Electromagnetics II - 5EPB0

5EPB0 Electromagnetics II

• Area: Electromagnetics and Waves
• Topics: Vector Calculus · Maxwell Equations · Plane Waves and Reflection · Transmission Lines · Waveguides · Antennas and Radiation · S-Parameters and Impedance Matching
• Related courses: Electromagnetics I - 5EPA0 · Components in Wireless Technologies - 5XTC0 · Photonics - 5XTB0

General notes

Topic: Vector Calculus
Plane wave - Wikipedia
result of a gradient is a vector field, while the result of a divergence is a scalar field.

decoupling: express transverse components in longitudinal. (4 transverse components based on 2 longitudinal…)
fundamental unknown usually: basically one of 6 variables
solve W.E→for fundamental unknown…

Waves in time and one spatial dimension

Summary

⇐>

Special cases:

• matched → no reflection →
• open circuit → total reflection with same polarity →
• short circuit → total reflection with inverted polarity →

Detailed notes

DON’T FORGET THE MINUS SIGN WHEN DECOMPOSING FOR CURRENT I-
LEARN CIRCUIT ANALYSIS BETTER
Uniform Plane Waves: A 3D Journey to the Simplest Electromagnetic Wave Propagation
Transmission Lines: Part 1 An Introduction
Maxwell Equations
But how exactly do the voltage and current propagate through transmission lines?
Characteristic impedance
What is the difference between characteristic impedance and input impedance in transmission lines
Transmission coefficient
Reflection coefficient
Z is the specific impedance over which the power is measured.

Uniform plane waves

Topics: Maxwell Equations · Plane Waves and Reflection

source free region
divergence: curl:



uniform plane waves only depend on time, one cartesian coordinate
two coupled systems (right hand rule (thumb: E field, middle: Z, point: H)): and

voltage current amplitudes (another way to represent a coupled system do (right hand rule))
and direction and time dependent
and

TL equations for uniform plane wave:
and
wave equation: or


at steady state TL is ignored

-: progressive wave; +: regressive wave

since



impedance is frequency dependent because is.

TEM-wave propagation along transmission lines (Transverse)

Topic: Transmission Lines



characteristic impedance (only when waves propagated not during steady state ): and also:


lumped components:

Wave reflection

composition: can compose
decomposition: can decompose pro(re)gressive wave
reflection coefficient
and so
and
and
and

when OC there is also no forward backward propagating waves

when there sudden OC → compensation
losses lower the amplitude instead of infinite propagation/reflection.

so time of propagation

otherwise reflected/absorbed
time domain reflection (TDR): used for cable testing: quick pulse with switch.

Transmission lines reflection and transmission

Topic: Transmission Lines

Summary

Detailed notes

SUMMARY

Boundary conditions for penetrable media & Reflection and Transmission & Bounce diagrams

incident, reflected, transmitted field

on the boundary of plane dielectric layers:

therefor:

example:
where
reflection takes place
when its steady use load impedance since TL is short circuit:





steady state no characteristic impedance, since no waves propagate. TL is short-circuited.
For circuit theory see exam question example in folder…

If there is complex load impedance there is phase and amplitude modulation of the reflected waves.

Transverse field vectors & Electromagnetic Power Balance

Poynting vector - YouTube




not useful to take a snapshot of poynting vector (relation of electromagnetic wave in time), thats why we usually find the average over time.

lumped elements represent transmission lines. not actual inductances/capacitances.
at least 2 conducts needed. (coaxial cable)

Poynting’s theorem

Poyting’s theorem: separate secondary and primary (source) current densities


maximize . Where Poynting vector:
stored electromagnetic energy in (electric/magnetic field)

extra

for transverse field: helmholz decomposition
laplace equation…
voltage and current normalization

for electromagnetic power balance: coulomb force + lorentz force

!!! moment in time just after the switch has been activated, but before possible reflections from the end of the line have returned:

Time-harmonic waves

Summary

Detailed notes

now instead of
its
constant complex voltage amplitudes not depending on t,z

where

in a loss-less environment





A-matrix the first one
A matrix is decomposition of medium 1 and composition of medium 2

Power Calculation:
introduces a stub to have impedance matching.

Scattering parameters - Wikipedia

Scattering matrix, phase and group speed, dispersion

Topic: S-Parameters and Impedance Matching

Summary

quarter-wavelength transformer optimal conditions:

Scattering matrix or s-matrix descries how incoming waves are transformed into outgoing waves after interacting with a system.

Transfer matrix or m-matrix relates waves amplitudes at different points within the system.

Detailed notes

only works for small bandwidth one frequency range. To match impedance.




allows us to infer →

where carrier wave angular frequency and slowly varying hull

PART 2

Plane waves in arbitrary directions - reflection and refraction at planar interfaces

Topic: Plane Waves and Reflection

slide info

Summary

wavenumber:

determine p or s polarization if p polarized then secondary conditions satisfied

if the the phase difference is not between 2 components then elliptical

The brewster’s angle is the angle of incidence where light will be perfectly transmitted through a surface with no reflection. The critical angle of internal reflection is when a wave is completely reflected 100% at a certain angle.

reflection and transmission coefficients:
s-polarized


p-polarization:

’

snells law:

Plane Waves II

other

Wavenumber - Wikipedia

wavenumber, refers to the spatial frequency of a wave over a specific unit distance.
A wave number is related to but different from frequency, which is temporal in nature.

Elementary plane wave (EPW): General Solution:


wave number: where is speed of propagation
wave impedance:

propagation in space and time… !!!
and and !!!
dispersion diagram but here we only use the real propagation coefficient so

three types of waves:
the wave number is, in general, a complex number:

wave polarization:

• Wave polarization: time-dependent orientation of the E-field vector at a fixed point in space.
• For instance, in case of a travelling wave in the (depends on direction)
• 
  

if the the phase difference is not between 2 components then elliptical

what is plane in a plane wave: surface of constant amplitude, surface of constant phase… (for most common plane waves these 2 are the same plane).

plane waves in arbitrary directions:
EPW with arbitrary direction:


!!! WAVE VECTOR AND WAVENUMBER DIFFERENT

curl identity applied below:

EPW: conditions for A and k

\text{this leads to dispersion relation: }\vec{k} \cdot \vec{k}=\omega^2\epsilon \mu=k^2$

for the 2nd one in the array this identity was used:

Converting From E to H

Based on format: **

Reflection and Refraction

Topic: Plane Waves and Reflection
only flat discontinuity is considered…
boundary conditions essential… are solutions to integral form Maxwell equations
Boundary Conditions:

• only tangential component of electric field at the surface…
• while magnetic tangential field can be discontinuous as it can induce current at the surface which compensates for it…
• normal of electric field can be discontinuous as charges that accumulate at the surface can compensate for it…
• while normal for the magnetic field is continuous: magnetic field in = magnetic field out… one of the Maxwell definitions…
• for BC keep in mind sign change based on how you choose normal…

oblique incidence medium 1:

• no fields in PEC(metallic surface)…
• tangential components near metals vanish…
• the reason we dont use the 2 boundary condition which equal is because we dont know those components on PEC…
• For component since its normal to surface compared to its conditions are the opposite…
• normal component of wave vector reflects (not tangential), tangential component of electric field reflects not the normal one…
  
  

for H think of right hand rule: middle: k, thumb: E, pointing: H

Deriving Polar Formulas

: Euler Formula

for sine:
for cosine:

• standing waves are obtained by superposition of plane waves when they are in certain shape…
• fourth type of wave:

• traveling power wave everywhere in the x direction…
• at boundary power is 0 then it grows and then again 0
• the circled thing is distructive interference example if you are near car that could impact your fm radio…

Wave guide: Conducting plates at , lossless medium
Additional boundary condition:
this requires:
Waveguide - Wikipedia

• p polarized(right diagram) E field parallel to plane of incidents, while s(right diagram) is perpendicular
• x component vanishses in field component since itsn ormal to the surface and equal 0 when we apply boundary conditions…

• z component only at surface continuous…

reflection and transmission coefficients:

s-polarization:
if both mediums are dielectric materials ()
we use for both mediums


p-polarization:
by exchanging for and for

Brewster angle and Critical angle

angle at which reflection and refraction are at angle means reflected light is polarized to a certain polarization
Physics 53 Polarization (5 of 5) Brewster’s Angle - YouTube


Substitution leads to a linear equation for
Solving the equation, it gives:

5.

