High Efficiency Onboard Charger Based on Two-Stage Circuit

Abstract

On-board charger (OBC) is key part of electric vehicles. Limited to space and weight, design objectives of OBC are high power-density and high efficiency. Two-stage circuit is commonly used for 3.3 kW OBC, interleaved power factor correction (ILPFC) is utilized for power factor correction and DC bus voltage regulation, LLC resonant converter is utilized for voltage and power regulations. In this paper, the relationships between the internal parameters and efficiency of ILPFC are studied by discrete iterative method, and the internal parameters are optimized to improve ILPFC`s efficiency. Meanwhile, the relationships between the resonant parameters and efficiency of LLC converter are also studied by fundamental harmonic approximation method to optimize the efficiency in wide charging voltage. A 3.3 kW OBC prototype is developed to verify the effectiveness and correctness of the optimal method, the power factor and total harmonic distortion at full-load state are about 99.99% and 2.98% with the charging voltage ranging from 230 to 430 V, respectively.

1 Introduction

Limited to space and weight, main design objectives of OBC are high power density [1,2,3,4,5,6], high efficiency [7,8,9], high power factor (PF) [10], and low total harmonic distortion (THD) [11,12,13]. Typical OBC is based two-stage circuit [14,15,16], AC/DC stage is used for power factor correction [17], DC/DC stage is utilized to regulate charging power and voltage [18]. Traditional single phase boost circuit has high current stress [19], and large inductor [20]. On contrary, interleaved boost power factor correction (ILPFC) has little current ripple, small inductor [21], high efficiency, and high power-density [20]. LLC converter has merits of soft-switching [22], high efficiency [23, 24], and wide voltage [25], then ILPFC + LLC converter is suitable for OBC [26]. However, the efficiency is affected by switching frequency, boost inductors, and DC link voltage [27], which determines conduction loss and switching loss, and the above losses are difficult to analyze by traditional method.

To improve OBC`s efficiency, there are commonly three methods: (a) high performance devices [14, 18, 28,29,30], (b) modified circuit [6, 22, 31,32,33], and (c) improved control [1, 34, 35]. High performance devices is the most direct approach, literatures [14, 28, 29] use SiC and GaN devices to improve the efficiency, however, presently, the cost of these devices is much higher than Si-based devices, which is the biggest obstacle for industrial application. To improve the efficiency, literature [10] proposed variable dc-link voltage control and ensure LLC converter at the resonant frequency in wide voltage range, however, which brings complex control to PFC. Literature [26] designed an OBC with considering efficiency and volume, however, the peak efficiency of LLC converter is 95.4% and needs further improvement. Hybrid control adopted in [1] and [2] can improve the peak efficiency of LLC converter to 96%, but the control is complex and not good for applications. Ultrawide dc-link voltage in [36] is also useful for efficiency, but the volume of passive components maybe increased because the large current at low charging voltage. Multi-resonant frequency is introduced in [37] to widen the charging voltage range, but the extra components results in the complex hardware, low reliability, and difficulty to design the multi-parameters. The above methods are useful to improve the efficiency, but it is hard to deal with the contradictions between hardware`s cost, complexity, reliability, control, implementation, and so on.

To improve the efficiency in wide charging voltage range without increasing hardware cost and control complexity, a discrete iterative method is proposed to estimate and optimize ILPFC`s efficiency. Moreover, LLC converter`s loss and efficiency in wide charging voltage range are also estimated and optimized based on fundamental harmonic approximation (FHA). The OBC`s efficiency in wide charging voltage range is optimized by designing the internal parameters of ILPFC and LLC converters.

2 ILPFC

Typical topology for OBC includes two stages, as Fig. 1 shows. AC/DC is used for PFC, DC/DC is used for output voltage and power regulation. Actually, the topology in Fig. 2 is commonly adopted. V_AC, i_AC and V_BUS are input voltage, input current and DC link voltage. L₁, L₂ are boost inductors, L₁ = L₂ = L_b. D₁ ~ D₄ are rectifier diodes. Q₁ ~ Q₂ and D₅ ~ D₆ are MOSFETs and freewheeling diodes. S₁ ~ S₂ are control signals of Q₁ ~ Q₂, they have a phase difference of 180 degree. i_L1 ~ i_L2 and i<sub>Q`</sub> ~ i<sub>Q2</sub> are the currents of boost inductor and MOSFET. i<sub>s,PFC</sub> is bridge current and i<sub>s, PFC</sub> = i<sub>L1</sub> + i<sub>L2</sub>. C<sub>B</sub> is link capacitor. Q<sub>3</sub> ~ Q<sub>6</sub> and D<sub>7</sub> ~ D<sub>10</sub> are LLC converter`s MOSFETs and diodes. L_m, L_r and C_r are magnetizing inductor, resonant inductor and capacitor. v_ab and v_cd are primary and secondary-side ac voltages. i_r and i_o are resonant and output currents. V_o is charging voltage, C_o is output capacitor.

Fig. 1

Diagram of on-board charger

Fig. 2

Topology of OBC with two-stage circuit

Losses of EMI filters, rectifier bridge, boost inductors, MOSFETs and diodes of ILPFC are denoted as P_{E, loss}, P_{b, loss}, P_{Lb, loss}, P_{d, PFC, loss} and P_{m, PFC, loss}. f_{s, PFC} is switching frequency, the sampling period is t_{s, PFC} = 1/f_{s, PFC}. v_a is discrete as Σv_k, k is sampling series, the sampling time is Σt_k. In a sampling period, v_a(t) can be seen as constant value. i_L1(t), v_a and S₁ are shown in Fig. 3 [17,18,19].

