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**ENE 428 Microwave Engineering**

Lecture 7 Waveguides RS

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**Review Impedance matching to minimize power reflection from load**

Lumped-element tuners Single-stub tuners Microstrip lines The most popular transmission line Knowing the characteristic impedance and the relative dielectric constant of the material helps determine the strip line configuration and vice versa. Attenuation conduction loss dielectric loss radiation loss

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**A pair of conductors is used to guide TEM wave**

Microstrip Parallel plate Two-wire TL Coaxial cable

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The use of waveguide Waveguide refers to the structure that does not support TEM mode, bring up “the cutoff frequency”

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**General wave behaviors along uniform guiding structures (1)**

The wave characteristics are examined along straight guiding structures with a uniform cross section such as rectangular waveguides We can write in the instantaneous form as We begin with Helmholz’s equations: assume WG is filled in with a charge-free lossless dielectric

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**General wave behaviors along uniform guiding structures (2)**

We can write and in phasor forms as and

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**Use Maxwell’s equations to show and in terms of z components (1)**

From we have

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**Use Maxwell’s equations to show and in terms of z components (2)**

We can express Ex, Ey, Hx, and Hy in terms of z-component by substitution so we get for lossless media = j,

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**Propagating waves in a uniform waveguide**

Transverse ElectroMagnetic (TEM) waves, no Ez or Hz Transverse Magnetic (TM), non-zero Ez but Hz = 0 Transverse Electric (TE), non-zero Hz but Ez = 0

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**Transverse ElectroMagnetic wave (TEM)**

Since Ez and Hz are 0, TEM wave exists only when A single conductor cannot support TEM

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**Transverse Magnetic wave (TM)**

From We can solve for Ez and then solve for other components from (1)-(4) by setting Hz = 0, then we have Notice that or j for TM is not equal to that for TEM .

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Eigen values We define Solutions for several WG problems will exist only for real numbers of h or “eigen values” of the boundary value problems, each eigen value determines the characteristic of the particular TM mode.

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**Cutoff frequency From The cutoff frequency exists when = 0 or or**

We can write

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**a) Propagating mode (1) or and is imaginary Then**

This is a propagating mode with a phase constant :

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**a) Propagating mode (2) Wavelength in the guide,**

where u is the wavelength of a plane wave with a frequency f in an unbounded dielectric medium (, )

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a) Propagating mode (3) The phase velocity of the propagating wave in the guide is The wave impedance is then

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**b) Evanescent mode or Then Wave diminishes rapidly with distance z.**

ZTM is imaginary, purely reactive so there is no power flow.

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**Transverse Electric wave (TE)**

From We can solve for Hz and then solve for other components from (1)-(4) by setting Ez = 0, then we have Notice that or j for TE is not equal to that for TEM .

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TE characteristics Cutoff frequency fc,, g, and up are similar to those in TM mode. But Propagating mode f > fc Evanescent mode f < fc

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Ex1 Determine wave impedance and guide wavelength (in terms of their values for the TEM mode) at a frequency equal to twice the cutoff frequency in a WG for TM and TE modes.

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