Design of a High-Power High-Efficiency Multi-Receiver Wireless Power Transfer System

This paper proposes a new control method to regulate the power flow into multiple receivers. This system consists of one transmitter controller and three receiver controllers. They work independently to decide the power distribution with their combined operation. The simulated and experimental models have been built, and the experimental results are in good agreement with the theoretical analysis results. The proposed method is robust, flexible, and generalizable, and can be employed under various wireless charging conditions.


Introduction
Wireless power transfer (WPT) is a technology that enables power transfer in a convenient and safe manner without any physical contact. WPT is widely used in many areas, such as cell phones, household appliances, industrial robots, and electric vehicles. There are many WPT system that can charge multiple receivers with a single transmitter. Tightly coupled inductive technology based on wireless power consortium (WPC or Qi) or loosely coupled magnetic resonant technology based on air fuel has been widely used in multi-receiver WPT systems, where receivers are powered simultaneously.
However, there are some challenges with the power distribution of multi-receiver WPT systems. First, receivers close to the transmitter tend to absorb more power than the ones far away [15], whereas the power demand of each receiver should be decided by its load instead of receiver distance. Second, many factors, such as power demand, receiver distances and receiver number may dynamically change. Third, the power capacity of the transmitter becomes more insufficient to meet the demands with the increase in receiver number. In most cases, the peak power demand is much higher than the average power demand [16]. Therefore, the transmitter may be overloaded if the receivers absorb more power than the capacity of the transmitter.
Recently, some methods have been proposed to control the power distribution in multi-receiver WPT systems. In [16], the frequency is used as the communication channel for the entire system. Receivers are configured to different priority levels. When the transmitter is on the verge of overloading, the power of all the receivers is decreased simultaneously. Reference [17] proposes a power distribution scheme that can be adapted A typical multi-receiver WPT system is shown in Figure 1. The transmitter is powered by a DC power supply. The current in the transmitting coil produces magnetic field around the transmitting coil, voltage is induced in all receiving coils, and then power will be delivered to all receivers simultaneously. When a receiver needs more power, the transmitter sees impedance changes. Since all receiver coils are parallel, the cross-coupling can usually be neglected [22,23], the reflected impedance Z i can be derived as [23].
where ω is the operating angular frequency; M (i) is the mutual inductance between the TX coil and the i-th RX coil; Z ri is the equivalent impedance seen though RX. Under resonance condition, the load impedance seen by the input is The output power from the input is first transferred to the i-th RX coil, and then to the load. Then, the power received by the i-th load can be calculated to be

Transmitter Analysis
The schematic of the transmitter is shown in Figure 2. A high-frequency (HF) inverter converts the DC power to HF AC power, and then the power is driven into the transmitting compensation circuit through the coils. The inverter adopts a full bridge configuration, and the amplitude of its output is controlled by phase-shift angle α. The waveform of inverter output voltage Uinv is a square waveform which consists of an infinite number of harmonics. When the rms of the fundamental component of Uinv (which is defined as Uinv (1)) is considered, the relationship between Uinv (1) and phase-shift angle α is The LCC circuit is employed for the transmitter as the compensation circuit, which consists of three components La, C1a, and C1b. The matching condition of the LCC circuit is [5]: LCC circuit Full bridge inverter U inv(1) Figure 1. Schematic of the multi-receiver WPT system.

Transmitter Analysis
The schematic of the transmitter is shown in Figure 2. A high-frequency (HF) inverter converts the DC power to HF AC power, and then the power is driven into the transmitting compensation circuit through the coils.

Transmitter Analysis
The schematic of the transmitter is shown in Figure 2. A high-frequency (HF) inverter converts the DC power to HF AC power, and then the power is driven into the transmitting compensation circuit through the coils. The inverter adopts a full bridge configuration, and the amplitude of its output is controlled by phase-shift angle α. The waveform of inverter output voltage Uinv is a square waveform which consists of an infinite number of harmonics. When the rms of the fundamental component of Uinv (which is defined as Uinv (1)) is considered, the relationship between Uinv(1) and phase-shift angle α is The LCC circuit is employed for the transmitter as the compensation circuit, which consists of three components La, C1a, and C1b. The matching condition of the LCC circuit is [5]: The inverter adopts a full bridge configuration, and the amplitude of its output is controlled by phase-shift angle α. The waveform of inverter output voltage U inv is a square waveform which consists of an infinite number of harmonics. When the rms of the fundamental component of U inv (which is defined as U inv (1) ) is considered, the relationship between U inv(1) and phase-shift angle α is The LCC circuit is employed for the transmitter as the compensation circuit, which consists of three components L a , C 1a , and C 1b . The matching condition of the LCC circuit is [5]: where ω is the angular frequency and L 1 is the self-inductance of transmitting coil. Then, the rms value of transmitting coil current I 1 is [24]: Electronics 2021, 10, 1308 4 of 20 The transmitting coil current is proportional to U inv (1) , irrespective of the load impedance. Compared with other common compensation circuits such as series compensation circuit, LCC compensation circuit has advantage of constant current characteristics, which is helpful to decouple receivers. The duty cycle is controlled under constant frequency and it is done easily.

