Figure 6.8: Voice coil schematic representation
Figure 6.9:
Voice coil GeeplusTM Voltage Current [A] R/L Resistance [Ω] Inductance [mH]
0.59 V 0.409 190.1 1.46 7.7
1.25 0.904 147.6 1.38 9.3
2.03 1.403 120.9 1.45 11.9
2.86 1.945 120.4 1.47 12.2
3.67 2.537 120.5 1.45 11.9
4.52 3.187 106.2 1.42 13.4
Table 6.3: Voice coil tting parameters
(a) Back electromagnetic force trend (b) Back electromagnetic zoom
(c) Back electromagnetic force after sign
change (d) Voice coil velocity
(e) Voice coil damping
Figure 6.10: Voice coil intrinsic damping characterization
Figure 6.11: Voice coil fully rectracted position
Figure 6.12: 0 - 4.52 Voltage step curves example
Figure 6.13: Simulink®Voice coil circuit
Figure 6.14: Simulink®RL circuit estimator
instead, sent to a third block that represents a MultibodyTMof the voice coil that simulates the real behaviour of the damper and from which the real voice coil velocity is extrapolated ( gure 6.15).
If we analyse gure 6.13 and gure 6.14 it is possible to see that in the voice coil circuit there are two lookUp tables: they simulate the variation of the resistance and of the inductance as a function of the position of the mover of the voice coil with respect to the stator. The RL circuit, instead, requires a single value of resistance and inductance: they are set as described in section 6.4.2. In addition, in the MultibodyTM, it is possible to set a mechanical disturbance on the voice coil through a Chirp signal.
In order to see the accuracy of the RL estimator, two simulations are performed.
In the rst one, a voice coil disturbance is set at almost 10 second but it is not considered noise on the current that, for example, can aect a physical measurement. In gure 6.16 are represented the voice coil real velocity (de-rived from the MultibodyTM, blue line) and the RL estimated one (orange line): even if the disturbance is followed by the estimator and the
magni-Figure 6.15: MultibodyTMvoice coil
tude is estimated quite well, there is discrepancy in the phase that is well represented in the Bode in gure 6.17: the phase dierence increases with the frequency reaching 2° of shift.
In the second simulation, in addition to the disturbance, is also taken into account the current noise. Figure 6.18 and gure 6.19 show the results: with respect to the previous case, the magnitude estimation gets worse and also the phase shift is increased reaching peaks of almost -11° and 8°.
6.5.2 Kalman Filter estimator
Since good results were not achieved with the RL estimator, a new one based on the Kalman Filter was implemented. The Kalman lter algorithm com-bines the information from the predictions and measurements to provide the best possible estimate of the new states [40]: based on Gaussian probability density functions, it corrects the states using two covariance matrices, Q and R; matrix Q takes into account the noise on the states, while matrix R takes into account the noise on the measurements, given, for example, by sensors.
For our system, we have to write the mechanical (equation 6.10) and the electrical domain equations (equation 6.5).
m¨x = −c ˙x − mg + Kmi (6.10)
Figure 6.16: MultibodyTMvelocity and RL estimated velocity without noise
Figure 6.17: Magnitude and phase Bode of RL estimator without noise
Figure 6.18: MultibodyTMvelocity and RL estimated velocity with noise
Figure 6.19: Magnitude and phase Bode of RL estimated velocity with noise
Figure 6.20: Simulink®Kalman Filter
Rewritten in state space representation, they become:
¨x
di dt
= −mc Kmm
−KLm −RL
˙x i
+−1 0 0 L1
g e(t)
(6.11)
˙
xKF = Ax + BuKF (6.12)
¨x i
=−mc Kmm
0 1
˙x i
+−1 0
0 0
g e(t)
(6.13)
yKF = Cx + DuKF (6.14)
Using Matlab®, matrices A, B, C, D are discretized and inserted in the Kalman Filter block in Simulink®with uKF and yKF ( gure 6.20) from which the estimated voice coil velocity and current are extrapolated. This block requires also the Q and R matrices that, through a trial and error procedure, are set as follows:
Q =0.1 0 0 0.001
(6.15)
R =0.001 0 0 0.001
(6.16) As before, the current and the force are obtained from the voice coil circuit and the latter is sent to the MultibodyTMcircuit. Even in this case, two simulations are performed. In the rst simulation, are taken into account the RL variation and a voice coil disturbance without noise on the current and on the acceleration. The results are represented in gure 6.21 and gure 6.22: especially from the Bode diagram, there is a clear improvement in results compare to the RL estimator both for the magnitude and the phase, even if the Kalman lter is not able to follow the impulse force, since it is not a gaussian disturbance.
Figure 6.21: MultibodyTMvelocity and KF estimated velocity without noise
Figure 6.22: Magnitude and phase Bode of KF estimator without noise
Figure 6.23: MultibodyTMvelocity and KF estimated velocity with noise
In the second simulation, it is added noise to the current and acceleration measurements. Also in this case, better results can be achieved ( gure 6.23 and gure 6.24).
