The structure of the proposed MTMR WPT system using double side LCC (DS-LCC) compensation topology is shown in Figure 1. There are three transmitter coils and three receiver coils in the MTMR WPT system, and there is an independent LCC compensation network with an independent inverter before each coil to provide high-frequency current. The receiver with the LCC compensation network connects in parallel with an independent rectifier to provide power for the load. In the DS-LCC compensation network, when the mutual inductance value of the system is stable, the current flowing through the load maintains constant. The transferred power of the MTMR WPT system is effectively increased because of its anti-offset properties.

The equivalent circuit of the MTMR WPT system is shown in Figure 2. U _i is the source of each inverter. L _i1, L _i2, C _i1, C _i2, C _pi, and C_si are the compensation inductance and capacitance of each LCC compensation network. L _pi and L _si are inductance of the coupled coil. In addition, since there is a coupling mutual inductance between every coil, M _{xx_ij} is defined as the mutual inductance (x = p (transmitter) or x = s (receiver)). U _xxi is the induced voltage generated between coils. According to Kirchhoff’s Law, the voltage can be expressed as: { U ˙ p p i = ∑ k = 1 k = i − 1 j ω M p p _ i k I ˙ p k + ∑ k = i + 1 k = n j ω M p p _ i k I ˙ p k U ˙ p s i = ∑ k = 1 k = i − 1 j ω M p s _ i k I ˙ s k + ∑ k = i + 1 k = n j ω M p s _ i k I ˙ s k (1)

Based on the previous analysis, it is obvious that the current flowing through the coils is still constant. However, the output current of the inverter is expressed as: I ˙ 1 i = U ˙ p p i + U ˙ p s i j ω L 1 i (2)

It can be seen that the output current of the inverter will increase with the mutual inductance between transmitter coils. Therefore, additional series capacitors have to be added to achieve decoupling between the transmitter coils. The designed equivalent circuit of the MTMR WPT system with compensation capacitors is shown in Figure 3. In order to cancel the corresponding induced voltage, the added series compensation capacitance is expressed as: C e i = 1 ( n − 1 ) ω 2 M p p (3)

In each transmitter LCC compensation network, the compensation capacitor in series with the coil is: C p i ′ = 1 ω 2 [ L p − L 1 + ( n − 1 ) M p p ] (4)

The induced voltage generated in the coils of the cross-coupling inductance between the transmitter coils can be eliminated. The compensation capacitor reduces the output current of the inverter and provides a favorable guarantee for the development and design of the MTMR WPT system.

In order to improve the transmission power and efficiency of the system, a WPT system with three transmitter coils and three receiver coils is taken as the research objective to analyze the mutual inductance and misalignment of MTMR coils. The plane layout of the circular multi-coil is shown in Figure 4. The coupling between each two receiving coils or transmitting coils can be decoupled so that their mutual inductance does not have to be concerned. When there is an offset between the transmitter and receiver coils, the mutual inductance between every two coils can be obtained as: M i j = μ 0 N i N j 4 π ∫ 0 2 π ∫ 0 2 π r i r j ( cos ⁡ θ ⁡ cos ⁡ φ ⁡ cos ⁡ α + sin ⁡ θ ⁡ sin ⁡ φ ) r i j d θ d φ (5) where r _ij is the distance between the center points of the two coils. N _x (x = i or j) is the number of turns of the coil, and r _x (x = i or j) is the radius of the coil.