Receiver Analysis
The receiver consists of a receiving coil, a compensation circuit, a rectifier, and a regulator, as shown in Figure 3. The induced voltage U 2 in receiving coil can be regarded as a voltage source in series with receiver coil, and its rms value can be calculated as: where M(i) is the mutual inductance between the transmitting coil and the coil of receiver #i, RL(i) is the load resistance of receiver #i. Po(i) can be regulated by the two parameters indicated in (15):α and D(i). In a single receiver WPT system, only one parameter needs to be changed to achieve desired power regulation, with the other one fixed. The total output power of the receivers Po_sum can be expressed as follows: where n is the number of receivers. The efficiency of the entire system η is defined as: where Pi is the input power of transmitter, which can be derived as: Figure 3. Schematic of the receiver. The input current of the rectifier I 2 is distorted because of the non-linear characteristics of the rectifier. The average power supplied by U 2 is:

Regulator Rectifier
The rectifier input waveform is shown in Figure 4. The output current of the rectifier can be expressed using Fourier harmonic expansion as [25].
Electronics 2021, 10, 1308 6 of 21 For a multi-receiver WPT system, the power distribution control strategy should satisfy two basic principles. First, the transmitter has limited power capacity, and it should not overload at any given time. Second, the power demand by each receiver should be satisfied as much as possible. When the power capacity of the transmitter meets the demand of all the receivers, every receiver should receive the required amount of power. When the power capacity of transmitter is lower than the total power demand of the receivers, and receivers should absorb less power than their demands to prevent overload-  Then, the input power of rectifier P rect_i can be derived using sinusoidal trigonometric functions calculus [26]. Since the power is calculated within a time period, only the DC term is not zero and P rect_i can be calculated to be: where U rect_o is the rectifier output voltage. The power losses of the rectifier is very small and, therefore, can be neglected, and the average power supply can be calculated as: The relationship between U 2 and U rect_o can be derived from (10): The regulator is a half-bridge converter, and its output voltage U o is where D is the duty cycle of the regulator. By combining (4), (6), (11), (12) and (13), the output power to the load P o can be derived as: It can be noted from (14) that P o can be controlled by α and D.
In our multi-receiver WPT system, the transmitting LCC circuit could minimize the effect of cross coupling due to its constant current characteristic of the transmitting coil [20]. The output power satisfies (14) and can be rewritten as a function of the control variable α and duty cycle of the ith regulator D (i) : where M (i) is the mutual inductance between the transmitting coil and the coil of receiver #i, R L(i) is the load resistance of receiver #i. P o(i) can be regulated by the two parameters indicated in (15):α and D (i) . In a single receiver WPT system, only one parameter needs to be changed to achieve desired power regulation, with the other one fixed. The total output power of the receivers P o_sum can be expressed as follows: where n is the number of receivers. The efficiency of the entire system η is defined as: where P i is the input power of transmitter, which can be derived as: Electronics 2021, 10, 1308 6 of 20 For a multi-receiver WPT system, the power distribution control strategy should satisfy two basic principles. First, the transmitter has limited power capacity, and it should not overload at any given time. Second, the power demand by each receiver should be satisfied as much as possible. When the power capacity of the transmitter meets the demand of all the receivers, every receiver should receive the required amount of power. When the power capacity of transmitter is lower than the total power demand of the receivers, and receivers should absorb less power than their demands to prevent overloading.

The Proposed Structure Analysis
Our proposed structure employs double-side control and it is comprised of two types of controllers. The first type is the transmitter controller. There is only one transmitter controller in the multi-receiver WPT system, and it is located at the transmitting side to regulate the total power of the WPT system and to prevent overloading the DC power supply. The second type is the receiver controller. Every receiver has an individual receiver controller. Based on the circuit topology, the system provides adaptive output based on the load requirement in each receiver. The current in the transmitting coil produces magnetic field around the transmitting coil, and voltage is induced in all receiving coils, and then power is delivered to all receivers simultaneously. The regulators control the output power of each load are based on the design requirements. Power distribution comes from mutual inductions and actual DCs loads.
The structure of the proposed transmitter controller is shown in Figure 5. This system contains a feedback loop with a PI controller and phase-shifted signal generator. The voltage and current values are sampled in to calculate the input power P i . This value is subtracted with the instructed power value P i * to obtain the error power P i_err . Then the error power is normalized in the domain (0, 1) to feedback the phase-shifted signal generator, which controls the gate driver of the HF invertor. Additional anti-trigonometric square root function is integrated to increase the stability and robustness of the controller.
The output of the PI controller represents the normalized power pi_norm, which is the ratio of the Pi to Pi_max. Combining (18) and (19), pi_norm can be derived as: Since pi_norm is between 0 and 1, the output of the PI controller is limited from 0 to 1. By rewriting (20), the phase shift angle of the inverter α can be derived as:  Therefore, a math function block is inserted after the PI controller to calculate the phase shift angle α. Finally, a phase-shifted signal generator produces the control signal for the inverter, and the gate driver drives the semiconductor switches in the inverter.
The transmitter controller regulates the inverter input power to prevent overloading of the DC power supply. Two possible states may appear for the transmitter controller. In state 1, Pi* is less than Pi_max, while in state 2, the instruction input power Pi* is greater than or equal to Pi_max, as shown in Figure 6.   Figure 5. Scheme of transmitter controller.
The transmitter controller would control the power of the entire system such that the instruction input power is not exceeded. The instruction input power is set to be slightly lower than the power capacity of the DC power supply to preserve redundancy power of the supply. First, the input voltage and input current of the inverter are sampled through a sampling circuit, and the input power of the inverter is measured by multiplying the input voltage and t current. The error between the instructed and measured power is calculated It can be seen from (18) that the DC power supply has the maximum output power P i_max when the inverter phase shift angle α is equal to 0, that is: The output of the PI controller represents the normalized power p i_norm , which is the ratio of the P i to P i_max . Combining (18) and (19), p i_norm can be derived as: Since p i_norm is between 0 and 1, the output of the PI controller is limited from 0 to 1. By rewriting (20), the phase shift angle of the inverter α can be derived as: Therefore, a math function block is inserted after the PI controller to calculate the phase shift angle α. Finally, a phase-shifted signal generator produces the control signal for the inverter, and the gate driver drives the semiconductor switches in the inverter.
The transmitter controller regulates the inverter input power to prevent overloading of the DC power supply. Two possible states may appear for the transmitter controller. In state 1, P i * is less than P i_max , while in state 2, the instruction input power P i * is greater than or equal to P i_max , as shown in Figure 6.
Since pi_norm is between 0 and 1, the output of the PI controller is limited from 0 to 1. By rewriting (20), the phase shift angle of the inverter α can be derived as: Therefore, a math function block is inserted after the PI controller to calculate the phase shift angle α. Finally, a phase-shifted signal generator produces the control signal for the inverter, and the gate driver drives the semiconductor switches in the inverter.
The transmitter controller regulates the inverter input power to prevent overloading of the DC power supply. Two possible states may appear for the transmitter controller. In state 1, Pi* is less than Pi_max, while in state 2, the instruction input power Pi* is greater than or equal to Pi_max, as shown in Figure 6.

