RectangularWaveGuides by H v K KUMAR

8/10/2019 RectangularWaveGuides by H v K KUMAR

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MICROWAVE ENGINEERING

Presented ByHima Venkata Kishore Kumar Maddukuri

BE(EIE),M.Tech(ECE(I&CS))ECE DEPT HOD,KSIT


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GENERAL DEFINITIONGENERAL DEFINITION

A transmission line can be defined asA transmission line can be defined as a device for a device for propagating or guiding energy from one point to propagating or guiding energy from one point to another another .. The propagationThe propagationof energy is for one of twoof energy is for one of twogeneral reasons:general reasons:


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2.2. Information transferInformation transfer examples are telephone, radio,examples are telephone, radio,

and fibreand fibre- -optic links (in each case the energy propagatingoptic links (in each case the energy propagatingdown the transmission line is modulated in some way).down the transmission line is modulated in some way).


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Example 1.2Example 1.2 - - Phase difference between the ends of a cable.Phase difference between the ends of a cable.

Determine the phase difference between the ends of:Determine the phase difference between the ends of:

(a) a 10m length of mains cable for a 50Hz electricity(a) a 10m length of mains cable for a 50Hz electricitysupplysupply

(b) a 10m length of coaxial cable carrying a 750MHz TV(b) a 10m length of coaxial cable carrying a 750MHz TVsignalsignal

N.B. one wavelength corresponds to one complete cycleN.B. one wavelength corresponds to one complete cycleor wave, and hence to a phase change of 360or wave, and hence to a phase change of 360 or 2or 2 radians. So the phase change over a distanceradians. So the phase change over a distance ll is justis just

360360 ll // (or 2(or 2 ll // radians)radians)


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PRACTICAL DEFINITIONPRACTICAL DEFINITION

We have to treat a conducting system as a transmission lineWe have to treat a conducting system as a transmission line if theif thewavelength of the signal propagating down the line is less than orwavelength of the signal propagating down the line is less than orcomparable with the length of the linecomparable with the length of the line

Assoc iated with transmiss ion lines there may be: Assoc iated with transmiss ion lines there may be:

Propagation lossesPropagation lossesDistortionDistortion

Interference due to reflection at the loadInterference due to reflection at the loadTime delaysTime delaysPhase changesPhase changes


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Some different types of transmission lines:Some different types of transmission lines:

Radio linkRadio linkwi th antennaswi th antennas

22--wire linewire line

(dc)(dc)22--wire linewire line

(ac)(ac)Coaxial lineCoaxial line(dc, ac, rf)(dc, ac, rf)

MicrostripMicrostripline (rf)line (rf)

Rectangular Rectangular waveguidewaveguide

(rf)(rf)OpticalOptical

fibre (light)fibre (light)

CrossCrosssectionsection


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APPLICATIONS OF MICROWAVE ENGINEERING

Antenna gain is proportional to the electrical size of the antenna. Athigher frequencies, more antenna gain is therefore possible for a givenphysical antenna size, which has important consequences forimplementing miniaturized microwave systems.

More bandwidth can be realized at higher frequencies. Bandwidth iscritically important because available frequency bands in theelectromagnetic spectrum are being rapidly depleted.

Microwave signals travel by line of sight are not bent by theionosphere as are lower frequency signals and thus satellite andterrestrial communication links with very high capacities are possible.

Effective reflection area (radar cross section) of a radar target isproportional to the targets electrical size. Thus generally microwavefrequencies are preferred for radar systems.


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Various molecular, atomic, and nuclear resonances occur at

microwave frequencies, creating a variety of unique applications inthe areas of basic science, remote sensing, medical diagnostics andtreatment, and heating methods.

Today, the majority of appl ications of microwaves are related to radarand communication systems. Radar systems are used for detecting andlocating targets and for air traffic contro l systems, missile trackingradars, automobile collision avoidance systems, weather prediction,motion detectors, and a wide variety of remote sensing systems.

