An Introduction to Electromagnetic Field Theory (CHRISTIAN GALLAI)

8/10/2019 An Introduction to Electromagnetic Field Theory (CHRISTIAN GALLAI)

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AN INTRODUCTION TO ELECTROMAGNETIC FIELD

THEORY

CHRISTIAN G ALLAI

MCGIL L U NIVERSITY, MONTREALQC, CANADA

FAL L 2011


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Christian Gallai An Introduction to Electromagnetic Field Theory Fall 2011

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Contents

1. Introduction to Transmission Lines ....................................................................................................... 7

1.1. Telegraphers Equations ............................................................................................................... 7

2. Lossless Propagation ............................................................................................................................. 7

2.1. Wave Equation .............................................................................................................................. 7

2.2. Lossless Wave Equation ( )........................................................................................... 73. Harmonic Waves on Lossless Lines ....................................................................................................... 8

3.1. Time Harmonic Representation .................................................................................................... 8

3.2. Phasor Representation.................................................................................................................. 8

4. Transmission Line Equations in Phasor Form ....................................................................................... 9

4.1. Propagation Constant ................................................................................................................ 94.2. Attenuation Coefficient ............................................................................................................. 94.3. Signal Strength .............................................................................................................................. 9

4.4. Impedance of Lossy Lines ............................................................................................................. 9

5. Power Transmission ............................................................................................................................ 10

5.1. Instantaneous Power .................................................................................................................. 10

5.2. Time-Average Power ................................................................................................................... 10

6. Wave Reflection .................................................................................................................................. 10

6.1. Reflection and Transmission Coefficients ................................................................................... 10

6.2. Power Delivered to the Load ...................................................................................................... 10

7. Voltage Standing Wave Ratio .............................................................................................................. 11

7.1. Incident and Reflected Waves .................................................................................................... 11

7.2. Maxima and Minima ................................................................................................................... 11

7.3. Voltage Standing Wave Ratio (VSWR) ........................................................................................ 12

7.4. Open and Short Circuit Lines....................................................................................................... 12

8. Input Impedance ................................................................................................................................. 12

8.1. Forward and Backward Travelling Waves ................................................................................... 12

8.2. Wave Impedance ........................................................................................................................ 13

8.3. Input Impedance at ........................................................................................................ 13


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8.4. Power Transferred to the Load: .................................................................................................. 13

8.5. Input Current and Voltage .......................................................................................................... 13

8.6. Half Wave and Quarter Wave Lines ............................................................................................ 14

8.7.

For Short and Open Circuit Loads ........................................................................................ 14

9. Smith Charts ........................................................................................................................................ 14

9.1. Normalized Load Impedance ...................................................................................................... 15

9.2. Short Circuit, Matched Load, and Open Circuit Points ............................................................... 15

9.3. R and X Circles ............................................................................................................................. 15

10. Single Stub Impedance Matching ................................................................................................... 15

10.1. Single Stub Calculations for Lossless Lines .............................................................................. 16

10.2. Double Stub Matching ............................................................................................................ 16

11. Transient Signals on Transmission Lines ......................................................................................... 17

11.1. Voltage as a Function of Time ................................................................................................. 17

11.2. Current as a Function of Time ................................................................................................. 18

12. Pulses and Initially Charged Lines ................................................................................................... 19

12.1. Pulse Representation .............................................................................................................. 19

13. Power Transfer to a Load Calculation Methods.............................................................................. 19

13.1. Method # 1 .............................................................................................................................. 19

13.2. Method # 2 .............................................................................................................................. 20

13.3. Power Efficiency ...................................................................................................................... 21

14. EM Wave Propagation in Free Space .............................................................................................. 21

14.1. Maxwells Equations ............................................................................................................... 21

14.2. Source Free Wave Equations ............................................................................ 2114.3. Homogeneous Vector Helmholtz Equations ........................................................................... 22

14.4. Plane Waves ............................................................................................................................ 22

14.5. Harmonic Waves ..................................................................................................................... 22

Magnetic Fields ....................................................................................................................... 2314.7. Transverse Electromagnetic Wave.......................................................................................... 2315. Wave Propagation in Dielectrics ..................................................................................................... 23