LCC Circuit
Gate Driver State 1: P i * < P i_max . Considering a scenario at time t 0 , P i is greater than P i *, the error power P i_err is negative, the PI controller decreases P i_norm , α increase, and, therefore, P i decreases. On the contrary, if P i is less than P i * at t 0 , P i_err would be positive, and the PI controller would increase P i_norm , and α would decrease, and therefore P i would be increased. The PI controller would adjust its output only if P i is not equal to P i *. Finally, in state 1, P i will be equal to P i * in the steady state. Moreover, there is no static error due to the integral part in the PI controller.
State 2: P i * ≥ P i_max . If P i is less than P i_max at time t 0 , then P i must also be less than P i *. Similar to state 1, P i_err is positive, and the PI controller would try to increase P i_norm to increase P i . When P i_norm increases to 1 at t 1 , P i is equal to P i_max but still less than P i *. Therefore, the PI controller tends to continue to increase P i_norm . However, P i_norm is limited to be no greater than 1, so the P i_norm would remain equal to 1, then the transmitter controller is saturated. Finally, in the steady state, P i will be equal to P i_max and would not increase to P i *. In summary, under steady state, P i is always less than P i * and P i_max under the control of the transmitter controller, and can be calculated as Each receiver has an individual receiver controller to regulate its output for the load. This receiver controller is designed for constant-voltage output. The structure of the receiver controller is shown in Figure 7. The input to the receiver controller is the instructed The error between U (o)i * and U o(i) is fed into the PI controller, and the output of which is the duty cycle D (i) for the regulator. D (i) is limited between 0 and 1. The PWM generator would generate the PWM control signal for the regulator, and the semiconductor switches are driven by the gate driver. Noting that once the instructed output voltage U (o)i * is determined, the equivalent instruction output power P o(i) * can be determined by: Electronics 2021, 10, 1308 9 of 21  Figure 8b shows how the receiver controller works in state 2. If Uo(i) is less than Uo(i)_max at t0, Uo(i)_err will be positive, and the PI controller would try to increase D(i). When D(i) increases to 1 at t1, the output voltage will be Uo(i)_max while the output power is Po(i)_max. Uo(i)_err would still remain positive. However, D(i) is limited to be no greater than 1. Thus, the output voltage will no longer increase, which means that the receiver controller is saturated. Finally, in the steady state, the output voltage will remain Uo(i)_max in state 2.  It can be seen that when duty cycle D (i) equals to 1, the receiver has the maximum power: Similar to the transmitter controller, each receiver controller has two states:  Figure 8b shows how the receiver controller works in state 2. If Uo(i) is less than Uo(i)_max at t0, Uo(i)_err will be positive, and the PI controller would try to increase D(i). When D(i) increases to 1 at t1, the output voltage will be Uo(i)_max while the output power is Po(i)_max. Uo(i)_err would still remain positive. However, D(i) is limited to be no greater than 1. Thus, the output voltage will no longer increase, which means that the receiver controller is saturated. Finally, in the steady state, the output voltage will remain Uo(i)_max in state 2. In general, under the control of the receiver controller, the output voltage of the receiver in the steady state can be described as: The controllers at the transmitter and receiver sides regulate the power in their own ways. The operation principles of the transmitter and receiver controllers are relatively simple. However, the interaction between controllers is more important.
It is shown that Po(i)_max varies with Pi_norm. However, Pi_norm is controlled by the transmitter controller. Thus, the regulation of the transmitter controller will change Po(i)_max. The output power of a receiver is: when Po(i)_max is changed by the transmitter controller, the receiver controller will try to maintain the output voltage, as well as the output power by adjusting D(i). Therefore, the transmitter controller will interfere with the receiver controllers.  Figure 8b shows how the receiver controller works in state 2. If U o(i) is less than U o(i)_max at t 0 , U o(i)_err will be positive, and the PI controller would try to increase D (i) . When D (i) increases to 1 at t 1 , the output voltage will be U o(i)_max while the output power is P o(i)_max . U o(i)_err would still remain positive. However, D (i) is limited to be no greater than 1. Thus, the output voltage will no longer increase, which means that the receiver controller is saturated. Finally, in the steady state, the output voltage will remain U o(i)_max in state 2.
In general, under the control of the receiver controller, the output voltage of the receiver in the steady state can be described as: The controllers at the transmitter and receiver sides regulate the power in their own ways. The operation principles of the transmitter and receiver controllers are relatively simple. However, the interaction between controllers is more important.
It is shown that P o(i)_max varies with P i_norm . However, P i_norm is controlled by the transmitter controller. Thus, the regulation of the transmitter controller will change P o(i)_max . The output power of a receiver is: when P o(i)_max is changed by the transmitter controller, the receiver controller will try to maintain the output voltage, as well as the output power by adjusting D (i) . Therefore, the transmitter controller will interfere with the receiver controllers.
Similarly, P i_max varies with D in each receiver. Although D is controlled by the receiver controllers, the regulation of the receiver controllers will change P i_max . For the transmitter, the power absorbed from the DC power supply is: The transmitter will try to maintain P i by adjusting P i_norm when any receiver controller changes P i_max . Therefore, the receiver controllers will also interfere with the transmitter controller.
Both the transmitter and receiver controllers will participate in the regulation process before the system enters the steady state, with the two types of controllers affecting each other during the regulation. The power distribution is the result of the interaction between the controllers.
The following relationship is derived by combining (16) and (17): From (20), the ratio of P o_max between the receivers can be calculated as: where i and j are the serial numbers of receivers. Under the assumption that the efficiency is constant and by differentiating (28) and (29) with respect to P i , the following equations can be derived: Figure 9 shows an example illustrating the power distribution between the receivers. In this example, a multi-receiver WPT system with three receivers is presented. The power distribution of the entire system in the steady state as P i *, which varies from low to high. The blue lines indicate the variations in P i and P o_sum with P i *. The gray area between P i and P o_sum represents the power loss P loss . The other lines indicate the variation in the output power of the receivers with P i *.
other during the regulation. The power distribution is the result of the interaction between the controllers.
The following relationship is derived by combining (16) and (17): From (20), the ratio of Po_max between the receivers can be calculated as: : : where i and j are the serial numbers of receivers. Under the assumption that the efficiency is constant and by differentiating (28) and (29) with respect to Pi, the following equations can be derived: Figure 9 shows an example illustrating the power distribution between the receivers. In this example, a multi-receiver WPT system with three receivers is presented. The power distribution of the entire system in the steady state as Pi*, which varies from low to high.

