MSc Lecture 2,_RF&ME

7/28/2019 MSc Lecture 2,_RF&ME

1/91

Lecture 2

TE 402RF & Microwave Engineering

Engr. Ghulam Shabbir


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Electromagnetic Theory


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Analysis Techniques in Microwave Theory

In general, circuit theory is not applicable to

microwave problems.

Circuit theory is derived from Maxwells equations

based on certain assumptions about the fields within

the circuit elements.

Specifically, the circuit elements must be small relative

to wavelength for circuit equations to be valid.

In this sense, microwave components must be modeled

by distributed elements, not lumped elements. For this reason, we must use field theory solutions

(Maxwells equations) for microwave applications.


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Maxwells Equations

Instantaneous, symmetric form


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Maxwells Equations

The quantities of magnetic current densityM and magneticcharge density m are nonphysical and included in the

symmetric forms of Maxwells equation for mathematical

convenience. These magnetic sources may be used to simplify the

mathematics of particular problems involving actual

electric currents and charges.



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Maxwells Equations

The flux and field quantities are related by the

constitutive relations:



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Maxwells Equations


The instantaneous Maxwells equations are valid, given

any type of time-dependence for the electromagnetic

fields.

Most applications in microwave engineering involve

fields which have a sinusoidal (harmonic) time-dependence.

This harmonic time-dependence allows us to simplify

Maxwells equations by writing them in terms of phasors

just like we use in circuit analysis.


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Maxwells Equations



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Maxwells Equations


According to Eulers identity, we may write the equation

above as


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Maxwells Equations



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Maxwells Equations

Phasor form

Relation of instantaneous quantities to phasor

quantities ...


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Maxwells Equations

Complex Permittivity and Permeability

In order to account for dielectric and magnetic losses in

media where time-harmonic electromagnetic fields exist,

we may define a complex permittivity and permeability.

In the case of dielectrics, we may combine the

conductivity losses with the dielectric losses according

to Maxwells equations. The conduction current density J in a given medium is

defined by


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Maxwells Equations


We may write a single equation which includes dielectric

and conductor losses by incorporating the complex

permittivity and the conduction current equation into

Amperes law:

since

&


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Maxwells Equations



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Maxwells Equations

Material Classifications


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Maxwells Equations

Electromagnetic Field Boundary Conditions

Knowledge of how the components of an electromagneticfield behave at the interface between two different media is

important in the solution of many problems in microwave

engineering.

A simple interface between two media is shown below.

The vector n is defined as the unit normal to the interface

pointing into region 2.


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Maxwells Equations

Electromagnetic Field Boundary Conditions

The general boundary conditions are:

Note that the individual components of the vector fields

are discontinuous at the interface by an amount equal tothe respective surface current or charge on the boundary.

In most applications, we do not encounter all of the

surface sources.

These general boundary conditions can be specialized to

problems involving specific combinations of materials.

n( ) defines the normal

components of the vector

n ( ) defines the tangential

components of the vector


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Maxwells Equations

Interface Between Two Lossless Dielectric Materials

Thus, the normal components of electric and magnetic

flux and the tangential components of electric and

magnetic field are continuous across a lossless dielectricinterface.

M ll E ti


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Maxwells Equations

Perfect Electric Conductor (PEC)

Note that the tangential electric field is always zero on the

surface of a PEC.

The tangential magnetic field on a PEC is equal to the

surface current while the normal electric flux is equal tothe surface char e.

M ll E ti


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Maxwells Equations

Perfect Magnetic Conductor (PMC)

Note that the tangential magnetic field is always zero on the

surface of a PMC.

The tangential electric field on a PEC is equal to the negative of

the surface magnetic current while the normal electric flux ise ual to the surface ma netic char e.


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Electromagnetic Waves


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Taking the curl of (1) and inserting (2) yields

while taking the curl of (2) and inserting (1) yields

in (5) and (6) gives


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However, the divergence terms in (7) and (8) are zero in

the source-free region. This gives the wave equations

(Helmholtz equations) for the electric and magnetic

field.

The wave equations for the Eand Hshow that energy

will propagate away from a time-varying

electromagnetic source in the form of electromagnetic

waves.

i


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Plane Waves

Plane waves are the most commonly encountered wave

types in electromagnetic applications and are the easiest

to define mathematically.

