### I. Introduction

### II. Modeling of MIMO Wireless Power Transfer System

*M*transmitters and

*N*receivers. All the loops have some inductances and resistances. We assume that each loop is loaded with a capacitor to resonate at a specific design frequency. In addition, we assume that the

*M*transmitters are excited by voltage sources and that

*N*receivers have extra load resistances to receive transferred powers. Once the sizes and positions of the transmitters and receivers are specified, the coupling coefficients can be determined in a straightforward manner [14, 15]. Then, the WPT system can be analyzed in various aspects. The general MIMO WPT system is commonly analyzed using Z-matrix formulation [16, 17]. The Z matrix as a function frequency is given by

*M*elements in the column matrix [

*V*] are the supplied voltages. The supplied voltages

*V*

*(= |*

_{n}*V*

*| ∠*

_{n}*θ*

*,*

_{n}*n*= 1, 2, ···,

*M*) may be complex numbers in general. The principal diagonal elements of [

*Z*] are expressed as

*n*= 1, 2, ···,

*M*and

*n*=

*M*+ 1,

*M*+ 2, ···,

*M*+

*N*, where is the common angular resonant frequency of each loop,

*R*

*’s are the loop resistances,*

_{n}*L*

*’s are the loop self-inductances,*

_{n}*ω*

_{0}

*L*

*’s are the reactance slope parameters of the loop resonator, and*

_{n}*R*

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

_{Ln}*Z*] are given by

*m*,

*n*= 1, 2, ···,

*M*,

*M*+ 1, ···,

*M*+

*N*, where

*M*

_{m}_{,}

*and*

_{n}*k*

_{m}_{,}

*’s are the mutual inductances and coupling coefficients between two loops, respectively [18].*

_{n}*I*] = [

*Z*]

^{−1}[

*V*]. Then, the total input power and the received power at each receiver are given by

*n*

^{th}receiver is defined as

*V*

*’s,*

_{n}*R*

*’s, and*

_{n}*R*

*’s for a MIMO system, are specified, the WPT efficiencies can be easily evaluated at a design frequency*

_{Ln}*ω*

_{0}using (5)–(8). Commonly, we aim to maximize (7) or (8). For this purpose, we may decide the values of

*R*

*’s by either an analytical or some numerical optimization methods. In this study, we focus on SIMO WPT systems and obtain closed-form solutions for the optimal loads.*

_{Ln}*M*= 1,

*N*= 1), the WPT efficiency

*η*

_{2}is compactly expressed [19] as

*b*

*(=*

_{2}*R*

*/*

_{L2}*R*

*) is the load deviation factor with respect to the optimum (*

_{L2,opt}*R*

*), which is [20]*

_{L2,opt}*b*

_{2}= 1 (

*R*

*=*

_{L2}*R*

*) in (9), the system is at the optimum state and the efficiency touches the upper bound. When*

_{L2,opt}*b*

_{2}> 1 or

*b*

_{2}< 1, the system is in the under-coupled or the over-coupled region, respectively, and the efficiencies are lower than the maximum. Explicitly, the efficiency degrades symmetrically with the deviation of

*b*

_{2}from unity in a log scale. For example, with

*b*

_{2}= 10 or 1/10, the efficiency degrades in a same rate from (9). Likewise, whether when

*b*

_{2}→ ∞ (open-circuited or under-coupled limit) or when

*b*

_{2}→ 0 (short-circuited or over-coupled limit),

*η*

_{2}→ 0. Lastly, when

*F*

_{21}→ ∞ or 0,

*η*

_{2}→ 1 or 0 irrespective of

*b*

_{2}as long as

*b*

_{2}has a finite value.

### III. Various Configurations in SIMO System

### 1. SIMO System with M = 1 and N = 2

*M*= 1 and

*N*= 2 shown in Fig. 2. One Tx loop and two Rx loops may be randomly positioned according to the required system configuration. If the coupling coefficients between the two receiving loops (

*k*

_{23}) are negligibly small compared with

*k*

_{21}and

*k*

_{31}, we may assume

*k*

_{23}= 0 and can express the efficiencies for Rx 2 (

*η*

_{2}) and for Rx 3 (

*η*

_{3}) as

*β*

_{2}and

*β*

_{3}, which are the normalized load resistances defined by

*R*

_{L}_{2}/

*R*

_{2}and

*R*

_{L}_{3}/

*R*

_{3}, respectively.

