GRAPHING TOOL FOR DC MOTORS AND PROPELLERS

GRAPHING TOOL FOR DC MOTORS AND PROPELLERS V1

There are a few calculators and programs for calculation of DC motors performance and propellers for quadcopters. However most of them are designed to match specific motor to specific propeller and are not very easy to to use. The goal of this development is easy-to-use plotter to study behavior of motor and propellers in general. The highlights of this tool are:

- Several plots can be drawn simultaneously
- Parameters can be set to change automatically
- Any value can be used as an X axis
- Plots can be viewed vs throttle position (v1 allows use different values as X axis)
- Simulation can be done with altering propeller properties and diameter
- Plot can be saved/shared via “generate link with parameters”

Without propeller (“propeller” checkbox unchecked) the plots show all possible values of motor parameters in a full possible range of operation.

When the motor is loaded with a propeller there is only a single value for *rotation speed*, *current, power, torque etc* at a fixed *voltage *(this voltage corresponds to 100% throttle). This point is a solution of *M _{motor}=M_{prop}* or

*P*equation and is shown with white rectangular label on the plots. If (“torque” and “rotor torque”) or (“power” and “rotor power”) are plotted and a “same for similar” checkbox is checked (makes corresponding torque and power scales the same) this point is clearly seen as a graphical solution of the equation.

_{motor}=P_{prop }When values are plotted vs “throttle” position we assume that PWM linearly change voltage with throttle; and for each of these voltages the above solutions at M_{motor}=M_{prop} give corresponding ω, I, M, H, η T values.

Propeller constants can be defined by 4 different ways, in any case all thrust related calculations include efficiency of the propeller (typical 0.4-0.6, see below). In general, we can assume that *Cp* and *Ct* are linearly depend on number of blades and pitch angle.

The following equation can be useful when motor’s constants are calculated (e.g for rewinding)

where *N* is a number of loops, *l* is a motor’s length, *r* is an internal radius of the motor, *B* is a density of magnetic flux, and ω is a rotation speed.

This tool does not take into account FOM (efficiency) of the ESC.

*Theory*

DC motor equations are discussed in many places, however (imho) most of them are not very clear and logical. Below I show my approach, may be it will be more convenient for someone.

*Motor:*

The following gives interconnection between rotation speed, voltage, current, motor efficiency etc.

- Voltage is equal to the sum of potential drop at the internal resistance of the motor and a back emf (Kirchoff’s law):

(1)

*k _{emf}* has

*V/(rad*sec*units, it is related to frequently used inverse coefficient

^{-1})*K*

_{v}*(rpm/V)*as

- Electric power applied to the motor is being spend to mechanical power only by part; mechanical power can be calculated as a product of “useful” voltage to “useful” current:

(2)

*IR* is a voltage drop at the motor’s active resistance and *I** _{0}* is no-load current (related to iron and friction loses).

- Torque is M=P
_{m}/ω (physics law) and, taking into*account*(1):

(3)

- FOM of the motor (motor efficiency) is a ratio of mechanical power to electrical power:

(4)

- Heat is a Joule’s heat:

(5)

(1)-(5) show values in all possible range of motor’s operation: for example, (1) defines maximal current *I**=**V**/**R*, reachable when load is so high, that motor stops and ω=0. On the other side current should be higher than *I** _{0}* (eq. (3)).

To experimentally get all possible values, the motor’s load should vary from 0 (free spinning) to the value when motor stalls.

*Propeller:*

To rotate a propeller the following mechanical power P_{m} should be applied (ref_1):

(6)

*C _{p}* (

*power coefficient*) depends on the propeller’s geometry and density of air.

Thrust of the propeller can be written as:

(7)

*C _{t}* is called

*thrust coefficient*.

*С*and

_{p}*С*are connected via dependence of thrust on mechanical power and propeller’s diameter:

_{t}(8)

Any propeller can be characterized by FOM (Figure of Merit, or propeller’s efficiency), only part of mechanical power is being converted to thrust. Typical values of quadcopter’s propellers are in the range of 0.4-0.6 (bigger prop gives higher FOM). This η_{prop} should be taken into account in all thrust related calculations, and *P* in (7) should be replaced with *P _{eff}*

*=*

*η*

_{prop}*η*

_{motor}***

*P*

_{m}Sometimes *Cp* and *Ct* are used without dependence on diameter and rotation speed is expressed as (rpm/1000), this is convenient when specific propellers are being compared. In this case *P _{m}=C_{pe}*(rpm/1000)^{3}* and

*T=С*

_{te}*(rpm/1000)^{2}.Ref 1.

https://m-selig.ae.illinois.edu/pubs/DetersSelig-2008-AIAA-2008-6246-MicroProps.pdf