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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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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:
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