ELECTRIC MOTORS &
GENERATORS
Andrew Holliday
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
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
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).
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.
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
Electric Motors
• Current, and thus force, is in opposite directions on
opposite ends of the coil
• Creates torque on the coil
Electric Motors
• When the coil is pulled "flat" by the magnetic force, the
direction of the current must be reversed:
Electric Motors
• This reverses the direction of the force
• Momentum continues the rotation, and the new force
accelerates the rotation
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
Electric Generators
• As the loop makes a rotation, the direction of current
reverses
• This produces alternating current
Electric Generators
• In my generator, coil is the "stator", magnet is the "rotor"
• Circuit demonstrates how the current alternates
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
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...
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)
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:
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
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
Back-EMF for square coil
We will calculate the peak back-EMF of a square coil.
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
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
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?
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
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
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
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
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
What loop shape is most efficient?
• Which loop shapes give most efficient conversion? Why?
• To find the answer, need more physics: