 − y3 , P2 = y3 − y1 , P3 = y1 − y2 Q1 = x3 − x2 , Q2 = x1 − x3 , Q3 = x2 − x1 Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 58 Finite Element Method (3) ∇ 2V = V1 ( x , y ) V ( x, y) V2 V3 V4 http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html • Discretizing the solution region into a finite number of elements, • Obtaining governing equations for a typical elements, • Combining all elements in the solution, & • Solving the system of equations obtained Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 59 Finite Element Method (10) T We = ε [Ve ] C ( e) [Ve ] N T W = ∑ We = ε [V ] [C ][V ] e =1 V1 V 2 [V ] = V3 ⋮ Vn Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 60 Finite Element Method (11) T W = ε [V ] [C ][V ] C11 C12 C13 C C22 C23 21 [C ] = C31 C32 C33 C41 C42 C43 C51 C52 C53 C11(1) + C11( 2) (1) C 31 [C ] = C21( ) (1) ( 2) C + C 21 31 C14 C24 C34 C44 C54 C15 C25 C35 C45 C55 1 1 C11 = C11(1) + C11( ) C12(2 ) C12(1) + C13(2 ) C33(1) C32(1) 0 (2) C22 + C11( 3) C23(1) C32( ) + C31( 3) C22(1) + C33( ) + C33( 3) C21(3) C23(3) C C13(1) ( 2) 23 +C (3) 13 C12(3) C32(3) C22(3) C22 = C33(1) C44 = C22(1) + C33(2) + C33(3) C14 = C41 = C12(1) + C13(2) C 23 = C32 = Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 61 Finite Element Method (3) ∇ 2V = V1 ( x , y ) V ( x, y) V2 V3 V4 http://imagine.inrialpes.fr/people/Francois.Faure/htmlCourses/FiniteElements.html • Discretizing the solution region into a finite number of elements, • Obtaining governing equations for a typical elements, • Combining all elements in the solution, & • Solving the system of equations obtained Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 62 Finite Element Method (12) ax + b = ∇ 2V = T W = ε [V ] [C ][V ] ∂W ∂W ∂W ∂W = =⋯ = =0↔ = 0, ∂V1 ∂V2 ∂Vn ∂Vk → Vk = − Ckk d ax + bx + c =0 dx k = 1, 2, , n n ∑ VC i =1,i ≠ k i ki Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 63 Finite Element Method (13) Ex Node x 0.5 3.1 5.0 2.8 y 1.0 0.4 1.7 2.0 Cij(e) = P2 Q3 − P3Q2 ( PP + Q Q ), A = i j 4A i j P1 = y − y3 , P2 = y3 − y1 , P3 = y1 − y2 y V =0 Q1 = x3 − x2 , Q2 = x1 − x3 , Q3 = x2 − x1 Element 1: P1 = 0.4 − 2.0 = −1.6; P2 = 2.0 −1.0 = 1.0; P3 = 1.0 − 0.4 = 0.6 Q1 = 2.8 − 3.1 = −0.3; Q2 = 0.5 − 2.8 = −2.3; Q3 = 3.1 − 0.5 = 2.6 1.0 × 2.6 − 0.6( −2.3) A= = 1.99 P P + Q1Q2 (−1.6)1.0 + (−0.3)(−2.3) C12(1) = = = −0.1143 ×1.99 ×1.99 0.3329 −0.1143 −0.2186 C (1) = −0.1143 0.7902 −0.6759 −0.2186 −0.6759 0.8945 V = 100 x 4 3 2 2 Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 64 Finite Element Method (14) Ex Node x 0.5 3.1 5.0 2.8 y 1.0 0.4 1.7 2.0 Cij(e) = P2 Q3 − P3Q2 ( PP + Q Q ), A = i j 4A i j P1 = y − y3 , P2 = y3 − y1 , P3 = y1 − y2 y V =0 Q1 = x3 − x2 , Q2 = x1 − x3 , Q3 = x2 − x1 Element 2: P1 = 1.7 − 2.0 = −0.3; P2 = 2.0 − 0.4 = 1.6; P3 = 0.4 −1.7 = −1.3 Q1 = 2.8 − = −2.2; Q2 = 3.1 − 2.8 = 0.3; Q3 = 5.0 − 3.1 = 1.9 1.6 × 1.9 − ( −1.3)0.3 A= = 1.715 P P + Q2Q2 1.6 ×1.6 + 0.3 × 0.3 C22(2) = 2 = = 0.3863 × 1.715 ×1.715 0.7187 −0.1662 −0.5525 C ( 2) = −0.1662 0.3863 −0.2201 −0.5525 −0.2201 0.7726 V = 100 x 4 3 2 2 Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 65 Ex Finite Element Method (15) 0.3329 C (1) = −0.1143 −0.2186 0.7187 C ( 2) = −0.1662 −0.5525 −0.1143 −0.2186 0.7902 −0.6759 −0.6759 0.8945 −0.1662 −0.5525 0.3863 −0.2201 −0.2201 0.7726 y 3 1 V =0 V = 100 x C11(1) (1) C21 C = [ ] (1) C31 C12(1) (1) C22 + C11(2) C21(2) (1) C32 + C31(2) C12(2) (2) C22 (2) C32 −0.2186 C13(1) 0.3329 −0.1143 (1) C23 + C13(2) −0.1143 1.5089 −0.1662 −1.2284 = ( 2) −0.1662 0.3863 −0.2201 C23 (1) (2) C33 + C33 −0.2186 −1.2284 −0.2201 1.6671 Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 66 Ex Finite Element Method (16) −0.2186 0.3329 −0.1143 −0.1143 1.5089 −0.1662 −1.2284 [C ] = −0.1662 0.3863 −0.2201 − 0.2186 − 1.2284 − 0.2201 1.6671 Vk = − Ckk y 3 n ∑ VC i=1, i ≠k i ki V =0 V = 100 V = − (V1C12 + V3C32 + V4C42 ) C22 → V4 = − (V1C14 + V2 C24 + V3 C34 ) C44 x (k +1) (k) (k ) V = − [(0( − 0.1143) + 100( − 0.1662) + V ( − 1.2284)] = 11.0146 + 0.8141 V 4 1.5089 → V (k +1) = − [0(−0.2186) + V ( k ) ( −1.2284) + 100(−0.2201)] = 13.2026 + 0.7368V ( k ) 2 1.6671 Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 67 Ex Finite Element Method (17) V2( k +1) = 11.0146 + 0.8141V4(k ) ( k +1) = 13.2026 + 0.7368V2(k ) V4 V2(0) = V4(0) = + 100 = 50 y 3 1 V =0 V = 100 V2(1) = 11.0146 + 0.8141V4(0) = 11.0146 + 0.8141× 50 = 51.7196 (1) (0) V4 = 13.2026 + 0.7368V2 = 13.2026 + 0.7368 × 50 = 50.0426 x V2( 2) = 11.0146 + 0.8141V4(1) = 11.0146 + 0.8141 × 50.0426 = 51.7543 ( 2) (1) V4 = 13.2026 + 0.7368V2 = 13.2026 + 0.7368 × 51.7196 = 51.3096 V2(3) = 11.0146 + 0.8141V4( 2) = 11.0146 + 0.8141× 51.3096 = 52.7857 (3) ( 2) V4 = 13.2026 + 0.7368V2 = 13.2026 + 0.7368 × 51.7543 = 51.3352 Poisson's & Laplace's Equations - sites.google.com/site/ncpdhbkhn 68 ... Reflection & Dispersion XV Guided Waves & Radiation Poisson' s & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson s & Laplace’s Equations Poisson s Equation Laplace’s Equation Uniqueness... Solution of Poisson s Equation Product Solution of Laplace’s Equation Finite Difference Method Finite Element Method Poisson' s & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson' s... Poisson' s & Laplace's Equations - sites.google.com/site/ncpdhbkhn Poisson s Equation (3) Ex Find the Laplacian of the following scalar fields: a ) A = xy z b) B = c) C = cos2ϕ ρ 20 sin θ r3 Poisson' s