Unified Torque Expressions of AC Machines. Qian Wu

Transcription

1 Unified Torque Expressions of AC Machines Qian Wu

2 Outline 1. Review of torque calculation methods. 2. Interaction between two magnetic fields. 3. Unified torque expression for AC machines. Permanent Magnet (PM) machine; Synchronous Reluctance Machine (SynRM); Induction Machine (IM); 4. Conclusion.

3 Torque Calculation Methods Numerical method with FEA. Maxwell stress tensor ( B n B t 2μ ) in the air gap region.

4 Torque Calculation Methods Based on the energy conversion theory i u Ri d dt Li v ui Ri 2 d i dt Power balance uidt Ri 2 dt di Energy balance d i Enclosed area = stored field energy i

5 Torque Calculation Methods Similar for an electrical machine u d R i dt qs s qs qs r, el ds For example, the q-axis, stator side winding analysis: P inq i qs u qs R s i 2 qs u ds s ds ds r, el qs i d R i dt qs d dt qs r, el ds i qs Copper loss Input power of the q-axis winding Rate change of the stored field energy Output mechanical power P mec, dq r, el ds i qs qs i ds

6 Torque Calculation Methods Another way to utilize the energy conversion theory Uniform air gap Sinusoidal MMF, B waveforms Similar MMF on the stator and rotor rotor stator F s1 Ni Ni

7 Torque Calculation Methods So finding the air gap average co-energy density air-gap total MMF (fundamental component) F 2 2 sr1 Fs1 Fr 1 2F F 1 cos s1 r sr rotor stator air-gap actual field intensity H ag, peak 2 F g sr1 Average co-energy density assuming sinusoidal air-gap field 2 2 H ag, peak 2 F 4 g sr1 Average of sin square function gives

8 Torque Calculation Methods So the torque is obtained as By accounting the volume of the air, the total co-energy W co, ave 4 F sr1 g 2 DLg By differentiation the co-energy, we obtain the torque

9 Torque expression for comparison 1. Some observations Classical torque equation may involve different inductances and e.g. PM flux linkage direct comparison is not so obvious We experience winding current excited magnetic field (MMF), permanent magnet field (PMSM) and salient rotor magnetic field modulation effects (sync. Reluctance motor). 2. An ideal torque expressions for AC machines. Applying the same principle. Intuitive understanding of torque production mechanism. Torque expressions with the same geometrical parameters and physical quantities.

10 Torque Calculation Methods So steps to take Turn all other magnetic field into winding current excited magnetic field (MMF) Using a uniform airgap

11 Magnet equivalent magnetic field Consider a permanent magnet Air gap B waveform or winding MMF waveform It is possible to replace the magnet with winding MMF for producing the same air gap B waveform

12 Magnet equivalent magnetic field For a magnet J and B have the same unit: [T] The relationship J B H r B B H pm

13 Magnet equivalent magnetic field Convenient expression J B H B B H r pm r J B H pm J M B M H M: moment (not a constant as well) M B r at H = J is not a constant; slightly decrease when H increases (Only is involved)

14 Magnet equivalent magnetic field The PM magnetic field N N S g h For example: h = 1 mm Br = 1.2 (T) Mh = 955 (A.turns) M B r B g N S Mh h g g h The MMF does not decrease when the magnet becomes narrower (neglecting the leakage flux) MMF Mh

15 Bn (T) Magnet equivalent magnetic field Magnet MMF vs. winding MMF.7 fundamental component of Bn slot-opening stator PM stator pole pair number

16 Magnetic Coupling (MC) 1. Torque production of magnetic coupling could be typically explained with the interaction between two magnetic fields. 2. Models: 4. Magnetic field from permanent magnets (fundamental component) Inner PMs: B n1_in = k B μ Outer PMs: B n1_out = k B μ Mh in g+h in +h out sin (Pθ ωt + β in ) Mh out g+h in +h out sin (Pθ ωt + β out ) M: magnetization intensity; k B : B n1 waveform factor considering the magnetizing direction of PMs.

17 Magnet equivalent magnetic field The PM magnetic field F = l a B n I N N S Current in the main air gap field N S N S Distributed current density layer

18 Torque Evaluation of MC 2. Equivalent current k(θ) to replace the PMs on out side of MC: satisfying k(θ) dθ = B n1 θ μ (g + h in + h out ) (ensuring identical MMF) yielding k θ = k B PMh out cos (Pθ ωt + β out ) (Peak value k m θ = k B PMh out ) Thus: Equivalent current is sinusoidally distributed along the inner surface of outer iron. Equivalent current is rotating at the same speed of PM in space. Peak value of equivalent current is determined by the MMF of the magnetic field source and the pole pair number.

