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Wound-field Doubly Salient Generator (WFDSG) has the outstanding advantages of simple structure, high reliability and controllable excitation. However, there exists obvious torque ripple in generator’s input torque, which has restricted its applications. Based on the concept of electromagnetic torque of WFDSG, the nonlinear formula under no load condition has been derived. Furthermore, it is proved to be reasonable theoretically in comparison with linear and approximate nonlinear formulas. Based on the 12/8 salient prototype, the correctness of nonlinear formula has been verified by comparing the results of virtual displacement method with the results of finite element analysis. At last, the simulation results have been validated by experiments.

WFDSG is a new type of brushless electrical machine, whose structure and manufacturing process are simple with high reliability, low cost. Thus, WFDSG has an excellent potential in generator of automotive and airplane [

At present, the study of torque ripple is mainly focus on Doubly Salient Permanent-magnet Generator, but rarely on WFDSG [_{em} for simplification) is the nonlinear function of rotor position, exciting current and winding current. Paper [

To simplify the analysis, a 12/8 salient prototype in lab will be studied in this paper. The lamination and major dimension of the WFDSG are shown in

The three-phase armature current is zero with no load. There is exciting current i_{f} if it is excited by voltage source. From paper [_{em} of the three-phase WFDSG can be given as

where θ refers to rotor position angle, W’ (i_{f}, θ) refers to magnetism co-energy, which is the function of i_{f} and θ. If the rotor has a certain value of position angle θ_{n}, the exciting current has the value of if_{n}. In this way, Equation (1) can be written as a differential operator

As long as the difference of exciting current if_{n} at θ_{n+}_{1} and θ_{n} is a constant, the ratio of the different magnetism co-energy ΔW’(if_{n}, θ_{n}) (ΔW’ for simplification) and Δθ is the T_{em}. This method can be called virtual displacement method. In this way, the area of ΔW’ determines the accuracy of the different methods.

Parameter name | Parameter value | Parameter name | Parameter value |
---|---|---|---|

Number of stator teeth | 12 | Number of rotor teeth | 8 |

Stator pole width | 14.6 mm | Rotor pole width | 14.6 mm |

Stator pole height | 37.1 mm | Rotor pole height | 19 mm |

Stator inner diameter | 222.9 mm | Rotor inner diameter | 144.2 mm |

Stator outer diameter | 330 mm | Rotor outer diameter | 222.2 mm |

Core length | 120 mm | Length of air gap | 0.35 mm |

Change the Equation (1) into the integration of flux linkage

where Ψ_{f} refers to self-inductance flux linkage in the exciting coils. It’s a function of i_{f} and θ. Replacing Equation (1) with Equation (3) and the T_{em} nonlinear equation can be written as

It can be found from Equation (3) and Equation (4) that the corresponding ΔW’ at certain position angle θ_{n} is

The area of ΔW’ is shown as dash area in _{n}, magnetic energy operation position is point a (if_{n}, Ψf_{n}), at the position angle θ_{n+}_{1}, magnetic energy operation position is point b (if_{n}, Ψf_{n+}_{1}). The area enveloped by oab is ΔW’, whose independent variable is if_{n}, integrating by Ψf_{n+}_{1} − Ψf_{n}.

Paper [_{em} with no load:

Meanwhile, at the rotor position angle θ_{n}, the formula of T_{em} can be written as differential operator:

Combined with Equation (2) and Equation (7), the ΔW’ can be written as:

The area enveloped by ΔW’ is shown in

The magnetic energy operation positions of approximate nonlinear formula and nonlinear formula are same: at the position θ_{n}, corresponds with point a (if_{n}, Ψf_{n}), namely (if_{n}, if_{n}Lf_{n}). At the position θ_{n+}_{1}, corresponds with point b (if_{n}, Ψf_{n+}_{1}), namely (if_{n}, if_{n}Lf_{n+}_{1}). The only difference is approximate nonlinear formula has the area of ΔW’ enveloped by Δoab, not oab. From the graph it can be found that as long as Δoab is small enough, the area of Δoab can be equal to oab approximately. Meanwhile, it can be found that the saturation gets deeper with increasing of if, the difference between Δoab and oab gets larger.

Paper [_{em} formula with no load.

Referring to the method introduced in 2.2, the linear formula of ΔW’ is written as:

The magnetic energy operation position of this method is different from those of methods mentioned before. The position θ_{n} corresponds the point a (if_{n}, Ψf_{n}), namely (if_{n}, if_{n}Lf_{n}). The position θ_{n+}_{1} corresponds the point b (if_{n+1}, if_{n+}_{1} Lf_{n+}_{1}). _{n} < if_{n+}_{1} while _{n} > if_{n+}_{1}.

In Equation (10), if_{n} Lf_{n} refers to ordinate of the magnetic energy operation position a while if_{n} Lf_{n+}_{1} doesn’t refers to the magnetic energy operation position b. The magnetic energy operation position should be the point c, which is the cross point of ob and if = if_{n}. Thus, the area of Δoac should be ΔW’. It is obvious that the area of Δ oac is different from that of oab. It is same as the situation when if_{n} > if_{n+}_{1}.

All the mentioned above is based on nonlinear magnetic circuit. As for linear magnetic circuit, it’s easy to get the same results with the nonlinear, approximate nonlinear and linear formulas. _{em}.

