Constant Output-Voltage Design for Bi-Directional Wireless Power Transfer System with Multiple Stages

: For the application of wireless power transfer (WPT) technology in a robot (like the snake robot), the power is supposed to be transferred to each device across multiple robot joints. This paper proposes a multi-stage bi-directional WPT (MB-WPT) system that not only provides power to multiple loads but also increases the power transfer distance. Besides, the last stage can charge for the preceding stages by reverse power transfer. The constant output voltage can be achieved whether the power is transmitted in a forward or reverse direction, and di ﬀ erent output voltages for each stage can be achieved to satisfy the respective voltage requirement through the parameter design method. The validity of the theoretical analysis and the feasibility of the system are veriﬁed by experiments.


Introduction
Based on the principle of magnetic induction, wireless power transfer (WPT) technology realizes non-contact transmission of energy from the transmitter to the receiver. Currently, WPT technology is widely used in electric vehicles, kitchen appliances, household equipment, mining equipment, implantable medical devices, and so on. WPT technology simplifies the structure of power transfer system, improves its performance, and makes the appliances more compact [1][2][3][4]. For robot applications, the flexibility and convenience of power supply can be improved by using wireless power transfer technology. As for the snake robot application, it is necessary to provide power for multiple loads. If the power can be transferred through robot joints to multiple loads, the risks of corrosion, twisting or damage due to insulation can be reduced. The flexibility and robustness of robot joint rotation can be improved. Furthermore, if the last stage can charge for the preceding stages which have power shortage, it can extend the working time of the snake robot. Therefore, the multi-stage bi-directional power transmission can improve the performance of the snake robot. In a WPT system, both the magnetic field and efficiency decrease rapidly with increased distance [5,6]. Through adding repeaters between the transmitter and the receiver, the transmission distance increases significantly, the transmission power and efficiency are enhanced, and the system performance is improved [7][8][9]. A battery-charging system with one repeater is proposed to expand the charging area [10]. To improve the efficiency of the system, the distance between the coils and the operation frequency are optimized [11]. The frequency splitting caused by a medium frequency and an odd number of repeaters in the system is also analyzed [12]. It also evaluates the output power and efficiency of the system by optimizing the position and number of the repeaters. A new relay coil is proposed to increase the position flexibility. The previous studies indicate that the frequency of a domino WPT system is relatively high. To reduce the power loss of the coils, a domino WPT system with a relatively low operation frequency is designed [13]. It is demonstrated that the coaxial straight domino system with unequal spacing has a higher efficiency than the system with equal

Proposed MB-WPT System
A three-stage system is taken as an example to analyze the proposed MB-WPT system, and the schematic diagram is shown in Figure 1. Aiming at the application of moving equipment like the robot, normally, a static power input stage (Stage 1) which is connected to the DC power supply is required to charge a robot battery. The remaining stages (Stage 2 and Stage 3) are mounted on the robot. In this paper, the power transfer among multiple stages on the robot should be bi-directional to distribute energy among multiple devices on the robot. In the forward power transmission, the power is transferred from power gird (Stage 1) to moving equipment which includes Stage 2 and Stage 3. The DC voltage of the power source is expressed as U dc1 . A high-frequency full-bridge inverter working at a fixed frequency is utilized to produce a square-wave voltage source for the following resonant tanks. The operation frequency is f, and the angular frequency is ω (ω = 2πf ). In the MB-WPT system, the battery is used as the load at each stage. If the robot leaves the power source and the preceding stages have power shortage, the system works in Mode B, where the power is transferred from Stage 3 to Stage 2. It is not necessary to transfer power from the moving equipment to the power gird. Therefore, the charging process (forward power transfer process) covers Stage 1, Stage 2 and Stage 3, while the reverse power transfer process only covers Stage 2 and Stage 3. In Mode B, the battery of the last stage charges for the preceding stages by reverse power transmission and the voltage of the battery is expressed as U dc3 . To analyze the system, some fundamental definitions are proposed as follows: the resistances of each primary coil and each secondary coil are expressed as R pi and R si , and the mutual inductance is expressed as M i . There are three stages in the system which can be divided into three categories. LCCL topology (L p11 -C p11 -C p12 -L p12 ) is adopted in Stage 1, LC series topology (L s2 -C s2 ) is adopted in Stage 3, and Stage 2 adopts both topologies (L p21 -C p21 -C p22 -L p22 and L s1 -C s1 ). Each joint consists of a primary coil of the preceding stage and a secondary coil of the subsequent stage. The impedances of the loads are expressed as R 2 and R 3 , and the equivalent AC impedances of the loads are expressed as R l2 and R l3 . The relation between R li and R i is R li = 8R i /π 2 (i = 2, 3).  The relationship between the AC voltage Uaci after inversion and the DC voltage Udci is as follows:

