Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Antenna
Toolbox™ uses two types of coordinate system: *rectangular
coordinate system* and *spherical coordinate system *.

Antenna
Toolbox uses the *rectangular coordinate
system* to visualize antenna or array geometry. The toolbox
uses the *spherical coordinate system * to visualize
antenna radiation patterns.

Visualize the geometry of a default `monopoleTopHat`

antenna
from the antenna library.

m = monopoleTopHat; show(m);

The toolbox displays the top-hat monopole antenna in the *rectangular* or *Cartesian* coordinate
system.

The *rectangular* coordinate system also
called *Cartesian* coordinate system specifies
a position in space as an ordered 3-tuple of real numbers, `(x,y,z)`

,
with respect to the origin `(0,0,0)`

.

You can view the 3-tuple as a point in space, or equivalently as a vector in three-dimensional Euclidean space. When viewed as a vector in space, the coordinate axes are basis vectors and the vector gives the direction to a point in space from the origin. Every vector in space is uniquely determined by a linear combination of the basis vectors. The most common set of basis vectors for three-dimensional Euclidean space are the standard unit basis vectors:

$$\{[1\text{\hspace{0.05em}}\text{\hspace{0.17em}}0\text{\hspace{0.05em}}\text{\hspace{0.17em}}0],[0\text{\hspace{0.05em}}\text{\hspace{0.17em}}1\text{\hspace{0.05em}}\text{\hspace{0.17em}}0],[0\text{\hspace{0.05em}}\text{\hspace{0.17em}}0\text{\hspace{0.05em}}\text{\hspace{0.17em}}1]\}$$

Any three linearly independent vectors define a basis for three-dimensional space. However, the Antenna Toolbox assumes that the basis vectors you use are orthogonal.

The standard distance measure in space is the *l ^{2}* norm,
or Euclidean norm. The Euclidean norm of a vector [

$$\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$$

The Euclidean norm gives the length of the vector measured from
the origin as the hypotenuse of a right triangle. The distance between
two vectors [*x**0** **y**0** **z**0*]
and [*x**1** **y**1** **z**1*]
is:

$$\sqrt{{({x}_{0}-{x}_{1})}^{2}+{({y}_{0}-{y}_{1})}^{2}+{({z}_{0}-{z}_{1})}^{2}}$$

Visualize the radiation pattern of the default `monopoleTopHat`

antenna.

m = monopoleTopHat; pattern(m,75e6);

The toolbox displays the radiation pattern of the top-hat monopole
using *spherical * coordinate system represented
by azimuth and elevation angles.

The *spherical * coordinate system defines
a vector or point in space with a distance *R* and
two angles. You can represent the angles in this coordinate system:

Azimuth and elevation angles

Phi (Φ) and theta (θ) angles

*u*and*v*coordinates

The *azimuth angle* is the angle
from the positive *x*-axis to the vector's
orthogonal projection onto the *xy* plane,
moving in the direction towards the y-axis. The azimuth angle is in
the range –180 and 180 degrees.

The *elevation angle* is the angle
from the vector's orthogonal projection on the *xy* plane
toward the positive *z*-axis, to the vector.
The elevation angle is in the –90 and 90 degrees.

The φ angle is the angle from the positive *x*-axis
to the vector's orthogonal projection onto the *xy* plane,
moving in the direction towards the y-axis. The azimuth angle is between
–180 and 180 degrees.

The θ angle is the angle from the positive *z*-axis
to the vector’s orthogonal projection on the *xy* plane
measured clockwise. The θ angle is in the range 0 and 180 degrees.

These angles are an alternative to using azimuth and elevation angles to express the location of point in a unit sphere.

You can define *u* and *v* in
terms of φ and θ:

$$\begin{array}{l}u=\mathrm{sin}\theta \mathrm{cos}\varphi \\ v=\mathrm{sin}\theta \mathrm{sin}\varphi \end{array}$$

In terms of azimuth and elevation angles, the *u * and *v* coordinates
are:

$$\begin{array}{l}u=\mathrm{cos}el\mathrm{sin}az\\ v=\mathrm{sin}el\end{array}$$

The values of *u * and *v* satisfy
the inequalities:

$$\begin{array}{l}-1\le u\le 1\\ -1\le v\le 1\\ {u}^{2}+{v}^{2}\le 1\end{array}$$

The φ and θ angles in terms of *u* and *v* are:

$$\begin{array}{l}\mathrm{tan}\varphi =u/v\\ \mathrm{sin}\theta =\sqrt{{u}^{2}+{v}^{2}}\end{array}$$

The azimuth and elevation angles in terms of *u* and *v* are:

$$\begin{array}{l}\mathrm{sin}el=v\\ \mathrm{tan}az=\frac{u}{\sqrt{1-{u}^{2}-{v}^{2}}}\end{array}$$

Convert rectangular coordinates to spherical coordinates (*az*, *el*, *R*)
using:

$$\begin{array}{l}R=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}\\ az={\mathrm{tan}}^{-1}(y/x)\\ el=\mathrm{tan}{}^{-1}(_{z}^{/}{}_{\sqrt{{x}^{2}+{y}^{2}}}{}^{)}\end{array}$$

Convert spherical coordinates (*az*, *el*, *R*)
to rectangular coordinates using:

$$\begin{array}{l}x=R\mathrm{cos}(el)\mathrm{cos}(az)\\ y=R\mathrm{cos}(el)\mathrm{sin}(az)\\ z=R\mathrm{sin}(el)\end{array}$$

where:

*R*is the distance from the antenna*el*and*az*are the azimuth and elevation angles

[1] Balanis, C.A. *Antenna Theory: Analysis and
Design*. 3rd Ed. New York: Wiley, 2005.

Was this topic helpful?