This articles intends to give a simplified view of and electric brushless motor. Because not all possible parameters are considered, the presented model will not be very accurate, but will suffice to compare motors with different characteristics.

The main application focus will be on quadcopters, or any other type of multicopter. The mechanical lift power that keeps a multicopter hovering heavily depends on the motor and on the propellers, which are covered in an different article. Depending on the motor-propellers combination the flying behavior of a multicopter drastically changes. Using a fast motor with little propellers allow for a fast and acrobatic flight, while a strong motor and big propellers change result in a stable flight with strong lift for extra payload.

Also, the input current of a motor depends on the propeller, which translates into mechanical load that needs to be driven. It is to be noted that more current at the same voltage corresponds to a bigger propeller.

#### Brief

RPM = Kv · (Vin - Rm·Iin)

T = Kt · (Iin - Io)

Motors with a lover Kv usually have a greatter Kt, for the same output power.

Eta = [(Vin - Iin · Rm)·(Iin - Io)] / [Vin · Iin]

#### Motor parameters

Brushless motors (DC motors) can by modeled by a series of parameter which determine it's behavior and performance. This model does not yield a 100% accurate result, since all motor-constants will be considered invariable by the operating conditions of the motor or external agents. Therefore this model will fail predicting some situation, like when operating the motor at very low angular velocity or at high temperatures. This parameters are:

- Variables depending on operating conditions
- Vin - Input voltage
- Iin - Input current
- Pin - Input power
- Pout - Output power
- RPM - Angular velocity in Revolution Per Minute

- Motor constants
- Rm - Armature Resistance
- Io - No-load current
- Kv - Angular speed to voltage constant
- Kt - Torque to current constant

- Motor limits
- Angular velocity limit
- Torque limit
- Temperature limit

#### Variables depending on operating conditions

These variables represent the operating conditions of the motor.

### Input voltage

How much voltage is applied by the Electric Speed Controller (ESC) to the motor. This is usually the variable the user can adjust to control motor speed and torque.

### Input current

How much current is entering the motor. This variable depends on the load the motor is driving, increasing as the required output torque increases.

### Input power

How much power is being transfered to the motor.

### Output power

How much useful power is being done by the motor. The difference between input power and output power is transformed into wasted heat.### Angular velocity

It measures how fast the motor is spinning in RPM. From now on the angular velocity will be referred to with this same term, RMP.

#### Motor constants

This parameters are usually provided in the motor's datasheet. They are actually not constant, but their variations under normal circumstances is small enough to be disregarded for a first estimation of motor performance.

### Armature resistance (Rm)

The windings of the motor have a non-zero resistance, although usually low (in the range of 100 - 300 mOhm). This resistance causes a voltage drop due to the input current (Ohm's law), so that not all input voltage is usefully used. Also, this resistance is the main source of waisted energy in form of heat.

### No load current (Io)

It measures the minimum amount of current the motor needs to start spinning when no load is being driven, and represents how much torque is lost due to inner friction, rotor inertia, etc. Therefore, better quality motors have a lower no-load current.

### Angular speed to voltage constant (Kv)

This constant determines the RPM of the motor for a determined input voltage.

Altought it may not sound logical at first, the load that is being driven has no impact on the RPM. It ony depends on the input voltage. The clue lies in the fact that increasing the load at a given input voltage (and thus a given RPM) will increase the input current. So driving a big propeller requires more power than driving a small propeller, because the needed current is greater (remember W = V·I). How much more power can be calculated the formula in the propeller article propeller article.

Yet, that the RPM solely depends on the voltage is only partially true. Due to the armature resistance, when the input current increases the lost voltage drop at the resistance also increases. This reduces the voltage which is actually used to create the magnetic field that drives the motor. So, the RPM can be calculated as follows:

RPM = Kv·(Vin - Rm·Iin).

