# 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:

(1) $r=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

Unit vectors of the coordinate system:

(2)

## Cylindrical coordinates around Z axis $\rho$ , $\phi$ , $z$

Observation point:

(3) $r=\left[\begin{array}{c}\rho \mathrm{cos}\phi \\ \rho \mathrm{sin}\phi \\ z\end{array}\right]$

Unit vectors of the coordinate system:

(4)

## Spherical coordinates $r$ , $\vartheta$ , $\phi$

Observation point:

(5) $r=\left[\begin{array}{c}r\mathrm{sin}\vartheta \mathrm{cos}\phi \\ r\mathrm{sin}\vartheta \mathrm{sin}\phi \\ r\mathrm{cos}\vartheta \end{array}\right]$

Unit vectors of the coordinate system:

(6)

## Cylindrical coordinates around X axis $r$ , $\phi$ , $x$

Observation point:

(7) $r=\left[\begin{array}{c}x\\ \rho \mathrm{cos}\phi \\ \rho \mathrm{sin}\phi \end{array}\right]$

Unit vectors of the coordinate system:

(8)

## Cylindrical coordinates around Y axis $r$ , $\phi$ , $y$

Observation point:

(9) $r=\left[\begin{array}{c}\rho \mathrm{cos}\phi \\ y\\ -\rho \mathrm{sin}\phi \end{array}\right]$

Unit vectors of the coordinate system:

(10)

## Conical coordinates around the Z axis $\phi$ , $z$

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

(11) $r\left(z\right)={r}_{0}+\frac{\Delta r}{\Delta z}\cdot \left(z-{z}_{0}\right)$
where $z$ is within the range ${z}_{0}\dots {z}_{0}+{n}_{z}\cdot \Delta z$ .

Observation point:

(12) $r=\left[\begin{array}{c}\left({r}_{0}+\frac{\Delta r}{\Delta z}\cdot \left(z-{z}_{0}\right)\right)\cdot \mathrm{cos}\phi \\ \left({r}_{0}+\frac{\Delta r}{\Delta z}\cdot \left(z-{z}_{0}\right)\right)\cdot \mathrm{sin}\phi \\ z\end{array}\right]$

Unit vectors of the coordinate system:

(13)