Reverse transmission
If Equation (2) is satisfied, the equivalent impedances of all resonant networks exhibit resistance. The relationship between the AC voltage U aci after inversion and the DC voltage U dci is as follows: If Equation (2) is satisfied, the equivalent impedances of all resonant networks exhibit resistance.

Mode A
The equivalent circuit of Mode A is shown in Figure 2. When the system works in Mode A, the power is transferred from Stage 1 to Stage 2 and Stage 3. The output voltages, output power of each stage and the efficiency are shown in Table 1.

Mode B
The equivalent circuit of Mode B is shown in Figure 3. When the system works in Mode B, the battery of the Stage 3 charges for the preceding stage by reverse power transmission. The output voltage, output power and the efficiency are shown in Table 2.  The output voltage of R l2 The output voltage of R l3 In Table 1, Z aeq2 , A and B can be expressed as

Mode B
The equivalent circuit of Mode B is shown in Figure 3. When the system works in Mode B, the battery of the Stage 3 charges for the preceding stage by reverse power transmission. The output voltage, output power and the efficiency are shown in Table 2.

Mode B
The equivalent circuit of Mode B is shown in Figure 3. When the system works in Mode B, the battery of the Stage 3 charges for the preceding stage by reverse power transmission. The output voltage, output power and the efficiency are shown in Table 2.

Rl2
Reverse transmission    The output voltage of R l2 Output power Table 2, C can be expressed as

Output Characteristic Analysis
In the multi-stage WPT system, it is necessary to guarantee that each stage has a steady output performance. The load and the mutual inductance may vary in the application of a WPT system. The output characteristics of the system are analyzed from the variation of loads and mutual inductances, and the calculated results and analysis are as follows.

Analysis of the Output Voltage
In Mode A, because the resistances of coils are smaller than the loads, assuming that the resistances of the coils are ignored, the output voltage at each stage is Equations (7) and (8) indicate that the output voltages are independent of the loads. Similarly, in Mode B, the output voltage U bo2 can be simplified if the resistance of each coil is negligible, and it is expressed as According to Equation (9), when the system works in Mode B, the output voltage is independent of the load.
The calculated results and analysis under both power transfer directions are as follows. Figure 4 shows the relationship between the output voltages and the loads when the system works in Mode A. As can be seen from Figure 4, when the load resistances R 2 and R 3 both increase Energies 2020, 13, 3739 6 of 14 from 10 to 30 Ω, U ao2 varies from 29.5 to 31.7 V and U ao3 varies from 25.7 to 28.7 V, and in other words, both output voltages vary slightly. It can be understood that the output voltage at each stage is independent of the loads under the condition of Mode A.  The relationship between the output voltages and the mutual inductances in Mode A is shown in Figure 5. In the design of a robot MB-WPT system, the relative position of the primary coil and secondary coil at each stage varies little, so that the variation of mutual inductance is also small. As can be seen from Figure 5a, Uao2 varies slightly with the increase in M2, but it is determined by M1. When M1 and M2 increase from 6 to 7 μH, Uao2 increases from 25 to 29.8 V. Figure 5b shows the variation of Uao3 with the mutual inductances. When both M1 and M2 increase from 6 to 7 μH, the minimum of Uao3 is 20 V, and the maximum of Uao3 is 26.9 V. The relationship between the output voltages and the mutual inductances in Mode A is shown in Figure 5. In the design of a robot MB-WPT system, the relative position of the primary coil and secondary coil at each stage varies little, so that the variation of mutual inductance is also small. As can be seen from Figure 5a, U ao2 varies slightly with the increase in M 2 , but it is determined by M 1 . When M 1 and M 2 increase from 6 to 7 µH, U ao2 increases from 25 to 29.8 V. Figure 5b shows the variation of U ao3 with the mutual inductances. When both M 1 and M 2 increase from 6 to 7 µH, the minimum of U ao3 is 20 V, and the maximum of U ao3 is 26.9 V.  Figure 6 shows the variation of the output voltage with the mutual inductance and the load in Mode B. It can be seen that Ubo2 is independent of the load, but it decreases with the increase in M2. When M2 increases from 6 to 7 μH, Ubo2 decreases from 35 to 29 V.  Figure 6 shows the variation of the output voltage with the mutual inductance and the load in Mode B. It can be seen that U bo2 is independent of the load, but it decreases with the increase in M 2 . When M 2 increases from 6 to 7 µH, U bo2 decreases from 35 to 29 V.

