Optimization of the parameters of an electric propulsion

Accustomed to thermal flight, we find ourselves very helpless to choose a suitable engine when we want to learn electric flight.

Some research on the net was, in my case, not satisfactory. Indeed, the only information I could find there is an article by Ken Myers and a table of Calculation by Claude Gueniffey on the site electron libre which, on the one hand is not easy to use, and on the other hand does not explain the observed phenomena.

This study was therefore done in order to fill the void before which we find ourselves. The result of this theoretical part is an Excel table, named Calpfon, which after a few clicks on the mouse gives all the useful results. The table above can be downloaded in Excel format compatible with Open Office (click here )

Part 1: Calculation of engine parameters:

We know the manufacturer characteristics which are:

• Kv = number of turns per volt (rev / volt)
• Ra = Internal resistance of the motor (Ohm)
• Io = no load current (A)

We are looking for a given battery voltage U (volt) and as a function of the current I (A) consumed by the motor:

• Its rotation speed V (rpm)
• The power supplied by the motor, Psortie (Watts)
• Its efficiency E (%)

We will also determine:

• The Ibloc current (A) absorbed by the motor if it is blocked
• Its maximum efficiency Emax (%).
• Current consumption at maximum efficiency Imaxeff (A)

Process to follow:

1. We fix the voltage of the battery U and we look for the parameters for a current I varying from 0 to n
   
2. Calculation of the speed of rotation V;
   This is given by the formula: V = Kv [U – (I x Ra)] 
   We note that, since Kv and U are constant, the speed decreases with increasing current. This decrease is all the greater, the greater the internal resistance Ra is.
   
3. Calculation of the Input Power Pinput; 
   This is given by the formula: Pinput = U x I
   It therefore increases with the current supplied.
   
4. Calculation of the Power available at the output of the motor Pout: 
   This is equal to the product of the battery voltage U minus the voltage lost in the internal resistance Ra by the useful current.
   Therefore the supplied current, minus the necessary current to run the motor at no load Io.
   We obtain the formula:  Pout = ( U – ( Ra x I ))( I – Io )

The power curve therefore has the following form:

5. Calculation of the efficiency of motor E:
   Efficiency is the ratio of output power to input power. It is therefore not constant. So we have :  E = Pout / Pinput
   From which : E = ( U – ( Ra x I ))( I – Io ) / UI

The efficiency curve therefore has the following form:

It can be seen that the efficiency increases with the current and then decreases. It is therefore interesting to determine the value of the current I for which the efficiency is maximum. For this it is necessary to derive the function E (I) then to seek the value of I for which it is canceled.

dE / dI = -Ra.U.I^2 + U^2.Io / U^2.I
or
dE / dI = 0 si U^2.Io = -Ra.U.I^2
d’où
Imaxeff = Racine carrée( U.Io / Ra )

This current value known, it is simple to calculate the value of the maximum efficiency.

6. Calculation of the current consumed by the motor in blocking situation:
   It is also the current that will be consumed at the start-up (for a short time). In this situation, the motor becomes equivalent to a resistor whose value is Ra. 
   So we have: Ibloc = U / Ra

2nd part: Calculation of the operating point

Process to follow:

1. We determine the propeller by choosing:

• Its diameter and its pitch.
• The calculation method:
  • Abbott , (the result is the same whatever the propeller)
  • Boucher, (the result takes into account the propeller parameters)

2. If a reduction gear must be used, it indicates:

• Its reduction ratio
• Its efficiency (the latter must be estimated in the absence of manufacturer data).

3. It is now appropriate to calculate the power absorbed by the propeller as a function of its rotation speed. For this, we use:

• Abbott’s formula: Pabs = Pas x Dia^4 x V^3 x 5,33 x 10^15 Where the pitch and diameter are in inches, the speed in rpm and the power in W.
• Or that Boucher’s formula: Pabs = a x Pas x Dia^4 x V^3 where:
  • “a” = 1.31 for propellers Master Airscrew, Top Flite, Zinger etc,
  • “a” = 1.18 for propellers APC

Ces formules ont été établies par expérimentation, On trouvera plus d’informations a ce sujet dansThese formulas have been established by experimentation, More information on this subject can be found in the article by  Ken Myers

4. The operating point is established by drawing the curves:

• Power output from the motor (taking into account the possible reduction gear),
• Power absorbed by the propeller,
• Power consumed by the battery.

This determines the maximum speed of rotation of the propeller and the power consumed by the battery for this speed of rotation. The Excel table takes care of calculating this power, so it is not necessary to determine it by the graph. We deduce the current consumed by the battery by the formula: I = P / U
Autonomy is thus obtained: Autonomy =( Battery capacity / I ) x ( 6 / 100 )
Where:

• The autonomy is in minutes,
• Battery capacity in mA / h,
• Current I in A.

5. Remarks : 
   All these results are based on an empirical formula (propeller performance) and above all on a library of engine characteristics from manufacturer data which is not necessarily exact. It is therefore carful to verify the simulation results by real measurements. However, this allows you to get an idea of the performance of a propulsion unit for a given aircraft before launching into production.