15.1. Propagation Constant ............................................................................................................. 24

15.2. Waves in Lossless Dielectrics .................................................................................................. 24

15.3. Impedance in a Dielectric ........................................................................................................ 24


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15.4. Refractive Index ...................................................................................................................... 25

15.5. Approximations for Low Loss Dielectrics ................................................................................ 25

16. The Loss Tangent............................................................................................................................. 25

16.1. Review Concepts ..................................................................................................................... 25

16.2. The Loss Tangent ..................................................................................................................... 25

16.3. Attenuation and Phase Coefficients ....................................................................................... 26

16.4. Average Power Loss ................................................................................................................ 26

16.5. Conductors and Insulators ...................................................................................................... 26

17. Power and Energy Transport .......................................................................................................... 26

17.1. The Poynting Vector ................................................................................................................ 27

17.2. Instantaneous Power Density ................................................................................................. 27

17.3. Time Average Power Density and Flux .................................................................................... 27

17.4. Example 1 ................................................................................................................................ 27

17.5. Power Flux Through a Medium ............................................................................................... 28

18. Waves in Good Conductors: The Skin Effect ................................................................................... 28

18.1. Skin Depth ............................................................................................................................... 28

18.2. Impedance in Good Conductors ............................................................................................. 29

18.3. Frequency Dependent Resistance .......................................................................................... 29

18.4. Surface Resistance .................................................................................................................. 29

19. Polarization ..................................................................................................................................... 29

19.1. Linear Polarization (P-State) ................................................................................................... 29

19.2. Circular Polarization ................................................................................................................ 29

19.3. Elliptical Polarization ............................................................................................................... 30

20. Reflection and Dispersion of Waves ............................................................................................... 30

20.1. Boundary Conditions ............................................................................................................... 31

20.2. Reflection and Transmission Coefficients ............................................................................... 31

20.3. Total Electric and Magnetic Fields in Region 1 ....................................................................... 31

20.4. Standing Wave Zeroes and Maxima........................................................................................ 31

20.5. Transmitted and Reflected Power .......................................................................................... 32

21. Standing Waves and Plane Wave Reflection .................................................................................. 32

21.1. Total Wave Amplitude ............................................................................................................ 32

21.2. Maximum Amplitude .............................................................................................................. 32


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21.3. Minimum Amplitude ............................................................................................................... 32

21.4. Standing Wave Ratio ............................................................................................................... 33

22. Wave Reflection at Multiple Interfaces .......................................................................................... 33

22.1. Review of Waves Incident on an Interface ............................................................................. 33

22.2. Wave Impedance at a Dielectric Boundary ............................................................................. 34

22.3. Dielectric Coating Antireflection ............................................................................................. 34

22.4. Radome Antireflection ............................................................................................................ 34

23. Wave Propagation in Arbitrary Directions ...................................................................................... 35

23.1. Transverse EM Waves ............................................................................................................. 35

24. Wave Reflection at Oblique Incidence ............................................................................................ 35

24.1. Parallel and Perpendicular Polarization .................................................................................. 35

24.2. Boundary Conditions ............................................................................................................... 36

24.3. Snells Law ............................................................................................................................... 36

24.4. Field Amplitudes ..................................................................................................................... 36

24.5. Fresnel Coefficients for Refractive Index ................................................................................ 36

25. Special Cases of Wave Reflection ................................................................................................... 37

25.1. Total Internal Reflection ......................................................................................................... 37

25.2. Brewster Angle ........................................................................................................................ 37

26. Dispersive Materials and Group Velocity ........................................................................................ 38

26.1. Dispersion ............................................................................................................................... 38

26.2. Phase and Group Velocity ....................................................................................................... 38

26.3. Calculation of Dispersion ........................................................................................................ 38

27. Electromagnetic Waves in Transmission Lines ............................................................................... 39

27.1. Parallel Plate Transmission Line .............................................................................................. 39

27.2. Power Transmitted ................................................................................................................. 39

27.3. Impedances ............................................................................................................................. 40

27.4. Ratio of Electric to Magnetic Field .......................................................................................... 40

27.5. Resistance ............................................................................................................................... 40

27.6. Loss .......................................................................................................................................... 40

27.7. Phase Velocity ......................................................................................................................... 40

28. Basic Waveguide Operation ............................................................................................................ 40

28.1. Disadvantages of Transmission Lines ...................................................................................... 40


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28.2. From Transmission Line to Waveguide ................................................................................... 41

28.3. Phase Constant ....................................................................................................................... 41

28.4. Cut-Off Frequency ................................................................................................................... 41

28.5. Cut-Off Wavelength ................................................................................................................ 42

28.6. Example 1 ................................................................................................................................ 42

29. Phase and Group Velocity in the Parallel Plate Waveguide............................................................ 43

29.1. Group Velocity Dispersion and Delay...................................................................................... 43

30. Wave Equations in Parallel Plate Waveguides ................................................................................ 43

30.1. General Field Solution ............................................................................................................. 44

30.2. Waves at Cut-Off ..................................................................................................................... 44

30.3. Magnetic Field (TE Mode) ....................................................................................................... 44

30.4. Impedance .............................................................................................................................. 45

30.5. TM Modes ............................................................................................................................... 45

31. Rectangular Waveguides ................................................................................................................ 45

31.1. TM Modes: Propagation Constant and Cut-Off Frequency .................................................... 45

31.2. TE Modes: Propagation Constant and Cut-Off Frequency ...................................................... 46

31.3. TE10 Mode .............................................................................................................................. 46

31.4. Power in a Waveguide ............................................................................................................ 46

32. Dielectric Slab Waveguides ............................................................................................................. 47

32.1. Fresnel Equations For Reflection ............................................................................................ 47

32.2. Phase Shift Expressions ........................................................................................................... 47

32.3. Self-Consistency Equation ....................................................................................................... 47

33. Introduction to Antennas ................................................................................................................ 48

33.1. What Does an Antenna Do? .................................................................................................... 48

33.2. Radiation Resistance ............................................................................................................... 48

33.3. Antenna Pattern and Directivity ............................................................................................. 48

33.4. Elemental Dipole Summary ..................................................................................................... 49

33.5. Elemental (Hertzian) Dipole .................................................................................................... 49

33.6. Spherical Coordinates Definitions ........................................................................................... 49

33.7. Time-Varying Field Patterns .................................................................................................... 49

33.8. Retarded Potential .................................................................................................................. 50

33.9. Time Harmonic Solution .......................................................................................................... 50


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33.10. Vector Magnetic Potential Due To Elemental Dipole ............................................................. 50

33.11. A in Spherical Coordinates ...................................................................................................... 50

33.12. H and E Fields .......................................................................................................................... 50

33.13. Far Field Regime ...................................................................................................................... 51

33.14. Typical H-Plane Patterns ......................................................................................................... 51

33.15. Radiated Power ....................................................................................................................... 51

33.16. Radiation Resistance ............................................................................................................... 51


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1. Introduction to Transmission Lines

At low frequencies we use a lumped circuit model

As frequencies increase, wave propagation becomes important

This occurs when

o The circuit dimensions are larger than the propagation time for voltage and current

transients

o The circuit dimensions are several wavelengths or less

1 1 Telegraphers Equations

2. Lossless Propagation

If a line has no resistance and no conductance we describe it as lossless

The speed of signal propagation on such a line is given by This is the speed of light on the line

Voltage and current are related to each other by the line impedance

2 1

Wave Equation

2 2

Lossless Wave Equation ( )


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3.

Harmonic Waves on Lossless Lines

When there is only one frequency present we can represent the signal as a cosine wave

The phasor representation is the complex time independent form

To obtain the time dependent measurable voltage, multiply the phasor by and take the realpart3 1

Time Harmonic Representation

|| ||

||

Where is the angular frequency, is the initial phase, and is the phase constant

3 2

Phasor Representation

||

Where ||is the complex amplitude. For the time-independent phasor form we have:


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Two voltage waves with equal frequencies and opposite amplitudes are propagating in opposite

directions on a transmission line. Determine the total voltage as a function of time and position:

[] 4.

Transmission Line Equations in Phasor Form

4 1

Propagation Constant In order to deal with lossy lines, we introduce a new parameter, the propagation constant

. We use this

to represent the spatial wave evolution:

4 2 Attenuation Coefficient Since is a positive real number, wave amplitude decays as the wave travels to the right (positive z). Itsunits are Nepers/m (Np/m).

