# Using the Hall Effect for Monitoring the Starter Condition in Motor Vehicles
## 2 Analysis of the Magnetic Circuit and Distribution  
The magnetic circuit of the starter can be divided into two basic parts:  
* stator--a stationary part;
* rotor--a movable part.  
The two parts are separated by an air gap called the armature gap. In the stator part of the magnetic circuit, the following elements can be distinguished: a pole shoe, a main pole core (pole body), an air gap between a core and a yoke (when a core and a yoke are made of separate parts), and a yoke [13].  
In the rotor part, the elements of the magnetic circuit are the following: armature teeth and the armature core. The size of the armature gap has a significant impact on the characteristics and dimensions of the machine. On the one hand, it is advisable to make the size of the \(\delta\) gap smaller because it causes the decrease in the magnetic tension of the gap. At the same time, the dimensions of the machine may become smaller because less space is necessary to accommodate excitation winding. This makes it possible to decrease the height of the pole and consequently to make the outer diameter of the machine smaller at a constant armature diameter. Excitation losses are becoming smaller as well. However, at small values of \(\delta\), the magnetic field of the machine changes when there are load variations, which has a negative impact on commutation and leads to its unstable work [14]. The size of the armature gap is usually not constant along the armature perimeter because the arch of the pole shoe may be of different shapes depending on the required distribution of magnetic field in the gap. Most frequently, the gap is constant and concentric along the rotor (about 0.6\(\div\)0.7 of the pole piece width), and, near the pole horns, it increases to 2 \(\delta\).  
If the gap was the same along the whole arch of the pole, a rapid change in magnetic induction on the pole horns would occur. It would cause more losses in teeth steel and in armature winding and the increase of noise generated by the machine [15; 16; 17].  
The properties of electric machines depend on the distribution of the radial constituent B\({}_{r}\) of magnetic induction in the air gap between the stator and the rotor [18]. The course of function \(B_{r}=\mathrm{f}(\alpha)\), describing the distribution of induction depending on angle \(\alpha\), is called the field distribution or the magnetic field curve (Figure 1).  
Figure 1: The cross section of stator I and Rotor II cores: (**a**) cylindrical layout; (**b**) equivalent flat layout.  
Field distribution depends on the following factors [19]:  
1. the current distribution along the inner circuit of the stator core and the outer circuit of the rotor core,
2. the distribution of magnetization vector of permanent magnets,
3. the shape of the air gap between the stator and the rotor, and
4. the magnetic tension in stator and rotor cores.  
Even at drastic simplifications consisting in neglecting magnetic tension in the core and assuming smooth by-gap surfaces of the armature and the poles, it is difficult to formulate analytical dependency describing induction distribution in the gap depending on the dimensions of the magnetic circuit and the winding. Such difficulties are the result of the complex shape of magnetic circuit elements. In the analysis of induction distribution in the electric machine, there are many additional simplifications.  
It is assumed that the magnetic field in the gap is flat, i.e., the winding wires are parallel to the rotor rotation axis; in the core there are no radial passages; the deformation of field distribution on the core edges is neglected [20]. The gap length in radial direction is so small in comparison with the other core dimensions that, for the purpose of analyzing field distribution, the cylindrical layout can be replaced with the flat one after outspreading the core which has been cut open (Figure 1b).