EEN320 - Power Systems I ( ) Part 7: Introduction to rotating - PowerPoint PPT Presentation

EEN320 - Power Systems I ( Συστήματα Ισχύος Ι ) Part 7: Introduction to rotating machines Dr Petros Aristidou Department of Electrical Engineering, Computer Engineering & Informatics Last updated: April 6, 2020

Today’s learning objectives After this part of the lecture and additional reading, you should be able to . . . . . . explain the basic principles of electromechanical energy conversion; 1 . . . explain the fundamental principles of rotating machines. 2 , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 2/ 60

Outline Basic rotating machines principles 1 Machine stator and rotor 2 Power flows, efficiency and losses 3 Synchronous machine characteristics 4 Synchronous motor 5 , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 3/ 60

1 Outline Basic rotating machines principles 1 Machine stator and rotor 2 Power flows, efficiency and losses 3 Synchronous machine characteristics 4 Synchronous motor 5 , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 4/ 60

1 Basic rotating machine components Stator ( Στάτης ) Gap ( διάκενο ) Rotor ( δρομέας ) , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 5/ 60

1 Lorentz Force Law review From electromagnetics, we know that Lorentz Force Law: F = q ( E + v × B ) where: F is the force (newtons) on a particle of charge q (coulombs) in the presence of electric and magnetic fields E is the electric field in volts per meter B is the magnetic field in teslas v is the velocity of the particle q relative to the magnetic field, in meters per second. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 6/ 60

1 Lorentz Force Law review Ignoring the electric field: Fitzgerald, A. E., Kingsley, C., & Umans, S. D. (2003). Electric machinery. McGraw-Hill. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 7/ 60

1 Application of Lorentz Force Law on a rotor A non-magnetic rotor ( δρομέας ) containing a single-turn coil is placed in a uniform magnetic field of magnitude B 0 generated by the stator (see later), as shown below. The coil sides are at radius R and the wire carries current I . Find the θ -directed torque as a function of rotor position. Assume that the rotor is of length ℓ . Adapted from ”Fitzgerald, A. E., Kingsley, C., & Umans, S. D. (2003). Electric machinery. McGraw-Hill”. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 8/ 60

1 Application of Lorentz Force Law on a rotor The force per unit length (in N) acting on the wire is given by F = I × B N For wire of length ℓ and current I is given as: F = − IB 0 ℓ sin( α ) For two wires: F = − 2 IB 0 ℓ sin( α ) The total torque (in Nm) is then: T = − 2 IRB 0 ℓ sin( α ) And, if we assume a rotation α = ω t : T = − 2 IRB 0 ℓ sin( ω t ) , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 9/ 60

1 Application of Lorentz Force Law on a rotor Since there is a current flowing, the rotor also produces a magnetic field. The magnetic flux density B R generated by the rotor due to the current I is: B R = µ H R = µ I G where G depends on the geometry of the rotor loop. For a circular one then G = 2 R . For a rectangular one, G depends on the length-to-width ratio. So, we get: T = AG µ B R B 0 sin( α ) where A = 2 R ℓ is the area of the wire on the rotor if assumed rectangular. We can rewrite as: T = kB R B 0 sin( α ) = kB R × B 0 where k = AG /µ is a factor depending on the machine construction. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 10/ 60

1 Basic rotating machines principle Conceptual explanation: A magnetic north and south poles can be associated with the stator and 1 rotor of a machine due to the current flows; Similar to a compass needle trying to align with the earth’s magnetic 2 field, these two sets of fields attempt to align; If one of the fields (stator or rotor) rotates, the other ones tries to ”catch 3 up”. Torque is associated with their displacement from alignment: In a motor, the stator magnetic field rotates ahead of that of the rotor, ”pulling” on it and performing work In a generator, the rotor magnetic field rotates ahead of that of the stator, ”pulling” on it and performing work , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 11/ 60

1 Basic rotating machines principle The torque in a real machine, depends on: The strength of the rotor magnetic field; 1 The strength of the external (stator) magnetic field; 2 The sin of the angle between them; and, 3 A constant depending on the construction of the machine. 4 , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 12/ 60

2 Outline Basic rotating machines principles 1 Machine stator and rotor 2 Power flows, efficiency and losses 3 Synchronous machine characteristics 4 Synchronous motor 5 , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 13/ 60

