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 ?