## 4 Mathematical Model
### Motor
The estimation equation for viscous friction coefficient is as shown in Equation (21) and it is derived from Equations (11), (16), and (18). Where \(I_{0}\) is a no-load current and \(V_{a_{0}}\) is an applied voltage under no-load current measurement conditions. These values are usually given in the specification of the motor. Figure 4 shows a tendency of the viscous friction coefficient relative to the motor size constant. Data for this trend analysis were obtained based on the product specifications of the major motor manufacturers for multicopter UAV [17; 18; 19; 20]. Consequently, the estimated model for the viscous friction coefficient of the BLDC motor for multicopter UAVs was obtained, as shown in Equation (22). \[\tau_{m}=J_{rotor}\cdot\frac{d\omega_{m}}{dt}+B_{m}\cdot\omega_{m}+Q_{prop} \tag{9}\]3  
\[\tau_{m}=J_{rotor}\cdot\frac{d\omega_{m}}{dt}+B_{m}\cdot\omega_{m}+Q_{prop} \tag{9}\]4  
\[\tau_{m}=J_{rotor}\cdot\frac{d\omega_{m}}{dt}+B_{m}\cdot\omega_{m}+Q_{prop} \tag{9}\]5  
On the otherhand, during actual operation, the performance of the motor is affected by a number of factors. Among the various factors that degrade performance, the most rep  
\[\tau_{m}=J_{rotor}\cdot\frac{d\omega_{m}}{dt}+B_{m}\cdot\omega_{m}+Q_{prop} \tag{9}\]6 Motor size constant vs viscous friction coefficient. reaction. Figure 5 is a simple diagram of the typical armature reaction of a two-pole DC motor, showing that the main field flux is affected by the armature flux, which results in an magnetic neutral line being misaligned from a geometric neutral line [21]. In the case of motors for general industrial machinery, where the operation range is usually set as the design value and does not deviate much from that the design value, this is offset by installing compensation winding in order to match the expected level of the armature reaction. Thus, in many cases, the effect of the armature reaction is often considered to be negligible. However, in motors for multicopter UAVs, notably, almost all products are not designed with consideration for the armature reaction. This is because motors for multicopter unmanned aerial vehicles are driven by speed control methods, which makes it difficult to reflect specific points, such as design values, owing to their wide operating range. In addition, because aircraft parts are sensitive to weight, the motor is designed to be as light and simple as possible; therefore, the installation of compensation winding is difficult. In addition, the effects of armature reactions are particularly noticeable in high-power conditions; thus, the larger the platform and motor size, the more necessary it is to consider armature reactions more carefully. In motors that ere used in small multicopter platforms, the effect of the armature reaction is usually observed in the short range near full throttle conditions. In contrast, in large-sized motors where a high-voltage, high-current drive is based, it can be seen that the armature reaction is prominent in the wide range starting from the intermediate throttle. However, because armature reactions are influenced by complex factors, such as stator shape and air gap, number of winding turns in coils, and magnetic flux in magnets, the analytical model for armature reactions is also very complex and difficult to express explicitly [22]. Thus, the armature reaction is usually considered to be simplified into additional voltage drop factors, as is included in Equation (8). According to the related studies that are referenced in this paper, it is mainly expressed as the product of the n-order polynomial for the motor current and the motor speed [23; 24; 25]. In conclusion, in this study, a function of the squared motor current and the motor speed was used as the model, such as Equation (23). Where \(K_{ar}\) is the armature reaction constant, which can be used to estimate the appropriate value through comparison with the experimental data.