INTRODUCTION
Synchronous Rectifier (SR) has been widely adopted in the low output voltage applications to reduce the conduction loss of the output rectifier (Lim et al., 2010). Therefore, SR has become a research hotspot associated with the ACDC converter of all kinds of switching power supply.
At present, researches on synchronous rectification are mainly concentrated in buck converter (Stankovic et al., 2012), forward converter (Coban and Cadirci, 2011), flyback converter (Kim et al., 2014), halfbridge converter (Jeong, 2008), pushpull converter. However, the application of SR in a resonant converter has been rarely studied.
Analyzing the existing synchronous rectifier converters, theirs rectifier section generally work in hard switching state. Therefore, when the switching frequency of SR is high, large switching losses are caused and the system efficiency is reduced. At the same time, it is easy to influence the main circuit operation of converter and reduce the system performance because synchronous rectification of the existing converters usually is adopted selfdriving mode.
However, compared with other converters, resonant converter has the advantages of small volume, high transmission efficiency. In particular, capacitor voltage and inductor current in the resonant tank are sine wave, so that a wide range of switching network worked in Zero Voltage Switching (ZVS) or Zero Current Switching (ZCS) can be easy to achieve. Further analysis shows that, due to the existence of resonant tank, the traditional selfdriving cannot be adopted to achieve the SR output for resonant converter.
In this study, the resonant converter circuit based on Inductive Power Transfer (IPT) (Li et al., 2012) is as a research object. By using its own resonant characteristic, a novel control method that ACDC section of the resonant converter works in ZVS SR state is proposed.
This study is organized as follows. ZVS operation principle of SR for the IPT resonant converter is presented. Then, the state space model of the resonant converter is established. In order to quickly and accurately calculate the ZVSoperating cycle of SR, the fixed point function about it is established by using a stroboscopic mapping method which is based on the state space model. Moreover, operating principle of control circuit for ZVS SR is also presented. Finally, experimental results are conducted to verify that control method proposed in this study is effective for the resonant converter circuit based on IPT.
ZVS OPERATION PRINCIPIE OF SR
In this study, the IPT resonant converter circuit used synchronous rectifier is showed in Fig. 1 which consists of a primary and a secondary circuit. In the primary circuit, the switching network (S_{1}, S_{4}and S_{2}, S_{3}) convert the dc voltage source E_{dc} to high frequency ac that drives an LCL resonant tank consisting of L_{i}, C_{P} and L_{P}. Therefore, a highfrequency ac can be generated in the primary coil L_{P}.

Fig. 1:  Circuit diagram of the IPT resonant converter used SR 
Due to the magnetic field coupling, a highfrequency ac voltage is induced in the secondary coil L_{s} which is then completely tuned by the parallel capacitor Cs. In the secondary circuit, the two switching pairs (Q_{1}, Q_{4} and Q_{2}, Q_{3}) constitute a fullbridge SR network. L_{f} and C_{o} constitute a filter link, R_{L} is the load resistance and M is the mutual inductance coupling value between primary coil and secondary.
In order to simplify the analysis, following assumptions are made for the analyses of operating modes:
•  All MOSFETs are considered to be ideal 
• 
The input voltage source E_{dc} is ideal 
• 
The filter inductor L_{f} current is continuous 
Critical operation waveforms of the system are given in Fig. 2, for steadystate operation. v_{AB }is the input voltage of the primary fullbridge converter. i_{i} is the flowing current of inductor L_{i}. v_{Cs} is the voltage across capacitor Cs. v_{Qi}(i = 1, 2, 3, 4) is the drainsource voltage of switch Q_{i} in the secondary SR network. G_{i}(i = 1, 2, 3, 4) is the drive voltage of switch Q_{i} in the secondary SR network.
Meanwhile, so as to clearly analyze operation modes of the circuit shown in Fig. 1, time t_{0} is assumed to be the starting point of one operational cycle and stipulate positive direction of the dc energy E_{dc} injected is that switches S_{1} and S_{4} are turned on. When Switches Q_{1 }and Q_{4} are turned on, this is SR output positive direction of the AC voltage v_{Cs}. So, effective circuits for each mode in one operational cycle are given in Fig. 3.
Mode 1 (t_{0}t_{1}): The state of primary circuit is that dc energy is injected from negative direction. Switches S_{2} and S_{3} are turned on when S_{1} and S_{4} are turned off. The state of secondary circuit is that the AC voltage v_{Cs} is output from positive direction of the synchronous rectifier.