Four Typical Case Study in the Proposed Structure
Four typical stages in this example are analyzed in detail, and each specific case is represented by in Figure 10.
Case 1: P i * is low enough, and the transmitter controller will set P i_norm to a low level to ensure that P i is equal to P i *. Consequently, P o_max of each receiver will be lower than P o *, and all the receiver controllers will work in state 2. P o of every receiver is equal to P o_max under the control of the receiver controller. The above analysis shows that P i = P i * and P o = P o_max . Substituting the above relationships into (30), the slope of P o can be derived based on (22) as:

R L(3)
, i= 1, 2, 3 (31) With the slope information of each case, the power distribution in this system in the steady state can be calculated. Receivers Receivers This example has three receiver and four cases, and the number of cases ncase is determined by the number of receivers: The total charging power is non-cooperative game, the Karush-Kuhn-Tucker could be applied to optimize the gaming relation among the receivers. The power can be inserted into the Lagrangian function as Receivers Case 2: When the system already works in case 1 and then increases P i * until one of the receiver controllers transfers to state 1, the system would transfer to case 2. The transmitter controller will increase P i_norm to increase P i , thereby increasing P o_max of every receiver. For the receiver #3, P o(3)_max will increase beyond P o(3) *. Therefore, the controller of receiver #3 would enter state 1 and P o(3) is equal to P o(3)_max . The receiver controllers of receivers #1 and #2 remain in state 2. To sum up, in case 2, P i = P i *, P o(1) = P o(1)_max , P o(2) = P o(2)_max , and P o(3) = P o(3) *. Substituting the above relationships into (31), the slope of P o in case 2 can be derived as: Case 3: When the system already works in case 2 and then further increases P i * until another receiver controller transfers to state 1, the system would transfer to case 3. The transmitter controller will further increase P i_norm to increase P i , thereby increasing P o_max of every receiver. In this example, the controller of receiver #2 will transfer to state 1. Meanwhile, the receiver controller of receiver #1 remains in state 2 and receiver controller of receiver #3 remains in state 1. To sum up, in case 3, P i = P i *, P o(1) = P o(1)_max , P o(2) = P o(2) *, and P o(3) = P o(3) *. Substituting the above relationships into (30), the slope of P o in case 3 can be derived as: Case 4: When P i * is set high enough, the system will enter case 4. In this case, P i_norm will be saturated at 1 under the control of the transmitter controller, which means that the transmitter works in state 2. Because P i_norm maintains its maximum value of 1, the P o_ max values of each receiver reach their maximum values and are higher than P o *. Consequently, all the receiver controllers will work in case 1. To sum up, in this case, P i = P i_max and P o = P o *. Substituting the above relationships into (26), the slope of P o in case 4 can be derived as: With the slope information of each case, the power distribution in this system in the steady state can be calculated.
This example has three receiver and four cases, and the number of cases n case is determined by the number of receivers: The total charging power is non-cooperative game, the Karush-Kuhn-Tucker could be applied to optimize the gaming relation among the receivers. The power can be inserted into the Lagrangian function as Then the partial derivatives can be derived as The KKT candidate point can be solved as Since the Lagrangian function is convex, the Hessian of the Lagrangian function is always positive definite and the derivative is the global optimal point. Then, the calculated power divisions are achieved through the duty cycle control of the DC-DC converters in the receivers. The proposed algorithm is linear, and is proportional to O(n).
For a given receiver, the combined Langrangian function can be calculated as where λ i is the Lagrange multiplier. Under the Karush-Kuhn-Tucker conditions, the optimization problem can be given as Then, the charging power of each receiver is competing under the balanced condition at the power of p total , therefore, Then, the distributed power is unique. In our application, the implementation of competitive mechanism is realized by searching λ i when the receivers need to compete on the limited total available transmitting power p total , as shown in the flow chart in Figure 11. Then, the distributed power is unique. In our application, the implementation of competitive mechanism is realized by searching λi when the receivers need to compete on the limited total available transmitting power ptotal, as shown in the flow chart in Figure  11. The power distribution is the result of the interplay among the receivers. The total power is limited by the transmitter, and all the receivers compete for more power until their demands are met. Receivers with higher mutual inductance and low load resistance have an advantage. Finally, there will be an equilibrium. The result is that some receivers get sufficient power, while some do not. The balance itself is the result of power distribution. With the increase in the transmitter power capacity, more and more receivers can obtain the required power, and they do not compete for more power. They maintain their existing power, while others still compete for more power.
In addition, the power distribution depends on the mutual positions of elements too. So, constant, defined, and identical position between elements should be expected. Figure  12 gives the magnetic field distribution of the coils. Since the receiving coils are aligned diagonally, magnetic induction lines from one receiving coil hardly pass through the other two receiving coils. Therefore, the cross-coupling among the receiving coils can be ne- The power distribution is the result of the interplay among the receivers. The total power is limited by the transmitter, and all the receivers compete for more power until their demands are met. Receivers with higher mutual inductance and low load resistance have an advantage. Finally, there will be an equilibrium. The result is that some receivers get sufficient power, while some do not. The balance itself is the result of power distribution. With the increase in the transmitter power capacity, more and more receivers can obtain the required power, and they do not compete for more power. They maintain their existing power, while others still compete for more power.
In addition, the power distribution depends on the mutual positions of elements too. So, constant, defined, and identical position between elements should be expected. Figure 12 gives the magnetic field distribution of the coils. Since the receiving coils are aligned diagonally, magnetic induction lines from one receiving coil hardly pass through the other two receiving coils. Therefore, the cross-coupling among the receiving coils can be neglected.

