### I. Introduction

### II. Modeling of a SIMO WPT System

### 1. Formulation of a SIMO WPT System

*L*

*) and resistances (*

_{i}*R*

*). Each coil is loaded with a capacitor (*

_{i}*C*

*) for resonance at a specific design frequency. The transmitter is excited by a voltage source (*

_{i}*V*

_{0}), and the

*N*receivers have load resistances (

*R*

*) to receive transferred powers. Once the sizes and positions of the transmitter and receivers are specified, the coupling coefficients between any two loops can be determined using the ratio of magnetic flux lines from one loop to the other [17]. The WPT system can then be analyzed in terms of the efficiencies for each receiver and for the total system. For (*

_{Li}*N*+ 1) loops of the SIMO WPT system in Fig. 1, we can write (

*N*+ 1) equations using KVL. They can be rearranged using a Z-matrix formulation. The Z-matrix as a function frequency is given by:

*V*

_{0}in the column matrix [

*V*] is the supplied voltage of the transmitter. The principal diagonal elements of [

*Z*] at the resonant design frequency are given by:

*R*

*s are the loop resistances and*

_{i}*R*

*’s are the load resistances for the receivers. The other elements of [*

_{Li}*Z*] are given by:

*k*

*s are the coupling coefficients between two loops. When all the elements of the Z-matrix are given, the current on each loop is determined by [*

_{ij}*I*] = [

*Z*]

^{−1}[

*V*], in which all couplings between any two loops including the mutual are all considered. The input power and the received power at each receiver can then be evaluated using:

### 2. Derivation of the Optimum Load and Efficiency in SIMO

*i*

^{th}receiver at the resonant frequency is defined and expressed as:

*S*

_{i0}|

^{2}/(1-|

*S*

_{00}|

^{2}) using EM simulations. The efficiencies using the two methods always agree well. If the coupling coefficients between any two receivers shown in Fig. 1 are negligible (

*k*

*→ 0,*

_{ij}*i*≠

*j*for

*i*,

*j*= 1, 2,···,

*N*), the ratio of the currents on the

*i*

^{th}receiver and the transmitter is expressed in an analytic form, as in [18]:

*F*

*and*

_{i}*β*

*are defined as the*

_{i}*figure of merit*and the

*normalized load resistance*defined by:

*k*

*is the coupling coefficient between the transmitter and*

_{i}*i*

^{th}receiver [17], and

*Q*

*is the quality factor of the*

_{i}*i*

^{th}receiver.

*R*

*. The*

_{Li}*system efficiency*(or

*total efficiency*) is defined and given by:

*N*normalized loads.

*N*partial differential equations must be solved simultaneously. The normalized load resistance

*β*

*for*

_{i}*i*= 1, 2,···,

*N*satisfying (13) turned out to be identical as a constant

*β*

*for all receivers [11], although the figures of merit (*

_{opt}*F*

*s) given by (10) are different. They are:*

_{i}*overall figure of merit*of a SIMO system (

*F*) as:

### 3. Numerical Examples with One Transmitter and Two Receivers (N = 2)

*F*

_{1}= 16.2 and

*F*

_{2}= 10.7. The efficiency of individual Rx

*η*

*and the total system efficiency*

_{i}*η*

*can be controlled by the (normalized) load*

_{t}*β*

*. From Fig. 2, we can analyze how selectively power can be distributed to each Rx in the following scenarios.*

_{i}*β*

_{2}→ ∞) to suppress any power flow into Rx2 and re-direct that power to Rx1. Simultaneously, the load value of Rx1 should be optimized to receive the maximum power. This conjecture is consistent with the result in Fig. 2. Explicitly, when

*β*

_{1}is about 16.2 and

*β*

_{2}→ ∞,

*η*

_{1}is maximized to 88.4% and

*η*

_{2}decreases to zero in Fig. 2(a) and (b).

