**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: