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2.9.2. kl =, tg kl = 1. ZZ = :

Z , { .

. , , . .

Z Z02 (..). / Z03, Z Z Z 01 03 Z= Z01:

Z p Z03 = Z01 Z02 :

11111111111111 , , (..), , , . , / . , .

2.10. T- ZZ . T- Z Z 2 H (..). :

Z1 +2ZZ + ZZ2(Z1 + Z) Z1 + ZZ = Z1 + = :

Z1 + Z2 + Z 1 + Z Z1 + Z , Z:

Z + jZ0 tg kl Z = :

j 1 + Z tg kl Z , Z1 + 2ZjZ0 tg kl = ZZ1 + Zj tg kl = :

Z0 Z1 + Z Z1 Z( ):

Z1 = jZ0 tg kl kl =(2n +1) Z2 = ; jZ0 csc kl Z1 = ; jZ0 ctg kl kl =2n :

Z2 = jZ0 csc kl , , .

2.11. () 2.11.1. U ; = :

U l , U(l) = U e;jkl jkl U(l) = U e U ;(l) = e; 2jkl = ;e; 2jkl:

U , j ;j = const,.. , 2kl.

2.11.2. , (.. ..).

, . , Z ;:

Z ; Z ; Z0 Z0 z ; ; = = = :

Z + Z0 Z z + + Z . . . j ;j 1, . ; Z - R 0 . . X = const R = const .

X = const X > -1 X=0 +Z=0 Z= R = const X < X = const R = const. , Z0. - . - l ' = 2kl = 4 :

.

, , ( ) . , , 180o, X B R G. , Y0.

2.12. TEM , , , { .

, . source to to load R= . .

00 00 00 11 11 11 00 00 00 11 11 11 . 00 00 00 11 11 11 ( ) X X Qi = Cik Uk Us = Psm Qm k m Qi { i- , Us { s- , Cik { ( i = k), Psm { . :

Q = C U U = P Q Q = fQig U = fUsg C = fCik g P = fPsmg { .

C P C P = 1 { ,.. C P { :

; P = C.

, X = Lik Ik i k { , i- i , Lik { . = L I:

, :

dU = ; j!L I dz dI = ; j!C U dz ( ).

:

d2U dI = ; j!L = ; !2LC U dz2 dz LC { ( L C).

LC, x i L P.

k 000 111 111 000 111 i k i = k (..).

a k- , i- y Ui:

a Z i Uk = ; Eix dx Eix { x- , i- . Eix ~ Eix = rot Hi x j!" ~ Hi { , i- . Hz =0, @Hiy 1 @Hiy ~ rot Hi = ; Eix = ; :

x @z j!" @z , a a Z Z 1 @Hiy 1 @ i Uk = dx = Hiy dx:

j!" @z j!" @z 0 R i ; Hiy dx = { , i- , k i k- . = LkiIi, k 1 @Ii i Uk = ; Lki :

j!" @z @Ii @z , @Q @I + = 0:

@t @z @Ii = ; j!Qi :

@z i Uk, 1 Lki i Uk = ; Lki (;j!Qi) = Qi:

j!" " " =, v { v . , i Uk = v2 Lki Qi:

P, Pki = v2 Lki i = k:

i = k. P P = v2L L = :

v , d2U ! = ; PC U = ; k2U:

dz2 v d2U + k2U = 0:

dz Ui, Ui,.. . jkz U = Ae;jkz + Be U, A, B { . ;

1 dU jkz I = ; L; 1 = Z0 ;1 Ae;jkz ; Be j! dz jkz Z0I = Ae;jkz ; Be :

! L = Z0 (Z0 { , k ).

, .

z = 0, :

U(0) = A + B Z0I(0) = A ; B:

A B ,..

z = l:

jkl U(l) = Ae;jkl + Be jkl Z0I(l) = Ae;jkl ; Be U(l) + Z0I(l) jkl A = e U(l) ; Z0I(l) B = e;jkl :

A B U (0) I(0), :

U(0) = U (l) cos kl + jZ0I(l) sin kl I(0) = jZ0 ; 1U(l) sin kl + I(l) cos kl:

, , , . , .

, , .

2.13. . p p = Z1 Y1 = (j!L1 + R1 )(j!C1 + G1 ) = s R1 G= ;!2L1C1 1 + 1 + = j!L1 j!Cs p R1 G= j! L1C1 1 + 1 + :

j!L1 j!C R1 !L1 G1 !C1, ( ) p 1 R1 G1 1 R1 G1 j! L1C1 1 + + + ; = 2j! L1 C1 8!2 L1 C( ) 1 R1 G1 2 1 R1 G= jk 1 + ; + + = jk0 + 8!2 L1 C1 2 Z0 Yr r L1 C Z0 =, Y0 = { C1 L .

(v = !=k0) ,..

. , , :

0 1 R1 GU = U0 e; z e;jk z = + :

2 Z0 Y . . , R1 G= :

L1 C p R1 R = j! L1C1 1 + = jk + :

j!L1 Z (R1 ), , ( ). , R1 !L1 R = :

2ZR1 -, .

, . , R1 G =. , L1 Cp L1=C1,.. .

. .

: d1 =0.6 , d2=4.0 Z0=75 f=1000.

l R1 =0:0175, S l { (1 ) S { , 2 (S = d1, { -).

- f=1 000 2:1 10; 3 .

: R1 4:46 /, R1 0:65 /, R1 5:1 /.

5:= =0:031 / =0:031 8:69 = 0:29 /.

2 3.

3.1. . E- H- x , y . , z (..) .

- , .

~ ~ ~ ~ rot E = ; j! H rot H = j!"E:

, :

~ ~ ~ ~ E + k2E = 0 H + k2H = 0:

Ez Hz { , @2Ez Ez + + k2Ez = @z@2Hz Hz + + k2Hz = @z { , .

:

Ez = E (x y) Ek (z) Hz = H(x y) Hk (z):

, d2Ek ; Ek = 0 E + (k2 + )E = dzd2Hk ; Hk = 0 H + (k2 + )H = dz { .

, z z : e. = jk, , .

, , ~ Et = 0 Hn = Et { , Hn { .

. , z ( z ; ):

@Ez ;j! Hx = + Ey @y @Ez ;j! Hy = ; Ex ;

@x @Ey @Ex ;j! Hz = ;

@x @y @Hz j!"Ex = + Hy @y @Hz j!"Ey = ; Hx ;

@x @Hy @Hx j!"Ez = ; :

@x @y Ex, Ey, Hx, Hy Ez, Hz. Ey, . @Ez @Hz (!2 " + )Hx = j!" ;

@y @x ( !2 " = k2) j!" @Ez @Hz Hx = ; :

2 k2 + @y k2 + @x j!" @Ez @Hz Hy = ; ;

2 k2 + @x k2 + @y j! @Hz @Ez Ex = ; ;

2 k2 + @y k2 + @x j! @Hz @Ez Ey = ; :

2 k2 + @x k2 + @y Ez Hz :

Ez + k2Ez = Hz + k2Hz = @ , , = ;, @z d2Ez d2Ez + + (k2 + )Ez = dx2 dyd2Hz d2Hz + + (k2 + )Hz = 0:

dx2 dy , Ex, Ey, Hx Hy.

:

Ez = (k2 + ) (x y) e; z Hz = (k2 + ) (x y) e; z (x y), (x y) { . , , + (k2 + ) = x y + (k2 + ) = x y , k2 + = g2, + g2 = x y + g2 = 0:

x y :

@ @ Hx = j!" ; e; z @y @x @ @ Hy = ;j!" ; e; z @x @y @ @ Ex = ;j! ; e; z @y @x @ @ Ey = j! ; e; z @x @y , ~ H = j!" rot ( ~ e; z) + grad div ( ~ e; z) + k2 e; z ~ z0 z0 z~ E = ; j! rot ( ~ e; z) + grad div ( ~ e; z) + k2 e; z ~ z0 z0 z ~ { z.

z, grad ( ) = grad + grad ~ ~ ~ rot ( F) = rot F + grad F ~ ~ ~ div ( F) = div F + grad F rot ( ~ e; z ) = e; z grad ~ z0 zgrad div ( ~ e; z ) = ; e; z grad + ~ :

z0 z0e; z , ~ H = j!" e; zgrad ~ + (k2 + )e; z ~ ; e; zgrad z0 z~ E = ; j! e; zgrad ~ + (k2 + )e; z ~ ; e; zgrad :

z0 z , , .

~ = ~ e; z ~ 0 = ~ e; z | z0 z .

, , (z) .

. ~ Et Hn .

~ Et z , ,..

. . Ez = 0 =0 C.

~ Es, ~ ~ ~ s n, n (.

.). Ey ( y s, x { n) @ @ 1111111111111 Es = j! ; :

@n @s =0 C, , Es = 0, , @ =@n =0 C.

, Hn =0 C. Hn Hx:

@ @ Hn = ; j!" ; :

@s @n , n (x) s (y) = 0 C @ = 0 C @n + g2 = x y + g2 = x y g2 = k2 +.

: . . , z- , -, z- .

, g2 { . . , , .

, . Z I @ ( + grad grad ) dS = dC @n S C S { , C { , .

. Z I @ ( + jgrad j2) dS = dC = @n S C . , = ;g2 .

, , R jgrad j2 dS S R g2 = j j2 dS S , g2 > 0. , .

, 2 2 : g1 g2 : : : gn : : :. , 1. .

, , , S:

Z Z 2 dS = 0 dS = 0 gn = gm:

n m n m S S :

Z I @ m @ n ( ; ) dS = ( ; ) dC:

n m m n n m @n @n S C Z ( ; ) dS = 0:

n m m n S 2 = ;gn, = ;gm, n n m m Z 2 (gn ; gm) dS = 0:

n m S 2 gn = gm, Z dS = 0:

n m S , Z dS = 0:

n m S ,.. n m , .

, ,..

, , ( ).

n , gn:

p 2 = gn ; k2 = gn ; k2:

n n , { n ,.. k2 > gn:

:

!2 " > gn gn ! > = !n | p " n- . , p = j = j k2 ; gn:

n n .

, 2 = k = n n { , { n , ".

, (gn = ):

n n s 1 1 = ;

2 n n = r :

n 1 ;

n :

! ! n c v = = = r > c:

n 1 ;

n , , , (E TM, 0) (H TE, 0) . ( ( ( ( ~ ~ ~ ~ , { Ene) Hne) { Enh) Hnh).

, , Z Z Z ~ ~ ~ ~ ~ ~ En Em dS = 0 Hn Hm dS = 0 (En Hm) ~ dS = zS S S 2 gn = gm. .

, .

H- :

~ E = ; j! e; zgrad ~ z~ H = ; e; z grad + (k2 + ) e; z ~ z0:

, ~ ~ E H = grad (grad ~ = 0:

z0) , ( ) ~ ~ E = Zh H ~ z Zh = j! =. Zh :

p ! =" Zh = = v = p :

2 1 ; = n E- ~ E = ; e; z grad + (k2 + ) e; z ~ z~ H = j!" e; z grad ~ z0:

~ ~ E H . , ~ ~ E = Ze H ~ z Ze = =j!". Ze :

s r Ze = = = 1 ; :

!" v " " n , , gn ! > !n = p " gn , g1. , .

, , .

3.2. 3.2.1. H- H- (..) y @2 @+ + g2 = a b @x2 @y@ =0.

b @n x a (x y) = X(x) Y(y) :

1 @2X 1 @2Y + = ; gX @x2 Y @y ,.. :

@2X @2Y 2 + gx X = 0 + gy Y = @x2 @y2 gx + gy = g2.

X = A cos gxx + B sin gxx Y = C cos gyy + D sin gyy:

dX : x =0 =0, , dx B = 0 X = cos gxx dY y =0 = 0, , dy D = 0 Y = cos gyy dX x = a =0,.. sin gxa = 0, dx n gxa = n gx = n = 0 1 2 : : :

a dY y = b =0,.. sin gyb =0, dy m gyb = m gy = m = 0 1 2 : : : :

b (x y):

n m (x y) = cos x cos y a b .

gnm 2 n m gnm = + n = 0 1 2 : : : m = 0 1 2 : : : :

a b , n m , = const .

( ):

2 2 = = s s = nm 2 2 2 gnm n m n m + + a b a b.. .

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