BLDC

$$v_{d\text{'}}=\frac{2}{3}\left(\left(v_a-R{i}_a-E_a\right)\cos {\theta }_e+\left(v_b-R{i}_b-E_b\right)\cos \left({\theta }_e-\frac{2\pi }{3}\right)+\left(v_c-R{i}_c-E_c\right)\cos \left({\theta }_e+\frac{2\pi }{3}\right)\right)$$

$$v_{q\text{'}}=-\frac{2\pi }{3}\left(\left(v_a-R{i}_a-E_a\right)\sin {\theta }_e+\left(v_b-R{i}_b-E_b\right)\sin \left({\theta }_e-\frac{2\pi }{3}\right)+\left(v_c-R{i}_c-E_c\right)\sin \left({\theta }_e+\frac{2\pi }{3}\right)\right)$$

$$\begin{array}{l}{E}_a=P{\omega }_m\frac{\partial {\psi }_a}{\partial {\theta }_e}\\ E_b=P{\omega }_m\frac{\partial {\psi }_b}{\partial {\theta }_e}\\ E_c=P{\omega }_m\frac{\partial {\psi }_c}{\partial {\theta }_e}\end{array}$$

$$\begin{array}{l}{i}_a=i_{d\text{'}}\cos {\theta }_e-i_{q\text{'}}\sin {\theta }_e\\ i_b=i_{d\text{'}}\cos \left({\theta }_e-\frac{2\pi }{3}\right)-i_{q\text{'}}\sin \left({\theta }_e-\frac{2\pi }{3}\right)\\ i_c=i_{d\text{'}}\cos \left({\theta }_e+\frac{2\pi }{3}\right)-i_{q\text{'}}\sin \left({\theta }_e+\frac{2\pi }{3}\right)\end{array}$$

$$\begin{array}{l}{\omega }_e=P{\omega }_m\\ \frac{d}{dt}{i}_{d\text{'}}=\frac{1}{L_d}{v}_{d\text{'}}+\frac{L_q}{L_d}P{\omega }_m{i}_{q\text{'}}\end{array}$$

$$\frac{d}{dt}{i}_{q\text{'}}=\frac{1}{L_q}{v}_{q\text{'}}-\frac{L_d}{L_q}P{\omega }_m{i}_{d\text{'}}$$

$$T_e=1.5P\left((L_d-L_q)i_{d\text{'}}{i}_{q\text{'}}\right)+P\left(\frac{\partial {\psi }_a}{\partial {\theta }_e}{i}_a+\frac{\partial {\psi }_b}{\partial {\theta }_e}{i}_b+\frac{\partial {\psi }_c}{\partial {\theta }_e}{i}_c\right)$$

$$\begin{array}{c}\frac{d}{dt}{\omega }_m=\frac{1}{J}\left(T_e-T_f-F{\omega }_m-T_m\right)\\ \frac{d{\theta }_m}{dt}={\omega }_m\end{array}$$