Plane wave - the electric and magnetic field of a plane

wave lie in the plane which is perpendicular to thedirection of wave propagation

(Direction ofE His the direction of wave propagation)

Uniform Plane wave - the electric and magnetic fields of a

uniform plane wave are uniform in the plane which isperpendicular to the direction of propagation

(Magnitude ofEand Hvary only in the direction of wave

propagation)

El i W


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Example - Uniform Plane Wave

El i W


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The uniform plane wave for this example has onlyo az-component of electric field and

o anx-component of magnetic field

o which are both functions of only y

The vector Laplacian operator (2 ) which appears in thewave equations for E and H may be expanded in

rectangular coordinates as

El t ti W


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Given the vector Laplacian definition, the wave equations forE and H reduce to

where the partial derivatives have been replaced by pure

derivatives given that the field components are functions ofonly one variable.

Note that the right hand side of the equations above is the

zero vector.

El t ti W


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Thus, by equating the vector components on both sides ofthe equation, we may write scalar equations for Ezand Hx

The general solutions to these D.E.s are

where E1 , E2 , H1, and H2are constants.

El t ti W


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The instantaneous forms of the wave field components are

and

El t ti W


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The direction of propagation for the plane wave may bedetermined by investigating the points of constant phase

on the waves.

Given the +aytraveling wave of our example, the constants

E1and H1must be zero so that

El t ti W


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Plane Wave Parameters

The velocity of propagation (vp) of the plane wave is found

by differentiating the position of the point of constant

phase with respect to position.

El t ti W


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The radian frequency of the plane wave is defined by

With the wave traveling at a velocity of vp , it takes oneperiod (T) for the wave to travel one wavelength ().

The wavenumber definition in terms of shows that the all

waves see a phase change of 2 radians per wavelength.

El t ti W


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Plane waves have the characteristic that:

The ratio of the electric field to magnetic field at any point

is a constant

which is related to the constitutive parameters of the

medium.

This property can be illustrated by using Maxwells

equations with our example of plane wave.

El t ti W


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Plane Waves in Lossy Media

A plane wave loses energy as it propagates through a lossymedium.

A medium is defined as a lossy medium if it is characterized

by any or all of the following loss mechanisms:

Dielectric and magnetic losses are typically small and can be

neglected for most materials.

However, conduction losses can be significant for commonly

encountered materials.



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Given the same +y-directed uniform plane waveassumed in the lossless example, the differential

equations governing the plane wave field components in

the lossy medium are

which have general solutions of the form



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Poyntings Theorem

The corresponding phasor form of the Poynting vector S is



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Poyntings Theorem

Given a volume Venclosed by the surface S which contains

electric and magnetic sources Jand M, Poyntings theorem

for the volume may be written as



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Poyntings Theorem

Poyntings theorem states that the complex power produced by

the sources is equal to the power transmitted out of the volume

plus that dissipated in the form of heat (through conductor,

dielectric and magnetic losses) plus the 2 times the net reactive

stored energy.

M ll E ti


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Maxwells Equations Equations in point (differential) form of time-varying

0

,

,

,

B

D

Jt

DH

Mt

B

E

EquationContinuity,0

t

J

( 0, 0)E M

Generally, EM fields and sources vary with space (x, y, z) and time (t) coordinates.

Equations in integral form

, Faraday's Law

,Ampere's Law

, Gauss's Law

0, No free magnetic charge

C S

C S

S

S

BE dl ds

t

DH dl ds I

t

Dds Q

Bds

,

Divergence theorem

,

Stokes' theorem

v s

s c

A A ds

A A dl

Time-Harmonic Fields


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Where MKS system of units is used, and

E: electric field intensity, in V/m.

H : magnetic field intensity, in A/m.

D : electric flux density, in Coul/m2.

B : magnetic flux density, in Wb/m2.

M : (fictitious) magnetic current density, in V/m2.

J : electric current density, in A/m2

.: electric charge density, in Coul/m3.

ultimate source of the electromagnetic field.

Q : total charge contained in closed surface S.

I : total electric current flow through surface S.