Fig. 2:  Critical steadystate operation waveforms 
Switches Q_{1 }and Q_{4} are reversely turned on when Q_{2} and Q_{3 }are turned off. In this mode, its equivalent circuit is shown in Fig. 3a.
Mode 2 (t_{1}t_{2}): The state of primary circuit is that energy is injected from positive direction. Switches S_{1} and S_{4} are turned on when S_{2 }and S_{3} are turned off. The state of secondary circuit is that the AC voltage v_{Cs} is output from positive direction of the synchronous rectifier. Switches Q_{1 }and Q_{4} are reversely turned on when Q_{2} and Q_{3} are turned off. In this mode, its equivalent circuit is shown in Fig. 3b.

Fig. 3(ad): 
Equivalent circuits for each operation mode (a) Mode 1, (b) Mode 2, (c) Mode 3 and (d) Mode 4 
Mode 3 (t_{2}t_{3}): The state of primary circuit is that energy is injected from positive direction. Switches S_{1} and S_{4} are turned on when S_{2 }and S_{3} are turned off. The state of secondary circuit is that the AC voltage v_{Cs} is output from negative direction of the synchronous rectifier. Switches Q_{2 }and Q_{3} are reversely turned on when Q_{1} and Q_{4} are turned off. In this mode, its equivalent circuit is shown in Fig. 3c.
Mode 4 (t_{3}t_{4}): The state of primary circuit is that energy is injected from negative direction. Switches S_{2 }and S_{3} are turned on when S_{1 }and S_{4} are turned off. The state of secondary circuit is that the AC voltage v_{Cs} is output from negative direction of the synchronous rectifier. Switches Q_{2 }and Q_{3} are reversely turned on when Q_{1} and Q_{4} are turned off. In this mode, its equivalent circuit is shown in Fig. 3d.
Through the above analysis, the switching states of two primary switching pairs (S_{1}, S_{4 }and S_{2}, S_{3}) are changed, when the current of inductance L_{i} is zero. This time that their switching states are changed between turned on and off, the current flowing through them are zero.Hence, the primary switching inverter works in zero current soft switching state.
In the secondary circuit, the voltage v_{Cs }of the capacitor C_{s }is a sinusoidal voltage. It is obvious to know that when v_{Cs }is equal to zero, the drain to source voltage v_{Qi} of the four switches (Q_{1}, Q_{2}, Q_{3} and Q_{4}) are also equal to zero. Therefore, ZVS SR output of the AC voltage v_{Cs} can be achieved by controlling the switching pairs (Q_{1}, Q_{4} and Q_{2}, Q_{3}) turned on or off when v_{Cs }is equal to zero.
In summary, the primary fullbridge converter working in zero current soft switching state and the secondary fullbridge SR network working in zero voltage soft switching state can be achieved in the IPT resonant converter.
STEADY STATE MODELING
In order to carry out the numerical analysis of the circuit shown in Fig. 1, a steady state mathematical model of the system is necessary to be established.
Assume that the switching period of the primary fullbridge inverter shown in Fig. 1 is T under steadystate conditions. Then the input voltage of the primary resonant tank can be described as:
where, m is zero or a positive integer.
In the secondary circuit shown in Fig. 1, switch transitions of the fullbridge SR network are determined by the polarity change of the resonant voltage v_{Cs}. Consequently, a sign function Sgn(v_{Cs}) can be used to represent operation of the fullbridge SR network:
After such that representations, the circuit diagram shown in Fig. 1 is simplified as an equivalent circuit shown in Fig. 4.
According to Kirchhoff’s current and voltage laws, the differential equations of the equivalent circuit as shown in Fig. 4, can be presented as follows:
where, Δ = L_{S}L_{P}M^{2}. By choosing x = [i_{i} v_{Cp} i_{Lp} i_{Ls} v_{Cs} i_{Lf }v_{Co}]^{T }and u = [E_{dc}] as the state vector and the input vector of the system, respectively. According to the differential Eq. 39, the system can be described by the following state space model:
where, i represent the figures of operation modes shown in Fig. 3:

