#### Abstract

The manuscript proposes a quadratic gain bidirectional converter (QGBC). The proposed converter is operated in two different stages—step-up (motoring) and step-down (regenerative braking)—which can be employed in electric vehicle (EV) applications. The converter is operated in the continuous inductor current mode (CICM). For regenerative braking (RB) of permanent magnet brushless DC (PMBLDC) motor, the self-inductance of the motor is exploiting to step up the back electromagnetic force (EMF) of the motor to extract the energy even at low rotor speed. The design parameters of the converter are selected as battery voltage , output voltage , output power , and switching frequency . The design system is simulated using MATLAB/Simulink. Finally, a 1 kW prototype is developed for validation and performance analysis of the converter. The converter operates at maximum efficiency of 95% during step-up operation of the converter. A DSP microcontroller TMS320F28335 is used to control the switches of the converter in the experimental setup.

#### 1. Introduction

The advancement of technology and standards of living continues to enhance. Nowadays, most of the works depend on electrical energy. Many commercial applications, such as uninterrupted power supply, electric vehicles (EVs), and microgrids. Batteries and bidirectional DC-DC converters (BDCs) have the main role for these systems. In recent years, the whole world moving towards the electric vehicle as fossil fuel are devastating and concern of environmental effect. In the vehicle, the motor drive and power electronic converter should be capable of bidirectional energy interference with the battery and motor drive system. A high gain BDC is proposed in [1, 2]. The high gain BDCs are used in electric vehicles (EVs) as they convert low voltage to high voltage and recovery of energy during regenerative braking (RB) which will enhance the driving range of the vehicle. Figure 1 explicate the electrical system of an EV which consists of a battery, BDC, DC-AC converter, electrical machine, and transmission system. Many high gain converter topologies have been proposed that can be categorized in different ways: isolated and nonisolated circuits. The transformer-based and isolated topology [3–5], a transformer or coupled inductor, is used for high gain. The problem with coupled-inductor or transformer has leakage inductance. To overcome the problem of leakage inductance, a snubber circuit is used. The nonisolated BDC topologies [6, 7] are simple in design and controlling without galvanic isolation.

Due to the simple design and without galvanic isolation, the converter size, and low cost, the nonisolated BDC is most preferred. Due to circuit parasitic components, a high duty cycle, and voltage stress on switches, this converter has a restricted voltage gain. To get a high voltage gain, different topologies have been proposed a few years back [8] to reduce voltage stress. A coupled inductor is used, but due to the presence of leakage inductance which increases the voltage stress, additional clamping circuit is used to reduce the voltage stress. A multiport transformerless converter [9, 10] is for EV application. In this converter, only three switches are used for bidirectional buck-boost operation with dual input and dual output ports, but the voltage gain is same as the conventional boost converter. For the converter for relatively high voltage conversion ratio, there are several coupled-inductor bidirectional converters presented in [11, 12]. In an integrated coupled-inductor converter low ripple current, low voltage stress and soft switching can be achieved by employing phase-shift angle modulation. A dual-switch buck/boost converter [13] has minimal power loss and excellent efficiency; however, the switch is under a high voltage stress. In [14, 15], a switched quasi-Z-source BDC is presented by alteration in the place of the main power switch in a conventional two-level quasi-Z-source BDC for improvement of conversion efficiency, but it requires more number of the power switch. To get higher voltage gain without wider duty cycle operation, a voltage multiplier circuit (VMC) [16] is proposed. The number of diodes required for 2N − 1 combinations for N diode-capacitor VMC and N fold of voltage gain are described in details. A switched-capacitor interleaved [17, 18] bidirectional converter for wide voltage conversion ratio in EVs is presented in this work. In interleaved bidirectional converter number of power switches, components and size are large for high voltage conversion ratio.

There are different resonant converters used for wide voltage gain and higher efficiency [19, 20]. In this fixed frequency, the phase-shift control technique is used, and soft switching is applied for minimum switching loss. There are different topologies of quadratic voltage gain converter proposed in [21, 22]. This quadratic converter has a ripple current on the low voltage side which affects the performance of the battery/supercapacitor. To reduce the ripple current on the low voltage side, a coupled inductor-based quadratic converter [23] is described. In the presence of leakage inductance of coupled inductor, the voltage stress on switches will increase and voltage gain is the same as the quadratic converter. To show the RB, the reversal of power flow from load to source is established. For reversal of power flow, the controlling of the voltage source inverter (VSI) to the power is flowing from load to source. The method for RB of brushless DC motor-based EV is done in [24, 25]. In this RB, we boost the back electromagnetic force (EMF) of PMBLDC machine to DC bus voltage with the help of self-inductance of PMBLDC machine. Another technique for controlling VSI during RB is described in [26]. In this, a T-S sliding-mode approach control technique is used for RB in the electric vehicle. A bidirectional buck-boost converter-based permanent magnet DC machine drive system for RB in an EV is presented in [27].