Fig. 3

Waveforms of i_L(t), v_a(t) and S₁ in: a CCM, b BCM, and c DCM in half line period

2.1 Current Waveforms of IBPFC

2.1.1 Current Waveforms in CCM

In a half line period, as Fig. 3a shows, i_L1(t) is continuous and the average is 0.5i_AC(t). In the kth switching period, i_L1(t) linearly increases when d_k − 1 = 1 and decreases when d_k − 1 = 0. I_{k, start} and I_{k, end} are the starting and ending values of i_L1(t). i_{L1, kr} and i_{L1, kf} are waveforms when d_k − 1 = 1 and d_k − 1 = 0. t_k − 1 and t_k are the starting time and ending time. I_k,turnoff is turn-off current of Q₁, T_{k, r} and T_{k, f} are the turn-on and turn-off time. i_{m1, k}(t), i_{d5, k}(t) and i_{L1, k}(t) are currents of Q₁, D₅ and L₁. Duty cycle of Q₁ is [20]

$$d_{k} = (V_{BUS} - v_{k} )/V_{BUS}$$

(1)

During t_k − 1 ≤ t ≤ t_k − 1 + d_kt_{s, PFC}, i_L1,k(t) raises in the slope of v_k/L. i_m1,k(t) = i_L1,k(t), and i_d5,k(t) = 0. During t_k − 1 + d_kt_s,PFC ≤ t ≤ t_k, i_L1,k(t) decreases in the slope of (V_BUS − v_k)/L. i_m1,k(t) = 0. i_d5,k(t) = i_L1,k(t). Because I_k,start = I_{k − 1,end}, then i_L1(t), i_m1(t) and i_d5(t) in the kth switching period in CCM can be expressed as (2), where T_k,f = (1 − d_k)t_s,PFC. T_k,r = d_kt_s,PFC. Because I_{k+1, start} = I_k,end, then i_L1(t), i_m1(t) and i_d5(t) can be expressed as (3), where f_line is the line frequency. By the same principle, i_m2(t), i_d6(t) and i_L2(t) in CCM can be also obtained.

$$\left\{ {\begin{array}{*{20}l} {I_{k,start} = I_{k - 1,end} } \hfill {t = t_{k} } \hfill \\ {i_{m1,k} (t) = I_{k,start} + v_{k} [t - (k - 1)t_{s,PFC} ]/L_{b} } \,\, \,\,\, \hfill {t_{k - 1} < t \le t_{k - 1} + d_{k} t_{s,PFC} } \hfill \\ {i_{m1,k} (t) = 0} \hfill {t_{k - 1} + d_{k} t_{s,PFC} < t \le t_{k} } \hfill \\ {i_{d5,k} (t) = 0} \hfill {t_{k - 1} < t \le t_{k - 1} + d_{k} t_{s,PFC} } \hfill \\ i_{d5,k} (t) = I_{k,start} + v_{k} T_{k,r} /L_{b} \\ - (V_{BUS} - v_{k} )[t - (k - 1)t_{s,PFC} - T_{k,r} ]/L_{b} \\ \hfill {t_{k - 1} + d_{k} t_{s,PFC} < t \le t_{k} } \hfill \\ {I_{k,turnoff} = I_{k,start} + T_{k,r} /L_{b} } \hfill {t = t_{k - 1} + d_{k} t_{s,PFC} } \hfill \\ {I_{k,end} = I_{k,start} + [v_{k} T_{k,r} + (V_{BUS} - v_{k} )T_{k,f} ]/L_{b} } \hfill {t = t_{k} } \hfill \\ \end{array} } \right.$$

(2)

$$\left\{ \begin{gathered} i_{m1} (t) = \sum\limits_{k = 1}^{{2f_{s} /f_{line} }} {\left\{ {i_{m1,k} (t)} \right\}} \hfill \\ i_{d5} (t) = \sum\limits_{k = 1}^{{2f_{s} /f_{line} }} {\left\{ {i_{d5,k} (t)} \right\}} \hfill \\ i_{L1} (t) = i_{m1} (t) + i_{d5} (t) \hfill \\ \end{gathered} \right.$$

(3)

2.1.2 Currents Waveforms in BCM

As Fig. 3b shows, in BCM, at the starting and ending points, i_L1(t) = 0, the duty cycle is d_k = (V_BUS − v_k)/V_BUS. At t = t_k − 1 and t = t_k, i_L1(t) = 0, and I_k,start = I_k,end = I_{k − 1,end} = I_k+1,start = 0. During t_k − 1 ≤ t ≤ t_k − 1 + d_kt_s,PFC, i_L1,k(t) raises in the slope of v_k/L, i_m1,k(t) = i_L1,k(t), i_d5(t) = 0. During t_k − 1 + d_kt_s,PFC < t < t_k, i_L,k(t) decreases in the slope of (V_BUS − v_k)/L. i_m1,k(t) = i_L1,k(t). i_d5,k(t) = 0. i_m1(t), i_d5(t) and i_L1(t) in the kth period are

$$\left\{ {\begin{array}{*{20}l} {I_{{k,start}} = 0} \hfill & {t = t_{{k - 1}} } \hfill \\ {i_{{m1,k}} (t) = I_{{k,start}} + v_{k} [t - (k - 1)t_{{s,PFC}} ]/L_{b} } \hfill & {t_{{k - 1}} < t \le t_{{k - 1}} + d_{k} t_{{s,PFC}} } \hfill \\ {i_{{m1,k}} (t) = 0} \hfill & {t_{{k - 1}} + d_{k} t_{{s,PFC}} < t \le t_{k} } \hfill \\ {i_{{d5,k}} (t) = 0} \hfill & {t_{{k - 1}} < t \le t_{{k - 1}} + d_{k} t_{{s,PFC}} } \hfill \\ \begin{gathered} i_{{d5,k}} (t) = I_{{k,start}} + v_{k} T_{{k,r}} /L_{b} \\ - (V_{{BUS}} - v_{k} )[t - (k - 1)t_{{s,PFC}} - T_{{k,r}} ]/L_{b} \\ \end{gathered} \hfill & {t_{{k - 1}} + d_{k} t_{{s,PFC}} < t \le t_{k} } \hfill \\ {I_{{k,turnoff}} = I_{{k,start}} + T_{{k,r}} /L_{b} } \hfill & {t = t_{{k - 1}} + d_{k} t_{s} } \hfill \\ {I_{{k,end}} = I_{{k + 1,start}} = 0} \hfill & {t = t_{k} } \hfill \\ \end{array} } \right.$$

(4)

2.1.3 Currents Waveforms in DCM

As Fig. 3c shows, in the kth switching period, i_L1,k(t) is 0 before the end, during t_k − 1 ≤ t < t_k − 1 + d_kt_s,PFC, i_L1,k(t) increase in the slope of v_k/L. i_m1,k(t) = i_L1,k(t). i_d5,k(t) = 0. Q₁ is turned off at t = t_k − 1 + d_kt_s,PFC. During t_k − 1 + d_kt_s,PFC ≤ t < t_k,PFC + T_k,r + T_k,f, i_L1,k(t) decreases in the slope of (V_BUS − v_k)/L. i_d5(t) = i_L1,k(t) = 0 at t = t_k − 1 + T_k,r + T_k,f. During t_k − 1 + T_k,r + T_k,f ≤ t < t_k, i_L1,k(t) is in DCM, i_L1,k(t) = i_d5,k(t) = i_m1,k(t) = 0. And d_k in DCM and the average current I_{L1_k_av} are give by (5) and (6) [21], where P_o,PFC is output power. i_m1(t), i_d5(t) and i_L1(t) in kth switching period in DCM are expressed as (7). Once i_m1(t), i_d5(t) and i_L1(t) are obtained, RMS values of i_m1(t), i_d5(t) and i_L1(t) can be calculated by (8), where I_L1, I_d5, I_m1 and I_s,PFC are RMS currents of L₁, D₅ and M₁, respectively.

$$d_{k} = \frac{{2\sqrt {L_{b} f_{s,PFC} P_{o,PFC} } }}{{\sqrt 2 V_{AC} }}\sqrt {1 - \frac{{v_{k} }}{{V_{BUS} }}}$$

(5)

$$I_{L1\_k\_av} = \sqrt 2 P_{o,PFC} \sin (2\pi f_{line} t_{s,PFC} k)/(2V_{AC} )$$

(6)

$$\left\{ {\begin{array}{*{20}l} {I_{k,start} = 0} \hfill & {t = t_{k - 1} } \hfill \\ {i_{m1,k} = I_{k,start} + v_{k} [t - (k - 1)t_{s,PFC} ]/L_{b} } \hfill & {t_{k - 1} < t \le t_{k - 1} + d_{k} t_{s,PFC} } \hfill \\ {i_{m1,k} = 0} \hfill & {t_{k - 1} + d_{k} t_{s,PFC} < t \le t_{k} } \hfill \\ {i_{d5,k} (t) = 0} \hfill & {t_{k - 1} < t \le t_{k - 1} + d_{k} t_{s,PFC} } \hfill \\ \begin{gathered} i_{d5,k} (t) = I_{k,start} + v_{k} T_{k,r} /L_{b} \\ - (v_{k} - V_{BUS} )[t - (k - 1)t_{s,PFC} - d_{k} t_{s,PFC} ]/L_{b} \\ \end{gathered} \hfill & {t_{k - 1} + d_{k} t_{s,PFC} < t \le t_{k} } \hfill \\ {I_{k,turnoff} = I_{k,start} + T_{k,r} /L_{b} } \hfill & {t = t_{k - 1} + d_{k} t_{s,PFC} } \hfill \\ {I_{k,end} = I_{k + 1,start} } \hfill & {t = t_{k} } \hfill \\ \end{array} } \right.$$

(7)

$$\left\{ \begin{gathered} I_{L1} = \left[ {\frac{1}{{2f_{line} }}\int_{0}^{{1/(2f_{lins} )}} {i^{2}_{L1} (t)dt} } \right]^{0.5} \hfill \\ I_{d5} = \left[ {\frac{1}{{2f_{line} }}\int_{0}^{{1/(2f_{lins} )}} {i^{2}_{d5} (t)dt} } \right]^{0.5} \hfill \\ I_{m1} = \left[ {\frac{1}{{2f_{line} }}\int_{0}^{{1/(2f_{lins} )}} {i^{2}_{m1} (t)dt} } \right]^{0.5} \hfill \\ I_{s,PFC} = \left\{ {\frac{1}{{2f_{line} }}\int_{0}^{{1/(2f_{lins} )}} {\left[ {i_{L1} (t) + i_{L2} (t)} \right]^{2} dt} } \right\}^{0.5} \hfill \\ \end{gathered} \right.$$

(8)

2.2 Efficiency of ILPFC

Losses of EMI, rectifier, inductors, MOSFETs and diodes are denoted as P_EMI, P_b,loss, P_Lb,loss, P_{m, PFC,loss} and P_{d, PFC,loss}, respectively.