*F*

_{21}and

*F*

_{31}are the figures of merit given by

*or*

*η*

_{2}in (11), we see that as

*β*

_{3}→ ∞ (open-circuited) or

*F*

_{31}→ 0 (Rx 3 placed far away from Tx 1, for example), the role of Rx 3 vanishes and (11) can be shown to reduce to (9) [19, 20]. The total efficiency

*η*

*is the sum of (11) and (12), and is given by*

_{t}*β*

_{2,}

*with which*

_{opt}*η*

_{2}is maximized for a fixed

*β*

_{3}, we require

*β*

_{2,}

*depends on*

_{opt}*β*

_{3}. Especially when

*β*

_{3}→ ∞ (open-circuited) or

*F*

_{31}→ 0 (Rx 3 placed far way), the system practically behaves like a SISO and

*β*

_{2,}

*becomes*

_{opt}*η*

_{3}for a fixed

*β*

_{2}is

*η*

*(13) is maximized, the following two equations should be satisfied simultaneously:*

_{t}*β*

_{2,}

*is equal to*

_{opt}*β*

_{3,}

*. For some illustrations with respect to the SIMO system in Fig. 2, we assume that*

_{opt}*r*

_{1}= 15 cm,

*r*

_{2}=

*r*

_{3}= 5 cm (which is chosen to represent the size of typical mobile phones), the radius of each loop ring (

*r*

*) is 0.2 cm, and the resonant frequency (*

_{ring}*f*

_{0}) is 6.78 MHz.

*β*

_{2}and

*β*

_{3}using (11), (12), and (13). The center positions of one Tx and two Rx loops are (0, 0, 0), (0, −15, 10), and (0, 7.5, 25), respectively. The Q-factors of Tx 1, Rx 2, and Rx 3 are 730, 630, and 630, respectively. For this example,

*F*

_{21}= 11 and

*F*

_{31}= 5.8. Fig. 3(a) and (b) show that when

*β*

_{2}is about 4 and

*β*

_{3}→ ∞,

*η*

_{2}is maximized to 0.82. Meanwhile, when

*β*

_{2}→ ∞ and

*β*

_{3}is approximately 2,

*η*

_{3}is maximized to 0.52. The maximum total efficiency

*η*

*in Fig. 3(c) is about 0.85 when*

_{t}*β*

_{2}and

*β*

_{3}are about 12, as expected by (19). Note that while the efficiencies

*η*

_{2}and

*η*

_{3}significantly vary depending on

*β*

_{2}and

*β*

_{3}, the total efficiency

*η*

*is rarely sensitive as long as they are greater than approximately 4. This is an interesting property of SIMO systems since the efficiency for each receiver may be properly distributed considering the charged battery energy levels of the receivers while the total remains unchanged.*

_{t}*η*

_{2},

*η*

_{3}, and

*η*

*for different center positions (0, 10,*

_{t}*v*) of loop 3 (Rx 3) in Fig. 2. For this example, the centers of loop 1 (Tx 1) and loop 2 (Rx 2) are fixed at (0, 0, 0), (0, −10, 15), respectively. At each position of Rx 3,

*β*

_{2,}

*and*

_{opt}*β*

_{3,}

*given by (19) are shown in Fig. 4. Alternatively,*

_{opt}*β*

_{2}and

*β*

_{3}can be optimized to maximize the fitness function

*η*

_{t}(8) using the GA in MATLAB based on (1)–(7). The optimized

*β*

_{2}and

*β*

_{3}are in good agreement with

*β*

_{2,}

*and*

_{opt}*β*

_{3,}

*obtained by (19). This demonstrates the validity of (19). In the GA optimization of this study, all the coupling effects are considered.*

_{opt}### 2. SIMO System with one Tx and N Rx’s (Same Receiving Loops Are Symmetrically Positioned)

*N*Rx loops are identical and symmetrically located around one Tx loop. From the symmetry of the system, the figure of merit (

*F*=

*kQ*) between the Tx and Rx loops should also be identical. When the coupling coefficients between the Rx loops are negligibly small, the total efficiency

*η*

*and the efficiency for each loop*

_{t}*η*can be expressed as

*b*(=

*R*

*/*

_{L}*R*

_{L}_{,}

*) is the load deviation factor from the optimum load resistance, given as*

_{opt}*N*= 1, (20) and (21) reduces to (9) and (10), respectively.

*R*

*) for different number of receivers. Again, the solutions from GA optimization are consistent with the closed-form solution in (21), validating the formulations. As the separation between the Tx and Rx loops increases, the figure of merit (*

_{L,opt}*F*) and the optimum load

*R*

_{L}_{,}

*usually become very small, as exemplified in Table 1. Ideally, these small optimum resistances can be transformed to the 50 Ω impedance of typical receivers using a feeding loop. In practice, we may use a multi-turn Rx loop to increase the optimum resistance enough to make it directly match practical receivers [21].*

_{opt}*M*= 1,

*N*= 4) with four symmetrically placed receivers. The Tx loop is centered at the origin. The vertical and horizontal distances of each Rx loop from the origin are

*v*and

*h*, respectively. Fig. 5(b) shows the fabricated SIMO system based on Fig. 5(a). Additional feeding loops for the Rx loops were used for 50-Ω matching to measure the

*S-*parameters using a network analyzer. Table 2 shows the approximated [21, 22] and measured values of circuit elements. The measured values were obtained using the Keysight E5063A VNA. The measured resistances were about two times the theoretical ones. This has been a usual phenomenon due to oxidization of copper wires [23]. On the other hand, the inductances and capacitances are in close agreement. The measured

*Q*of the transmitting loop and receiving loops are approximately 330 and 234, respectively.