4 3

Signal Strength

|| 4 4 Impedance of Lossy Lines

||


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5. Power Transmission

Time-average power is always positive when net power is flowing along a positive axis

Instantaneous power can be zero, or negative

Power loss is often measured in dB/m

5 1 Instantaneous Power

5 2 Time-Average Power

||||

6. Wave Reflection

Changes of impedance give rise to reflections

The ratio of incident voltage to reflected voltage is given by the reflection coefficient

The fraction of power reflected is given by the square of this value

The reflection coefficient can be complex

6 1 Reflection and Transmission Coefficients

The reflection coefficient defines the ratio of the reflected voltage to the incident voltage

There is a corresponding transmission coefficient that defines the ratio of the load voltage to the

incident voltage

6 2 Power Delivered to the Load

||


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|||| 7. Voltage Standing Wave Ratio

When the load is not matched to the line, reflections occur

These interfere with the incident wave, producing a mixture of standing and travelling waves

The location of the standing wave minima and maxima are a function of the load impedance

The ratio of maximum to minimum amplitude is called the voltage standing wave ratio

7 1 Incident and Reflected Waves

7 2 Maxima and Minima

If , , we have:

If , , we have:


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7 3 Voltage Standing Wave Ratio (VSWR)

|| ||7 4 Open and Short Circuit Lines

8. Input Impedance

Due to reflections the ratio of voltage to current changes along the line

This can be expressed by the wave impedance

At the input we call this the input impedance

The input impedance determines the power that can be delivered to the load

A half-wave line has an input impedance equal to the load impedance

A quarter-wave line can be used for impedance matching

Short and open circuits have purely reactive input impedances

8 1 Forward and Backward Travelling Waves


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8 2

Wave Impedance

8 3 Input Impedance at

8 4

Power Transferred to the Load:

8 5 Input Current and Voltage


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8 6

Half Wave and Quarter Wave Lines

If we look at the expression for input impedance we see that when we have: This is called a half-wave line, the input impedance is always equal to the load impedance. When we

have we have:

This is called a quarter-wave line. It can be used for impedance matching. In order to perform

impedance matching we add a line that is a quarter wavelength long with an impedance :

8 7 For Short and Open Circuit Loads

9. Smith Charts


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The Smith Chart is a graphical representation of the complex quantities involved in transmission

line calculations

It plots the reflection coefficient as a function of load impedance

It plots the normalized input impedance as a function of distance from the load

It also allows the standing wave ratio to be determined

9 1 Normalized Load Impedance

9 2 Short Circuit, Matched Load, and Open Circuit Points

9 3 R and X Circles

R-circles represent the load resistance

X-Circles represent the load reactance

10. Single Stub Impedance Matching

Impedance matching is necessary for maximum power transfer

Simple impedance matching can be achieved through the use of a quarter-wave length of line of

intermediate impedance in front of the load


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This is not generally the most suitable technique

Improved impedance matching is achieved through the use of a stub: a parallel circuit which has

an input admittance equal to that of the line

A more advanced technique is to employ a double stub

10 1 Single Stub Calculations for Lossless Lines

10 2 Double Stub Matching


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Fix one stub in parallel across the load

Place the other at fixed distances (1/8, wavelength, etc) Two variable lengths and permit the real and imaginary parts to be matched

11.

Transient Signals on Transmission Lines

Transients cannot be modelled the same way as harmonic signals

In a lossless line edges travel at the phase velocity

Forward and backward going waves build up on the line

Eventually the load current and voltage match the low-frequency solution

Reflection diagrams show the reflected edges graphically as a plot of or 11 1

Voltage as a Function of Time

( )


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11 2

Current as a Function of Time


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12. Pulses and Initially Charged Lines

Pulses can be modeled as the combination of a positive and negative edge

An initially charged line can be used to generate a pulse

12 1 Pulse Representation

{ 13.

Power Transfer to a Load Calculation Methods

13 1 Method # 1

The first method is based on calculating the power flowing into the line:

1.

Power atthe load:


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2.

In steady state this is equal to the power deliveredby the generator at the input:

3.

To calculate

and

(the voltage and current at the input) we can apply the standard circuit laws

at the generator:

4.

Note that is the input impedance (calculated in the usual way). This takes the line length intoaccount.

13 2

Method # 2

The second method makes use of the reflection coefficient.

1.

Again, power at the load:

2.

Now recognize that load voltage and current are given by (Where and are the forward-goingvoltage and current amplitudes):

3.

So power at the load is:

|| ||4.

We still need to find :


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13 3

Power Efficiency

We can write the power efficiency (ratio of power delivered to the load to the forward-going power) as:

|| 14. EM Wave Propagation in Free Space

Electromagnetic waves are a self-consistent solution to Maxwells equations

Time harmonic waves have a single frequency

We can represent them via phasor notation

For a time harmonic wave, we can obtain a set of wave equations that are time independent

These are the Helmholtz equations

14 1 Maxwells Equations

Constitutive relations:

14 2

Source Free Wave Equations With and the Maxwell equations become:


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This gives us the source free wave equations:

14 3

Homogeneous Vector Helmholtz Equations

14 4 Plane Waves

Lets assume that the E-field only has an x-component and only varies in z:

14 5 Harmonic Waves


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Where is the angular frequency, is the wavenumber (), is the initial phase, is theamplitude, is the phase velocity, is the period, and is the phase.14 6 Magnetic Fields

Magnetic fields and Electric fields are related through:

The existence of an electric field implies the existence of a magnetic field. For the Harmonic wave,

travelling in the z direction, we have the Electric field in the x direction of:

This gives us a magnetic field of:

14 7 Transverse Electromagnetic Wave

Where

is the impedance of free space:

15. Wave Propagation in Dielectrics


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A lossy medium absorbs energy from an EM Wave

Lost energy is converted to heat

All real media are lossy

Loss depends on frequency

Loss can be due to:

o

Conduction: moving charges against resistance requires energy

o Damping: forces in a dielectric

Waves are attenuated in a lossy medium

15 1 Propagation Constant

15 2 Waves in Lossless Dielectrics

For the lossless dielectric

:

The wavelength is reduced!

15 3

Impedance in a Dielectric

The impedance in a medium is also modified from that of a vacuum:

This means that the ratio of the electric to the magnetic field will change.


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15 4

Refractive Index 15 5 Approximations for Low Loss Dielectrics

16.

The Loss Tangent

Loss can arise from

o Conduction (in a metal)

o Damping (in a lossy dielectric)

We can represent this with a complex dielectric constant

Loss can be modeled as if a conduction current existed

The angle between the displacement current and the actual current is called the loss tangent

The magnitude of this parameter provides an estimate of loss Large loss tangent high absorption

16 1 Review Concepts

For a lossless medium we have:

For a lossy medium we have:

16 2 The Loss Tangent


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Use the ratio of conduction to displacement current to determine the degree of loss:

16 3 Attenuation and Phase Coefficients

16 4 Average Power Loss

We know that in a conducting region, the average power dissipated by current flow is:

16 5 Conductors and Insulators

When the loss tangent is small :o

Low loss

o Insulator/dielectric

When the loss tangent is large :o

High loss

o Conductor

17. Power and Energy Transport


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EM Waves transport energy

The direction of power flow is given by the Poynting vector

The Poynting vector describes the power density In isotropic media this is the same direction as the wave vector

We can distinguish between instantaneous and time average power Time average power is related to field amplitude via:

||

17 1

The Poynting Vector

17 2 Instantaneous Power Density

|| [ ( )]17 3 Time Average Power Density and Flux

|| 17 4 Example 1

The transmitter mast on Mount Royal is specified as . The IEEE safety limit for RF power is . How close can you stand to the transmitter? How much power reaches us here? What is theelectric field in the classroom due to the transmitter?

Area:

Power Density:


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17 5 Power Flux Through a Medium

||

( )18. Waves in Good Conductors: The Skin Effect

In conductors, wave amplitude falls by within the skin depth The attenuation goes up with the square root of frequency

Skin depth falls with the square root of frequency The power lost is the same as if all the current were flowing within one skin depth

18 1

Skin Depth

Attenuation is very rapid in a conductor. Skin depth measures the distance over which a wave isattenuated by a factor of :


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18 2

Impedance in Good Conductors

18 3 Frequency Dependent Resistance

18 4

Surface Resistance

19.

