\section{Theory of electromagnetic radiation}


Antennae are engineered devices to transmit and receive data by means of electromagnetic radiation. This is produced by an alternating current through an inductor (the antenna), and electromagnetic radiation interacts with conductors, hence how antennae receive signals.

\subsection{Maxwell's equations}

\[ \nabla \times \mathbf{E} = - \frac{d\mathbf{B}}{t} \]
\[ \nabla \times \mathbf{H} = \frac{d\mathbf{D}}{t} + \mathbf{J}\]
\[ \nabla \cdot \mathbf{D} = \rho \]
\[ \nabla \cdot \mathbf{B} = 0 \]

COnsider a constant $\mathbf{J}$, then $\nabla \times \mathbf{H}$ is constant and hence $\nabla \times \mathbf{E}$ is $\mathbf{0}$.

No curent, static electric field
Constant current, static magnetic field
Linearly increasing current, linearly varying magnetic field and static electric field
Quadratically increasing current, quadratically+statically varying magnetic field and linear electric field


As we can see, with polynomially varying current, $\nabla$ always vanishes. However with sinuosids this is not the case.
The oscillations ensure that the magnativ and electric fields give rise to eachother perpetually; this is the self propagating EM wave.

\begin{itemize}
reflection
creation of surface currents
\end{itemize}

THe geometry, orientation and composition of an antenna has an effect on how much it relects or allows surface currents.






\subsection{Wave equations}


Suppose there existed a hypothetical 'magnetic current' such that
\[ \nabla \times \mathbf{E} = - \frac{d\mathbf{B}}{t} - \mathbf{J}_{B} \]
\[ \nabla \times \mathbf{H} = \frac{d\mathbf{D}}{t} + \mathbf{J}_{E}\]

By using Maxwell's equations, some assumptions on differentiability of the electric and magnetic fields, and a common cross product identity, we can prove the following.


\[\nabla^2 \mathbf{H} =  \varepsilon \mu \mathbf{H}'' + \varepsilon \mathbf{J}_{B}' - \nabla \times \mathbf{J}_{E}\]
\[\nabla^2 \mathbf{E} =  \varepsilon \mu \mathbf{E}'' + \mu \mathbf{J}_{E}' - \nabla \times \mathbf{J}_{B}\]

Derivative ' is taken with respect to time.

It is particularly useful to study wave equations and maxwell's equations with time harmonic electric/magnetic fields.




\section{Electromagnetic wave propagation}

\subsection{Maxwell's laws in integral form}
\subsection{Maxwell's laws in static, non-source, and dynamic cases}


convolution with permeability and permittivity


\subsection{Phasors}

LTI systems without constant solving of ODEs
\[e^{i\theta} = \cos\theta + i \sin\theta\]


Time harmonic functions satisfy the following
\[ \mathbf{F}' = i \omega \mathbf{F}\]

\subsection{Helmholtz wave equation}

From Maxwell's equtions, one can derive the following

\[\nabla^2 \mathbf{H} =  \mu \varepsilon \mathbf{H}''\]
\[\nabla^2 \mathbf{E} =  \mu \varepsilon \mathbf{E}''\]

If we assume that our fields are time harmonic (which they are in the case of AC current), this culminates in solving the following PDE; the Helmholtz equation.

\[\nabla^2 \mathbf{H} =  - \mu \varepsilon \omega^2 \mathbf{H}\]
\[\nabla^2 \mathbf{E} =  - \mu \varepsilon \omega^2 \mathbf{E}\]

where we call $k=\omega \sqrt{\mu \varepsilon}$ the wavenumber.
Furthermore we call $v_p= \frac{\omega}{k} = \frac{1}{\sqrt{\mu \varepsilon}}$ the phase velocity (speed of wave propagation)


Physics aside, this is mathematically evaluating this
\[\nabla^2 \mathbf{F} = -k^2 \mathbf{F}\]
where $\mathbf{F}(\mathbf{x},t)$ is a function of space-time and $\nabla^2$ is the vector Laplacian

We can solve this PDE by Separation of variables (Fourier transforms also do the trick).



Make the arbitrary definition
\[\nabla \times \mathbf{A} = \mathbf{B}\]

\[\nabla \times (\mathbf{E}+ \frac{ \partial \mathbf{A}}{\partial t} = \mathbf{0}\]
\[ -\nabla V = \mathbf{E}+ \frac{ \partial \mathbf{A}}{\partial t} = \mathbf{0} \]
Then we apply the \emph{Lorenz Gauge condition}
\[\nabla \cdot \mathbf{A} = -\mu \varepsilon  \frac{d\mathbf{V}}{dt}\]


\subsection{Boundary conditions}


When considering how Maxwell's equations hold between different medium, we have to 'switch' the fields midway through the integrals in Maxwell's equations.

We introduce some approximations (given this is an engineering course, approximations are all the rage)



\section{Electromagnetic plane waves}

Plane waves are:
transverse electromagnetic waves, as expected (direction of propagation orthogonal to electric and magnetic field).
invariant over a plane normal to propagation (for all points on some plane normal to propagation, the electric and magnetic field returns the same vectors; i.e magnetic and electric field is determined only by the direction of propagation)



\subsection{Intrinsic impedance}
we know that electric,magnetic fields along the direction of propagation are all orthogonal.

Applying faraday's law, we can write the wave equation solution for an electric field in terms of the magnetic field (and vice versa). We find that $[\mathbf{E}]^{\pm}_x = \pm \frac{\omega \mu }{k}[\mathbf{H}]^{\pm}_{y}$

Intrinsic impedance is the ratio of the electric to magnetic field magnitude $\sqrt{\frac{\mu}{\varepsilon}}$, which in plane waves is constant as we can see from our previous proporitonal equality.