In order to get the analytical solution, it is necessary to get the area of ΔW’ in _{n} can be divided into m parts, if_{n}, if_{n}_{2}, if_{n}_{3}…if_{nm}. Each parts equals to if_{n}/m. All these points divides the curve into m parts, S_{1}, S_{2}, S_{3}…S_{m}, as shown in

If every part of S can be calculated, the summation of all these parts is ΔW’. When the current is if_{n}, the corresponding flux linkage of θ_{n+}_{1} and θ_{n} are Ψf_{(n+1)1} and Ψf_{n1}. If m is large enough, the area of S_{1} can be replaced with (if_{n} − if_{n2}) (Ψf_{(n+1)1} − Ψf_{n1}), namely (if_{n}/m) (Ψf_{(n+1)1} − Ψf_{n1}). Thus,

Replaced with Equation (2), the analytical solution of T_{em} is given as

To get the solution of Equation (12), it is necessary to get the information of every rotor position and corres-

Formula name | Formula | Applications |
---|---|---|

Linear | Linear magnetic condition | |

Approximate nonlinear | Linear magnetic condition and the saturation of magnetic circuit is not deep condition | |

Nonlinear | Both linear magnetic and nonlinear magnetic conditions |

ponding Ψ_{f} with every current point. The combined simulation of Maxwell and Simplorer is adopted. The simulation condition is shown as follows: motor operates with no load, voltage exciting with 20 V (internal resistor 2 Ω, exciting current 10 A), rotate speed is 60 rpm, 10 uH inductance measured from prototype. Simulation cycle is 4 with 120 sampling points in each cycle. Δθ = 45˚/120 = 0.375˚ since the mechanical angle is 45˚ in each cycle.

In order to make sure that the exciting current changes from 0 to if_{n} evenly at certain position θ_{n}, it is necessary to input the rotate speed and exciting current according to the table. The Data_If in

shows the rotate speed impact signal. Only when exciting current changes to the end value that the rotate speed gives out a value, which makes the rotor changes Δθ. At other time, the rotate speed is zero.

After the simulation, the relationship between T_{em} and rotor position angle can be derived from the calculation results by using Equation (12).

From the result in

The 12/8 salient prototype WFDSG in lab is used to test the torque. The generator is controlled with Siemens servo controller, 1FT 6 Permanent Magnet Synchronous Motor (PMSM) and NI test control system. The torque of WFDSG is measured with HBM sensor. This sensor can get the torque waveform signal under 200 Hz, which is suitable in the 24 Hz situation. The experiment platform is shown in

(1) The test torque signal also matches the simulation at the frequency of 24 Hz (3pn/60).

(2) The zero point of T_{em} matches the cross-zero of the phase voltage (point a corresponds point b in

(3) The max point in T_{em} correspond the positive natural commutation point in phase voltage (point c corresponds point d in

(4) The min point in T_{em} correspond the negative natural commutation point in phase voltage (point e corresponds point f in

However, there is still some difference between the simulation and test results, as shown in _{em} is zero while the test result is −1.61 Nm. Besides, the waveform of the simulation is much smooth than that of test result, where there is a lot of high frequency harmonic.

Name | Maximum | Minimum | Peak to Peak | Average |
---|---|---|---|---|

Experiment torque (Nm) | 16.20 | −17.10 | 33.30 | −1.61 |

Simulation torque (Nm) | 17.89 | −18.35 | 36.24 | −0.08 |

Experiment phase voltage (V) | 1.20 | −1.20 | 2.40 | 0.02 |

Simulation phase voltage (V) | 1.12 | −1.12 | 2.24 | 0.00 |

Based on the simulation and test result, the difference is mainly due to the following items:

(1) Theoretical error

The torque of simulation is T_{em}. However, in the experiment, the torque is input torque, which is not only the T_{em} but also contains other kinds of torque:

where Jdω/dt refers to the flywheel torque, J refers to the rotary inertia, ω refers to the rotor angular speed, contributing to the high frequency harmonic in every cycle. Bω refers to the friction torque, B is equivalent to the mechanical friction of motor bearings, rotor and air friction and heat loss, together with the friction coefficient. If the rotate speed of the motor is constant, then the average value of Bω is a constant too. That’s why the average torque in the test is negative.

(2) Model error

The FEM model in the simulation can’t simulate the real condition totally. Only based on the following conditions then FEM can be applied.

1) Punching isotropic core material, and the magnetization curve is a single value, namely ignoring hysteresis.

2) External motor magnetic field is negligible.

3) Using Cartesian coordinates, which is a two-dimensional field analysis.

(1) This paper compares the theoretical difference in nonlinear formula, approximate nonlinear formula and linear formula, emphasizing the correctness of nonlinear formula.

(2) Get the analytical solution of T_{em} nonlinear formula by adopting virtual displacement method. Verify the correctness by the combined simulation of Maxwell and Simplorer.

(3) The simulation result and test result match each other well by WFDSG torque prototype under no load condition. Explain the difference between simulation and test.

(4) Get the analytical T_{em} solution of WFDSG under no load condition, which offers a base for further analysis on T_{em} analytical solution with load.

Fan Yang,Jin Ma,Haoji Xu, (2015) Study on Torque Ripple of Wound-Field Doubly Salient Generator with No Load. World Journal of Engineering and Technology,03,111-118. doi: 10.4236/wjet.2015.33C017