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### Lecture09

Course: EEL 3216, Spring 2011
School: FSU
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Word Count: 930

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Machine AC Fundamentals Generators Motors 3-phase 1-phase Synchronous Induction Fundamentals of Power Systems Lecture 9 1 AC Machines Systematic Fundamentals Principle of generating a rotating magnetic field Magnetomotive force and induced voltage Principle of generating a torque in machines Power flow in AC machines Synchronous Machines Construction and operating principle Steady state model and phasor...

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Machine AC Fundamentals Generators Motors 3-phase 1-phase Synchronous Induction Fundamentals of Power Systems Lecture 9 1 AC Machines Systematic Fundamentals Principle of generating a rotating magnetic field Magnetomotive force and induced voltage Principle of generating a torque in machines Power flow in AC machines Synchronous Machines Construction and operating principle Steady state model and phasor diagram Single generator operation Induction (Asynchronous) Machines Fundamentals of Power Systems Lecture 9 Page 1 2 Space Vector Notation Three windings, aa ,bb cc, separated 120 mechanically carry a set of balanced three-phase currents (separated 120 electrically) Space vector (in spatial xy plane) y Represents the spatial direction of a vector quantity (H, B, possibly time dependent) Not to be confused with phasor diagram (time dependent scalar in Re/Im plane)! Excitation from winding aa alone Lecture 9 Fundamentals of Power Systems 3 Rotating Magnetic Field Symmetric 3~ current system iaa ' (t ) I M sin( t ) ibb ' (t ) I M sin( t 120 ) icc ' (t ) I M sin( t 240 ) H (t ) i (t ) B H B(t ) i (t ) In linear region of the magnetic material (iron) the magnetic filed is proportional to the winding current Due to spatial arrangement the three windings produce B-fields in three different directions Magnitudes shown here are for the positive peak of the corresponding phase b and c current Fundamentals of Power Systems Lecture 9 Page 2 4 Observations at t = 0 Baa ' BM sin(0) 0 Bbb ' BM sin( 120 ) 120 Bcc ' BM sin( 240 ) 240 electrical, time domain Bnet Baa ' Bbb ' 3 BM 120 2 3 BM 240 2 120 mechanical, spatial domain Bcc ' 3 BM 2 90 Lecture 9 Fundamentals of Power Systems 5 Observations at t = 90 Baa ' BM sin(90 ) Bbb ' BM sin( 30 ) 120 Bcc ' BM sin(210 ) 240 Bnet Baa ' Bbb ' BM 0 Bcc ' 1 BM 120 2 1 BM 240 2 3 BM 0 2 It appears that with evolving time a vector of constant magnitude Bnet = 1.5 BM rotates clockwise in the machine Fundamentals of Power Systems Lecture 9 Page 3 6 Rotating Magnetic Field - Proof Baa ' BM sin( t ) Bbb ' BM sin( t 120 ) 120 Bcc ' BM sin( t 240 ) 240 Bnet Baa ' Bbb ' y Bbb Baa x Bcc ' Bcc Decomposition of all three space vectors Bnn into their spatial x components and y components Baa ' Bbb ' Bcc ' Baa ' BM sin( t ) 0 Bbb ' 3 Bcc ' 1 B sin( t 120 ) 2M 1 B sin( t 240 ) 2M 3 2 BM sin( t 120 ) 2 BM sin( t 240 ) Lecture 9 Fundamentals of Power Systems 7 Rotating Magnetic Field - Proof 3 1 sin( t ) 2 sin( t 120 ) 2 sin( t 240 ) cos( t ) x components 2 cos( t ) y components Baa ' BM sin( t ) Bbb ' 1B 2M 1 sin( t ) 2 3 Bcc ' 1B 2M 1 sin( t ) 2 3 Baa ' Bnet ( x, y, t ) 3 1 sin( t ) 2 2 2 0 cos( t ) Bbb ' 3 cos( t ) Bcc ' 3 1.5BM sin( t )x 2 BM 1 sin( t ) 2 3 2 BM 1 sin( t ) 2 3 2 2 cos( t ) cos( t ) 1.5BM cos( t )y This is a vector of constant Bnet magnitude = 1.5 BM rotating clockwise with the angular frequency . In the chosen reference frame Bnet points in the y direction for t = 0. Fundamentals of Power Systems Lecture 9 Page 4 8 Rotating Magnetic Field - Equivalent The magnetic field BS generated by a set of symmetrical three-phase AC currents at angular power frequency e flowing in a set of circularly arranged (symmetrical) three-phase windings is equivalent to a permanent magnet rotating with the angular speed m = in a 2-pole machine e Winding aa closes within 180 mechanically Lecture 9 Fundamentals of Power Systems 9 Number of Poles >2 The rotating magnetic field BS in an P - pole machine rotates with an angular frequency of m = e/(P/2) Therefore, the mechanical speed of the rotating magnetic field can only be reduced by increasing the number of pole-pairs Often used in machine theory: Number of pole-pairs p = P/2 so the number of poles become P = 2p The mechanical speed is often given in rounds-per-minute (rpm) n 2 60 f e P Fundamentals of Power Systems 120 f e P Winding aa closes within 90 mechanically e e Lecture 9 Page 5 P 2 P 2 m , m fe P fm 2 Mechanical angle m = mt 10 Speed Reversal by Sequence Change Altering two phases (e.g. b and c) changes the sequence from abc to acb x components y components Baa ' BM sin( t ) Bbb ' 1B 2M 1 sin( t ) 2 3 Bcc ' 1B 2M 1 sin( t ) 2 3 Baa ' Bnet ( x, y, t ) cos( t ) Bbb ' 3 cos( t ) Bcc ' 3 2 2 0 1.5BM sin( t )x 2 BM 1 sin( t ) 2 3 2 BM 1 sin( t ) 2 3 2 2 cos( t ) cos( t ) 1.5BM cos( t )y This is a vector of constant magnitude Bnet = 1.5 BM rotating counterclockwise with the angular frequency . In the chosen reference frame Bnet points in the +y direction for t = 0. Lecture 9 Fundamentals of Power Systems 11 Generating Torque Forces on Conductors The rotating field BS of the stator interacts with currents flowing in the rotor BS is distributed sinusoidal around the circumference of the stator/rotor BS ( ) Direction of rotor current i defines vector l BS sin( ) Force and torque on a rotor conductor of length l carrying a current i F Find ind il B Direction of torques ind rF ilBS sin( ) 2rilBS sin( ), CCW Fundamentals of Power Systems Lecture 9 Page 6 12 Generating Torque Interacting B-fields Currents in the rotor generate their own magnetic field (C const) BR HR Ci Therefore, the torque ind 180 2rilBS sin( ) can be written as ind kBR BS sin( ) with the machine constant k 2rl C This is equivalent to ind kBR BS BS kBR Bnet Bnet BR Lecture 9 Fundamentals of Power Systems 13 Generating Torque Permanent Magnet Rotor Using only the rotor field BR, the total field Bnet, and the torque angle the torque becomes ind kBR Bnet sin( ) This observation is important for synchronous machines (permanent magnet and electromagnet) Fundamentals of Power Systems Lecture 9 Page 7 14
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