6.5.3 Comparison between RL and KF estimator
The inacurate RL estimation of the velocity is due to the uncertainty on the values of resistance and inductance: in fact, since the estimator needs only a value of resistance and inductance, they were put equal to the mean value calculated as in section 6.4.2. However, this leads to the introduction of an error in the estimation as the two quantities actually vary depending on the position of the mover with respect to the stator. Furthermore, the shift phase is very dangerous for the control strategy, because it means that the controller would lag behind the change in the system it is supposed to control (in this case the speed of the voice coil) leading to even more unexpected behaviour. The Kalman lter use improves the results because it is able to compensate the uncertainty of the resistance and inductance, except when an impulse force is applied.
Figure 6.24: Magnitude and phase Bode of KF estimator with noise
Chapter 7
Conclusions and further studies
Two very important aspects for the feasibility of the project Hyperloop T ransportation T echnologies were illustrated in this thesis: a quasi-static analysis using a test bench was performed and a dynamic analysis.
In order to better analyse the experimental results, load cells, aluminum disk and copper rim were characterized. For the rst ones, was modelled the interference between the measurements in the two directions and computed numerically the interference matrix used to correct the results.
Regarding the aluminum rotor and the copper rim, was noticed that they are not planary: the rst one has a peak of 0.5 mm while the second one has more irregular surface with peaks of -0.5 to 0.3 mm.
Furthermore, the procedure to create the 90° Halbach array needed for the experiment was explained and the related sources of non idealities were illus-trated such as pad inclination with respect to the copper rim, eddy currents, no negligible edge eects, air gap uncertainty due to pad vibrations, copper rim not recti ed and aluminum rotor not balanced.
One of the main chapter focuses on the quasi-static experiment: it was per-formed at ωref equal to 400 rpm and with 9 dierent airgaps. The behaviour of the lift and drag forces were extrapolated. The lift force increases with the speed and its magnitude increases with smaller gaps; the drag force, instead, increases up to the electromagnetic pole which is around 163 rpm and then decreases. Furthermore, the lift to drag ratio was computed: it increases with the speed, but it has not a constant behaviour for all the airgaps (for higher airgaps we have found higher ratio values); in addition, using COM-SOL simulations scaled factor equal to 4 was derived and consenquently, the scaled Fland Fdare proportional to the scale factor2 while the velocity is proportional to the scale factor.
The last chapter concentrates on the dynamic part: the tting analysis was performed on the experiment data obtained in quasi-static conditions follow-ing the multiple branches approach. The number of branches that minimizes the error between the experimental data and the lumped-parameter model
is equal to 2. The root locus, instead, showed a velocity instability equal to 6.92 m/s without damping, while a value of optimal damping equal to 367 Ns/m.
Then, the use of a voice coil as a damper was justi ed and an attempt was made to characterise its most important quantities such as resistance, inductance, and intrinsic damping, which do not remain linear during its operation time. In fact, we have seen that this device has a moving part made by aluminum so, when it moves, it generates eddy currents: after a few steps, an intrinsic damping value was found equal to 207 Ns/m. The resistance and inductance values were, instead, extrapolated when the voice coil was energized with dierent voltage steps and held in its fully retracted position: a single value was calculated with an arithmetic mean.
Finally, two possible control solutions were implemented: the goal is to es-timate accurately the voice coil velocity. The rst one uses a RL estimator based on the RL circuit equation of the voice coil: this technique introduces a shift phase between the real and the estimated velocity that cannot be neglected because it would led to a dangerous behaviour of the controller.
The second solution, instead, uses a Kalman Filter estimator: with this tech-nique better results can be achieve both for the magnitude (except when an impulse force is applied since it is not a gaussian disturbance) and the phase.
Future studies could focus on analysing dierent con gurations of Halbach arrays and see if they could achieve better results in terms of lift and drag curves. Furthermore, dierent control strategies could be implemented to reach more e cient controls.
Appendix A
2 DOF state-space
representation for stability analysis
x = id,1 iq,1 ... id,Nb iq,Nb z˙p zp z˙s zs
T
(A.1) A = Ael Aep
Ape Am
(A.2)
Ael=
−ωp,1 ω 0 ... 0 0
−ω −ωp,1 0 ... 0 0
... ... ... ... ... ...
0 0 0 ... −ωp,Nb ω
0 0 0 ... −ω −ωp,Nb
(A.3)
Aep=
Λ0
γL1e
−zp,0
γ 0 0 0
0 ωγLΛ0
1e
−zp,0
γ 0 0
... ... ... ...
Λ0
γLNbe−zp,0γ 0 0 0
0 ωγLΛ0
Nbe−zp,0γ 0 0
(A.4)
Ape =
−2Λ0 γmpe
−zp,0
γ 0 ... −2Λγm0
p e
−zp,0
γ 0
0 0 ... 0 0
0 0 ... 0 0
0 0 ... 0 0
(A.5)
Am =
−mcs
p −ksm+kus
p
cs
mp
ks
mp
1 0 0 0
cs
ms
ks
ms −mcs
s −mks
s
0 0 1 0
(A.6)
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