FDTD(Finite Difference Time Domain) Maxwell FDTD FDTD FDTD (FFT) FDTD CP(Contour-Path)-FDTD i

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1 FDTD GD191

2 FDTD(Finite Difference Time Domain) Maxwell FDTD FDTD FDTD (FFT) FDTD CP(Contour-Path)-FDTD i

3 FDTD FDTD CP-FDTD FDTD FDTD CP-FDTD FDTD CP-FDTD θ = tan θ = tan ii

4 1 1.1 FDTD FDTD Finite Difference Time Domain Method FDTD [1][2] FDTD FDTD FDTD FDTD FDTD FDTD (FFT) FDTD 3 FDTD 1

5 FDTD FDTD FDTD FDTD (EMC) EMC FDTD PHS LAN IC (CAD) FDTD / 1.2 FDTD FDTD E[V/m] H[A/m] D[C/m 2 ], B[T ], ρ[c/m 3 ], J[A/m 2 ] 2

6 roth(r, t) = D(r, t) + J(r, t) t (1.1) B(r, t) rote(r, t) = t (1.2) divb(r, t) = 0 (1.3) dibd(r, t) = ρ(r, t) (1.4) (1) (2) (3) (4) FDTD (3)(4) (1)(2) (1)(2) Yee 3 FDTD x (i, j, k) =H n 1 2 x (i, j, k) t µ {En z (i, j, k +1) En z (i, j 1,k) y En y (i, j, k +1) En y (i, j, k 1) } (1.5) z Ex n+1 (i, j, k) =Ex n (i, j, k)+ t 1 2 ɛ {Hn+ z (i, j +1,k) H n+ 1 2 z (i, j, k) y 1 Hn+ 2 y (i, j, k +1) H n+ 1 2 y (i, j, k) } (1.6) z H n+ 1 2 ( t 2 ) 1.3 CP-FDTD FDTD CP(Contour Path) FDTD[3][4][5] 2.6 H z 3

7 Hz node Ex, Ey node 1.1 : CP-FDTD CP- FDTD [6][7] 1.2 4

8 : 45 FDTD CP-FDTD CP-FDTD 3 FDTD 1 2 θ = tan θ = tan

9 2 45 FDTD CP-FDTD CP-FDTD 45 CP 2.1 FDTD FDTD 2 6

10 z b a c h y x 2.1 : 2.1 : 7.5cm cells x = y = z =5.0mm 20,000 PML 8 7

11 2.1.2 FDTD thin wire approach 2.2 FDTD r r E y,e z,h x 1/r E z,h x H y z H y,e z E x H y (x, y, z) = H y (i + 1 2,j,k+ 1 2 )( x 2 ) 1 x {1+c 1(z (k + 1 ) z)} (2.1) 2 E x (x, y, z) = E x (i + 1 2,j,k) x 1,z = k z 2 x = E x (i + 1 2,j,k+1) x 1,z =(k + 1) z (2.2) 2 x E z (x, y, z) = 0,x= i x = E z (i +1,j,k+ 1 2 ){1+c 2(z (k + 1 ) z)},x=(i + 1) x (2.3) 2 r0 H y E z ( i, j, k + 1 ( i +, j, k ) 2 1 ) 2 E x 1 ( i +, j, k + 1) 2 E z ( i + 1, j, k + 1 ) 2 1 E x ( i +, j, k) : 8

12 (2.1) c E d l = t s B d S (2.4) H y t (i + 1 2,j,k+ 1 2 )= 1 µ 0 z {E x(i + 1 2,j,k) E x(i + 1 2,j,k+1)} 2 + µ 0 xln x E z (i +1,j,k+ 1 r 0 2 ) (2.5) (2.5) H n+ 1 2 y (i + 1 2,j,k )=Hn 2 y (i + 1 2,j,k+ 1 2 ) + t µ 0 z {En x (i + 1 2,j,k) En x (i + 1 2,j,k+1)} + 2 t µ 0 xln x r 0 E n z (i +1,j,k+ 1 2 ) (2.6) H n+ 1 2 y (i 1 2,j,k )=Hn 2 y (i 1 2,j,k+ 1 2 ) + t µ 0 z {En x (i + 1 2,j,k) En x (i + 1 2,j,k+1)} 2 t µ 0 xln x Ez n (i 1,j,k+ 1 r 0 2 ) (2.7) H n+ 1 2 x (i, j + 1 2,k )=Hn 2 y (i, j + 1 2,k+ 1 2 ) t µ 0 z {En y (i, j + 1 2,k) En y (i, j + 1 2,k+1)} 2 t µ 0 yln y Ez n (i, j +1,k+ 1 r 0 2 ) (2.8) H n+ 1 2 x (i, j 1 2,k )=Hn 2 x (i, j 1 2,k+ 1 2 ) t µ 0 z {En y (i, j 1 2,k) En y (i, j 1 2,k+1)} + 2 t µ 0 yln y r 0 E n z (i, j 1,k+ 1 2 ) (2.9) r 0 x, y z x, y z n µ0 9

