To deal with the major challenges of embedded sensor networks, we consider the use of magnetic fields as a means of reliably transferring both information and power to embedded sensors. We focus on a power allocation strategy for an orthogonal frequencydivision multiplexing system to maximize the transferred power under the required information capacity and total available power constraints. First, we consider the case of a coreceiver, where information and power can be extracted from the same signal. In this case, we find an optimal power allocation (OPA) and provide the upper bound of achievable transferred power and capacity pairs. However, the exact calculation of the OPA is computationally complex. Thus, we propose a lowcomplexity power reallocation algorithm. For practical consideration, we consider the case of a separated receiver (where information and power are transferred separately through different resources) and propose two heuristic power allocation algorithms. Through simulations using the Agilent Advanced Design System and Ansoft High Frequency Structure Simulator, we validate the magneticinductive channel characteristic. In addition, we show the performances of the proposed algorithms by providing achievable
ƞ

C
regions.
Sensors can be embedded in a wide range of dense media, including water, soils, and masonry walls, which can be used in many applications, such as home networks and underground sensor networks. Sensors for use in home networks can also be embedded in the walls of buildings for the purposes of convenience and aesthetics [1]. Furthermore, sensors can be buried underground to monitor soil conditions or to provide information about earth movements [2]. However, the embedding of sensors presents two main challenges: effective communication between sensors and practical power supply. Therefore, a solution is needed to deal with these obstacles so that, in reallife practices, embedded sensors can operate more effectively and for longer periods of time.Magnetic induction (MI) communication is emerging as a promising technology that can allow embedded sensors to communicate with each other. Traditionally, wireless communications have relied on the use of electromagnetic (EM) radiation. However, EM waves are not appropriate for the transfer of information in dense media due to the three major problems of high path loss, dynamic channel variation, and large antenna size [3]–[4]. On the other hand, MI communication uses the MI of coil antennas to transfer information. The magnetic permeability of media such as water or soil is similar to that of air [2]. This means that magnetic fields experience a lower propagation loss than EM waves in dense media. Therefore, magnetic fields are more appropriate than EM waves as a mechanism of communication for embedded sensor networks. Recently, many research groups are actively investigating MI communication, including path loss, capacity, and MI waveguide techniques [2]–[10].Batteries are not a suitable power source for embedded sensors because their replacement is unlikely to be straightforward. One solution is the wireless transfer of power, an established technology that can be used to eliminate the need for a wired power connection. The efficient transfer of power is possible over a range of several meters using coupled magnetic resonance [11]–[18].Both MI communication and wireless power transfer can be achieved using magnetic fields. Therefore, it should be possible to transfer both information and power simultaneously through magnetic fields. To maximize the transfer of power, it is best to use one sinusoid at the resonant frequency that has the highest power transfer efficiency. However, a sinusoid with a bandwidth of zero has a communication capacity of zero. Therefore, there is a tradeoff between communication capacity and power transfer efficiency, with respect to bandwidth. However, the following works considered EM waves as the means for transferring information and power, not magnetic fields; as a result, the properties of magneticinductive channels were not reflected [19]–[21].In this paper, we consider the simultaneous optimization of information and power transfer so as to enable the practical use of embedded sensors. In practice, a low data rate is generally satisfactory for the transfer of information in embedded sensor networks [2]. On the other hand, for the successful operation of sensors, it is important to ensure a stable power supply. Therefore, our approach is focused on maximizing the transferred power, while ensuring the required information capacity. Our contributions can be described as follows. First, we present a magneticinductive channel model and a wireless information and power transfer system. We also verify the characteristics of the magneticinductive channel using the Agilent Advanced Design System (ADS) and the Ansoft High Frequency Structure Simulator (HFSS). Based on the optimization problem, we consider the case of a coreceiver (where information and power are transferred simultaneously through the same resource) to obtain the upper bound of achievable transferred power and capacity pairs. Here, we use an optimization technique to find the optimal allocated power, which is found by adjusting the water level of each subchannel. We also derive the conditions for the existence and boundedness of the optimal solution. The calculation of the optimal power allocation (OPA) is computationally complex. So, we propose a lowcomplexity power reallocation algorithm, which finds a nearOPA with a reduced computational complexity. However, it is difficult in practice to implement in the case of a coreceiver. Thus, we consider only the case of a separated receiver, where information and power are transferred separately through different resources. In this case, we propose two heuristic power allocation algorithms — one based on frequency division and the other on time division. We also describe the ƞC regions, which show the achievable power transfer efficiency and information capacity. Here, we compare the performance of the proposed algorithms with that of the optimal solution and that of a conventional equal power allocation. Carrying out simulations, we show that the proposed algorithms have a significant performance gain with respect to power transfer efficiency.
