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Prof. D. R. Wilton Notes 19 Waveguiding Structures Waveguiding Structures ECE 3317 [Chapter 5]

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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

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Transmission lines Lossless: 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 E z or H z components of the fields (TEM z ) Properties (always real: = 0 )

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Fiber-Optic Guide Properties Has a single dielectric rod Can propagate a signal only at high frequencies Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both E z and H z components of the fields

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Fiber-Optic Guide (cont.) Two types of fiber-optic guides: 1) Single-mode fiber 2) Multi-mode fiber Carries a single mode, as with the mode on a waveguide. Requires the fiber diameter to be small relative to a wavelength. Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).

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Multi-Mode Fiber http://en.wikipedia.org/wiki/Optical_fiber Higher index core region

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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 E z or H z component of the fields (TM z or TE z ) Properties http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)

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Waveguides (cont.) Cutoff frequency property (derived later) (wavenumber of material inside waveguide) (wavenumber of material at cutoff frequency c ) (propagation) (evanescent decay) In a waveguide: We can write

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Field Expressions of a Guided Wave All six field components of a wave guided along the z -axis can be expressed in terms of the two fundamental field components E z and H z. Assumption: (This is the definition of a guided wave. Note for such fields, ) A proof of the “statement” above is given next. Statement: "Guided-wave theorem"

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Field Expressions (cont.) Proof (illustrated for E y ) or Now solve for H x :

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Field Expressions (cont.) Substituting this into the equation for E y yields the result Next, multiply by

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Field Expressions (cont.) Solving for E y, we have: This gives us The other three components E x, H x, H y may be found similarly.

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Field Expressions (cont.) Summary of Fields

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Field Expressions (cont.) TM z Fields TE z Fields

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TEM z Wave To avoid having a completely zero field, Assume a TEM z wave: TEM z Hence,

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TEM z Wave (cont.) Examples of TEM z waves: In each case the fields do not have a z component! A wave in a transmission line A plane wave E H coax x y z E H S plane wave

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TEM z Wave (cont.) Wave Impedance Property of TEM z Mode Faraday's Law: Take the x component of both sides: The field varies as Hence, Therefore, we have

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TEM z Wave (cont.) Now take the y component of both sides: Hence, Therefore, we have Hence,

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TEM z Wave (cont.) These two equations may be written as a single vector equation: The electric and magnetic fields of a TEM z wave are perpendicular to one another, and their amplitudes are related by . Summary: The electric and magnetic fields of a TEM z wave are transverse to z, and their transverse variation as the same as electro- and magneto-static fields, rsp.

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TEM z Wave (cont.) Examples coax x y z E H S plane wave microstrip The fields look like a plane wave in the central region. x y d z

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Waveguide In a waveguide, the fields cannot be TEM z. Proof: (property of flux line) Assume a TEM z field (Faraday's law in integral form) y x waveguide E PEC boundary A B C contradiction!

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Waveguide (cont.) In a waveguide, there are two types of fields: TM z : H z = 0, E z 0 TE z : E z = 0, H z 0

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