2.1. Circuit Structure and Mode Analysis
The prototype of the six-switch IPIFR–WPT system is shown in
Figure 1. In
Figure 1, a unidirectional power supply comprises an isolation diode
D0 and a DC power supply
Edc. The purpose of a one-way power supply is to ensure that energy entering the system does not return to the power supply. Switches
S1–
S4 form a full-bridge converter to connect power E
dc to the system and inject energy into the system. Switches
S5 and
S6 let the primary compensation capacitor
Cp be isolated from or connected to the system. Diodes
D1–
D6 are the bypass diodes of switches
S1–
S6, wherein bypass diodes
D1–
D4 do not function during the entire working process due to their different directions from isolation diode
D0; the bypass diodes
D5 and
D6 are used for continuous current conduction during the resonance process and can isolate
Cp from the system. The self-inductances of the primary and secondary coils are
Lp and
Ls, while
M is the mutual inductance of the coils.
Rp and
Rs are the parasitic resistance of the primary and secondary coils, respectively. The load
R0 consumes the energy received by the secondary circuit after being rectified by the back-end rectifying circuit of the secondary circuit. Supposing the inductive characteristics and the voltage drop of the back-end rectifier circuit are ignored, in that case, the equivalent load
RL can be used to convert the load resistance
R0 to the input terminal of the rectifier circuit, and the conversion relationship is as follows:
Figure 2 is the schematic diagrams of the operation modes of the six-switch IPIFR–WPT system. When the system operates, there are eight operating modes. Mode 1 connects the power supply
Edc to the system in a forward direction, a forward power injection mode. Modes 2–4 are resonant modes, also called free resonance processes, due to the isolation of the power supply
Edc from the system. Note that Modes 2 and 4 are transitional modes, both of which are unidirectional resonance processes. According to the symmetry of the full-bridge converter operation, Mode 5 reversely connects the power supply
Edc to the system as a reverse power injection mode. Furthermore, Modes 6–8 are resonant processes opposite to the working processes of Modes 2–4.
Figure 3 shows the waveform diagram of the source current
i1, the primary capacitor voltage
up, and the primary inductor current
ip of the six-switch IPIFR–WPT system under steady-state operation. According to
Figure 2 and
Figure 3, the operation modes of the six-switch IPIFR–WPT system can be analyzed as follows:
Mode 1 [t0, t1]: This is a forward power injection mode with a duration of ξ1 = t1–t0. Before the time t0, the switches S1 and S4 have been turned on in advance, and the switches S2 and S3 remain turned off. At this time, the primary capacitor voltage up is greater than Edc, forcing D0 to be turned off, isolating the power supply Edc from the system, thereby preventing the energy from being injected into the system. Until the time t0, the voltage up drops to equal to Edc, causing diode D0 to turn on, and the current ip flows from the branch D6–Cp–S5 to the switches S1 and S4 that have been turned on in advance. During the period [t0, t1], while the primary capacitor Cp is isolated from the system, the DC power supply Edc directly injects energy into the primary inductor Lp.
Mode 2 [t1, t2]: This is a transitional mode of the first free resonance process. At the moment t1, the switches S1 and S4 turn off, and at the same time, the current ip will continuously conduct through the bypass diode D6. At this time, the turning off of switches S1 and S4 meets the ZVS condition. In Mode 2, the primary inductor Lp and the primary compensation capacitor Cp form a resonant cavity and begin passive free resonance. The duration of Mode 2 is very short and can generally be taken as 1–2 μs, even a few hundred nanoseconds.
Mode 3 [t2, t3]: This is a bidirectional free resonant mode. Since the switch S6 turns on at t2, the primary current ip can achieve bidirectional free resonance in the resonant network Lp–Cp. During Mode 3, after the value of the current ip resonates to less than zero, the flow direction of ip in the branch is S5–Cp–S6. After that, at time t3, the switch S5 turns off, and the current ip can still continuously conduct through the bypass diode D5. The switch S5 turns off to meet the ZVS condition. At the same time, at t3, the switches S2 and S3 turn on. Due to the capacitance voltage up being less than −Edc at this time, the isolation diode cannot be turned on, and the current cannot be exchanged from the branch D5–Cp–S6 to the switches S2 and S3. That is, before and after the turn-on of S2 and S3, the current flowing through them is zero, satisfying the zero current switching (ZCS) condition.
Mode 4 [t3, t4]: In this mode, the system remains in a state of free resonance. In addition, the current ip continuously conducts through the branch D5–Cp–S6 during the time span until the time of t4, when the value of up rises to equal to −Edc through resonance. At time t4, because the switches S2 and S3 have turned on in advance at t3, the current ip can be naturally switched from the branch D5–Cp–S6 to the switches S2 and S3, thereby switching to Mode 5.
Mode 5 [t4, t5]: This is a reverse power injection mode, which is reversed with Mode 1. Due to the natural commutation of the current ip from the branch D5–Cp–S6 to the switches S2 and S3 that have turned on before t4, the power supply Edc is reversely connected to the system and injects energy into the primary inductor Lp.
