# Chapter 2 of FEM of Antennas
The main objective of antenna analysis is to predict the **radiation patterns** and **input impedance**.
## 2.1 Governing equations in frequency domanin
$$\begin{align}
\nabla \times \mathbf{E}&=-j\omega\bar{\mu}\cdot\mathbf{H}-\mathbf{M}_{imp} \tag{2.1.1}\\
\nabla \times \mathbf{H}&=j\omega\bar{\varepsilon}\cdot \mathbf{E} + \mathbf{j}_{imp} \tag{2.1.2}\\
\nabla \cdot (\bar{\varepsilon}\cdot\mathbf{E})&=-\dfrac{1}{j\omega}\nabla \cdot \mathbf{J}_{imp} \tag{2.1.3}\\
\nabla \cdot (\bar{\mu}\cdot\mathbf{H})&=-\dfrac{1}{j\omega}\nabla \cdot \mathbf{M}_{imp} \tag{2.1.4}\\
\hat{n}\times\mathbf{E}&=0 \quad\mathbf{r}\in S_{PEC} \tag{2.1.5}\\
\hat{n}\times\nabla\times\begin{pmatrix}\mathbf{E} \\ \mathbf{H} \end{pmatrix} &+ jk_0\hat{n}\times\hat{n}\times \begin{pmatrix}\mathbf{E} \\ \mathbf{H} \end{pmatrix} \approx 0 \quad \mathbf{r}\in S_0 \tag{2.1.6}
\end{align}$$
* The current is excited by a current source with electric current density denoted by $\mathbf{J}_{imp}$.
* The antenna my contain an anisotropic materials carachterized by permittivity or permeability tensors, denoted as $\bar{\varepsilon}=\varepsilon_0\bar{\varepsilon}_r$ and $\bar{\mu}=\mu_0\bar{\mu}_r$, respectively.
* $\mathbf{M}_{imp}$ denotes the magnetic current density of an impressed magnetic current. (It doesn't exist in reality! However it is used to ease the computational process of speciall problems.)
* (2.1.6) is an approximation to *Sommerfeld radiation condition* where $k_0=\omega\sqrt{\varepsilon_0\mu_0}$, free-space wavenumber. $S_0$ is the outer boundary.
* $S_{PEC}$ denotes the perfect electrically conducting surface of the anenna.

## 2.2 Deriving vector equations for $\mathbf{E}$
$$\begin{align}
\nabla \times \mathbf{E}&=-j\omega\bar{\mu}\cdot\mathbf{H}-\mathbf{M}_{imp}\\
\nabla \times \mathbf{E}&=-j\omega\bar{\mu}_r\mu_0\cdot\mathbf{H}-\mathbf{M}_{imp}\\
\bar{\mu}_r^{-1}\nabla \times \mathbf{E}&=-j\omega\mu_0\mathbf{H}-\bar{\mu}_r^{-1}\mathbf{M}_{imp}\\
\nabla \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})&=-j\omega\mu_0\nabla \times\mathbf{H}-\nabla \times (\bar{\mu}_r^{-1}\mathbf{M}_{imp})\\
\nabla \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})&=-j\omega\mu_0(j\omega\bar{\varepsilon}_r\varepsilon_0\cdot \mathbf{E} + \mathbf{j}_{imp})-\nabla \times (\bar{\mu}_r^{-1}\mathbf{M}_{imp})\\
\nabla \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})&=\omega^2\bar{\varepsilon}_r\varepsilon_0\mu_0\cdot \mathbf{E} -j\omega\mu_0 \mathbf{j}_{imp}-\nabla \times (\bar{\mu}^{-1}\mathbf{M}_{imp})\\
\end{align}$$
then by assuming $k_0=\omega\sqrt{\varepsilon_0\mu_0}$ and $Z_0=\sqrt{\mu_0/\varepsilon_0}$ as free-space wavenumber and intrinsic impedance:
$$\nabla \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\cdot \mathbf{E} = -jk_0Z_0\mathbf{j}_{imp}-\nabla \times (\bar{\mu}^{-1}\mathbf{M}_{imp}) \quad \mathbf{r}\in V \tag{2.2.1}$$

and for the boundary condition
$$\hat{n}\times\nabla\times\mathbf{E} + jk_0\hat{n}\times\hat{n}\times \mathbf{E} \approx 0 \quad \mathbf{r}\in S_0 \tag{2.2.2}$$

