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Rectangular Waveguides

- Dr. S. Cruz-Pol
- INEL 6216
- University of Puerto Rico
- Mayagüez

Waveguide components

Waveguide to coax adapter

Rectangular waveguide

E-tee

Waveguide bends

Figures from www.microwaves101.com/encyclopedia/w

aveguide.cfm

More waveguides

http//www.tallguide.com/Waveguidelinearity.html

Uses

- To reduce attenuation loss
- High frequencies
- High power
- Can operate only above certain frequencies
- Acts as a High-pass filter
- Normally circular or rectangular
- We will assume lossless rectangular

Rectangular WG

- Need to find the fields components of the em

wave inside the waveguide - Ez Hz Ex Hx Ey Hy
- Well find that waveguides dont support TEM

waves

http//www.ee.surrey.ac.uk/Personal/D.Jefferies/wg

uide.html

Rectangular Waveguides Fields inside

- Using phasors assuming waveguide filled with
- lossless dielectric material and
- walls of perfect conductor,
- the wave inside should obey

Then applying on the z-component

Fields inside the waveguide

Substituting

Other components

- From Faraday and Ampere Laws we can find the

remaining four components

So once we know Ez and Hz, we can find all the

other fields.

Modes of propagation

- From these equations we can conclude
- TEM (EzHz0) cant propagate.
- TE (Ez0) transverse electric
- In TE mode, the electric lines of flux are

perpendicular to the axis of the waveguide - TM (Hz0) transverse magnetic, Ez exists
- In TM mode, the magnetic lines of flux are

perpendicular to the axis of the waveguide. - HE hybrid modes in which all components exists

TM Mode

- Boundary conditions

From these, we conclude X(x) is in the form

of sin kxx, where kxmp/a, m1,2,3, Y(y) is

in the form of sin kyy, where kynp/b, n1,2,3,

So the solution for Ez(x,y,z) is

Figure from www.ee.bilkent.edu.tr/microwave/prog

rams/magnetic/rect/info.htm

TM Mode

- Substituting

TMmn

- Other components are

TM modes

- The m and n represent the mode of propagation and

indicates the number of variations of the field

in the x and y directions - Note that for the TM mode, if n or m is zero, all

fields are zero. - See applet by Paul Falstad
- http//www.falstad.com/embox/guide.html

TM Cutoff

- The cutoff frequency occurs when
- Evanescent
- Means no propagation, everything is attenuated
- Propagation
- This is the case we are interested since is when

the wave is allowed to travel through the guide.

Cutoff

attenuation

Propagation of mode mn

fc,mn

- The cutoff frequency is the frequency below which

attenuation occurs and above which propagation

takes place. (High Pass) - The phase constant becomes

Phase velocity and impedance

- The phase velocity is defined as
- And the intrinsic impedance of the mode is

Summary of TM modes

Related example of how fields look Parallel

plate waveguide - TM modes

0 a x

TE Mode

- Boundary conditions

From these, we conclude X(x) is in the form

of cos kxx, where kxmp/a, m0,1,2,3, Y(y)

is in the form of cos kyy, where kynp/b,

n0,1,2,3, So the solution for Ez(x,y,z) is

Figure from www.ee.bilkent.edu.tr/microwave/prog

rams/magnetic/rect/info.htm

TE Mode

- Substituting
- Note that n and m cannot be both zero because the

fields will all be zero.

TEmn

- Other components are

Cutoff

attenuation

Propagation of mode mn

fc,mn

- The cutoff frequency is the same expression as

for the TM mode - But the lowest attainable frequencies are lowest

because here n or m can be zero.

Dominant Mode

- The dominant mode is the mode with lowest cutoff

frequency. - Its always TE10
- The order of the next modes change depending on

the dimensions of the guide.

Summary of TE modes

Variation of wave impedance

- Wave impedance varies with frequency and mode

h

hTE

h

hTM

0

fc,mn

Example

- Consider a length of air-filled copper X-band

waveguide, with dimensions a2.286cm, b1.016cm

operating at 10GHz. Find the cutoff frequencies

of all possible propagating modes. - Solution
- From the formula for the cut-off frequency

Example

- An air-filled 5-by 2-cm waveguide has
- at 15GHz
- What mode is being propagated?
- Find b
- Determine Ey/Ex

Group velocity, ug

- Is the velocity at which the energy travels.
- It is always less than u

http//www.tpub.com/content/et/14092/css/14092_71.

htm

Group Velocity

- As frequency is increased, the group velocity

increases.

Power transmission

- The average Poynting vector for the waveguide

fields is - where h hTE or hTM depending on the mode

W/m2

W

Attenuation in Lossy waveguide

- When dielectric inside guide is lossy, and walls

are not perfect conductors, power is lost as it

travels along guide. - The loss power is
- Where aacad are the attenuation due to ohmic

(conduction) and dielectric losses - Usually ac gtgt ad

Attenuation for TE10

- Dielectric attenuation, Np/m
- Conductor attenuation, Np/m

Dielectric conductivity!

Waveguide Cavities

- Cavities, or resonators, are used for storing

energy - Used in klystron tubes, band-pass filters and

frequency meters - Its equivalent to a RLC circuit at high

frequency - Their shape is that of a cavity, either

cylindrical or cubical.

Cavity TM Mode to z

TMmnp Boundary Conditions

From these, we conclude kxmp/a kynp/b kzpp/

c where c is the dimension in z-axis

c

Resonant frequency

- The resonant frequency is the same for TM or TE

modes, except that the lowest-order TM is TM111

and the lowest-order in TE is TE101.

Cavity TE Mode to z

TEmnp Boundary Conditions

From these, we conclude kxmp/a kynp/b kzpp/

c where c is the dimension in z-axis

c

Quality Factor, Q

- The cavity has walls with finite conductivity and

is therefore losing stored energy. - The quality factor, Q, characterized the loss and

also the bandwidth of the cavity resonator. - Dielectric cavities are used for resonators,

amplifiers and oscillators at microwave

frequencies.

A dielectric resonator antenna with a cap for

measuring the radiation efficiency

Univ. of Mississippi

Quality Factor, Q

- Is defined as

Example

- For a cavity of dimensions 3cm x 2cm x 7cm

filled with air and made of copper (sc5.8 x 107) - Find the resonant frequency and the quality

factor for the dominant mode. - Answer

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