2.2.1 Input EMI Filter Losses of ILPFC

As Fig. 4 shows, EMI includes capacitor C_E and inductor L_e. If i_AC(t) is sine wave, then i_EMI and P_E,loss are (9) and (10), where r_Le and r_Ce are equivalent series resistance (ESR) of C_E and L_E. I_AC and I_EMI are the RMS currents of L_E and C_E.

$$i_{EMI} (t) = i_{s,PFC} (t) - \sqrt 2 P_{o,PFC,full} \sin (2\pi f_{line} t)/V_{AC}$$

(9)

$$P_{E,loss} = r_{Le} I_{AC}^{2} + r_{Ce} I_{EMI}^{2}$$

(10)

Fig. 4

EMI`s: a equivalent circuit, and b waveforms

2.2.2 Power Device Losses of ILPFC

The power devices` loss includes bridge loss P<sub>b,loss</sub>, MOSFET loss P<sub>m,PFC,loss</sub>, and diode loss P<sub>d,PFC,loss</sub>, where V<sub>b,F</sub> and V<sub>d,F</sub> are the voltage drop of bridge and diodes. R<sub>ds,on</sub> is the MOSFETs` on-resistor. C_FB and t_fall are the MOSFETs` parasitic capacitance and fall time. I<sub>rr</sub> and t<sub>rr</sub> are the reverse recovering current and reverse recovery time of diodes. P<sub>m,PFC,c</sub> and P<sub>m,PFC,s</sub> are the MOSFETs` conduction loss and switching loss. P_d,PFC,c and P_d,PFC,s are the diodes` conduction loss and switching loss.

$$\left\{ {\begin{array}{*{20}l} {P_{b,c} = V_{b,F} I_{s,PFC} } \hfill \\ {P_{m,PFC,c} = 2R_{ds,on} I_{m1}^{2} } \hfill \\ {P_{d,PFC,c} = 2V_{d,F} I_{d5} } \hfill \\ {P_{m,PFC,s} = 2\sum\limits_{k = 1}^{{0.5f_{s} /f_{line} }} {\frac{{I_{m1,turnoff}^{2} t_{fall}^{2} }}{{6C_{FB} }}} } \hfill \\ {P_{d,PFC,s} = v_{o} I_{rr} t_{rr} f_{s} } \hfill \\ {P_{b,loss} = P_{b,c} } \hfill \\ {P_{m,PFC,loss} = P_{m,s} + P_{m,c} } \hfill \\ {P_{d,PFC,loss} = P_{d,s} + P_{d,c} } \hfill \\ \end{array} } \right.$$

(11)

2.2.3 Boost Inductor Losses of ILPFC

Boost inductor`s loss is given by (12), where ${S}_{{L}_{b}}$, ${N}_{{L}_{b}}$ and ${l}_{{L}_{b}}$ are the effect conducting area, turn number and average winding length. ρ<sub>T</sub> is the conductor`s resistivity. The ILPFC`s efficiency can be expressed as (13), where P_other is mainly the driving loss. P_b,c, P_m,PFC,c and P_d,PFC,c are affected by I_L1+L2, I_m1 and I_d5. P_m,PFC,s, P_d,PFC,s are effected by I_m1,k,off and I_rr, t_rr. I_s,PFC, I_m1, I_d5 and I_EMI are affected by V_AC, V_BUS, L and f_s,PFC.

$$P_{{L_{b} ,loss}} = (I_{L1}^{2} + I_{L2}^{2} )\rho_{T} l_{{L_{b} }} N_{{L_{b} }} /S{ = }2I_{L1}^{2} \rho_{T} l_{{L_{b} }} N_{{L_{b} }} /S_{{L_{b} }}$$

(12)

$$\eta_{PFC} = \frac{{P_{o,PFC} }}{{P_{o,PFC} + P_{E,loss} + P_{{L_{b} ,loss}} + P_{m,PFC,loss} + P_{d,PFC,loss} + P_{other} }}$$

(13)

3 LLC Resonant Converter

3.1 Waveforms of LLC Resonant Converter in P Mode

To simplify the analysis, following assumptions are taken: (a) all components are ideal devices, (b) V_o is a constant value, (c) duty cycle of Q₃ ~ Q₆ is 0.5. i_r, i_m, i_s, v_ab and v_cd at the resonant frequency are shown in Fig. 5. i_r(t) is sinusoidal waveform. i_m(t) increases in the slope of V_BUS/L_m during 0 ≤ t < 0.5t_r,LLC and decreases in the slope of − V_BUS/L_m during 0.5t_r,LLC ≤ t < t_r,LLC.