Microwave communication systems handle a large fraction of thewor lds international and other long haul telephone, data and televisiontransmissions.

Most of the currently developing wireless telecommunications systems,such as direct b roadcast satellite (DBS) television , personalcommunication systems (PCSs), wireless local area networks (WLANS),cellular video (CV) systems, and global pos itioning satellite (GPS)systems rely heavily on microwave technology.


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A SHORT HISTORY OF MICROWAVE ENGINEERING

Modern electromagnetic theory was formulated in 1873 by James ClerkMaxwell solely from mathematical considerations.

Maxwells formulation was cast in its modern form by Oliver Heaviside,during the period 1885 to 1887.

Heinrich Hertz, a German professor of physics understood the theorypublished by Maxwell, carried out a set of experiments during 1887-1891that completely validated Maxwells theory of electromagnetic waves.

It was only in the 1940s (World War II) that microwave theory receivedsubstantial interest that led to radar development.

Communication systems using microwave technology began to developsoon after the birth of radar. The advantages offered by microwave systems, wide bandwidths and

line of sight propagation, provides an impetus for the continuingdevelopment of low cost miniaturized microwave components.


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Waveguides

ByH V k KUMAR

BE(EIE),M.Tech(ECE(I&CS))


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WaveguideWaveguide

crosscrosssectionsection

rectangularrectangularwaveguideswaveguides


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MODES OF PROPAGATIONMODES OF PROPAGATIONThe energy propagating down a transmission lineThe energy propagating down a transmission line

propagates as a wavepropagates as a wave . Different modes of propagation. Different modes of propagation(i.e. different patterns of(i.e. different patterns of E E andand HH fields) are possible.fields) are possible.These fall into two categories:These fall into two categories:

TETE TRANSVERSE ELECTRICTRANSVERSE ELECTRICTMTM TRANSVERSE MAGNETICTRANSVERSE MAGNETIC

TEM Modes:TEM Modes: In the special caseIn the special casewherewhere EE andand HH are both transverseare both transverse(i.e. at r ight angles) to the direction(i.e. at r ight angles) to the direction

of energy flow, the mode is termedof energy flow, the mode is termed TEMTEM..EE andand HH will also be at r ight angles to each other.will also be at r ight angles to each other.

TEMTEM TRANSVERSE ELECTROMAGNETICTRANSVERSE ELECTROMAGNETIC

TE modeTE mode


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1. LINES PROPAGATING TEM MODES1. LINES PROPAGATING TEM MODES: :There is noThere is no E E oror HH field in the direction of propagation.field in the direction of propagation.

twintwin--wire, coaxial, stripline and (approximately)wire, coaxial, stripline and (approximately)microstrip lines are in this group.microstrip lines are in this group.

2. LINES PROPAGATING TE OR TM MODES:2. LINES PROPAGATING TE OR TM MODES:EE oror HH have components in the direction of energyhave components in the direction of energyflow.flow.

The kinds of mode that can propagate down a lineThe kinds of mode that can propagate down a linedepend on the geometry and materials of the line.depend on the geometry and materials of the line.

Transmission lines can be classif ied into 2 groupsTransmission lines can be classif ied into 2 groupsaccording to the type of mode that normallyaccording to the type of mode that normallypropagates down them.propagates down them.


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17

Transmission Lines and

WaveguidesWaveguide and other transmissionlines for the low-loss transmission ofmicrowave power.Early microwave systems relied onwaveguide and coaxial lines fortransmission line media.Waveguide: high power-handlingcapability, low loss, but bulky andexpensive


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Cont.,Coaxial line: high bandwidth,convenient for test applications,difficult medium in which to fabricatecomplex microwave components.Planar transmission lines: stripline,microstrip, slotline, coplanarwaveguide compact, low cost,easily integrated with active devices


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At frequencies below the cutoff frequency ofa given mode, the propagation constant ispurely imaginary, corresponding to a rapidexponential decay of the fields. cutoff orevanescent modes.TMn mode propagation is analogous to ahigh-pass filter response.The wave impedance pure real for f > f c, pureimaginary for f < f c.The guide wavelength is defined the distance

between equiphase planes along the z-axis.


