## 2 Boost Converter-Fed BLDC Machine Model  
The boost converter-fed PM BLDC machine drive system is supplied by a voltage source \(U\), as shown in Figure 1. A boost converter containing five elements, namely coil \(L\), two capacitors \(C_{d}\) and \(C_{s}\), diode \(D\), and the metal-oxide semiconductor field-effect transistor (MOSFET) \(T9\), is used to feed the machine. A large capacity, small size aluminum electrolytic capacitor is a superior choice for \(C_{d}\) to store the energy, while \(C_{s}\) should be a small series inductance and resistance polypropylene capacitor that has the favorable characteristic of high frequency peak voltage absorption. A protective circuit with power resistance \(R\) and MOSFET \(T7\) is introduced to discharge when the bus pumping voltage reaches a certain value. The MOSFET \(T8\) is used to cut off the bus power supply when a serious fault happens in the machine. The three-phase full bridge consists of six MOSFETs (\(T1\)-\(T6\)). Meanwhile, \(D1\)-\(D6\) are six diodes used to provide a continuous flow circuit to the windings of the machine. Figure 2 shows the operating principle of a Y-connected windings BLDC motor in a clockwise cycle, including the conduction sequence, current flow direction, waveforms of induced electromotive forces (EMFs) and currents, along with hall signals (\(HA\), \(HB\), \(HC\)). An electrical cycle of a BLDC motor can be divided into six different switching states. In each state, a different pair of MOSFETs conducts and two phases energize while leaving the third floated. In the presented work, the following assumptions are used for simplification: (1) the air gap magnetic field distribution is square wave, and magnetic hysteresis, saturation and eddy losses are ignored [15]; (2) Y-connected stator windings are symmetrical, and their resistance and inductance are constant. A group of first-order differential equations with state variables is used to set up the simulation model of the machine. The state equations of a BLDC machine can be obtained by the algebraic transformation of the conventional differential equation model. The fourth-order state equation of the BLDC motor can be expressed as  
\[\dot{x}=Ax+Bu, \tag{1}\]  
\[x=[i_{A}\ i_{B}\ i_{C}\ \Omega]^{\mathsf{T}}, \tag{2}\]  
\[u=[u_{A}\ u_{B}\ u_{C}\ T_{L}]^{\mathsf{T}}, \tag{3}\]  
Figure 1: The topology of a boost converter-fed brushless DC (BLDC) motor. Figure 2: Operating principle in the clockwise cycle. \[A=\left[ \begin{array}{cccc}-\frac{R}{L-M}&0&0&-\frac{p\psi_{pm}(\theta)}{L-M}\\ 0&-\frac{R}{L-M}&0&-\frac{p\psi_{pm}(\theta-\frac{2\pi}{3})}{L-M}\\ 0&0&-\frac{R}{L-M}&-\frac{p\psi_{pm}(\theta-\frac{4\pi}{3})}{L-M}\\ \frac{p}{l}\psi_{pm}(\theta)&\frac{p}{l}\psi_{pm}(\theta-\frac{2\pi}{3})&\frac {p}{l}\psi_{pm}(\theta-\frac{4\pi}{3})&-\frac{B_{V}}{l}\\ \end{array} \right], \tag{4}\]  
\[B=\left[ \begin{array}{cccc}\frac{1}{L-M}&0&0&0\\ 0&\frac{1}{L-M}&0&0\\ 0&0&\frac{1}{L-M}&0\\ 0&0&0&-\frac{1}{l}\\ \end{array} \right], \tag{5}\]  
where \(i_{A}\), \(i_{B}\) and \(i_{C}\) are the phase-current, \(u_{A}\), \(u_{B}\) and \(u_{C}\) are the phase voltages, \(R\) is the phase resistance, \(L\) is the self-inductance, \(M\) is the mutual inductances between phases, \(\psi_{pm}\) is the permanent magnet flux-linkage, \(\theta\) is the rotor position angle, \(T_{L}\) is the load torque, \(\Omega\) is the angular velocity of rotation, \(p\) is the number of pole-pairs, \(J\) is the moment of inertia and \(B_{V}\) is the coefficient of viscosity. In Equation (4), the PM flux linkage \(\psi_{pm}(\theta)\) is only a function of \(\theta\), which can be regarded as a coefficient of the equation. The \(\psi_{pm}(\theta)\) has the same shape with induced EMF, as shown in Figure 2, and its magnitude can be obtained from induced EMF by utilizing a finite element field analysis or test. The waveform data of \(\psi_{pm}(\theta)\) are stored in central processing unit (CPU) and the lookup table (LUT) method is adopted to obtain them according to the angle detected by a position sensor.