comm (701) lecture #3 1 - german university in cairoeee.guc.edu.eg/courses/communications/comm701...

GUC (Dr. Hany Hammad) 10/5/2016

COMM (701) Lecture #3 1

© Dr. Hany Hammad, German University in Cairo

Lecture # 4

• Rectangular waveguide (Cont.)

• Power Losses

• Circular waveguide

• Spherical Coordinates.

• Basic Antenna Parameters

– Patterns

– Beam Area

– Radiation Intensity

– Beam Efficiency

– Directivity

– Efficiency

– Gain

– Bandwidth

– Antenna Field Zones


Wave impedance vs frequency for TE and TM modes

21 cgTM

21 c

gTE

2 TMTE gg




Wave propagation in the guide

zj

z

c

y ea

xH

ak

jE

sinmax2

a

xzj

a

xzj

z

c

eeHak

max22

x

zy

xz aa

a ˆˆ

xz aa

a ˆˆ

Two plane TEM waves

21

,

c

g

c

c

g

k

2,

2

f

v

k

2

ck

a

11 tantan

cktan

zja

xj

a

xj

z

c

ej

eeH

ak

j

2max2

c

g

21 c

c

Note



v pv gv

Medium velocity Phase velocity Group velocity

Velocities

rorok

v

1

pv

1gv

2211 cc

gp

vffv

211

cg vv

21tan

c

c

c

21 c

1

21cos ff

vvv

c

p

21cos ffvvv cg

2vvv gp 21cos ff

vvv

c

p

pv

v

gv





pv

pvgv

g

2

Energy travels in a waveguide at the group velocity, which will always be less than the speed of light. The phase velocity could be larger than the speed of light.

cos

vvp cosvvg

1cos vvp vvg

2vvv gp


Higher order modes




Power Calculations

For the dominant mode 0 yzx HEE

zj

z

c

y ea

xH

ak

jE

sinmax2

zj

zz ea

xHH

cosmax

zj

z

c

x ea

xH

ak

jH

sinmax2

akc

zj

z

c

ea

xH

k

j

sinmax

zj

z

c

ea

xH

k

j

sinmax

maxyE

maxxH

2

EHEP

SdHEPav

2

1zadxdySd ˆ

a b

zzz

cc

av adxdyaa

xH

k

j

k

jP

0 0

22

maxˆˆsin

2

1

a b

z

c

dxdya

xH

k0 0

2

max2

2cos1

2

1

2

1

a

ab

z

c a

a

x

xyHk

0

00

2

max2 2

2sin

4

1

baH

kz

c

2

max24

1


Power Calculations

2

2

2

2

2

22

2

g

cg

c

g

g

ccTETEkk

2

2

max4

g

cgzav TE

Hab

P

2

1

2

2

max

214

c

c

c

zav Hab

P

2

2

2

max 14

cc

zHab

c

c

c

Power imaginary (non-propagating)

No Power

Power is real (propagating)




Lossy Waveguides

• Two types of losses:

– Dielectric Losses. (d)

– Conductor Losses. (c)

cd (attenuation constant)

zjz

zz ea

xeHH

cosmax

Attenuation Term


Dielectric losses

For lossless 22

ckk

For lossy 22

ckkjj

jjj

vk 11

tan (Loss tangent)

2222 1 cc kjjkjjj

Square both sides




Dielectric losses

2222 2 ckjj Equating both sides

2222

ck

2

Real

Imaginary

For propagation 22

222

ck 22222

cc kkk

22

gTE

d

212 c

d


Conductor losses

z

oePP 2

av

z

oL PePdz

dPP 22 2

(Rate of change)

total

L

P

P

2

We have calculated the total power, we need to calculate the total power loss

Boundary condition KaHH n

1221

ˆ 1

2

1H

2H

nH1

tH1

tH 2

nH 2

12ˆ

na

Assume two is the conductor (waveguide walls)

KaH n

121

ˆ KHan

121

ˆor

zzyyxx aKaKaKK ˆˆˆ

For the dominant mode we have Hx, and Hz




Conductor losses

Conductor

Air

y

x

Hz

Hx

za

yaxa

KHan

121

ˆ

xzzzy aHaHaK ˆˆˆ

zj

zzx exa

HHK

cosmax

Hz

y = 0

ya

Kx

w

lR

w

l

w

l

A

l

A

lR s

111

dw

dlRR s

RIP2

2

1 zKI xx

z

xRR s

xzRKz

xRzK

z

xRzKP sxsxsx

22222

2

1

2

1

2

1

xRKz

Psx

2

2

1(y=0) xRK

z

Psx

2 (y=0 & y=b)

zw

y

xl

RIVIP2*

2

1

2

1


Conductor losses

Hz

xa

x = 0

Ky

KHan

121

ˆ

yzzzx aHaHaK ˆˆˆ

zj

zy

zj

zzy

eHK

ea

HHK

max

max 0cos

z

yl

zKI yy z

yRR s

yRK

z

Psy

2

2

1

yRKz

Psy

2(x = 0 & x = b)

(x = 0)




Conductor losses

zj

z

c

z ea

xH

k

jK

sinmaxHx

ya

Kz y = 0

KHan

121

ˆ

dxdw y

zl

xKI zx x

zRR s

xRKz

Psz

2

2

1

xRKz

Psz

2( y = 0 & y = b )

( y = 0 )


Conductor losses

a

sz

b

sy

a

sx dxRKdyRKdxRKdz

dP

0

2

0

2

0

2

b a

z

c

z

a

zs dxa

xH

kdyHdx

a

xHR

dz

dP

0 0

22

max2

22

max

0

22

max sincos

22 2

22

max

a

kb

aHR

dz

dP

c

zs

b

k

aHR

c

zs 2

22

max 12

2

2

max4

g

cgzav TE

Hab

P

2

2

2

2

2

2

max

2

22

max

1

2121

1

2

12

2

c

cs

g

c

g

c

g

s

g

cgz

c

zs

c

abRab

R

Hab

bk

aHR

P

dzdP

TE

TE

21 c

g

gTE

ac 2




Conductor losses


Surface Currents




Surface Currents


Surface Currents




Circular Waveguide

• Circular Waveguides

– TM modes.

– TE modes.

– Losses.

– Modes.


Circular Waveguide (TM-modes) Hz=0

r = a

dielectric

Conductor

022 zz EkE

2

2

2

2

2

2 11

z

Assume

zEE zz ,, )()()( zZ

011 2

2

2

2

2

2

z

zzz Ekz

EEE

0

)()(11 2

2

2

2

2

2

Zk

z

ZZZ

02

2

2

2

2

2

Zk

z

ZZZ

Z

0111 2

2

2

2

2

2

k

z

Z

Z

0222 kkc

(Cylindrical Coordinates)

2

ck

2





2

2

21

z

Z

Z02

2

2

Z

z

Z

zjzj eCeCZ 21

Forward Backward

zjeCzZ 1)(

011 22

2

2

2

k

01 22

2

2

ck

2

2

21m

)cos()sin()cos()( 343 omCmCmC Constants

)sin(sin)cos(cos)cos()( 333 mCmCmC ooo

3C 4C

)cos()( 3 omC

The cross section is circular and accordingly it is symmetrical for any value of o

0o mC cos)( 3



0222

ckm

2

2

0222

2

22

mkc

cmcm kYCkJC 65)(

Bessel Function of the 1st type Bessel Function of the 2nd type

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

)0(zE

cm kJC5)(

06 C

comm (701) lecture #3 1 - german university in cairoeee.guc.edu.eg/courses/communications/comm701...

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