## 2.3 Extracting the weak form
The weak form is driven by multiplying the (2.2.1) by appropriate testing function $\mathbf{T}$ and integration over $V$.
$$\int_V \mathbf{T}\cdot\left[\nabla \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\cdot \mathbf{E}\right]dV
=-\int_V \mathbf{T}\cdot \left[jk_0Z_0\mathbf{j}_{imp}+\nabla \times (\bar{\mu}^{-1}\cdot\mathbf{M}_{imp})\right]dV \tag{2.3.1}$$

Focusing on the LHS:
$$\begin{align}
\int_V \mathbf{T}\cdot\left[\nabla \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\cdot \mathbf{E}\right]dV
&=\int_V \left[\mathbf{T}\cdot\nabla \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\mathbf{T}\cdot \mathbf{E}\right]dV\\
&=\int_V \left[(\nabla \times \mathbf{T})\cdot \bar{\mu}_r^{-1} \cdot (\nabla \times \mathbf{E})-\nabla\cdot\left[\mathbf{T} \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})\right]-k_0^2\bar{\varepsilon}_r\mathbf{T}\cdot \mathbf{E}\right]dV\\
&=\int_V \left[(\nabla \times \mathbf{T})\cdot \bar{\mu}_r^{-1} \cdot (\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\mathbf{T}\cdot \mathbf{E}\right]dV
-\oint_{S_0 \cup S_{PEC}}\hat{n}\cdot\left[\mathbf{T} \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})\right]dS\\
&=\int_V \left[(\nabla \times \mathbf{T})\cdot \bar{\mu}_r^{-1} \cdot (\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\mathbf{T}\cdot \mathbf{E}\right]dV
-\oint_{S_{PEC}}\hat{n}\cdot\left[\mathbf{T} \times (\bar{\mu}_r^{-1}\nabla \times \mathbf{E})\right]dS
-\oint_{S_0}\hat{n}\cdot\left[\mathbf{T} \times (\nabla \times \mathbf{E})\right]dS\\
&=\int_V \left[(\nabla \times \mathbf{T})\cdot \bar{\mu}_r^{-1} \cdot (\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\mathbf{T}\cdot \mathbf{E}\right]dV
-\oint_{S_{PEC}}(\hat{n}\times\mathbf{T}) \cdot \bar{\mu}_r^{-1}\cdot (\nabla \times \mathbf{E})dS
-\oint_{S_0}(\hat{n}\times\mathbf{T}) \cdot (\nabla \times \mathbf{E})dS\\
&=\int_V \left[(\nabla \times \mathbf{T})\cdot \bar{\mu}_r^{-1} \cdot (\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\mathbf{T}\cdot \mathbf{E}\right]dV
-\oint_{S_{PEC}}(\hat{n}\times\mathbf{T}) \cdot \bar{\mu}_r^{-1}\cdot (\nabla \times \mathbf{E})dS
+\oint_{S_0}(\mathbf{T}\times\hat{n}) \cdot (\nabla \times \mathbf{E})dS\\
\end{align}$$
Last term of the above equation can be replaced by the term driven from (2.2.2)
The weak form of (2.2.2) drives from
$$\begin{align}
\oint_{S_0}\mathbf{T}\cdot[\hat{n}\times\nabla\times\mathbf{E}]dS &+ \oint_{S_0} jk_0\mathbf{T}\cdot[\hat{n}\times\hat{n}\times \mathbf{E}]dS \approx 0\\
\oint_{S_0}(\mathbf{T}\times\hat{n})\cdot(\nabla\times\mathbf{E})dS &\approx -\oint_{S_0} jk_0\mathbf{T}\cdot[\hat{n}\times\hat{n}\times \mathbf{E}]dS \\
&\approx -jk_0 \oint_{S_0} (\mathbf{T}\times\hat{n})\cdot(\hat{n}\times \mathbf{E})dS \\
&\approx +jk_0 \oint_{S_0} (\hat{n}\times\mathbf{T})\cdot(\hat{n}\times \mathbf{E})dS \\
\end{align}$$

Finally (2.3.1) becomes
$$\int_V \left[(\nabla \times \mathbf{T})\cdot \bar{\mu}_r^{-1} \cdot (\nabla \times \mathbf{E})-k_0^2\bar{\varepsilon}_r\mathbf{T}\cdot \mathbf{E}\right]dV
+jk_0 \oint_{S_0} (\hat{n}\times\mathbf{T})\cdot(\hat{n}\times \mathbf{E})dS
=\oint_{S_{PEC}}(\hat{n}\times\mathbf{T}) \cdot \bar{\mu}_r^{-1}\cdot (\nabla \times \mathbf{E})dS
-\int_V \mathbf{T}\cdot \left[jk_0Z_0\mathbf{j}_{imp}+\nabla \times (\bar{\mu}^{-1}\cdot\mathbf{M}_{imp})\right]$$