Fig. 5

Waveforms of i_r, i_m, i_s, u_ab, and u_cd at f_s,LLC = f_r,LLC

The resonant currents i_r(t) and magnetizing current i_m(t) are given by (14) and (15), where I_r,peak and ω_r are the peak resonant current and angular frequency. θ is phase difference between i_r(t) and v_ab. ω_r = 2πf_r = (L_rC_r)^−0.5.

$$i_{r} (t) = I_{r,peak} \sin (\omega_{r} t - \theta )$$

(14)

$$i_{m} (t) = \frac{{V_{BUS} (4t - t_{r,LLC} )}}{{4L_{m} }}{ = }\frac{{V_{BUS} (4f_{r} t - 1)}}{{4f_{r1} L_{m} }}$$

(15)

Because i_r(t) = i_m(t) at t = 0, then θ can be given by (16). The input energy ${E}_{{V}_{BUS}}$ from V_BUS and the absorbed energy ${E}_{R}$ by load can be obtained by (17) and (18), where N is transformer`s turn ratio. Because ${E}_{{V}_{BUS}}$=${E}_{R}$, I_r,peak can be obtained by solving Eqs. (17) and (18), and θ can be calculated by solving Eq. (19) and (16). Moreover, because i_s(t) = N[i_r(t) − i_m(t)], then teh primary and secondary RMS currents I_p,RMS and I_s,RMS, and the turn-off current I_turnoff can be given by (21).

$$\theta = \arcsin \left[ {\frac{{V_{BUS} t_{s,LLC} }}{{4L_{m} I_{r,peak} }}} \right] = \arcsin \left[ {\frac{{V_{BUS} }}{{4L_{m} I_{r,peak} f_{r} }}} \right]$$

(16)

$$E_{{V_{BUS} }} = \frac{{I_{r,peak} V_{BUS} }}{{2\pi f_{r} }}\sqrt {\frac{{V_{BUS}^{2} - 16I_{r,peak}^{2} L_{m}^{2} f_{r}^{2} }}{{I_{r,peak}^{2} L_{m}^{2} f_{r}^{2} }}}$$

(17)

$$E_{R} = P_{o,LLC} t_{s,LLC} = \frac{{V_{o}^{2} }}{{Rf_{r} }} = \frac{{V_{BUS}^{2} }}{{N^{2} Rf_{r} }}$$

(18)

$$I_{r,peak} = \frac{{V_{BUS} }}{{4L_{m} N^{2} Rf_{r} }}\sqrt {4\pi^{2} L_{m}^{2} f_{r}^{2} + N^{4} R^{2} }$$

(19)

$$\theta { = }\arcsin \left[ {\frac{{N^{2} R}}{{\sqrt {N^{4} R^{2} { + }4\pi^{2} L_{m}^{2} f_{r}^{2} } }}} \right]$$

(20)

$$\left\{ \begin{gathered} I_{p,RMS} = \frac{{V_{BUS} }}{{4\sqrt 2 L_{m} N^{2} Rf_{r} }}\sqrt {4\pi^{2} L_{m}^{2} f_{r}^{2} - N^{4} R^{2} } \hfill \\ I_{s,RMS} = \sqrt {2f_{r} \int\limits_{0}^{{0.5/f_{r} }} {\left[ {i_{r} (t) - i_{m} (t)} \right]^{2} N^{2} dt} } \hfill \\ = \frac{{V_{BUS} \sqrt {5\pi^{2} N^{4} R^{2} - 48N^{4} R^{2} + 12\pi^{4} L_{m}^{2} f_{r}^{2} } }}{{4\sqrt 6 L_{m} f_{r} NR}} \hfill \\ I_{turnoff} = \frac{{V_{BUS} t_{r} }}{{4L_{m} }}{ = }\frac{{V_{BUS} }}{{4f_{r} L_{m} }} \hfill \\ \end{gathered} \right.$$

(21)

3.2 Losses of LLC Resonant Converter

Losses of LLC converter includes MOSFETs’ loss P_m,LLC, transformer’s loss P_T, resonant inductor’s loss P_Lr, and diodes’ loss. The total loss and efficiency of LLC converter are given by (26) and (27). In a half switching period, part of the reactive energy is transferred from V_BUS to resonant tank during θ/(2πf_r) ≤ t < 1/f_r and returns from the resonant tank to V_BUS during 0.5/f_r ≤ t < 0.5/f_r + θ/(2πf_r) in the next half period. The reactive energy W_Q,Vin in a switching period is given by (28), and the reactive power Q_LLC and power factor PF_LLC are given by (29) and (30). Based on FHA method, V_o can be obtained by (31) [22], where Q is the quality factor, and k = L_m/L_r. The selection of f_r and L_m is based on the required P_total,LLC, and C_r can be determined by (33), then L_r can be further obtained.

$$\left\{ \begin{gathered} P_{Q,LLC,s} = {{I_{turnoff}^{2} t_{fall}^{2} } \mathord{\left/ {\vphantom {{I_{turnoff}^{2} t_{fall}^{2} } {C_{FB} }}} \right. \kern-\nulldelimiterspace} {C_{FB} }} \hfill \\ P_{Q,LLC,c} = 2I_{p,RMS}^{2} R_{ds,on} \hfill \\ P_{Q,LLC,loss} = P_{Q,LLC,s} + P_{Q,LLC,c} \hfill \\ \end{gathered} \right.$$