II. System Model and MagneticInductive Channel
A block diagram that illustrates our wireless information and power transfer system using magnetic resonance is shown in Fig. 1. Here, r_{t} and r_{r} are the radii of the transmitter coil (Tx coil) and receiver coil (Rx coil), respectively. The two coils lie along a single axis and are separated by a distance, d. Also, Fig. 1 shows its equivalent circuit model, which contains the effect of inductive coupling between the two coils. The Tx coil is connected to an alternating voltage source, V_{S}, with an angular frequency, ω. The Rx coil is connected to the load resistance, R_{L}. The selfinductances of the Tx and Rx coils are L_{t} and L_{r}, respectively; r_{lt} and r_{lr} are the internal resistances of the coils; and C_{t} and C_{r} are the capacitances that make the two coils resonate at the same resonant frequency. The angular resonant frequency ω_{o} can be defined as follows:$${\omega}_{\text{o}}=2\text{\pi}{f}_{\text{c}}=\frac{1}{\sqrt{{L}_{\text{t}}{C}_{\text{t}}}}=\frac{1}{\sqrt{{L}_{\text{r}}{C}_{\text{r}}}},$$where f_{c} is a resonant frequency. In the Tx coil, a sinusoidal current, i_{t}(ω), is generated by the voltage source V_{S}. Then, i_{t}(ω) induces another sinusoidal current, i_{r}(ω), in the Rx coil. This implies that the wireless link between the two coils is generated through inductive coupling. And, this mechanism makes it possible to transfer information and power wirelessly. The MI between the coils can be represented by the following coupling coefficient, k, such that k = M/(L_{t}L_{r})^{1/2}, where M is the mutual inductance. Also, k can be approximated as a function of the distance between the coils [8], as follows:$$k(d)=\frac{{r}_{\text{t}}^{2}{r}_{\text{r}}^{2}}{\sqrt{{r}_{\text{t}}{r}_{\text{r}}}{(\sqrt{{d}^{2}+{r}_{\text{t}}^{2}})}^{3}}.$$In addition, we consider a multicarrierbased wireless information and power transfer system, such as an orthogonal frequencydivision multiplexing (OFDM) system [22]. In this system, the resonant frequency is used as a central frequency for the transfer of information, and the total bandwidth of the frequency band is B. This frequency band is divided into N subchannels, each of which has the same bandwidth size (B/N).
Using the equivalent circuit model, we can derive the equivalent input impedance Z_{in}, which reflects the effect of coupling seen in the Tx coil.$${Z}_{\text{in}}={r}_{\text{lt}}+\text{j}\omega {L}_{\text{t}}+1/\text{j}\omega {C}_{\text{t}}+\frac{{k}^{2}{\omega}^{2}{L}_{\text{t}}{L}_{\text{r}}}{{r}_{\text{lr}}+\text{j}\omega {L}_{\text{r}}+1/\text{j}\omega {C}_{\text{r}}+{R}_{\text{L}}}.$$Here, we define the function f(ω) for the Tx coil as f(ω) = jωL_{t} + 1/jωC_{t}; then, the first derivation of f(ω) is obtained as f'(ω) = j(L_{t} + 1/ω^{2}C_{t}). Using the firstorder Taylor series expansion, f(ω) near ω_{o} can be approximated as follows:$$\begin{array}{l}f(\omega )=f({\omega}_{\text{o}})+{f}^{\prime}({\omega}_{\text{o}})(\omega {\omega}_{\text{o}})\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=0+\text{j}({L}_{\text{t}}+\frac{1}{{\omega}_{\text{o}}{}^{2}{C}_{\text{t}}})(\omega {\omega}_{\text{o}})\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\text{j}2(\frac{\omega {\omega}_{\text{o}}}{{\omega}_{\text{o}}})(\frac{{\omega}_{\text{o}}{L}_{\text{t}}}{{r}_{\text{lt}}}){r}_{\text{lt}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\text{j}2(\Delta \omega )({Q}_{\text{t}}){r}_{\text{lt}},\end{array}$$where the quality factor of Tx, Q_{t} = ωoL_{t}/r_{lt}, denotes the strength of the mutual coupling near the resonant frequency. Using a similar method to that used in obtaining (4), the quality factor of Rx can be found as Q_{r} = ω_{o}L_{r}/r_{lr}. Then, Z_{in} can be approximated as$$\begin{array}{l}{Z}_{\text{in}}={r}_{\text{lt}}(1+\text{j}2\Delta \omega {Q}_{\text{t}})+\frac{{k}^{2}{\omega}^{2}{L}_{\text{t}}{L}_{\text{r}}}{{r}_{\text{lr}}(1+\text{j}2\Delta \omega {Q}_{\text{r}})+{R}_{\text{L}}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}={Z}_{\text{in,\hspace{0.17em}1}}+{Z}_{\text{in,\hspace{0.17em}2}}.