Spherical Coordinates

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Understanding Spherical Coordinates is a must for the practicing antenna engineer. You are probably familiar with Cartesian Coordinates - a position (point P) can be specified by a triplet like (x,y,z) where x is the distance from the origin to the point along the X-axis, and so on (see Figure 1). Spherical coordinates use a different coordinate system, one with spherical symmetry, which makes it very useful in engineering and physics in certain problems.

cartesian coordinate system

Figure 1. A point P defined in the Cartesian Coordinate System.

The point P could be specified relative to the same origin in a different coordinate system. Spherical coordinates utilize three distinct coordinates:

R - the magnitude of the distance between the origin and the point (always positive)

polar angle in spherical coordinates - angle between the z-axis and the vector from the origin to the point (ranges from 0 to 180 degrees)

azimuth angle in spherical coordinates - angle between the x-axis and the projection of the point onto the x-y plane (ranges from 0 to 360 degrees)

Any point specified in Cartesian coordinates as (x,y,z) can be re-expressed in spherical coordinates via the following transformation:

transformation from cartesian to spherical coordinates

The above might look complicated, but after you've worked with it for a while it makes a lot of sense. The point P=(0,6,5) can be evaluated in spherical coordinates as:

cartesian point converted to spherical coordinates

The coordinates are illustrated in Figure 2:

point illustrated in spherical coordinates

Spherical coordinates are popular for antennas, because we often are only interested in the antennas response in a particular direction, not how far away something is (radiation patterns die off as 1/R^2 for all antennas in the far field). In Cartesian coordinates, 3 variables need specified to determine the direction from the origin, and it is not intuitive. For spherical coordinates, once it is understood, the polar angle and the azimuth angle can be readily used.

For practice, make sure the following table makes sense. I give a set of rectangular coordinates on the left, and the corresponding spherical coordinates on the right.

Table I. Conversion of Cartesian Coordinates to Spherical (these should all make sense).

Cartesian (X,Y,Z)Spherical (R, , ) [angles in degrees]
(1, 0, 0)(1, 90, 0)
(0, 1, 0)(1, 90, 90)
(-1, -1, 0)(1.414, 90, 225)
(0, 0, 1)(1, 0, 0) [note: not unique here - could be anything. Why is that?]
(1, 1, 1)(1.73, 45, 45)
(0, 0, -1)(1, 180, 0) [not unique again]
(0, 0, 0)(0, 0, 0) [ and could be any number!]
(-5, 0, 0)(5, 90, 180)
(2, -4, -4)(6, 153, -135)
That is an overview of spherical coordinates. They come up a lot in the study of electromagnetics and physics, and can aid in understanding and solving of certain problems. They arise in antenna engineering most often in regards to radiation patterns, which specify how an antenna radiates versus angle (or direction) away from the antenna.

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