Analysis of the Efficiency
In the WPT system, the efficiency is worthy of attention. The calculated results and analysis of the variation of the efficiency with the mutual inductances and the loads are as follows. The variation of the efficiency η with the mutual inductances and the loads in Mode A are shown in Figure 7. Figure 7a shows the variation of η with the increase in the loads. η increases with the increase in R 2 and R 3 . When the loads R 2 and R 3 increase from 10 to 30 Ω, the efficiency η increases from 87% to 94.5%. Figure 7b shows the variation of η with the increase in the mutual inductances. It can be observed that η varies slightly with the increase in M 1 , but it decreases with the increase in M 2 . When the mutual inductances M 1 and M 2 increase from 6 to 7 µH, the efficiency η decreases from 83.7% to 82%.  Figure 6 shows the variation of the output voltage with the mutual inductance and the load in Mode B. It can be seen that Ubo2 is independent of the load, but it decreases with the increase in M2. When M2 increases from 6 to 7 μH, Ubo2 decreases from 35 to 29 V.

Analysis of the Efficiency
In the WPT system, the efficiency is worthy of attention. The calculated results and analysis of the variation of the efficiency with the mutual inductances and the loads are as follows.
The variation of the efficiency η with the mutual inductances and the loads in Mode A are shown in Figure 7. Figure 7a shows the variation of η with the increase in the loads. η increases with the increase in R2 and R3. When the loads R2 and R3 increase from 10 to 30 Ω, the efficiency η increases from 87% to 94.5%. Figure 7b shows the variation of η with the increase in the mutual inductances. It can be observed that η varies slightly with the increase in M1, but it decreases with the increase in M2. When the mutual inductances M1 and M2 increase from 6 to 7 μH, the efficiency η decreases from 83.7% to 82%.     In the calculation model, when R 2 increases from 10 to 30 Ω, M 2 increases from 6 to 7 µH, and the efficiency η increases from 91.5% to 97.2%.

Loss Analysis of Each Stage
The loss of the system is mainly caused by the resistance of the coils. The loss of each stage is analyzed under the variation of the loads and the mutual inductances. The calculated results are as follows.
The variation of the loss P Loss with the loads and the mutual inductances in Mode A is shown in Figure 9. According to the characteristic of the LCCL topology, the current in the L p12 of Stage 1 is constant. Therefore, the loss in Stage 1 is constant with the variation of the loads and the mutual inductances. Figure 9a shows the variation of P Loss with the increase in the loads. When the loads R 2 Energies 2020, 13, 3739 8 of 14 and R 3 increase from 10 to 30 Ω, the loss in Stage 1 is 1.7 W, the loss in Stage 2 decreases from 23 to 3.7 W, and the loss in Stage 3 decreases from 6 to 0.9 W. Figure 9b shows the variation of P Loss with the increase in the mutual inductances. When M 1 and M 2 increase from 6 to 7 µH, the loss in Stage 1 is 1.7 W, the loss in Stage 2 increases from 13 to 23.2 W, and the loss in Stage 3 increases from 3.1 to 5.6 W.  Figure 8 shows the variation of the efficiency η with the mutual inductance and the load when the system works in Mode B. It can be seen from Figure 8 that η increases with the increase in R2 and M2. In the calculation model, when R2 increases from 10 to 30 Ω, M2 increases from 6 to 7 μH, and the efficiency η increases from 91.5% to 97.2%.

