Editing VHF omnidirectional range (section)

===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.