L4_Supplemental - SIMPLIFIED EXPRESSIONS FOR...

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Unformatted text preview: SIMPLIFIED EXPRESSIONS FOR Z—PROPAGA’I‘ING WAVES Assume Eg(:1;,y,z) : E($,y) (3—72 Hg(m, y, z) : H(J:, 6—73 Where from NIaXWCI].’s equations (“g” refers to guided wave) V X ES!) : ijwan, V X Hg : +jweEg Expanding these we have 8E? , 8H3 . 3y +7193; : ——jw,u,H$ 3y + ’yfly — jWEE‘T 8E3 . 8H3 , 7E1,” + ax — +lijLHy —*y.[ a; —— am — jaieEy 5E1, 5E , 8H " (9H — a: : e f]; y — a: : * EN (9m 8y Jw'u 81 (9y 3%”: L which can be further manipulated to express the a: and y components in terms of the 2 components. We obtain H m _i [811; M M0132 _' 1'13 7 323 J (r 331 1 . J 2 2 . i . | r \ fl ' Hy — Hz (7 8y I Jwe 83:) h k +7” 1 I ,— E fl 1 8E2. an; y _ 1’13 7 5y ELM 3:8 TM Waves (Hz 2 0) m [12 8y _ "y y i 2102352 Hy— h2 82: .1 7/ (3E; by 2 *7 ’ h) 3:0 E2 Hy : _l 8. h? 6y NOTE: ZTM : Ear/Hy : —Ey/H$ = 7/wa : wave impedance 7E 77 = Z : «nu/e . Also, H 2 (l/ZTM)2 X E (for the transverse components only). TE Waves (E2 2 0) ’7 (9H; 7 ’12 ’f Hmzm—w m—w ,— E1: E1 112 33: h? (Mu J ‘1 ‘ 7 (3H; H 3 —.— y b? 8y Ex 2 _i% QHZ [12 (3y ‘, jam. (9H; b x —- g h? 053 NOTE: ZTE I “fly/Hm : Egg/Hy : jam/7 : wave impedance 7% n : Z 2 . Also for the transverse components E : WZTEQ X H]. E Waves and H Waves in Cartesian Coordinates Consider a time—harmonic field in a source—free homogeneous region. Let 82EZ dz2 8sz :yzEz and 822 :vsz. (1) Then it is convenient to use EZ and H ,3 as generating functions to determine the other field compo- nents as follows. Let hzzyz+to2pexyz+k2 (2) E Waves H Waves wit: (‘0 Hx : flog}; 88—? E2 2 0 (5) aZ = o Hy = (7) A more general field can be expressed as the sum of an E wave and an H wave. Still more generality is obtained by considering the field to be the sum or integral of E and H waves having different values of 7. Some of the solutions of (1) are: COShWZ): Sinh('YZ), 8W: 6—72 (8) come), sinrszZ), eszZ’ [jazz (9) where y : jBZ. If the field does not satisfy (1), the Fourier transform may be applied to express the field as a spectrum of waves that do satisfy (1). Thus, the field in a homogeneous source—free region is determined by EZ and Hg. However, uniqueness is spoiled by the possible existence of TEM waves having EZ : 0 and HZ : 0. SIMPLIFIED EXPRESSIONS FOR X ~PROPAGATING WAVES Assume (“‘g” indicates a guided wave) where from Maxwell’s equations Expanding these we have anZ as), _ , 8Hz an}, _ _ __,_ _ _. Hx w _ _ z ' E ay dz 1°)“ By dz m x 8E _ 3H a—flhl 2 w 103w, “523+sz : jCOSE} 3E de VEy+ a— : ijHz FYI—131+ T); — “JmEEz These can be further manipulated to express the z and )2 components in terms of the x components. We obtain 1 BHX 8E Hy: ——< —-ujon—JC—) TE and TM Fields Separable in the Cylindrical Coordinate System The harmonic electromagnetic fields listed below satisfy Maxwell’s equations in a homogeneous source~free region. TE Fields TM Fields _ '00 EP = ij‘uqu’Z Ep : CR’cpz’ 1 Eq, : C jmyR’qnz E¢ : CERCD’Z’ EZ = 0 EZ : CBZRCDZ Hp : CR’tDZ’ HP 2 Gig—FIRth 1 H4) : CERCD’Z’ Hq, = —j0)eCR’<1)Z HZ = CBZRCDZ HZ = 0 C denotes an arbitrary constant. The time dependence 61"” is understood. R is a function of p only, (I) is a function of q) only, and Z is a function of z only. Primes indicate differentiation with respect to p, (I) or z. The functions satisfy the following differential equations: where and kp and h are constants. Note: R’ = kpjglkpp) ifR 2 Jm(kpp) R’ = kpNMkpp) if R 2 Nmfkpp) : NmUcpp) R’ = kagdpp) ifR : Hm(kpp) Some solutions of these differential equations are listed below. R(p) E Jm(ka) t 0050714)) Z(Z) : COSUSZZ) Mnkpp) sin(m¢) sin(Bzz) Hmkpp) give em Wisp) e‘jm‘l 6‘th If [i : 0, the radial function is R(p) : pim. Note the definitions: Jm(-) = Bessel function of order m : Neumann function of order m Hm1 : mth order Hankel function of the 1st kind Hm” ( : mth order Hankei function of the 2nd kind ...
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