\end{array}$$From (5), the power transfer efficiency (PTE) at a frequency of ω can be expressed as$$\begin{array}{l}{\eta}_{\omega}=\frac{{Z}_{\text{in,\hspace{0.17em}2}}}{{Z}_{\text{in,\hspace{0.17em}1}}+{Z}_{\text{in,\hspace{0.17em}2}}}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\frac{{R}_{\text{L}}}{{r}_{\text{lr}}(1+\text{j}2\Delta \omega {Q}_{\text{r}})+{R}_{\text{L}}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{1}{\left[1+\frac{{Q}_{\text{L}}(1+\text{j}2\Delta \omega {Q}_{\text{r}})}{{Q}_{\text{r}}}\right]\left[1+\frac{1}{{\sigma}^{2}}{\left(1+\frac{{Q}_{\text{r}}}{{Q}_{\text{L}}(1+\text{j}2\Delta \omega {Q}_{\text{r}})}\right)}^{2}\right]}.\end{array}$$Here, Q_{L} = ω_{o}L_{r}/R_{L} and σ are the distancedependent figure of merit (FOM), which are defined as follows:$$\sigma =k\sqrt{\frac{{Q}_{\text{t}}{Q}_{\text{r}}}{(1+\text{j}2\Delta \omega {Q}_{\text{t}})(1+\text{j}2\Delta \omega {Q}_{\text{r}})}}.$$Substituting (2) and (7) into (6), we can see that ƞ_{ω} is proportional to 1/d^{6} as the path loss of the MI channel for communication [4]. In particular, the reactance terms are zero at the resonant frequency; therefore, the PTE at ω_{o} is reduced to$${\eta}_{{\omega}_{\text{o}}}=\frac{1}{1+\frac{{Q}_{\text{L}}}{{Q}_{\text{r}}}\left[1+\frac{1}{{\sigma}^{2}}{\left(1+\frac{{Q}_{\text{r}}}{{Q}_{\text{L}}}\right)}^{2}\right]}.$$Then the optimal load quality factor, Q_{L}^{*}, which is required to maximize ƞ_{ωo}, can be found from the condition ∂ƞ_{ωo} / ∂Q_{L} = 0.$${Q}_{\text{L}}{}^{*}=\frac{{Q}_{\text{r}}}{\sqrt{1+{Q}_{\text{t}}{Q}_{\text{r}}{k}^{2}}}.$$By substituting (9) into (8), we can obtain the maximum PTE.$$\begin{array}{l}{\eta}_{{\omega}_{\text{o}}}{}^{*}=\frac{1}{1+\frac{1}{\sqrt{1+{\sigma}^{2}}}\left[1+{\left(\frac{1}{\sigma}+\sqrt{1+\frac{1}{{\sigma}^{2}}}\right)}^{2}\right]}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{\sqrt{1+{Q}_{\text{t}}{Q}_{\text{r}}{k}^{2}}1}{\sqrt{1+{Q}_{\text{t}}{Q}_{\text{r}}{k}^{2}}+1}.\end{array}$$In (10), the PTE is proportional to σ, thus, to maximize the PTE, σ should be large. To increase σ, large Q and k are required. In particular, if the condition σ^{2} > 1 is satisfied, then this is called a strong coupling regime [11]–[12]. In addition, in a magneticinductive channel, the subchannel close to ω_{o} has a high PTE.The PTE ƞ_{ω} indicates how much power is transmitted to the load resistor of the Rx coil at a frequency of ω; therefore, the received power can be expressed as p_{ω}ƞ_{ω}, where p_{ω} is the transmitted power. Similarly, in the case of communication, information is also transferred to the load resistor of the Rx coil through the wireless channel. In this case, the received signal power can be expressed as p_{ω}h_{ω}^{2}, where h_{ω}^{2} is the channel gain of ω. In addition, the power transfer efficiency and the path loss of the MI channel for communication have a similar attenuation tendency, which is proportional to 1/d^{6}. The relationship between the channel gain and the PTE (for a given value of ω) may, therefore, be defined by ƞ_{ω} = h_{ω}^{2} in the wireless information and power transfer system [21]. Then the capacity of subchannel i can, therefore, be expressed as c_{i} = log_{2} (1 + p_{i}ƞ_{i}/N_{0}), where p_{i} is the power allocated to subchannel i and N_{0} is the noise power of each subchannel.
III. Coreceiver Case
First, we consider the case of a coreceiver that combines a receiver for information detection (RxID) and a receiver for power extraction (RxPE); that is, the RxID and RxPE use the same resource. This means that it is possible to observe information and extract power simultaneously from the same signal. In the coreceiver case, we can find an OPA and provide the upper bound of achievable transferred power and capacity pairs. In addition, we propose a suboptimal algorithm that has low complexity.