Time Harmonic Fields

0

,,

,

B

DJDjH

MBjE

When steady-state condition is considered, phasor representations of

Maxwells equations can be written as : (time dependence by multiply e -jt)

2: Displacement current density, in A/m EM wave propagatiomD

In free space In isotropic materials


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Constitutive Relations

Question:2(6) equations are not enough to solve 4(12) unknownfield components

In free space

HB

ED

0

0,

where 0 = 8.85410-12 farad/m is the permittivity of free space.0 = 410

-7 Henry/m is the permeability of free space.

In isotropic materials

(e.g. Crystal structure and ionized gases)

3 3 3 3,

x x x x

y y y y

z z z z

D E B H

D E B HD E B H

)1(,)(

);1(,

0"'

0

0"'

0

mm

ee

jHPHB

jEPED

wherePe

is electric polarization,Pm

is magnetic polarization,

e

is electric susceptibility, and m

is magnetic susceptibility.

Complexand

The negative imaginary part ofand account for loss in medium (heat).


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, Ohm's law from an EM field point of view

=

= ' ( " )

= ( ' " )

"tan , Loss tangent

'

J E

H j D J

j E E

j E E

j j j E

where is conductivity (conductor loss),

is loss due to dielectric damping,(+ ) can be seen as the total effective conductivity,

is loss angle.

In a lossless medium, and are real numbers.Microwave materials are usually characterized by specifying the real

permittivity, =r0,and the loss tangent at a certain frequency.

It is useful to note that, after a problem has been solved assuming a

lossless dielectric, loss can easily be introduced by replaced the real with

a complex .

Example1.1 : In a source-free region, the electric field intensity is given as


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follow. Find the signal frequency?

V/m4 )3( yxjezE Solution :

)3(

0)3(

0

0

412

400

1 yxj

yxj

eyx

e

zyx

zyx

jHHjE

)3(

002

)3(

0

)3(

0

0

0

16

0412

1

yxj

yxjyxj

ez

ee

zyx

zyx

jE

EjH

8

2 0 00 0

16 24 6 10 rad/s

Boundary Conditions


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Boundary Conditions

2121

2121

,

,,

HnHnEnEn

BnBnDnDn

Fields at a dielectric interface

Fields at the interface with a perfect conductor (Electric Wall)

S

S

JHnEn

BnDn

,0

,0,

Magnetic Wallboundary condition (not really exist)

0

,

,0

,0

Hn

MEn

Bn

Dn

S

tyconductiviAssumed

It is analogous to the relations between voltage and current at the end ofa short-circuited transmission line.

It is analogous to the relations between voltage and current at the end ofa o en-circuited transmission line.

H l h lt (V t ) W E ti


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Helmholtz (Vector) Wave Equation

In a source-free, linear, isotropic, and homogeneous

medium

0

,022

22

HH

EE

is defined the wavenumber, or propagation constant

, of the medium; its unit are 1/m.

Plane wave in a lossless medium

( ) ,

1( ) [ ],

jkz jkzx

jkz jkzy

E z E e E e

H z E e E e

k

Solutions of above wave equations

H

E

k

is wave impedance, intrinsic impedance of medium.

In free space, 0=377.

Transverse Electromagnetic Wave

(TEM)

x yE H z

,

EjH

HjE

is phase velocity defined as a fixed phase point on 1dz


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)tan1()(1 '"'

jjjjjjj

is phase velocity, defined as a fixed phase point on

the wave travels.

In free space, vp=c=2.998108 m/s.

kdtvp

f

vv

k

pp

22

is wavelength, defined as the distance between twosuccessive maximum (or minima) on the wave.

Plane wave in a general lossy medium

In wave equations, jk for following conditions.

-1: Complex propagation constant (m )

: Attenuation constant(Np/m;1Np/m=8.69dB/m), : Phase constant(rad/m)

21s

is skin depth or penetration depth, defined as the

amplitude of fields in the conductor decay by an amount

1/e or 36.8%, after traveling a distance of one skin depth.

Good conductor

Condition: (1) >> or (2) >>

Scattering Parameters (S Parameters)


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Scattering Parameters (S-Parameters)

Consider a circuit or device inserted

into a T-Line as shown in the Figure.