*β*

_{1}→ ∞ (Rx1 is open) and

*β*

_{2}is about 10.7,

*η*

_{1}and

*η*

_{2}reach zero and 82.9%, respectively. Obviously, the maximum

*η*

_{1}(88.4%) is greater than the maximum

*η*

_{2}(82.9%) because Rx1 is inherently more strongly coupled to Tx than Rx2;

*F*

_{1}(= 16.2) is greater than

*F*

_{2}(= 10.7).

*η*

*. The maximum total efficiency in Fig. 2(c) is about 90.2% when*

_{t}*η*

_{1}= 63.1% and

*η*

_{2}= 27.1%.

*β*

_{1}and

*β*

_{2}, the total efficiency

*η*

*is nearly invariant from the maximum of 90.2%. This implies the power can be distributed among multiple receivers in almost any ratio users desire, with minimum loss in total efficiency.*

_{t}### III. Control of Power Distribution Ratio

### 1. Load Resistance for Desired Power Distribution

*α*

_{1},

*α*

_{2}, ···,

*α*

*) for the receivers as follows:*

_{N}*F*

*s (for*

_{i}*i*= 1, 2,···,

*N*) are all known,

*α*

*is determined by the choices of*

_{i}*β*

*s (for*

_{i}*i*= 1, 2,···,

*N*). In particular, when

*β*

*=*

_{i}*β*

*(for*

_{opt}*i*= 1, 2,···,

*N*) (14),

*α*

*in (18) becomes:*

_{i}*α*

_{i}_{,}

*’s (for*

_{opt}*i*= 1, 2,···,

*N*) are the optimum power distribution ratio to maximize the total efficiency (12). If the desired power distribution ratio (

*α*

*’s for*

_{i}*i*= 1, 2,···,

*N*) is different from the optimum power distribution ratio (

*α*

_{i}_{,}

*’s for*

_{opt}*i*= 1, 2,···,

*N*), the total efficiency may become somewhat lower than the maximum depending on the degree of the difference of the optimum (12).

*α*

*’s) is specified first in (18), the load values,*

_{i}*β*

*s, of the receivers can be obtained numerically. However, the exact analytic solution in a simple form has not been found yet.*

_{i}*β*≈

*β*(or

*β*>> 1), which is indeed the case for most practical WPT problems. This leads to the conclusion that to achieve the desired

*α*

*,*

_{i}*β*

*is:*

_{i}*i*from 1 to

*N*. The receiver load solution (21) can be used as a formula determined by only

*F*(15),

*F*

*(10), and the desired*

_{i}*α*

*. In Sections III-B and III-C, we will show that the accuracy of (21) is high enough for most practical problems unless the desired power distribution ratio (*

_{i}*α*

_{1},

*α*

_{2},···,

*α*

*) deviates too much from its optimum (*

_{N}*F*

_{1}

^{2}/

*F*

^{2},

*F*

_{2}

^{2}/

*F*

^{2},···,

*F*

_{N}^{2}/

*F*

^{2}) (19).

*β*

*/*

_{i}*β*

*) as a function of the desired*

_{opt}*α*

*and*

_{i}*F*

_{i}^{2}/

*F*

^{2}(=

*α*

_{i}_{,}

*). With this universal curve, we can determine the normalized load resistance*

_{opt}*β*

*(or*

_{i}*R*

*/*

_{Li}*R*

*) to realize the desired*

_{i}*α*

*(for*

_{i}*i*= 1, 2,···,

*N*) once the figures of merit (

*F*

*s) for any specific SIMO WPT system are known.*

_{i}*F*

_{1}= 16.2 and

*F*

_{2}=

*F*

_{3}= 11.4,

*F*= 22.9 using (15),

*α*

_{1,}

*=*

_{opt}*F*

_{1}

^{2}/

*F*

^{2}= 0.5,

*α*

_{2,}

*=*

_{opt}*α*

_{3,}

*= 0.25 from (19),*

_{opt}*β*

*= 22.9 from (14) or (16), and*

_{opt}*η*

*(*

_{t}*β*

*) = 91.6% from (17). If the desired*

_{opt}*α*

_{1},

*α*

_{2}, and

*α*

_{3}are 0.7, 0.2, and 0.1, we can obtain

*β*

_{1}= 16.4,

*β*

_{2}= 28.6, and

*β*

_{3}= 57.3 using (21) and

*η*

*= 91% using (12).*

_{t}### 2. Numerical Validation by Circuit Simulations (N = 2)