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g > = 2 /k, the wavelength of a

plane wave in the material.The phase velocity and guidewavelength are defined only for apropagation mode, for which is real.


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Rectangular Waveguides

ByH V k KUMAR


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Waveguide components

Figures from: www.microwaves101.com/encyclopedia/waveguide.cfm

Rectangular waveguide Waveguide to coax adapter

E-teeWaveguide bends


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More waveguides

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


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UsesTo reduce attenuation loss

High frequenciesHigh power

Can operate only above certainfrequencies

Acts as a High-pass filter Normally circular or rectangular

We will assume lossless rectangular


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Rectangular Waveguides:

Fields insideUsing phasors & assuming waveguide

filled with

lossless dielectric material andwalls of perfect conductor,

the wave inside should obey

ck

H k H

E k E

22

22

22

where

00


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Then applying on the z-component

2

22

2

2

2

2

2

:obtainwewherefrom

)()()(),,(

:Variablesof Separationof method bySolving

0

k Z Z

Y Y

X X

z Z yY x X z y x E

E k z E

y E

x E

'' '' ''

z

z z z z

022 z z E k E


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Fields inside the waveguide

0

00

:sexpressionin theresultswhich

2

2

2

2222

2

Z Z

Y k Y X k X

k k k

k Z Z

Y Y

X X

''

y''

x

''

y x

'' '' ''

z z

y y

x x

ecec z Z

yk c yk cY(y) xk c xk c X(x)

65

43

21

)(

sincossincos

22222 y x k k k h


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Substituting

z z

y y

x x

ecec z Z

yk c yk cY(y)

xk c xk c X(x)

65

43

21

)(

sincos

sincos

)()()(),,( z Z yY x X z y x E z

z y y x x z

z y y x x z

z z y y x x z

e yk B yk B xk B xk B H

e yk A yk A xk A xk A E

z

ecec yk c yk c xk c xk c E

sincossincos

,field magneticfor theSimilarly

sincossincos

:direction-intravelingwaveat thelookingonlyIf

sincossincos

4321

4321

654321


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Other componentsFrom Faraday and Ampere Laws we can find the

remaining four components:

22222

22

22

22

22

y x

z z y

z z x

z z y

z z x

k k k h

where

y H

h x E

h j

H

x

H

h y

E

h

j H

x H

h j

y E

h E

y

H

h

j

x

E

h E

*So once we know E z and H z, we canfind all the otherfields.


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Modes of propagationFrom these equations we can conclude:

TEM ( E z=H z=0 ) cant propagate.

TE ( E z=0 ) transverse electricIn TE mode, the electric lines of flux areperpendicular to the axis of the waveguide

TM ( H z=0 ) transverse magnetic, E z existsIn TM mode, the magnetic lines of flux areperpendicular to the axis of the waveguide.

HE hybrid modes in which all componentsexists


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TM ModeBoundaryconditions: ,a x E

,b y E

z

z

0at0

0at0

Figure from: www.ee.bilkent.edu.tr/~microwave/programs/magnetic/rect/info.htm

z y y x x z e yk A yk A xk A xk A E sincossincos 4321

z j y x z e yk xk A A E sinsin42

From these, we conclude: X(x) is in the form of sin k x x ,

where k x=m /a , m=1,2,3,Y(y) is in the form of sin k y y,

where k y=n /b, n =1,2,3,So the solution for E z(x,y,z) is


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TM ModeSubstituting

222

sinsin

b

n

a

mh

where

e yb

n xa

m E E z jo z

22

k


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TMmnOther components are

x E

h j

H

y E

h j

H

y E

h E

x E

h E

z y

z x

z y

z x

2

2

2

2

zo y

zo x

zo y

zo x

eb yn

a xm

E a

mh

j H

eb yn

a xm

E bn

h j

H

eb yn

a xm

E b

nh

E

eb yn

a xm

E am

h E

sincos

cossin

cossin

sincos

2

2

2

2

0

sinsin

z

z jo z

H

e yb

n x

am

E E


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TM modesThe m and n represent the mode of propagationand indicates the number of variations of thefield in the x and y directionsNote that for the TM mode, if n or m is zero, allfields are zero.See applet by Paul Falstad

http://www.falstad.com/embox/guide.html


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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 totravel through the guide.

222

222

bn

am

k k k y x

22

222

121

or

0thenWhen

bn

am

f

jb

na

m

c

c

0and When22

2

b

n

a

m

0 and When22

2

jb

na

m


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Cutoff The cutoff frequency is the frequencybelow which attenuation occurs and above

which propagation takes place. (High Pass)

The phase constant becomes

2222 1'

f f

bn

am c

22

2'

bn

amu

f mnc

f c,mn

attenuation Propagation

of mode mn


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Phase velocity and impedanceThe phase velocity is defined as

And the intrinsic impedance of the modeis

f uu p p

2'

2

1' f f

H

E

H E c

x

y

y

xTM


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Summary of TM modesWave in the dielectricmedium

Inside the waveguide

/'

'/' u

2

1' f f c

TM

2

1

'

f f c

/

1'2

f f

uc

p

2

1' f f c

f u /''

/1'/' f u


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Related example of how fields look:Parallel plate waveguide - TM modes

axm

sin AE z zt je

0 a xm = 1

m = 2

m = 3xz a

E z


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TE ModeBoundaryconditions: ,a x E

,b y E

y

x

0at0

0at0

Figure from: www.ee.bilkent.edu.tr/~microwave/programs/magnetic/rect/info.htm

z j y x z e yk xk B B H coscos31

From these, we conclude: X(x) is in the form of cos k x x ,

where k x=m /a , m=0,1,2,3,Y(y) is in the form of cos k y y,

where k y=n /b, n =0,1,2,3,So the solution for E z(x,y,z) is

z y y x x z e yk B yk B xk B xk B H sincossincos 4321


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TE ModeSubstituting

Note that n and m cannot be both zerobecause the fields will all be zero.

222

againwhere

coscos

bn

am

h

e yb

n

a

xm H H z jo z


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TE mnOther components are

zo y

zo x

zo y

zo x

eb yn

a xm

H b

nh j

H

eb yn

a xm

H am

h j

H

eb yn

a xm

H a

mh

j E

eb yn

a xm

H bn

h j

E

sincos

cossin

cossin

sincos

2

2

2

2

0

coscos

z

z jo z

E

e yb

n x

am

H H

y H

h H

x H

h H

x H

h j

E

y H

h j

E

z y

z

x

z y

z x

2

2

2

2


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Cutoff The cutoff frequency is the sameexpression as for the TM mode

But the lowest attainable frequencies arelowest because here n or m can be zero.

22

2'

bn

amu

f mnc

f c,mn

attenuation Propagation

of mode mn


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Dominant ModeThe dominant mode is the mode withlowest cutoff frequency.Its always TE 10The order of the next modes changedepending on the dimensions of the

guide.


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Summary of TE modesWave in the dielectricmedium

Inside the waveguide

/'

'/' u

2

1

'

f f c

TE

2

1

'

f f c

/

1'2

f f

uc

p

2

1' f f c

f u /''

/1'/' f u


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Variation of wave impedanceWave impedance varies withfrequency and mode

TE

TM

f c,mn


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

waveguide, with dimensions a=2.286cm,b=1.016cm operating at 10GHz. Find thecutoff frequencies of all possible propagatingmodes.

Solution:From the formula for the cut-off frequency

22

2'

bn

amu

f mnc


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Example An air-filled 5-by 2-cm waveguide has

at 15GHzWhat mode is being propagated?Find

Determine E y/E x

V/m 50sin40sin20 z j z e y x E


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Group velocity, ugIs the velocity at whichthe energy travels.

