# A System Identification Technique Using Bias Current Perturbation for the Determination of the Magnetic Axes of an Active Magnetic Bearing

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**ε**. The technique analyzes the AMB system’s response to the perturbation of bias currents in conjunction with a magnetic circuit model to infer the center and axes positions. The end result of the technique is a set of transformation equations that map the geometric coordinates reported by the AMB system to an effective coordinate system. Once the transformation equations are established, bias perturbation is no longer necessary and an analytical approach to system identification of the bearing’s magnetic field results.

_{n}#### Literature Review

## 2. Materials and Methods

#### 2.1. The Multi-Point Method System Identification Approach

_{top}and i

_{bottom}) are recorded once the rotor is stabilized to establish an initial data point. Multiple data points are obtained by increasing current via small incremental changes in bias, thus “perturbing” the system.

_{top}and g

_{bottom}are functions of rotor position and are calculated as:

_{o}is the nominal (manufacturer’s) air gap, and x is the displacement of the target from the bearing’s effective center. For a given geometry, material and coil current, Equation (1) has two unknowns, F

_{magnetic}and x. The MPM method recognizes that separate current datasets that result from modification of the bias current must correspond to the same bearing force and rotor position due to the control system. For any two pairs of equations, a single unknown value x may be determined that corresponds to the same reaction, or force, at the same rotor position set point. Consider input bias setting i

_{bias}

_{,1}resulting in output currents (i

_{top}

_{,1}, i

_{bottom}

_{,1}). Substitution of currents into Equation (1) leads to:

_{bias}

_{,2}), where we assume i

_{bias}

_{,2}> i

_{bias}

_{,1}:

_{1}= F

_{2}, therefore [3]:

_{bias}

_{2}> i

_{bias}

_{1}occurs [3]. The cosine term in Equation (8) accounts for horseshoe pairs oriented at an angle θ′ from the vertical.

_{1}and F

_{2}are each equal to the actual magnetic bearing force applied to the rotor to keep it levitated at the rotor position set point. Using the modified approach, Prins reports bearing reaction forces applied to the shaft with measurement accuracies within 3% in a stationary rotor [2].

#### 2.2. Experimental Approach

#### 2.3. Experimental Signal Flow

_{v}, and is the rotor displacement along the v axis (μm). Similarly, x

_{w}is the rotor displacement along the w axis (μm).

_{v}, x

_{w}and biases i

_{v,bias}, i

_{w,bias}, current injection occurs at the top and bottom actuators, providing signal perturbation. The PID controller receives a feedback error signal by subtracting the desired set point (in volts) from the position sensor voltage.

#### 2.4. Bearing Rotor Space Geometry

_{v,geo}and x

_{w,geo}) are shown in Figure 4 for Quadrant 1. The horizontal axis corresponds to the geometric v axis, and the vertical axis corresponds to the geometric w axis, which in reality are oriented 45 degrees with respect to the vertical. Rotor Quadrants 2, 3 and 4 are partitioned similarly for a total of 65 geometric coordinates.

_{o}, is assumed to be equal to 762 µm, but due to manufacturing tolerances, rotor-stator misalignment, and environment conditions, g

_{o}may vary from this value. It is also assumed that position sensor alignment coincides with the centerline of each magnetic horseshoe. Any deviation from these ideal conditions reduces the positional accuracy needed to calculate bearing reaction values.

#### 2.5. Reaction Measurement Using Geometric Coordinates and Geometric Set Points

_{v}(35.36 µm) and x

_{w}(35.36 µm) and the bias current was set to 1.5 A, the control system responded with the actuator currents shown in Table 2.

_{o}, b). The process of moving the rotor to a location specified by controller-reported geometric coordinates and observing the associated actuator currents was repeated for several locations within the AMB working space. The force model described by Equation (1) was applied to each case to predict bearing reaction force. The results are shown for each case in Table 3. Notice that x

_{v}and x

_{w}coordinate values shown in Columns 5 and 6 of Table 3 are the same as corresponding set point coordinates shown in Columns 7 and 8. This is done to illustrate that no set point coordinate transformation has yet occurred.

#### 2.6. Transformation Equations

#### 2.7. Rotational Transformation

_{v,geo}, x

_{w,geo}) represents a typical coordinate in the first rotor space quadrant rotated to the final position E′ (x

_{v,rot}, x

_{w,rot}). Points H and D are perpendicular projections of Point E′ onto the rotated v and w rotated axes, respectively. Points B, C, F, J and I are additional points used to derive expressions for x

_{v,rot}and x

_{w,rot}[3].