Loss Analysis of Each Stage
The loss of the system is mainly caused by the resistance of the coils. The loss of each stage is analyzed under the variation of the loads and the mutual inductances. The calculated results are as follows.
The variation of the loss PLoss with the loads and the mutual inductances in Mode A is shown in Figure 9. According to the characteristic of the LCCL topology, the current in the Lp12 of Stage 1 is constant. Therefore, the loss in Stage 1 is constant with the variation of the loads and the mutual inductances. Figure 9a shows the variation of PLoss with the increase in the loads. When the loads R2 and R3 increase from 10 to 30 Ω, the loss in Stage 1 is 1.7 W, the loss in Stage 2 decreases from 23 to 3.7 W, and the loss in Stage 3 decreases from 6 to 0.9 W. Figure 9b shows the variation of PLoss with the increase in the mutual inductances. When M1 and M2 increase from 6 to 7 μH, the loss in Stage 1 is 1.7 W, the loss in Stage 2 increases from 13 to 23.2 W, and the loss in Stage 3 increases from 3.1 to 5.6 W.  Figure 10 shows the variation of the loss PLoss with the load and the mutual inductance when the system works in Mode B. It can be seen from Figure 10a that the loss in Stage 2 increases with the increase in R2, and it decreases with the increase in M2. Figure 10b illustrates that the loss in Stage 3 decreases with the increase in R2 and M2. When R2 increases from 10 to 30 Ω, M2 increases from 6 to 7 μH, the loss in Stage 2 varies from 2.1 to 3.1 W, and the loss in Stage 3 varies from 12.5 to 0.5 W.

Parameter Design Method
When the power is transmitted in the forward and reverse directions, it is necessary to maintain the output voltage constant. Most importantly, the output voltage at each stage should satisfy the respective voltage requirement.
From Equations (7)-(9), it is seen that the output voltages are insensitive to the variation of the  Figure 10 shows the variation of the loss P Loss with the load and the mutual inductance when the system works in Mode B. It can be seen from Figure 10a that the loss in Stage 2 increases with the increase in R 2 , and it decreases with the increase in M 2 . Figure 10b illustrates that the loss in Stage 3 decreases with the increase in R 2 and M 2 . When R 2 increases from 10 to 30 Ω, M 2 increases from 6 to 7 µH, the loss in Stage 2 varies from 2.1 to 3.1 W, and the loss in Stage 3 varies from 12.5 to 0.5 W.  Figure 10 shows the variation of the loss PLoss with the load and the mutual inductance when the system works in Mode B. It can be seen from Figure 10a that the loss in Stage 2 increases with the increase in R2, and it decreases with the increase in M2. Figure 10b illustrates that the loss in Stage 3 decreases with the increase in R2 and M2. When R2 increases from 10 to 30 Ω, M2 increases from 6 to 7 μH, the loss in Stage 2 varies from 2.1 to 3.1 W, and the loss in Stage 3 varies from 12.5 to 0.5 W.

Parameter Design Method
When the power is transmitted in the forward and reverse directions, it is necessary to maintain the output voltage constant. Most importantly, the output voltage at each stage should satisfy the respective voltage requirement.

Parameter Design Method
When the power is transmitted in the forward and reverse directions, it is necessary to maintain the output voltage constant. Most importantly, the output voltage at each stage should satisfy the respective voltage requirement.
From Equations (7)- (9), it is seen that the output voltages are insensitive to the variation of the loads, but they can be determined by L pi and M i (i = 1, 2). If the output voltage ratio of Stage i and Stage i + 1 is λ i , then L pi can be designed as L pi = λ i M i (i = 1, 2), and C pi1 and C pi2 can be set as follows: The parameter design flowchart of topology is shown as follows.
The design steps of the topology parameter are illustrated in Figure 11. Through the above parameter design method, the different output voltage of each stage can be achieved to satisfy the respective voltage requirement. Select the resonance capacitor Cpi1 and Csi according to Equation (2) Select the resonance capacitor Cpi2 according to Equation (10) End Figure 11. The flowchart of the topology parameter design.

Experimental Verification
To verify the proposed method, an experimental demonstration platform that was used for charging the snake robot is shown in Figure 12. There were two operation modes in the MB-WPT system, Mode A and Mode B. The LCCL-LC-LCCL-LC topology structure was adopted in the system. The primary and secondary coils were designed as a cylindrical structure with the diameters of 50 mm and 45 mm, respectively, and their height is 50 mm. The experimental parameters are listed in Table 3.

