Doubly salient reluctance machine or, as it is also called, switched reluctance machine. [Pyrhönen et al 2008]

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1 Doubly salient reluctance machine or, as it is also called, switched reluctance machine [Pyrhönen et al 2008]

2 Pros and contras of a switched reluctance machine Advantages Simple robust rotor with a small mass of inertia Simple stator winding with short coil ends => easy manufacturing, small restive losses and easy cooling Good torque to volume ratio particularly at low speeds Weak electromagnetic coupling between the phases => fault tolerant Disadvantages Torque ripple is large particularly at low speeds Requires a small air gap to produce a large torque => problems in manufacturing Torque ripple and/or small airgap cause noise and vibration problems No conventional calculations rutines => design relies on electromagnetic FEA

3 Energy analysis of an SR motor y D C W = y 0 idy B A Let us consider a linearized SR motor supplied by a rectangular current pulse. The current pulse is switched on when the rotor pole is half a pole pitch from the position of the stator pole (see the first slide). The current and flux increase from zero O to point A. The rotor pole turns so that it is along the stator pole while the current is kept constant, line AB. O i The current pulse is switched off and current and flux linkage go to zero along line BO. The energy supplied from the stator coil to the rotor is the area of triangle OAB.

4 Number of phases A three-phase machine does not produce a good torque waveform. Four- or five-phase machines are better. [Pyrhönen et al 2008]

5 Torque of a switched reluctance motor One of the basic assumptions of the two-axis model is that the flux density is sinusoidally distributed in the air gap. This is not a good approximation for a switched reluctance machine. The torque can be calculated based on the principle of virtual work.

6 Torque using the principle of virtual work Energy balance of a circuit model of an electromechanical system dwe = iedt = dwf + dwm Using flux linkage y and electromagnetic torque T idy = dw f + Tdg where e and i are the electromotive force and current of the circuit, W f is the energy of the electromagnetic field, W m is the mechanical work done by the system and dg is virtual angular displacement of the rotor.

7 Principle of virtual work Taking the current i and angle g as free variables, we can write y = y g ( g) ( i, ) Wf = Wf i, After some mathematical analysis, we come to the equation for the torque T = W ( i g) c, j where W c is the co-energy of the system. c (, g) = y (, g) - (, g) W i i i W i f

8 Principle of virtual work II Another alternative is to use the flux linkage as a free variable instead of the current. This leads to a torque equation T( y, g) =- W ( y g) f, g For the details of deriving these equations, see for instance, [Krause & Wasynczuk 1989].

9 Measured current versus flux linkage curves [Pyrhönen et al 2008] Magnetic saturation of the poles has a strong effect on the operation and characteristics of a switched reluctance machine. Reluctance difference between the rotor positions g = 0 and g = 90 should be as large as possible to get a large torque. Rotation angle g is used as a parameter

10 Torque of a switched reluctance machine A test of the virtual-work principle using finite element analysis. Rotor in two slightly different positions. Rotor in position A g = 30 degrees Rotor in position B g = 30.5 degrees

11 Flux linkage versus current from FEA model Flux linkage [Wb] Current [A] Position A Position B Energy of the system Wf = H db dv V Ł 0 ł = = V y 0 Ł B A 0 idy J da dv ł Numerical integration from the figure on the left.

12 Energy and coenergy from current and flux linkage Numerical integration using the trapezoid rule W f Energy y = idy Wc = ydi = iy - idy i 0 0 Coenergy y Energy [J] Coenergy [J] Current [A] Current [A] Position A Position B Position A Position B

13 Starting torque Torque from numerical derivation of co-energy with respect to the rotor position angle T» W ca g -W -g cb A B i= const. Torque [Nm] g A g B = 0.5 degrees Current [A]

14 Comparison with the torque from Maxwell s stress Torque [Nm] Current [A] Maxwell's stress; pos. A Virtual work Maxwell's stress; pos. B

15 Flux-switching machines Both the three-phase armature winding and permanent magnets are in the stator. The rotor is like in a switched reluctance machine. The outmost ring in the figure is there for allowing to model the PM leakage flux in the FE analysis. In practice, it would be a non-ferromagnetic frame. There are many variations of the stator structure. The one shown here is a so called E-core type stator.

16 Flux-switching machine with an E-core type stator Stator coils around the PM poles Circumferentially magnetized NdFeB magnets Rotor as in a switched reluctance machine

17 Axial-flux flux-switching machine The flux-switching principle can also be used in an axial flux machine. Xiping Liu et al 2012

18 Pros and contras of a flux-switching machine Advantages Simple robust rotor with a small mass of inertia Simple stator winding with short coil ends => easy winding to manufacture, small restive losses and easy cooling Good torque to volume ratio Disadvantages No conventional calculations rutines => design relies on electromagnetic FEA Large losses in the rotor core and permanent magnets because of flux variation at a relatively high frequency [Pang et al 2008]

19 Flux produced by the magnets at two rotor positions Coil A The motion of the rotor changes the flux linkages of the stator coils. In the first figure, the flux through coil A is relatively large. In the second one, it is almost zero.

20 No-load voltages of the windings When the rotor of the E-core machine is rotated at 1500 rpm, a 250 Hz voltage is induced in the winding. If we count the number of pole pairs from the rotor to be 5, the produced frequency is twice the frequency that a conventional 10-pole synchronous machine would produce. Putting this in another way, the slip of the rotor, if calculated in the conventional way, is equal to 0.5. There are significant harmonics in the voltages and the torque ripple is very large. The main dimensions of this machine are the same as those of the machines compared in Lecture 10, thus the rated torque should be around 250 Nm.

21 Operating characteristics of the E-core machine Stator current [A] Displacement factor Resistive stator loss [W] Shaft power [kw] Shaft power [kw] Shaft power [kw] PM loss [W] Stator loss [W] Rotor loss [W] Shaft power [kw] Shaft power [kw] Shaft power [kw]

22 Loss problem When simulating the results of the previous slide, the machine was supplied from a 250 Hz three-phase PWM voltage source. The rotor rotated at 1500 rpm. The displacement factor is good and resistance of the winding small. This means small resistive losses in the winding. The losses of the permanent magnets are rather large. They could be reduced by dividing the magnet into still smaller parts insulated from each other. The main problem is the core loss, particularly, in the rotor. To reduce it, 0.2 mm high-frequency electrical steel was used. This material has about the smallest losses one can find from the markets but it does not seem to be good enough material for this application. In the flux-switching principle, one switches a large flux density at a relatively high frequency and causes too large losses in the core of the machine.

23 References Pyrhönen J., Jokinen T., Hrabovcova V., Design of rotating electrical machines, John Wiley & sons 2008, 512 p. Krause P.C., Wasynczuk O., Electromechanical Motion Devices. McGraw-Hill Book Co., New York 1989, 423 p. ISBN S E Rauch, and L J Johnson, Design principles of the flux-switch alternators, AlEE Trans., 74llI, p , Xiping Liu, Chen Wang, Aihua Zheng, Operation principle and topology structures of axial fluxswitching hybrid excitation synchronous machine. Journal of International Conference on Electrical Machines and Systems Vol. 1, No.3, pp.312 ~319, Pang, Y.; Zhu, Z.Q.; Howe, D.; Iwasaki, S.; Deodhar, R.; Pride, A., Investigation of iron loss in flux-switching PM machines, 4th IET Conference on Power Electronics, Machines and Drives, pp