^{1}

^{*}

^{1}

^{*}

Three speed controllers for an axial magnetic flux switched reluctance motor with only one stator, are described and experimentally tested. As it is known, when current pulses are imposed in their windings, high ripple torque is obtained. In order to reduce this ripple, a control strategy with modified current shapes is proposed. A workbench consisting of a machine prototype and the control system based on a microcontroller was built. These controllers were: a conventional PID, a fuzzy logic PID and a neural PID type. From experimental results, the effective reduction of the torque ripple was confirmed and the performance of the controllers was compared.

The radial magnetic flux version of switched reluctance motors (SRM) has been widely used in many variable speed industrial applications and some advantages have been reported: high torque output, wide range of operating speed, geometrical simplicity, reliability and robustness [

The switched reluctance motor (SRM) analyses are complex because of their doubly salient pole structure and nonlinear magnetic characteristics. The developed torque is a nonlinear function of the currents applied to the stator windings and their inductances, which depend on the rotor position. Nevertheless, with an appropriate control system, minimum ripple torque can be obtained.

From literature, it can be observed that many works are related to radial flux SRM; however few works are related to axial flux SRM [

For ordinary radial flux SRM, the mutual inductances of the stator windings are considered small [

When voltages or current pulses are imposed in the stator windings, the level of torque ripple is high, if compared with other kind of motors, which is the primary disadvantage of SRM as it contributes to the vibration and acoustic noise. For this reason, the imposition of different current shapes to reduce the torque ripple appears as an interesting solution. In this work, the procedure to obtain alternatives shapes is described and the effectiveness of this strategy is experimentally tested in an AFSRM prototype. Even more, in order to evaluate the performance of different kinds of speed controllers, three PID types were implemented: a conventional, a fuzzy logic and a neural based approach. Although in different levels the experimental results confirm the reduction of torque ripple, there are also differences with respect to the execution time and the speed response.

The prototype has six poles in the stator, corresponding to three phases (“a”, “b” and “c”), and four poles in the rotor. It is a 3-phase 6/4 poles AFSRM, shown in

The characteristics of the 3-phase 6/4 AFSRM prototype are shown in

Parameter | Value |
---|---|

Outer diameter of rotor and stator | 126 mm |

Inner diameter of rotor and stator poles | 63 mm |

Shaft diameter | 40 mm |

Air gap | 1.9 mm |

Stator pole width | 34 mm |

Rotor pole width | 26 mm |

Stator and rotor poles arc | 40˚ |

Poles radial length | 31.5 mm |

Stator yoke thickness | 5 mm ^{ } |

Stator pole area (axial cross-section) | 1039 mm^{2} |

Rotor yoke thickness | 17 mm |

Number of turns per stator pole | 175 |

Turn wire | 24 AWG |

Coil resistance | 2.3 W |

Stator and rotor cores material | Steel SAE-1020 |

Motor shaft material | Stainless steel |

Each stator pole has one coil, so there are six coils named “a1”, “a2”, “b1”, “b2”, “c1” and “c2”.

The stator and rotor cores were solid because the objective of this work was the torque ripple minimization and not the loss reduction.

Considering that the motor core is operating in the linear region, the net electromagnetic torque T_{e} for the three phase AFSRM prototype is found as:

where: L and M stand for the self and mutual inductances.

The rotor position angle q_{a}, for phase “a”, is considered as 45˚ when the rotor pole was completely overlapping with the stator pole, as depicted in

The q_{b} and q_{c} angles correspond to rotor angular positions for phases “b” and “c”, respectively. The relationships between these angles are:

