ene 428 microwave engineering

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ENE 428Microwave

Engineering

Lecture 1 Uniform plane waves

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Syllabus

•Asst. Prof. Dr. Rardchawadee Silapunt, [email protected]•Lecture: 9:30pm-12:20pm Tuesday, CB41004

12:30pm-3:20pm Wednesday, CB41002 •Office hours : By appointment•Textbook: Applied Electromagnetics by Stuart M. Wentworth (Wiley, 2007)

mailto:[email protected]

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Homework 20% Midterm exam 40% Final exam 40%

Grading

Vision Providing opportunities for intellectual growth in the context of an engineering discipline for the attainment of professional competence, and for the development of a sense of the social context of technology.

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Course overview

• Maxwell’s equations and boundary conditions for electromagnetic fields

• Uniform plane wave propagation• Waveguides• Antennas• Microwave communication systems

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Introduction

• From Maxwell’s equations, if the electric field is changing with time, then the magnetic field varies spatially in a direction normal to its

orientation direction

• A uniform plane wave, both electric and magnetic fields lie in the transverse plane, the plane whose normal is the direction of propagation

• Both fields are of constant magnitude in the transverse plane, such a wave is sometimes called a transverse electromagnetic (TEM) wave.

BBBBBBBBBBBBBBEBBBBBBBBBBBBBBH

http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=52




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Maxwell’s equations

0

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

BBBBBBBBBBBBBBBBBBBBBBBBBBBB


v

DH J

t

BE

t

D

B

(1)

(2)

(3)

(4)

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Maxwell’s equations in free space

= 0, r = 1, r = 1

0

0



EH

t

HE

t

Ampère’s law

Faraday’s law

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General wave equations

• Consider medium free of charge where• For linear, isotropic, homogeneous, and

time-invariant medium,

(1)

(2)

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB EH E

t

BBBBBBBBBBBBBBBBBBBBBBBBBBBB HE

t

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General wave equations

Take curl of (2), we yield

From

then

For charge free medium

( )

BBBBBBBBBBBBBBBBBBBBBBBBBBBB HE

t

2

2

( )

BBBBBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBBBBB

BBBBBBBBBBBBBBE

E E EtEt t t

2 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

A A A

22

2

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

E EE E

t t

0 BBBBBBBBBBBBBBE

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Helmholtz wave equation

22

2

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB E EE

t t

22

2

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB H HH

t t

For electric field

For magnetic field

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Time-harmonic wave equations

• Transformation from time to frequency domain

Therefore

j

t

2 ( ) BBBBBBBBBBBBBBBBBBBBBBBBBBBBs sE j j E

2 ( ) 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBs sE j j E

2 2 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBs sE E

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Time-harmonic wave equations

or

where

This term is called propagation constant or we can write

= +j

where = attenuation constant (Np/m) = phase constant (rad/m)

2 2 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBs sH H

( ) j j

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Solutions of Helmholtz equations

• Assuming the electric field is in x-direction and the wave is propagating in z- direction

• The instantaneous form of the solutions

• Consider only the forward-propagating wave, we have

• Use Maxwell’s equation, we get

0 0cos( ) cos( )

BBBBBBBBBBBBBBz z

x xE E e t z a E e t z a

0 cos( )

BBBBBBBBBBBBBBz

xE E e t z a

0 cos( )

BBBBBBBBBBBBBBz

yH H e t z a

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Solutions of Helmholtz equations in phasor form

• Showing the forward-propagating fields without time-harmonic terms.

• Conversion between instantaneous and phasor form

Instantaneous field = Re(ejtphasor field)

0

BBBBBBBBBBBBBB

z j zs xE E e e a

0

BBBBBBBBBBBBBB

z j zs yH H e e a

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Intrinsic impedance

• For any medium,

• For free space

x

y

E jH j

0 0

0 0

120 x

y

E EH H

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Propagating fields relation

1


BBBBBBBBBBBBBBBBBBBBBBBBBBBBs s

s s

H a E

E a H

where represents a direction of propagationa

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Propagation in lossless-charge free media

• Attenuation constant = 0, conductivity = 0

• Propagation constant

• Propagation velocity

– for free space up = 3108 m/s (speed of light)

– for non-magnetic lossless dielectric (r = 1),

1

pu

p

r

cu

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Propagation in lossless-charge free media

• intrinsic impedance

• wavelength

2

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Ex1 A 9.375 GHz uniform plane wave is propagating in polyethelene (r = 2.26). If the amplitude of the electric field intensity is 500 V/m and the material is assumed to be lossless, finda) phase constant

b) wavelength in the polyethelene

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c) propagation velocity

d) Intrinsic impedance

e) Amplitude of the magnetic field intensity

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Propagation in dielectrics• Cause

– finite conductivity– polarization loss ( = ’-j” )

• Assume homogeneous and isotropic medium

' "( ) BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBH E j j E

" '[( ) ] BBBBBBBBBBBBBBBBBBBBBBBBBBBBH j E

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Propagation in dielectrics

" effDefine

From2 ( ) j j

and2 2( ) j

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Propagation in dielectrics

We can derive2

( 1 1)2

2

( 1 1)2

and 1

.1 ( )

j

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Loss tangent

• A standard measure of lossiness, used to classify a material as a good dielectric or a good conductor

"

' 'tan

eff

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Low loss material or a good dielectric (tan « 1)

• If or < 0.1 , consider the material

‘low loss’ , then

1

2

(1 ).2

jand

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Low loss material or a good dielectric (tan « 1)

• propagation velocity

• wavelength

1

pu

2 1

f

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High loss material or a good conductor (tan » 1)

• In this case or > 10, we can

approximate

1

2 f

45 .

jje

therefore

2

1 1)

and

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High loss material or a good conductor (tan » 1)

• depth of penetration or skin depth, is a distance

where the field decreases to e-1 or 0.368 times of

the initial field

• propagation velocity

• wavelength

1 1 1m

f

pu

22

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Ex2 Given a nonmagnetic material having r = 3.2 and = 1.510-4 S/m,

at f = 3 MHz, find a) loss tangent

b) attenuation constant

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c) phase constant

d) intrinsic impedance

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Ex3 Calculate the followings for the wave with the frequency f = 60 Hz propagating in a copper with the conductivity, = 5.8107 S/m: a) wavelength

b) propagation velocity

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c) compare these answers with the same wave propagating in a free space

ene 428 microwave engineering

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