Editing VHF omnidirectional range (section)

==Technical specification==
The VOR signal encodes a morse code identifier, optional voice, and a pair of navigation tones. The radial azimuth is equal to the phase angle between the lagging and leading navigation tone.

===Constants===
{| class="wikitable"
|+ Standard<ref name="FRS2001"/> modulation modes, indices, and frequencies
|-align="center"
! Description
! Formula
! Notes
! Min
! Nom
! Max
! Units
|-align="center"
|rowspan="4" align="left"| ident
| rowspan="2"| ''i''(''t'')
| on || || 1 || ||
|-align="center"
| off || || 0 || ||
|-align="center"
| ''M''<sub>i</sub>
| align="left"|A1 modulation index || ||0.07|| ||
|-align="center"
| ''F''<sub>i</sub>
| align="left"|A1 subcarrier frequency|| ||1020|| ||Hz
|-align="center"
|rowspan="2" align="left"| voice
| ''a''(''t'')
| ||−1|| ||+1||
|-align="center"
| ''M''<sub>a</sub>
| align="left"|A3 modulation index || ||0.30|| ||
|-align="center"
|align="left"|navigation
| ''F''<sub>n</sub>
| align="left"|A0 tone frequency || ||30|| ||Hz
|-align="center"
|align="right"|variable
| ''M''<sub>n</sub>
| align="left"|A3 modulation index || ||0.30|| ||
|-align="center"
|rowspan="3" align="right"|reference
| ''M''<sub>d</sub>
| align="left"|A3 modulation index || ||0.30|| ||
|-align="center"
| ''F''<sub>s</sub>
| align="left"|F3 subcarrier frequency|| ||9960|| ||Hz
|-align="center"
| ''F''<sub>d</sub>
| align="left"|F3 subcarrier deviation|| || 480|| ||Hz
|-align="center"
|rowspan="2" align="left"|channel
| ''F''<sub>c</sub>
| align="left"|A3 carrier frequency||108.00|| ||117.95||MHz
|-align="center"
| ||align="left"|carrier spacing || 50|| || 50||kHz
|-align="center"
|align="left"|speed of light
| ''C'' || || ||299.79|| ||Mm/s
|-align="center"
|align="left"|radial azimuth
| ''A'' ||relative to magnetic north ||0 || ||359 ||deg
|}

===Variables===
{| Class="wikitable"
|+ Symbols
|-align="center"
! Description
! Formula
! Notes
|-align="center"
|rowspan="3" align="left"|time signal left
| ''t'' ||center transmitter
|-align="center"
| ''t''<sub>+</sub>(''A'',''t'') ||higher frequency revolving transmitter
|-align="center"
| ''t''<sub>−</sub>(''A'',''t'') ||lower frequency revolving transmitter
|-align="center"
|rowspan="3" align="left"|signal strength
| ''c''(''t'') ||isotropic
|-align="center"
| ''g''(''A'',''t'')||anisotropic
|-align="center"
| ''e''(''A'',''t'')||received
|}

===CVOR===
[[Image:vor conventional.gif|frame|alt=F3 (colour background) changes the same in all directions; A3 (greyscale foreground) pattern rotates N->E->S->W-> |Conventional VOR<br />red(F3-) green(F3) blue(F3+)<br />black(A3-) grey(A3) white(A3+)]]
The conventional signal encodes the station identifier, {{math|''i''(''t'')}}, optional voice {{math|''a''(''t'')}}, navigation reference signal in {{math|''c''(''t'')}}, and the isotropic (i.e. omnidirectional) component. The reference signal is encoded on an F3 subcarrier (colour). The navigation variable signal is encoded by mechanically or electrically rotating a directional, {{math|''g''(''A'',''t'')}}, antenna to produce A3 modulation (grey-scale). Receivers (paired colour and grey-scale trace) in different directions from the station paint a different alignment of F3 and A3 demodulated signal.

<math display="block">
\begin{array}{rcl}
e(A,t) & = & \cos( 2 \pi F_c t ) ( 1 + c(t) + g(A,t) ) \\
 c(t) & = & M_i \cos ( 2 \pi F_i t ) ~ i(t) \\
 & + & M_a ~ a(t) \\
 & + & M_d \cos ( 2 \pi \int_0^t ( F_s + F_d \cos ( 2 \pi F_n t ) ) dt ) \\
g(A,t) & = & M_n \cos ( 2 \pi F_n t - A ) \\
\end{array}
</math>

===DVOR===
[[Image:vor doppler.gif|frame|alt=A3 (grey-scale background) changes the same in all directions; F3 (colour foreground) pattern revolves N->W->S->E->|Doppler VOR<br/>{{fontcolor|red|'''red'''}}(F3-) {{fontcolor|lightgreen|'''green'''}}(F3) {{fontcolor|blue|'''blue'''}}(F3+)<br/>'''black'''(A3-) {{fontcolor|darkgrey|'''grey'''}}(A3) {{background|silver| {{fontcolor|white|'''white'''}} }}(A3+)<br/>USB transmitter offset is exaggerated<br/>LSB transmitter is not shown]]
The doppler signal encodes the station identifier, <math>\ i(t)\ ,</math> optional audio voice, <math>\ a(t)\ ,</math> navigation variable signal in <math>\ c(t)\ ,</math> and the isotropic (i.e. omnidirectional) component. The navigation variable signal is A3 modulated (greyscale). The navigation reference signal is delayed, <math>\ t_{+}\ , t_{-}\ ,\ </math> by electrically revolving a pair of transmitters. The cyclic doppler blue shift, and corresponding doppler red shift, as a transmitter closes on and recedes from the receiver results in F3 modulation (colour). The pairing of transmitters offset equally high and low of the isotropic carrier frequency produce the upper and lower sidebands. Closing and receding equally on opposite sides of the same circle around the isotropic transmitter produce F3 subcarrier modulation, <math>\ g(A,t) ~.</math>

<math display="block">
\begin{array}{rcl}
 t & = & t_{+}(A,t) - \frac{\ R\ }{ C }\ \sin\!\left(\ 2 \pi\ F_n\ t_{+}(A,t) + A\ \right) \\
 t & = & t_{-}(A,t) + \frac{\ R\ }{ C }\ \sin\!\left(\ 2 \pi\ F_n\ t_{-}(A,t) + A\ \right) \\
\\
e(A,t) & = & \left(\ 1 + c(t)\ \right)\ \cos\!\left(\ 2 \pi\ F_c\ t\ \right) ~+~ g(A,t) \\
\\
 c(t) & = & M_i \cos\!\left(\ 2 \pi\ F_i\ t\ \right)\ i(t) ~+~ M_a\ a(t) ~+~ M_n \cos\!\left(\ 2 \pi\ F_n\ t\ \right) \\
\\
g(A,t) & = & \tfrac{1}{2}\ M_d\ \cos\!\left(\ 2 \pi \left( F_c + F_s \right)\ t_{+}(A,t)\ \right)
\\
 & + & \tfrac{1}{2}\ M_d\ \cos\!\left(\ 2 \pi \left( F_c - F_s \right)\ t_{-}(A,t)\ \right) \\
\end{array}
</math>

where the revolution radius <math>\ R = \frac{\ F_d\ C\ }{\ 2 \pi\ F_n\ F_c\ }\ </math> is {{nobr|{{math| 6.76 ± 0.3}} m .}}

The transmitter acceleration <math>\ 4 \pi^2\ F_n^2\ R\ </math> (24,000&nbsp;g) makes mechanical revolution impractical, and halves ([[gravitational redshift]]) the frequency change ratio compared to transmitters in free-fall.

The mathematics to describe the operation of a DVOR is far more complex than indicated above. The reference to "electronically rotated" is a vast simplification. The primary complication relates to a process that is called "blending".{{citation needed|date=April 2011}}

Another complication is that the phase of the upper and lower sideband signals have to be locked to each other. The composite signal is detected by the receiver. The electronic operation of detection effectively shifts the carrier down to 0&nbsp;Hz, folding the signals with frequencies below the Carrier, on top of the frequencies above the carrier. Thus the upper and lower sidebands are summed. If there is a phase shift between these two, then the combination will have a relative amplitude of {{nobr| {{math| 1 + cos ''φ''}} .}} If {{mvar|φ}} was {{math|180°}}, then the aircraft's receiver would not detect any sub-carrier (signal A3).

