## 4 Mathematical Model
### Esc  
The ESC uses switching circuits that consist of multiple metal oxide semiconductor field-effect transistors (MOSFET) to convert DC electricity from the power source into a three-phase waveform for motor control [29; 30]. There are two typical control methods for motors for multicopter UAVs: trapezoidal commutation and field oriented control(FOC). The conventionally used method is trapezoidal commutation. The switching circuit consists of at least six MOSFETs depending on the ESC capacity, as shown in Figure 6. The three-phase waveform is controlled by switching the high and low side MOSFETs, and the applied voltage to the motor is regulated from the input voltage by pulse width modulation (PWM). This results in a conduction loss and a switching loss due to the on/off operation of the MOSFET, which are the main power loss factors for the ESC. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Magnet Type** & **Ceramic** & **SmCo** & **AlNiCo** & **NdFeB** \\ \hline \(\alpha_{mag}\) [/\({}^{\circ}\)C] & \(-\)0.0020 & \(-\)0.0004 & \(-\)0.0002 & \(-\)0.0012 \\ \hline \(T_{max}\) [\({}^{\circ}\)C] & 300 & 300 & 540 & 150 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Magnet temperature coefficient and temperature limit for various magnet types. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Conductor Material** & **Silver** & **Gold** & **Copper** & **Aluminum** \\ \hline \(\alpha_{con}\) [/\({}^{\circ}\)C] & 0.0038 & 0.0037 & 0.0040 & 0.0043 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Conductor temperature coefficient for various materials. Figure 7 shows PWM signal waveforms for high and low side MOSFET control and three phase voltages in the ideal trapezoidal commutation. Figure 8 shows actual three phase voltage waveforms according to the PWM duty ratio (\(d\)) of the output voltage, which was measured by an experiment driving the actual motor(T-motor U5 motor with Flame 40A ESC, 3S1P LiPo battery) under the no-load condition. As the PWM duty ratio increases, it shows that the width of the pulse that forms the phase voltage waveform, which may seem like noise, is widening. Finally, when the PWM duty ratio reaches 1, it shows that the traces of pulse disappear from the phase voltage and only the phase transition of the neat trapezoidal waveform takes place. For most multicopter dedicated ESCs, the PWM duty ratio directly refers to the motor throttle command from the flight controller. On the other hand, the shorter cycle of trapezoidal waveforms that are inversely proportional to the PWM duty ratio is due to the increased commutation frequency as the motor speed increases. Consequently, the value of the output voltage (\(V_{out}\)), which is the time average of the actually applied voltage to the motor, can be calculated, as shown in Equation (26). \[V_{out}=d\cdot V_{in} \tag{26}\]  
Figure 6: Equivalent circuit of the electric speed controllers (ESC). Figure 7: Pulse width modulation (PWM) signal and phase voltage waveform for ideal trapezoidal commutation. The total power consumption of the ESC can be approximated by the sum of the conduction loss (\(P_{esc.cd}\)), switching loss (\(P_{esc.sw}\)), and integrated circuit (IC) operating power(\(P_{esc.ic}\)), as shown in Equation (27) [11, 31, 32, 33, 34]. \[P_{esc}=P_{esc.cd}+P_{esc.sw}+P_{esc.ic} \tag{27}\]  
The conduction loss of the MOSFET is calculated from the output current and on-state resistance. For the ESC, the output current is the current consumption of the motor. To obtain the total conduction loss of the ESC, the number of MOSFETs in the on-state must be considered according to the commutation sequence and PWM duty ratio that determines the time of the on-state. Thus, the total conduction loss is calculated, as shown in Equation (28), where \(R_{esc.on}\) is the ESC total on-state resistance. \[P_{esc.cd}=d\cdot I_{m}^{2}\cdot R_{esc.on} \tag{28}\]  
The switching loss is due to a gate charge of the MOSFET.