It is always less than u

sm

f f

uu cg rad/mrad/s

1'/1

2

2'uuu g p

zo y ea

xm H

ah j

E

sin2

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


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Group Velocity

As frequency is increased,the group velocity increases .


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Power transmissionThe average Poynting vector for the waveguidefields is

where = TE or TM depending on the mode

z E E

H E H E H E

y x

x y y xave

2

Re21

Re21

22

***P

a

x

b

y

y x

aveave dxdy E E

dS P0 0

22

2P

[W/m 2]

[W]


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Attenuation in Lossy

waveguideWhen dielectric inside guide is lossy, and wallsare not perfect conductors, power is lost as ittravels along guide.

The loss power is

Where c+ d are the attenuation due to ohmic(conduction) and dielectric lossesUsually c >> d

zoave ePP

2

aveave

L PdzdP

P 2


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Attenuation for TE 10Dielectric attenuation, Np/m

Conductor attenuation, Np/m

2

12

'

f f c

d

210,

210,

5.0

1'

2 f

f

ab

f

f b

R c

c

sc

Dielectricconductivity!


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Waveguide CavitiesCavities, or resonators, areused for storing energy

Used in klystron tubes,band-pass filters andfrequency metersIts equivalent to a RLCcircuit at high frequencyTheir shape is that of acavity, either cylindrical orcubical.


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Cavity TM Mode to z

:obtainwewherefrom

)()()(),,(:Variablesof Separation bySolving

z Z yY x X z y x E z

zk c zk c z Z

yk c yk cY(y)

xk c xk c X(x)

z z

y y

x x

sincos)(

sincos

sincos

65

43

21

2222 z y x k k k k where


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TMmnp Boundary Conditions

,c z E E

,a x E

,b y E

x y

z

z

0at,0

0at0

0at0From these, we conclude:

k x=m /ak y=n /bk z=p /c

where c is the dimension in z-axis

2222

2

sinsinsin

c p

bn

am

k

wherec

z p

b

yn

a

xm E E o z c


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Resonant frequencyThe resonant frequency is the samefor TM or TE modes, except that thelowest-order TM is TM110 and thelowest-order in TE is TE 101 .

222

2'

c p

bn

amu

f r


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Cavity TE Mode to z

:obtainwewherefrom

)()()(),,(

:Variablesof Separation bySolving z Z yY x X z y x H z

zk c zk c z Z

yk c yk cY(y)

xk c xk c X(x)

z z

y y

x x

sincos)(

sincos

sincos

65

43

21

2222 z y x k k k k where


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TE mnp Boundary Conditions

,b y E

,a x E

,c z H

x

y

z

0at,0

0at0

0at0From these, we conclude:

k x=m /ak y=n /bk z=p /c

where c is the dimension in z-axis

c y pb yna xm H H o z sincoscos c


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Quality Factor , QThe cavity has walls with finiteconductivity and is therefore losingstored energy.The quality factor, Q, characterized theloss and also the bandwidth of the

cavity resonator.Dielectric cavities are used forresonators, amplifiers and oscillators atmicrowave frequencies.


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A dielectric resonator antennawith a cap for measuring theradiation efficiency

Univ. of Mississ ippi


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Quality Factor , QIs defined as

2233

22

101

2

TEmodedominantFor the

101 caaccababcca

QTE co f

where

101

1

LPW

latione of oscil y per cyclloss energ

stored ge energyTime avera Q

2

2


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ExampleFor a cavity of dimensions; 3cm x 2cm x 7cm filled with

air and made of copper ( c=5.8 x 10 7)Find the resonant frequency and the quality factor

for the dominant mode. Answer:

GHz f r 44.571

20

31

2103

22210

6

9 106.1)1044.5(

1

co

378,56873737322

723732233

22

101 TE Q

GHz f r 970

21

31

2103 22210

110


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66

3.7 Stripline A planar-type of transmission line thatlends itself well to microwaveintegrated circuitry andphotolithographic fabrication.Since stripline has 2 conductors and ahomogeneous dielectric, it can support

a TEM wave.The stripline can also support higherorder TM and TE modes, but these areusually avoided in practice.


