The Course

In winter 2015, as the 2nd course of my UofT M.Eng (course based engineering master’s), I took Advanced Antenna Theory (ECE1229H1S).

Notes for that course (205 pages) are available in the following formats:

This course was taught by Prof. G.V. Eleftheriades,.

The official course description at the time was:

This course deals with the analysis and design of a range of antennas. Topics addressed include: definitions of antenna parameters; vector potentials; solutions to the inhomogeneous wave equation; principles of duality and reciprocity; wire antennas; antenna arrays; phased arrays; synthesis techniques for discrete and continuous line sources; integral equations and solutions using the method of moments; field equivalence principle; aperture antennas; antenna measurement techniques; diffraction; horn antennas; reflector antennas; microstrip antennas; reflectarrays; electrically small antennas; and broadband antennas.

I think it is pretty pretentious to call this course “Advanced”. It really didn’t cover too much of the material in the text (Balanis), much of which could probably be described as advanced, as it is an extremely thorough text.  I found that the toughest part of this course was the barrage of nomenclature (dB vs dBi, directivity, …) that was tossed out at the beginning. After that, a lot of the material was fairly straightforward.

This was the first place that I encountered any use of magnetic currents and charges in all the physics and engineering that I’d studied (which is usually described only as an interesting theoretical construct related to magnetic monopoles). This posed a bit of a challenge for me, because I didn’t know how to represent that in my favorite form of Maxwell’s equation (i.e.: the one-equation geometric algebra form). It turns out that such currents and charges fit into the GA formalism very naturally, and I detailed that in my book, Geometric Algebra for Electrical Engineers (this course is the reason I included magnetic charges and currents in that book.) My first musings along those lines can be found in these class notes.

This course was taught off of slides, which, as well as being annoying and hard to follow, makes note taking very difficult.  A consequence of this, unlike many of the other class notes collections I have assembled, is that this set of notes is not standalone, and probably useless without the textbook. The biggest value that my notes offer to future students of this course is Appendix A, “Prof. Eleftheriades’ handwriting decoder ring” (one page.)

I used a hodge-podge of computer algebra and plotting methods for this course and my notes.  My notebooks in those languages can be found here:

  • Mathematica.
  • Julia — I was using a very early version of Julia (long before version 1.0.0), so I wouldn’t be surprised if my notebooks no longer work.
  • Matlab.


Should you wish to actively contribute typo fixes (or additions, editing, …) to this book, you can do so by contacting me, or by forking your own copy of the associated git repositories and building the book pdf from source, and submitting a subsequent merge request.

git clone [email protected]:peeterjoot/latex-notes-compilations.git peeterjoot
cd peeterjoot

submods="figures/ece1229-antenna julia matlab mathematica ece1229-antenna latex"
for i in $submods ; do
git submodule update --init $i
(cd $i && git checkout master)

export PATH=`pwd`/latex/bin:$PATH

cd ece1229-antenna

I reserve the right to impose dictatorial control over any editing and content decisions, and may not accept merge requests as-is, or at all. That said, I’ll probably not refuse reasonable suggestions or merge requests.


  • Copyright
  • Document Version
  • Dedication
  • Preface
  • Contents
  • List of Figures
  • Antenna Theory
  • 1 Fundamental parameters of Antennas
  • 1.1 Poynting vector
  • 1.2 Typical far-field radiation intensity
  • 1.3 Field plots
  • 1.4 dB vs dBi
  • 1.5 Trig integrals
  • 1.6 Polarization vectors
  • 1.7 Phasor power
  • 1.8 Radar cross section examples
  • 1.9 Scattering from a sphere vs frequency
  • 1.10 EIRP
  • 1.11 Free space impedance
  • 1.12 Notation
  • 1.13 Problems
  • 2 Maxwell’s equations
  • 2.1 Review
  • 2.2 Constitutive relations
  • 2.3 Boundary conditions
  • 2.4 Linear time invariant
  • 2.5 Green’s functions
  • 2.6 Tangential and normal field components
  • 2.7 Energy momentum conservation
  • 2.8 Duality transformation
  • 2.9 Reciprocity theorem
  • 2.10 Notation
  • 2.11 Problems
  • 3 Linear wire antennas
  • 3.1 Magnetic Vector Potential
  • 3.2 Plots of infinitesimal dipole radial dependence
  • 3.3 Electric Far field for a spherical potential
  • 3.4 Magnetic Far field for a spherical potential
  • 3.5 Plane wave relations between electric and magnetic fields
  • 3.6 Transverse only nature of the far-field fields
  • 3.7 Duality transformation of the far field fields
  • 3.8 Vertical dipole reflection coefficient
  • 3.9 Horizontal dipole reflection coefficient
  • 3.10 Resolving fields into components parallel to the reflecting plane
  • 3.11 Image theorem
  • 3.12 Problems
  • 4 Antenna arrays
  • 4.1 Chebyscheff polynomials
  • 4.2 Problems
  • 5 Aperature antennas
  • 5.1 Problems
  • 6 Microstrip antennas
  • 6.1 Problems
  • Geometric Algebra formalism
  • 7 Electric sources
  • 7.1 Maxwell’s equation in GA phasor form
  • 7.2 Preliminaries. Dual magnetic form of Maxwell’s equations
  • 7.3 Constructing a potential representation
  • 7.4 Maxwell’s equation in Four vector form
  • 7.5 Helmholtz equation directly from the GA form
  • 8 Magnetic sources
  • 8.1 Dual-Maxwell’s equation in GA phasor form
  • 8.2 Preliminaries. Dual magnetic form of Maxwell’s equations
  • 8.3 Constructing a potential representation
  • 8.4 Maxwell’s equation in Four vector form
  • 8.5 Helmholtz equation directly from the GA form
  • 9 Electric and magnetic sources
  • 9.1 Space time split
  • 9.2 Covariant form
  • 9.3 Trial potential solution
  • 9.4 Lorentz gauge application to Helmholtz
  • 9.5 Recovering the fields
  • 10 Reciprocity theorem
  • 11 Relation to tensor form
  • 12 Parallel projection of electromagnetic fields
  • Fields and Waves
  • 13 Fields and Waves
  • 13.1 coupled wave equation in cylindrical coordinates
  • 13.2 Impedance transformation
  • 13.3 Problems
  • Appendices
  • A Prof. Eleftheriades’ handwriting decoder ring
  • B Mathematica notebooks
  • C Julia notebooks
  • D Matlab notebooks
  • Bibliography


  • ece1229.V0.2.3 (June 2, 2020)
    • Convert to 6×9″ format for
    • General cleanup.
    • Reduce figure sizes.
    • Convert 3D Plots to greyscale. This reduces the pdf size from 23Mb to 5.4Mb
    • Remove blank lines before and after equations.
    • Remove \parts (move Geometric Algebra chapters to the appendix.)
    • Update email address to: [email protected]
    • Fix hyphenation issue in listings and pdfsubject copy and paste error (was modelling multi-physics.)