Assume that the current flowing through one of three coils is I ˙ p , when ignoring the internal resistance of the coil, according to the superposition principle, the output currents Isi from the receiver compensation network are: { I ˙ s 1 = 1 L 2 ( I ˙ p a M a x + I ˙ p b M b x + I ˙ p c M c x ) = I ˙ p L 2 ( M a x + M b x + M c x ) I ˙ s 2 = 1 L 2 ( I ˙ p a M a x + I ˙ p b M b x + M c x ) = I ˙ p L 2 ( M a y + M b y + M c y ) I ˙ s 3 = 1 L 2 ( I ˙ p a M a x + I ˙ p b M b x + I ˙ p c M c x ) = I ˙ p L 2 ( M a z + M b z + M c z ) (6)

For the DS-LCC compensation network, the receiver output current is constant, so the total current I ˙ s flowing through the load can be expressed as: I ˙ s = I ˙ s 1 + I ˙ s 2 + I ˙ s 3 = I ˙ p L 2 ∑ i = a c ∑ j = x z M i j (7)

Therefore, the influence of the offset on the mutual inductance can be evaluated by the sum of mutual inductance ( ∑ i = a c ∑ j = x z M i j ). Taking the structure shown in Figure 4, assuming the axial distance between the centers is 15 cm, and the angles are 0°, 5°, and 10°, when the lateral distance between the coil centers gradually increases, the variation of mutual inductance is shown in Figure 5. The coils in the simulation are single-turn with a metallic material. The mutual inductance is utilized as the Y-ordinate, for it is an instinctive and straightforward index to indicate the performance of different coil offsets. It can be seen from Figure 5 that angular offset has little influence on mutual inductance. Therefore, the mutual inductance variation due to a certain lateral distance offset is the main factor affecting the system efficiency.

In order to balance the transmission efficiency and anti-offset performance of the MTMR WPT system, the PSO algorithm (Hasanzadeh and Vaez-Zadeh, 2013) is used to reasonably allocate the parameters of the system to find the optimal parameter setup of the system. The optimization goal can be summarized in the following three aspects: efficiency, anti-offset performance, and cost of the MTMR WPT system. The evaluation standard of the anti-offset performance is that under the same coil offset, the higher efficiency of the system equals better performance. The optimization conditions include: the current of each coil is less than the rated current I ₀, and the total outer diameter of the multiple coils is less than the radius limit r ₀. Furthermore, the efficiency of the WPT system is related to the mutual inductance and internal resistance between the coils. Therefore, the optimization goal can be written as: max:   f ( x ) = ( M m i s s ( x ) , M ( x ) , R 0 ( x ) , P ( x ) )      s . t .                 I i ( x ) ≤ I 0                  i = 1,2 , ⋯ q                         r j ( x ) ≤ r 0                  j = 1,2 , ⋯ p (8) where M m i s s ( x ) , M ( x ) represents the anti-offset performance and they are expressed as mutual inductance. R 0 ( x ) and P ( x ) are the function of parasitic resistance and cost. In addition, x = ( n t , d c , r c , n s ) represents the decision amount of the objective function, where n t , d c , r c and n s represents the number of turns of the coil, the distance between two coils, the outer radius of the coil, and the number of Litz wire strands. In a word, optimization is to achieve efficient operation of the system at a low cost.

In the actual WPT system, due to the inconsistent parameters of each element in the circuit, it is difficult to equalize the inductance and capacitance of each resonance compensation. As a result, assume that the tolerance between each element is α, according to Figure 4, the actual parameter is obtained as: { C 1 = ( 1 + α ) C C 2 = ( 1 − α ) C L 1 = ( 1 + α ) L L 2 = ( 1 − α ) L ω = 1 / L C (9)

The difference between the transmit coil currents can be derived from Kirchhoff’s law: I ˙ c = I ˙ p 1 − I ˙ p 2 = 1 A 2 + B 2 { [ 2 B M 2 ω ( α 2 − 2 α + 2 ) + A R 0 D 1 ] U ˙ 1 − [ 2 B M 2 ω ( α 2 + 2 α + 2 ) + A R 0 D 2 } U ˙ 2 ] − j A 2 + B 2 { [ 2 A M 2 ω ( α 2 − 2 α + 2 ) + B R 0 D 1 ] U ˙ 1 − [ 2 A M 2 ω ( α 2 + 2 α + 2 ) + B R 0 D 2 ] U ˙ 2 } (10) where A and B are expressed by mutual inductance and α. As can be seen from Eq. 10, there is a certain difference in the current through each transmitting coil, and the vector diagram of the current flowing through each coil is shown in Figure 6A. There are differences in both amplitude and angle in the current vector diagram. However, when there is no difference in the system or the difference is minimized according to the control strategy, the current distribution shown in Figure 6B can be achieved. In this case, the current through each coil can be kept consistent in both phase and amplitude, and there is no reactive power. The losses caused by reactive currents in the compensation network will be reduced by the synchronization strategy.