13 r thin wire approath r0 =0.1mm r0 =0.1mm Return loss [db] mea. r0=1mm r0=0.6675mm r0=0.4mm r0=0.1mm Frequency [GHz] 2.3 : 2.2 : mea. r0 =1mm r0 =0.6675mm r0 =0.4mm r0 =0.1mm

14 (Non-Uniform Mesh)[8] Yee Yee 1 FDTD 2.4 Lz1 z x y V L Lz2 (a) Fine Mesh Lx,y Coarse Mesh (b) 2.4 : 2.1 x = y = z =5mm x = y = z =1mm (L x L y L z1,2 ) 11

15 ( mm) ( x = y = z =5mm) Return loss [db] Mea. FDTD Non Uniform Mesh Frequency [GHz] 2.5 : 12

16 FDTD CP-FDTD CP E 0 FDTD CP CP-FDTD CP-FDTD (2.3) (2.1)(2.2) FDTD H = 1 E (2.10) t µ E = 1 H (2.11) t ε E d l = µ H d S (2.12) c t s C S H E CP-FDTD 2.6(a) ABEF H n+ 1 2 z CP-FDTD (a) (b) S ABF 0.5d x d y S ACDE (a) (b) (c) (d) Courant (2.13) c 13

17 t = 0.9 c (2.13) ( 1 d x ) 2 +( 1 d y ) 2 +( 1 d z ) 2 H n+ 1 2 x (i, j + 1 2,k )=Hn 2 x (i, j + 1 2,k+ 1 2 ) t + µ(s ABF + S BCDEF ) {En z (i, j +1,k+ 1 2 ) l ED E n y (i, j + 1 2,k+1) l DC E n z (i, j, k) (l CB + l BA )} (2.14) C D E B A F A B C F (a) (b) E D D E C F B A F A B E D C (c) (d) 2.6 : CP-FDTD 14

18 2.2.2 CP-FDTD CP-FDTD FDTD 2.7 Yee 2.7 (H x1,h x2 ) k+1 y C1 x z z k Hx1 C2 S1 S2 Hx2 j Ey j+1 y Ez 2.7 : CP C1,C2 (2.15),(2.16) H n+ 1 2 x1 (i, j + 1 2,k )=Hn 2 x1 (i, j + 1 2,k+ 1 2 ) t µs 1 {Ey n (i, j + 1 t,k+ 1) y} + { Ez n (i, j, k + 1 ) z} (2.15) 2 µs 1 2 H n+ 1 2 x2 (i, j + 1 2,k )=Hn 2 x2 (i, j + 1 2,k+ 1 2 ) t µs 2 { Ey n (i, j + 1 t,k) y} + {Ez n (i, j +1,k+ 1 ) z} (2.16) 2 µs (a) 2.8 V 15

19 E n z n Vz (0, 0, 0) = z V z z z z (2.17) E z V = z y x 2.8 : (b) V z y V 1 V 2,V 1,V 2 I 1,I 2. z E = V ( y) ( z) y x 2.9 : 16

2.2.4 45 2.10(a) 7.5cm 36 36cm 2.10(b) 0 2.10(c) CP 2.

20 (a) 7.5cm 36 36cm 2.10(b) (c) CP 2.10(d) CP 1 V Measurement (b) Stair-stepped V V1 V2 (c) Gap-feed (d) Vector-feed 2.10 : 17

21 CP-FDTD 0 Return loss [db] Mea. Stair-stepped Gap feed Vector feed Frequency[GHz] 2.11 : 18

22 2.3 : : [GHz] [ ]

23 [db] [deg.] : zx [db] [deg.] 2.13 : yz Mea. Stair-stepped Gap Feed Vector Feed 20

24 yz z 5 5 x y 5 5 fine : y = z =1mm coarse : x = y = z =5mm 2.14 : 45 yz y, z x x (a) x +x 5 (b) 0 +x 5 (c) x 1mm 5mm 21

25 2.16 x x x z (a) Non Uniform Mesh -x~+x y z (b) Non Uniform Mesh 0~+x y (c) Uniform Mesh fine : x = y = z =1mm coarse : x = y = z =5mm : x Return loss [db] Mea. Vector feed Non Uniform Mesh -x~+x Non Uniform Mesh 0~+x Uniform Mesh Frequncy [GHz] 2.16 : x 22

