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### 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!
- Iron cores in coils
- 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:

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