In order to improve the output power, reduce the power loss and achieve effective capacitance compensation, a decoupling algorithm based on active and reactive currents is proposed. The algorithm can synthesize the current direction as the reference direction, and decompose the vector of each coil current into active current and reactive current respectively. A vector diagram of the active and reactive currents decomposed by the coil current is shown in Figure 7.

First, assume that the resultant current is the reference signal. Then the kth coil current can be expressed as I ˙ p k = 1 2 I a k + j 1 2 I r k , where the I a k and I r k represent the magnitudes of the active and reactive current components of the kth coil. Shift the phase of synthetic current to achieve a π/2 delay, and then the quadrature parameters can be obtained as i p t = I m ⁡ sin ( ω t + π 2 ) = I m ⁡ cos ( ω t ) . Second, define virtual active and reactive components as: [ P k Q k ] = [ Re [ I ˙ p ⋅ ( I ˙ p k ) ∗ ] Im [ I ˙ p ⋅ ( I ˙ p k ) ∗ ] ] = [ I m I a k 2 − I m I r k 2 ] (11)

In fact, the virtual can be used to indicate the consistency of the direction and magnitude of the coil current in practice. Then multiply the i _p and the virtual components, and eliminate the doubled signal in the result: [ i a k i r k i a ] = [ I m k I m ⁡ cos ( ϕ k ) 2 I m k I m ⁡ sin ( ϕ k ) 2 I m 2 2 ] (12)

The resultant current amplitude I m is 2 i a . Finally, the current of each coil is decomposed into active and reactive components. The amplitudes of the active and reactive components of each coil can be expressed as: [ P k Q k ] = [ i a k i r k ] = [ I m k I m ⁡ cos ( ϕ k ) 2 I m k I m ⁡ sin ( ϕ k ) 2 ] (13)

The magnitude can be expressed as: [ I a k I r k ] = [ 2 i a k i a 2 i r k i a s ] (14)

In the control strategy, when there is parameter deviation in the circuit, the working state of the inverter (i.e., the phase and amplitude of the fundamental voltage output by the inverter) can be adjusted to compensate for the differences. In addition, the inverter phase and duty cycle can be adjusted according to the active and reactive components in the coil current. The transmitter’s current synchronization strategy will provide effective detection values for subsequent phase locking and current sharing. Compared with the zero-crossing comparator, the algorithm requires less accurate sampling of the current, so it is more suitable for high-frequency and high-power power occasions.

Combined with the actual requirements of the WPT system for EVs, the coupling structure and its parameters are designed according to the coupling structure optimization algorithm proposed in Section 2. First, the operation frequency of the proposed MTMR WPT system is 85 kHz, and the outer dimension radius of the coil is less than 1 m. In addition, according to the parameters of Litz wire, the constraints such as wire diameter are formulated. Then, the conditions and functions such as mutual inductance and internal resistance are brought into the optimization algorithm. The Pareto frontier based on multi-objective optimization is shown in Figure 9. It can be seen that when the number of turns of the coil is fixed, the small difference in the coil radius does not change the system efficiency intensely. When the coil radius is constant, the number of turns of the coil will have a great influence on the efficiency and cost. The optimization result on coil parameters is shown in Table 1.

Based on optimization results, finite element analysis is carried out to evaluate the magnetic field distribution for two cases: no offset between coils and 10 cm offset between coils. The simulation result is shown in Figure 10. When there is no offset, the magnetic field distribution is relatively uniform, but with a horizontal offset between coils, the magnetic field distribution will be distorted, and it is more distinct in the edge area of coils. For example, there is a point A in Figure 10, and the magnetic field around point A is very different in Figures 10A, B. Moreover, the parameters of the system can be measured from the simulation, such as the self-inductance, internal resistance, and mutual inductance.

