General Theory on Motors

There are 2 kinds of motors, AC motors and DC motors. In this course, we are going to focus on DC motors only. So, the following discussions focus mainly on DC motors.

There are several kinds of DC motors, examples are stepper motors, servos, brushed/brush-less motors.

Stepper motors: The inputs of a stepper motor are signal pulses and the shaft of a stepper motor moves between discrete postions proportional to pulses. If the load of the motor is not too great, open-loop control is usually used to control the motor. Stepper motors are used in disk drive head positioning, plotters, and numerous other applications.

Servo motors: The input of a servo motor is a voltage value and the output shaft of the servo motor is commanded to a particular angular position according to the input voltage. Servo motors are used in radio control airplanes to control the position of wing flaps and similar devices.

DC motors: The input of a DC motor is current/voltage and its output is torque (speed).

How does a motor work?

Let's consider a permanent magnet brushed motor. The piece connected to the ground is called the stator and the piece connected to the output shaft is called the rotor. The inputs of the motor are connected to 2 wires and by applying a voltage across them, the motor turns.

The torque of a motor is generated by a current carrying conductor in a magnetic field. The right hand rule states that if you point your right hand fingers along the direction of current, I, and curl them towards the direction of the magnetic flux, B, the direction of force is along the thumb.

Now, imagine a loop of wire with some resistance is inserted between the two permanent magnets. The following diagrams show how the motor turns:

Diagram showing how the motor works

Relationship between the Torque and the angle the loop made with the magnet.

You might be able to notice that the direction of rotation is changing every half cycle. To keep it rotating in the same direction, we have to switch the current direction. The process of switching current is called commutation. To switch the direction of curent, we have to use brushes and commutators. Commutation can also be done electronically (Brushless motors) and a brushless motor usually has a longer life. The following diagram shows how brushes and commutators work.

We could also have several commutators and loops. The total torque generated is the sum of all the torques from each of the loops added.

Motor with several commutators and loops

So, the torque is proportional to the current through the windings,

T = kI where T is the torque, I is the current, and k is a constant

The wire coils have both a resistance, R, and an inductance, L. When the motor is turning, the current is switching, causing a voltage,

V = L dI/dt

This voltage is known as the back-emf(electromotive force), e.

If the angular velocuty of the motor is w, then e = kw (like a generator)

This voltage, e, is working against the voltage we apply across the terminals, and so,

(V- kw) = IR where I = T/R

which implies (V-kw) = (T/k) R

The maximum or stall torque is the torque at which w = 0 or T = kV/R, and

The stall or starting current, I = V/R

The no load speed, w = V/k, is the maximum speed the motor can run. Given a constant voltage, the motor will settle at a constant speed, just like a terminal velocity.

If we plot w = V/k - (T/k^2)R, we can get the speed-torque curve:


Here are the different units for the torque, current and voltage

Torque:, Nm (=kgm/s^2*m), kgfm(=9.8 times Nm), gfcm, mNm, etc.

Current: Amperes(Amps), mA

Voltage: Volts

mechanical power = T*w(Nm/sec) = 1 watt

electrical power = VI = 1 volt * amp = 1watt



Now, let's look at a data sheet of a motor. On the data sheet, specifications such as voltage, no load speed, stall torque, starting current, efficienct of power conversion are shown. (To look at a data sheet, click here)

From the data sheet, we can find that some of the motors run too fast with too little torque for robotic applications. To handle the problem, we can to gear down the motor.

There are lots of types of gears (e.g. planetory, spur, HD, worm, etc.) (Gears alone can cost 100's of dollars!) but our main concern is the Gear Ratio of the gear that we use. Let's take a look at the following diagram:

In the above diagram, the output shaft of the motor is attached a 5-teeth gear and a 10-teeth gear is placed right next to the 5-teeth gear with their teeth touching each other. In this example, the gear ratio, G, is 2 (=10/5). Typically, higher gear ratios, such as 10:1, 50:1, are used in practice.

If we have a gear ratio of G, then

T' = GT

w' = w/G

The above are just part of the issues we concern about a DC motor. There are other issues such as backlash, efficiency, etc. You can refer to the book, Moblie Robots, by Jones & Flynn, for more information on these issues.
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