We can refer to this circuit or deviceas a two-port network.

The behavior of the network can be

completely characterized by its

scattering parameters (S-parameters),or its scattering matrix, [S].

Scattering matrices are frequently

used to characterize multiport

networks, especially at high

frequencies.

They are used to represent microwave

devices, such as amplifiers and

circulators, and are easily related to

concepts of gain, loss and reflection.

11 12

21 22

S SS

S S

Scattering matrix



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The scattering parameters represent

ratios of voltage waves entering and

leaving the ports (If the samecharacteristic impedance, Zo, at all ports

in the network are the same).

1 11 1 12 2.V S V S V

2 21 1 22 2

.V S V S V

11 121 1

21 222 2

,S SV V

S SV V

In matrix form this is written

.V S V

2

1

11

1 0V

VS

V

1

1

12

2 0V

VS

V

1

2

22

2 0V

VS

V

2

2

21

1 0V

VS

V



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Properties:

The two-port network is reciprocal

if the transmission characteristics

are the same in both directions

(i.e. S21 = S12).

It is a property of passive circuits

(circuits with no active devices or

ferrites) that they form reciprocal

networks.

A network is reciprocal if it is equal

to its transpose. Stated

mathematically, for a reciprocal

network

,t

S S

11 12 11 21

21 22 12 22

.

tS S S S

S S S S

12 21S SCondition for Reciprocity:

1) Reciprocity

i A li i


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Microwave Applications

Wireless Applications TV and Radio broadcast

Optical Communications

Radar

Navigation

Remote Sensing

Domestic and Industrial Applications

Medical Applications

Surveillance

Astronomy and Space Exploration


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Radar System Comparison

Radar Characteristic wave mmwave optical

tracking accuracy poor fair good

identification poor fair good

volume search good fair poor

adverse weather perf. good fair poor

perf. in smoke, dust, ... good good fair

Mi E Di i i


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Microwave Engr. Distinctions 1 - Circuit Lengths:

Low frequency ac or rf circuits

time delay, t, of a signal through a device

t = L/v T = 1/f where T=period of ac signal

but f =v so 1/f= /v

so L , I.e. size of circuit is generally much

smaller than the wavelength (or propagation

times or phase shift 0) Microwaves: L

propagation times not negligible

Optics: L

Microwave Distinctions


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Microwave Distinctions

2 - Skin Depth:

degree to which electromagnetic field

penetrates a conducting material

microwave currents tend to flow along the

surface of conductors so resistive effect is increased, i.e.

R RDC a / 2 , where

= skin depth = 1/ ( fo

cond

)1/2

where, RDC = 1/ ( a2cond)

a = radius of the wire

R waves in Cu >R low freq. in Cu


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Microwave Engr. Distinctions

3 - Measurement Technique

At low frequencies circuit properties

measured by voltage and current

But at microwaves frequencies, voltages

and currents are not uniquely defined; so

impedance and power are measured rather

than voltage and current

Circuit Limitations


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Circuit Limitations

Simple circuit: 10V, ac driven, copper wire,

#18 guage, 1 inch long and 1 mm indiameter: dc resistance is 0.4 m,L=0.027H

f = 0; XL = 2 f L 0.18 f10-6 =0

f = 60 Hz; XL 10-5 = 0.01 m f = 6 MHz; XL 1

f = 6 GHz; XL 103 = 1 k

So, wires and printed circuit boards cannot beused to connect microwave devices; we needtransmission lines, waveguides, striplines, andmicrostrip


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High-Frequency Resistors Inductance and resistance of wire resistors

under high-frequency conditions (f 500MHz):

L/RDC a / (2 )

R /RDC a / (2 ) where, RDC = /( a

2cond)

a = radius of the wire

= skin depth = 1/ ( focond)-1/2


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Reference: Ludwig & Bretchko, RF Circuit Design


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High Frequency Capacitor

Equivalent circuit consists of parasitic lead

conductance L, series resistance Rs describing

the losses in the the lead conductors and

dielectric loss resistance Re = 1/Ge (in parallel)with the Capacitor.

Ge = C tan s, where

tan s = (/diel) -1 = loss tangent


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Transit Limitations

Consider an FET

Source to drain spacing roughly 2.5 microns

Apply a 10 GHz signal: T = 1/f = 10-10 = 0.10 nsec

transit time across S to D is roughly 0.025 nsec

or 1/4 of a period so the gate voltage is low

and may not permit the S to D current to flow


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Ref: text by Pozar

Wi l C i ti


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Wireless Communications

Options

Sonic or ultrasonic - low data rates, poor

immunity to interference

Infrared - moderate data rates, but easilyblocked by obstructions (use for TV remotes)

Optical - high data rates, but easily

obstructed, requiring line-of-sight RF or Microwave systems - wide bandwidth,

reasonable propagation


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Cellular Telephone Systems (1)

Division of geographical area into non-overlapping hexagonal cells, where each

has a receiving and transmitting station

Adjacent cells assigned different sets ofchannel frequencies, frequencies can be

reused if at least one cell away

Generally use circuit-switched publictelephone networks to transfer calls

between users


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Cellular Telephone Systems (2)

Initially all used analog FM modulation anddivided their allocated frequency bandsinto several hundred channels, AdvancedMobile Phone Service (AMPS)

both transmit and receive bands have 832, 25kHz wide bands. [824-849 MHz and 869-894MHz] using full duplex (with frequencydivision)

2nd generation uses digital or PersonalCommunication Systems (PCS)

Satellite systems


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Satellite systems

Large number of users over wide areas

Geosynchronous orbit (36,000 km aboveearth)

fixed position relative to the earth

TV and data communications Low-earth orbit (500-2000 km)

reduce time-delay of signals

reduce the need for large signal strength

requires more satellites

Very expensive to maintain & often needsline-of sight

Global Positioning Satellite


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Global Positioning Satellite

System (GPS) 24 satellites in a medium earth orbit (20km)

Operates at two bands, L1 at 1575.42 and L2at 1227.60 MHz , transmitting spread

spectrum signals with binary phase shiftkeying.

Accurate to better that 100 ft and withdifferential GPS (with a correcting known basestation), better than 10 cm.


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Frequency choices

availability of spectrum

noise (increases sharply at freq. below 100

MHz and above 10 GHz)

antenna gain (increases with freq.)

bandwidth (max. data rate so higher freq.

gives smaller fractional bandwidth)

transmitter efficiency (decreases with freq.)

propagation effects (higher freq, line-of sight)


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Propagation

Free space power density decreases by 1/R2

Atmospheric Attenuation

Reflections with multiple propagation pathscause fading that reduces effective range, data

rates and reliability and quality of service

Techniques to reduce the effects of fading areexpensive and complex

Antennas


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Antennas

RF to an electromagnetic wave or the inverse

Radiation pattern - signal strength as a function

of position around the antenna

Directivity - measure of directionality

Relationship between frequency, gain, and size

of antenna, = c/f

size decreases with frequency

gain proportional to its cross-sectional area \ 2

phased (or adaptive) array - change direction of

beam electronically

R iM th


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berikutnyacoordinatesystemsUntuk

zx

anmenghasilkx/partialolehndidefisikaygPerubahan

CsinABA

lainnyar terhadapsatu vectoprojeksi

product,dotatauscalar:cosABA

on vectorsinterseksiBdanAMisalkan

Review

zyy

x

B

B

Math


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sungai)dimengalirygdaun(pusaranrotation

(Russian)ROTor;)()A(

flowoutwardnet:Divergence;A

changeofrate:gradient;u

(Space)ruangdalambervariasi

z)y,u(x,uscalarmemilikifieldsebuahjika

z Curl

y

A

x

A

zA

yA

xA

zz

uy

y

ux

x

u

zxy

zyx


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theorem(batu)Stokes;)(

theoremDivergence;)(

0curlofdivor

0)()(;)()(

0gradientofcurlor0;0

s

vs

dsAdA

dVAdsA

CCCBACBA

uAA


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Maxwells Equations

Gauss

No Magnetic Poles Faradays Laws

Amperes Circuit LawtDJH

tBEB

D

/

/0

Characteristics of Medium


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Characteristics of Medium

Constitutive Relationships

npropagatioofdirectionzconstant,phase

constantonattentuati,jwhere

z)-texp(jtoalproportionHE,

plasmaferrites,exceptscalars,,

surfacesonsonotitself,mediumin the0,J

sAssumptionCurrentConvectiveJJJJE,J

tyPermeabiliMagnetic,H,B

yPermitivitDielectric,ED

vv,cc

ro

,or


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Fields in a Dielectric Materials