Results
An experimental multi-receiver WPT system has been built to verify the validity of the theoretical analysis.
The experimental setup is shown in Figure 13. It has one transmitter and three individual receivers. The model of all semiconductor switches in the system is Infineon IHW50N65R5, and the model of receiver rectifier diodes is CREE D3065C5.

Results
An experimental multi-receiver WPT system has been built to verify the validity of the theoretical analysis.
The experimental setup is shown in Figure 13. It has one transmitter and three individual receivers. The model of all semiconductor switches in the system is Infineon IHW50N65R5, and the model of receiver rectifier diodes is CREE D3065C5.
The coil structure is shown in Figure 14. The transmitting coil has three slots designed for three receivers. Its winding consists of eight turns of Litz wire. All the receiving coils have the same structure and size. They are square-shaped with ferrite bricks over them to augment the mutual inductance. The gap between the transmitting and receiving coils is 5 cm.  The coil structure is shown in Figure 14. The transmitting coil has three slots designed for three receivers. Its winding consists of eight turns of Litz wire. All the receiving coils have the same structure and size. They are square-shaped with ferrite bricks over them to augment the mutual inductance. The gap between the transmitting and receiving coils is 5 cm. The configuration of this experimental setup is listed in Table 1. ωLa of the LCC compensation circuit at the transmitting side is designed to be 10 ohms. The designated parameters of the compensation circuit are derived from the analysis.
The switching frequency of the receiver regulator is 50 kHz. The control algorithm is realized by programming using DSP controllers, with every transmitter and receiver having an individual DSP controller. The loads of the receivers were electric loads in the constant resistance (CR) mode. A power analyzer is connected to record the powers. In the experiment, the gate driver UCC21520 and PI controller dsp28335 have been used. The current and voltage signals are sampled and differentially amplified by AMC1302. The signal is collected through AD7682 and then transmitted back to the DSP. The static experiment was conducted to verify the validity of the theoretical analysis, where the input power was set at the maximum value. The parameters of the controllers and the loads are shown in Table 2. In this experiment, all the receivers were active. Pi* was swept from low to high. Each time Pi* changed, the controllers would regulate the power and, thus, the system would enter steady state again. The three receivers are terminated with 12 Ω, 14 Ω, and 20 Ω loads, respectively.
The result of the static experiment is shown in Figure 15. The optimal efficiency in the simulation is equal to 88%. With the increase in Pi*, Pi is always lower than Pi*, indicating that Pi is limited under the regulation of the transmitter controller. The total output The configuration of this experimental setup is listed in Table 1. ωL a of the LCC compensation circuit at the transmitting side is designed to be 10 ohms. The designated parameters of the compensation circuit are derived from the analysis. The switching frequency of the receiver regulator is 50 kHz. The control algorithm is realized by programming using DSP controllers, with every transmitter and receiver having an individual DSP controller. The loads of the receivers were electric loads in the constant resistance (CR) mode. A power analyzer is connected to record the powers. In the experiment, the gate driver UCC21520 and PI controller dsp28335 have been used. The current and voltage signals are sampled and differentially amplified by AMC1302. The signal is collected through AD7682 and then transmitted back to the DSP.
The static experiment was conducted to verify the validity of the theoretical analysis, where the input power was set at the maximum value. The parameters of the controllers and the loads are shown in Table 2. In this experiment, all the receivers were active. P i * was swept from low to high. Each time P i * changed, the controllers would regulate the power and, thus, the system would enter steady state again. The three receivers are terminated with 12 Ω, 14 Ω, and 20 Ω loads, respectively.