_{v,rot}and x

_{w,rot}as functions of x

_{v,geo}, x

_{w,geo}and λ are:

_{v}and λ

_{w}, a mean value (λ) is used in final empirical transformation equations.

#### 2.8. Scale and Displacement Transformation

_{v}or s

_{w}< 1) or lengthening (s

_{v}or s

_{w}> 1) due to variations in the AMB magnetic field. Scale parameters s

_{v}and s

_{w}are determined from the slope of linear regression plots of geometric vs. effective coordinates. Scaling is required for both axes and varies as a function of the distance from the effective center, but due to the small variations observed, average values of s

_{v}and s

_{w}are used in final transformation equations.

_{v}in the v direction and b

_{w}in the w direction, as shown in Figure 6, where Point E″ represents the final transformed coordinate location. Namely:

_{v,rot}and x

_{w,rot}are the same for all rotor space coordinates. Therefore, Equations (14) and (16) may be applied at any location in the bearing’s operational space to obtain transformed coordinates.

## 3. Results

#### 3.1. Locating the Effective Origin

_{n}is introduced and defined as:

_{v}

_{,1}= 12.98 μm, x

_{w}

_{,1}= 56.06 μm, and effective origin coordinates are x

_{v,eff}= 0, x

_{w,eff}= 0. Error is thus equal to:

_{n}tends to decrease as coordinates returned from successive iterations approach the effective origin. Set points for the next iteration are established by subtracting coordinates returned from iteration n from set points for iteration n−1:

_{v,setpt,n}= x

_{v,setpt,n}

_{−1}− x

_{v,n}

_{−1}

_{v,setpt,}

_{2}= 50.00 − 12.98 = 37.02 µm

_{w,setpt,}

_{2}= 50.00 − 56.08 = −6.08 µm

_{n}may be thought of as the magnitude of a vector equal to the MPM coordinate to effective coordinate distance for iteration n. Applying the error vector in this manner allows for the adjustment of both v and w set points simultaneously.

_{v,setpt,final}= 41.49 µm

_{w,setpt,final}= −5.57 µm

#### 3.2. Effective Coordinate Axes

_{v}= 69.19 µm and x

_{w}= −9.11 µm. Initial set points are established in this case by adding 25 μm to 41.49 μm for the v coordinate and 0 μm to −5.57 µm for the w coordinate. Using Equation (18), the error for the third iteration is:

_{3}is within 1.44 μm, set points 69.19 μm and −9.11 μm represent the location of this effective coordinate with respect to the geometric origin. Performing similar computations, additional coordinates along the v and w axes result as shown as solid squares in Figure 9.

## 4. Empirical Transformation Equations

^{2}= 0.71):

_{v}= tan

^{−1}(−0.0464) = −2.657° (cw)

^{2}= 1.00):

_{w}= tan

^{−1}(45.893) − 90 = −1.2483° (cw)

_{v}and λ

_{w}.

_{v}and s

_{w}are found by plotting geometric coordinates vs. effective coordinates for each rotor axis. The slope of the straight-line yields the scale factor for each axis [11]. Figure 10 shows the plot x

_{v,geo}vs. x

_{v,eff}, along with information on the linear regression fit. Similar results may be shown for the w axes.

_{v}is established to be 1.0701, and s

_{w}is 1.0698. Rotation, scale and displacement parameters may now be inserted into Equations (14) and (16) to obtain final empirical transformation equations. For the v axis:

_{v,trans}= (x

_{v,geo}cos 1.953° + x

_{w,geo}sin 1.953°) 1.0701 + 40.078

_{v,trans}= 1.06947 x

_{v,geo}+ 0.036468 x

_{w,geo}+ 40.078

_{w,trans}= −0.036458 x

_{v,geo}+ 1.06868 x

_{w,geo}− 9.8416

#### Reaction Measurement using Corrected Set Points

_{v}= 13.53 N

_{w}= 13.74 N

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

A | area of two pole faces (2 faces per horseshoe) = 2.992 × 10 ^{−4} m^{2} |

F_{magnetic} | force due to magnetic source (N) |

L_{i} | flux path through rotor and stator (0.045 m) [2] |

N | number of wire turns for two actuator coils (two coils per horseshoe) = 248 |

b | L _{i}/μ_{r} = manufacturer’s equivalent air gap based on magnetic reluctance of magnetic material (15.0 × 10^{−6} m) |