Fig. 4:  Equivalent circuit of the IPT resonant converter used SR 
DETERMING ZVSOPERATING CYCLE OF SR
The ZVS operating point of the SR circuit must be obtained to make fullbridge SR network of the secondary circuit work at zero voltage soft switching state.
The system matrix A_{1}, A_{2}, A_{3}, A_{4} are invertible, so that the analytical solution of Eq. 1) has the following format:
where, x_{0} = x(t)_{t = 0} and φ(t) = e^{At}x(t).
As can be seen from Fig. 2 and 3, during one operational cycle, the final value of each mode equals to the initial value of the next mode. Therefore Eq. 11 can be extended into four stroboscopic mapping equations to cover all the four modes (Liu et al., 2011):
where, I is a 7order unit matrix and x_{0} is the initial value of the circuit state vector.
Under steadystate conditions, the state vector repeats periodically which means the state vector results in a fixed point x* is described as follows:
Substitute Eq. 1315 into 12, the fixed point x* can be expressed as follows:
where, α = (Iφ_{4}φ_{3}φ_{2}φ_{1})^{1}.
Corresponding to the ZVS condition of the fullbridge SR network, the voltage v_{Cs} must be zero at the switching instants. Then from Eq. 17, ZVS operation cycle (T_{ZVS}) of the fullbridge SR network can be obtained as follows (Tang et al., 2009; Sun et al., 2011):
where, Y = [0 0 0 0 1 0 0] is a selection matrix for the state variable v_{Cs}.
OPERATING PRINCIPIE OF CONTROL CIRCUIT
In order to achieve zero voltage soft switching state operation of the secondary fullbridge SR network, the working operation of control circuit as shown in Fig. 5.
The operation processes of the control circuit are described as follows. First of all, the AC voltage v_{Cs} through the sampling circuit, a weak signal (v_{1}) with the same frequency can be obtained. The phase of the signal (v_{1}) must be compensated, because the control circuit operation can lead to a delay of the signal. To precisely control switching pairs (Q_{1}, Q_{4} and Q_{2}, Q_{3}) turned on or off, the zerocrossing detection circuit is designed for detecting of the AC voltage v_{Cs}. Therefore, after the previous circuit, the digital signal (G_{in2}) digital signal can be obtained. G_{in2 }is used as the input of driving circuit for switches (Q_{2}, Q_{4}). In order to achieve the two SR switching pairs (Q_{1}, Q_{4} and Q_{2}, Q_{3}) working in zero voltage soft switching state, the two drive voltage pairs (G_{1}, G_{4} and G_{2}, G_{3}) must be complementary.