In this paper, a quadratic gain bidirectional converter (QGBC) is connected to a load as shown in Figure 2. The load consists of a VSI fed PMBLDC motor, and the motor is coupled to a pulley and belt. The proposed converter has the following advantages: high voltage gain at a low duty cycle, fewer power switches, a simple construction, and easy control. Section 2 describes the topology and converter’s operation, and Section 3 gives the details of converter design. The simulation and experimental results are described in Section 4, and the conclusion of the work is given in Section 5.

#### 2. Topology and Operation of Converter

The proposed QGBC with battery and load is illustrated in Figure 2. Here, is input voltage, is voltage across capacitor , is output voltage, is inductor current of , and is inductor current of . In the proposed converter, it consists of four power switches , and with antiparallel diode. The converter’s operation in the step-up switches and is controlled, and converter’s operation in the step-down switches and are controlled. The load on the converter consists of VSI fed PMBLDC Motor with a pulley and belt in the converter’s step-up operation.

##### 2.1. Step Up or Motoring Mode Operation

The converter’s step-up operation is described in two different stages. In the step-up mode, energy is transferred from battery to load through controlled switches and . The working waveform of converter’s operation in the step-up mode is described in Figure 3. Stage I : in this stage, the and switches are ON and and switches are OFF for the time interval of . The inductor current is increasing, and energy is transferred to inductor from battery. The converter’s operation in this stage is shown in Figure 4(a). Capacitor supplies energy to load, inductor current increases, and capacitor charges. Stage II : in this stage, all the switches , , , and are OFF for time interval . The stored energy in inductor and inductor is transferred to load. During this interval, both inductors’ current decreases. The converter’s operation in this stage is shown in Figure 4(b). Upon application of Volt-sec balance principle on inductor and , the following equation is obtained:

**(a)**

**(b)**

By eliminating from equations (1) and (2), the output voltage during step-up operation is obtained in equation (3) quadratic in nature. Voltage across capacitor is obtained as

##### 2.2. Step Down or Regenerative Braking Mode Operation

The step-down mode operation of converter is described in two different stages. In RB operation of converter, the energy is transferred from load to battery by controlling switches and . The working waveform of the converter during regenerative braking is described in Figure 5. Stage I : in this stage, the switches and are ON and switches and are OFF during the time interval . During this interval, both inductors’ current increases, and energy is transferred from higher voltage to lower voltage. The converter’s operation in this stage is shown in Figure 6(a). Stage II : in this stage, all switches , , , and are OFF during the interval . During this interval, the stored energy in inductor is transferred to load and the energy stored in inductor is transferred to capacitor . The converter’s operation in this stage is shown in Figure 6(b).

**(a)**

**(b)**

By eliminating from equations (4) and (5), the output voltage of the converter during the step-down mode is obtained in equation (6) quadratic in nature.

##### 2.3. Working of VSI during Regenerative Braking of PMBLDC Motor

In the RB (step-down), the energy has to flow from the PMBLDC machine to the battery. During RB, the PMBLDC machine acts as a generator, and kinetic energy stored in the vehicle can be transferred to the battery by reversal of current flow. Only by controlling the converter can it not siphon the mechanical energy from the PMBLDC machine, and the machine is needed to be operated in the II quadrant. Instead of direct rectification, a back EMF boosting technique is applied in this work during regenerative braking. The self inductances of the PMBLDC machine are charged by shorting all three phases together. All the switches on the high side are turned OFF, and switches on the low side are turn ON and OFF together. When all the switches are on the low side, the energy is stored in the self-inductance of the PMBLDC machine, and all switches are OFF; the stored energy in these inductances is transferred to the converter’s output capacitor . The equivalent circuit of two active phases of the VSI braking and switching PWM of VSI during regenerative phase is shown in Figures 7(a) and 7(b), respectively.

**(a)**

**(b)**

#### 3. Converter Design and Stability Analysis

##### 3.1. Converter Design Consideration

The converter design is done as per its step-up and step-down operations, and the converter operates in boundary condition mode for step-up. In the boundary condition mode, the minimum value of inductor is zero at the beginning of stage I. The working waveform of the converter in boundary condition is shown in Figure 8(a). In boundary condition operation, the current in inductor is continuous. The inductor’s computation is dependent on the current ripple.