 1. OPA
Before proposing the OPA strategy, which considers the simultaneous transfer of information and power, we derive two power allocation strategies — one for maximizing transferred power and the other for maximizing information capacity.Proposition 1. Total power allocation (TPA) on a subchannel that has the highest PTE, p_{i}^{TPA}, maximizes the transferred power for a given total available power. Then p_{i}^{TPA} is given by$$\begin{array}{l}{p}_{{i}_{\text{max}}}^{\text{TPA}}={P}_{\text{s}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{i}_{\text{max}}=\underset{i}{\text{max}}\text{\hspace{0.17em}}{\eta}_{i},\\ {p}_{i}^{\text{TPA}}=0\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}otherwise},\end{array}$$where P_{S} is the total available power. The transferred power and capacity achieved by the TPA are defined as ƞ^{TPA} = P_{S}ƞ_{max} and C^{TPA} = log_{2} (1 + P_{S}ƞ_{max}/N_{0}), respectively.Proof. The proof of Proposition 1 is simple and trivial; therefore, it is omitted here [20]. ■Proposition 2. Waterfilling power allocation (WFPA), p_{i}^{WF}, maximizes the sum of the capacities of the subchannels for a given total available power. Then p_{i}^{WF} is given by$${p}_{i}^{\text{WF}}={\left(\frac{1}{{\lambda}_{\text{WF}}\mathrm{ln}2}\frac{{N}_{0}}{{\eta}_{i}}\right)}^{+},$$where we define (x)^{+} = max(0, x) and λ_{WF} satisfies
∑ i=1 N p i WF
= P_{S}. The transferred power and capacity achieved by the WFPA are defined as ƞ^{WF} =
∑ i=1 N p i WF
ƞ_{i} and
C WF = ∑ i=1 N log 2 (1+ p i WF η i / N 0 ) ,
respectively.Proof. The proof of Proposition 2 is well known; therefore, it is also omitted here [23]. ■Proposition 1 suggests that the total available power must be allocated to the subchannel that has the highest PTE, to maximize the transferred power. However, this method cannot guarantee the required information capacity. On the other hand, Proposition 2 demonstrates that the WFPA can maximize the information capacity, but it does not consider the wireless power transfer. The analysis of Propositions 1 and 2 leads us to formulate the following optimization problem, whose objective is to maximize the transferred power while preserving the required information capacity within the total available power constraint, as follows:$$\begin{array}{l}\underset{p}{\text{maximize}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{i=1}^{N}{p}_{i}{\eta}_{i}}\\ \text{subject\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}(s.t.)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{i=1}^{N}{c}_{i}\text{\hspace{0.17em}\hspace{0.17em}}\ge \text{\hspace{0.17em}\hspace{0.17em}}{C}_{\text{m}}},\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{i=1}^{N}{p}_{i}\text{\hspace{0.17em}\hspace{0.17em}}\le \text{\hspace{0.17em}\hspace{0.17em}}{P}_{\text{S}}},\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{p}_{i}\text{\hspace{0.17em}\hspace{0.17em}}\ge \text{\hspace{0.17em}\hspace{0.17em}}0\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\forall i,\end{array}$$where p is the set of power allocated to subchannels, such that p≜(p_{1}, ... , p_{N}), and C_{m} is the required information capacity.Theorem 1. If C_{m} > C^{TPA}, then the optimal solution of (13) is given by$${p}_{i}^{*}={\left(\frac{{\mu}^{*}}{({\lambda}^{*}{\eta}_{i})\mathrm{ln}2}\frac{{N}_{0}}{{\eta}_{i}}\right)}^{+}.$$In addition, λ^{*} and μ^{*} satisfy the following conditions:$$\sum _{i=1}^{N}{\mathrm{log}}_{2}(\begin{array}{l}\\ \\ \underset{}{}\end{array}\text{}1+\frac{{\left(\frac{{\mu}^{*}}{({\lambda}^{*}{\eta}_{i})\mathrm{ln}2}\frac{{N}_{0}}{{\eta}_{i}}\right)}^{+}\text{}\cdot {\eta}_{i}}{{N}_{0}}\text{}\begin{array}{l}\\ \\ \underset{}{}\end{array})}={C}_{\text{m}},$$$$\sum _{i=1}^{N}{\left(\frac{{\mu}^{*}}{({\lambda}^{*}{\eta}_{i})\mathrm{ln}2}\frac{{N}_{0}}{{\eta}_{i}}\right)}^{+}}={P}_{\text{S}};$$otherwise, p_{i}^{*} =p_{i}^{TPA}.Proof. First, we find the OPA when C_{m} > C^{TPA}. The problem in (13) is a convex optimization problem. Therefore, to find an optimal solution, we consider its Lagrangian function given by (17), where λ and μ are nonnegative Lagrange multipliers.