# Balance Control of Brushless Direct Current Motor Driven Two-Rotor UAV
## 2 Brushless Direct Current Motor
### Mathematical Model of the BLDCM  
Figure 2 shows the equivalent electrical circuit of the BLDCM. Three-phase electricity is used to power this equivalent circuit. Other waveforms, such as square waves, can be  
Figure 1: A 2-Rotor UAV example for load carrying operations.  
applied in place of a sinusoidal source, provided that the peak voltage stays within the motor's maximum voltage limit [19; 20].  
Mathematical equations to be used in modeling can be obtained by using the electrical equivalent circuit of a star-connected three-phase BLDCM. Phase voltages are given in Equations (1)-(3).  
\[v_{a}=R\;i_{a}+L\frac{d\;i_{a}}{dt}+e_{a} \tag{1}\]  
\[v_{b}=R\;i_{b}+L\frac{d\;i_{b}}{dt}+e_{b} \tag{2}\]  
\[v_{c}=R\;i_{c}+L\frac{d\;i_{c}}{dt}+e_{c} \tag{3}\]  
Interphase voltages are given in Equations (4)-(6).  
\[v_{ab}=R(i_{a}-i_{b})+L\frac{d(i_{a}-i_{b})}{dt}+e_{a}-e_{b} \tag{4}\]  
\[v_{bc}=R(i_{b}-i_{c})+L\frac{d(i_{b}-i_{c})}{dt}+e_{b}-e_{c} \tag{5}\]  
\[v_{ca}=R(i_{c}-i_{a})+L\frac{d(i_{c}-i_{a})}{dt}+e_{c}-e_{a} \tag{6}\]  
Here, \(R\) is the stator phase resistance, \(L\) is the stator phase inductance, \(e_{a}\), \(e_{b}\) and \(e_{c}\) are the back-EMF values, \(i_{a}\), \(i_{b}\) and \(i_{c}\) are the phase currents that belongs to the winding stated in subscript, \(v_{a}\), \(v_{b}\) and \(v_{c}\) are the phase voltages, and \(v_{ab}\), \(v_{bc}\) and \(v_{ca}\) are the phase-to-phase voltages [2].  
Equation (7) provides the phase current relationship.  
\[i_{a}+i_{b}+i_{c}=0 \tag{7}\]  
If the interphase voltages are rearranged using Equation (7), the expressions given in Equations (8) and (9) are obtained. Thus, the voltage expressions between phases are reduced to two equations.  
\[v_{ab}=R(i_{a}-i_{b})+L\frac{d(i_{a}-i_{b})}{dt}+e_{a}-e_{b} \tag{8}\]  
\[v_{bc}=R(i_{a}+2\;i_{b})+L\frac{d(i_{a}+2\;i_{b})}{dt}+e_{b}-e_{c} \tag{9}\]  
According to [PERSON]'s second law of motion, the relationship between electromagnetic moment (\(T_{e}\)) and motor speed (\(\omega_{m}\)) is given in Equation (10). The relationship between motor speed and mechanical angle (\(\theta_{m}\)) is given in Equation (11).  
\[T_{e}-T_{L}=J\frac{d\;\omega_{m}}{dt}+B\;\omega_{m} \tag{10}\]  
Figure 2: Electrical equivalent circuit of three-phase star-connected BLDCM.  
\[\omega_{m}=\frac{d\theta_{m}}{dt} \tag{11}\]  
Here, \(T_{L}\) is the load moment, \(J\) is the moment of inertia, \(B\) is the damping constant. The back-EMF expressions are given in Equations (12)-(14).  
\[e_{a}=\frac{k_{e}}{2}\,\omega_{m}\,F(\theta_{e}) \tag{12}\]  
\[e_{b}=\frac{k_{e}}{2}\,\omega_{m}\,F(\theta_{e}-\frac{2\pi}{3}) \tag{13}\]  
\[e_{c}=\frac{k_{e}}{2}\,\omega_{m}\,F(\theta_{e}-\frac{4\pi}{3}) \tag{14}\]  
The resulting total moment expression is given in Equation (15).  
\[T_{e}=k_{t}[F(\theta_{e})i_{a}+F(\theta_{e}-\frac{2\pi}{3})i_{b}+F(\theta_{e} -\frac{4\pi}{3})i_{c}] \tag{15}\]  
Here, \(k_{e}\) and \(k_{t}\) are the back-EMF constant and the electromagnetic moment constant, respectively. The relationship between the electrical displacement (\(\theta_{e}\)) and the rotor's mechanical angle (\(\theta_{m}\)) is given in Equation (16).  
\[\theta_{e}=\frac{P}{2}\theta_{m} \tag{16}\]  
Here, \(P\) is the number of poles of the motor. The function \(F\), given in Equation (17), provides the back-EMF's trapezoidal waveform based on the rotor position [21].  
\[F(\theta_{e})=\begin{cases}1,&0\leq\theta_{e}\leq\frac{2\pi}{3}\\ 1-\frac{6}{\pi}(\theta_{e}-\frac{2\pi}{3}),&\frac{2\pi}{3}\leq\theta_{e}\leq \pi\\ -1,&\pi\leq\theta_{e}\leq\frac{5\pi}{3}\\ -1+\frac{6}{\pi}(\theta_{e}-\frac{5\pi}{3}),&\frac{5\pi}{3}\leq\theta_{e}\leq 2 \pi\end{cases} \tag{17}\]