## 4 Mathematical Model
### Esc
The gate charge is the amount of charge that needs to be injected into the gate electrode to turn on the MOSFET completely. While the gate of the MOSFET is fully open, the power on the MOSFET will be lost as much as the gate charge is charged. Conversely, while the gate of the MOSFET is closed, the loss is caused by the gate that is not fully closed until the gate charge is completely discharged. Here, the time that is required to charge and discharge the gate charge it is called the rise and fall time, which depends on the structural and material characteristics of the MOSFET. Power MOSFETs for high-speed switching typically have a rise and fall time in the tens of nanoseconds. The input voltage of the ESC is adjusted to the applied  
Figure 8: Actual three phase voltage waveform measured by experiment. voltage via the high frequency PWM, as shown in Figure 8. The time that is required for the gate of MOSFET to open and close is actually extremely short, but the total amount cannot be ignored if the MOSFET operates at a very high frequency, like ESCs. Switching loss of the ESC can be computed while using Equation (29), where \(t_{r}\) and \(t_{f}\) are the rise time and fall time of the MOSFET, respectively; and, \(V_{t}\) is the terminal voltage of the battery pack as the input voltage on the ESC. \(f_{sw}\) is the switching frequency, which varies depending on the ESC product and settings, and it is commonly 8 kHz, 16 kHz, and 32 kHz. In this study, 16 kHz was used as the default value. \[P_{esc.sw}=\frac{1}{2}\cdot V_{t}\cdot I_{m}\cdot(t_{r}+t_{f})\cdot f_{sw} \tag{29}\]  
The model of on-state resistance and the default rise/fall time were derived based on typical values on the rated condition given in the datasheets for MOSFET chips. To this end, some major commercial ESC products have been investigated and analyzed. Table 3 lists the specifications for sample commercial multicopter-dedicated ESCs and the MOSFETs used in these products [17; 35; 36; 37]. Note that, for the data on the MOSFET, because of that datasheets for the ESC did not provide any relevant information, it was obtained by disassembled the actual sample ESC products. Additionally, the detailed specifications of the MOSFETs were referenced from the specifications for each of the chips. Figure 9 shows the relationship between the allowable maximum continuous current and on-state resistance, arranged from the specifications of the sample ESCs outlined in Table 3, and the estimation model is shown in Equation (30). Where \(I_{z,esc}\) is the current-carrying capacity of the ESC, similar to the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Item** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Manufacturer & T-motor & T-motor & T-motor & T-motor & CobraMotor & STM \\ Product & Flame60A HV & Flame100A LV & Flame100A HV & Flame200A HV & MR60 & ESC001V1 \\ Max. Current & 60 A & 100 A & 100 A & 200 A & 60 A & 20 A \\ Peak Current & 80 A & 120 A & 120A & 240 A & 75 A & 30 A \\ Max. Voltage \({}^{(4)}\) & 52.2 V & 34.8 V & 60.9V & 60.9 V & 26.1 V & 26.1 V \\ Max. Power \({}^{(5)}\) & 3132 W & 3480 W & 6090 W & 12180 W & 1566 W & 522 W \\ MOSFET & TPRR4008 NH & FDR3833L & 119D15 NN08N5 & TPRH480 ANH & TPCA8057-H & STL160N4F7 \\ No. of MOSFETs & 18 & 30 & 6 & 42 & 18 & 6 \\ \(R_{max/data}\)\({}^{(6)}\) & 3.3 m\(\Omega\) & 2.4 m\(\Omega\) & 1.1 m\(\Omega\) & 3.7 m\(\Omega\) & 2.6 m\(\Omega\) & 2.1 m\(\Omega\) \\ \(R_{max}\)\({}^{(4)}\) & 2.2 m\(\Omega\) & 0.96 m\(\Omega\) & 2.2 m\(\Omega\) & 1.057 m\(\Omega\) & 1.733 m\(\Omega\) & 4.2 m\(\Omega\) \\ \(t_{min}\)\({}^{(5)}\) & 5.7 mSec & 4.7 mSec & 32.0 mSec & 9.6 mSec & 4.3 nSec & 6.6 nSec \\ \(t_{fall}\)\({}^{(6)}\) & 11.0 mSec & 4.2 nSec & 28.0 mSec & 13.0 mSec & 6.3 nSec & 5.7 nSec \\ \hline
**Item** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline Manufacturer & Hobbywing & Hobbywing & Hobbywing & Hobbywing & Hobbywing & Hobbywing \\ Product & XRotor10A & XRotor15A & XKotor20A & XRotor25A & XRotor40A Pro & XRotor50A Pro \\ Max.