# Brush DC Motor Basics

## Brush DC Motor Drive

Brushed DC ironless motors are found in a large variety of products and applications such as medical, robotics, factory automation, security and access, civil aviation and aerospace products.

The conventional ironcore brushed DC motor is greatly surpassed by the ironless technology. The main advantages of this unique concept are: absence of iron losses, low friction and a good thermal dissipation, resulting in a very efficient motor - meaning a perfect choice for battery-operated equipment. The design of the low inertia rotor enables very high acceleration and fast reaction time. Finally the linear torque-speed characteristics make the motor very easy to drive.

This article is a brief technical introduction on ironless miniature DC motors. It is intended to help engineers better understand the brushed DC ironless motor basics and to help them to select the best motor for their application.

### 1. THE BASIC EQUATIONS OF THE BRUSH DC IRONLESS MOTORS: A MOTOR WITH IRONLESS ROTOR CAN BE REPRESENTED BY THE FOLLOWING SIMPLIFIED DIAGRAM.

The voltage induced in the rotor is proportional to the angular velocity of the rotor: (See Figure 1)
Ui = k * ω

k: torque constant         ω: angular velocity

So the resulting equation is:
U   =      R      *       I       +      k    *     ω
(V)         (Ώ)           (A)         (Nm/a)    (rd/s)

The particularity of the ironless DC motor is such that the speed and torque function are linear. The speed is proportional to the voltage and the torque being proportional to the current:

T = k * ( I – IO)
(Nm) (Nm/A) (A) (A)
I: armature current IO: no load current

### 2. HOW TO DETERMINE THE MECHANICAL POWER, THE ELECTRICAL POWER AND THE EFFICIENCY?

The mechanical power developed by the motor is equal to the sum electrical power given to the motor and the power dissipated(heat): Pelect = Pmech + PJ  (See Figure 2)

The efficiency is defined by the ratio of the mechanical power and the electrical power:
η = Pmech / Pelect

The efficiency of a brushed DC coreless motor may reach up to 90%

Remember: the highest efficiency is obtained at high speed

### 3. UNDERSTAND THE EQUATIONS - FOUR THINGS TO REMEMBER:

(See Figure 3)

#1: The current in the motor is proportional to the motor torque.

#2: The speed of the motor is proportional to supply voltage (U).

#3: The maximum efficiency is obtained at high speed.

#4: The maximum mechanical power reaches its maximum when the load torque is equal to half the stall torque.

### 4. HOW TO DETERMINE THE ROTOR TEMPERATURE AND THE ROTOR RESISTANCE AT THIS TEMPERATURE:

(Brushed DC coreless motor maximum coil temperature is usually rated to 155° C.)

R22 * I2  * (Rth1 + Rth2) * (1-22 α) + Ta
Tr = ––––––––––––––––––––––––––––
1 - α * R22 * I2  * (Rth1 + Rth2)

R = R22 * (1 + α * ΔTemp)

Tr : temperature of the rotor (°C)
R22: motor resistance at 22° C (Ohms) - catalog value
I: current (A)
Rth1: thermal resistance rotor/body (°C/W) -catalog value
Rth2: thermal resistance body/ambient (°C/W) -catalog value
α: thermal coefficient of the resistance of copper (0.0039/°C)
Ta : ambient temperature (°C)
R: resistance (Ohms)
ΔTemp = Tr -22

### 5. HOW TO DETERMINE THE TIME CONSTANT OF THE SYSTEM AND THE STARTING TIME OF A BRUSHED DC IRONLESS MOTOR (VOLTAGE DRIVEN):

τ = τM * ( 1 + JL / JM)                   t = τ * ln (ω1 / (ω1 -ω) )

τ: time constant of the motor + load (ms)
τM: time constant of the motor alone (ms) – catalog value
JL: inertia of the load (kgm2)
JM: inertia of the motor (kgm2) – catalog value
t: starting time (ms)
ω1 : angular velocity obtain after an infinite time (rd/s)
ω: angular velocity (rd/s) after a time = t

### 6. THE PORTESCAP BRUSHED DC IRONLESS TECHNOLOGY IN ONE GLANCE:

(See Figure 4)