*N*) of Rx loops in the SIMO system. The resonant frequency (

*f*

_{0}) is 6.78 MHz,

*r*

_{1}= 15 cm,

*r*

_{2}=

*r*

_{3}=

*r*

_{4}=

*r*

_{5}= 5 cm, the radius of each loop (

*r*

*) is 0.2 cm,*

_{ring}*v*= 7.5 cm, and

*h*= 15 cm in connection with Fig. 5.

*N*) increases, the total efficiency (

*η*

*) increases to some extent. The efficiency (*

_{t}*η*

*) for each Rx loop is roughly the total efficiency (*

_{n}*η*

*) divided by the number of Rx loops*

_{t}*N*. The efficiencies obtained by the numerical calculations using the GA method, closed-form expressions, and measurements are all in good agreement. In Table 3, we summarize the coupling coefficients between the Tx and Rx loops and between two adjacent Rx loops for different numbers (

*N*) of receivers from 1 to 4 with the configurations described in Figs. 5 and 6. As we have first assumed, the coupling coefficients between the Tx and Rx loops are considerably larger than those between the two adjacent Rx loops. This explains why the efficiencies obtained from the closed-form expressions (20) and (21), neglecting the coupling coefficients between the two adjacent Rx loops, agree well with those obtained from the numerical optimization and measurements.

### 3. SIMO System with M = 1 and N = 4 (Same Receiving Loops, But Not Symmetrically Positioned)

*r*

_{1}= 15 cm,

*r*

_{2}=

*r*

_{3}=

*r*

_{4}=

*r*

_{5}= 5 cm, the radius of the loop ring is 0.2 cm,

*h*= 15 cm, and the Rx loops are placed right above the Tx loop with a vertical distance of

*v*.

*η*

*obtained using GA as the vertical distance*

_{t}*v*increases. The system has been optimized to maximize

*η*

*’s for each case. The total efficiency for case A is shown to decrease approximately from 88% to 50% as*

_{t}*v*increases from 2 cm to 24 cm. The total efficiency for case C is 3%–10% lower than that for case A due to the effect of the stronger mutual couplings between the Rx loops. In Fig. 8(b),

*η*

_{2}(=

*η*

_{5}) for case C is shown to be the highest, and

*η*

_{3}(=

*η*

_{4}) for case C is shown to be the lowest and to approach zero when

*v*is about 25 cm. The differences of

*η*

_{2}and

*η*

_{3}increase as

*v*increases for case B. This is due to the effects of the mutual coupling between the Rx loops becoming relatively stronger as

*v*increases.

*R*

*) are obtained from GA and plotted as a function of*

_{L,opt}*v*for the three cases. The optimum load resistances

*R*

_{L}_{2}

*(=*

_{,opt}*R*

_{L}_{5,}

*) for all of the three cases are shown to decrease as expected when*

_{opt}*v*increases. On the other hand,

*R*

_{L}_{3,}

*(=*

_{opt}*R*

_{L}_{4,}

*) for case C shows a tendency to increase due to the effect of the strongest mutual coupling between Rx loops.*

_{opt}*v*. However, as

*v*increases from 1 cm to 25 cm, the coupling coefficient between the Tx loop and any of the Rx loops (

*k*

_{21}=

*k*

_{31}=

*k*

_{41}=

*k*

_{51}) decreases from 0.0357 to 0.006. The absolute value of

*k*

_{32}for case C is the largest (= 0.0654), which is even larger than

*k*

_{n}_{1}(= 0.0357) for

*n*= 2–5. Therefore, the efficiency

*η*

_{3}(=

*η*

_{4}) for case C is the lowest.

*h*

*. For this case,*

_{5}*r*

_{1}= 15 cm,

*r*

_{2}=

*r*

_{3}=

*r*

_{4}=

*r*

_{5}= 5 cm, and the radius of the Rx loop ring is 0.2 cm. In Fig. 11(a) and (b), the efficiencies and the optimum load resistances obtained using GA are shown as a function of

*h*

_{5}. The optimum load resistances have been obtained using the GA to maximize the total efficiencies (

*η*

*) for each*

_{t}*h*

_{5}with all mutual couplings considered. In Fig. 11(a), we can see that

*η*

*’s do not change much as*

_{t}*h*

_{5}increases from 0 to 26 cm. It only decreases from 87% to 80%. However, the efficiency of each receiver does change significantly. Obviously,

*η*

_{5}decreases most dramatically from 64% to a few percent. Instead,

*η*

_{2}(=

*η*

_{4}) and

*η*

_{3}are shown to increase from about 10% to 26%. One important observation is that the total efficiency can be maintained rather steady while the efficiency for each receiver changes considerably.