Flux Biasing Structure

This section discusses the design of the flux biasing structure, which includes two rows of permanent magnets and a stator backiron in each stator assembly. The main design parameters for the flux-biasing system include: (1) permanent magnet height ℎ𝑝𝑚, i.e. the magnet’s dimension in the vertical direction; (2) permanent magnet

thickness 𝑡𝑝𝑚, i.e. the magnet’s dimension in the magnetization direction; (3) the

bias flux air gap length 𝑔𝑏 (same with yaw control flux air gap). Other variables, such

as separation distance between the bottom of magnets and the top of motor stator, motor flux air-gap, yaw control stator height, and stage backiron thickness, can also influence the bias flux’s performance through changing the leakage reluctance of the bias flux.

In the design of the flux biasing structure, we sweep the design parameters using three-dimensional finite element simulation via Ansys Maxwell. First, we set the bias flux air gap length at 2 mm, which is a typical value for our linear stage, and then we swept the height and the thickness of the bias magnet. In this simulation, other geometric parameters for the stage are set as follows: hysteresis secondary thickness is 9 mm, hysteresis secondary width is 40 mm, stator back iron thickness is 14 mm, stage back iorn thickness is 6 mm, motor stator total height 56 mm, motor stator width is 40 mm, and stage total height is 75 mm. Figure 5-12 shows the calculated passive stiffness in the vertical direction and the negative stiffness in the lateral direction under different magnet geometries, and Figure 5-13 shows the ratio between the negative stiffness and the passive stiffness with respect to the magnet dimension. It can be seen that while both negative stiffness and passive stiffness increase as the magnet’s thickness and height increase, minimizing the ratio between the negative stiffness and the passive stiffness would prefer a design with large magnet thickness and small magnet height. In our linear stage design, we selected to use a magnet thickness of 25.4 mm (1 inch) and a magnet height of 6.35 mm (1/4 inch). This selection is close to the optimal design in terms of the ratio between the negative stiffness and the passive stiffness, while satisfying a minimum passive

(a) (b)

Figure 5-12: Simulated flux bias structure performance with different permanent mag- net dimensions. (a) Passive stiffness in 𝑧-direction (vertical). (b) Negative stiffness in 𝑥-direction (lateral).

Figure 5-13: Simulated ratio between the negative stiffness and passive stiffness with respect to the dimension of biasing permanent magnet.

stiffness requirement of 2×104_{N/m, which provides a vertical mode natural frequency}

of 10 Hz. We selected magnets with such dimension since they are available off-the- shelf, therefore no custom-made magnets are required, which typically have higher cost and take a longer lead time for manufacturing.

The next step is to set the air gap length for the bias flux path. Figure 5-14 shows the simulated vertical-directional restoring force generated by the permanent magnet bias fluxes when the stage is displaced from the equilibrium position in the vertical

-3 -2.5 -2 -1.5 -1 -0.5 0 -20 0 20 40 60 80 100 120 Airgap 1.5 mm Airgap 1.75 mm Airgap 2 mm Airgap 2.25 mm Airgap 2.5 mm Weight of Stage

Figure 5-14: Simulated vertical-directional restoring force at different bias flux air gap lengths. Weight of the moving stage is plotted with dashed line.

Table 5.3: Simulated bias flux performance at different air gaps.

𝑔𝑏𝑖𝑎𝑠 Δ𝑠𝑎𝑔 𝑘𝑧 𝑓𝑛𝑧 |𝑘𝑥| 𝑓𝑛𝑥 1.5 mm 0.7 mm 4.8 × 104 N/m 16.2 Hz 5.1 × 105 N/m 51 Hz 1.75 mm 0.9 mm 3.8 × 104 N/m 14.0 Hz 3.7 × 105 N/m 44 Hz 2 mm 1.0 mm 2.7 × 104 _{N/m 12.1 Hz 2.6 × 10}5 _{N/m 37 Hz} 2.25 mm 1.2 mm 2.0 × 104 N/m 10.3 Hz 2.0 × 105 N/m 32 Hz 2.5 mm 1.5 mm 1.5 × 104 N/m 9.1 Hz 1.6 × 105 N/m 29 Hz

direction at different air gap lengths. Note that the stage’s weight is compensated with the vertical directional restoring force. As a result, the gravity-induced sag of the moving stage can change with respect to the air gap length. Table 5.3 shows the simulated gravity-induced sag distance ∆𝑠𝑎𝑔, passive stiffness in vertical direction 𝑘𝑧,

negative stiffness in lateral direction |𝑘𝑥|, natural frequency in vertical direction 𝑓𝑛𝑧,

and unstable natural frequency in lateral direction 𝑓𝑛𝑥 at different air gap lengths. In

our final design, we selected an air gap length of 2 mm for the bias flux path due to a trade-off between the negative stiffness and passive stiffness.

Table 5.4: Simulated bias flux performance with the selected parameters. Parameter Description Variable Value

Air gap flux density 𝐵𝑏𝑖𝑎𝑠 0.45 T

Gravity induced sag ∆𝑠𝑎𝑔 1.0 mm

Passive stiffness in 𝑧-direction 𝑘𝑧 2.7 × 104 N/m

Passive stiffness in roll direction 𝑘𝜃𝑦 112 Nm/rad

Passive stiffness in pitch direction 𝑘𝜃𝑥 72 Nm/rad

Negative stiffness in lateral direction |𝑘𝑥| 2.6 × 105 N/m

Negative stiffness in yaw direction |𝑘𝜃𝑧| 527 Nm/rad

dimensional finite element method with the selected design parameters, and the re- sults are shown in Table 5.4. Note that in the final linear stage system, both passive and negative stiffnesses can further increase when the effect of the motor flux is in- cluded.