ε | parameter to quantify the error between the bearing force obtained from force transducers and the force obtained from an MPM iteration |

g_{o} | air gap between rotor and magnetic horseshoe when rotor and stator geometric centers are concentric |

g_{top} | air gap between top of rotor and inside face or stator |

g_{bottom} | air gap between bottom of rotor and inside face or stator |

i | total current in single magnetic horseshoe |

i_{top} | total current at top horseshoe |

i_{bottom} | total current at bottom horseshoe |

λ | angle between the geometric v and w axes and the transformed v and w axes, respectively |

R | Magnitude of the vector resultant of the magnetic forces along the v and w axes |

θ | angle between the vertical axis and the v or w magnetic axes |

θ′ | angle between the v or w magnetic axes and the axes of the position sensor |

x_{v,n} | v axis coordinate returned from MPM iteration n |

x_{v,eff} | desired v axis effective coordinate |

x_{w,n} | w axis coordinate returned from MPM iteration n |

x_{w,eff} | desired w axis effective coordinate |

x_{v,geo}, x_{w,geo} | measured from the geometric center of the bearing located at the intersection of v and w sensor axes. Geometric coordinates are based on the assumption that the magnetic field is in perfect alignment with the rotor geometric center, magnetic axes and positional sensor axes. |

## References

- Marshall, J.; Kasarda, M.; Imlach, J. A Multi-Point Measurement Technique for the Enhancement of Force Measurement with Active Magnetic Bearings. ASME J. Eng. Gas Turbine Power
**2003**, 125, 90–94. [Google Scholar] [CrossRef] - Prins, R. System Identification and Calibration Techniques for Force Measurement in Active Magnetic Bearing. Ph.D. Thesis, Virginia Tech., Blacksburg, VA, USA, 2005, Section 1.1 (unpublished). [Google Scholar]
- Prins, R.J.; Kasarda, M.E.F.; Bates, S.C. A System Identification Technique Using Bias Current Perturbation for Determining Effective Rotor Origin of Active Magnetic Bearings. ASME J. Vib. Acoust.
**2007**, 129, 317–322. [Google Scholar] [CrossRef] - Kasarda, M. An Overview of Active Magnetic Bearing Technology and Applications. Shock Vib. Dig.
**2000**, 32, 91–99. [Google Scholar] [CrossRef] - Gahler, C.; Forch, P. A Precise Magnetic Bearing Exciter for Rotordynamic Experiments. In Proceedings of the Fourth International Symposium on Magnetic Bearings, Zurich, Switzerland, 23–26 August 1994. [Google Scholar]
- Rantatalo, M.; Aidanpaa, J.; Goransson, B.; Norman, P. Milling Machine Spindle Analysis Using FEM and Non-Contact Spindle Excitation and Response Measurement. Int. J. Mach. Tools Manuf.
**2007**, 47, 1034–1045. [Google Scholar] [CrossRef] - Auchet, S.; Chevrier, P.; Lacour, M.; Lipinski, P. A New Method of Cutting Force Measurement Based on Command Voltages of Active Electro-Magnetic Bearings. Int. J. Mach. Tools Manuf.
**2004**, 44, 1441–1449. [Google Scholar] [CrossRef] - Aenis, M.; Knopf, E.; Nordmann, R. Active Magnetic Bearings for the Identification and Fault Diagnosis in Turbomachinery. Mechatronics
**2002**, 12, 1011–1021. [Google Scholar] [CrossRef] - Hussien, A.A.; Yamada, S.; Iwahara, M.; Okada, T.; Ohji, T. Application of the Repulsive-Type Magnetic Bearing for Manufacturing Micromass Measurement Balance Equipment. IEEE Trans. Magn.
**2005**, 41, 3802–3804. [Google Scholar] [CrossRef] - Plonus, M.A. Applied Electromagnetics; Section 10.7; McGraw-Hill: New York, NY, USA, 1978. [Google Scholar]
- Spangler, D. Application of a Bias Current Perturbation Method for Determining Effective Gaps in Magnetic Bearings Utilizing an Error Vector Concept; Mechanical Engineering Master of Engineering Final Report; Virginia Tech: Blacksburg, VA, USA, 2011. [Google Scholar]
- MBRotor Research Test Stand (no. 893-0005-001-05/98). In Hardware User’s Guide, version 1.0; Section 2.4; Revolve Magnetic Bearings Inc.: Calgary, AB, Canada, 1998.