· for q_{a} ≤ 30˚

· for 30˚ < q_{a} ≤ 60˚

· for 60˚ < q_{a} ≤ 90˚

The relationship among the three phases self and mutual inductances are: L_{a}_{2a2}(θ_{a}) = L_{a}_{1a1}(θ_{a}); L_{b}_{1b1}(θ_{b}) = L_{a}_{1a1}(θ_{b}); L_{b}_{2b2}(θ_{b}) = L_{b}_{1b1}(θ_{b}); L_{c}_{1c1}(θ_{c}) = L_{a}_{1a1}(θ_{c}); L_{c}_{2c2}(θ_{c}) = L_{c}_{1c1}(θ_{c}); M_{a}_{2a2}(θ_{a}) = M_{a}_{1a2}(θ_{a}); M_{a}_{2b1}(θ_{a}) = M_{a}_{1b2}(θ_{a}); M_{a}_{2b2}(θ_{a}) = M_{a}_{1b1}(θ_{a}); M_{a}_{2c1}(θ_{a}) = M_{a}_{1c2}(θ_{a}); M_{a}_{2c2}(θ_{a}) = M_{a}_{1c1}(θ_{a}), M_{b}_{1a1}(θ_{a}) = M_{a}_{1b1}(θ_{a}); M_{b}_{1a2}(θ_{a}) = M_{a}_{2b1}(θ_{a}); M_{b}_{1b2}(θ_{b}) = M_{a}_{1a2}(θ_{b}); M_{b}_{1c1}(θ_{b}) = M_{a}_{1b1}(θ_{b}); M_{b}_{1c2}(θ_{b}) = M_{a}_{1b2}(θ_{b}); M_{b}_{2a1}(θ_{a}) = M_{a}_{1b2}(θ_{a}); M_{b}_{2a2}(θ_{a}) = M_{a}_{2b2}(θ_{a}); M_{b}_{2b1}(θ_{b}) = M_{b}_{1b2}(θ_{b}); M_{b}_{2c1}(θ_{b}) = M_{b}_{1c2}(θ_{b}); M_{b}_{2c2}(θ_{b}) = M_{b}_{1c1}(θ_{b}); M_{c}_{1a1}(θ_{a}) = M_{a}_{1c1}(θ_{a}); M_{c}_{1a2}(θ_{a}) = M_{a}_{2c1}(θ_{a}); M_{b}_{2b1}(θ_{b}) = M_{b}_{1b2}(θ_{b}); M_{c}_{1b1}(θ_{b}) = M_{b}_{1c1}(θ_{b}); M_{c}_{1b2}(θ_{b}) = M_{b}_{2c1}(θ_{b}); M_{c}_{1c2}(θ_{c}) = M_{a}_{1a2}(θ_{c}); M_{c}_{2a1}(θ_{a}) = M_{a}_{1c2}(θ_{a}); M_{c}_{2a2}(θ_{a}) = M_{a}_{2c2}(θ_{a}); M_{c}_{2b1}(θ_{b}) = M_{b}_{1c2}(θ_{b}); M_{c}_{2b2}(θ_{b}) = M_{b}_{2c2}(θ_{b}); and M_{c}_{2c1}(θ_{c}) = M_{c}_{1c2}(θ_{c}).

The self and mutual inductances were estimated based on a simulation model using a three dimensional finite element method of the “ANSYS Multiphysics” software [

In _{a}_{1a1} and the mutual inductances are named M_{a}_{1a2}, M_{a}_{1b1}, M_{a}_{1b2}, M_{a}_{1c1} and M_{a}_{1c2}, respectively.

As shown in

From _{a}_{1b1}/L_{a}_{1a1}) is around 0.25 near 20˚.

In this way, in order to provide a constant electromagnetic torque for all rotor positions, appropriated reference currents should be designed. From previous tests in open loop [

· phases “a” and “b” from rotor position 0˚ to 15˚;

· only phase “a” from rotor position 15˚ to 30˚;

· phases “a” and “c” from rotor position 30˚ to 45˚;

· only phase “c” from rotor position 45˚ to 60˚;

· phases “b” and “c” from rotor position 60˚ to 75˚;

· only phase “b” from rotor position 75˚ to 90˚.

This cycle repeats each 90˚.

On the other hand, to define a reference torque without harmonics, it was considered as that generated when a current of 3A is imposed in phase “a”, while the rotor position was 30˚. In this situation a net electromagnetic torque of 0.126 Nm was calculated.

Next, for every one degree, the current values of the phase “a” were calculated as:

· from rotor position 15˚ to 30˚

· from rotor position 0˚ to 15˚ a function that starts in zero and it then assumes the value calculated with (17) in the rotor position of 15˚, was used

· from rotor position 30˚ to 45˚ the I_{a} current values are calculated from quadratic Equation (19) considering that I_{c} assumes the values of I_{a} currents obtained from (18). For example, the I_{c} value for 35˚ is equal to the I_{a} value for 5˚

The reference currents of phases “c” and “b” are shifted by 30˚ and 60˚ relative to phase current “a”.

All these reference currents are shown in

In order to evaluate the performance of different kinds of speed controllers, three PID types were implemented: a conventional, a fuzzy logic and a neural based approach.

In

In this case, the PID gains were adjusted by simulations using MatLab^{®} Simulink software, resulting in gains proportional, integral and derivative, of 16, 3 and 1, respectively.

The controller output was limited to the interval of [0, 1].

In

In the case of the fuzzy controller, the rules were based on the observation of the simulation results of the PID controller that use the modified current shapes. The inputs of the fuzzy logic controller were speed error, derivative of speed error, integral of speed error and angular rotor position. The output variables are the three phase’s reference currents. In the actual implementation, this strategy was translated to a table.

The fuzzy logic characteristics used in MatLab^{®} Simulink simulations are: minimum “And” Method, maximum “Or” Method, minimum “Implication”, maximum “Aggregation” and centroid “Defuzzification”.

The input variable speed error (“E”) has three linguistic values: “negative”, “zero” and “positive”. The corresponding membership functions are:

· trapezoidal type

· triangular type

· trapezoidal type

The input variable derivative of speed error (“CE”) has three linguistic values: “negative”, “zero” and “positive”. The corresponding membership functions are:

· triangular type

· triangular type

· trapezoidal type

The input variable integral of speed error (“IE”) has three linguistic values: “negative”, “zero” and “positive”. The corresponding membership functions are:

· triangular type

· triangular type

· triangular type

The input variable angular rotor position (“Pos”) has ten linguistic values: “P1”, “P2”, “P3”, “P4”, “P5”, “P6”, “P7”, “P8”, “P9” and “P10”, whose corresponding membership functions are shown in

The output variables are the three phase’s reference currents. They have four linguistic values: “zero” (“Z”), “low” (“L”), “medium” (“M”) and “high” (“H”). The corresponding membership functions for phase “a” current are shown in

Take into account the number of membership functions, there are 270 (3 × 3 × 3 × 10) fuzzy rules. Its structure is similar to: if (E is “P”) and (CE is “P”) and (IE is “N”) and (Pos is “P5”) then (I_{a} is “M”) (I_{b} is “Z”) (I_{c} is “H”).

