electric far field

Updated notes for ece1229 antenna theory

March 16, 2015 ece1229 No comments , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

I’ve now posted a first update of my notes for the antenna theory course that I am taking this term at UofT.

Unlike most of the other classes I have taken, I am not attempting to take comprehensive notes for this class. The class is taught on slides which go by faster than I can easily take notes for (and some of which match the textbook closely). In class I have annotated my copy of textbook with little details instead. This set of notes contains musings of details that were unclear, or in some cases, details that were provided in class, but are not in the text (and too long to pencil into my book), as well as some notes Geometric Algebra formalism for Maxwell’s equations with magnetic sources (something I’ve encountered for the first time in any real detail in this class).

The notes compilation linked above includes all of the following separate notes, some of which have been posted separately on this blog:

Duality transformation of the far field fields.

February 27, 2015 ece1229 No comments , , , ,

We’ve seen that the far field electric and magnetic fields associated with a magnetic vector potential were

\begin{equation}\label{eqn:dualFarField:40}
\BE = -j \omega \textrm{Proj}_\T \BA,
\end{equation}
\begin{equation}\label{eqn:dualFarField:60}
\BH = \inv{\eta} \kcap \cross \BE.
\end{equation}

It’s worth a quick note that the duality transformation for this, referring to [1] tab. 3.2, is

\begin{equation}\label{eqn:dualFarField:100}
\BH = -j \omega \textrm{Proj}_\T \BF
\end{equation}
\begin{equation}\label{eqn:dualFarField:120}
\BE = -\eta \kcap \cross \BH.
\end{equation}

What does \( \BH \) look like in terms of \( \BA \), and \( \BE \) look like in terms of \( \BH \)?

The first is

\begin{equation}\label{eqn:dualFarField:140}
\BH
= -\frac{j \omega}{\eta} \kcap \cross \lr{ \BA – \lr{\BA \cdot \kcap} \kcap },
\end{equation}

in which the \( \kcap \) crossed terms are killed, leaving

\begin{equation}\label{eqn:dualFarField:160}
\BH
= -\frac{j \omega}{\eta} \kcap \cross \BA.
\end{equation}

The electric field follows again using a duality transformation, so in terms of the electric vector potential, is

\begin{equation}\label{eqn:dualFarField:180}
\BE = j \omega \eta \kcap \cross \BF.
\end{equation}

These show explicitly that neither the electric or magnetic far field have any radial component, matching with intuition for transverse propagation of the fields.

References

[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley & Sons, 3rd edition, 2005.