**(a)**

**(b)**

Let us consider the current in inductor is ; thus, the change in current is described as

Assuming a linear variation and consider , the following equation is obtained:

When the switch is turned ON, considering , , and , the maximum current of inductor is given as

To avoid the boundary condition mode, it must satisfy these conditions:

Substituting equations (3), (4), and (11) into equation (10), the critical value of inductor is a function of duty cycle, *D*

Assuming constant output voltage and switching frequency , the value of duty cycle (*D*) which gives maximum value of . The value of *D* is obtained by calculating first derivative of equation (12) with respect to *D* and equals to zero, the *D* is obtained as 1/2. Therefore, the value of is given as

Similiar methods can be used to calculate the value of inductor . The critical value of inductor can be obtained as

The value of *D* is obtained to maximize by calculating the first derivative of equation (14) with respect to *D* and equals to zero. Then, the value of *D* is obtained as 1/3.

Consider linear variation, capacitor can be calculated based on output current.

When switch is turned ON, the value of , , and using equation (17), the capacitor is calculated as

The value of *D* is obtained to maximize by calculating the first derivative of equation (18) with respect to *D* and equals to zero. Then, the value of *D* is obtained as 1/3.

The value of capacitor is calculated as

The value of duty cycle *D* is obtained to maximize by calculating the first derivative of equation (22) with respect to *D* and equals to zero. Then, the value of *D* is obtained as 1/3.

The voltage stress on switches , , and during step-up and step-down operation of converter are obtained as

The different stages for the discontinuous conduction mode (DCM) in step-up operation are explained as follows. In DCM, there are four stages, the first stage is alike to CCM and no need to be explained it again. In second stage, when all the energy stored in is transferred to load and current through become zero at before next switch is ON. In the third stage, all the energy stored in is transferred to capacitor and current through becomes zero at before switch is ON. Figure 8(b) shows the working waveform of the four stages in DCM. As per this, at steady state, the average diode current over one switching period is equal to .

The voltage across and considering step-up and DCM operation

The voltage gain in DCM is calculated as

The equivalent load resistance as and substituting (17) into (25), and using (25), the duty is given by

From working waveform of Figure 8(b), it is observed from the DCM operation that it is required that

Thus, the critical value of the duty cycle to avoid the converter operates in DCM is given by

##### 3.2. Stability Analysis

The stability of any system is defined as the ability to produce bounded output when a bounded input is applied to it. The general state space representation of a system is given by

The state space equation of the converter in step-up for ON and OFF condition is

Now, applying the state-space averaging technique,

Small signal model is derived by applying perturbations, and the overall state and output equation of the converter is obtained as2

After substituting all passive elements value from Table 1 and duty cycle *D* = 0.51, duty to output voltage transfer function *G*(*s*) is obtained as follows:

From the above transfer function, it is observed that the all the poles and two zeros lie on the left half and one zero lies on the right half of S-plane of in root locus. Transfer function of a PI controller is

Ziegler–Nichols’ closed loop tunning method is used to tune the PI controller, the value of and substituted in (36), and therefore, the closed loop transfer function (CLTF) of the converter is obtained as follows:

The magnitude and phase plot is obtained from CLTF are shown in Figure 9. In magnitude plot, there two up-down glitch, and it occurs due to two pair of complex conjugate pole (*P*_{1,2} = −26.8 ± *j*6243, *P*_{3,4} = −99.1 ± *j*821) and one pair of conjugate zero (*Z*_{1,2} = −860 ± *j*4544) in the open loop transfer function *G*(*s*) of the converter. From Bode diagram, one can see phase margin (PM = 40.8°) and gain margin (GM = 2.81 dB), so it can say that the system is stable.

#### 4. Computational and Experimental Results

##### 4.1. Computational Results

The bidirectional quadratic converter along with VSI for the PMBLDC machine is simulated in MATLAB/Simulink. The vehicular load is simulated through a pulley and belt during motoring operation and an inertial load of 0.1 kg·m^{2} during regenerative braking operation. The PMBLDC machine having parameters is given in Table 2. The proposed converter is simulated and designed for 1 kW power at an output voltage of 200 V with battery voltage at 48 V and switching frequency of 20 kHz. The DC-DC converter and inverter switching PWM generated in MATLAB simulink are illustrated in Figures 10(a) and 10(b), respectively. The parameters of the converter are calculated, and its value is given in Table 1. The type of battery used in experimental setup and its parameters is given in Table 3.

**(a)**

**(b)**

The system is simulated for 5 s with 0 s to 3 s in step-up (motoring) and 3 s to 5 s in the step-down (RB) mode. The steady-state inductor’s current in the converter’s step-up operation is shown in Figure 11(a), and the step-down operation is shown in Figure 11(b). In step up, when switches and are ON, both inductor’s current increases and OFF both inductor’s current decreases. Similarly, in step down, when switches and are ON, both inductor’s current increases, and when it is OFF, both inductor’s current decreases and both inductor current is negative. The steady-state output voltage , capacitor , capacitor voltage , and battery voltage during step-up operation of converter are shown in Figure 12. The battery voltage , capacitor voltage , and output at duty cycle 51% can be seen in Figure 12.