2 Three-phase machine stator Assume now a three-phase stator ( στάτης ) with windings aa ′ , bb ′ , cc ′ as shown in the figure below: Chapman, S.J. (2005). Electric machinery fundamentals (4e). McGraw-Hill. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 14/ 60

2 Three-phase machine stator: magnetic field We feed the three coils with (in Ampere): I aa ′ ( t ) = I M sin( ω t ) I bb ′ ( t ) = I M sin( ω t − 120 ◦ ) I cc ′ ( t ) = I M sin( ω t − 240 ◦ ) Which generates magnetic field intensity (in Ampere-turns/m): H aa ′ ( t ) = H M sin( ω t ) 0 ◦ H bb ′ ( t ) = H M sin( ω t − 120 ◦ ) 120 ◦ H cc ′ ( t ) = H M sin( ω t − 240 ◦ ) 240 ◦ The direction of the field is shown on the figure and given by the ”right-hand rule”. The phase shown at the end is the spacial degrees. The magnitude changes sinusoidally but direction same. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 15/ 60

2 Three-phase machine stator: magnetic field The flux density is given by B = µ H (in Tesla): B aa ′ ( t ) = B M sin( ω t ) 0 ◦ B bb ′ ( t ) = B M sin( ω t − 120 ◦ ) 120 ◦ B cc ′ ( t ) = B M sin( ω t − 240 ◦ ) 240 ◦ where B M = µ H M . , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 16/ 60

2 Three-phase machine stator: magnetic field Examples: ω t = 0 ◦ B net = B aa ′ + B bb ′ + B cc ′ √ � √ � 3 � 3 � 120 ◦ + = 0 + 2 B M 2 B M 240 ◦ − = 1 . 5 B M − 90 ◦ ω t = 90 ◦ B net = B aa ′ + B bb ′ + B cc ′ � − 1 � � − 1 � = B M 0 ◦ + 120 ◦ + 2 B M 2 B M 240 ◦ = 1 . 5 B M 0 ◦ , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 17/ 60

2 Three-phase machine stator: magnetic field In the general case 1 : � � sin( ω t )ˆ x − cos( ω t )ˆ B net = 1 . 5 B M y How about changing the rotation of the field? → We swap the current in two of the phases: � � sin( ω t )ˆ x + cos( ω t )ˆ B net = 1 . 5 B M y 1 Try to prove this using the trigonometric relations used in part 2. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 18/ 60

2 Three-phase machine stator: two-pole This is equivalent to a two-pole The magnetic poles complete one full (north-south) field rotating: mechanical rotation for every one electrical cycle: f e = f m ω e = ω m Chapman, S.J. (2005). Electric machinery fundamentals (4e). McGraw-Hill. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 19/ 60

2 Three-phase machine stator: four-pole This is equivalent to two two-pole (north-south) field rotating: The magnetic poles complete one full mechanical rotation for every two electrical cycle: θ e = 2 θ m , f e = 2 f m , ω e = 2 ω m Chapman, S.J. (2005). Electric machinery fundamentals (4e). McGraw-Hill. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 20/ 60

2 Three-phase machine stator: P -pole In general for a P -poles machine: θ e = P 2 θ m (rad) f e = P 2 f m (Hz) ω e = P 2 ω m (rad/s) n = 120 f e (rounds per minute) P , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 21/ 60

2 Three-phase machine rotor type In general, there are two type of rotors: (a) cylindrical or nonsalient-pole ( κυλινδρικός δρομέας ) (b) salient-pole ( έκτυπους πόλους ). Chapman, S.J. (2005). Electric machinery fundamentals (4e). McGraw-Hill. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 22/ 60

2 Three-phase machine rotor type Salient-pole is characteristic of hydroelectric generators because hydraulic turbines operate at lower speeds → lower speeds requires higher number of poles → salient poles are better mechanically for large number of poles. Steam and gas turbines operate better at high speeds and are commonly two- or four-pole cylindrical-rotor. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 23/ 60

2 Three-phase machine rotor type Salient pole field windings create the Cylindrical rotor needs an uneven magnetic field. The construction of the distribution of the conductors to poles generates a sinusoidal field. generate a sinusoidal magnetic field. Fitzgerald, A. E., Kingsley, C., & Umans, S. D. (2003). Electric machinery. McGraw-Hill. , ΕΕΝ 320 — Dr Petros Aristidou — Last updated: April 6, 2020 24/ 60