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ENE 428Microwave

Engineering Lecture 1 Introduction, Maxwell’s equations, fields in media, and boundary conditions

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Syllabus•Asst. Prof. Dr. Rardchawadee Silapunt, rardchawadee.sil@kmutt.ac.th•Lecture: 9:30pm-12:20pm Tuesday, CB41004

12:30pm-3:20pm Wednesday, CB41002 •Office hours : By appointment•Textbook: Microwave Engineering by David M. Pozar (3rd edition Wiley, 2005)• Recommended additional textbook: Applied Electromagnetics by Stuart M.Wentworth (2nd edition Wiley, 2007)

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Homework 10% Quiz 10% 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.

10-11/06/51RS

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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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• 300Microwave frequency range ( MH 300z – GHz)

• MMMMMMMMM MMMMMMMMMM MMM MMMMMMMMMMM MMMMMMMMts.

• MMMMMM MMMMMMM MMMMMMMM MMMMMMMMMMMMMM MMM invalid.

• MMMMMMMMM MMM MMMM MM MMMMMMM MMMMM’MM MMMMMMMMM ( MMM )

Introduction

E

H

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

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• 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

• Knowledge of fields in media and boundary conditions allows useful applications of material properties to microwave components

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

EH

Introduction (2)

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Maxwell’s equations

0v

BE MtDH J t

D

B

(1)

(2)

(3)(4)

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Maxwell’s equations in free space

= 0, r = 1, r = 1

0 = 4x10-7 Henrys/m0 = 8.854x10-12 farad/m

0

0

EHtHE t

Ampère’s law

Faraday’s law

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Integral forms of Maxwell’s equations

(1)

(2)

(3)

0 (4)

C S

C S

S V

S

E dl B dSt

H dl D dS It

D d s dv Q

B d s

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Fields are assumed to be sinusoidal or harmonic, and time dependence with steady-state conditions

( , , ) cos( ) xE A x y z t a

• Time dependence form:

• Phasor form:( , , ) j

s xE A x y z e a

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Maxwell’s equations in phasor form

0

S

S

v

E j B M

H J j D

D

B

(1)(2)(3)(4)

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Fields in dielectric media (1)• An applied electric field causes the polarization

of the atoms or molecules of the material to create electric dipole moments that complements the total displacement flux,

where is the electric polarization. • In the linear medium, it can be shown that

• Then we can write

E

D

20 /eD E P C m

eP

0 .e eP E

0 0(1 ) .e rD E E E

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Fields in dielectric media (2)• may be complex then can be complex and

can be expressed as

• Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments.

• The loss of dielectric material may be considered as an equivalent conductor loss if the material has a conductivity . Loss tangent is defined as

' ''j

''tan .'

e

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Anisotropic dielectrics• The most general linear relation of

anisotropic dielectrics can be expressed in the form of a tensor which can be written in matrix form as

.x xx xy xz x x

y yx yy yz y y

z zx zy zz z z

D E ED E ED E E

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Analogous situations for magnetic media (1)• An applied magnetic field causes the

magnetic polarization of by aligned magnetic dipole moments

where is the electric polarization. • In the linear medium, it can be shown that

• Then we can write

H

20 ( ) /mB H P Wb m

mP

.m mP H

0 0(1 ) .m rB H H H

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Analogous situations for magnetic media (2)• may be complex then can be complex and

can be expressed as

• Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments.

' ''j

m

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Anisotropic magnetic material

• The most general linear relation of anisotropic material can be expressed in the form of a tensor which can be written in matrix form as

.x xx xy xz x x

y yx yy yz y y

z zx zy zz z z

B H HB H HB H H

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Boundary conditions between two media

Ht1

Ht2

Et2

Et1Bn2

Bn1

Dn2

Dn1

n

2 1

2 1

Sn D D

n B n B

2 1

2 1

S

S

E E n M

n H H J

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Fields at a dielectric interface

2 1

2 1

1 2

1 2.

n D n D

n B n B

n E n E

n H n H

• Boundary conditions at an interface between two lossless dielectric materials with no charge or current densities can be shown as

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Fields at the interface with a perfect conductor

0

0

0.

S

n D

n B

n E M

n H

• Boundary conditions at the interface between a dielectric with the perfect conductor can be shown as

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General plane wave equations (1)

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

time-invariant medium, assuming no free magnetic current,

(1)

(2)

EH E t

HE t

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General plane wave equations (2)

Take curl of (2), we yield

From

then

For charge free medium

( )

HE t

2

2

( )

EE E EtE t t t2

A A A

22

2

E EE Et t

0 E

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Helmholtz wave equation

22

2

E EE t t

22

2

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

jt

2 ( ) s sE j j E

2 ( ) 0 s sE j j E

2 2 0 s 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 s 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( )

z z

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

0 cos( )

z

xE E e t z a

0 cos( )

z

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

z j zs xE E e e a

0

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 0120 x

y

E EH H

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Propagating fields relation

1

s s

s s

H a E

E a H

where represents a direction of propagationa