$$\Lambda (\bm{p},\lambda ,\mu )\triangleq {\displaystyle \sum _{i=1}^{N}{p}_{i}{\eta}_{i}}\text{\hspace{0.05em}}+\text{\hspace{0.05em}}\lambda \left({P}_{\text{S}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\displaystyle \sum _{i=1}^{N}{p}_{i}}\right)+\mu \left({\displaystyle \sum _{i=1}^{N}{c}_{i}}\text{\hspace{0.05em}}{C}_{\text{m}}\right).$$From the Lagrangian function, we define the following Karush–Kuhn–Tucker (KKT) conditions:$$\frac{\mu}{\mathrm{ln}2}\times \frac{{\eta}_{i}}{{N}_{0}}\times \frac{1}{1+\frac{{p}_{i}{\eta}_{i}}{{N}_{0}}}+{\eta}_{i}\lambda =0,$$$$\mu \left({\displaystyle \sum _{i=1}^{N}{c}_{i}}{C}_{\text{m}}\right)=0,$$$$\lambda \left({P}_{\text{S}}{\displaystyle \sum _{i=1}^{N}{p}_{i}}\right)=0,$$$$\text{and}\lambda \ge 0,\text{\hspace{0.17em}\hspace{0.17em}}\mu \ge 0,\text{\hspace{0.17em}\hspace{0.17em}}\bm{p}\succcurlyeq 0.$$From the KKT condition (18), it is possible to find the optimal allocated power on each subchannel, which can then be represented as (14). Also, λ^{*} and μ^{*} can be obtained from complementary slackness conditions. For the maximum transferred power, both the required information capacity and the total available power constraints should be tight. This implies that the equalities
∑ i=1 N log 2 (1+ p i * η i / N 0 )− C m =0
and
P S − ∑ i=1 N p i * =0
should hold for positive numbers, λ and μ, in KKT conditions (19) and (20). Therefore, λ^{*} and μ^{*} satisfy (15) and (16).When C_{m} ≤ C^{TPA}, μ needs to be zero to satisfy (19) since
∑ i=1 N c i − C m
is always greater than zero. The problem of (13) is the same as that of Proposition 1; thus, p_{i}^{*} is the same as p_{i}^{TPA}. ■Proposition 3. The condition λ ≥ ƞ_{max} must hold so as to obtain a bounded optimal solution in (13).Proof. Proposition 3 can be proved by a contradiction. Let us consider the Lagrangian dual function of (13), which is defined as$$g(\lambda ,\mu )=\underset{\bm{p}\text{\hspace{0.17em}}\succcurlyeq 0}{\mathrm{max}}\text{\hspace{0.17em}}\Lambda (\bm{p},\lambda ,\mu ).$$Then, (22) can be simplified by discarding the constant terms related to λ and μ.$$\underset{\bm{p}\text{\hspace{0.17em}}\succcurlyeq 0}{\mathrm{max}}\text{\hspace{0.17em}}{\displaystyle \sum _{i=1}^{N}{p}_{i}({\eta}_{i}\lambda )}+\mu {\displaystyle \sum _{i=1}^{N}{c}_{i}}.$$Suppose that λ < ƞ_{max}. Then, the optimal value of (23) is unbounded when the p_{i} of the subchannel i with the maximum power transfer efficiency ƞ_{max} goes to infinity. Thus, the assumption λ < ƞ_{max} is a contradiction; hence, the proof is completed. ■We can also find the upper bound of μ for a given λ. From the relation P_{S} > p_{i}, μ_{max} can be determined by ln 2 × (λ–ƞ_{max}) × (P_{S}+ N_{0}/ƞ_{max}). Thus, μ lies in the interval (0, μ_{max}).Since (13) is a convex optimization problem, μ^{*} and λ^{*} can be found by updating μ and λ jointly through a gradient algorithm. Therefore, p_{i}* can also be found using an iterative method. The algorithm used to find p_{i}^{*} can be summarized in Algorithm 1. Here, α and β are step sizes that are sufficiently small to allow convergence, ε is given by a small positive number, and λ_{init} is the initial value of λ. If we denote I_{α} and I_{β} asAlgorithm 1. Optimal power allocation. 1: Initialization: k = 1, λ_{k} = λ_{init}2: If C_{m} ≤ C^{TPA}, set p_{i}^{*} = p_{i}^{TRA}. Else,3: Repeat4: Initialization: k' = 1, μ_{k'} = μ_{max}5: Repeat6: Find p_{i} from (14) for all i = 1, 2, …, N7: Update ${\mu}_{k\text{'}+1}={\mu}_{k\text{'}}\alpha \left({\displaystyle {\sum}_{i=1}^{N}{c}_{i}{C}_{\text{m}}}\right)$8: Until ${\mu}_{k\text{'}+1}{\mu}_{k\text{'}}\text{\hspace{0.17em}\hspace{0.17em}}\le \epsilon $9: Update ${\lambda}_{k+1}={\lambda}_{k}\beta \left({P}_{\text{S}}{\displaystyle {\sum}_{i=1}^{N}{p}_{i}}\right)$10: Until ${\lambda}_{k+1}{\lambda}_{k}\text{\hspace{0.17em}\hspace{0.17em}}\le \epsilon $the number of iterations for the convergence of μ and λ, respectively, then the computational complexity involved in finding p_{i}^{*} is O(I_{α } I_{β}).