(22)

$$P_{T,loss} = I_{p,RMS}^{2} r_{p,w} + I_{s,RMS}^{2} r_{s,w}$$

(23)

$$P_{{L_{r} ,loss}} = I_{p,RMS}^{2} r_{{L_{r} }}$$

(24)

$$P_{d,LLC,loss} = 2V_{F} I_{s,RMS}$$

(25)

$$P_{total,LLC,loss} = P_{{L_{r} ,loss}} + P_{Q,LLC} + P_{T,loss} + P_{d,LLC} = \frac{{V_{BUS}^{2} \left\{ {C_{FB} \left[ \begin{gathered} 3\pi^{2} R^{2} N^{4} r_{p} + \left( {5\pi^{2} - 48} \right)N^{6} R^{2} r_{s,w} + 12\pi^{4} \left( {r_{p} + N^{2} r_{s,w} } \right)f_{r}^{2} L_{m}^{2} \hfill \\ + {{8\sqrt 6 \pi N^{3} RV_{F} f_{r} L_{m} \sqrt {12\pi^{4} L_{m}^{2} f_{r}^{2} + \left( {5\pi^{2} - 48} \right)N^{4} R_{2} } } \mathord{\left/ {\vphantom {{8\sqrt 6 \pi N^{3} RV_{F} f_{r} L_{m} \sqrt {12\pi^{4} L_{m}^{2} f_{r}^{2} + \left( {5\pi^{2} - 48} \right)N^{4} R_{2} } } {V_{BUS} }}} \right. \kern-\nulldelimiterspace} {V_{BUS} }} \hfill \\ \end{gathered} \right] + 6\pi^{2} N^{4} R^{2} t_{fall}^{2} L_{m}^{2} } \right\}}}{{96\pi^{2} C_{FB} N^{4} R^{2} f_{r}^{2} L_{m}^{2} }}$$

(26)

$$\eta_{LLC} = {{P_{o,LLC} } \mathord{\left/ {\vphantom {{P_{o,LLC} } {\left( {P_{o,LLC} + P_{total,LLC,loss} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {P_{o,LLC} + P_{total,LLC,loss} } \right)}}.$$

(27)

$$W_{{Q,V_{in} }} = \frac{{V_{BUS}^{2} }}{{4\pi L_{m} N^{2} f_{r} R}}\left( {\sqrt {4\pi^{2} L_{m}^{2} f_{r}^{2} + N^{4} R_{2} } - \sqrt {4\pi^{2} L_{m}^{2} f_{r}^{2} } } \right)$$

(28)

$$Q_{LLC} = f_{r} W_{{Q,V_{in} }} = \frac{{V_{BUS} }}{{4L_{m} f_{r} }}\left\{ {\arcsin \left( {\frac{{N^{2} R}}{{\sqrt {N^{4} R^{2} - 4\pi^{2} L_{m}^{2} f_{r}^{2} } }}} \right)\left[ {2\sin^{2} \left( {\pi f_{r} } \right) - 1} \right] + \frac{{2L_{m} f_{r} }}{{N^{2} R}}\sin^{2} \left[ {\frac{\pi }{2}\arcsin \left( {\frac{{N^{2} R}}{{\sqrt {N^{4} R^{2} - 4\pi^{2} L_{m}^{2} f_{r}^{2} } }}} \right)} \right]} \right\}$$

(29)

$$PF_{LLC} = \frac{{P_{o,LLC} }}{{\sqrt {P_{o,LLC}^{2} + Q_{LLC}^{2} } }} = \frac{{4\pi L_{m} N^{2} f_{r} RP_{o,LLC} }}{{\sqrt {N^{4} R^{2} V_{BUS}^{2} - 4\pi L_{m} f_{r} V_{BUS}^{4} \sqrt {4\pi^{2} L_{m}^{2} f_{r}^{2} + N^{4} R^{2} } + 8\pi^{2} L_{m}^{2} f_{r}^{2} V_{BUS}^{2} + 16\pi^{2} L_{m}^{2} N^{4} f_{r}^{2} R^{2} P_{o,LLC}^{2} } }}$$

(30)

$$V_{o} = \frac{{V_{BUS} }}{{N\sqrt {\left[ {1 + \frac{1}{k}\left( {1 - \frac{{f_{r}^{2} }}{{f_{s,LLC}^{2} }}} \right)} \right]^{2} + Q^{2} \left( {\frac{{f_{s,LLC} }}{{f_{r} }} - \frac{{f_{r} }}{{f_{s,LLC} }}} \right)^{2} } }}$$

(31)

$$Q = {{\pi^{2} \sqrt {L_{r} /C_{r} } } \mathord{\left/ {\vphantom {{\pi^{2} \sqrt {L_{r} /C_{r} } } {\left( {8N^{2} R} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {8N^{2} R} \right)}}$$

(32)

$$C_{r} = {1 \mathord{\left/ {\vphantom {1 {4\pi^{2} f_{r}^{2} L_{r} }}} \right. \kern-\nulldelimiterspace} {4\pi^{2} f_{r}^{2} L_{r} }}$$

(33)

4 Design of On-Board Charger

4.1 Design of ILPFC

The design flow of ILPFC is shown in Fig. 6. Once V_AC, f_PFC, L_b and V_BUS are determined, P_E,loss, P_b,loss, P_m,PFC,loss, P_d,PFC,loss and ${P}_{{L}_{b},loss}$ are calculated by the proposed method, and η_PFC can be improved by optimizing f_s,PFC, L_b, V_BUS. The design of L_b can refer to [23].

Fig. 6

Design flow of ILPFC

The relationships between η_PFC and V_AC, f_s,PFC, L_b and V_BUS are presented in the Fig. 7, where α_PFC is the load factor. Figure 7a shows that all loss reduces with the increase of V_AC. Figure 7b shows that almost all loss decreases with the increase of L_b except for P_Lr,loss, because large L_b brings large ESR and copper loss. Figure 7c shows that almost all loss decreases with the increase of f_s,PFC except for P_m,PFC,loss because of large switching loss. Figure 7d shows that almost all loss increases with the increase of V_BUS except for P_d,PFC,loss because of large diodes` current ripple. Figure 7e shows that η_PFC increases with the increase of V_AC. Figure 7f, g present that the influence of L_b on η_PFC and the influence of f_s,PFC on η_PFC are nonlinear. Figure 7h shows that the influence of V_BUS on η_PFC is almost linear. High V_BUS can improve η_PFC, but which leads to high voltage stress.

