# CG % MAC Formula

#### Sebastian Kluge

I have a question about CG % MAC. The CG stuff makes me crazy .

I have many variables for the CG Calculation from aircraft.cfg and the calculated Total Weight and Total Arm for the B737-800 (FSX default).

Now the question is, what is the fsx ( or AirWrench ) internal formula for the CG % MAC value? I have tried a lot of formula find via Google. But I never have the same resulat as FSX or AirWrench ( B737-800 with EmptyWeight 35.95 % CG MAC).

Also I have try this formula in the AVSIM Documentation "fs_flight_dynamics_v1.0" , but not with the same result as FSX or AirWrench.

As I understood in the "flight dynamic document" is the challange to get the LEMAC ( leading edge MAC ). The basic formula for the CG % is:

Distance to CG - Distance to LEMAC = Distance CG to LEMAC
and
(Distance CG to LEMAC / MAC) * 100 = CG % MAC

For Example Default B737-800, variables given:

wing_span = 117.42 ft
wing_sweep = 27.2°
wing_root_chord = 21.0 ft
wing_area = 1344.0 ft
MAC 14.10 ft ( calculated, same result as AirWrench )

The Calculation: OK 14,41 ft LEMAC, then I will calculate the CG % MAC from empty weight of the B737-800
Arm = 9 ft

Then the formula for CG% MAC: 9 ft - 14,41 ft = -5,41 ft
And: (-5,41 ft / 14,10 ft) * 100

= -38,64 CG % MAC

This result is not the same as FSX and AirWrench empty weight CG ( 35.95 % ).
Can anyone see the mistake in the calculation ? Thanks in advance for your help !

Best regards

Sebastian

#### Sebastian Kluge

Dear Roy,

Is it true, that the LEMAC is the key for the calculation?
The problem that I have is, that I dont now the right way .

Is my posted formula the right way, to calculated the LEMAC ? Which formula do you use in your example to get it?
And last question, on the Default Learjet45 the empty_weight_cg_pos is -35.79 ft. When i will get the CG % MAC from the empty weight:

-35.70 ft - LEMAC ft
------------------- x 100
MAC ft

Is it true ?

Just for understanding, I write a tool like "FS Passenger", just better and free. And my math skills for this complex stuff are limited .

Sebastian

#### jx_

Your math works, but the biggest thing I see missing here is reference datum. If both are not using identical reference datum, the CG of the airplane will be calculated in the wrong place.

Distance to CG - Distance to LEMAC

you are really using:

Distance FROM datum to CG - Distance FROM datum to LEMAC

So if MSFS and Airwrench have a different datum, you will easily have a discrepancy.

Once the numbers are entered into the aircraft.cfg correctly, MSFS does produce accurate results. If you are able to confirm the datum, I would then look into what Airwrench is using to calculate these numbers. Is there a config file for each airplane somewhere or is it pulling from the aircraft/cfg?

#### Sebastian Kluge

Dear jx_

Would you sacrifice your time to give me an example of a calculation. I think I made a mistake in the calculation , which I do not see.

Values for Lear45:

reference_datum_position=34.5, 0, 0
empty_weight_CG_position=-35.79, 0, 0
wing_area=312.0
wing_span=45.8
wing_root_chord=9.9
wing_sweep=13.4
wing_pos_apex_lon=-31.5
MAC = 7,28 ft

LEMAC = ?
CG % = ?

I pull the variables from the aircraft.cfg.

Sebastian #### jx_

using your numbers as an example, you would need to know if the LEMAC is from the datum or some other point.

For example, datum is 34.5 feet ahead of the model design center (which can be anywhere). In MSFS, it is where you see the red + in top down view and where the center of the airplane loads in a parking spot.

From that datum point, CG is 35.79 feet aft, or (34.5 + -35.79) = -1.29 feet aft from the red +

The LEMAC and CG distance on the REAL airplane may have a different datum. For example, some use the tip of the nose, while I have seen others that use some point in space ahead of the nose ( 12 inches forward from the nose, etc.)

Calibrate your model to the real numbers and your CG should align well. Calibrate Airwrench the same and it should be able to calculate the numbers correctly.

