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Electromagnetic and Electromechanical Engineering Principles Notes 03 Basic Machines

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Electromagnetic and Electromechanical Engineering Principles Notes 03 Basic Machines Electromagnetic and Electromechanical Engineering Principles Notes 03 Basic Machines Marc T. Thompson, Ph.D. Thompson Consulting, Inc. 9 Jacob Gates Road Harvard, MA 01451 Phone: (978) 456-7722 Email: [email protected] Website: http://www.thompsonrd.com © Marc Thompson, 2006-2008 Portions of these notes reprinted from Fitzgerald, Electric Motors, 6th edition Electromechanics Basic Machines 3-1 Course Overview --- Day 3 Electromechanics Basic Machines 3-2 Electrodynamic Levitation Demo • AC levitation with 60 Hz excitation from autotransformer Reference: Marc Thompson, “Electrodynamic Magnetic Suspension --- Models, Scaling Laws and Experimental Results,” IEEE Transactions on Education, vol. 43, no. 3, August 2000, pp. 336-342 Electromechanics Basic Machines 3-3 One Method to Calculate Lift Force • We can deduce force acting on the coil by making measurement of inductance of coil at different heights Electromechanics Basic Machines 3-4 Levitation Example Reference: Marc Thompson, “Electrodynamic Magnetic Suspension --- Models, Scaling Laws and Experimental Results,” IEEE Transactions on Education, vol. 43, no. 3, August 2000, pp. 336-342 Electromechanics Basic Machines 3-5 Circular Coil Above Conducting Aluminum Plate • Flux density plots at DC and 60 Hz via FEA • At 60 Hz, currents induced in plate via magnetic induction create lift force DC 60 Hz Electromechanics Basic Machines 3-6 Levitation Example • Coil inductance vs. coil height above aluminum plate at 60 Hz Reference: Marc Thompson, “Electrodynamic Magnetic Suspension --- Models, Scaling Laws and Experimental Results,” IEEE Transactions on Education, vol. 43, no. 3, August 2000, pp. 336-342 Electromechanics Basic Machines 3-7 Levitation Example • Curve fit for inductance: • Magnetic stored energy: • Levitation force: Reference: Marc Thompson, “Electrodynamic Magnetic Suspension --- Models, Scaling Laws and Experimental Results,” IEEE Transactions on Education, vol. 43, no. 3, August 2000, pp. 336-342 Electromechanics Basic Machines 3-8 Levitation Example Reference: Marc Thompson, “Electrodynamic Magnetic Suspension --- Models, Scaling Laws and Experimental Results,” IEEE Transactions on Education, vol. 43, no. 3, August 2000, pp. 336-342 Electromechanics Basic Machines 3-9 AC Magnetic Suspension Scaling Laws • Scaling law for zero speed EDS 2 P ∝ M 3 • Coil mass M = 0.35 kg; P = 257 W at h = 10 mm • Full scale suspension: For M = 1000 kg, P = 51 kW • Other tradeoffs: – Excitation frequency vs. guideway thickness Electromechanics Basic Machines 3-10 Example and Case Study: Solenoid Design • Follows Example 3.10 in Fitzgerald (pp. 153-155) • Note that dimension x is the distance from the top of the plunger to the top of the coil • As this dimension x varies, axial force on the plunger varies as well Electromechanics Basic Machines 3-11 Strategy • Use reluctance model • Find flux Φ • From flux, find flux linkage λ • From flux linkage, find inductance as a function of plunger position x • From L(x), find force Electromechanics Basic Machines 3-12 Reluctance Model • Reluctance of steel is very small if the steel doesn’t saturate Path length in direction of field ℜ = Areaμ() of fluxperpendicular path to field Electromechanics Basic Machines 3-13 Reluctance Model g ap: gUpper ℜ upper = μo()d π x g ap: gerLowℜlower = μo()d π a Electromechanics Basic Machines 3-14 Flux : Flux NI Φ = ℜupper +lower ℜ Electromechanics Basic Machines 3-15 Flux Linkage linFlukaxge: NI2 =λ N Φ = ℜupper +lower ℜ Electromechanics Basic Machines 3-16 as a Function of Plunger PositionInductance e: anctInduc λ N 2 L= = Iℜupper +lower ℜ g g ⎛ g ⎞ a+ x ⎜ ⎟⎛ ⎞ ℜupper + ℜlower = +⎜ = ⎟⎜ ⎟ μ πo dx μo da π⎝ o da μ⎠⎝ πx ⎠ ⎛μdaN π 2 ⎞⎛ x ⎞ ⎛ x ⎞ L() x ⎜ o ⎟ L = ⎜ ⎟⎜ ⎟ = o ⎜ ⎟ ⎝ g ⎠⎝a+ x⎠ ⎝a+ x⎠ Electromechanics Basic Machines 3-17 Find Force se icfore thngndifie, ctancudn the inowu koOnce y :easy I2 dL() x f = x 2 dx I 2 d ⎛ x ⎞ = Lo ⎜ ⎟ 2 dx ⎝a+ x⎠ I 2 a = L 2 o()a+ x2 Electromechanics Basic Machines 3-18 Solenoid Design Example Dimensions of solenoid design example Item Description Dimension a Height of solenoid backiron arm 12.5 mm h Height of solenoid center backiron arm 10 mm d Radius to backiron 10 mm g Airgap 0.5 mm rcoil Coil mean radius 7.75 mm x Variable offset of plunger top from top of 5 mm (variable) coil rplunger Plunger radius 9.5 mm Electromechanics Basic Machines 3-19 Design Example --- Analytic Result • For NI = 354 A-turns (DC) • Coil power dissipation ∼ 3.5 Watts Example solenoid design 4.5 4 3.5 3 2.5 2 Force, [N] Force, 1.5 1 0.5 0 024681012 x, [mm] Electromechanics Basic Machines 3-20 Simulation • Trust, but verify • 2D axisymmetric finite-element analysis program