26 FDTD CP CP CP CP x y, z 23

27 3 FDTD 2 CP θ = tan 1 2 θ = tan CP-FDTD (a) 3.1(a)(b) 2 (a) (b) FDTD 3.1(a)(b) ( a AB,AH,EF,FG z b AB,BC,EG,AG z y ) 24

28 H G I H A B F K E J A G F E B z C D K L C D (a) (b) x y (i,j,k) 3.1 : (b) FDTD 2 3.1(a)(b) 3.1(a) 3.1(b) E z (i, j, k + AB 2 z ) = E y(i, j, k 1 2 ) E z (i, j, k +1 AH 2 z ) = E y(i, j, k ) (3.1) E z (i, j +1,k+ EF 2 z ) = E y(i, j +1,k 1 2 ) E z (i, j +1,k+1 FG 2 z ) = E y(i, j +1,k+ 3 2 ) (3.2) E z (i, j, k + BC 2 z ) = E y(i, j, k 1 2 ) E z (i, j, k +1 AB 2 z ) = E y(i, j, k ) (3.3) (3.1) (3.3) 3.1 ABCDEFA BCDEB 25

29 H x (i, j +1/2,k 1/2) E y E y (i, j + 1 2,k) = E z(i, j, k 1 2 ) AB y + E z(i, j +1,k 1 2 ) EF y ) (3.4) 3.1(b) BCDEFGB BCDEGB H x (i, j+1/2,k+1/2) E y (i, j+1 EG/2 y, k+ 1) E y (i, j +1 EG 2 x,k+1) = E z(i, j +1,k+ 1 2 ) EF EG (3.5) (3.3)(3.4) 3.1(a) H x (i, j+1/2,k 1/2) ABCDEFA H n+ 1 2 x (i, j + 1 2,k )=Hn 2 y (i, j + 1 2,k 1 2 )+ t µ 0 z y {Ey n (i, j + 1 2,k 1) y + En z (i, j + 1 2,k 1 EF )(1 + 2 z ) y Ez n (i, j, k 1 AB )(1 + ) z} (3.6) 2 z H x (i, j +1/2,k+1/2) AFGHA 3.1(b) H x (i, j +1/2,k+1/2) BCDEFGB H x (i, j +1/2,k+3/2) ABFHIA (3.1) (3.4) 3.1(a) H y (i, j +1/2,k+ 1/2) E z (i, j, k + AB/2 y) E z (i, j, k +1 AH/2 y) H n+ 1 2 y (i + 1 2,j,k+ 1 2 )=Hn 1 2 y (i + 1 2,j,k+ 1 2 )+ t µ 0 z x {[Ex n (i + 1 2,j,k+1) En x (i + 1,j,k+ 1)] x 2 Ez n (i +1,j,k+ 1 2 ) z + En z (i, j, k 1 2 )AB +Ez n (i, j, k + 3 )AH} (3.7) 2 H y (i 1/2,j,k 1/2),H y (i +1/2,j+1,k+1/2) H y (i 1/2,j+1,k 1/2) 3.1(b) H y (i +1/2,j,k+1/2),H y (i 1/2,j,k+1/2) H z (i +1/2,j+1/2,k+ 1),H z (i 1/2,j+1/2,k+1) 26

30 V y z V y z z (a)vector feed 1 x y (b)vector feed : y z θ y z (3.8) y z x = y = 1 tanθ z V y = V z (3.8) 2 y z, y z (3.9) x = y = z V y = 1 tanθ V z (3.9) (3.10) E tan,y z 27

31 Etan = E z + E y = ( V )2 =( V z z ) 2 +( V y y ) 2 (3.10) (3.8),(3.9) (3.10) 1 y z V y = V z = 2 y z V y = tanθ 1+(tanθ) V 2 tanθ 1+(tanθ) V (3.11) 2 1 2(1 + (tanθ) 2 ) V V z = 1 2 2(1 + (tanθ) 2 ) V (3.12) CP-FDTD θ = tan 1 2 θ = tan θ = tan θ = tan L θ 3.3 : θ = tan

32 3.4 E y (i, j + BA 2 y,k+1) E y(i, j +1 BC 2 y,k+1) E y(i, j 1,k+1) E y (i, j +1,k+1) E z E y (i, j 1,k+1) l AB z 3.4 H x H z k+2 k+1 D E z H x A B C H E y x k E H x E z F H x j-1 j j+1 j : E z (i, j, k ) = E y(i, j 1 2,k+1) l AB z (3.13) E y (i, j + BA 2 y,k+1) = E y(i, j 1,k+ 1) 2 (3.14) E y (i, j +1 BC 2 y,k+1) = E y(i, j + 3,k+ 1) 2 (3.15) H n+ 1 2 x (i, j + 1 2,k+ 1 2 )=Hn 1 2 y (i, j + 1 2,k+ 1 2 ) t µ 0 z y {En y (i, j + 1 2,k) y Ey n (i, j + 3 2,k+1)l BC + Ez n (i, j +1,k+ 1 ) z} (3.16) 2 29

33 H n+ 1 2 z (i + 1 2,j ,k)=Hn 2 z (i + 1 2,j+ 1 t,k) 2 µ 0 z y {[E n x (i + 1 2,j,k) En x (i + 1 2,j+1,k)] x +Ey n (i +1,j+ 1 2,k) y + En y (i, j 1 2,k) y 2 E y (i, j ,k) y 2 } (3.17) θ = tan θ = tan y z V y V y z V z y =0.25mm, z =0.5mm (3.11) V y V z 2 V 2 y z 5 V y V y z V z y =0.5mm, z =0.5mm (3.12) V y 10 V 10 V z 10 V CP

34 0 Return loss [db] Mea. Stair-stepped Gap feed Vector Feed Freqency [GHz] 3.5 : θ = tan θ = tan :

35 0 Return loss [db] Mea. Stair-stepped Gap feed Vector Feed Freqency [GHz] 3.6 : θ = tan

36 0 Return loss [db] Mea. Vector Feed 1 Vector Feed Freqency [GHz] 3.7 : θ = tan : θ = tan 1 2 mea. 1 2 [GHz] [ ]

37 0 [db] [deg.] 3.8 : θ = tan 1 2zx 0 [db] [deg.] 3.9 : θ = tan 1 2yz Mea. Vector Feed 1 Vector Feed 2 34

38 θ = tan 1 2 x y z x z Return loss [db] Mea. Uniform mesh Non Uniform Mesh -x~+x Non Uniform Mesh x= Freqency [GHz] 3.10 : θ = tan 1 2 fine : x = z =1mm, y =0.5mm coarse : x = z =5mm, y =2.5mm 35

39 3.4 θ = tan θ = tan L θ 3.11 : θ = tan y z V y V y z V z y =0.5mm, z =0.25mm (3.11) V y V z 2 V 2 y z 5 V y V y z V z y =0.5mm, z =0.5mm (3.12) V y 10 V 5 V z 10 V , Return loss [db] Mea. Stair-stepped Gap Feed Vector Feed Freqency [GHz] 3.12 : θ = tan

40 0 Return loss [db] Mea. Stair-stepped Gap Feed Vector Feed Freqency [GHz] 3.13 : θ = tan Return loss [db] Mea. Vector Feed 1 Vector Feed Freqency [GHz] 3.14 : θ = tan : θ = tan mea. 1 2 [GHz] [ ]

41 [db] [deg.] : θ = tan zx [db] [deg.] 3.16 : θ = tan yz Mea. Vector Feed 1 Vector Feed 2 38

42 Mea. Uniform mesh Non Uniform Mesh -x~+x Non Uniform Mesh x= : θ = tan fine : x = y =1mm, z =0.5mm coarse : x = y =5mm, z =2.5mm θ = tan θ = tan x θ = tan

43 3.5 FDTD 2 CP 2 θ = tan 1 2 θ = tan CP x

44 4 FDTD r 0 =0.4mm CP x y z 0.71 CP 1 2 θ = tan 1 2 θ = tan

45 OB 42

46 [1], FDTD,, [2], FD-TD, ( 17/18 ). [3] T.G.Jurgens, A.Taflove, K.Umashankar and T.G.Moore, Finite-Difference Time- Domain Modeling of Curved Surfaces, IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Vol40, No.4, APPIL [4],, FDTD,, Vol.100, No.340AP ,2000. [5],, CP F FDTD,, AP , pp [6],, CP FDTD,, AP ( ). [7] K.Yamamoto and H.Iki, Slant Wire Models Using Accurate Correction Techniques in FDTD Method, International Conference on Power Systems Transients- IPST2003. [8],, FDTD,, AP98-8, pp.7-12, May

47 1.,,, FDTD,, B-1-77, T.Lu, N.Michishita and H.Arai, Oblique Voltage Excitation for CP-FDTD, 2004 International Symposium on Antennas and Propagation, 2C4-1, Sendai, Japan, Aug ,,, FDTD,, B-1-76,

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main.dvi FDTD S A Study on FDTD Analysis based on S-Parameter 18 2 7 04GD168 FDTD FDTD S S FDTD S S S S FDTD FDTD i 1 1 1.1 FDTD.................................... 1 1.2 FDTD..................... 3 2 S 5 2.1 FDTD