# Investigation of Regenerative Braking Performance of Brushless Direct Current Machine Drive System
## 3 Proposed Braking Strategy Based on Critical Duty Ratio Determination  
In the proposed study, the model of BLDC machine inverter reduced to a single-switch boost converter circuit is used to elucidate the regenerative braking operation. Single switch is capable of producing required braking torque and better energy recovery in mid to high-speed range [14]. The main principle during regenerative braking is to change the switching scheme to send the recovered energy back to battery, as it can be seen in Figure 1, instead of using regular motor drive switching. The circuit operation includes two states of basic voltage boosting operation: Storing phase and energizing phase. In the regenerative braking operation at the energy storing state, the bottom MOSFETs, \(S_{4}\), \(S_{6}\), and \(S_{2}\) are switched accordingly to form a braking current waveform for providing proper braking performance. A nearly trapezoidal back electromotive force (EMF) is rectified and boosted by the motor driver converter for charging battery. When a bottom switch is off, the related top and bottom diodes transfer the power, i.e., current, to the battery. Figure 2 shows one operation interval of the converter, when \(S_{6}\) and \(D_{4}\) are on, the energy stored in the boosting inductor, i.e., the machine winding inductance, due to the terminal induced voltage. Then the current due to the sum of \(E_{L}\) and the inductor voltage is sent to the battery through \(D_{1}\) and \(D_{6}\). \(E_{L}\) is the phase-to-phase or line-induced voltage during the generator operation of the BLDC motor and boosting the motor converter's operation, given as:  
\[E_{L}=2E_{max}=e_{a}-e_{b}=e_{b}-e_{c}=e_{c}-e_{a} \tag{3}\]  
where \(E_{max}\) is the maximum value of phase induced voltage. For trapezoidal back EMF motors, \(E_{L}\) is almost a constant voltage for each \(60^{\circ}\) overlapping of the phase induced voltages. For each \(60^{\circ}\) of commutation interval of the BLDC motor, \(E_{L}\) can be assumed a DC voltage then the converter is operated as a cascade connected rectifier - boost converter. \(e_{a}\), \(e_{b}\), and \(e_{c}\) are the phase-induced voltages of the phase a, phase b, and phase c, respectively. \(E_{L}\) can be assumed at its maximum for \(60^{\circ}\) intervals, coinciding with the energizing intervals of the BLDC motor. Thus, in the study, the same \(60^{\circ}\) conduction intervals provided by Hall sensors are used to define the range in which \(E_{L}\) is the highest then the boosting operation by a specified transistor is implemented. By investigating the nature of regenerative braking, it is known that the boosted-induced voltage must compensate all non-ideal voltage drops in the principal circuit [10]. This boosting action is modeled by using a non-ideal boost converter, which is demonstrated in Figure 2 as a lump-parameter equivalent circuit. The non-ideal and parasitic components reduce the boosting action in their extent. For this reason, all nonlinear components should be taken into account while analysis of the circuit and modeling of the whole system are being derived. These analyses are crucial for the regenerative braking performance in order to increase the efficiency and to provide comfortable braking. The stator windings are used as the storing inductors in the boost-operated converter during  
Figure 1: The circuit operation includes two states of basic voltage boosting operation: storing mode and energizing mode. Figure 2: Boost converter equivalent circuit of a powertrain. regeneration. Naturally, the inductance value of those windings is higher than that of a conventional boost converter because the main intention is to design a BLDC motor stator winding, although the slotless topology of the motor diminishes the winding inductance to some extent. The main source of loss is the resistance value of the boosting inductor formed by stator windings because of the high number of winding turns. The approximate solutions are made by assuming small-ripple inductor current.