used • Finite Element Method Magnetics (FEMM) by Dave Meeker at Foster-Miller Electromechanics Basic Machines 3-21 Simulation --- 2D Axisymmetric FEA Electromechanics Basic Machines 3-22 Simulation --- 2D Axisymmetric FEA • Shown for x = 5 mm • FEA shows fx = 1.6 Newtons, Pdiss = 3.5 Watts Electromechanics Basic Machines 3-23 Comparison of Analytic vs. FEA • Note that FEA shows lower force than that predicted analytically • Probably due to saturating iron reducing B and hence force Example solenoid design 4.5 4 3.5 3 2.5 2 Force, [N] Force, 1.5 1 0.5 0 024681012 x, [mm] Electromechanics Basic Machines 3-24 Introduction to Rotating Machines • Elementary concepts and terminology • AC machines overview • DC machines overview • MMF of distributed windings • Magnetic fields in rotating machines • Generated voltage • Torque • Linear machines Electromechanics Basic Machines 3-25 Motor Terminology • Armature winding --- windings that carry AC currents • Stator --- stationary part of the motor • Rotor --- rotating part of the motor • DC machine --- armature winding found on the rotor • AC machine --- armature winding typically on the stator • Field winding --- windings that carry DC current and produce main operating flux Electromechanics Basic Machines 3-26 Types of Machines • AC machines • Synchronous motor • Induction motor • DC machines Electromechanics Basic Machines 3-27 Simple 2-Pole Synchronous Generator • This is a “salient pole” machine. • Field winding excited by brushes contacting slip rings • It advantageous to have the high power armature winding mounted to the stator for cooling purposes Electromechanics Basic Machines 3-28 Flux Density and Generated Voltage • Idealized analysis assuming sinusoidal magnetic flux density in airgap varying with angle θa and flux produced by field winding only • As the rotor rotates, there is a dλ/dt which results in a sinusoidal armature voltage Electromechanics Basic Machines 3-29 Flux Density and Generated Voltage • What is rotation speed needed to supply 60 Hz in a 2-pole machine? • Frequency in Hz = rotation speed … hence “synchronous” generation occurs. 2-pole synchronous machine rotates at 3600 RPM to produce 60 Hz f606060⋅ ⋅ ω == =3600 RPM s p/2 2/2 Electromechanics Basic Machines 3-30 Four-Pole Single Phase Synchronous Generator • This is also a “salient pole” machine • Generated frequency is twice that of the two-pole machine • 4-pole synchronous machine rotates at 1800 RPM to produce 60 Hz ⎛ poles ⎞ θ ae = ⎜ ⎟θ a ⎝ 2 ⎠ θae = electrical angle θa = mechanical angle Electromechanics Basic Machines 3-31 Distribution of Airgap Flux in 4-Pole Gen. • Note that electrical angle is twice the mechanical angle, since there are 2 N-S electrical pole pairs per rotor rotation Electromechanics Basic Machines 3-32 PM Machine • 4 pole machine; field provided by NdFeB magnets Electromechanics Basic Machines 3-33 PM Machine • 4 pole machine; field provided by NdFeB magnets Electromechanics Basic Machines 3-34 Nonsalient Pole Machine • 2-poles; windings are distributed in slots in the rotor • Suitable for high-speed generator Electromechanics Basic Machines 3-35 Three-Phase Generators 2-pole 4-pole Y-connection of windings Electromechanics Basic Machines 3-36 Induction Machines • Stator winding is excited with AC currents, as in a synchronous machine • AC flows in the rotor windings; rotor currents are produced by magnetic induction (i.e. transformer action) into short- circuited rotor windings • We can think of an induction machine as a type of transformer where power is transformed between rotor and stator, with a change of frequency and flow of mechanical power • Induction motor is the most common motor Electromechanics Basic Machines 3-37 Typical Induction Motor Torque-Speed Curve • The rotor of an induction machine does not rotate synchronously with the stator field; it “slips” ω −ω s = s r ω = synchronous speed ω s s ωr = rotor speed • Note that at full “synchronous speed” the motor generates zero torque Electromechanics Basic Machines 3-38 DC Machines • Armature winding of a DC generator is on the rotor with current conducted from it by brushes • Field winding is on the stator and is excited by DC current Electromechanics Basic Machines 3-39 Elementary DC Machine with Commutator Electromechanics Basic Machines 3-40 DC Machine Characteristics • Brushes cause a rectification Electromechanics Basic Machines 3-41 MMF of Concentrated Full-Pitch Winding • This is an extreme example • MMF is “magnetomotive force” • Remember, by Ampere’s law a current forces H to flow • A current in a wire causes a change in the MMF • Note first harmonic of airgap MMF (Fag1) Electromechanics Basic Machines 3-42 MMF of Concentrated Full-Pitch Winding Fundamental component of airgap MMF ⎛ 4 ⎞⎛ NI ⎞ Fag1 = ⎜ ⎟⎜ ⎟cosθ a ⎝ π ⎠⎝ 2 ⎠ Peak value of airgap MMF ⎛ 4 ⎞⎛ NI ⎞ ()Fag1 = ⎜ ⎟⎜ ⎟ peak ⎝ π ⎠⎝ 2 ⎠ Electromechanics
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