Polarization

Defined by direction of electric field

Different states:

o Unpolarized light: no constant E-field direction

o Linear polarization: E-field always remains in the same direction

o Circular polarization: E-field direction rotates in a circle; amplitude remains constant

o Elliptical polarization E-field direction and magnitude traces an ellipse

19 1

Linear Polarization (P-State)

General linear polarization can be described as a vector sum of vertical and horizontal linear

polarizations:

( )Angle:

Magnitude:

19 2

Circular Polarization


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Right Circular:

[ ]Left Circular:

[ ]Sum:

19 3

Elliptical Polarization

General case where :

20. Reflection and Dispersion of Waves

All power is reflected from a perfect conductor

There is no net transfer of power and the Poynting vector is zero

The wave in front of the conductor is a standing wave

Nodes of the standing wave are spaced at a wavelength

The transmitted and reflected power must add to the incident power


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20 1

Boundary Conditions

At the boundary, the tangential E-field is continuous:

20 2

Reflection and Transmission Coefficients

20 3

Total Electric and Magnetic Fields in Region 1

These are standing waves.

20 4

Standing Wave Zeroes and Maxima

The electric field has zeroes at:

The electric field has maxima at:


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20 5 Transmitted and Reflected Power

The transmitted power is equal to:

||While the reflected power is equal to:

||21. Standing Waves and Plane Wave Reflection

When partial reflection occurs, partial standing waves will also arise

The standing wave ratio tells about the degree of reflection

The standing wave peaks are spaced by one half a wavelength

The phase of the standing wave pattern is a function of the reflection coefficient

21 1 Total Wave Amplitude

Calculate the total wave amplitude in region 1:

( )21 2 Maximum Amplitude

Maximum amplitude occurs when the two terms are in phase:

| | ||

21 3 Minimum Amplitude

Minimum amplitude occurs when the forward and reflected waves are 180 degrees out of phase:


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| | ||

21 4 Standing Wave Ratio

As with transmission lines, the standing wave ratio tells us how much reflection is occurring. The

standing wave Ratio is equal to: | || | || ||

22. Wave Reflection at Multiple Interfaces

The effective impedance at the surface of a multiple dielectric stack is a function of the

impedances and spacings of all the layers and also of the wavelength of the incident wave

The reflection and transmission coefficients are calculated from the effective impedance

If the effective impedance matches the incident medium impedance there will be no reflection

Two important antireflection configurations:

o wave layer with impedance intermediate between air and substrate

o

wave layer (radome) surrounded by a single medium

22 1 Review of Waves Incident on an Interface


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22 2

Wave Impedance at a Dielectric Boundary

22 3 Dielectric Coating Antireflection

22 4 Radome Antireflection


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23. Wave Propagation in Arbitrary Directions

E and H are orthogonal

E and H are related through the impedance of the medium

Direction of propagation is described by the wave vector The plane of constant phase is perpendicular to the wave vector E and H are perpendicular to the wave vector

A wavefront is a plane of constant phase

The wave vector defines the direction of the wave

Wavefronts are orthogonal to the wave vector

Plane waves have planar wavefronts

A plane EM wave is called a transverse electromagnetic wave (TEM)

23 1 Transverse EM Waves

We can write the wave as:

24. Wave Reflection at Oblique Incidence

For oblique incidence at a dielectric boundary, waves are reflected and refracted

The angle of refraction is given by Snells law

The reflection and refraction coefficients are functions of the incident wave polarization

We describe the wave polarization as being either parallel to the plane of incidence (p-

polarization) or perpendicular to the plane of incidence (s-polarization)

24 1 Parallel and Perpendicular Polarization


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24 2 Boundary Conditions

We know that at the boundary tangential E-fields are conserved. Tangential E-fields are in the z-

direction, therefore at the boundary: 24 3 Snells Law Giving us that . And Snells refraction law:

24 4 Field Amplitudes

24 5

Fresnel Coefficients for Refractive Index


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25. Special Cases of Wave Reflection

Under certain conditions of oblique incidence we can have either total reflection or total

transmission

Total internal reflection occurs for both polarizations when a wave passes from a low index

medium to a higher index medium and when the angle is greater than the critical angle

At the Brewster angle, only s-polarized light is reflected, and all p-polarized light is transmitted

25 1 Total Internal Reflection

Is there a condition for which we get 100% reflection at an interface? The reflection coefficient for p-

polarization is:

So the condition for total reflection is given by the critical angle :

Thus we require that . So the wave must start in a higher index material. This is also described astotal internal reflection.

