A method, in a displacement device, for controlling the movement of a moveable stage relative to a stator, the method comprising:
- providing a stator comprising a plurality of electrically conductive coils, the plurality of electrically conductive coils shaped to provide:
- a first plurality of coil traces generally elongated in a stator-x direction and distributed over at least a first portion of a first layer;
- a second plurality of coil traces generally elongated in a stator-y direction and distributed over at least a second portion of a second layer, the stator-y direction non-parallel to the stator-x direction;
- the first and second portions of the first and second layers overlapping one another in a stator-z direction, the stator-z direction generally orthogonal to both the stator-x direction and the stator-y direction;
- providing a moveable stage comprising one or more magnet arrays, the moveable stage moveable relative to the stator within a two-dimensional working region, the one or more magnet arrays comprising a first magnet array, the first magnet array comprising a plurality of first magnetization segments, each first magnetization segment having a corresponding first magnetization direction;
- controllably driving a plurality of currents into the plurality of electrical coils to thereby cause the moveable stage to track a desired position, (xr,yr), within the working region, where xr is a desired position of the moveable stage in the stator-x direction in the working region and yr is a desired position of the moveable stage in the stator-y direction in the working region;
- wherein controllably driving the plurality of currents into the plurality of electrical coils comprises:
- driving first currents, ix, in at least some of the first plurality of coil traces and driving second currents, iy, in at least some of the second plurality of coil traces;
- the first currents, ix, comprising a plurality of first current phases, ikx, each first current phase, ikx, driven into a corresponding one of the at least some of the first plurality of coil traces and phase-adjacent first current phases driven into spatially adjacent ones of the at least some of the first plurality of coil traces, where:
- k is an integer from 0 to np−1 representing a first phase index;
- and
- np is a number of different effective phases in the first currents, ix;
- each first current phase, ikx, is determined based on a function F(yr), where F(yr) is a spatially periodic function of yr in the stator-y direction over the working region, with a first spatial period, λ1;
- each first current phase, ikx, in a particular one of the at least some of the first plurality of coil traces is out of phase, as a function of yr, with one or more phase-adjacent first current phases in one or more spatially adjacent ones of the at least some of the first plurality of coil traces by a first spatial phase difference
- λ
- 1
- 2
-
- n
- p
- ;
- the second currents, iy, comprising a plurality of second current phases, ijy, each second current phase, ijy, driven into a corresponding one of the at least some of the second plurality of coil traces and phase-adjacent second current phases driven into spatially adjacent ones of the at least some of the second plurality of coil traces, where:
- j is an integer from 0 to mp−1 representing a second phase index;
- and
- mp is a number of different effective phases in the second currents, iy;
- each second current phase, ijy, is determined based on a function G(xr), where G(xr) is a spatially periodic function of xr in the stator-x direction over the working region, with a second spatial period, λ2;
- and
- each second current phase, ijy, in a particular one of the at least some of the second plurality of coil traces is out of phase, as a function of xr, with one or more phase-adjacent second current phases in one or more spatially adjacent ones of the at least some of the second plurality of coil traces by a second spatial phase difference
- λ
- 2
- 2
-
- m
- p
- .