Mode 6 [t5, t6]: This is a transitional mode of the second free resonance process, which is symmetric to Mode 2 and has a short duration. At time t5, the switches S2 and S3 turn off. Since the current ip will be continuously conducted through the bypass diode D5, turning off switches S2 and S3 meets the ZVS condition. In Mode 6, the primary inductor Lp and the primary compensation capacitor Cp form a resonant network and begin free resonance.
Mode 7 [t6, t7]: This is a bidirectional free resonant mode symmetric to Mode 3. At t6, the switch S5 turns on. Due to the capacitance voltage up greater than Edc, the current ip continuously conduct through the bypass diode D5 without commutation to S1 and S4. Therefore, the turn-on of switch S5 satisfies the ZVS condition. Then the primary current ip can achieve bidirectional free resonance in the resonant network Lp–Cp during [t6, t7].
Mode 8 [t7, t8]: This is a transitional mode symmetric to Mode 4. At t7, switch S6 turns off while turning on switches S1 and S4. Because of the voltage up being greater than Edc, D0 is turned off, and the current ip continuously conducts through D6 without commutation to S1 and S4. Therefore, S6 meets the ZVS condition, while the opening of S1 and S4 satisfies the ZCS condition. The resonant network maintains resonance during [t7, t8]. Until t8, the voltage up is equal to Edc, and the current ip naturally commutates from the resonant network to S1 and S4. Meanwhile, the system operates into Mode 1.
2.2. Calculation of System Soft Switching Operating Point
Based on the mode analysis of the six-switch IPIFR–WPT system in the previous section, the switching conditions for its eight operation modes are shown in
Figure 4. The eight operation modes of a six-switch IPIFR–WPT circuit can be classified as four cyclic operating processes, namely, the forward power injection process, the first free resonance process, the reverse power injection process, and the second free resonance process. Among them, Mode 1 and Mode 5 are forward and reverse power injection processes, respectively. Modes 2, 3, and 4 jointly constitute the first free resonance process, and Modes 6, 7, and 8 jointly constitute the second free resonance process.
Figure 4 shows no switch operates when the free resonance process switches to the power injection process. The switching condition is to rely on the voltage
up to form a voltage clamp with the power supply voltage
Edc through the diode
D0. In Mode 8, the switch groups
S1 and
S4 have turned on, but at this time, the voltage
up is greater than
Edc, and the system remains in the free resonance process. Meanwhile,
ip > 0, that is,
up will continue to decrease until
up drops to equal to
Edc, and the diode
D0 reaches the critical conduction condition. At the same time, the current is commutated from branch
S6–
Cp–
S5 to branch
S4–
Edc–
D0–
S1. The system switches from a free resonance process to a forward power injection process, and the voltage
up on the primary capacitor
Cp will maintain the voltage clamp at
Edc. Similarly, when switching from Mode 4 to Mode 5, it is necessary to clamp the voltage
up on the primary capacitor
Cp to −
Edc, and the free resonance process completes the switching to the reverse power injection process.
For convenience, the switching conditions
β1,
β2,
β3, and
β4 between the four operation processes are shown in
Figure 4 and written in parallel:
Figure 5 shows the equivalent circuit diagram of the main operation process of the six-switch IPIFR–WPT system. Since the state space model of the system during the forward power injection process and the reverse power injection process are entirely consistent, the equivalent circuit model is shown in
Figure 5a. However, the bus voltages during the forward power injection and reverse power injection processes are different due to the conduction switch sets of the full-bridge converters, and their input voltages are
Edc and −
Edc, respectively. At the same time, the system models for the first and second free resonance processes are also entirely consistent, with their equivalent circuit models shown in
Figure 5b, and there is no power input in the free resonance process.
During the power injection process in
Figure 5a, the primary compensation capacitor
Cp is isolated from the system by the switches
S5 and
S6 being turned off. The DC power supply directly charges the primary inductance
Lp through full-bridge forward conduction (switches
S1 and
S4) or reverse conduction (switches
S2 and
S3), with input voltages of
Edc or −
Edc, respectively. At the same time, due to the isolation diode
D0, the current
i1 only flows in one direction,
i1 > 0, and energy can only be injected into the system from the power source and cannot be returned to the power source. In the free resonance process shown in
Figure 5b, the DC power source is isolated from the system, while the inductance
Lp and capacitance
Cp in the primary circuit are directly connected to form a resonance circuit. Hence, through mutual inductance coupling, electrical power is transferred from the primary circuit to the secondary circuit.