Guided waves

Topic: Waveguides

Summary

Resonant Cavities

• TE Modes:
  • Fundamental equation:
    
    IF TM
  • Solution with boundary conditions:
    
    
  • Cutoff frequencies:
    
    cutoff frequency, corner frequency, or break frequency is a boundary in a system’s frequency response at which energy flowing through the system begins to be reduced (attenuated or reflected) rather than passing through.
    n=2 means two half-wavelengths fit along the y-axis.

The indices m, n, and p in the context of resonant cavities represent the mode numbers, which correspond to the number of half-wavelengths of the standing wave pattern along each of the three spatial dimensions (x, y, and z) inside the cavity.

Dielectric Slab Waveguides

• Waveguide Structure:

  • For , Total Internal Reflection (TIR) occurs.
  • Fields in each region:
    

• TM Modes:

  • Inside the slab:
    
    
    
  • ODD modes:
    
    
  • EVEN modes:
    
    

• TE Mode Summary:

  • Similar expressions for TE modes can be derived by replacing with and adjusting boundary conditions.

• Key Relationships:

  • Evanescent field decay constant:
    
  • Phase condition for modes:
    
    
  • Single mode condition:
    
    

Parallel-Plate Waveguide

Topic: Waveguides
Waveguides Explained - YouTube

Slab Waveguide Explained - YouTube
Origin of TE and TM Modes - YouTube

• ampere, faraday law, must be imaginary otherwise wave decays wont work.

• 
  

• polarisation in terms of wave guides

• fundamental unknwon: able to find all other of the total 6 components

• Since TE waves…

• decoupling:

• 
  

• TM Waves:

• 
  

• TEM Waves:

• 
  

Rectangular Waveguides

Topic: Waveguides

• The rectangular guide will support the TE and TM modes, but it will not support a TEM mode.
• so the graph
• before only plates on top bottoms, now on left and right…
• certain combination of there is a certain frequency ()…
• 
  
• when TE/TM have same cutoff frequency: degenerate mode

• must exceed to achieve real value for so mp propagates…
• cutoff for both is same frequency and wavelength…

Resonant Cavities and Dielectric Slab Waveguides

• for decoupling, only cavity formulas given (closed book) look and remember how those equations for parallel and rectangular those look like…
• Evanescent field - Wikipedia
  
• so modes of TE, TM also split into (odd/even)
• standing wave inside, and 2 evanescent waves outside

Resonant Cavities

fully enclosed space… producing standing waves inside… no direction of propagation




even though now no propagation in (z) we still state it (standard).

Dielectric Slab

• not using metal but plastic, with
• assume no dependence for each homogeneous region :
  
  for outside
  solutions expressed in TM/TE modes below is analysis for TM:
  where
  
  
  ODD
  EVEN
  
  
  

FINALLY FOR TE MODE SIMILAR:

single mode condition

wavelength range of single mode operation

Radiation

Topic: Antennas and Radiation

Summary

see behavior of far field and how dipole electric properties behave ()

Radiated Power
Electric Field Calculation

The Electric Point Dipole

Topic: Antennas and Radiation

we can eliminate E

Point Dipole

• The solution is a spherical, two-dimensional wave: the fields only depend on r and θ.
• The solution is a TM-wave: there is no magnetic field component in the direction of propagation.
• The wave propagates radially in all directions.

Magnetic point dipole & Antenna Specifications

Topic: Antennas and Radiation
Near and far field - Wikipedia
where antenna radiates: radiation pattern…


solving magnetic dipole:

• working out fields
• using duality
• using potentials

duality

pick solutions for electric dipole and replace:

• 
  

potentials

antennas

far field, radiation pattern, radiation resistance, directive gain

far field

radiation pattern

as shown on the picture of the board its a polar diagram

radiation resistance

as derived on the board, its needed to have a proper antenna circuit, we only care about the real part…

directive gain

The directive gain of an antenna is the ratio of the maximum power flux in the far field, and total power divided by (i.e. the power radiated by an isotropic radiator).

Dipole Arrays

Topic: Antennas and Radiation
controlling distance and phase…