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67

Figure 3.22 (p. 137)Stripline transmission line. ( a ) Geometry. ( b) Electric andmagnetic field lines.


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68

Figure 3.23 (p. 138)Photograph of a stripline circuit assembly, showing fourquadrature hybrids, open-circuit tuning stubs, and coaxialtransitions. Courtesy of Harlan Howe, Jr. M/A-COM Inc.


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69

3.8 MicrostripMicrostrip line is one of the most popular types oftransmission lines, primarily because it can befabricated by photolithographic process and iseasily integrated with other passive and active

microwave devices.Microstrip line cannot support a pure TEM wave.In most practical applications, the dielectricsubstrate is electrically very thin (d


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70

Figure 3.25 (p. 143)Microstrip transmission line. ( a ) Geometry. ( b) Electric andmagnetic field lines.


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71

Formulas for Propagation Constant,Characteristic Impedance and Attenuation

The effective dielectric constant of amicrostrip line:

The characteristic impedance of amicrostrip line is

1 1 1

2 2 1 12 /r r

e d W

0

60 8ln for / 14

120 for / 1

/ 1.393 0.667 ln( / 1.444)

e

e

d W W d W d

Z

W d W d W d


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72

Given Z 0, and r , the strip width is

where

The attenuation due to dielectric loss

2

8 for / 2

2 2 1 0.61

1 ln(2 1) ln( 1) 0.39 for / 22

A

A

r

r r

eW d

eW d

B B B W d

0

0

1 1 0.110.2360 2 1

3772

r r

r r

r

Z A

B Z

0 ( 1) tan

2 ( 1)r e

d

e r

k


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73

The attenuation due to the conductor losswhere is the surface

resistivity of the conductor.0

sc R Z W

0 / 2s R


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Evanescent Wave below Cutoff We have assumed propagation of theform e- jzand found that for thewaveguide above cutoff (f>fc).

This equation is valid only for f > fcorWhat happens below cutoff whencondition (my7.42) is not satisfied? In

this case wave propagation is of theform e- zwith an attenuation constant.


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Such a wave is called an evanescent

wave. It does not carry any power inthe z-direction butconsists of EM fields that decayexponentially in the z-direction.Awaveguide below cutoff supports only the evanescent wave. Asection of cutoff waveguideacts like an

attenuator


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WAVE PROPAGATION

Many guided waveconcepts can beexplained byunbounded TEM

waves reflected off w/gwalls.

a) y-polarized TEM plane wave propagates in the +zdirection. (b)Wavefrontview of the propagating wave.


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We take two identical y-polarized TEM waves, rotate oneby +and the other by as shown in (a), and combine

them in (b).


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(a) Replacingadjacent zerofield lines withconductingwalls, we get anidenticalfield patterninside. (b) The

u+ wave frontsfor a supportedpropagationmode areshown for anarbitrary angle .(c) The velocityof thesuperimposedfields, or groupvelocity, is uG.


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We see that a is determined by and , where f = uu/.Considerdistance AC:


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NB: phase velocity can begreater than speed of light


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Waveguide Impedance


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Figure 7-11ab (p. 356)TM11 field dis tribution inside a rectangular waveguide. Adjacent to the left-columncontour plot s are conventional plots taken across the middle of the guide. Thecontour plot has been modified with heavier lines representing l arger magnitudes.


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Figure 7-12 (p. 358)The TM11 Ezplo ts of MATLAB 7.2. This i sa black and white rendition of plots that

will appear in co lor when you run theprogram.The contour plot has been modifiedwith heavier lines representing largermagnitudes.

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RectangularWaveGuides by H v K KUMAR