## 2 Equivalent Dynamic Mathematical Model of BLDCM  
The human power generation system uses a three-phase BLDCM. The generator equivalent circuit is shown in Figure 1[10]. The Resistance of the Coil is _Rs_; the self-Inductance of the Coil and Mutual Inductor of the Coil are \(L\) and \(M\); _e\({}_{a}\)_, _e\({}_{b}\)_ and _e\({}_{c}\)_ represent the electromotive reaction of the generator; _i\({}_{a}\)_, _i\({}_{b}\)_ and _i\({}_{c}\)_ represent the phase current; _v\({}_{a}\)_, _v\({}_{b}\)_ and _v\({}_{c}\)_ are the terminal voltages. The dynamic equation of the three-phase BLDCM is expressed as follows: [11]\[ \begin{bmatrix}V_{a}\\ V_{b}\\ V_{c}\end{bmatrix} = \begin{bmatrix}e_{a}\\ e_{b}\\ e_{c}\end{bmatrix} - \begin{bmatrix}R_{s}&0&0\\ 0&R_{s}&0\\ 0&0&R_{s}\end{bmatrix}\begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\end{bmatrix} - \begin{bmatrix}L-M&0&0\\ 0&L-M&0\\ 0&0&L-M\end{bmatrix}\begin{bmatrix}i_{a}\\ 0\\ i_{c}\end{bmatrix}  \tag{1}\]  
The magnetic flux density distribution of the driven BLDCM in the air gap is a trapezoidal wave as shown in Figure 2.  
The three-phase inner electric potential waveform in Figure 2 can be expressed as Equations (2) to (4), where \(K_{s}\) is the generator inner electric potential constant; \(\omega_{e}\) is the angular-rate of the rotor which is proportional to the mechanical rotation speed and pole of the generator; \(\lambda\) is the magnetic flux linkage of the rotator side to stator side. The converter BLDCM power \(P_{g}\) can be deduced from the three-phase bridge rectifier circuit waveform timing diagram at fixed DC load, expressed as Equation (5). Figure 1: Generator equivalent circuit. Figure 2: Three-phase rectifier circuit waveform timing diagram at fixed DC load. \[e_{a}=\left\{ \begin{aligned} K_{g}\omega_{e}\lambda,0\leq \omega_{e}t\leq\frac{2\pi}{3}\\ \frac{-6K_{g}\omega_{e}\lambda}{\pi}(\omega_{e}t-\frac{5\pi}{6}), \frac{2\pi}{3}\leq\omega_{e}t\leq\pi\\ -K_{g}\omega_{e}\lambda,\pi\leq\omega_{e}t\leq\frac{5\pi}{3}\\ \frac{6K_{g}\omega_{e}\lambda}{\pi},\frac{5\pi}{3}\leq\omega_{e}t \leq 2\pi\\ e_{b}(\omega_{e}t)=e_{a}(\omega_{e}t-\frac{2\pi}{3})\\ e_{c}(\omega_{e}t)=e_{a}(\omega_{e}t+\frac{2\pi}{3})\\ P_{g}=e_{a}i_{a}+e_{b}i_{b}+e_{c}i_{c}\end{aligned} \right. \tag{2}\]  
The relationship between the generator inner electric potential and converter control current is observed, the BLDCM three-phase commutation sequence \(S_{a}(t)\), \(S_{b}(t)\), \(S_{c}(t)\) can be defined by Equations (6) to (8) [12]:  
\[S_{a}(t)=\overset{\sim}{\underset{n=0}{\sum}}\left[u(\omega_{e}t-2n\pi)-u( \omega_{e}t-\frac{2\pi}{3}-2n\pi)-u(\omega_{e}t-\pi-2n\pi)+u(\omega_{e}t-\frac {5\pi}{3}-2n\pi)\right] \tag{6}\]  
\[S_{b}(t)=S_{a}(t-\frac{2\pi}{3\omega_{e}}) \tag{7}\]  
\[S_{c}(t)=S_{a}(t-\frac{4\pi}{3\omega_{e}}) \tag{8}\]  
where u(\(t\)) is the unit step function, and n is an integer. The three-phase current distribution is observed, the equivalent DC armature current \(i_{\omega_{e}}(t)\) of the BLDCM is defined by Equation (9). \[i_{eq}(t)=\frac{1}{2}\left[S_{a}(t)S_{b}(t)S_{c}(t)\right]\left[i_{a}(t)i_{b} (t)i_{c}^{\prime}(t)\right]^{T} \tag{9}\]  
\[V_{eq}(t)=\frac{1}{2}\left[S_{a}(t)S_{b}(t)S_{c}(t)\right]\left[V_{a}(t)V_{b} (t)V_{c}(t)\right]^{T} \tag{10}\]  
The equivalent DC inner electric potential \(e_{eq}\) and transfer power \(P_{m}\) of the BLDCM can be deduced from Equations (2) to (5) and (9) and (10), expressed as follows:  
\[e_{eq}=K_{g}\omega_{e}\lambda \tag{11}\]  
\[P_{g}=K_{g}\omega_{e}\lambda i_{eq} \tag{12}\]  
The equivalent DC dynamics equation of the BLDCM can be obtained by substituting Equations (9) to (12) in Equation (1), and the dynamics equation is expressed by Equation (13). \[V_{eq}=e_{eq}-R_{s}i_{eq}-(L-M)\frac{di_{eq}}{dt} \tag{13}\]  
According to the above mathematical model, the human powered generator can be simulated using PSIM, the P-V characteristics curves at different rotation speeds is shown in Figure 3.  
As can be seen, the pedal powered exercise bike generates different MPPs at different speeds.