## 2 Structure and Working Principle  
The proposed short-stroke Lorentz-force-driven planar motor, shown in Figure 1, has a moving-magnet scheme, in which the magnet part is the mover (secondary) and the coil part is the stator (primary). The primary is composed of an array of coils, and the secondary is composed of permanent magnets and two back-irons. The four square coils in the primary are controlled by four separate controllers. Compared with round-shaped coils, the effective edges of the square coils in the magnetic field can be fully used, which increases the force level. Actually, the proposed planar motor is combined by four Lorentz-force-driven units. The motor can realize three-degree-of-freedom motion through controlling the current combination of the four driving units. As shown in Figure 2, the Lorentz-force-driven unit consists of three parts: upper magnets, lower magnets, and one square coil. The double-sided structure is adopted to increase the air-gap magnetic field.  
The features of the short-stroke planar motor are as follows:  
* The working principle is the same as voice coil motors, which is based on the Lorentz force. Compared with three-phase planar motors, with the proposed motor it is easier to achieve high-precision positioning.
* There is no electrical connection and cable disturbance in the moving parts due to the moving-magnet scheme. Additionally, the temperature rise of the motor can be easily controlled because the heat source is in the stator.
* There is no cogging force and force ripple for this slotless motor with air-core coils, which is beneficial for servo performance improvement.  
Figure 1: Basic configuration of the Lorentz-force-driven planar motor.  
Figure 2: Lorentz-force-driven unit. (PM: Permanent Magnet).  
The Lorentz-force-driven unit can be seen as a linear voice coil motor with one square coil. According to the principle of Lorentz force, a force will be generated once an energized conductor is put in the magnetic field. The amplitude and direction of the force is decided by the magnetic flux density, conductor current, and their directions. The Lorentz force equation is written as  
\[\mathbf{F}=N\times I\times(\mathbf{L}\times\mathbf{B}) \tag{1}\]  
where \(N\), \(I\), \(\mathbf{L}\), and \(\mathbf{B}\) are the number of turns, current, conductor length, and magnetic flux density, respectively.  
When the magnetic flux density and the conductor are orthogonal, Equation (1) can be written, in the scalar form, as  
\[F=N\mathbf{B}\mathbf{L} \tag{2}\]  
Upon considering Equation (2), the motor force is dependent on the parameters of the coils and the magnetic field. Due to the correlative relations among different parameters, the optimization design should be taken to achieve the optimal performance of the motor. When conductors move in the magnetic field, the back EMF (Electromotive Force) will be generated, which is in direct proportion to the moving velocity. The power should provide enough current to meet the force target, and conquer the back EMF and the leakage inductance simultaneously. The back EMF equation is written as  
\[E=N\mathbf{B}\mathbf{L}\mathbf{v} \tag{3}\]  
where \(\mathbf{v}\) is the moving velocity.  
Figure 3 shows the working principle of the Lorentz-force-driven planar motor. The colors (or the stripe in the black-white version) represent the current directions. The blue one (horizontal stripe) represents anticlockwise current, and the red one (slant stripe) represents clockwise current. The arrows represent the force acting on each Lorentz-force-driven unit. Through different force combinations, the planar motor can move in the \(x\), \(y\) axis and rotate about the \(z\) axis within a small angle.