According to
Figure 5, set the state variable as the voltage
up on the primary capacitor
Cp, the current
ip on the primary coil inductance
Lp, the current
is on the secondary coil inductance
Ls, and the voltage
uo on the secondary capacitor
Cs, that is,
x = [
up,
ip,
is,
uo]
T. Because the secondary capacitance
Cs and the equivalent load resistance
RL are in parallel, the voltage
uo can also be considered as the equivalent output voltage. Assuming the system input is
u = [
Edc], differential equations can be listed as follows for each state variable according to Kirchhoff’s voltage law and current law:
Forward power injection process:
First free resonance process:
Reverse power injection process:
Second free resonance process:
where
For a linear time-invariant system Σ: (
A,
B,
C,
D), if the system matrix
A is invertible, and when the initial time is
t0, the corresponding time domain solution can be expressed as follows [
24]:
where Φ(
t) = exp{
At},
I is the identity matrix,
xzi(
t) is the zero input response,
xzt(
t) is the zero state response, and
x0 is the initial state of the system at the initial time
t0.
Since the system matrices
A1 and
A3 of the power injection process are irreversible, it is impossible to bring them into Equation (11) directly. Furthermore, the voltage
up on the primary capacitor
Cp is clamped at
Edc during the power injection process. Hence, the differential equation of
up can be rewritten as follows:
Note that Equations (2) and (3) only hold during the power injection process. The system matrices
A1,
A3 and input matrices
B1,
B3 of the power injection process will be rewritten to be reversible as follows:
According to Equation (11), if the initial state of the four linear time-invariant systems at their respective initial time t0 is x0, the time domain solutions of the subsystems can be expressed as follows:
Forward power injection process:
First free resonance process:
Reverse power injection process:
Second free resonance process:
Due to the symmetry of the two energy resonance processes and the two free resonance processes, the time interval experienced by their subsystems is the same, so the time interval between the two power injection processes is assumed to be
ξ1. Set the time interval between two power injection processes as
ξ2. Let the initial value of the system in the nth cycle be
xn, and the terminal values of the four subsystems be
xn1,
xn2,
xn3, and
xn+1, respectively, where
xn+1 is the terminal value of the system in the
nth cycle and also the initial value of the system in the (
n + 1) cycle. The expressions for the terminal values
xn1,
xn2,
xn3, and
xn+1 of each subsystem are as follows:
According to the modal analysis and the symmetry of the full-bridge system, when the system’s operating state reaches a steady state, there is
xn = −
xn2 =
xn+1. Generally,
xn =
xn+1 is called a fixed point, and the mapping relationship between
xn and
xn+1 becomes a fixed point mapping. To simplify the iterative process, take
xn = −
xn2 for fixed point mapping calculations, and the following equation can be obtained:
After transforming and bringing
ξ2 = 1/2
T −
ξ1 for Equation (23), we can obtain the following equation:
According to the modal analysis and subsystem switching conditions, the initial value of the system needs to meet the boundary
β4:up =
Edc and
ip > 0. Namely, the element up in the fixed point
xn is constant, and the following equation can be expressed:
Therefore, the soft switching operation point of the six-switch IPIFR–WPT system is the solution of Equation (25), and we can construct the following function
g(
ξ1) to solve Equation (25):
The parameters of the six-switch IPIFR–WPT system are shown in
Table 2. The curve of the function
g(
ξ1) with
T = 110 μs is shown in
Figure 6. There are two results of
g(
ξ1) = 0, that is,
ξ1a = 11.11 μs and
ξ1b = 36.02 μs. Then, we bring
ξ1a and
ξ1b into (20) to solve the corresponding fixed point
xn, listed in
Table 3.
The fixed point
xn is both the initial and final values of the steady-state period. Moreover, the fixed point
xn must satisfy the boundary condition
β4 (
up =
Edc and
ip > 0). In
Table 3, when taking
ξ1a = 11.11 μs, the value of fixed point
xn satisfies the boundary condition
β4., and when taking
ξ1b = 36.02 μs,
ip = −3.272 A < 0 does not meet the boundary condition
β4. Therefore, the root
ξ1b = 36.02 μs could be abandoned (
xn = −
xn2 =
xn+1).
After calculating the fixed point
xn and the transition state quantity
xn1 of the six-switch IPIFR–WPT system, according to the proposed symmetry of the full-bridge converter, there are
xn = −
xn2 =
xn+1,
xn1 = −
xn3. Then, in order to more intuitively represent the trade-off between the two results mentioned above, by substituting
xn and
xn1 obtained in
Table 3 into Equations (19)–(22), the curves of state variables in the complete steady-state cycle can be obtained, and the curves of current
ip and voltage
up in them can be plotted in
Figure 7.
As can be seen from
Figure 7, taking
ξ1a = 11.11 μs, the simulation waveform is consistent with the mode analysis, and all boundary conditions can be satisfied within a steady-state cycle. However, taking
ξ1b = 36.02 μs, the calculated operating waveform is inconsistent with the expected, and the boundary conditions
β2 (
up = −
Edc and
ip < 0) and
β4 (
up =
Edc and
ip > 0) cannot be satisfied, so that
ξ1b can be abandoned.