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As a clarification at the beginning: I'm talking about non-GPS, pure VOR-DME based RNAV devices here.

On RNAV devices that use just the selected VOR-DME, not GPS, the phantom radio station is set up by a desired bearing from the VOR-DME station and a distance. Then you can select a desired radial from that phantom station (instead of the VOR-DME station), and the deviation from that radial is then indicated by the CDI.

Given

  • the aircraft position relative to the station as radial $R_a$ in degrees and distance $D_a$ in nautical miles
  • the desired phantom station position relative to the VOR-DME station as radial $R_s$ in degrees and distance $D_s$ in nautical miles
  • the selected radial of that phantom station $R_{cdi}$ in degrees

how would you calculate the deviation $D_{cdi}$ from the selected radial $R_{cdi}$ of the phantom station ?

EDIT:

This is what I have tried so far:

  • $x_a = D_a * cos(rad(R_a))$, $x_a$ being the X position of the aircraft relative to the VOR-DME
  • $y_a = D_a * sin(rad(R_a))$, $y_a$ being the Y position of the aircraft relative to the VOR-DME
  • $x_s = D_s * cos(rad(R_s))$, $x_s$ being the X position of the selected RNAV phantom station relative to the VOR-DME
  • $y_s = D_s * sin(rad(R_s))$, $y_s$ being the Y position of the selected RNAV phantom station relative to the VOR-DME
  • $D_{as} = \sqrt {(x_a - x_s) ^ 2 + (y_a - y_s) ^ 2}$, $D_{as}$ being the distance between the selected phantom station and the aircraft in nautical miles
  • $B = deg(atan2(y_a - y_s, x_a - x_s))$, $B$ being the bearing of the selected phantom station to the aircraft in degrees
  • $D_r = sin(rad(R_s - B)) * D_{as}$, $D_r$ being the deviation of the aircraft from the selected radial of the selected phantom station

I've tried using these formulas, but I ended up with nonsense values for $D_{as}$ - 0 if $D_s$ was 90, close to zero for smaller values and 90 for $D_s = 0$. I must say that I was never good in trigonometry … so please, can you tell me what is wrong with theses formulas ?

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  • $\begingroup$ It’s just triangles and trig. What have you tried. How far have you got? $\endgroup$
    – Jim
    Feb 2, 2022 at 3:43
  • $\begingroup$ @Jim I edited my question. $\endgroup$
    – TheEagle
    Feb 2, 2022 at 14:23
  • $\begingroup$ Maybe you could get an answer on Math SE? Seems like a better fit… $\endgroup$ Feb 2, 2022 at 15:50
  • $\begingroup$ @MichaelHall maybe you are right … but anyways it looks like I just made a mistake somewhere as it's giving plausible values now. $\endgroup$
    – TheEagle
    Feb 2, 2022 at 16:36

1 Answer 1

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To keep it simple and hide the trig here we can do it with vectors.

Using your subscript nomenclature:

Define Va as the vector from the VOR to the aircraft. (The VOR/DME gives this directly in polar coordinates (Radial, distance).

Define Vs as the vector from the VOR to the phantom station. This is a given when setting up the phantom station.

Define Vsa as the vector from the phantom station to the aircraft.

Then: Va = Vs + Vsa

And Vsa = Va - Vs

The CDI shows angular deviation from the selected radial. Define a unit vector vcdi for the selected CDI radial.

The dot product can be used to get the angle:

vcdiVsa = ‖vcdi‖ ‖Vsacos α

Where α is now the angular deviation of the aircraft to the selected CDI radial.

But ‖vcdi‖ ( a unit vector) is 1.

So vcdiVsa = ‖Vsacos α

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  • $\begingroup$ Could you please add an example calculation, say, for $V_a = (135°, ~14.14 nm)$ and $V_s = (90°, 10 nm)$ (so $V_{sa} = (45°, ~4.14 nm)$) and $V_{cdi} = (180°, 1)$ - because I don't get how to calculate the CDI needle deflection from $V_{sa}$ and $V_{cdi}$. $\endgroup$
    – TheEagle
    Mar 27, 2022 at 11:15

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