Power Magnet Devices: A Multi-Objective Design Approach Chapter 8: Distributed Windings and Rotating Electric Machinery S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Stator construction S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Position measurements S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Poles S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Electrical angles and positions S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Discrete description of distributed windings S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Developed diagram S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Continuous description of distributed windings S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Symmetry conditions S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Converting discrete to continuous S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Converting discrete to continuous S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Converting discrete to continuous S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Continuous to discrete S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings End conductors S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings End conductors S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Winding arrangements S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.1 Describing Distributed Windings Winding arrangements S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Winding functions S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Calculating the discrete winding function S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Continuous winding function S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Example 8.2A (part 1). Finding winding function for S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Example 8.2A (part 2). Find winding function for S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Continued… S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Continued some more … S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.2 Winding Functions Comparison of winding functions S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.3 Air Gap Magneto Motive Force Consider Ampere’s law on path below S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.3 Air Gap Magneto Motive Force MMF drop definitions S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.3 Air Gap Magneto Motive Force Defining stator, rotor, and total MMF source as It can be shown that S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.3 Air Gap Magneto Motive Force Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.3 Air Gap Magneto Motive Force Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.3 Air Gap Magneto Motive Force Derivation (continued some more) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.3 Air Gap Magneto Motive Force Derivation (and some more) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.3 Air Gap Magneto Motive Force A simpler result and airgap fields S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.4 Rotating MMF Suppose S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.4 Rotating MMF Then S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.4 Rotating MMF Now suppose The stator MMF is given by S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.4 Rotating MMF This reduces to Meaning S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.4 Rotating MMF Meaning (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.5 Flux Linkage and Inductance Leakage and magnetizing flux linkage Leakage and magnetizing inductance S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.5 Flux Linkage and Inductance Calculating magnetizing flux linkage Calculating magnetizing inductance S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.5 Flux Linkage and Inductance Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.5 Flux Linkage and Inductance Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.5 Flux Linkage and Inductance Derivation (still continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.6 Slot Effects and Carter’s Coefficient Carter’s coefficient specifies an effective airgap to account for slot effects For slotted stator For slotted stator and rotor S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.6 Slot Effects and Carter’s Coefficient Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.6 Slot Effects and Carter’s Coefficient Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.6 Slot Effects and Carter’s Coefficient Derivation (continued again) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.6 Slot Effects and Carter’s Coefficient Derivation (and again) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Leakage inductance matrix S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Slot leakage S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance End leakage S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Total leakage inductance S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Slot leakage paths 1-4 S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Derivation of Ps1,1 S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Derivation of Ps1,1 S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Slot leakage paths 5-7 S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Derivation of Psl,5 S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance Derivation of Psl,5 S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance End leakage permeance S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance End leakage permeance Pel,1 S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.7 Leakage Inductance End leakage permeance Pel,2 S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.8 Resistance Volume of conductor in slots: Volume of conductor in end-turns: Total volume: Total length: Resistance: S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Park’s transformation to the rotor reference frame S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Park’s transformation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Transformation of a balanced set S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Transformation of voltage equations. Suppose we have S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Transformation of voltage equations. In qd0 variables, we have S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Derivation (continued again) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Transformation of flux linkage equations. Suppose we have S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Transformation of flux linkage equations. In qd0 variables we obtain S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Partial derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Partial derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Transformation of power. In abc variables we have In qd0 variables power may be expressed S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.9 Introduction to Reference Frame Theory Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.10 Expressions for Torque Using an energy based approach it can be shown that S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.10 Expressions for Torque Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.10 Expressions for Torque Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.10 Expressions for Torque Derivation (continued some more) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.10 Expressions for Torque A field approach to calculating torque. Using field analysis, it can be shown that S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.10 Expressions for Torque Derivation S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.10 Expressions for Torque Derivation (continued) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8.10 Expressions for Torque Derivation (continued some more) S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach