Analysis of Efficiencies for MultipleInput MultipleOutput Wireless Power Transfer Systems
 Author: Kim Sejin, Lee Bomson
 Publish: Journal of electromagnetic engineering and science Volume 16, Issue2, p126~133, 30 Apr 2016

ABSTRACT
Wireless power transfer (WPT) efficiencies for multipleinput multipleoutput (MIMO) systems are formulated with a goal of achieving their maximums using Z matrices. The maximum efficiencies for any arbitrarily given configurations are obtained using optimum loads, which can be determined numerically through adequate optimization procedures in general. For some simpler special cases (singleinput singleoutput, singleinput multipleoutput, and multipleinput singleoutput) of the MIMO systems, the efficiencies and optimum loads to maximize them can be obtained using closedform expressions. These closedform solutions give us more physical insight into the given WPT problem. These efficiencies are evaluated theoretically based on the presented formulation and also verified with comparisons with circuit and EMsimulation results. They are shown to lead to a good agreement. This work may be useful for construction of the wireless Internet of Things, especially employed with energy autonomy.

KEYWORD
Coupling Coefficient , Optimum Load Resistance , Transfer Efficiency , Wireless Power Transfer , Z Matrix

I. INTRODUCTION
Recently, the wireless power transfer (WPT) technology has become more important, and products related to singleinput singleoutput (SISO) WPT systems have been globally commercialized with the example of wireless charging pads [1]. To expand the WPT market, regulatory issues regarding WPT commercialization and standardization are being finalized [2]. The WPT systems with multiple transmitters or multiple receivers have also been studied [3–5]. In [3], multipleinput singleoutput (MISO) WPT systems were investigated using closedform solutions for the maximum WPT efficiencies. An analysis of the WPT efficiencies considering the multipleoutput system was provided in [4]. Multipleinput multipleoutput (MIMO) WPT systems including a repeater with a misalignment angle were examined in [5].
Still, there are many problems left in WPT systems, such as low efficiency, realization, stability, adaptability to mobility, EMI/EMC/EMF, and so on. In this paper, the WPT problem for general MIMO systems is formulated to achieve maximum efficiencies based on optimum loads. The maximum efficiencies of the systems can be obtained systematically for any number of transmitters and receivers. With this formulation, the optimum loads for maximum efficiencies can always be found at least numerically through a proper systematic optimization process. For some special cases such as SISO, SIMO, and MISO, closedform expressions for the efficiencies and optimum loads for maximum efficiencies are derived and examined with various examples. The validity of circuit and EMsimulated efficiencies are compared with the theoretical efficiencies.
II. MIMO WIRELESS POWER TRANSFER SYSTEM
Fig. 1 shows a WPT system consisting of transmitters with a number of
M and receivers with a number ofN . They are assumed to be made of loops that have some inductancesL . Each loop is loaded with a capacitor for resonance at a specific angular frequencyω _{0}. Then, theZ matrix for the WPT system is given byIn (1), the elements in the column matrix [
V ] are the supplied voltages. The supplied voltages of the receiving loops are assumed to be zero. The diagonal and offdiagonal elements of the Z matrix as a function of angular frequencyω are expressed asand
respectively. In (2),
R_{m} (m = 1, 2, ⋯,M +N ) is the loss resistance of each loop, andR_{Ln} (n =M + 1,M + 2, ⋯,M +N ) is the load resistance of each receiver. In (3),k_{mn} is the coupling coefficient [6].At the resonant angular frequency (
ω _{0}), the diagonal and offdiagonal elements in [Z] are shown to become purely resistive and purely imaginary, respectively. The current is determined by [I ] = [Z ]^{1}[V ]. Then, the total input power and the received power atn ^{th} receiver are readily obtained byand
The WPT efficiencies for the