Experimental Verification
To verify the proposed method, an experimental demonstration platform that was used for charging the snake robot is shown in Figure 12. There were two operation modes in the MB-WPT system, Mode A and Mode B. The LCCL-LC-LCCL-LC topology structure was adopted in the system. The primary and secondary coils were designed as a cylindrical structure with the diameters of 50 mm and 45 mm, respectively, and their height is 50 mm. The experimental parameters are listed in Table 3.

Mode A
When the power is transferred in the forward direction, the variations of output voltages and input current with the loads are shown in Figure 13. U ao2 and U ao3 are the output voltages of Stage 2 and Stage 3, respectively, I Lp11 is the current in the L p11 , and it is also the input current of the system. When R 3 = 30 Ω, and R 2 varies from 30 to 10 Ω, U ao2 varies from 31.2 to 28.1 V, U ao3 varies from 27.1 to 25 V, and I Lp11 varies from 2.16 to 3.87 A. ∆U ao2 = 3.1V, ∆U ao3 = 2.1 V. It can be concluded that output voltages vary little with the variations of loads. With the decrease in the R 2 , the output power increases, and the input power increases. Therefore, the input current I Lp11 increases. Similar to Figure 13a, Energies 2020, 13, 3739 10 of 14 when R 3 = 10 Ω, and R 2 varies from 30 to 10 Ω, U ao2 varies from 28.6 to 27.9 V, U ao3 varies from 24.6 to 23.9 V, and I Lp11 varies from 3.19 to 4.68 A. ∆U ao2 = 0.7 V, ∆U ao3 = 0.7 V. When R 2 = 30 Ω, and R 3 varies from 30 to 10 Ω, U ao2 varies from 31.2 to 28.6 V, U ao3 varies from 27.1 to 24.6 V, and I Lp11 varies from 2.16 to 3.19 A. ∆U ao2 = 2.6 V, ∆U ao3 = 2.5 V. When R 2 = 10 Ω, and R 3 varies from 30 to 10 Ω, U ao2 varies from 28.1 to 27.9 V, U ao3 varies from 25 to 23.9 V, and I Lp11 varies from 3.87 to 4.68 A. ∆U ao2 = 0.2 V, ∆U ao3 = 1.1 V. From the above experimental results, it can be seen that the output voltages are insensitive to the variation of the loads. The experimental results are consistent with the theoretical analysis.
When the power is transferred forward, the variation of output voltages and input current with the rotation of secondary coils, and the rotation schematic of the secondary coils are shown in Figure 14. The rotation angle of the secondary coil is marked as θ i . When both secondary coils rotate from 0 • to 360 • , the fluctuation of the output voltages is very small. That is because the variation of the mutual inductances with the rotation of the secondary coil is very small.

Mode B
When the power is transferred in the reverse direction, the variations of output voltage and input current with the load are shown in Figure 15. U bo2 is the output voltage of Stage 2, I cs2 is the current in the C s2 , and it is also the input current of the system. When R 2 varies from 30 to 10 Ω, U bo2 varies from 34.9 to 32.8 V, and I cs2 varies from 1.19 to 3.6 A. ∆U bo2 = 2.1 V. It can be concluded that the output voltage varies little with the variations of the load when the power is transferred reversely. The experimental results are consistent with the theoretical analysis. Select the resonance capacitor Cpi1 and Csi according to Equation (2) Select the resonance capacitor Cpi2 according to Equation (10) End Figure 11. The flowchart of the topology parameter design.

