y
e
xe
x
VOR
VOR
R
VOR
G
y
(east) YE
Reference
VOR radial
Figure 5.11: Geometry of VOR navigation
radial, that can be entered using an ‘Omni Bearing Selector’. The reference bearing
is called ‘Course Datum’ (CD), while the bearing on which the aircraft is actually
flying is denoted by QDR (a term used in radio telephony). The course deviation
angle GVOR, which is equal to the angle between the reference bearing and the actual
bearing, is shown on the cockpit instrument; a typical value for a full-scale deflection
is GVOR = 10.
The bearing information can also be coupled to an automatic control system, allowing
the airplane to automatically fly from or to the VOR along a reference radial,
or to automatically intercept a VOR bearing. Although many autopilots still offer
such functionality today, modern airliners and business aircraft normally have more
advanced Area Navigation systems which use information of multiple VOR stations
and other navigation equipment to follow arbitrary routes between arbitrary waypoints.
In this report we will consider tracking of VOR radials only.
68 Chapter 5. Radio-navigation, sensors, actuators
In order to compute VOR signals for simulation purposes we need to know the exact
positions of the VOR station and the aircraft in relation to the Earth-fixed reference
frame. If the horizontal position of the VOR ground station is defined by the coordinates
(xVOR, yVOR) and the horizontal position of the aircraft by (xe, ye), we can
find the following equations for the QDR and the course deviation angle GVOR (see
figure 5.11):
QDR = arctan
ye − yVOR
xe − xVOR
(5.23)
GVOR = CD − QDR (5.24)
We also need to know whether the aircraft flies toward the VOR beacon, or away
from it. This information is visualized in the cockpit by means of a ‘TO-FROM indicator’,
which verifies the difference between the airplane’s heading y and the QDR:
|y − QDR| > 90 ) ‘TO’ (5.25)
|y − QDR| < 90 ) ‘FROM’
(In figure 5.11 we can observe that the aircraft is flying away from the VOR beacon.
This corresponds with the above given relations since y − QDR 40, which is
indeed smaller than 90.)
5.2.2 VOR coverage and the Cone of Silence
The ground distance RVOR can be used to determine whether the aircraft flies in
an area where the VOR signals can be received with appropriate reliability. This
distance is equal to:
RVOR =
q
(xe − xVOR)2 + (ye − yVOR)2 (5.26)
If the aircraft flies in a certain area in the direct neighbourhood of the VOR transmitter,
the signals are not accurate. This area is formed by a cone with a top-angle of
approximately 80 to 120 degrees, the so-called Cone of Silence, which has been shown
in figure 5.12. The aircraft flies outside the cone of silence if [5]:
x arctan
H − HVOR
RVOR
90 − (40 to 60) (5.27)
where H is the altitude of the aircraft and HVOR is the elevation of the VOR station.
These altitudes are expressed in [m] above sea level; H − HVOR represents the height
of the airplane above the VOR station.
Table 5.3 gives the maximum coverage of the VOR signals as a function of the
height above ground level, measured in two different flight tests [7]. Using the MATLAB
function POLYFIT to fit a polynomial to the data from this table, the following
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