Electric motors and generators operate based on the principles of electromagnetic induction. Motors convert electrical energy to mechanical motion, while generators convert mechanical motion to electrical energy. They both work by using magnetic fields to exert forces on electric currents or moving charges. The direction of these forces causes rotation, which is then harnessed to power other devices or generate electricity from mechanical inputs. Back electromotive force (EMF) is induced any time a coil moves through a magnetic field, and it ensures the conservation of energy as devices switch between motor and generator functions. However, real-world motors and generators have various inefficiencies that reduce their performance below the theoretical maximum.

1. ELECTRIC MOTORS &
GENERATORS
Andrew Holliday

2. Motors and Generators
• Simple devices that use basic principles of electromagnetic
theory
• Technologically important
• Motors drive everything from hybrid cars to vibrating phones.
• Most electrical power is provided by generators
• Work on the same principles: converting between
mechanical and electrical energy using the magnetic force

3. Electric and Magnetic Fields
• Electric and magnetic fields are vector fields
• A vector has magnitude and direction
• A vector field describes a vector for every point in space

4. Electric Force
• An electric field exerts force on electric charge.
• The force is in the direction of the field - charges get pushed in
the direction of the field.
• F = qE (q is magnitude of charge).

5. Magnetic Force
• Magnetic fields exert force on moving charges
• Force is perpendicular to field and to velocity
• Units of Gauss: 1 G = 1 N*s/C*m
• FB = qvBsin(ϴ) : ϴ is the angle between v and B.

6. Electric Motors
• In a motor, current passes through a coil of wire in a
magnetic field
• Magnetic field exerts force on charges moving in the coil

7. Electric Motors
• Current, and thus force, is in opposite directions on
opposite ends of the coil
• Creates torque on the coil

8. Electric Motors
• When the coil is pulled "flat" by the magnetic force, the
direction of the current must be reversed:

9. Electric Motors
• This reverses the direction of the force
• Momentum continues the rotation, and the new force
accelerates the rotation

10. Electric Generators
• In a generator, we rotate the rotor from "outside"
• Wire moves in opposite directions on either side of loop
• Opposite forces on either side create voltage around loop

11. Electric Generators
• As the loop makes a rotation, the direction of current
reverses
• This produces alternating current

12. Electric Generators
• In my generator, coil is the "stator", magnet is the "rotor"
• Circuit demonstrates how the current alternates

13. Motors and Generators
• Different designs: magnet can be either rotor or stator
• Some motors use an electromagnet instead of a permanent
magnet
• All designs operate on the same principle described here
• Charges moving relative to a magnetic field are
pushed perpendicular to their motion and the field

14. Back-EMF and Symmetry
• Motors and generators are basically the same
• In some cases, a single device is used as both a motor and
a generator
• Gas turbines, hybrid electric cars (regenerative breaking)
• This symmetry is important for a deeper reason...

15. Back-EMF and Symmetry
• Guarantees conservation of energy
• Current through a motor's coil causes it to rotate
• A rotating coil in a magnetic field induces voltage!
• By the Right-Hand Rule, this voltage is always in the
opposite direction as the supplied voltage
• This is called back-EMF (ElectroMotive Force)

16. Back-EMF and Symmetry
• Likewise, current induced in a generator induces torque
• Torque opposes rotation of the generator
• These reaction forces always resist the applied forces
• This is required by the Maxwell-Faraday Equation:

17. Back-EMF and Symmetry
• Back-EMF is how energy is extracted from a voltage source
by a motor
• Without load, motor is allowed to accelerate
• Back-EMF increases with motor speed
• When back-EMF equals supplied voltage, there is no net
voltage, no current over the motor - it stops accelerating
• Since no current flows, no energy leaves the battery

18. Back-EMF and Symmetry
• Load on the motor extracts rotational energy
• Motor does not reach the same top speed, so back-EMF is
always less than supply voltage
• Heavier load => lower top speed => more current flows

19. Back-EMF for square coil
We will calculate the peak back-EMF of a square coil.

20. Back-EMF for square coil
Assume B and v are perpendicular:
• Force on charge: FB = qvB (v and B are perp.: drop sin(ϴ))
• Force per unit charge: FB/q = vB
• Work per unit charge over distance L: LFB/q = LvB
• This is the Back-EMF over a distance L

21. Back-EMF for square coil
Over the top and bottom edges of square coil:
• ϴ = 90 degrees, sin(90) = 1
• Speed of edge v = 2π*f*r = 2π*f*0.019 m
• 6 turns, so length L = 6*0.038 m
• B = 0 T over top edge, 0.083 T over bottom edge
• 6*2π*f*(0.019 m)*(0.038 m)*(0.083 T) = (0.0023 m2T)*f
• (0.0023 m2T)*f = Vback

22. Back-EMF for square coil
• What about the 3 turn coil?
• 3*2π*f*(0.019 m)*(0.038 m)*(0.083 T) = (0.0012 m2T)*f =
Vback
• At maximum speed, Vback should be equal for both coils
• So 6*2πfrLB = 6*2πfrLB
• 2f1 = f2
• Top speed of the 3 turn coil should be about twice that of the
6 turn coil. Is it?

23. Other coils
• What about the rectangular coils? Circular coils? Will they
be faster or slower?
• For rectangular coil, B = 0.047 T
• (0.0013 m2T)*f = Vback
• For circular coil, B = 0.140 T
• (0.0039 m2T)*f = Vback

24. Efficiency
• Vin = 2.7 V, but only applied half the time, so 1.35 V
• These frequencies are much lower than we'd expect
• These motors have very low efficiencies
• Efficiency is defined in terms of power, energy-per-time
• Efficiency n = Pout/Pin: ratio of input power to output power

25. Efficiency
• Ideally, mechanical power of a motor equals electrical input
power
• Electrical power of a generator equals mechanical input
power
• In reality, this never happens

26. Sources of Inefficiency
• Friction between the rotor and its joint
• Resistance and between electrons and the wire (resistance)
• Geometry - magnetic field, coil shapes don't maximize
torque on coil

27. Design considerations
• Number of turns: more turns give more torque, but also
more resistance
• Joints: sliding contacts have a lot of friction - some motors
apply current to loop by induction
• Geometry: vast variation in designs to maximize magnetic
force!
o Iron cores in coils
o Multiple coils, multiple magnets

28. What loop shape is most efficient?
• Which loop shapes give most efficient conversion? Why?
• To find the answer, need more physics: