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

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

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 )

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

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).

Multi-Mode Fiber Higher index core region

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

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

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"

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

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

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.

Field Expressions (cont.) Summary of Fields

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

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

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

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

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

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.

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

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!

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