**Figure 2.**Experimental rotor test stand configuration (photo: [2]).

**Figure 7.**Spatial relationship between geometric and effective origins. All values shown are in microns (µm).

Run | Sum of AMB Outboard Reaction (N) |
---|---|

1 | 19.4 |

2 | 19.5 |

3 | 19.7 |

4 | 19.3 |

5 | 20.6 |

6 | 20.1 |

7 | 19.5 |

8 | 19.8 |

9 | 19.8 |

10 | 19.8 |

Average | 19.8 |

Bias Current (Amp) | i_{v,top} | i_{w,top} | i_{v,bottom} | i_{w,bottom} | Bearing Reaction R (N) |
---|---|---|---|---|---|

1.5 | 1.768 | 1.670 | 1.228 | 1.305 | 22.28 |

**Table 3.**Reaction measurement using geometric coordinates and geometric current set points (bias current = 1.5 Amp).

Station | Quadrant | Polar Coordinate R (μm) | Polar Coordinate Θ (°) | x_{v} (μm) | x_{w} (μm) | x_{v,setpoint} (μm) | x_{w,setpoint} (μm) | Reaction R (N) | Percent Difference |
---|---|---|---|---|---|---|---|---|---|

1-origin | 1 | 0 | 90 | 0.00 | 0.00 | 0.00 | 0.00 | 22.02 | 11.5 |

3-positive (pos) w axis | 1 | 50 | 90 | 0.00 | 50 | 0.00 | 50 | 22.51 | 14 |

5-pos w axis | 1 | 100 | 90 | 0.00 | 100 | 0.00 | 100 | 22.95 | 16.2 |

11 (from Table 2) | 1 | 50 | 45 | 35.36 | 35.36 | 35.36 | 35.36 | 22.28 | 12.8 |

13 | 1 | 100 | 45 | 70.71 | 70.71 | 70.71 | 70.71 | 22.46 | 13.7 |

19-pos v axis | 1 | 50 | 0 | 50.0 | 0.00 | 50.0 | 0.00 | 22.06 | 11.7 |

21-pos v axis | 1 | 100 | 0 | 100 | 0.00 | 100 | 0.00 | 22.28 | 12.8 |

51-negative (neg) v axis | 2 | 50 | 180 | −50.0 | 0.00 | −50.0 | 0.00 | 21.84 | 10.6 |

53-neg v axis | 2 | 100 | 180 | −100 | 0.00 | −100 | 0.00 | 22.37 | 13.3 |

59 | 2 | 50 | 135 | −35.36 | 35.36 | −35.36 | 35.36 | 22.24 | 12.6 |

61 | 2 | 100 | 135 | −70.71 | 70.71 | −70.71 | 70.71 | 22.60 | 14.4 |

35-neg w axis | 3 | 50 | 270 | 0.00 | −50 | 0.00 | −50 | 21.44 | 8.56 |

37-neg w axis | 3 | 100 | 270 | 0.00 | −100 | 0.00 | −100 | 21.22 | 7.43 |

43 | 3 | 50 | 225 | −35.36 | −35.36 | −35.36 | −35.36 | 21.75 | 10.1 |

45 | 3 | 100 | 225 | −70.71 | −70.71 | −70.71 | −70.71 | 21.35 | 8.11 |

27 | 4 | 50 | 315 | 35.36 | −35.36 | 35.36 | −35.36 | 21.66 | 9.68 |

29 | 4 | 100 | 315 | 70.71 | −70.71 | 70.71 | −70.71 | 21.57 | 9.23 |

Average | 22.02 | 11.6% |

Bias Current (Amp) | i_{v,top} | i_{w,top} | i_{v,bottom} | i_{w,bottom} |
---|---|---|---|---|

1.3 | 1.579 | 1.500 | 1.015 | 1.086 |

1.5 | 1.738 | 1.646 | 1.259 | 1.331 |

1.7 | 1.904 | 1.797 | 1.495 | 1.569 |

**Table 5.**Application of Multi-Point Method (MPM) to determine the location of effective origin at (0, 0) μm.

Iteration (n) | x_{v,setpt,n} (μm) | x_{w,setpt,n} (μm) | x_{v,n} Returned from MPM Iteration, n (μm) | x_{w,n} Returned from MPM Iteration, n (μm) | ε_{n} |
---|---|---|---|---|---|

