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ECE 3317 Prof. Ji Chen Spring 2014 Notes 19 Waveguiding Structures 1 Waveguiding Structures A waveguiding structure is one that carries a signal (or power) from one point to another. There are three common types: Transmission lines Fiber-optic guides Waveguides 2 Transmission lines Properties Has two conductors running parallel Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is loss May or may not be immune to interference Does not have Ez or Hz components of the fields (TEMz) j R j L G jC k z j j R j L G jC j Lossless: kz LC k (always real: = 0) 3 Fiber-Optic Guide Properties Has a single dielectric rod Can propagate a signal at any frequency (in theory) Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields (“hybrid mode”) 4 Fiber-Optic Guide (cont.) Two types of fiber-optic guides: 1) Single-mode fiber Carries a single mode, as with the mode on a waveguide. Requires the fiber diameter to be small relative to a wavelength. 2) Multi-mode fiber Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect). 5 Multi-Mode Fiber Higher index core region max max http://en.wikipedia.org/wiki/Optical_fiber 6 Multi-Mode Fiber (cont.) Higher index core region max c max / 2 c nrod Assume cladding is air At left end of rod: nair sin max nrod sin c 2 sin max nrod cos c 7 Multi-Mode Fiber (cont.) Higher index core region max At top boundary with air: sin max nrod cos c nrod sin c nair sin 90o 1 nrod 1 sin 2 c nrod 1 1 / nrod 2 sin c 1 / nrod 2 sin max nrod 1 8 Properties Waveguide Has a single hollow metal pipe Can propagate a signal only at high frequency: > c The width must be at least one-half of a wavelength Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz) Inside microwave oven http://en.wikipedia.org/wiki/Waveguide_(electromagnetism) 9 Waveguides (cont.) Cutoff frequency property (derived later) In a waveguide: k We can write kc c kz k k 2 2 1/2 c kc constant (wavenumber of material inside waveguide) (definition of cutoff frequency) c : kz k 2 kc2 real (propagation) c : kz j kc2 k 2 imaginary (evanescent decay) 10 Field Expressions of a Guided Wave Statement: All six field components of a guided wave can be expressed in terms of the two fundamental field components Ez and Hz. "Guided-wave theorem" Assumption: E x, y, z E 0 x, y e jkz z H x, y , z H 0 x, y e jk z z (This is the definition of a guided wave.) A proof of this statement is given next. 11 Field Expressions (cont.) Proof (illustrated for Ey) H j E 1 H z H x Ey j x z or 1 H z Ey jk H z x j x Now solve for Hx : E j H 12 Field Expressions (cont.) E j H 1 E z E y Hx j y z 1 E z jk z E y j y Substituting this into the equation for Ey yields the result 1 H z 1 E z Ey jk z jk z E y j x j y Next, multiply by j j k 2 13 Field Expressions (cont.) This gives us H z E z k E y j jk z k z2 E y x y 2 or H z E z k k E y j x jkz y 2 2 z Solving for Ey, we have: j H z jkz E z Ey 2 2 2 2 k k z x k k z y The other three components Ex, Hx, Hy may be found similarly. 14 Field Expressions (cont.) Summary of Fields j H z jk z E z Ex 2 2 2 2 k k z y k k z x j H z jkz E z Ey 2 2 2 2 k k z x k k z y j E z jk z H z Hx 2 2 2 2 k k y k k z z x j E z jk z H z Hy 2 2 2 2 k k x k k z z y 15 TEMz Wave Assume a TEMz wave: Ez 0 Hz 0 To avoid having a completely zero field, k k 0 2 2 z Hence, TEMz kz k 16 TEMz Wave (cont.) Examples of TEMz waves: A wave in a transmission line (no conductor loss) A plane wave In each case the fields do not have a z component! kz k z S E H Coax H x Plane wave y E 17 TEMz Wave (cont.) Wave Impedance Property of TEMz Mode Faraday's Law: E j H Take the x component of both sides: The field varies as E z E y j H x y z E y x, y, z E y 0 x, y e jkz Hence, jk E y j H x Therefore, we have Hx k Ey 18 TEMz Wave (cont.) E j H Now take the y component of both sides: Hence, jk Ex jH y Therefore, we have Hence, E z E x j H y x z E x Hy k Ex Hy 19 TEMz Wave (cont.) Summary: Ey Hx Ex Hy These two equations may be written as a single vector equation: E zˆ H The electric and magnetic fields of a TEMz wave are perpendicular to each other, and the amplitudes of them are related by . 20 TEMz Wave (cont.) Examples z S E H Plane wave y H E x Coax E H The fields look like a plane wave in the central region. Microstrip “Quasi-TEM” (TEM-like at low frequency) 21 Waveguide In a waveguide, the fields cannot be TEMz. y PEC boundary A Proof: nˆ zˆ C E Assume a TEMz field x B waveguide B E dr 0 (property of flux line) C A E dr C S E dr 0 B B nˆ dS z dS 0 t t S contradiction! (Faraday's law in integral form) 22 Waveguide (cont.) In a waveguide (hollow pipe of metal), there are two types of fields: TMz: Hz = 0, Ez 0 TEz: Ez = 0, Hz 0 23