# Analysis and Modeling of Linear-Switched Reluctance for Medical Application

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^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Concept of the LVAD

#### 2.1. Presentation of the Actuator

#### 2.2. Kinetic Characteristics of the Motor

## 3. Dimensional Characteristics of the LVAD

#### 3.1. Prototype Dimensions and Characteristics

_{sy}, stator tooth length, h

_{st}, mover tooth length, h

_{mt}, mover yoke thickness, e

_{my}, external radius, R

_{ext}, air gap radius, R

_{g}, and valve radius, R

_{v}. The axial dimensions (x-axis) are: tooth width, a, and slot width, b.

_{s}and the tooth pitch of the mover, λ

_{m}, to be equal, the tooth width, a, and the slot width, b, to be equal, too:

_{sT}, is given by:

_{mT}, depends on the desired stroke, L, and is given by:

_{xT}is given by:

_{g}, is high. Moreover, increasing the mmf, Ni leads to an increase of the thrust, but also a proportional increase in copper loss (square increase).

Name | Abbreviation | Value (mm) |
---|---|---|

Valve radius | R_{v} | 12.5 |

Mover yoke thickness | e_{my} | 2 |

Mover tooth length | h_{mt} | 1.5 |

Air gap length | g | 0.2 |

Air gap radius | R_{g} | 16.1 |

Stator tooth length | h_{st} | 9 |

Stator yoke thickness | e_{sy} | 1.5 |

External radius | R_{ext} | 26.7 |

Tooth width | a | 2.9 |

Slot width | b | 2.9 |

Non-magnetic ring thickness | c | 1.45 |

Total length of the stator | L_{sT} | 80 |

Total length of the mover | L_{mT} | 140 |

Name | Abbreviation | Value |
---|---|---|

Number of phases | m | 4 |

Number of coils per phase | n | 2 |

Number of turns per slot | N | 155 |

Turn diameter | Ф_{turn} | 0.335 mm |

Slot area | A_{slot} = bh_{st} | 26.1 mm² |

Coil area | A_{coil} = NπФ_{turn}^{2}/4 | 13.66 mm² |

Slot fill factor | k_{fill} = A_{coil}/A_{slot} | 52.30% |

Characteristic | Value |
---|---|

Mass of the mover | m = 270.8 g |

Static dry friction force | f_{s} = 1.75 N |

Winding resistance (average value) | R = 8.5 Ω |

Unaligned Position inductance (average value) | L_{u} = 34.1 mH |

Aligned Position inductance (average value) | L_{a} = 44.6 mH |

Winding inductance (average value) | L = 39.4 mH |

Rate of change of inductance (average value) | ∆L/∆x = 10/2.9 |

#### 3.2. Input-Output Characteristics

_{out}, is given as the product of the kinetic thrust, F

_{xT}, and average speed, , as:

_{J}are the Joule losses.

_{t}, and its cross section, A

_{t}:

_{g}dimension in the previous expression):

_{sy}, compared to the external radius, R

_{ext}, the optimum ratio is 0.4 between the air gap radius, R

_{g}, and the external radius, R

_{ext}. For our prototype, the external radius, R

_{ext}, according to Equation (17), will be larger than 31 mm, considering the radius valve value, R

_{v}, which is equal to 12.5 mm. In order to limit the sheer size of the actuator (external radius inferior to 30 mm) and the mass of the mover, we have chosen the dimensions given in Table 1.

## 4. Control Principle

#### 4.1. Open Loop Control of the Motor

#### 4.1.1. Presentation

_{e}= 100 µs. The real position of the mover, called x

_{m}, measured by the sensor position, will be compared to the reference sinusoidal signal, x, of Equation (18). Figure 5 shows the open loop control of the motor. Concerning the use of Matlab Simulink and dSPACE tools for the control of SRM, we can cite the works [11,12] for variable speed pumping applications.

_{1}to U

_{4}, to the motor. In the “Sensor position” block, the DS2202 card is used to retrieve the real position of the mover, x

_{m}, measured by the Mitutoyo sensor position (analog to digital conversion). The other blocks of Figure 5 (presented in next section) are the control blocks, which are located in the PC (in the form of Matlab Simulink blocks).

#### 4.1.2. Determination of the Control Blocks

#### 4.1.2.1. “Position to Speed” Block

#### 4.1.2.2. “Motor Simplified Electric Motor” Block

_{1}to i

_{4}, as input and generates the duty cycles, k

_{1}to k

_{4}. The content of this block is based on Equation (19) for phase number k:

#### 4.1.2.3. “Current Signal Generator” Block

_{1}to i

_{4}, is performed by the “Current signal generator” block. This block is an “S-function” of Matlab Simulink. The inputs are the desired total force, F, and the reference signal position, x. Figure 6 illustrates the current supply sequence required for a reference signal position.

_{1}, i

_{2}, i

_{3}, i

_{4}allowing movement of the mover in the direction of increasing x. On the contrary, when the reference position decreases (negative speed), the current supply sequence generated is in the order i

_{4}, i

_{3}, i

_{2}, i

_{1}, allowing movement of the mover in the direction of decreasing x.

#### 4.1.2.4. “Speed to Force” Block

_{vF}, in order to simplify the control. Thus, the speed, v, and the desired total force, F, have the same evolution, which explains why the current waveform shown in Figure 6 is sinusoidal. A preliminary test was performed in order to determine the gain, K

_{vF}. During this experiment, one phase of the motor was supplied by a square wave signal of 1 A, and the position of the mover was recorded (cf. Figure 7).

_{vF}= 0.05, where v is expressed in mm/s and F in N.

