rs 1 ene 428 microwave engineering lecture 1 introduction, maxwell’s equations, fields in media,...

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ENE 428Microwave

Engineering

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

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Syllabus•Assoc. Prof. Dr. Rardchawadee Silapunt (Ann), rardchawadee.sil@kmutt.ac.th •Dr. Ekapon Siwapornsathain (Eric), sie4129@hotmail.com, Tel: 0814389024•Lecture: 9:00am-12:00pm Wednesday, AIT•Instructors at King Mongkut’s University of Technology Thonburi, BKK, Thailand•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.

10-11/06/51

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Course overview• Maxwell’s equations and boundary

conditions for electromagnetic fields• Uniform plane wave propagation• Transmission lines• Matching networks• Waveguides• Two-port networks• Resonators• Antennas• Microwave communication systems

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• Microwave frequency range (300 MHz – 300 GHz) ( = 1 mm – 1 m in free space)

• Microwave components are distributed components.

• Lumped circuit elements approximations are invalid.

• Maxwell’s equations are used to explain circuit behaviors ( and )

Introduction

DDDDDDDDDDDDDDE

DDDDDDDDDDDDDDH

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

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Lumped circuit model and distributed circuit model

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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

DDDDDDDDDDDDDDEDDDDDDDDDDDDDDH

Introduction (2)

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Point forms of Maxwell’s equations

(1)

(2)

(3)

(4)0

B

D

Jt

DH

Mt

BE

v

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The magnetic north can never be isolated from the south.

Magnetic field lines always form closed loops.

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Maxwell’s equations in free space

• = 0, r = 1, r = 1

0 = 4x10-7 Henrys/m0 = 8.854x10-12 Farads/ms = conductivity (1/ohm)(“constitutive parameters”)

0

0

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

EH

t

HE

t

Ampère’s law

Faraday’s law

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Integral forms of Maxwell’s equations

Note: To convert from the point forms to the integral forms, we need to apply Stoke’sTheorem (for (1) and (2)) and Divergence theorem (for (3) and (4)), respectively.

0

SdB

QdVSdD

ISdDt

ldH

SdBt

ldE

S

enc

VS

S

S

(1)

(2)

(3)

(4)

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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 DDDDDDDDDDDDDD

• Time dependence form:

• Phasor form:

( , , ) js xE A x y z e a

DDDDDDDDDDDDDD

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Maxwell’s equations in phasor form

(1)

(2)

(3)

(4)0

B

D

JDjH

MBjE

v

S

S

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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

EDDDDDDDDDDDDDD

DDDDDDDDDDDDDDD

20 /eD E P C m

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

ePDDDDDDDDDDDDDD

0 .e eP E DDDDDDDDDDDDDDDDDDDDDDDDDDDD

0 0(1 ) .e rD E E E DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

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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 E

D E E

D 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 magnetic polarization or magnetization.

• In the linear medium, it can be shown that

• Then we can write

HDDDDDDDDDDDDDD

20 ( ) /mB H P Wb m

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

mPDDDDDDDDDDDDDD

.m mP HDDDDDDDDDDDDDDDDDDDDDDDDDDDD

0 0(1 ) .m rB H H H DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

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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 H

B H H

B H H

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Boundary conditions between two media

Ht1

Ht2

Et2

Et1

Bn2

Bn1

Dn2

Dn1

n

2 1

2 1

Sn D D

n B n B

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

2 1

2 1

S

S

E E n M

n H H J

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

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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

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

• 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

DDDDDDDDDDDDDD

DDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDD

• 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)

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD EH E

t

DDDDDDDDDDDDDDDDDDDDDDDDDDDD HE

t

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General plane wave equations (2)

Take curl of (2), we yield

From

then

For charge free medium

( )

DDDDDDDDDDDDDDDDDDDDDDDDDDDD HE

t

2

2

( )

DDDDDDDDDDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDE

E E EtEt t t

AAA 2

0 E

2

22

t

E

t

EEE

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Helmholtz wave equation

22

2

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD E EE

t t

22

2

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD 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 ( ) DDDDDDDDDDDDDDDDDDDDDDDDDDDDs sE j j E

2 ( ) 0 DDDDDDDDDDDDDDDDDDDDDDDDDDDDs sE j j E

2 2 0 DDDDDDDDDDDDDDDDDDDDDDDDDDDDs 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 DDDDDDDDDDDDDDDDDDDDDDDDDDDDs 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( )

DDDDDDDDDDDDDDz z

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

0 cos( )

DDDDDDDDDDDDDDz

xE E e t z a

0 cos( )

DDDDDDDDDDDDDDz

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

DDDDDDDDDDDDDD

z j zs xE E e e a

0

DDDDDDDDDDDDDD

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

DDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDs s

s s

H a E

E a H

where represents a direction of propagationa