Note: Your math was fine. CG distance from LEMAC / MAC = CG%MAC

25% MAC = 25% CG%MAC

If MAC is ten feet, and CG is 5 feet aft of LEMAC, the airplane is 50% CG%MAC (unstable!)

#### Sebastian Kluge

ok, now my calculation procedure according to your instructions:

reference_datum_position=34.5, 0, 0
empty_weight_CG_position=-35.79, 0, 0
wing_area=312.0
wing_span=45.8
wing_root_chord=9.9
wing_sweep=13.4
wing_pos_apex_lon=-31.5
MAC = 7,28 ft ( checked with AirWrench )

Calculated values:

Distance FROM datum to CG = 34.5 + -35.79
Distance FROM datum to CG = -1.29 ft

Distance FROM datum to LEMAC = 34.5 + 4.14 ( LEMAC calulated via the formula from "flight dynamic 1.0" document )
Distance FROM datum to CG = 38.64 ft

CG % MAC = ( -1.29 - 38.64 ) / 7.28 ft * 100
CG % MAC = 548 % ( looks wrong )

You see my problem ? Which values are wrong, LEMAC? Wrong Distance?
On EASA TCDS for Learjet 45 the L.E. of MAC is given with 34.48 ft., but the calculation with this value is also not correct.

I make no plane, I create a tool like FSPassenger. And for the load and balance I need the correct CG% .

thank you for your patience !

#### mgh

I think you have confused your datums:

empty_weight_cg_position = -35.79 ft
wing_apex_pos_lon = -31.5 ft
MAC = 7.28 ft
LEMAC = 2.37 ft (from wing_apex_pos_lon)
LEMAC ref = -31.5-2.37 = 33.87 ft (from datum reference position)
MAC cg = 35.79-33.87 = 1.92 ft (from LE MAC)
MAC cg % = 100 *1.92/7.28 = 26.37%

According to FSInterrogate the position of Centre of Gravity (CoG) on the mean Aerodynamic Chord (MAC) is 26.76% for the default Lear45. This is FSUPIC offset #2EF8. Unfortunately, FSUIPC doesn't give the MAC itself.

#### Sebastian Kluge

Gentleman, I´ve got it !

The mistake was in the LEMAC formula and the LEMAC ref calculation.

Sebastian

#### Roy Holmes

Resource contributor
Airwrench gives 27.13% for the Empty CG Location as %MAC.
Although mgh was close, I suspect he did not calculate LEMAC for the small sweep angle of the Lear (13.4°)
When calculated for sweep, LEMAC is 2.3156. Substituting this value yields a CG %MAC of 27.125
Roy

#### mgh

I transcribed LE MAC as 2.32 ft instead of 2.37 ft which now gives 27.13%

FSInterrogate gives 27.11% for an empty default LearJet 45.

Last edited:

#### jx_

"You see my problem ? Which values are wrong, LEMAC? Wrong Distance?
On EASA TCDS for Learjet 45 the L.E. of MAC is given with 34.48 ft., but the calculation with this value is also not correct."

Assuming that your reference datum matched that of the TCDS, the to use would be -34.48 because the LEMAC is behind the datum. The datum position is always 0.

otherwise, glad to hear you got it working!

G

#### gr8guitar

##### Guest
Okay, I'm new to CG%, MAC%, etc. I downloaded gauges for Lear 45 (from http://www.miqrogroove.com/pro/software/) and in that file is a gauge (weightandbalance.xml). This gauge calculates MAC percent. I looked at the formula in the gauge, which is:
(A:CG PERCENT, number) 9.9 * 2.632 - 7.3 / 100 * (&gt;L:MAC_PERCENT, number). This = 2.82. I don't believe it's correct (per
http://www.nasascale.org/howtos/mac-calculator.htm). It's formula is: %MAC B.P. = ((CG-C) / MAC)*100.
If I plug in the numbers of (.27165 [per FSUIPC offset of 2EF8] - 2.632 [from xml] / 7.3 [from xml]* 100, then the result is = -32.2 %. Seems better. So, 1) where does 2.632 [from xml] and 2) LEMAC = 2.37 [from the above explanation] come from?