# Investigation of Regenerative Braking Performance of Brushless Direct Current Machine Drive System
## 3 Proposed Braking Strategy Based on Critical Duty Ratio Determination
The DC voltage gain solution is based on the inductor voltage variation along a switching period. Of course, a certain amount of loss is due to ESRs (equivalent series resistor) of DC-link capacitors. However, thanks to low a ESR and parallel connected DC-link capacitors, the loss caused by ESR is neglected for a steady-state DC solution. Another loss source is the ferromagnetic material of stator core, i.e., inductor core, this loss amount can be easily negligible but due to the slotless stator design. For the sake of simplicity of overall calculation and the implementation of control algorithm, those two loss components are not taken into account. For steady-state voltage equation, \(V_{d}\) is the DC bus voltage which is the boosted value of \(E_{L}\), the terminal back-EMF voltage. The most important performance parameter of the system is duty ratio which plays a role in providing enough current from boosted voltage to battery. Duty ratio has its maximum value that the boosted voltage decreases dramatically even as the duty ratio is increased as shown in Figure 3. Ultimately the battery charging current during regeneration, i.e., braking current, must be controlled effectively in order to use high performance regenerative braking. The second limitation is back-EMF. Since \(D_{max}\), which is the maximum duty cycle, depends on the (terminal resistance)/(battery internal resistance) boost conversion ratio, the internal resistance of the battery is an inevitable limitation. When the regenerative braking equivalent circuit operates at a continuous conduction mode (CCM), the circuit can be analyzed by using two switching states as shown in Figure 1. For both cases, all considerable parasitic and non-linear elements are included and the output voltage (\(V_{d}\)) and the load current (\(i_{d}\)) expressions are obtained as presented in Equations (4) and (5). \[V_{d}=\frac{f(D_{r})}{g(D_{r})}=\frac{E_{b}(D_{r}R_{T}+2R_{s}+2R_{D}+D_{r}R_{ D})+(1-D_{r})(R_{i}+R_{p})[E_{L}-(2-D_{r})V_{th}]}{(R_{i}+R_{p})(D_{r}-1)^{2}+(D_{r} R_{T}+2R_{s}+2R_{D}+D_{r}R_{D})} \tag{4}\]  
\[i_{d}=\frac{\frac{E_{b}(D_{r}R_{T}+2R_{s}+2R_{D}+D_{r}R_{D})+(1-D_{r})(R_{i}+R _{p})[E_{L}-(2-D_{r})V_{th}]}{(R_{i}+R_{p})(D_{r}-1)^{2}+(D_{r}R_{T}+2R_{s}+2R_ {D}+D_{r}R_{D})}}{R_{i}+R_{p}(1-e^{-t/(R_{p}C_{p})})} \tag{5}\]  
Figure 3: Ideal and non-ideal boosting curves. where \(V_{d}\), \(D_{r}\), \(V_{th}\), \(R_{i}\), \(R_{T}\), \(R_{D}\), \(E_{b}\), \(R_{s}\), \(R_{p}\), \(C_{p}\) are the battery voltage, specific duty ratio, diode voltage drop, total battery internal resistance, MOSFET on-resistance, diode on-resistance, battery open circuit voltage, stator resistance, total electrochemical polarization resistance, and total electrochemical polarization capacitance, respectively. The duty ratio for maximum voltage gain can be calculated by equating the denominator of Equation (4) to zero. Using the known values of Equation (4) the required duty ratio for the maximum regeneration effect is calculated:  
\[\frac{dg(D_{r})}{dD_{r}}=0 \tag{6}\]  
The essential performance parameter of the system is the specific duty ratio which provides a proper charging current for the battery pack. The value of \(D_{r}\) which is heavily dependent on the internal resistance of the battery has an important limitation. If non-ideal circuit parameters are not excluded, the PWM will be applied according to the steady-state and ideal machine equations. At this point, if the total internal resistance variation depending on the battery temperature is not known, an excessive current will charge the battery and cause the system to overheat and be damaged. Thus, in order to achieve a regenerative braking performance at maximum efficiency, the transfer functions including parasitic elements must be obtained. The line-to-output and control-to-output transfer functions of a boost converter model including parasitic elements are derived for regenerative braking operation at CCM.