 2. Power Reallocation Algorithm (PRA)
In OPA, the exact value of p_{i}^{*} can be obtained if α and β are sufficiently small to allow convergence, but this makes the values of I_{α} and I_{β} large. Larger values of I_{α} and I_{β} significantly increase the computational complexity involved in calculating the exact value of p_{i}^{*}. As a result, it can be difficult to find p_{i}^{*} in real time. Thus, based on the analytical results in Section III1, we propose a lowcomplexity PRA. This achieves nearoptimal performance, while substantially reducing the computational complexity so that power can be allocated in real time.In the OPA, μ* and λ* should be obtained jointly using a gradient algorithm to find p_{i}*. However, in the PRA, μ and λ are found separately. First, the values of μ and λ are initialized as μ_{max} and λ_{init}. Secondly, μ is found separately by the gradient algorithm for the given value of λ_{init}. In this step, the required information capacity is guaranteed because μ and λ_{init} satisfy condition (15), but P_{S} is not used fully. Therefore, in the third step, λ is also found by the gradient algorithm for the determined value of μ to satisfy condition (16). The water level (μ/(λ–ƞ_{i})ln2) in (14) increases as λ is updated by the gradient algorithm, so the remaining available power can be used fully. However, in this step, the achieved capacity is greater than the required information capacity C_{m}, so loss occurs in transferred power because supporting the exact value of C_{m} can increase transferred power. Thus, the fourth step is needed to satisfy both (15) and (16) by reassigning the allocated power; the process of which is described as follows:
1. A transmitter finds the subchanneliminthat achieves the minimum capacity among those subchannels whose capacity is greater than zero, such thatcmin =mini ciforci＞ 0.
2. The transmitter eliminates the allocated powerpiminin subchanneliminand reallocatespiminto the subchannelimaxthat has the highest PTE,ƞmax.
In general, the subchannel i_{max} uses a resonant frequency as its central frequency. If the sum capacity of the subchannels after the reallocation of power is still greater than C_{m}, then the aforementioned fourth step is repeated. Otherwise, p_{imin} is reduced by half to find the value that exactly guarantees C_{m}. Thus, p_{imin} can be determined using a bisection algorithm. The fourth step is terminated when C_{m} is guaranteed accurately or the total available power is allocated to subchannel i_{max}; that is when
 ∑ i=1 N c i − C m ≤ε
or p_{imax} = P_{S}, respectively. The case of p_{imin} = P_{S} occurs when C_{m} is sufficiently small, so C_{m} can be guaranteed even though the total available power is allocated to subchannel i_{max}. The process of the PRA is summarized in Algorithm 2.Algorithm 2. Power reallocation algorithm. 1: Initialization: k = 1, λ_{k} = λ_{init}, k' = 1, μ_{k'} = μ_{max}2: Repeat3: Find p_{i} from (14) for all i = 1, 2, …, N4: Update ${\mu}_{k\text{'}+1}={\mu}_{k\text{'}}\alpha \left({\displaystyle {\sum}_{i=1}^{N}{c}_{i}{C}_{\text{m}}}\right)$5: Until ${\mu}_{k\text{'}+1}{\mu}_{k\text{'}}\text{\hspace{0.17em}\hspace{0.17em}}\le \epsilon $6: Repeat7: Find p_{i} from (14) for all i = 1, 2, …, N8: Update ${\lambda}_{k+1}={\lambda}_{k}\beta \left({P}_{\text{S}}{\displaystyle {\sum}_{i=1}^{N}{p}_{i}}\right)$9: Until ${\lambda}_{k+1}{\lambda}_{k}\text{\hspace{0.17em}\hspace{0.17em}}\le \epsilon $10: Repeat11: Find the subchannel i_{min} and set P_{temp} = P_{imin}12: Repeat13: Set P_{imin} = P_{imin}− P_{temp} and P_{imax} = P_{imax}+P_{temp}14: Set P_{temp} = P_{temp} / 215: Until $\sum}_{i=1}^{N}{c}_{i}}\text{\hspace{0.17em}}\ge {C}_{\text{m$16: Until $\left{\displaystyle {\sum}_{i=1}^{N}{c}_{i}}\text{\hspace{0.17em}}{C}_{\text{m}}\right\le \epsilon $ or p_{imax} = P_{S}In short, the PRA finds the minimum size of B that guarantees C_{m} by eliminating P_{imin} in subchannel i_{min}. At the same time, the transferred power is efficiently increased while ensuring a minimum loss of capacity by reallocating P_{imin} to subchannel i_{max}. In addition, the computational