Instruction output voltage
The result of the static experiment is shown in Figure 15. The optimal efficiency in the simulation is equal to 88%. With the increase in P i *, P i is always lower than P i *, indicating that P i is limited under the regulation of the transmitter controller. The total output power P o_sum increases with the increase in P i *. The difference between P i and P o_sum is the power loss of the entire system P loss . The power distribution of this experiment is comprised of four cases: P o curves have different slopes in different cases. With the increase in P i *, all the receivers absorb more power until the output power P o reaches its equivalent instructed output power P o *. Finally, all the receivers reach their P o * as P i * is high enough.
power Po_sum increases with the increase in Pi*. The difference between Pi and Po_sum is the power loss of the entire system Ploss. The power distribution of this experiment is comprised of four cases: Po curves have different slopes in different cases. With the increase in Pi*, all the receivers absorb more power until the output power Po reaches its equivalent instructed output power Po*. Finally, all the receivers reach their Po* as Pi* is high enough.

Symbol Value
Load Resistance

Instruction output voltage
The comparison shows that the experimental result is consistent with the theoretical result. However, there are some errors between the measured and theoretical values. The errors originate from the following sources: (1) Theoretical result is obtained under the assumption that the efficiency is constant; in the practical system, the efficiency increases with the increase in the power. A dynamic experiment is performed to verify the flexibility of the proposed method, where the input power is increased dynamically from 0 to the maximum value. In this experiment, the working condition was set to simulate a practical working condition. In a practical working condition, the system parameters may vary dynamically. For example, the coil position, load resistance, and number of receivers may dynamically change while the system is running. The proposed control method is expected to handle these variations.
This experiment simulated the condition wherein the number of receiver changes from 2 to 3. The setup is the same as the setup in the static experiment. This procedure consists of two phases. In phase 1, only receivers #1 and #3 are operated in the WPT system The comparison shows that the experimental result is consistent with the theoretical result. However, there are some errors between the measured and theoretical values. The errors originate from the following sources: (1) Theoretical result is obtained under the assumption that the efficiency is constant; in the practical system, the efficiency increases with the increase in the power. (2) Voltage drops of the wire and semiconductor devices are not considered in the theoretical analysis. (3) The practical components of the compensation circuits and coil have tolerances. Thus, the system deviates from the resonance state. (4) The harmonic components are ignored in the theoretical analysis. Nonetheless, the theoretical analysis is in good agreement with the experimental result with acceptable errors.
A dynamic experiment is performed to verify the flexibility of the proposed method, where the input power is increased dynamically from 0 to the maximum value. In this experiment, the working condition was set to simulate a practical working condition. In a practical working condition, the system parameters may vary dynamically. For example, the coil position, load resistance, and number of receivers may dynamically change while the system is running. The proposed control method is expected to handle these variations.
This experiment simulated the condition wherein the number of receiver changes from 2 to 3. The setup is the same as the setup in the static experiment. This procedure consists of two phases. In phase 1, only receivers #1 and #3 are operated in the WPT system while receiver #2 is disabled. In phase 2, all the receivers are operated. First, the system works in phase 1 and is in steady state, then receiver #2 joins the WPT system and the system turns to phase 2. With the joining of receiver #2, the controllers regulate the power among the receivers. Finally, after the transient regulation of the controllers, the system would enter the steady state of phase 2. A power analyzer is employed to record the input and output power continuously with a period of 10 ms.
Under the theoretical power distribution condition in the steady state of phase 1, the optimal efficiency is 88%. As the condition of phase 2 is the same as that in the static experiment, the power distribution state is steady. The theoretical power distribution in both phases 1 and 2 can be determined. Figure 16 shows the power variation when the system moves from phase 1 to phase 2. At the beginning, the system is in the steady state of phase 1. The output power of receiver #2 was 0 as it is disabled, and the power only flows to receiver #3. As soon as the receiver #2 is activated at 43.2 s, the controller starts to increase the output power of receiver #2. The output power of receiver #3 decreases to the steady value in phase 2. The output power of receiver #3 decreases first, then increases, and finally maintains the same value. The regulation process lasts for approximately 3 s. Finally, the system returns to the steady state. In Table 3, the theoretical and measured power distributions in the steady state are compared. The measured power values match well with the theoretical power values with the maximum error of −5.50%.
Electronics 2021, 10, 1308 18 of 21 while receiver #2 is disabled. In phase 2, all the receivers are operated. First, the system works in phase 1 and is in steady state, then receiver #2 joins the WPT system and the system turns to phase 2. With the joining of receiver #2, the controllers regulate the power among the receivers. Finally, after the transient regulation of the controllers, the system would enter the steady state of phase 2. A power analyzer is employed to record the input and output power continuously with a period of 10 ms. Under the theoretical power distribution condition in the steady state of phase 1, the optimal efficiency is 88%. As the condition of phase 2 is the same as that in the static experiment, the power distribution state is steady. The theoretical power distribution in both phases 1 and 2 can be determined. Figure 16 shows the power variation when the system moves from phase 1 to phase 2. At the beginning, the system is in the steady state of phase 1. The output power of receiver #2 was 0 as it is disabled, and the power only flows to receiver #3. As soon as the receiver #2 is activated at 43.2 s, the controller starts to increase the output power of receiver #2. The output power of receiver #3 decreases to the steady value in phase 2. The output power of receiver #3 decreases first, then increases, and finally maintains the same value. The regulation process lasts for approximately 3s. Finally, the system returns to the steady state. In Table 3, the theoretical and measured power distributions in the steady state are compared. The measured power values match well with the theoretical power values with the maximum error of −5.50%.  The dynamic experimental result demonstrates that the proposed control method can handle the dynamic variation in the number of receivers. The power is redistributed under the combined action of all the controllers following a change in the number of receivers in the system. The control method demonstrates flexibility and robustness in the dynamic  The dynamic experimental result demonstrates that the proposed control method can handle the dynamic variation in the number of receivers. The power is redistributed under the combined action of all the controllers following a change in the number of receivers in the system. The control method demonstrates flexibility and robustness in the dynamic experiment. The controller works as expected even when some of the parameters vary while the system is in operation.