n ^{th} receiver and the whole receivers are simply given byand
respectively. Based on this formulation, the WPT efficiencies for any MIMO configurations with arbitrary loads
R_{Ln} (n =M + 1, ⋯,M +N ) can be evaluated. For design problems to maximize (6) or (7), the solutions forR_{Ln} (n =M + 1, ⋯,M +N ) can be obtained at least numerically using Genetic Algorithm (evolutionary algorithm), which is a proper optimization algorithm. For some special cases of MIMO WPT systems such as SISO, SIMO, and MISO, we also try to analyze them analytically if possible.When the number of transmitting and receiving loops is 1 (
M =N = 1, SISO), all solutions are already well known but are repeated here for readers’ convenience. The transmitted powerP_{in} , received powerP_{L} , and the WPT efficiency are given by [7]and
respectively, where the normalized load
β (=R_{L} /R _{2}) and the figure of meritF (=kQ ) are independent variables to determine the efficiencies.Q is the geometric mean ofQ _{1} andQ _{2} and each is given byω _{0}L /R . It is known that (10) can also be expressed aswhere
b (=R_{L} /R_{L,opt} =β /β_{opt} ) is the normalized load resistance referenced to the optimumR_{L,opt} with which the maximum efficiency is achieved.β_{opt} is the normalized optimum load resistance for maximum efficiency, given by [6,7]. Whenb = 1, the efficiency in (11) becomes maximum. Whenb is greater or less than 1, the twoloop WPT system becomes undercoupled or overcoupled, respectively, and the efficiencies decrease. It is interesting to observe that asF (=kQ ) becomes smaller,R_{L,opt} also becomes smaller. The condition ofR_{L,opt} =R _{2} whenF = 0 (two loops placed far apart) reminds us of the condition of the maximum received power of an Rx antenna in the far field. It is also notable that even though the role of Tx and Rx loops are interchanged, the efficiency remains the same with a similar use of the optimum load.III. ANALYSIS OF WIRELESS POWER TRANSFER FOR SPECIAL CASES
1. Singleinput MultipleOutput System (M = 1, N = 2)
Fig. 2 shows a WPT system consisting of one transmitter and two receivers. The center positions of each loop are assumed to be at (0, 0, 0), (0, 
h, v ), and (0,h, v ), respectively. We can analyze the system using a simple 3×3 Z matrix using (1)–(3).When the distance between the two receiving loops is large, the coupling coefficient
k _{32} is negligibly small and can be assumed to be zero. For this case, the WPT efficiency of each loop can be expressed asand
where
F_{n} _{1} is the figure of merit defined byQ_{n} is the quality factor given byω _{0}L_{n} /R_{n} , andβ_{n} is the normalized load resistance defined byR_{Ln} /R_{n} . The total WPT efficiencyη is the sum of (12) and (13) as defined in (7). The normalized optimum load resistancesβ _{2,}_{opt} andβ _{3,}_{opt} with which (12) and (13) are maximized can be shown to be given byand
respectively. When
β _{3}→∞ (open) in (14), loop 3 becomes inactive (undercoupled limit), (normalized optimum load for the SISO WPT system). Whenβ _{3}→0 (short, overcoupled limit), The solution ofR_{L} _{2} andR_{L} _{3} for the maximumη (=η _{2} +η _{3}) may also be numerically determined with GA.All the efficiencies and optimum loads evaluated and presented in this paper have been checked between the GA results based on (2)–(7) and circuit/EMsimulation results. The efficiencies and optimum loads have been found to be always in good agreement.
Fig. 3(a) and (b) show the maximum WPT efficiencies and the optimum load resistances for the case described in Fig. 2. The resonant frequency is 6.78 MHz,
r _{1} = 15 cm,r _{2} =r _{3} = 5 cm, andr_{ring} = 0.2 cm. The Qfactors for the loops withr _{1} = 15 cm andr _{2} =r _{3} = 5 cm are about 730 and 560, respectively.In Fig. 3(a), the efficiency
η (=η _{2} +η _{3}) are plotted as a function ofh forv = 3 cm, 5 cm, and 7.5 cm. Besides, for the purpose of comparison, we have also plottedη _{2} whenv = 3 cm without loop 3. It is shown that whenh is smaller than roughly the radius of the Tx loop 1 (15 cm),η _{2} without loop 3 is greater than 90%, and it is almost the same as the total efficiencyη (=η _{2} +η _{3}) with both loop 2 and loop 3 symmetrically placed.This means that when the Rx loops are properly placed in symmetry relative to the Tx loop, the nearly 100% efficiency can be shared by the receiving loops. When
h is roughly between 17 cm and 20 cm, the efficiencies drastically drop to zero. These zeroefficiency phenomena occur at positions where the upward magnetic flux lines crossing a receiving loop change their directions and fall down into the loop again such that the net magnetic flux crossing the loop is zero. This corresponds to the zerocoupling coefficients (k ).Beyond the zeroefficiency (or the zerocoupling coefficient) positions, the net magnetic flux lines on the receiving loops go downward (the negative
k changes only the direction of currents on the receiving loops), and the efficiencies are shown to increase to about 80% to 90% , but decrease again ash becomes large.In Fig. 3(b), we show
R_{L} _{2,}_{opt} (=R_{L} _{3,}_{opt} ) as a function ofv andh . When the distance between the Tx and Rx loops becomes large,R_{L,opt} may become too small to be realized. This can be handled with a multiturn Rx loop or an additional feeding loop or matching circuit [8–10] depending on the system requirements.Fig. 4 shows the coupling coefficients
k _{21},k _{31}, andk _{32} depending on the distancesv andh based on the same system configuration as described in Fig. 3. The coupling coefficientsk’ s were extracted based on [6]. These extracted coupling coefficients were checked to be the same as those obtained by [11]. As explained with Fig. 3(a),k _{21} andk _{31} (the coupling coefficients between the transmitting and receiving loops) are shown to be negative, zero, and positive depending on the loop 1 and loop 2 (or loop 3) positions. Due to the particular placements of loop 2 and loop 3 on a horizontal plane,k _{32} in Fig. 4(b) is always negative.Fig. 5 shows the coupling coefficients
k _{21} (=k _{31}),k _{32}, and k _{32}/k _{21} depending on the distanceh with the fixedv of 3 cm based on the system configuration as described in Fig. 4. By evaluating k _{32}/k _{21} depending on the horizontal separationh , we can determine the condition wherek _{32} may be ignored. We can see that since k _{32}/k _{21} is very small (less than 0.3) whenh > 10 cm andv = 3 cm, the solutions in (12)(15) are more accurate in this region.2. MultipleInput SingleOutput System (M = 2, N = 1)
Fig. 6 shows a MISO configuration consisting two transmitters and one receiver. The center positions of each loop are (0,
h , 0), (0,h , 0), and (0, 0,v ), respectively.Fig. 7 shows the maximum WPT efficiencies for the systemdescribed in Fig. 6. The resonant frequency is 6.78 MHz,
V _{1} =V _{2},r _{1} =r _{2} = 15 cm,r _{3} = 5 cm, andr_{ring} = 0.2 cm. The Qfactors for the loops withr _{1} =r _{2} = 15 cm andr _{3} = 5 cm are about 730 and 560, respectively.In Fig. 7, the maximum efficiency
η _{3} are plotted as a function ofh forv = 3 cm, 5 cm, and 7.5 cm. Besides, for the purpose of comparison, we have also plottedη _{3} without loop 2 whenv = 3 cm. It is shown that whenh is about 16 cm,η _{3} without loop 2 is approximately 90%, butη _{3} with loop 2 is somewhat less than 90%. Whenh is roughly between 17 cm and 20 cm, the efficiencies drastically drop to zero. Beyond the zeroefficiency (or the zerocoupling coefficient) positions, the efficiencies are shown to increase to about 55% to 85%, but decrease again ash becomes large.Fig. 8 shows the coupling coefficients
k _{21},k _{31}, andk _{32} depending on the distancesv andh under the same conditions as described in Fig. 7.Fig. 9 shows a MISO system (
M = 2,N = 1) with two vertically placed transmitters. The center positions of each loop are (0, 0,v ), (0, 0,v ), and (0, 0, 0), respectively.Fig. 10(a) and (b) show the WPT efficiency
η _{3} and coupling coefficients for the case described in Fig. 9. The resonant frequency is 6.78 MHz,V _{1} =V _{2},r _{1} =r _{2} = 15 cm,r _{3} = 5 cm, andr_{ring} = 0.2 cm. The Qfactors for the loops withr _{1} =r _{2} = 15 cm andr _{3} = 5 cm are about 730 and 560, respectively. In Fig. 10(a), the maximum efficienciesη _{3} with and without loop 2 are plotted as a function ofv . For the purpose of comparison, we have also plottedη _{3} without loop 2.When
v is smaller than 26 cm,η _{3} without loop 2 is somewhat greater thanη _{3} with loop 2. Whenv is larger than 26 cm, the opposite is true. Asv increases, the benefit of using two Tx loops is shown to be more significant.3. MultipleInput MultipleOutput System (M = 2, N = 2)
Fig. 11 shows a MIMO system (
M = 2,N = 2). Performances of this system can also be analyzed using the formulation (1)–(7).Fig. 12(a) and (b) show the WPT efficiencies and the optimum load resistances for the case described in Fig. 10. The resonant frequency is 6.78 MHz,
V _{1} =V _{2},r _{1} =r _{2} = 15 cm,r _{3} =r _{4} = 5 cm,r_{ring} = 0.2 cm, andh _{1} = 20 cm. The Qfactors for the loops withr _{1} =r _{2} = 15 cm andr _{3} =r _{4} = 5 cm are about 730 and 560, respectively.In Fig. 12(a),
η (=η _{3} +η _{4}) are plotted as a function ofv andh _{2}. It is shown that when 5 cm ≤v ≤ 15 cm and 10 cm ≤h _{2} ≤ 30 cm,η (=η _{3} +η _{4}) is well above 90%. In Fig. 12(b),R_{L,opt} is shown to be about 0.5 Ω whenv is 5 cm and 30 cm ≤h _{2} ≤ 35 cm.Fig. 13 shows the coupling coefficients
k _{21},k _{31},k _{32},k _{41},k _{42}, andk _{43} depending on the distancesv andh _{2} under the same conditions as described in Fig. 12. In Fig. 13(a),k _{21} (the coupling coefficients between the transmitting loops) is about 0.0014. In Fig. 13(c),k _{41} andk _{32} (the coupling coefficients between the transmitting and receiving loops that are located diagonally) are shown to be negative, zero, and positive, depending on the loop positions.Since the MIMO WPT systems are considered to be a natural extension of the SISO system, the overall WPT system efficiencies have been found to be well above 90% when the distances between the TX and Rx loops are roughly less than the radius of the Tx loops.
Fig. 14 shows a MIMO system (
M = 2,N = 2). Performances of this system can also be analyzed using (1)–(7).Fig. 15 shows the WPT efficiencies for the case described with all loops (MIMO) or without loop 2 and loop 4 (SISO) in Fig. 14. The resonant frequency is 6.78 MHz,
V _{1} =V _{2},r _{1} =r _{2} =r _{3} =r _{4} = 5 cm,r_{ring} = 0.2 cm. The Qfactors for the loops withr _{1} =r _{2} =r _{3} =r _{4} = 5 cm are about 560.In Fig. 15,
η (=η _{3} +η _{4}) (MIMO) andη _{3} (SISO) are plotted as a function ofv andh . It is shown that when 4 cm ≤v ≤ 8 cm,η (=η _{3} +η _{4}) (MIMO) andη _{3} (SISO) are well above 90%. It is shown thatη (=η _{3} +η _{4}) (MIMO) becomes smaller as the vertical separationv becomes larger.η (=η _{3} +η _{4}) (MIMO) is about 0.1–0.6 when 5 cm ≤h ≤ 10 cm and 14 cm ≤v ≤ 24 cm. It becomes smaller thanη _{3} (SISO) sinceη (=η _{3} +η _{4}) (MIMO) is affected by the coupling of the four loops. This implies that it is more effective to use a SISO system when 5 cm ≤h ≤ 10 cm than to use a MIMO system.Although the EM simulation results have been analyzed and discussed up to
M = 2 andN = 2 systems, MIMO WPT systems with more Tx and Rx loops are expected to have similar performance behaviors.In particular, for the WPT system with
M = 1 and arbitraryN , the efficiencies of the Rx loops have been found to be shared among them, and the total efficiency can be engineered up to 1 depending on the system configurations.IV. CONCLUSION
We formulated the MIMO WPT systems in terms of transfer efficiencies for each receiver and the whole receivers. The optimum loads of the receivers for the maximum WPT efficiencies have been shown to be found at least numerically based on the formulation. For some typical special cases of SISO, MISO, and SIMO system have been analyzed using circuit and EMsimulations together with the derived closedform solutions. In particular, the SIMO system has been shown to be effective in that the near 100% efficiency can be shared by the receiving loops. The derived closedform solutions have been demonstrated to give us plentiful physical insight for the systems. The results of this paper may be useful to construct WPT systems for Internet of Things requiring sensors with energy autonomy without batteries.