Experimental Verification
To verify the proposed method, an experimental demonstration platform that was used for charging the snake robot is shown in Figure 12. There were two operation modes in the MB-WPT system, Mode A and Mode B. The LCCL-LC-LCCL-LC topology structure was adopted in the system. The primary and secondary coils were designed as a cylindrical structure with the diameters of 50 mm and 45 mm, respectively, and their height is 50 mm. The experimental parameters are listed in Table 3.   10 Ω/30 Ω R 3 10 Ω/30 Ω -- varies from 30 to 10 Ω, Uao2 varies from 31.2 to 28.6 V, Uao3 varies from 27.1 to 24.6 V, and ILp11 varies from 2.16 to 3.19 A. ΔUao2 = 2.6 V, ΔUao3 = 2.5 V. When R2 = 10 Ω, and R3 varies from 30 to 10 Ω, Uao2 varies from 28.1 to 27.9 V, Uao3 varies from 25 to 23.9 V, and ILp11 varies from 3.87 to 4.68 A. ΔUao2 = 0.2 V, ΔUao3 = 1.1 V. From the above experimental results, it can be seen that the output voltages are insensitive to the variation of the loads. The experimental results are consistent with the theoretical analysis. (c) (d) When the power is transferred forward, the variation of output voltages and input current with the rotation of secondary coils, and the rotation schematic of the secondary coils are shown in Figure  14. The rotation angle of the secondary coil is marked as θi. When both secondary coils rotate from 0° to 360°, the fluctuation of the output voltages is very small. That is because the variation of the mutual inductances with the rotation of the secondary coil is very small.   (c) (d) When the power is transferred forward, the variation of output voltages and input current with the rotation of secondary coils, and the rotation schematic of the secondary coils are shown in Figure  14. The rotation angle of the secondary coil is marked as θi. When both secondary coils rotate from 0° to 360°, the fluctuation of the output voltages is very small. That is because the variation of the mutual inductances with the rotation of the secondary coil is very small.

Mode B
When the power is transferred in the reverse direction, the variations of output voltage and input current with the load are shown in Figure 15. Ubo2 is the output voltage of Stage 2, Ics2 is the current in the Cs2, and it is also the input current of the system. When R2 varies from 30 to 10 Ω, Ubo2 varies from   When the power is transferred reversely, the variation of output voltage and input current with the rotation of the secondary coil are shown in Figure 16. Similar to forward power transmission, when the secondary coil rotates from 0° to 360°, the fluctuation of the output voltage is very small. The output power and the efficiency are listed in Table 4.

Conclusions
The MB-WPT system proposed in this paper is mainly applied on the snake robot with a long power transfer distance. Besides, the system can satisfy the requirements of multiple loads and bidirectional power transmission. A three-stage MB-WPT system is taken as an example to analyze the proposed method, and the equivalent circuit model of the system is established on this basis. Then, the output characteristic of the system is analyzed by calculation and simulation. In both operation modes, Mode A and Mode B, the constant output voltage can be achieved, and a different output voltage for each stage can be achieved to satisfy the respective voltage requirement. Finally, the experimental results have validated the effectiveness of the proposed WPT system. When the power is transferred reversely, the variation of output voltage and input current with the rotation of the secondary coil are shown in Figure 16. Similar to forward power transmission, when the secondary coil rotates from 0 • to 360 • , the fluctuation of the output voltage is very small.  When the power is transferred reversely, the variation of output voltage and input current with the rotation of the secondary coil are shown in Figure 16. Similar to forward power transmission, when the secondary coil rotates from 0° to 360°, the fluctuation of the output voltage is very small. The output power and the efficiency are listed in Table 4.

Conclusions
The MB-WPT system proposed in this paper is mainly applied on the snake robot with a long power transfer distance. Besides, the system can satisfy the requirements of multiple loads and bidirectional power transmission. A three-stage MB-WPT system is taken as an example to analyze the proposed method, and the equivalent circuit model of the system is established on this basis. Then, the output characteristic of the system is analyzed by calculation and simulation. In both operation modes, Mode A and Mode B, the constant output voltage can be achieved, and a different output voltage for each stage can be achieved to satisfy the respective voltage requirement. Finally, the experimental results have validated the effectiveness of the proposed WPT system. The output power and the efficiency are listed in Table 4.

Conclusions
The MB-WPT system proposed in this paper is mainly applied on the snake robot with a long power transfer distance. Besides, the system can satisfy the requirements of multiple loads and bi-directional power transmission. A three-stage MB-WPT system is taken as an example to analyze the proposed method, and the equivalent circuit model of the system is established on this basis. Then, the output characteristic of the system is analyzed by calculation and simulation. In both operation modes, Mode A and Mode B, the constant output voltage can be achieved, and a different output voltage for each stage can Driving signal of the inverter in Stage i C i Filter capacitor of Stage i L pi2 , L si Primary and secondary coils of Joint i L pi1 , C pi1 , C pi2 , C si-1 Compensation inductance and capacitances of Stage i R pi , R si Resistances of the primary and secondary coils in Joint i f, w Operation frequency and angle frequency U aoi , U boi Output voltage when the power is transferred in both directions P out_a , P out_b Output power when the power is transferred in both directions M i Mutual inductance of Joint i η The efficiency of the system