*R*,

*L*, and

*C*) are 0.034 Ω, 0.533 μH, and 1.034 nF for the Tx loop and 0.017 Ω, 0.223 μH, and 2.472 nF for the Rx loop. The quality factors of the Tx and Rx loops are 668 and 559, respectively.

*k*

_{1}and

*k*

_{2}) are assumed to be identical at 0.0265; therefore,

*F*

_{1}=

*F*

_{2}= 16.2 using (10). Fig. 4(a) shows the realized (or achieved) power distribution ratios and the total efficiencies as a function of the desired

*α*

_{1}. Note that

*α*

_{2}= 1 –

*α*

_{1}.

*α*

_{1,}

*=*

_{opt}*α*

_{2,}

*= 0.5 ( = 16.2*

_{opt}^{2}/(16.2

^{2}+ 16.2

^{2})) from (19). When

*β*

_{1}=

*β*

_{2}=

*β*

*= 22.9 (16), the total efficiency is maximized to*

_{opt}*η*

*(*

_{t}*β*

*) of 91.64%. One can find that the power is distributed to Rx1 with an accurate ratio as we target. Fig. 4(a) also demonstrates that the closed-form formula (21) is more accurate than the formula presented in [14]. Furthermore, we can see that even when the mutual coupling between two receivers is considered with*

_{opt}*F*

_{21}/

*F*

_{1}= 0.5, the results do not show much difference.

*k*

_{1}and

*k*

_{2}) are assumed to be 0.0265 and 0.0175. Therefore,

*F*

_{1}and

*F*

_{2}are 16.2 and 10.7. The realized power distribution ratios and the total efficiencies are shown in Fig. 4(b) as a function of the desired

*α*

_{1}. The optimum power distribution ratio at Rx1 (

*α*

_{1,}

*) is 0.7 using (19). When*

_{opt}*β*

_{1}=

*β*

_{2}=

*β*

*= 19.4, the total efficiency is maximized to 90.21%. Unlike the first example, the realized power distribution ratio slightly deviates from the target ratio, especially when the target power distribution ratio is low. For example, when the target ratios*

_{opt}*α*

_{1}and

*α*

_{2}are 0.3 and 0.7, the realized ratios are about 0.34 and 0.66.

_{1}for various maximum total efficiencies of 0.99, 0.9, 0.8, and 0.7 (17) obtained when the overall figures (

*F*) of merit are 199, 19, 8.9, and 5.6, respectively. It is assumed that

*N*= 2 and

*F*

_{1}

^{2}/

*F*

^{2}=

*α*

_{1,}

*= 0.7 for all cases. Although the accuracy of the realized power distribution ratio slightly decreases as the desired*

_{op}*α*

_{1}deviates from its optimum, the proposed simple solution (21) is good enough for all practical applications. If a higher accuracy is required, the numerical method can be used to obtain the exact solution.

