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5.6 Helmholtz VCA 5 –

5.6 Helmholtz VCA 5 – ANALYSIS, CALCULATIONS AND EXPERIMENTS A first idea for optimizing the VCA based on the well known Helmholtz coil setup. The goal there is to elongate the magnet field’s peak for achieving an extended region with a constant field along the x-axis (compare fig. 4.6). This can be realized by two axially oriented coils separated in a particular distance. The reason for experimenting with a Helmholtz coil setup was the fact, that the generated force no longer depends on the magnet’s position expressed as FV CA(x) ∝ Bx(x) ⇒ FV CA ∝ Bx. (5.4) Since it was clear that the magnet’s and the coil’s axial origins must be displaced for a cylindric single coil VCA, its optimal positions had to be figured out for the Helmholtz VCA. This was mainly performed with means of the FEMM simulation software. Obviously, four variants are possible for wiring and coupling the coils of the Helmholtz VCA named as 1. serial – equal coupled, 2. serial – anti-coupled, 3. parallel – equal coupled, 4. parallel – anti-coupled. The terms serial and parallel point to the electrical wiring of the coils. Equal coupled and anti-coupled define, whether the coils are wound in the same or in the opposite orientation relative to the x-axis. Elongating the constant B-field will be optained with an equal coupled setup. The two coils thereby act as a long cylindric coil. But since it was clear, that the required coil lenght must exceed the magnet length for generating the required Lorentz force, no improvements can be achieved with a equal coupled Helmholtz coil compared to a single coil setup. However, the simulation shows, that an anti-coupled setup is capable to augment the generated force by approximately 80 % compared to a corresponding equal coupled coil. The |Bx(x)| distribution of an equal coupled Helmholtz coil has a behaviour comparable to that of a single coil with an extended summit. But for the anti-coupled Helmholtz coil, the |Bx(x)| distribution corresponds to the curve shown in figure 4.7. 38

5 – ANALYSIS, CALCULATIONS AND EXPERIMENTS Coils Magnet Material MULTOGAN Material NdFeB dC 0.125 mm dM 2 mm hC 1.8 mm dL 3 mm lC 2.1 mm BHmax 40 MGOe N 2 × 250 0.5 A IC Table 5.3: Parameters of the Helmholtz VCA simulation. A formula for calculating the force can be deduced using (4.20). However, the force can be directly determined with the FEMM software. A free parameter of a Helmholtz VCA is the clearance d between the coils. The figures 5.6 to 5.8 show the simulated curves for three different clearances comparing the forces of an equal coupled VCA (blue) and an anti-coupled VCA (red) with the parameters listed in table 5.3. The maximum force of the anti-coupled VCA will be reached when the magnet is precisely positioned in the middle of the coils. Compared to the maximum force of the equal coupled VCA, an absolute augmentation of 80 % can be achieved in the best case. This can be explaned when considering each coil as a single magnet. One pushes the permanent magnet and the other one pulls it respectively. The maximum force depends on the clearance when considering the negative peaks of the red curves 5 . Figure 5.9 shows the simulation result of Fmax(d). The optimal clearance was calculated with the Taylor approximation to dopt ≈ 1.3 mm. The highest mechanical force appears at the maximum deflection of the FH structure due to the spring force F (x) = cx according to (4.9). Therefore, it is reasonable that the axial magnet center shall then coincide with Fmax of the anti-coupled Helmholtz VCA. However, high acceleration forces occur during the deflection phase because of the fast switching mode requirement. These forces may exceed the spring force. Hence, an optimization of the total VCA energy EV CA was performed in terms of EV CA = x1 x2 FV CA(x)dx . (5.5) The required stroke for the MSTS amounts to f = 1.5 mm. Two cases of the magnet’s end points were analyzed for optimizing EV CA . For the first, Fmax shall be achieved 5 The negative force peak results of the magnet’s B-field orientation defined in the simulation. Thus, the magnet must be correctly oriented in the MSTS to achieve the desired movement. 39

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