Fig. 7

Relationship between loss and: a V_AC, b L_b, c f_s,PFC, d V_BUS, and between η_PFC with α_PFC for different: e V_ac, f L_b, g f_s,PFC, and h V_BUS

4.2 Design of LLC Resonant Converter

The design flow of LLC converter is shown in Fig. 8. If V_BUS, V_o,nor are determined, then N can be determined. Then the total losses can be figured out by (27), and f_r and L_m can be selected by the required P_total,LLC, and then L_r and C_r can be determined by voltage gain.

Fig. 8

Design flow chart of LLC converter

The surface map of P_total,LLC and η_LLC at f_s = f_r are shown in Fig. 9a, b for R_ds,on = 150 mΩ, ${r}_{{L}_{r}}$=25 mΩ, r_p,w = 25 mΩ, r_s,w = 22 mΩ, R = 31Ω, V_F = 0.7 V, t_fall = 6 ns, C_FB = 130 pF. Large f_r and L_m is useful to improve η_LLC but reduces the voltage gain, then f_r and L_m must be optimized. If required η_LLC is determined, target area of (f_r, L_m) can be searched in Fig. 9b. In this paper, L_r and C_r are designed as 45 µH and 75 nF. The peak voltage V_o,peak is approximate 433 V, as Fig. 9c shows, which satisfies the requirement V_o,max = 430 V. As Fig. 9d shows, if L_r and C_r are designed as 50 µH and 67.55 nF, V_o,peak = 418 V, the design result will miss the requirement.

Fig. 9

Surface map of: a P_total,LLC(f_r,L_m), b η_LLC(f_r,L_m), and curves of V_o(f_s,LLC) for: c L_r = 45 µH, C_r = 75 nF, and d L_r = 50 µH, C_r = 67.55 nF

5 Control of On-Board Charger

Average current control is used for ILPFC as shown in Fig. 10a. V_BUS,ref is the reference of V_bus. i_L1,ref and i_L2,ref are the references of i_L1 and i_L2, the duty cycles are Temp₁ and Temp₂. An offset d_offset is added to improve PF, the modified duty cycles are Temp₁ + d_offset and Temp₂ + d_offeset. K_d is the offset coefficient. As Fig. 11 shows, if L_b = 220 µH and f_s,PFC = 60 kHz, i_L1 is in DCM and i_AC is distorted near the zero crossing point for K_d = 0. If K_d = 1, i_L1 is in CCM and i_AC is a sinusoidal waveform. Here K_d is set as 0.8.

Fig. 10

Control of: a ILPFC, and b LLC converter

Fig. 11

Waveforms of: a i_L, and b i_AC for K_d = 0 and 1

I_o,ref and V_o,ref are the references of i_o and V_o. The charging curves is given in Fig. 12, if V_o > 275 V, the charging power is 3.3 kW, the converter works in constant voltage mode, otherwise, the converter works in constant current mode, and the maximum charging current is 12A.

Fig. 12

Charging curve of 3.3 kW OBC

6 Experiment Result and Analyses

The requirements are given in Table 1. A 3.3 kW prototype is developed, as Fig. 13 shows. Two same transformers are used for LLC converter, the primary and secondary windings are in series and paralleled, the turn ratio is 17:26. PQ35/35 DMR cores are used for ILPFC, PQ35/35 DMR95 cores are used for LLC converter. TI F28035 is selected for the controller, MOSFETs with types of IPP60R099C6 and IPP65R110CFD are used for Q₁ ~ Q₂ and Q₃ ~ Q₆. In Fig. 13a, conventional OBC is based on single-phase boost PFC and LLC converter, the parameters are L_r = 12 µH, C_r = 99 nF, and L_m = 88 µH, the size is 245 mm × 210 mm × 95 mm, the volume is 298 in³, and the power density is 11.1 W/in³. For this paper`s prototype, the size is 241 mm × 175 mm × 75 mm, as Fig. 13b shows, the volume and power density are 193 in³ and 7.1 W/in³, which is higher than conventional prototype.

Table 1 Key parameters of 3.3 kW OBC

Fig. 13

Pictures of: a conventional, and b the developed 3.3 kW prototypes

Measured i_ac, V_ac, i_L1, and i_L2 at the full-load, half-load and 20% load states are given in Fig. 14. The normal input voltage is 220 V with 50 Hz. At the full and half load states, i_ac is sinusoidal and has the same phase as V_ac, and the PFs are higher than 99.9% and 99.8%. At 20% load state, as Fig. 14c shows, i_L1 and i_L2 are in DCM.

Fig. 14

Measured V_ac, i_ac, i_L1, and i_L2 at: a full, b half, and c 20% load states

Figure 15 shows the measured waveforms of LLC converter. The maximum and minimum switching frequencies f_max,LLC and f_min,LLC are 117 kHz and 47.6 kHz, respectively.

Fig. 15

Measured i_r, v_ab, v_cd, and − v_g for V_o is: a 230 V, b 306 V, and c 430 V

Figure 16 presents the harmonic distribution at 20%, half, and full load states. The THDs at 20%, half, and full load states are 6.3%, 4.09%, and 2.98%, respectively. The PFs at 20%, half, and full load states are 96.99%, 99.91% 99.96%, respectively.