### Torque to current constant (Kt)

Similarly to the Kv constant, Kt measures the relation between the output torque and the input current. It is responsible of the fact that driving a bigger propeller (which requires more torque) causes the motor to demand more current.

Again, although the output torque is mainly determined by the input current, losses have to be considered. Because of the no-load current, the current sued to produce the effective output torque is reduced. The output torque can be estimated whith this formula:

T = Kt · (Iin - Io)

If the input current is below the output current the motor will not move, even when driving no load, as there is not even enough current to overcome the inner friction.

### Relation between Kv and Kt

Manufacturers can produce motors with a wide range of Kv and Kt, but the product of these two constants is always the same for a given size and characteristics of a brushless motor type.

That means that a motor with a lower Kv will spin at a lower RPM at a given voltage, but will be able to provide more trust at that speed because its Kt is bigger.

So, decreasing Kv requires more voltage to archive a determined speed, but requires less current for the same propeller. Also, increasing Kv requires less voltage but more current for the same propeller. With this consideration in mind, and knowing that the losses increase with current (due to Rm), it is preferably to use motors with a lower Kv.

#### Motor limits

A motor can not operate under all conditions. There are limits to its maximum temperature, velocity and torque.

### Temperature limit

The windings of the motor are usually made of copper, or other suitable metal. If the motor is pushed beyond its temperature limits these filaments melt and the motor becomes permanently destroyed.

### Angular velocity limit.

The RPM limit is imposed by the ability of the armature to sustain the centripetal forces without damage.

Working at RPMs under this limit is safe, and usually increases the efficiency of the motor.

### Torque limit

This limit is directly derived from the temperature limit. Because higher torque require more current, there is a limit where the generated heat in the windings (due to armature resistance) is high enough to get the motor past it's temperature limit.

The phenomenon of powering a motor but blocking the rotor is called stalling. When stalling all the voltage is applied to armature resistance, causing a high current. Stalling inmediately destroys a motor unless there is a current limit imposed by other elements in the circuit (like by the ESC or battery).

#### Efficiency

Efficiency is calculated as the relation between the input power and the useful mechanical output power. Because the mechanical power depends on the effective voltage and current, it can be written as follows:

Respecting current, efficiency hits a maximum where the losses due to Rm and Io cancel each other. With increasing current losses due to Rm increase, but those caused by Io decrease. So the maximum efficiency point for a given voltage is located where both losses are equal.

Eta = [(Vin - Iin · Rm)·(Iin - Io)] / [Vin · Iin]You can learn from this formula that efficiency decreases with with Rm and Io. Also than the maximum achievable efficiency increases as Vin does.

Respecting current, efficiency hits a maximum where the losses due to Rm and Io cancel each other. With increasing current losses due to Rm increase, but those caused by Io decrease. So the maximum efficiency point for a given voltage is located where both losses are equal.

#### Simulation

A
quick way to obtain an estimated idea of the performance of a
propeller-motor combination is to simulate. There are a few web-based
simulators which allow for this.

Here are some of them:

Here are some of them:

There are also non-web-based simulators, for example Drive Calculator by Christian Persson.

#### Considerations when selecting a motor

So as to conclude, some final considerations are presented.

- For a given motor working with a 3S battery (11.1V), changing the battery to 4S (14.7V) changes the flight behavior of the multicopter to more acrobatic. Because the maximum speed of the propellers increase, and so does the input current, smaller propellers have to be used to avoid overheating the motor.

- For medium sized quadcopters, this are the most common motor types and propellers depending on application. This serves only for orientation with a 3S battery.

- Acrobatic: 1000-1200 Kv w/ 8x4.5" propeller
- In-between: 900-1000 Kv w/ 10x4.5" propeller
- Stable and Lifting (for payload): 700-900 Kv w/ 12x4.5" propeller

- Because motors are mounted on the furthes point from the mass center fo the multicopter, its weight has an important impact on flight behaviour. Thus, it is important to use lightweight motors not only for the total weight.

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