Fig. 5:  ZVS control diagram of the fullbridge SR inverter 
So, invert G_{in2} and then G_{in1} is used as the input of driving circuit for switches (Q_{1}, Q_{3}).
EXPERIMENTAL STUDY
According to the Fig. 1 and operating principle of control circuit. The experimental circuit is built to validate the feasibility of control method proposed and the experimental parameters are shown in Table 1.
According to Eq. 18, the parameters shown in Table 1 are substituted into it. Therefore, the ZVS operation cycle (T_{ZVS}) of the fullbridge SR network can be obtained by solving M file in matlab. By solving Eq. 18, T_{ZVS} is equal to 13.21 μsec. It means that the frequency of the driving voltage (G_{1}, G_{2}, G_{3} and G_{4}) is 75.7 kHz.
According to the result of T_{ZVS}, MOSFET IRFB4110 is selected as the component for SR and control circuit of the experiment for SR is well designed.
The experimental waveforms of the input voltage (v_{AB}) of the primary fullbridge converter and the flowing current (i_{i}) of inductor L_{i} are showed in Fig. 6. From Fig. 6, it can be seen that the polarity of voltage (v_{AB}) is changed when the current i_{i }is equal to zero. Hence, the conclusion that the two primary switching pairs (S_{1}, S_{4 }and S_{2}, S_{3}) works at zero current soft switching state can be clearly obtained.
Figure 7 shows the experimental waveforms of the waveforms of v_{Cs}, G_{1} and G_{2}. From Fig. 7, it can be seen that the voltage v_{Cs} is a sine wave. Meanwhile, when v_{Cs} is greater than zero, switch Q_{1} is turned on owing to G_{1} driving but switch Q_{2} is turned on owing to G_{2} driving when v_{Cs} is less than than zero.
The experimental waveforms of v_{Cs}, v_{Q1}, v_{Q4}, G_{1} and G_{4} are shown in Fig. 8. From Fig. 8, it can be seen that switches Q_{1}, Q_{4} are simultaneously turned on or off because of the drive voltage G_{1}, G_{4}, respectively. At same time, it also can be found that the switching states of Q_{1} and Q_{4} are changed when the drain to source voltage of Q_{1} and Q_{4} is equal to zero.
Figure 9 shows the experimental waveforms of the drive voltage G_{2}, G_{3}, the AC voltage v_{Cs} and the drain to source voltage of Q_{2}, Q_{3}.

Fig. 6:  Experimental waveforms of the current i_{i} and voltage v_{AB} 
Table 1:  Experimental parameters 

As it can be seen from Fig. 9, the drive voltage G_{2}, G_{3 }simultaneously drive switches Q_{2}, Q_{3} which are turned on or off, respectively. Moreover, it can be found that the switching states of Q_{2 }and Q_{3} are changed when the drain to source voltage of Q_{2 }and Q_{3}is equal to zero.
Analyzing the Fig. 79, it can be seen that when v_{Cs} is in the positive half cycle, switches Q_{1}, Q_{4} are turned on and switches Q_{2}, Q_{3} are turned off.

Fig. 7:  Experimental waveforms of v_{Cs}, G_{1} and G_{2} 

Fig. 8:  Experimental waveforms of v_{Cs}, v_{Q1}, v_{Q4},G_{1} and G_{4} 
When v_{Cs} is in the negative half cycle, switches Q_{2}, Q_{3} are turned on and switches Q_{1}, Q_{4} are turned off. This means that the AC voltage v_{Cs} has been synchronously rectified. Moreover, all of the drain to source voltage of four switches Q_{1}, Q_{2}, Q_{3 }and Q_{4} are equal to zero when switching states of four switches are changed. This means that the secondary fullbridge SR network works in zero voltage soft switching state.
Therefore, the experimental results show that zero current soft switching state of the primary fullbridge and zero voltage soft switching state of the secondary fullbridge SR network have been achieved in the IPT resonant converter.

Fig. 9:  Experimental waveforms of v_{Cs}, v_{Q2}, v_{Q3}, G_{2} and G_{3} 
CONCLUSION
In this study, due to resonance characteristics of the resonant converter based on IPT, a novel control method has been proposed to achieve ZVS SR state operation for the ACDC network of the resonant converter. A steady state mathematical model has been established to determine the ZVS operating point of SR network which is based on the analysis about ZVS operation principle of SR and stroboscopic mapping method. The control circuit working operation has also been presented to guide the design of corresponding hardware circuit. Finally, the method proposed in this study has been verified by experimental results.