complexity of the PRA is O(I_{α}+I_{β}+Nlog_{2}N), where Nlog_{2}N is very small compared with I_{α} or I_{β}. This indicates that the PRA achieves a significant reduction in complexity compared with O(I_{α}I_{β}) when calculating p_{i}^{*} exactly and that it allows operation in real time.We now compare four methods, TPA, WFPA, OPA, and PPA, to explain the benefits of the proposed solutions. Figure 2 shows the allocated powers obtained using the four methods. When the TPA is used to allocate power, the total available power is allocated to only the subchannel with the highest PTE. This approach leads to a maximum transferred power using a minimum bandwidth. On the other hand, the WFPA uses a broader bandwidth than the TPA to achieve the maximum capacity. When the WFPA is used to obtain p_{i}^{WF}, the water level (1/λ_{WF}ln2) is the same for all subchannels and more power is allocated to subchannels that have a higher PTE. Conversely, the OPA can be executed by controlling the water level of each subchannel. When the OPA is used to obtain p_{i}^{*}, the water level (μ^{*}/(λ^{*}–ƞ_{i})ln2) varies for each subchannel. The subchannel with a higher PTE, therefore, has a higher water level. Thus, compared with p_{i}^{WF}, more power is allocated to the subchannel with higher PTE, while less power is allocated to the subchannel with lower PTE. This means that the OPA uses the minimum bandwidth needed to ensure the required information capacity by reducing the allocated power in the subchannel with lower PTE. At the same time, the OPA maximizes the transferred power by allocating more power to the subchannel with higher PTE instead of the subchannel with lower PTE. Also, we show that the allocated power of the PRA is almost the same as that of the OPA. This indicates that the PRA achieves a nearOPA with a significant reduction in complexity, compared with the OPA. The PTEcapacity (ƞC) regions that describe all achievable PTE and capacity pairs can be expressed as follows:$${R}_{\eta C}(\bm{p})\triangleq \left\{(\eta ,C):\eta \le {\displaystyle \sum _{i=1}^{N}{p}_{i}{\eta}_{i}},\text{\hspace{0.17em}\hspace{0.17em}}C\le {\displaystyle \sum _{i=1}^{N}{c}_{i}},\text{\hspace{0.17em}}{\displaystyle \sum _{i=1}^{N}{p}_{i}\le {P}_{\text{S}}},\bm{p}\text{\hspace{0.17em}}\succcurlyeq 0\right\}.$$
It is difficult to implement a coreceiver practically; thus, we consider a separatedreceiver case that separates an RxID and an RxPE for practical consideration. In the separatedreceiver case, the RxID and RxPE observe information and extract power separately from different resources. Here, we propose two heuristic power allocation algorithms based on frequency division and time division.
 1. Power Allocation Based on Frequency Division (PAFD)
Information and power can be transferred through different frequencies. In general, it is best to use only the subchannel that has the largest PTE for transferring power. Thus, we use the subchannel i_{max} for transferring power and use other subchannels for transferring information. When a PAFD is used in a separatedreceiver case, we can formulate the following optimization problem:$$\begin{array}{l}\underset{p}{\text{maximize}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{p}_{{i}_{\mathrm{max}}}{\eta}_{{i}_{\mathrm{max}}}\\ \text{subject\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}(s.t.)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{i=1,i\ne {i}_{\mathrm{max}}}^{N}{c}_{i}\text{\hspace{0.17em}\hspace{0.17em}}\ge \text{\hspace{0.17em}\hspace{0.17em}}{C}_{\text{m}}},\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{i=1}^{N}{p}_{i}\text{\hspace{0.17em}\hspace{0.17em}}\le \text{\hspace{0.17em}\hspace{0.17em}}{P}_{\text{S}}},\\ \text{and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{p}_{i}\text{\hspace{0.17em}\hspace{0.17em}}\ge \text{\hspace{0.17em}\hspace{0.17em}}0\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\forall i.\end{array}$$Theorem 2. If we assume that C_{m} is sufficiently small, such that C_{m} << C^{WF}, then the solution of (25) is given by$${p}_{i}^{\text{FD}}={\left(\frac{\mu}{\lambda \mathrm{ln}2}\frac{{N}_{0}}{{\eta}_{i}}\right)}^{+},$$$${p}_{{i}_{\mathrm{max}}}^{\text{FD}}={P}_{\text{S}}{\displaystyle \sum _{i=1,i\ne {i}_{\mathrm{max}}}^{N}{p}_{i}^{\text{FD}}}.