19 Torque Evaluation of MC 1. Equivalent model: current is located in the air gap magnetic field. 2. Electromagnetic force endured on the current at the position of θ. f(θ) = L a B n1 (θ)k(θ) (along the tangential direction) 3. Torque of MC: T = 2π K m = L a πd 2 Rf θ dθ = K m B n1m F n1m sin (β r β s ) (geometrical parameters) B n1m = k B μ Mh in g+h in +h out (peak value of fundamental magnetic field ) F n1m = k B PMh out (peak value of fundamental MMF for P pole pairs ) f δ = sin δ = sin β r β s (relative position between rotor and stator magnetic field)

20 torque (Nm) torque (Nm) Torque performance of MC 1. Torque comparison 2. Effect of pole pair number on torque FE calculation fundamental component theoretical analysis electrical degree torque per pole relative position (electrical degree) P=1 P=2 P=3 P=4 P=5 P T max Observations: A good agreement between the results from theoretical analysis and FE calculation. Neglecting flux leakage between poles, the maximum torque value of MC is in a linear proportion to the pole number, which may be explained by torque expression of T = K m B n1m F n1m f(δ).

21 Torque analysis of PM machine 1. Equivalent model: Exactly the same with that of MC, thus the obtained torque expression may be applied for PM machine. 2. Specific expression of each term: Stator MMF: F s1_pole (θ) = m 2 Thus total MMF: 4 1 JZSf fill k w π 2 F s1m = PF s1m_pole m(2p) sin (Pθ ωt + β s ) Rotor magnetic field: B r1 (θ) = μ k B Mh g+h sin (Pθ ωt + β r) Torque expression: T = K m B r1m F s1m f δ = (L a πd 2 ) μ k B Mh g + h JZSf fill k w 2π sin (β r β s )

22 torque (Nm) torque (Nm) Torque analysis of PM machine 1. Torque expression validation 2. Effect of pole pair number on torque Observation: FE calculation fundamental component theoretical analysis eletrical degree torque of PM machines P=2 P= rotor postion (mechanical degree) Maximum torque value of PM is determined by the main dimension (L a and D), the rotor magnetic field production capability of per pole (Mhand g), and stator total current. When neglecting flux leakage between poles, the pole pair number has no effect on the total torque.

23 1. Model Torque analysis of SynRM Features: slot-opening stator with winding current; salient rotor without magnetic field source; non-uniform air gap; 2. Equivalent model Features: identical stator configuration with that of original model; uniform air gap (g); rotor is assigned with current loading. 3. Conditions: Keep the stator configuration unchanged, including structure and armature current. Keep the air gap magnetic field unchanged.

24 Torque analysis of SynRM The derived torque equation T = L a πd 2 μ F s1mpole 2g q (1 g q g d ) F s1m sin (2β) 2 x angle between rotor and stator magnetic field Stator total MMF (similar to PMSM) Equivalent rotor magnetic field B rn1m = μ F s1m_pole 2g q 1 g q g d

25 torque (Nm) Torque analysis of SynRM Torque comparison torque versus relative position between rotor and stator MMFs fundamental FE FE calculation theoretical analysis relative position (mechnical degree)

26 Torque analysis of IM Expressions of the terms in torque expression Rotor magnetic field Stator MMF Dimensional factor B r1m = μ F r1m g F s1m = PF s1m_pole K m = L a πd 2 Position function f δ = sin (β r β s ) Torque calculation T = K m B r1m F s1m f δ = L a πd 2 (μ F s1m_pole )F 2g s1m sin2δ Equivalent rotor magnetic field: B r1m = μ F s1m_pole 2g

27 Conclusion 1. A unified torque expression is obtained for different AC machines, applying the same principle. 2. By FEM, the accuracy of this torque equation is validated; 3. Under the condition of the same stator configurations, only the peak value of rotor magnetic fields need to be compared for the torque comparison of AC machines. PM machine: B r1m = μ k B Mh g+h IM: B r1m = μ F s1m_pole 2g SynRM: B rn1m = μ F s1m_pole 2g q 1 g q g d