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EIE 332 Electromagnetics

Lecturer: Dr. W.Y.TamRoom no.: DE604Phone no.: 27666265e-mail: enwytam@polyu.edu.hk

web: www.en.polyu.edu.hk/~em/mypage.htmNormal Office hour: 9:00am – 5:30pm (Mon-Fri)

9:00am – 12:30pm (Sat)

Acknowledgement: I would like to thank Mr. K.Y. Tong to let me to modify his handouts for this subject.

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Assessment

Examination (open book) 60%

PracticalMini-project 15%• Week 9 to Week 13• Report, demonstration/presentation

Test 10%• Week 5 and Week 10

Quizzes 10%Assignments 5%

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Textbook

1. D. K. Cheng, Fundamentals of Engineering Electromagnetics, Addison Wesley, 1993.

2. D. K. Cheng, Field and Wave Electromagnetics, AddisionWesley, 1989.

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Why study Electromagnetics?

• High speed circuits - Microwave and high speed digital circuits

• Antenna - Wireless communication

• Optical communication - Light propagation in fibresElectromechanical machines

• Electromagnetic interference and compatibility

Introduction

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Electromagnetics started with the experimental observation of (i) forces between electric charges; (ii) forces between conductors carrying electric currents

In free space (vacuum), +Q -Q . . d

2

2

4 dQF

oπε=

where εo is permittivity of free space

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I1 R Force I2

Attraction force between two parallel wires with length ∆Lcarrying currents I1, I2 in the same direction:

2

221

4)(

RLIIF o

πµ ∆

=

where µo is the permeability of free space

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• Introduce the concept of FIELD to facilitate the manipulation of the above forces

• Electric field - generated by charges

Magnetic field - generated by currents

• Determine the forces acting on charges and currents placed in electric and magnetic FIELDS

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Reference

Transmission line (3 Weeks)– Chapter 8.1 Overview– Chapter 8.2 Generalized Transmission-line Equations– Chapter 8.3 Transmission-line Parameters– Chapter 8.4 Wave Characteristics on an Infinite Transmission

Line– Chapter 8.5 Wave Characteristics on Finite Transmission

Lines– Chapter 8.6 The Smith Chart– Chapter 8.7 Transmission-line Impedance Matching

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II. TRANSMISSION LINE

2.1 Introduction

• Any pair of wires and conductors carrying currents in opposite directions form transmission lines.

• Transmission lines are essential components in any electrical/ communication system. They include coaxial cables, two-wire lines, microstrip lines on printed-circuit-boards (PCB). (Note that at very high frequencies, any conductor on a PCB must be considered as transmission lines.)

• The characteristics of transmission lines can be studied by theelectric and magnetic fields propagating along the line. But in most practical applications, it is easier to study the voltages and currents in the line instead.

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Coaxial cable Two-wire transmission line

dielectric substrate

Microstrip line

conductor

ground shield

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Magnetic field

Electric field

Cross-section of a coaxial cable showing the electric and magnetic fields

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2.2 Revision of Travelling Waves

The equation

represents a wave travelling in the +z direction with constant amplitude A, where ω=2πf, β=2π/λ. Any point of constant phase P advances towards the +z direction with a phase velocity

βω

=dtdz

Similarly a wave represented by

travels in the -z direction.

)cos( ztAv βω −=

)cos( ztAv βω +=

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ωT = 0 P

sin βz( )

βz0 5 10 15

1

0

1

ωT = 2 P

sin 2 βz( )

βz0 5 10 15

1

0

1

ωT = 4 P

sin 4 βz( )

βz0 5 10 15

1

0

1

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Often the amplitude of a wave varies exponentially with distance. The equation for such a wave is:

( )ztAev z βωα −= − cos

If α is positive, the wave amplitude is attenuated exponentially as it travels in the +ve z direction. If αis negative, the wave amplitude increases exponentially as it travels in the +ve z direction.

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Phasor representation: The cosine function is often replaced by the complex exponential function.

( ) ( )ztjzz eAeztAe βωαα βω −−− ↔−cos

The wave phasor V is written as:

zjzeAeV βα −−=

after dropping the term .tje ω

zAe α− is the magnitude, and -βz is the phase angle.

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Equivalent circuit of an element section (length ∆z) of the transmission line: L, R are the distributed inductance and resistance (per unit length) of the conductor; C,G are the distributed capacitance and conductance (per unit length) of thedielectric between the conductors.

2.3 Voltage and Current Waves in general transmission lines

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Relation between instantaneous voltage v and current i at any point along the line:

tiLRi

zv

tvCGv

zi

∂∂

−−=∂∂

∂∂

−−=∂∂

For periodic signals, Fourier analysis can be applied and it is more convenient to use phasors of voltage V and current I.

( )

VCjGzI

ILjRzV

)( ω

ω

+−=∂∂

+−=∂∂

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IzI

Vz

V

22

2

22

2

γ

γ

=∂∂

=∂∂

Decoupling the above equations, we get

where γ is called the propagation constant, and is in general complex.

))(( CjGLjR ωωγ ++=

α is the attenuation constant, β is the phase constant.

βα j+=

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The general solutions of the second-order, linear differential equation for V, I are :

zz

zz

eIeII

eVeVVγγ

γγ

+−−+

+−−+

+=

+=

V+, V-, I+, I- are constants (complex phasors). The terms containing e-γz represent waves travelling in +z direction; terms containing e+γz represent waves travelling in -z direction.

zjzz eee βαγ −−− =α determines the attenuation along the line, and β determines the phase shift along the line.

Since

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where Zo is the characteristic impedance of the line, given by

CjGLjRZo ω

ω++

=

The current I can now be written as:

z

o

z

oe

ZVe

ZVI γγ +

−−

+−=

It can be shown that the ratio of voltage to current is given by:

oZIV

=+

+

oZIV

−=−

−

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2.4 Lossless transmission lines

In lossless transmission lines, the distributed conductor resistance R and dielectric conductance G are both zero.In this case the characteristic impedance is real and is equal to:

CLZo =

The propagation constant γ is also imaginary with:

LCjj ωβγ

α

==

= 0

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Expressing the waves in time-domain,

( ) ( )

( ) ( )ztZV

ztZV

zti

ztVztVztv

oo

βωβω

βωβω

+−−=

++−=−+

−+

coscos),(

coscos),(

The velocity with which a front of constant phase travels is called the phase velocity up.

In any transmission line, βω

=puλπβ 2

=

In lossless transmission line,

Therefore

LCωβ =

LCu p

1==

βω

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In a coaxial cable, (Chap.2)

(Chap.3)

=

ba

C ro

ln

2 επε

==

ba

IL o ln

2πµφ

oropu

µεε1

=So

εo – permittivity of vacuum

εr – relative permittivity (dielectric constant) of dielectric

µo – permeability of vacuum

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Example: Calculate the characteristic resistance Ro of a RG-58U coaxial cable which has a inner conductor of radius a=0.406 mm and a braided outer conductor with radius b=1.553 mm. Assume the dielectric is polyethylene with dielectric constant of 2.26.

Solution: The distributed capacitance and inductance of the cable can be calculated to be:

L = 0.268 µH/m

C = 93.73 pF/m

CLRo /= Ω= 47.53

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2.5 Reflections of time-harmonic waves:

Consider a transmission line of length l terminated by an arbitrary impedance ZL:

I

+ Zin V Zo ZL _ z=-l z=0 z

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At the load z=0, the voltage and current phasors can be written as:

( )−+

−+

−=

+=

VVZ

I

VVV

o

1)0(

)0(

Load impedance ZL=V(0)/I(0), so we can express the ratio of the backward to forward voltages as:

oL

oLL ZZ

ZZVV

+−

=≡Γ +

−

ΓL is called the load reflection coefficient if we consider V+ as the incident wave and V- as the reflected wave.

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One important effect of a transmission line is to transform the load impedance. Let’s find the input impedance looking into thetransmission line of length l.

lL

l

lL

l

oin

in

eVeVeVeVZlZ

lIlVlZ

γγ

γγ

−++

−++

Γ−Γ+

=−

−−

≡−

)(

)()()(

Replacing ΓL in terms of Zo and ZL,

)tanh()tanh()(

lZZlZZZlZ

Lo

oLoin γ

γ++

=−

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In lossless transmission line, γ= jβ giving:

)tan()tan()(

ljZZljZZZlZ

Lo

oLoin β

β++

=−

There are interesting applications when the length l is multiple of λ/4.

Example: Calculate the input impedance of a 1 m length of cable that is terminated in a load impedance of ZL=20Ω. Assume that the characteristic impedance of the line is 50Ω, its dielectric constant is 1.5 and the frequency of operation is50MHz.

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37.3205037.3502050)(

37.328.1tantan

28.12

×+×+

=−

==

===

jjlZ

l

fu

in

orop

β

µεεπωβ

Ω+= )2.507.87( j

Solution:

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2.6 Standing wave ratio

• In a lossless line, the amplitude of the forward (or backward) voltage remains constant as the wave propagates along z, only with a shift in the phase angle. The superimposition of the forward wave and backward wave results in a standing wave pattern.

• In a standing wave, there are positions at the line where the amplitude of the resultant voltage has maximum and minimum.

|V|max

|V|min

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• The voltage standing wave ratio (VSWR) is the ratio of the maximum and minimum voltage magnitudes. The distance between two successive maximums is equal to λ/2.

L

L

VV

VSWRΓ−Γ+

==11

min

max

• VSWR is useful to find the maximum voltage magnitude on the line due to reflection from the load. If Vinc is the incident voltage on the load,

12max +

=VSWR

VSWRVV inc

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2.7 Smith Chart: a convenient graphical means of determining voltages along transmission lines. It is essentially a plot of the complex reflection coefficient Γ(-l) at a point with input impedance Zin(-l) looking into the end of the transmission line.

oin

oin

ZlZZlZl

+−−−

=−Γ)()()(

Let the real and imaginary parts of Γ(-l) be Γr , Γi respectively, and z be the input impedance normalized by Zo.

11

)(

+−

=Γ

+==

zz

jxrZ

lZzo

in

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After some manipulations, it can be shown that:

( )22

2

22

2

111

11

1

=

−Γ+−Γ

+=Γ+

+−Γ

xx

rrr

ir

ir

• These equations define family of circles on the ( Γr , Γi ) plane corresponding to constant resistance r, and constant reactance x. The reflection coefficient at a point on the line with normalized input impedance z = r+jx is then the vector ending at the intersection point between the constant r and x circles.

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• In a lossless transmission line, there is no attenuation and a wave travelling along the line will only have a phase shift. So the reflection coefficient Γ(-l) at a point of distance lfrom the load at the end of the line is related to the load reflection coefficient ΓL by:

ljL el β2)( −Γ=−Γ

• It means the reflection coefficient has same magnitude but only a phase shift of 2 β l if we move a length l along the line ( Γ rotates clockwise on the Smith Chart when moving away from the load and anti-clockwise when moving towards the load).

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Im

Γ(-l)

ΓL

Re

constant

VSWR

2βl

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Example:

(a) If the reflection coefficient at a location on a transmission line of 100Ω characteristic impedance is Γ = 0.4+j0.2, use Smith chart to determine the input impedance Z at that location.

(b) A load of ZL = 50-j25 Ω is attached to the above line. Use Smith chart to find the input impedance at a distance l = 0.4λfrom the load.

Solution:

(a) Γ = 0.4+j0.2 = 0.45exp(j26.56o)

Find the above point Q in the Smith chart. It corresponds to the intersection of r=2.0 and x=1.0 circles.

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.Q

.P1

.P2

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Therefore the input impedance Zin=100 (2+j1.0)

= 200 + j100

(b) The normalized load impedance zL= 0.5-j0.25 is

represented by the point P1. To find Z at a distance l = 0.4λ rotate the point P1 to point P2 clockwise through a wavelength of 0.4λ. We find z=0.952-j0.77

Therefore the input impedance is 95.2-j77.0 Ω after multiplying by the characteristic impedance of 100Ω.

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END

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A transmission line always has two conductors and a dielectric between the two conductors. The conductors have a resistance and inductance in series. The dielectric has a capacitance and resistance in parallel. But all the resistance, inductance and capacitance are distributed in nature. It means we have to first represent a small elemental section of the line by the above equivalent circuit, and then assume the complete line is represented by an infinite number of such small elemental section connected together.

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You should be able to derive these equations from the equivalent circuit if you remember the following formulae for voltage/current in inductors and capacitors:

In time domain,

dttdvCti

dttdiLtv

)()(

)()(

=

=In an inductor,

In a capacitor,

Using phasors,

In an inductor,

In a capacitor, CVjILIjV

ωω

==

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The general expression for a travelling wave with a time-varying amplitude is:

)cos( ztAev z βωα −= −

In complex representation, the cos function is replaced

by so that )( ztje βω −

tjzjztjz eAeeAev ωβαβωα )()( +−−− ==In phasors, the term is understood, sotje ω

zzj AeAeV γβα −+− == )(

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ILjRzV

eVeVV zz

)( ω

γγ

+−=∂∂

+= −−+

Therefore we can write I as:

( )zz eVeVLjR

I γγ γγω

−−+ +−+

−=1

zz eIeI γγ −−+ +=On simplification, we can find I+, I- and hence the ratio of V+/I+, V-/I-.

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Let Vinc be the forward voltage incident on the load (at

z=0), and the load reflection coefficient φjLL eΓ=Γ

The voltage at any point on the transmission line is:zj

Linczj

inc eVeVV ββ Γ+= −

( )( )φββ +− Γ+= zjL

zjinc eeV

At maximum voltage points, πφββ nzz 2++=−( )LincVV Γ+= 1

max

At minimum voltage points, ππφββ nzz 2+++=−

( )LincVV Γ−= 1min

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EHF (30-300GHz) Radar, radio astronomy, remote sensing

SHF (3-30GHz) Radar, satellite, aircraft navigation

UHF (300MHz-3GHz) TV, radar, microwave oven, mobile phone

VHF (30-300MHz) TV, FM, mobile radio, air traffic control

HF (3-30MHz) Short wave broadcasting

MF (300kHz-3MHz) AM

LF (30-300kHz) Weather broadcast for air navigation

VLF (3-30kHz) Navigation and position location

ULF (300Hz-3kHz) Audio signals on telephone

SLF (30-300Hz) Ionospheric sensing, submarine communication

ELF (3-30Hz) Detection of metal objects