Near Field Coordinate Systems (FE card)

The coordinate systems for the FE card have specific definitions/conventions. Use the appropriate coordinate system for the application.

Cartesian coordinates x, y, z

Observation point:

r = [\begin{matrix} x \\ y \\ z \end{matrix}]

Unit vectors of the coordinate system:

\hat{x} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] \hat{y} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] \hat{z} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]

Observation point:

r = [\begin{matrix} ρ \cos φ \\ ρ \sin φ \\ z \end{matrix}]

Unit vectors of the coordinate system:

\hat{ρ} = [\begin{matrix} \cos φ \\ \sin φ \\ 0 \end{matrix}] \hat{φ} = [\begin{matrix} - \sin φ \\ \cos φ \\ 0 \end{matrix}] \hat{z} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]

Observation point:

r = [\begin{matrix} r \sin ϑ \cos φ \\ r \sin ϑ \sin φ \\ r \cos ϑ \end{matrix}]

Unit vectors of the coordinate system:

\hat{r} = [\begin{matrix} \sin ϑ \cos φ \\ \sin ϑ \sin φ \\ \cos ϑ \end{matrix}] \hat{ϑ} = [\begin{matrix} \cos ϑ \cos φ \\ \cos ϑ \sin φ \\ - \sin ϑ \end{matrix}] \hat{φ} = [\begin{matrix} - \sin φ \\ \cos φ \\ 0 \end{matrix}]

Observation point:

r = [\begin{matrix} x \\ ρ \cos φ \\ ρ \sin φ \end{matrix}]

Unit vectors of the coordinate system:

\hat{ρ} = [\begin{matrix} 0 \\ \cos φ \\ \sin φ \end{matrix}] \hat{φ} = [\begin{matrix} 0 \\ - \sin φ \\ \cos φ \end{matrix}] \hat{x} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]

Observation point:

r = [\begin{matrix} ρ \cos φ \\ y \\ - ρ \sin φ \end{matrix}]

Unit vectors of the coordinate system:

\hat{ρ} = [\begin{matrix} \cos φ \\ 0 \\ - \sin φ \end{matrix}] \hat{φ} = [\begin{matrix} - \sin φ \\ 0 \\ - \cos φ \end{matrix}] \hat{y} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]

This option is similar to the field calculation in cylindrical coordinates around the Z axis, where the radius $r$ changes with the height $z$ :

r (z) = r_{0} + \frac{Δ r}{Δ z} \cdot (z - z_{0})

where

z

is within the range

z_{0} \dots z_{0} + n_{z} \cdot Δ z

Observation point:

r = [\begin{matrix} (r_{0} + \frac{Δ r}{Δ z} \cdot (z - z_{0})) \cdot \cos φ \\ (r_{0} + \frac{Δ r}{Δ z} \cdot (z - z_{0})) \cdot \sin φ \\ z \end{matrix}]

Unit vectors of the coordinate system:

\hat{r} = [\begin{matrix} \cos φ \\ \sin φ \\ 0 \end{matrix}] \hat{φ} = [\begin{matrix} - \sin φ \\ \cos φ \\ 0 \end{matrix}] \hat{z} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]