*F*(15) for different desired

*α*

_{1}s of 0.3 and 0.5. The optimum power distribution ratio

*α*

_{1,}

*is fixed at 0.7. The accuracy of power distribution ratio increases as*

_{opt}*F*increases. This figure also demonstrates that the proposed method is good enough unless

*F*in (15) is extremely small. Notice also that the realized total efficiency

*η*

*(12) is only slightly lower than its maximum of*

_{t}*η*

*(*

_{t}*β*

*) (17) obtained when the desired*

_{opt}*α*

_{1}is identical to

*α*

_{1,}

*of 0.7 (21).*

_{opt}### 3. Validation by Full Wave Electromagnetic Simulation with Three Receivers (N = 3)

*f*

_{0}’s) of Tx and Rx loops are 6.78 MHz. The radii of Tx and Rx loops are 10 cm and 5 cm, respectively. The radii of the loop ring are 0.2 cm. Theoretical and EM-simulated circuit element values of Tx and Rx loops are summarized in Table 1. The quality factors of the Tx and Rx loops based on EM simulations are 614 and 506 at 6.78 MHz, respectively.

*S*-parameter obtained from EM simulations [19]. Figures of merit can be obtained using (10). Coupling coefficients and figures of merit between two loops are summarized in Table 2. The EM-simulated

*k*s agree well with the theoretical ones.

*F*

_{1},

*F*

_{2}, and

*F*

_{3}are 14.8, 10.4, and 10.4, respectively, using (10). For this example,

*α*

_{1,}

*=*

_{opt}*F*

_{1}

^{2}/

*F*

^{2}= 0.5 and

*α*

_{2,}

*=*

_{opt}*α*

_{3,}

*= 0.25 from (19).*

_{opt}*R*

*s) using (21) based on Tables 1 and 2 to realize a desired power distribution ratio (*

_{Li}*α*

*s) based on the assumptions of*

_{i}*α*

_{2}=

*α*

_{3}and

*α*

_{2}= 2

*α*

_{3}, respectively. Note that for each case,

*α*

_{1}+

*α*

_{2}+

*α*

_{3}= 1. As we can see in (21),

*R*

*decreases as the desired*

_{Li}*α*

*increases.*

_{i}*α*

_{1},

*α*

_{2}, and

*α*

_{3}are 0.5, 0.25, and 0.25 in Fig. 8(a), the load resistances (

*R*

_{L}_{1},

*R*

_{L}_{2}, and

*R*

_{L}_{3}) are the optimum values (

*R*

_{L}_{,}

*s) (16), which are about 0.39 Ω.*

_{opt}*R*

*, and EM simulation with and additional feeding loop (50 Ω) [20].*

_{Li}*α*

_{1}, with

*α*

_{2}=

*α*

_{3}. For this case, the maximum efficiency is 91.6%, and realized power distribution ratios are shown to be almost the same as the desired ones. Again, we can see that the realized

*α*

*s agree well with the desired ones for all cases.*

_{i}*α*

*s and total efficiency are plotted as a function of the desired*

_{i}*α*

_{1,}with

*α*

_{2}= 2

*α*

_{3}in Fig. 9(b). We can also see that the

*α*

*s are well realized, as we desire. For example, when the desired*

_{i}*α*

_{1},

*α*

_{2}, and

*α*

_{3}are 0.7, 0.2, and 0.1, respectively, they are realized as 0.69, 0.2, and 0.11. The results of EM simulations without and with a feeding loop [20] are also shown to be in good agreement with the results of circuit simulations.

*H*

*| distributions for the cases of desired*

_{z}*α*

_{1}= 0.5 and

*α*

_{2}=

*α*

_{3}= 0.25 and desired

*α*

_{1}= 0.7,

*α*

_{2}= 0.2, and

*α*

_{3}= 0.1. For these field distributions, a power source of 1 W was used in HFSS EM simulations. The |

*H*

*| field distributions in Fig. 10(a) and (b) well reflect the desired power distributions or efficiencies. The total efficiencies and power distribution ratios based on theory and EM simulations are in good agreement.*

_{z}*α*

*s using the simple closed-form solution (21) show good agreement with the desired ones for a wide range of applications.*

_{i}### IV. Conclusion

*N*receivers. There have also been some analytical or closed-form solutions, but their accuracy is far lower than the proposed solution. The proposed formula is expected to be implemented in future SIMO industrial applications due to its simple but relatively accurate property.