## 3 Sensitivity Analysis
### Input and Output Parameters  
The objective of this work is to analyse the effect of the most significant geometrical dimensions of the machine on its electromagnetic and vibration response. More specifically, the mean torque of the machine and the RMS value of the acceleration of the external surface of the stator were defined as the outputs of the sensitivity analysis, which were calculated by means of the analytical procedure described in Section 2.  
The torque and the vibration were selected as the outputs since they are two of the most representative parameters of the behaviour of an electric machine. On the one hand, the torque of the machine defines the output power it is able to provide, or in other words, its capacity. On the other hand, the vibration response of the machine is related to the comfort of the users and the fatigue of its components. For that reason, the ideal objective in an initial design stage would be to maximize the torque while minimizing the vibrations.  
The geometrical dimensions shown in Table 1 are the main dimensions that define the electric machine. However, not all those parameters are independent. Due to the existing dependencies between them, a uniform and independent distribution cannot be defined for all the input parameters, as it would result in an incorrect machine definition.  
In this work, the most influential parameters among all the geometrical dimensions were selected as independent. The internal diameter of the rotor (\(D_{\rm ri}\)) and the length of the stator (\(L_{\rm s}\)) affect the size of the machine, and thus the electromagnetic torque. The height of the magnets (\(l_{\rm m}\)) and the length of the air gap (\(g\)) are also critical parameters for the definition of a machine. Finally, the width of the stator yoke (\(h_{\rm sy}\)) defines its stiffness, and it is very influential on the vibration response. As a design restriction, it was decided to set the external diameter of the stator (\(D_{\rm se}\)) constant in order to limit the size of the motor.  
As a result, the rest of the input parameters were defined as dependent. The height of the teeth (\(h_{\rm t}\)) is related to the diameters of the stator and the rotor, according to Equation (38):  
\[D_{\rm se}=D_{\rm ri}+2(h_{\rm m}+g+h_{\rm t}+h_{\rm sy}) \tag{38}\]  
Moreover, so that the area of the slots is kept constant (in order to introduce the same amount of windings), the width of the slots (\(w_{\rm sl}\)) is related to the height of the teeth, as shown in Equation (39):  
\[w_{\rm sl}=\frac{A_{\rm sl,nom}}{h_{\rm t}} \tag{39}\]  
where \(A_{\rm sl,nom}\) is the nominal area of the slots.  
Finally, the width of the teeth (\(w_{\rm t}\)) is calculated considering the radius of the mid-height of the teeth, according to Equation (40):  
\[w_{\rm t}+w_{\rm sl}=\frac{\pi}{Q_{\rm s}}(D_{\rm ri}+2(h_{\rm m}+g)+h_{\rm t}) \tag{40}\]  
When defining the ranges of the input parameters, it was decided to take the nominal dimensions as a reference, and to define the lower and upper limits by applying a certain percentage. If this percentage is too high, inconsistent values might be obtained for the dependent parameters. Initially, a percentage of 10% was defined to ensure a correct machine definition in all the cases.