Fig. 16

Harmonic at 20%, half, and full load states

As Fig. 17a shows, measured η_PFC is lower than the theoretical efficiency, the error comes from the driving loss. η_PFC, η_LLC and η_overall of the developed prototype are higher than conventional OBC. The PFC and LLC converters’ peak efficiencies are about 97.3% and 97.4%. The overall and peak efficiencies are higher than 93.3% and 94.8% (Fig. 18).

Fig. 17

Curves of: a η_PFC for theory and test, b η_PFC for two prototypes, c η_LLC, and d η_overall for prototype

Fig. 18

Curves of relationship between power factor and load factor

Figure 19 shows the measured ripple factor of V_o, the ripple factor is calculated by 0.5ΔV_o/V_o, where ΔV_o is the peak-to-peak value of the voltage ripple, the voltage ripple factor decreases with the increase of V_o, which can be explained as follows: firstly, the voltage ripple of V_o comes from the DC-link voltage`s ripple, in this paper, the peak-to-peak ripple of V_BUS is about 20 V, because the voltage ripple can be seen as a AC voltage, it can be transferred to V_o by the transformer, and the transformer turn ratio is about 1.308, then the peak-to-peak voltage ripple transferred from V_BUS to V_o is about 15.3 V. Secondly, the voltage controller of LLC converter has the ability to adjust V_o even V_BUS is fluctuates, then the voltage ripple transferred from V_BUS to V_o can be reduced, in the measured results, the peak-to-peak voltage ripple of V_o ranges from 6 to 7 V in the range of V_o from 230 to 430 V, and the voltage ripple factor is ranging from 1.39 to 0.69% as shown in Fig. 19.

Fig. 19

Curves of relationship between voltage ripple factor and V_o

Table 2 gives the hardware cost of the main components. For ILPFC, the most cost is used for inductors and capacitors, and power devices` cost also takes a great part. For LLC converter, the most part cost is used for MOSFETs and transformer. The hardware cost of ILPFC`s main components is about 30.75 $, the hardware cost of LLC converter is about 40 $, the total cost of the main components is about 70.75$.

Table 2 Hardware cost of main components

The requirements are given in Table 1. A 3.3 kW prototype is developed, as Fig. 13 shows. Two same transformers are used for LLC converter, the primary and secondary windings are in series and paralleled, the turn ratio is 17:26. PQ35/35 DMR cores are used for ILPFC, PQ35/35 DMR95 cores are used for LLC converter. TI F28035 is selected for the controller, MOSFETs with types of IPP60R099C6 and IPP65R110CFD are used for Q₁ ~ Q₂ and Q₃ ~ Q₆. In Fig. 13a, conventional OBC is based on single-phase boost PFC and LLC converter, the parameters are L_r = 12 µH, C_r = 99 nF, and L_m = 88 µH, the size is 245 mm × 210 mm × 95 mm, the volume is 298 in³, and the power density is 11.1 W/in³. For this paper`s prototype, the size is 241 mm × 175 mm × 75 mm, as Fig. 13b shows, the volume and power density are 193 in³ and 7.1 W/in³, which is higher than conventional prototype.

Table 3 shows the comparison between the developed prototypes in literatures [17, 26], where THD_full, PF_full, η_PFC,full, η_LLC,full, η_OBC,full are the THD, PF, PFC`s efficiency, LLC`s efficiency, and overall efficiency at the full load. PD is the power density. The prototype in this paper has better performance in total efficiency, power density, THD and PF compared with the prototypes in literatures [17, 26], this is attributed to the proposed comprehensive design method to optimize OBC`s internal parameters.

Table 3 Comparisons between this work and Ref. [17, 26]

Table 4 shows the comparisons between this paper with literatures [1, 10, 36,37,38]. Compared with literatures [1, 10, 36,37,38], the developed prototype has higher efficiency and wider output voltage range. In literatures [1, 38], SiC-based diodes are adopted on the secondary-side, and the reverse recovery loss of SiC-based diodes is much lower than that of Si-based diodes adopted in this paper, that is to say, if the SiC-based diodes are replaced by Si-based diodes in literatures [1, 38], the advantage of efficiency in this paper over that in literatures [1, 38] will be further enhanced. Compared with this paper, the output voltage range in literature [36] is wider, however, which needs regulate dc-link voltage dynamically to ensure LLC converter always operates at the resonant frequency and sacrifices the efficiency in low output voltage region and increases the current ripple in high output voltage region. The comparison between literatures [37, 38] and the this paper is more fair, because the power is the same, but the comparison result further verified that the efficiency in this paper is higher.

Table 4 Comparisons between prototypes in this paper and references [1, 10, 36,37,38]

7 Conclusion

The loss and efficiency of ILPFC are analyzed by the proposed discrete iterative method. The loss, efficiency and voltage gain of LLC converter are analyzed by FHA method. And the efficiency of OBC is optimized by designing the internal parameters. The ILPFC`s efficiency is improved by about 1.1% compared with conventional scheme. The efficiency of LLC converter is improved by about 1.3% compared with conventional scheme. The PF and THD of the prototype are approximate 99.99% and 2.98%, respectively, the overall efficiency is improved by about 1% compared with conventional 3.3 kW OBC at the full load state.

In the future, the proposed method will be extended to optimize the efficiency of 6.6 kW and 11 kW bi-directional OBC with properly modification for actual topologies.