25 2 Brewster Angle

At the Brewster angle, which mean that all of the wave amplitude is transmitted:


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26. Dispersive Materials and Group Velocity

If the phase constant is a nonlinear function of frequency then signal propagation is frequency

dependent

This is called dispersion

The speed of information is called the group velocity

This is typically less than the phase velocity

Signals will be distorted

Energy also travels at the group velocity

26 1

Dispersion

In free space all frequencies travel at the same speedthe phase velocity

. However, in materials, the

phase velocity, and hence the phase constant is a nonlinear function of frequency.26 2 Phase and Group Velocity

Speed of the carrier = phase velocity :

Speed of the envelope = group velocity :

26 3

Calculation of Dispersion


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27. Electromagnetic Waves in Transmission Lines

The equations for inductance and resistance that we have assume that the current is confined

within a thin skin depth

At lower frequencies this assumption is no longer true

The textbook includes the equations for low frequency operation

TEM wave between conductors = Transmission line!

Non-TEM wave between conductors = waveguide!

27 1 Parallel Plate Transmission Line

Voltage:

Current:

Therefore, for the parallel plate transmission line we have:

27 2 Power Transmitted

||


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27 3

Impedances

27 4 Ratio of Electric to Magnetic Field

27 5 Resistance

27 6 Loss

27 7 Phase Velocity

The phase velocity is the same as that for a plane wave:

28.

Basic Waveguide Operation

EM waves can travel in other modes than TEM inside waveguides

Each mode travels at a given angle as a function of frequency and waveguide dimensions

Non-TEM modes cannot propagate below the cut-off frequency

The phase and group velocity of these waves are modified by the waveguide

28 1 Disadvantages of Transmission Lines

Transmission lines work well at moderate frequencies (MHzGHz depending on distance). Loss

increases with frequency, due to surface resistance. Once wavelength


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28 2

From Transmission Line to Waveguide

TEM wave between conductors:

Transverse Electric Wave:

Transverse Magnetic Wave:

28 3 Phase Constant

Stable propagation requires that a twice-reflected wave has same phase as an unreflected wave.

Condition for stable mode:

Phase constant:

28 4 Cut-Off Frequency

Phase constant:

Define the cut-off frequency as:


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So that the phase constant becomes:

Above cut-off phase constant is real

o

Propagation

Below cut-off phase constant is imaginaryo No Propagation

28 5

Cut-Off Wavelength

28 6 Example 1

What is the lowest frequency transverse mode that will propagate in a planar waveguide 5 mm thick,

filled with a dielectric material with relative permittivity 2.25? For a frequency 20% above this, calculatethe guide wavelength, phase velocity, and group velocity. What is the frequency of the next highest

mode?

TM1 and TE1 both have the same cut-off frequency:


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The next highest mode is :

29.

Phase and Group Velocity in the Parallel Plate Waveguide

The phase and group velocity are frequency dependent

At cut-off, phase velocity is infinite, and group velocity is zero

As frequency increases above cut-off, phase velocity decreases and group velocity increases

For the planar waveguide, we can model the electric field as interference of two waves

29 1 Group Velocity Dispersion and Delay

If the group velocity changes with frequency, then dispersion will occur. Since:

Waveguides are usually dispersive. Also, signals that travel in different modes will suffer from a group

delay difference:

30.

Wave Equations in Parallel Plate Waveguides

We have used the term mode without really defining it

A mode of a waveguide is a distribution of electric and magnetic fields which will propagate

along the waveguide without change except for a phase evolution term

i.e. it is a solution to the equation:


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As we have seen, for planar waveguides there are two families of solutions (TE and TM)

which are distinguished by their eigenvalues These are the TE1, TE2, , TEn and TM1, TM2, , TMn modes

30 1 General Field Solution

We derived the equation for the E-field for a TE mode in a parallel plate waveguide by considering wave

interference. However, we did not calculate the magnetic field. Is there a more general method to

obtain E and H fields for any type of waveguide? Yes there is! The field solution is:

30 2 Waves at Cut-Off

When , the waves are oscillating between the top and bottom plates as standing waves. Themode number is the number of half-wave cycles that fit between the plates.

30 3 Magnetic Field (TE Mode)

Now that we have the E-field we can find the H-Field:


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30 4

Impedance

30 5 TM Modes

We can follow a similar route to derive the equations for TM modes:

31. Rectangular Waveguides

Rectangular waveguides provide confinement in both x and y

We assume conducting boundary conditions

They are single conductor waveguides

Rectangular waveguides do not support TEM waves. Nor do any other single conductor

waveguides.

31 1

TM Modes: Propagation Constant and Cut-Off Frequency


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Cut-Off Frequency:

31 2 TE Modes: Propagation Constant and Cut-Off Frequency

Propagation constant and cut-off frequency the same as for TM modes:

31 3

TE10 Mode

Field values for the TE10 mode:

31 4 Power in a Waveguide

Power can be calculated using the Poynting vector:


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32.

Dielectric Slab Waveguides

32 1

Fresnel Equations For Reflection

Perpendicular amplitude reflection coefficient:

In the TIR regime we modify these expressions as follows:

32 2 Phase Shift Expressions

For Perpendicular Polarization (TE):

For Parallel Polarization (TM):

32 3

Self-Consistency Equation


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33.

Introduction to Antennas

The elemental dipole antenna is the basic element of all linear antennas

Its just a short length of wire!

Changing the current in it changes the external E and H fields

There is a lag between the current changing and the external fields changing

The information about the changing current travels as a wave into space

In the far-field, the waves look like plane waves

33 1 What Does an Antenna Do?

Transform electrical signals into electromagnetic waves, and vise versa

Converts oscillating electrons into photons

Power is transferred from the antenna to a distant object

Parallel plate transmission line with flared end:

33 2 Radiation Resistance

What is the load impedance of the flared transmission line?

Not infinite (open circuit), because no power would be broadcast from it

An antenna has a radiation resistance

This is the equivalent resistance that would dissipate the same amount of power as the antenna

is broadcasting

Efficient antennas have a high radiation resistance

33 3 Antenna Pattern and Directivity


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33 4 Elemental Dipole Summary

The elemental dipole antenna is the basic element of all linear antennas

Its just a short length of wire!

Changing the current changes the external E and H fields

There is a lag between the current changing and the external fields changing

The information about the changing current travels as a wave into space

In the far-field, the waves look like plane waves

33 5 Elemental (Hertzian) Dipole

Electric dipole with oscillating charges

Current flows from one pole to the other

33 6 Spherical Coordinates Definitions

33 7

Time-Varying Field Patterns

Both current and charge are time-dependent

We could calculate the radiation field using either:

o

The E-field (from charge)

o The H-field (from current)


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We will calculate the H-field

To do this we need to work out the retarded potential

33 8 Retarded Potential

The potential at a distance is retarded It is the potential due to the current at time

agoo is the speed at which information about the potential travels

33 9

Time Harmonic Solution

Current densityis time harmonic:

Time harmonic wave equation for potentials:

33 10 Vector Magnetic Potential Due To Elemental Dipole

For a thin short dipole we can write:

33 11

A in Spherical Coordinates

33 12 H and E Fields


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33 13 Far Field Regime

33 14 Typical H-Plane Patterns

H-plane patterns are often more complex than that of the simple elemental dipole

Antenna arrays have directed patterns

33 15 Radiated Power

Calculate this via the Poynting vector:

33 16 Radiation Resistance

Since the antenna is radiating power, it must present a resistive load to the source

We can introduce a radiation resistance

This is the resistance that would dissipate the same power

Since for an elemental dipole we have:

Therefore the radiation resistance is:

Documents

An Introduction to Electromagnetic Field Theory (CHRISTIAN GALLAI)