"Blending" describes the process by which a sideband signal is switched from one antenna to the next. The switching is not discontinuous. The amplitude of the next antenna rises as the amplitude of the current antenna falls. When one antenna reaches its peak amplitude, the next and previous antennas have zero amplitude.

By radiating from two antennas, the effective phase center becomes a point between the two. Thus the phase reference is swept continuously around the ring – not stepped as would be the case with antenna to antenna discontinuous switching.

In the electromechanical antenna switching systems employed before solid state antenna switching systems were introduced, the blending was a by-product of the way the motorized switches worked. These switches brushed a coaxial cable past 48 or 50&nbsp;antenna feeds. As the cable moved between two antenna feeds, it would couple signal into both.

But blending accentuates another complication of a DVOR.

Each antenna in a DVOR uses an omnidirectional antenna. These are usually Alford Loop antennas (see [[Andrew Alford]]). Unfortunately, the sideband antennas are very close together, so that approximately 55% of the energy radiated is absorbed by the adjacent antennas{{cn|date=October 2024}}. Half of that is re-radiated, and half is sent back along the antenna feeds of the adjacent antennas{{cn|date=October 2024}}. The result is an antenna pattern that is no longer omnidirectional. This causes the effective sideband signal to be amplitude modulated at 60&nbsp;Hz as far as the aircraft's receiver is concerned. The phase of this modulation can affect the detected phase of the sub-carrier. This effect is called "coupling".

Blending complicates this effect. It does this because when two adjacent antennas radiate a signal, they create a composite antenna.

Imagine two antennas that are separated by half their wavelength. In the transverse direction the two signals will sum, but in the tangential direction they will cancel. Thus as the signal "moves" from one antenna to the next, the distortion in the antenna pattern will increase and then decrease. The peak distortion occurs at the midpoint. This creates a half-sinusoidal 1500&nbsp;Hz amplitude distortion in the case of a 50&nbsp;antenna system, (1,440&nbsp;Hz in a 48&nbsp;antenna system). This distortion is itself amplitude modulated with a 60&nbsp;Hz amplitude modulation (also some 30&nbsp;Hz as well). This distortion can add or subtract with the above-mentioned 60&nbsp;Hz distortion depending on the carrier phase. In fact one can add an offset to the carrier phase (relative to the sideband phases) so that the 60&nbsp;Hz components tend to null one another. There is a 30&nbsp;Hz component, though, which has some pernicious effects.

DVOR designs use all sorts of mechanisms to try to compensate these effects. The methods chosen are major selling points for each manufacturer, with each extolling the benefits of their technique over their rivals.

Note that ICAO Annex&nbsp;10 limits the worst case amplitude modulation of the sub-carrier to 40%. A DVOR that did not employ some technique to compensate for coupling and blending effects would not meet this requirement.

===Accuracy and reliability===
The predicted accuracy of the VOR system is ±1.4°. However, test data indicates that 99.94% of the time a VOR system has less than ±0.35° of error{{Citation needed|date=August 2023}}. Internal monitoring of a VOR station will shut it down, or change over to a standby system if the station error exceeds some limit. A Doppler VOR beacon will typically change over or shut down when the bearing error exceeds 1.0°.<ref name="FRS2001"/> National air space authorities may often set tighter limits. For instance, in Australia, a Primary Alarm limit may be set as low as ±0.5° on some Doppler VOR beacons. {{Citation needed|date=April 2011}}

[[ARINC]] 711 – 10 January 30, 2002, states that receiver accuracy should be within 0.4° with a statistical probability of 95% under various conditions. Any receiver compliant with this standard can be expected to perform within these tolerances.

All radio navigation beacons are required to monitor their own output. Most have redundant systems, so that the failure of one system will cause automatic change-over to one or more standby systems. The monitoring and redundancy requirements in some [[instrument landing system]]s (ILS) can be very strict.