For the neural PID controller design, the usual procedure of off-line training was adopted. This was based in the data, with sampling frequency of 1 kHz, from the simulation results of the PID controller that use the modified current shapes.

The simulation with MatLab^{®} Simulink used 12,000 input data values (speed error, derivative of speed error, integral of speed error and angular rotor position) distributed as 6000 for training, 3000 for test and 3000 for validation. On the other hand, 12,000 output data values (currents in three phases) were used, obtained with the simulation of the PID conventional controller.

For neural network training the MatLab^{®} software was used (“nntool”). The neural network properties were: feed-forward back propagation network type, network training function that updates weight and bias values according to Levenberg-Marquardt optimization (TRAINLM), gradient descent with momentum weight and bias adaption learning function (LEARNGDM), mean squared error performance function (MSE), two layers: layer 1 using 36 neurons with hyperbolic tangent sigmoid transfer function (TANSIG) and layer 2 using 3 neurons with linear transfer function (PURELIN).

The training generated weight and bias values of the two layers that were used in the neural PID controller to calculate the references currents of the simulation. In the actual implementation, a correspondent table was prepared.

The hardware to control the AFSRM consists of a controller circuit, three power converter circuits, one for each phase, a position sensor circuit, three Hall Effect current measurement circuits, also one for each phase, a torque meter connected between the AFSRM shaft and the mechanical load shaft. The shaft of a DC machine was used as an inertial load. These components are arranged as shown in the block diagram of

The experimental hardware is shown in

The position sensor consists of an infrared optical circuit and an aluminum dish with 180 holes, fixed to the rotor structure. The signal from the sensor is transformed into a rectangular pulse stream with amplitude of 5 V,

which is sent to the controller circuit.

The actual phase currents are measured using commercial Hall Effect circuits. Their gain and zero adjustments were done in order to obtain 1 V for 1 A of phase current.

The power circuits consist of three asymmetric bridge converters and auxiliary circuits. In

As illustrated, the actual measured current is compared with the reference current coming from the controller circuit. If the real current is greater than the reference, the power circuit switches off the Mosfets of this phase in order to decrease the current in the coils.

The controller circuit consists of one microcontroller PIC 18F4680, programmed using the “C” language, that receives the rectangular pulses from the position sensor and calculates the reference currents of the three phases

in accordance with the controller type used, sending them to the power converter. Due to the 180 holes in the aluminum dish, the controller imposed the reference currents every 2˚.

In order to evaluate the torque ripple minimization strategy, four experiments were performed:

· current pulses were imposed and a conventional PID speed controller was used;

· proposed current waveforms, shown in

· a fuzzy logic PID speed controller was used;

· a neural PID speed controller was used.

In this case the current pulses were applied in the following sequence:

· phase “a” from rotor position 0˚ to 30˚;

· phase “c” from rotor position 30˚ to 60˚;

· phase “b” from rotor position 60˚ to 90˚.

The

The

In this case, the torque ripple in steady state was around 17.1% of the average torque.

The current waveform proposed as reference and the actual current in phase “a” are shown in

Note that the actual and reference currents are practically overlapping.

The

In this case, the torque ripple in steady state was around 2.1% of the average torque.

The actual currents in two phases of the AFSRM are shown in

In this case, the torque ripple in steady state was around 6.5% of the average torque.

The actual currents in two phases of the AFSRM are shown in

The

In this case, the torque ripple in steady state was around 2.0% of the average torque.

In this work, particular reference waveforms for the currents of an ASFRM in a closed loop speed control

system have been proposed and experimentally tested. Comparing the performance of a conventional PID speed controller using current pulses with a conventional PID speed controller, using the current waveforms proposed, it is confirmed that the level of torque ripple is reduced.

All the controllers were adjusted through simulations, using a motor model that is composed by self and mutual inductances obtained through 3D FEM simulations.

Comparing the performance of a conventional PID speed controller using the proposed current waveforms with a fuzzy logic PID speed controller, it was noted that the torque ripple for the first one is lower, which can be explained because the currents obtained with fuzzy logic controller are not exactly similar to those that would produce constant torque.

Nevertheless, in respect of the speed response, the fuzzy logic PID controller reaches the steady state in less time than the conventional PID controller. Even more, with respect to the implementation, the execution time of the fuzzy logic PID is lower than that used with the conventional PID controller but similar to that spent by the neural PID controller.

Finally, with respect to the neural PID controller, it notes similar torque ripple to the conventional PID controller but lower than that obtained using the fuzzy logic approach, which is expected as this was obtained from training based on the conventional PID controller results. However, this has a cost, which is the time spent in off-line training.