Concept Detail Motor Characteristics Advantages for the Applications
Ironless Rotor Low moment of inertia High acceleration, Ideal for incremental motion, linear speed-torque function, insensitive to shocks
No hysteresis and eddy current losses High efficiency, low losses from friction only. Ideal for battery operation
No magnetic saturation High peak torques without the risk of demagnetization
Central Stator Magnet High power per size and per weight Ideal for portable or small equipment or requiring small dimensions
Small Sized Bearings Low viscous damping High peak speeds, very low speed dependent losses
Low starting voltage
Precious Metal Commutation System Low friction, little electrical noise Low losses and wear, low electromagnetic interference
RatafenteTMSeries Copper-Graphic Commutation High current densities may be commutated High continuous and peak torques without the risk of demagnetizing the motor
Rated motor temperature up to 155oC Continuous torque is exceptionally high for motor size reducing the weight, dimensions and the cooling system
Very compact commutation system Excellent resistance to shocks and vibration
High torque to inertia ration High acceleration, short mechanical time constant

### 7. HOW TO SELECT THE APPROPRIATE BRUSH DC MOTOR? LET’S TAKE A LOOK, USING A MINIATURE AIR PUMP APPLICATION AS AN EXAMPLE.

A 6 volts, 0.6 A battery operated miniature air pump needs to have a flow range of 850 - 2500 cc/min which is equivalent to:
T = 3mNm of torque at 9’000 rpm (ω = 942.5 rd/s).

The requested mechanical power is:
Pmech = T * ω = 0.003 * 942.5 = 2.82 W

Portescap 16G brushed DC motor series is rated for 5W maximum output power.
Let consider the 16G88 -220E 1 (6v rated winding) (See Figure 5)

Winding Type -220P -220E -213E -211E -210E -208E
Measured Values
Measuring Voltage V 3 6 9 12 15 24
No-load speed rpm 10600 10900 8000 8700 9000 10400
Stall torque mNm (oz-in) 16 (2.3) 19.9 (2.8) 12.7 (1.80) 12.1 (1.71) 12.2 (1.73) 14.3 (2.0)
Avg. No-load Current mA 45 17 8 6.5 5.5 3.5
Typical Starting Voltage V 0.02 0.05 0.12 0.18 0.20 0.30
Max Reccomended Values
Max. continuous current A 2.0 1.21 0.55 0.42 0.35 0.25
Max. continuous torque mNm (oz-in) 5.2 (0.74) 6.3 (0.89) 5.8 (0.82) 5.4 (0.76) 5.4 (0.76) 5.4 (0.76)
Max. angular acceleration 103rad/s2 282 277 292 273 291 272
Intrinsic Parameters
Back-EMF constant V/1000 rpm 0.28 0.55 1.12 1.37 1.65 2.3
Torque constant mNm/A (oz-in/A) 2.67 (0.38) 5.3 (0.74) 10.7 (1.51) 13.1 (1.85) 15.8 (2.23) 22 (3.11)
Terminal resistance ohm 0.5 1.6 7.6 13 19.5 37
Motor regulation R/k2 103/Nms 70 58 66 76 79 77
Rotor inductance mH 0.01 0.045 0.15 0.26 0.40 0.72
Rotor inertia kgm210-7 0.8 0.91 0.8 0.8 0.74 .08
Mechanical time constant ms 5.6 5.3 5.6 6.1 5.8 6.7

The first step is to calculate the current which is supplied to the motor under the conditions described above.

T = k * ( I – IO ) ---> I = T / k + IO = 0.003 / 0.0053 + 0.017 = 0.583 A

The second step is to determine the supply voltage to get the requested speed.
9’000 rpm (942.5 rd/s)

U = R * I + k * ω = 1.6 * 0.583 + 0.0053 * 942.5 = 5.93 V < 6 V

Thus the motor will reaches the desired speed under the specified torque within the limitations of the battery.

Now we can determine the motor efficiency.

Pelect = U x I = 5.93 * 0.583 = 3.45 W

Efficiency* = Pmech / Pelect = 2.82 / 3.45 = 81%

Let's assume this pump needs to reach at least 5000 rpm in less than 15ms. (See Figure 6)
Load inertia: 1 x 10-7 kg.m2
Rotor inertia: 0.91 x 10-7 kg.m2

τ = τM * ( 1 + JL / JM) t = τ * ln (ω1 / (ω1 -ω))

τ = 5.3 * (1+1/0.91) = 11.12ms

t = 11.12 * ln (9000 / (9000-5000)) = 9ms < 15ms

The speed of the pumps will be 5000rpm after 9ms.

This excellent dynamic characteristic is due to the ironless rotor concept. The low moment of inertia of the rotor enables very high acceleration.

*The motor efficiency is above 80% which will contribute to very long battery life. Achieving such efficiency is only possible thanks to the Portescap state-of-the-art brushed DC ironless motor technology.

### Figure 1

U0 = power supply voltage, V
I = current, A
Vi = voltage (EMF) inducted in the rotor, V
R = winding resistance, Ohms

### Figure 3

η : efficiency; n: speed; P: power; I: current