The general philosophy followed is that no signal is preferable to a poor signal.

VOR beacons monitor themselves by having one or more receiving antennas located away from the beacon. The signals from these antennas are processed to monitor many aspects of the signals. The signals monitored are defined in various US and European standards. The principal standard is [[European Organisation for Civil Aviation Equipment]] (EuroCAE) Standard ED-52. The five main parameters monitored are the bearing accuracy, the reference and variable signal modulation indices, the signal level, and the presence of notches (caused by individual antenna failures).

Note that the signals received by these antennas, in a Doppler VOR beacon, are different from the signals received by an aircraft. This is because the antennas are close to the transmitter and are affected by proximity effects. For example, the free space path loss from nearby sideband antennas will be 1.5&nbsp;dB different (at 113&nbsp;MHz and at a distance of 80&nbsp;m) from the signals received from the far side sideband antennas. For a distant aircraft there will be no measurable difference. Similarly the peak rate of phase change seen by a receiver is from the tangential antennas. For the aircraft these tangential paths will be almost parallel, but this is not the case for an antenna near the DVOR.

The bearing accuracy specification for all VOR beacons is defined in the [[International Civil Aviation Organization]] [[Convention on International Civil Aviation]] Annex 10, Volume 1.

This document sets the worst case bearing accuracy performance on a Conventional VOR (CVOR) to be ±4°. A Doppler VOR (DVOR) is required to be ±1°.

All radio-navigation beacons are checked periodically to ensure that they are performing to the appropriate International and National standards. This includes VOR beacons, [[distance measuring equipment]] (DME), [[instrument landing system]]s (ILS), and [[non-directional beacon]]s (NDB).

Their performance is measured by aircraft fitted with test equipment. The VOR test procedure is to fly around the beacon in circles at defined distances and altitudes, and also along several radials. These aircraft measure signal strength, the modulation indices of the reference and variable signals, and the bearing error. They will also measure other selected parameters, as requested by local/national airspace authorities. Note that the same procedure is used (often in the same flight test) to check [[distance measuring equipment]] (DME).

In practice, bearing errors can often exceed those defined in Annex 10, in some directions. This is usually due to terrain effects, buildings near the VOR, or, in the case of a DVOR, some counterpoise effects. Note that Doppler VOR beacons utilize an elevated groundplane that is used to elevate the effective antenna pattern. It creates a strong lobe at an elevation angle of 30° which complements the 0° lobe of the antennas themselves. This groundplane is called a counterpoise. A counterpoise though, rarely works exactly as one would hope. For example, the edge of the counterpoise can absorb and re-radiate signals from the antennas, and it may tend to do this differently in some directions than others.

National air space authorities will accept these bearing errors when they occur along directions that are not the defined air traffic routes. For example, in mountainous areas, the VOR may only provide sufficient signal strength and bearing accuracy along one runway approach path.

Doppler VOR beacons are inherently more accurate than conventional VORs because they are less affected by reflections from hills and buildings. The variable signal in a DVOR is the 30&nbsp;Hz FM signal; in a CVOR it is the 30&nbsp;Hz AM signal. If the AM signal from a CVOR beacon bounces off a building or hill, the aircraft will see a phase that appears to be at the phase centre of the main signal and the reflected signal, and this phase center will move as the beam rotates. In a DVOR beacon, the variable signal, if reflected, will seem to be two FM signals of unequal strengths and different phases. Twice per 30&nbsp;Hz cycle, the instantaneous deviation of the two signals will be the same, and the phase locked loop will get (briefly) confused. As the two instantaneous deviations drift apart again, the phase locked loop will follow the signal with the greatest strength, which will be the line-of-sight signal. If the phase separation of the two deviations is small, however, the phase locked loop will become less likely to lock on to the true signal for a larger percentage of the 30&nbsp;Hz cycle (this will depend on the bandwidth of the output of the phase comparator in the aircraft). In general, some reflections can cause minor problems, but these are usually about an order of magnitude less than in a CVOR beacon.