0onconservatientergytoduenegative(heat)mediumin thelossforaccounts

magnitude)oforders4or(3dielectricgoodfor,

j)1(

EE)1(D

itysuceptibildielectric,EdensitymomentdipoleP

density)ntdisplacemeorfluxelectric(D0Jand

somagnetic,nonand,PEDAssume

eo

eo

eoe

oo


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Fields in a Conductive Materials

tantangentlosseffective

tyconductivieffectivetheiswhere

E)](j[E)jj(j

E))j(jj(E)j(j

EjEt

EE

t

DJH

easvaryfieldsEwhere,EJJ tjc

W E i


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Wave Equation

andbydescribedmediumin

wavesofconstantnpropagatio:

;H-H

;E-E

E))((

)H(E-E)(E)(

EjHH,-jE

jt/Consider

2

22

22

2

kdefine

similarly

jj

j

General Procedure to Find Fields in a


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Guided Structure

1- Use wave equations to find the z

component of Ez and/or Hz note classifications

TEM: Ez =Hz= 0

TE: Ez =0, Hz 0

TM: Hz =0, Ez 0 HE or Hybrid: Ez0, Hz 0



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Guided Structure

2- Use boundary conditions to solve for any

constraints in our general solution for Ez

and/or Hz

conductorofsurfacethetonormalnwhere

conductorperfectofsurfaceon0Hor,0Hn

JHn

/En

conductorperfectofsurfaceon0Eor0,En

n

s

t

s

Pl W i L l M di


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Plane Waves in Lossless Medium

directionzin themovingconstantkzt

))kzt(cos(E))kzt(cos(E)t,z(E

:domaintimein theor

eEeE)z(E0Ekz

E

0y/x/andEE

mediumlosslessain

realareandsincerealiskwhere0,EkE

x

jkzjkzxx

2

2

x2

x

22

Ph V l it


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Phase Velocity

cfv

fvf

vv

c

kdt

d

dt

dz

p

p

pp

oo

:spacefreein

or2

k

2

k2))k(z-t(-kz)-t(

maximasuccessive2betweendistance:Wavelength

m/sec1031

vspacefreein

1)

k

constant-t(v

a velocityatelspoint travphaseFixed

8

p

p

W I d


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Wave Impedance

E/Hork

where

)eEeE(

k

H

HjejkEejkE

yz

ExEz

zso;0

yx

Hjt

H-E:eqnsMaxwell'By

jkzjkz

y

y

jkzjkz

x

x

Plane Waves in a Lossy Medium


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Plane Waves in a Lossy Medium

kandjand

0,0note)j1(jj

complexnow,numberewav)j1(

0E)j1(E

E)E(E

)EEj(j)H(jEEEjHandHjE

22

22

2

W I d i L M di


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Wave Impedance in Lossy Medium

losseswithimpedancewavej

where

)eEeE(

j

H

)ztcos(edomaintimeeee

eEeE)z(E0Ez

E

0y/x/andxEEbeforeas

zz

y

zzjzz

zz

xx

2

2

x

2

x

Pl W i d C d t


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Plane Waves in a good Conductor

surfaceon theflowcurrentss,frequenciemicrowaveat

Au)Ag,Cu,(Al,metalsmostform1GHz,10at

depthskin/2/1

2/2/)j1(

/jj/jj

casepractical

s

s

2

Energy and Power


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Energy and Power

)WW(j2PP

sourcesbygeneratedpowerP

dvHH4/dvBHRe4/1W

dvEE4/dvDERe4/1W

lossasdissipatedoredtransmittbemaythat

powercarryandenergymagneticandelectricstore

thatfieldsupsetsenergyneticelectromagofsourceA

*

emo

s

v

*

v

*

m

v

*

v

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MSc Lecture 2,_RF&ME

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