1 | 50.00 | 50.00 | 12.98 | 56.08 | 57.54 |

2 | 37.02 | −6.08 | 29.73 | 2.94 | 29.88 |

3 | 7.29 | −9.02 | −33.08 | −6.86 | 33.78 |

4 | 40.37 | −2.16 | −0.9 | 1.39 | 1.66 |

5 | 41.27 | −3.55 | −0.23 | −2.32 | 2.33 |

6 | 41.5 | −1.23 | −0.43 | 2.37 | 2.41 |

7 | 41.93 | −3.6 | 2.28 | 0.81 | 2.42 |

8 | 39.65 | −4.41 | 3.27 | 3.49 | 4.78 |

9 | 36.38 | −7.9 | −6.64 | −3.66 | 7.58 |

10 | 43.02 | −4.24 | 0.34 | −4.57 | 4.58 |

11 | 42.68 | 0.33 | 9.98 | 7.84 | 12.69 |

12 | 32.7 | −7.51 | −2.89 | −2.39 | 3.75 |

13 | 35.59 | −5.12 | −6.09 | −2.6 | 6.62 |

14 | 41.68 | −2.52 | −1.57 | 1.01 | 1.87 |

15 | 43.25 | −3.53 | 9.62 | 3.04 | 10.09 |

16 | 33.63 | −6.57 | −5 | −1.28 | 5.16 |

17 | 38.63 | −5.29 | 3.97 | 1.56 | 4.27 |

18 | 34.66 | −6.85 | −6.83 | −1.28 | 6.95 |

19 | 41.49 | −5.57 | −0.92 | −0.67 | 1.14 |

Bias Current (Amp) | i_{v,top} | i_{w,top} | i_{v,bottom} | i_{w,bottom} | Bearing Reaction R (N) |
---|---|---|---|---|---|

1.3 | 1.590 | 1.588 | 1.008 | 0.998 | 19.37 |

1.5 | 1.751 | 1.744 | 1.251 | 1.232 | 19.28 |

1.7 | 1.923 | 1.909 | 1.482 | 1.458 | 19.25 |

Station | Quad. | x_{v} (μm) | x_{w} (μm) | x_{v,setpoint} (μm) | x_{w,setpoint} (μm) | Reaction R (N) | Percent Diff. |
---|---|---|---|---|---|---|---|

1-origin | 1 | 0.00 | 0.00 | 44.11 | −6.92 | 19.26 | −2.48 |

3-pos w axis | 1 | 0.00 | 50 | 46.59 | 46.51 | 19.13 | −3.15 |

5-pos w axis | 1 | 0.00 | 100 | 49.07 | 99.94 | 19.22 | −2.7 |

11 | 1 | 35.36 | 35.36 | 83.69 | 30.04 | 18.90 | −4.28 |

13 | 1 | 70.71 | 70.71 | 123.27 | 66.99 | 19.04 | −3.6 |

19-pos v axis | 1 | 50.0 | 0.00 | 97.6 | −8.08 | 18.90 | −4.28 |

21-pos v axis | 1 | 100 | 0.00 | 151.1 | −9.25 | 20.91 | 5.86 |

51-neg v axis | 2 | −50.0 | 0.00 | −9.38 | −5.75 | 19.30 | −2.25 |

53-neg v axis | 2 | −100 | 0.00 | −62.87 | −4.59 | 19.75 | 0.00 |

59 | 2 | −35.36 | 35.36 | 8.04 | 31.69 | 19.17 | −2.93 |

61 | 2 | −70.71 | 70.71 | −28.03 | 70.29 | 19.70 | −0.23 |

35-neg w axis | 3 | 0.00 | −50 | 41.63 | −60.35 | 19.17 | −2.93 |

37-neg w axis | 3 | 0.00 | −100 | 39.15 | −113.78 | 19.17 | −2.93 |

43 | 3 | −35.36 | −35.36 | 4.53 | −43.88 | 19.48 | −1.35 |

45 | 3 | −70.71 | −70.71 | −35.05 | −80.83 | 19.53 | −1.13 |

27 | 4 | 35.36 | −35.36 | 80.18 | −45.52 | 19.13 | −3.15 |

29 | 4 | 70.71 | −70.71 | 116.25 | −84.13 | 18.99 | −3.83 |

Average | 19.34 | −2.77% |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Spangler, D.; Prins, R.; Kasarda, M. A System Identification Technique Using Bias Current Perturbation for the Determination of the Magnetic Axes of an Active Magnetic Bearing. *Actuators* **2017**, *6*, 13.
https://doi.org/10.3390/act6020013

**AMA Style**

Spangler D, Prins R, Kasarda M. A System Identification Technique Using Bias Current Perturbation for the Determination of the Magnetic Axes of an Active Magnetic Bearing. *Actuators*. 2017; 6(2):13.
https://doi.org/10.3390/act6020013

**Chicago/Turabian Style**

Spangler, Dewey, Robert Prins, and Mary Kasarda. 2017. "A System Identification Technique Using Bias Current Perturbation for the Determination of the Magnetic Axes of an Active Magnetic Bearing" *Actuators* 6, no. 2: 13.
https://doi.org/10.3390/act6020013