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[Fig. 1.] Typical configuration of a MIMO WPT system.

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[Fig. 2.] A SIMO system (M = 1, N = 2) with two horizontally placed receivers. Each loop is located at (0, 0, 0), (0, h, v), and (0, h, v).

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[Fig. 3.] WPT efficiencies and optimum load resistances at 6.78 MHz (r1 = 15 cm, r2 = r3 = 5 cm, rring = 0.2 cm, Q1 = 730, Q2 = Q3 = 560). (a) WPT efficiencies depending on h, (b) optimum load resistance RL2,opt (= RL3,opt) to maximize the total efficiency η (= η2 + η3), as a function of v and h.

[Fig. 4.] Coupling coefficients at 6.78 MHz as a function of h and v (r1 = 15 cm, r2 = r3 = 5 cm, rring = 0.2 cm). (a) k21, k31 and (b) k32.

[Fig. 5.] k21, k31 and k32 and k32/k21 at 6.78 MHz as a function of h (r1 = 15 cm, r2 = r3 = 5 cm, rring = 0.2 cm, v = 3 cm).

[Fig. 6.] A MISO system (M = 2, N = 1) with two horizontally placed transmitters. Each loop is located at (0, h, 0), (0, h, 0), and (0, 0, v).

[Fig. 7.] WPT efficiencies at 6.78 MHz with V1 = V2 (r1 = r2 = 15 cm, r3 = 5 cm, rring = 0.2 cm, Q1 = Q2 = 730, Q3 = 560).

[Fig. 8.] Coupling coefficients at 6.78 MHz as function of h and v (r1 = r2 = 15 cm, r3 = 5 cm, rring = 0.2 cm, Q1 = Q2 = 730, Q3 = 560). (a) k21 and (b) k31, k32.

[Fig. 9.] A MISO system (M = 2, N = 1) with two vertically placed transmitters. Each loop is located vertically at (0, 0, v), (0, 0, v) and (0, 0, 0).

[Fig. 10.] WPT efficiencies η3 and coupling coefficients k21, k31, and k32 at 6.78 MHz with V1 = V2 depending on v (r1 = r2 = 15 cm, r3 = 5 cm, rring = 0.2 cm, Q1 = Q2 = 730, Q3 = 560). (a) WPT efficiencies, (b) k21, k31 and k32.

[Fig. 11.] A MIMO system (M = 2, N = 2) with two horizontally placed transmitters and receivers. Each loop is located at (0, h1, 0), (0, h1, 0), (0, h2, v), and (0, h2, v).

[Fig. 12.] WPT efficiencies and optimum load resistances at 6.78 MHz with V1 = V2 depending on h2 and v (r1 = r2 = 15 cm, r3 = r4 = 5 cm, rring = 0.2 cm, h1 = 20 cm, Q1 = Q2 = 730, Q3 = Q4 = 560). (a) WPT efficiencies η (= η3 + η4), (b) optimum load resistance RL3,opt (= RL4,opt).

[Fig. 13.] Coupling coefficients at 6.78 MHz with V1 = V2 as function of v and h2 (r1 = r2 = 15 cm, r3 = r4 = 5 cm, rring = 0.2 cm, h1 = 20 cm, Q1 = Q2 = 730, Q3 = Q4 = 560). (a) k21, (b) k31, k32, (c) k41, k42, and (d) k43.

[Fig. 14.] A MIMO system (M = 2, N = 2) with two horizontally placed transmitters and receivers. Each loop is located at (0, h, 0), (0, h, 0), (0, h, v), and (0, h, v).

[Fig. 15.] WPT efficiencies and optimum load resistances at 6.78 MHz with V1 = V2 depending on h and v (r1 = r2 = r3 = r4 = 5 cm, rring = 0.2 cm, Q1 = Q2 = Q3 = Q4 = 560), WPT efficiencies.