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## 4 – FUNDAMENTALS

4 – FUNDAMENTALS Figure 4.5 (right) shows a model of a cylindric single coil VCA without ferrite ring. The x-axis indicates the direction of the translational movement and coincides with FV CA, which follows with (4.15) to FV CA(x) = ICLCBx(x). (4.18) The coil’s magnet field Bx(x) induced by the current shows the same distribution like the field of the permanent magnet (fig. 4.6). Thus, the Lorentz force pushes or pulls the coil 9 on the x-axis depending on the current flow direction. It is reasonable that no force will be generated when the axial centers of the magnet and the coil conincides, because the magnet fields cannot cause a magnetic attraction or repulsion respectively. An axial displacement of this two components is necessary for optaining a functional cylindric single coil VCA. The magnet field distribution on the x-axis of a cylindric coil can be derived to BxC(ˆx) = µ0N ⎛ ⎝ 2lC lC 2 lC − ˆx 2 − ˆx 2 + DC 2 lC + 2 2 lC + ˆx 2 + ˆx ⎞ ⎠ 2 , DC 2 + 2 (4.19) whereas µ0 indicates the vacuum permeability, N the number of windings of the coil, lC the coil length and DC the coil diameter. Due to the same field line distribution, the magnet’s field BxM(x ∗ ) will be defined analogously to (4.19) with the translation x ∗ = x + a. Note that ˆx = x − a. The force distribution of a cylindric single coil VCA can now be calculated with the superposition of the two B-fields depending on the shifting parameter a, to FV CA(x, a) = ICLC(BxC(ˆx) − BxM(x ∗ )). (4.20) Anticipating to the following chapter, figure 4.7 shows a calculated force distribution for arbitrary defined values, which won’t be discussed here. In fact, the characteristic of the FV CA curve interests in terms of the optimal coil positioning relative to the magnet. The goal is to maximize the VCA force. The two optimal displacement positions can be calculted using the first derivative ∂FV CA(x, a) = 0. (4.21) ∂x It must be noted, that the this displacement optimization must be considered dy- namically due to the stroke, which the VCA shall perform for operation. Further investigations will be presented in chapter 5. 9 Evidentially, the magnet will be pushed or pulled when applying the moving magnet concept. 24

B x / a.u. 1.5 1 0.5 4 – FUNDAMENTALS 0 −5 −4 −3 −2 −1 0 x / a.u. 1 2 3 4 5 Figure 4.6: Calculated |Bx(x)| distribution of a cylindric single coil. F / N 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −8 −6 −4 −2 0 2 4 6 8 x 10 −3 −0.2 x / m Figure 4.7: Calculated Lorentz force distribution of a cylindric single coil VCA. 25