# Period-1 Motions and Bifurcations of a 3D Brushless DC Motor System with Voltage Disturbance
## 2 Analytical Solutions  
A typical electronically commutated motor is illustrated in Figure 1. Consider such a brushless DC motor in electrical dynamics [1] as  
\[\frac{\mathrm{d}\mathbf{I}(t)}{\mathrm{d}t}=\mathbf{L}^{-1}(\theta)[\mathbf{V}(t)-\mathbf{RI}( t)-(\frac{\partial L(\theta)}{\partial\theta}\mathbf{I}(t)+\frac{\partial\mathbf{ \Lambda}_{M}}{\partial\theta})\frac{d\theta}{dt}] \tag{1}\]  
where \(\mathbf{L}(\theta)\) is the matrix of inductance; \(\mathbf{V}(t)\) is the vector of voltages over the phase windings; \(\mathbf{I}(t)\) is the vector of phase current; \(\mathbf{R}\) is the matrix of resistance; \(\mathbf{\Lambda}_{M}\) is the flux linkage vector due to the permanent magnets, and \(\theta\) is the angular displacement. The mechanical dynamics of the motor can be expressed as  
\[\frac{\mathrm{d}\omega}{\mathrm{d}t}=\frac{T(\mathbf{I},\theta)-T_{l}(t)}{J} \tag{2}\]  
where \(\omega\) is the angular velocity of the rotor; \(J\) is the inertia; \(T(\mathbf{I},\theta)\) is the electromagnetic torque, and \(T_{l}(t)\) stands for the external torques imposed by friction, cogging, etc. Consider that a damping coefficient \(b\), \(T_{l}(t)\) has the following form  
\[T_{l}(t)=b\omega+T_{L} \tag{3}\]  
Figure 1: A schematic of a brushless DC motor. where \(T_{L}\) is the external torque excluding the one caused by damping. Assuming such an ideal smooth-air-gap brushless DC motor is subjected to sinusoidally distributed windings, the electromagnetic torque \(T(\mathbf{I},\theta)\)[21] is expressed in the following form  
\[T(\mathbf{I},\theta)=nk_{e}\sum_{k=1}^{3}i_{k}\cos[n\theta-\frac{2(k-1)\pi}{3}] \tag{4}\]  
where \(n\) is the number of permanent pole pairs; \(k_{e}\) is the permanent-magnet flux constant, and \(i_{k}(k=1,2\) and \(3)\) is the current. To better solve Equations (1) and (2), [PERSON]'s transformation is adopted and the equations are transformed into the rotating frame [13] as  
\[ \begin{array}{l}\frac{\mathrm{d}i_{q}}{\mathrm{d}t}=\frac{1}{L_{q}}\big{[}- Ri_{q}-n\omega(L_{d}i_{d}+k_{t})+u_{q}\big{]},\\ \frac{\mathrm{d}i_{d}}{\mathrm{d}t}=\frac{1}{L_{d}}\big{[}-Ri_{d}+n\omega L_{q }i_{q}+u_{d}\big{]}.\end{array}  \tag{5}\]  
where \(k_{t}=\sqrt{1.5}k_{e}\). The electromagnetic torque is  
\[T(i_{q},i_{d})=n[k_{i}i_{q}+(L_{d}-L_{q})i_{d}i_{q}] \tag{6}\]  
where \(i_{q}\) and \(i_{d}\) are quadrature-axis and direct-axis currents; \(u_{q}\) and \(u_{d}\) are quadrature-axis and direct-axis voltages; and \(L_{q}\) and \(L_{d}\) are the quadrature-axis and direct-axis fictitious inductance. Based on Equation (5), the mechanical dynamics in Equation (2) can be rewritten as [1]  
\[\frac{\mathrm{d}\omega}{\mathrm{d}t}=\frac{n}{\int}\big{[}k_{t}i_{q}+(L_{d}- L_{q})i_{q}i_{d}\big{]}-\frac{1}{\int}(b\omega+T_{L}), \tag{7}\]  
Define new variables as  
\[x_{1}=\frac{i_{q}}{\alpha_{1}},x_{2}=\frac{i_{d}}{\alpha_{2}}+\frac{k_{t}}{L _{d}\alpha_{2}}+\rho,x_{3}=\frac{i_{q}}{\alpha_{1}},\tau=\frac{tR}{L_{q}} \tag{8}\]  
where  
\[\alpha_{1}=\frac{-\delta k_{t}\pm\sqrt{\delta^{2}k_{t}^{2}-4\rho\delta\zeta L _{q}ba_{3}^{2}/R}}{2\rho\delta\zeta},\alpha_{2}=\delta\alpha_{1},\alpha_{3}= \frac{R}{nL_{q}},\zeta=L_{d}-L_{q},\delta=\frac{L_{q}}{L_{d}} \tag{9}\]  
and  
\[v_{q}=\frac{1}{\alpha_{1}R}u_{q},v_{d}=\frac{L_{q}}{\alpha_{2}RL_{d}}[u_{d}+R (\rho\alpha_{2}+\frac{k_{t}}{L_{d}})],\overline{T}_{L}=\frac{L_{q}}{J\alpha_ {3}R}T_{L},\sigma=\frac{L_{q}b}{JR},\eta=\frac{\zeta\alpha_{1}\alpha_{2}}{J \alpha_{3}^{2}} \tag{10}\]  
where \(\rho\) is a free parameter. Combining Equations (8)-(10), Equation (1) becomes  
\[\begin{array}{l}\frac{\mathrm{d}x_{1}}{\mathrm{d}t}=v_{q}-x_{1}-x_{2}x_{3}+ \rho x_{3},\\ \frac{\mathrm{d}x_{2}}{\mathrm{d}t}=v_{d}-\delta x_{2}+x_{1}x_{3},\\ \frac{\mathrm{dDefine the following vectors:  
\[\mathbf{F}(\mathbf{x},t)= \begin{pmatrix}v_{q}-x_{1}-x_{2}x_{3}+\rho x_{3}+Q_{0} \cos\Omega t\\ v_{d}-\delta x_{2}+x_{1}x_{3}\\ \sigma(x_{1}-x_{3})+\eta x_{1}x_{2}-\overline{T}_{L}\end{pmatrix} .