**(a)**

**(b)**

The voltage stress on switches , , , and during converter’s step-up operation are shown in Figure 13. are voltage stresses on switches and , respectively, and voltage stress on switch and are (Figures 13(a) and 13(b)). The speed characteristics of PMBLDC machine during motoring mode from time 0 s to 3 s are illustrated in Figure 14. From graphical descriptions, the speed of motor reaches up to 3600 rpm at time 3 s. The RB is applied to reduce the motor speed from 3600 rpm to 500 rpm as described in Figure 14.

**(a)**

**(b)**

The stator current of PMBLDC machine when it acts as motor can be seen in Figure 15(a) and when it act as generator can be seen in Figure 15(b).

**(a)**

**(b)**

##### 4.2. Hardware Results

A model of the proposed converter is designed and developed for the experimental validation as shown in Figure 16. The converter component’s values used in the prototype are the same as chosen in simulation parameters.

In this paper, the control strategy is realized by using the TMS320F28335 DSP microcontroller for proposed converter and driving PMBLDC machine in both motoring and RB. A PMBLDC machine coupled with a flywheel is used to emulate the RB action. To reduce the impulse torque condition and safety of the system, a pulley belt system is employed for mechanical coupling of the motor and inertial load. Figure 17 shows the system voltages in steady-state, battery voltage , capacitor voltage , and output voltage in step-up operation of converter.

The steady-state inductors current and during step-up operation of converter are shown in Figure 18. In step-up operation of converter, the steady-state inductor current and inductor current can be seen in Figure 18. The inductor current at discontinuous condition mode and boundary condition mode is shown in Figures 19(a) and 19(b), respectively. The PMBLDC machine is coupled with pulley, and in order to apply load torque, a belt which is connected through a digital weighing machine at the both end. The steady-state three phase stator current of PMBLDC machine in motoring mode during converter’s step-up operation is shown in Figure 20.

**(a)**

**(b)**

A load torque of 1.5 N-M is applied on PMBLDC machine through belt, and all three phase currents are flowing which is shown in Figure 20.

The voltage stress on switches and are shown in Figure 21 and switches and are shown in Figure 22 during converter’s step-up operation. In the step-down operation of the converter, the PMBLDC machine is loaded with a flywheel to show the power flow from load to battery during RB. A physical mode transition switched is used to change the mode of operation from motoring to RB and vice-versa. When RB is applied through the mode transition switch, the change in the inductor’s current and output voltage of the converter is shown in Figure 23.

Figure 24 show the inductor and current during step-down operation of converter, and it is negative which means power is flowing from load to source. The steady-state inductor current and inductor current during RB of PMBLDC machine and converter operate in step-down. The stator current of PMBLDC motor during RB is shown in Figure 25. A compression of different BDCs is done in Table 4. Efficiency curve of the converter in step-up and step-down operation is shown in Figures 26 and 27, respectively.

#### 5. Conclusion

A QGBC is designed, developed, and tested for RB application in EV. The power flow direction is controlled successfully by changing the working mode of the VSI and the QGBC. In the regenerative braking mode, the inertial load’s mechanical energy is changed to electrical energy and fed back to the battery, as evident from the results. A control strategy is implemented to boost the back EMF of the PMBLDC motor by controlling the VSI and using the self-inductance of the motor. The quadratic gain bidirectional DC-DC converter operates at maximum efficiency of 95% at an output voltage of 200 V during the converter’s step-up operation. The implemented strategy and the system configuration present a frugal and practical approach to eliminate the drawbacks of regenerative braking in the buck mode of BDC.

#### Abbreviations

: | Input voltage |

: | Output voltage |

: | Voltage across capacitor |

: | Diode |

: | Inductor |

: | Capacitor |

: | Switch |

D: | Duty ratio |

: | Inductor current |

: | Capacitor current |

J: | Moment of inertia |

: | Transistor |

: | Switching frequency |

: | ON time |

RB: | Regenerative braking |

E_{vs}: | Electric vehicles |

: | Switching time |

PMBLDC: | Permanent magnet brushless DC |

QGBC: | Quadratic gain bidirectional converter |

VSI: | Voltage source inverter |

EMF: | Electromotive force |

ICE: | Internal combustion engine |

BDCs: | Bidirectional DC to DC converters |

CICM: | Continuous inductor current mode. |

#### Data Availability

The data will be available on request to Prof. R. K. Saket ([email protected]).

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.