Finally, the transmitter and receiver coils are wound according to the optimization results, as shown in Figure 8B. The self-inductance and mutual inductance of the coils were measured using an impedance analyzer. The comparison between the experimental results and the simulation results is shown in Table 2. It can be seen that the WPT system with MTMR coils can obtain higher coupling coefficients even in the occurrence of offset, which helps to achieve higher transmission efficiency. And the actual measured values are basically consistent with the simulation results.

In order to verify the proposed transmitter current synchronization strategy, two experiments were carried out with different parameter setups.

First, when no control strategy is utilized, the currents in each coil cannot be completely consistent, and there are differences in the phase and amplitude of the currents. In addition, the current of one coil will be larger, so the losses on this coil are also higher than others.

Afterward, when the system adopts the control strategies, the current of each transmitter coil can be consistent in both phase and amplitude. At this time, the reactive component of each coil current corresponding to the combined current is substantially zero. In addition, the capacitance compensation method effectively decoupled the coils, so as to ensure each inverter maximizes its function.

Two experiments are carried out to validate the aforementioned analysis. First, the control board provides the same drive signal for three different inverters as shown in Figure 11A, which is an alternating square wave of 85 kHz. When the inverter output voltage is adjusted to 300 V, the waveform of the steady-state current of each coil is shown in Channel 2, Channel 3, and Channel 4 in Figure 11A. As can be seen from the waveform, the current flowing through each coil are different in phase and amplitude. Secondly, we add the proposed phase synchronization strategy to the control system. The experimental result under the same input voltage is shown in Figure 11B. With the control strategy proposed in this study, both the current phase and amplitude of each transmitter coil can be kept stable. The reactive component of each coil current is approximately zero, and the active component is basically identical. In addition, it is shown that the capacitor compensation measures adopted at the transmitter play a good role in cross-decoupling.

The overall output power of the system is 3.072 kW, and the overall system efficiency is 92.49%. Compared with the system without the proposed strategy, the system efficiency is improved by nearly 27.7%. Finally, several tests were carried out with different output power, and the experimental results are shown in Figure 12. To summarize, during the full power range, the efficiency can be improved with the proposed strategy, and the active and reactive current decomposition methods are effective.

Furthermore, in order to verify the anti-offset performance of the system, corresponding experiments were carried out. The experimental results of the MTMR WPT experimental system are compared with the experimental results of the STSR WPT system in the offset situation, and the results are shown in Figure 13. With the same coil distance and system power level, the transmission efficiency under different horizontal offset conditions is measured. It is obvious that the MTMR WPT system outperforms the STSR WPT system in anti-offset capability. When there is no offset, the efficiency of both systems can achieve 90% or higher results. However, with the increase in the offset distance, the drop rate of the STSR WPT system was significantly faster than the MTMR WPT system. When the offset distance is 40cm, the efficiency of the MTMR WPT system is about 1.45 times of the STSR WPT system. In a word, compared with the STSR WPT system, the proposed MTMR WPT system perform much better in offset situation.

In order to present the novelty of this study more comprehensively, a Table of comparison between conducted related works in literature is added to the study. Table 3 shows the comparison of experimental results between the multi-coil WPT system in recent years in the existing literature and the system designed in this study. As can be summarized from Table 3, low power WPT systems (usually <100 W) have higher degrees of freedom in design, which come in a variety of geometries and some geometries (e.g., cylindrical, spherical, etc.) have good anti-offset performance. However, in the scenario of EV charging, the shape of the coils is limited by the vehicle chassis. Without particular anti-offset design consideration, the wireless charging efficiency will drop significantly when misalignment happens. The coupling structure proposed in this study proves to have the much better anti-offset capability while maintaining high power output capability.