#### 4.1.3. Modeling of the Motor with Matlab Simulink

#### 4.1.3.1. Principle of Modeling

_{1}

^{*}to i

_{4}

^{*}, thanks to the “LSRM Electrical model” block. Then, from the mechanical equation, we construct the estimate of the measured position, x

_{m}

^{*}, thanks to the “LSRM Mechanical model” block.

#### 4.1.3.2. Electrical Modeling of the Motor

_{k}, the estimated current, i

_{k}

^{*}, can be reconstructed from:

_{m}

^{*}, comes from the next “LSRM Mechanical model” block. Figure 9 shows the block diagram of the electrical modeling of the phase, k.

#### 4.1.3.3. Mechanical Modeling of the Motor

_{k}

^{*}, will produce a driving force, F

_{mk}

^{*}, according to:

_{mk}

^{*}gives the estimated total motor force. As mentioned in Section 4.1.2, we assume a linear functioning of the magnetic circuit and a magnetic independence of the motor phases.

_{s}, and the mass, m, of the mover are given in Section 3.1. The viscous friction force is neglected. Figure 10 shows in block diagram form, the mechanical model of the motor based on Equation (23).

_{m}

^{*}, is calculated at time, t, the function, g(x) = dL/dx, takes into account this value to generate the position at the next time, t + T

_{e}. It is important to define the function, g(x), for the control and the modeling of the motor well. In [14,15], this function is used to elaborate the current control of an LSRM used for an elevator application.

_{k}changes linearly between the values, L

_{u}(unaligned teeth) and L

_{a}(aligned teeth). Use of this type of model is common, for example, in work given in [11] and [13].

_{2}, L

_{3}and L

_{4}, can be deduced from L

_{1}by translation. Figure 11 shows the inductance profiles, L

_{1}to L

_{4}, with respect to the position and the current sequence power supply associated with it. As the rate of change of inductance is assumed to be linear, the functions, g

_{k}(x) = dL/dx

_{k}, are portions of straight lines. The possible values of g

_{k}(x) are real constants, shown in Table 4.

−a < x < -a/2 | −a/2 < x < 0 | 0 < x < a/2 | a/2 < x < a | a < x < 3a/2 | 3a/2 < x < 2a | |
---|---|---|---|---|---|---|

g_{1}(x) | −10/2.9 | −10/2.9 | 10/2.9 | 10/2.9 | −10/2.9 | −10/2.9 |

g_{2}(x) | 10/2.9 | −10/2.9 | −10/2.9 | 10/2.9 | 10/2.9 | −10/2.9 |

g_{3}(x) | 10/2.9 | 10/2.9 | −10/2.9 | -10/2.9 | 10/2.9 | 10/2.9 |

g_{4}(x) | −10/2.9 | 10/2.9 | 10/2.9 | -10/2.9 | −10/2.9 | 10/2.9 |

_{k}(x) according to the position, x (cf. Table 4), and determines the effort of each phase, according to Equation (22).

_{k}(x) in the electrical modeling of the motor phase. For an inductance profile that is not linear without saturation, the modeling approach remains valid by giving the non-linear equation of the inductance profile with respect to the position.

#### 4.1.4. Results of the Open Loop Control

_{m}

^{*}, to the desired position, x (10 mm amplitude and 2 Hz frequency).

#### 4.2. Closed Loop Control of the Motor

_{m}*, this block will generate the appropriate control effort. This corrector includes an integral action and an anticipatory action, called feed-forward, which allows for compensation of the tracking deviations during the speed ramps, and the integral action allows for compensation of the effects of disturbing efforts. To implement all these elements, we used the reference speed, v, and the estimated measured velocity, v

_{m}*.

_{c}, which is provided to the “Current signal generator” block. We recall that this block must generate the current supply sequence, which allows the mover to achieve the desired position. It has the estimated measured position, x

_{m}*, as the second input. Figure 15 shows the implemented “Force Controller” block.

_{1}defines the proportional gain, which acts on the position error. The output of this gain is called the control speed, V

_{c}. The integrator gain is chosen equal to ¼ of K

_{1}to obtain a proper integral action of the controller. Then, the control speed is corrected by an anticipatory (feed-forward) action. The gain, K

_{2}, defines the proportional gain, which acts on the speed error. The output of the corrector is the control force, F

_{c}. Upstream of the integrator, a “magic-switch” has been inserted. When the switch is open (out = 0), the integral action is held. This block is an “S-function” of Matlab Simulink and has x, x

_{m}

^{*}and the control speed, V

_{c}, for inputs.

**Figure 16.**(

**a**) Flow chart of the “magic-switch”;(

**b**) Closed loop configuration: input (position x) and output (position x

_{m}

^{*}) waveforms.

_{c}was compared to the estimated speed, V

_{est}. When the estimated speed, V

_{est}, is too rapid compared to V

_{c}, then the switch is open (out = 0); otherwise, it will be closed (out = V

_{c}).

## 5. Conclusions

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**MDPI and ACS Style**

Llibre, J.-F.; Martinez, N.; Leprince, P.; Nogarede, B.
Analysis and Modeling of Linear-Switched Reluctance for Medical Application. *Actuators* **2013**, *2*, 27-44.
https://doi.org/10.3390/act2020027

**AMA Style**

Llibre J-F, Martinez N, Leprince P, Nogarede B.
Analysis and Modeling of Linear-Switched Reluctance for Medical Application. *Actuators*. 2013; 2(2):27-44.
https://doi.org/10.3390/act2020027

**Chicago/Turabian Style**

Llibre, Jean-Francois, Nicolas Martinez, Pascal Leprince, and Bertrand Nogarede.
2013. "Analysis and Modeling of Linear-Switched Reluctance for Medical Application" *Actuators* 2, no. 2: 27-44.
https://doi.org/10.3390/act2020027