Discussion
In a practical scenario, P i * of the transmitter controller should be the same or only slightly lower than the rated power of the DC power supply of the transmitter to prevent it from overloading. The P i * limits the maximum power of the entire system. P o * of the receivers should be the required power of their loads. However, the practical output power P o may be lower than the required power due to the limit of the transmitter power capacity. The experimental setup adopts a transmitting coil designed especially for three receivers. However, the transmitting coil design is not limited. Coils with any shape, size, number of receivers, etc., are acceptable for the proposed control method. Receivers with higher power demand should be equipped with bigger receiving coil to achieve higher mutual inductance. Otherwise, the receiver may face disadvantage in the competition.
The proposed method distributes power unequally, but the competition mechanism makes dynamic distribution possible. Receivers can join or quit the system freely, and the receiver power demand can also vary dynamically. No matter how the system varies, the system will redistribute the power automatically. This control method can handle massive receivers without reprogramming, and will not become complicated with the increase in the number of receivers. Therefore, the system can expand easily. Table 4 gives the comparison between the proposed method and some representative existing methods, and the advantages and disadvantages are listed. It demonstrates the superiority of the proposed method over previous methods. In the comparison, algorithm complexity is judged by the time and space consumed at a given input of size n [27]. The controller algorithms lower than O(nlogn), such as O(logn) and O(n), are defined as "simple" whereas those higher than O(nlogn), such as O(nˆk) and O(kˆn), are defined as "complicated". Circuit configuration complexity is based on equivalent circuit topology. If the topology structure satisfies Karp-Lipton condition [28], it is defined as "simple", otherwise it is defined as "complicated". Dynamic distribution is judged by whether the circuit can distribute power flexibly with the change of outside factors, such as load, receiver number, or charging distance. Scalability is judged by whether this system can be applied in other multi-receiver scenarios with only changing the parameters instead of system structure. This is mainly determined by circuit topology, operating frequency and dynamic distribution. The proposed method has a constant-voltage output characteristic. In the future, the configuration can be set to constant current output by modifying the compensation circuit and receiver controller to suit applications, such as charging batteries or supercapacitors.

Conclusions
A power distribution control method for a multi-receiver WPT system is proposed in this paper. By employing double-side control, the power from transmitter distributes to multiple receivers automatically. Receivers acquire power by competition, and the receivers with higher mutual inductance is easier to gain higher power. This procedure is dynamic, so the power will be redistributed when the system parameters varied. Hence, this method can handle various occasions such as the number of receivers changing, receiver power demands changing or coil position (mutual inductance) changing. For receivers, the received power will not exceed their demands, while the transmitter will not be overloaded at any given time. The control algorithm and main circuit configuration are simple, and the communication channel is not required. The increase in the number of receivers will not increase the complexity. Thus, the number of receivers can be increased easily. Static experiment with fixed system parameters indicates the power distribution result can be accurately predicted based on the circuit and controller parameters. Dynamic experiment with dynamically varying system parameters demonstrates the reliability, scalability, flexibility, and generality of this method. The proposed WPT system is compared with several others. The classification is derived based on algorithm theory and circuit topology. The characteristics are determined through mathematical calculations based on the results provided in those references.