$$Proof. Since the problem (25) is a convex optimization problem, we consider its Lagrangian function, given by (28), where λ and μ are nonnegative Lagrange multipliers.$$\Lambda (\bm{p},\lambda ,\mu )\triangleq {p}_{{i}_{\mathrm{max}}}{\eta}_{{i}_{\mathrm{max}}}+\lambda \left({P}_{\text{S}}{\displaystyle \sum _{i=1}^{N}{p}_{i}}\right)+\mu \left({\displaystyle \sum _{i=1,i\ne {i}_{\mathrm{max}}}^{N}{c}_{i}}{C}_{\text{m}}\right).$$By differentiating Λ(p, λ, μ) with respect to p_{i}, p_{i} can be obtained as (26). To find p_{imax}, we can rewrite (28) as
Λ(𝒑, λ) ≜ p i max ( η i max −λ) + λ( P S − ∑ i = 1, i ≠ i max N p i )
by eliminating constant terms. In addition, the Lagrangian dual function of (25) can be given by
g(λ)= max p ≽0 Λ(𝒑,λ)
. Here, the condition (ƞ_{imax}−λ) ≤ 0 should be met so that g(λ) has a bounded value. Otherwise, g(λ) is unbounded as p_{imax} tends to infinity. In addition, the dual problem of (25) is defined as
min λ ≽0 g(λ)
. It has a zero duality gap, so the following equality should be satisfied for the optimal primal and dual solutions:
p i max * ( η i max − λ * )=0.
Then, the dual problem can be rewritten as follows:$$\begin{array}{l}\underset{\lambda}{\text{minimize}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\lambda \left({P}_{\text{S}}{\displaystyle \sum _{i=1,i\ne {i}_{\mathrm{max}}}^{N}{p}_{i}}\right)\\ \text{subject\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}(s.t.)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\eta}_{{i}_{\mathrm{max}}}\lambda \le 0,\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\lambda \text{\hspace{0.17em}\hspace{0.17em}}\ge \text{\hspace{0.17em}\hspace{0.17em}}0.\end{array}$$Thus, from (29), the solution of λ is ƞ_{i}, and p_{imax} can be obtained as (27). ■The PAFD implies that the power is allocated to subchannels for transferring information at first to guarantee C_{m}, and then the remaining power is allocated to the subchannel for transferring power. When C_{m} = 0, the PAFD is the same as TPA; thereby, maximizing the transferred power. However, the PAFD cannot achieve the maximum information capacity C^{WF} even though total power is used for transferring information, because the subchannel i_{max} cannot be used for transferring information. Then, the ƞC regions of the PAFD can be expressed as follows:$${R}_{\eta \text{C}}^{\text{FD}}(\bm{p})\triangleq \left\{\begin{array}{l}(\eta ,C):\eta \le {p}_{{i}_{\mathrm{max}}}^{\text{FD}}{\eta}_{{i}_{\mathrm{max}}},\text{\hspace{0.17em}\hspace{0.17em}}\\ C\le {\displaystyle \sum _{i=1,i\ne {i}_{\mathrm{max}}}^{N}{\mathrm{log}}_{2}\left(1+\frac{{p}_{i}^{\text{FD}}{\eta}_{i}}{{N}_{0}}\right)},\text{\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{i=1}^{N}{p}_{i}\le {P}_{\text{S}}},\bm{p}\text{\hspace{0.17em}}\succcurlyeq 0\end{array}\right\}.$$
 2. Power Allocation Based on Time Division (PATD)
Information and power can be transferred at different moments in a time slot. It is best to use WFPA for transferring information, whereas it is best to use TPA for transferring power. Therefore, a PATD can be defined simply by using WFPA and TPA. Let ρ, 0 ≤ ρ ≤ 1, denote the ratio of transmission time allocated for transferring information. Then the WFPA is used firstly for transferring information until ρC^{WF} = C_{m}. After that, the TPA is used for transferring power during the remaining time (1 – ρ). Then the transferred power becomes (1 – ρ)ƞ^{TPA}. The PATD, therefore, ensures that for any given portion of a time slot, the initial part is used for the transfer of information until C_{m} is guaranteed and the remaining part is used for the transfer of power. When C_{m} = 0 and ρ = 0, the PATD can achieve the maximum transferred power ƞ^{TPA}. Also, it can achieve the maximum information capacity C^{WF} when ρ = 1. This means that the PATD is more efficient than the PAFD as C_{m} approaches C^{WF}. Then the ƞC regions of the PATD can be expressed as follows:$${R}_{\eta C}^{\text{TD}}(\bm{p})\triangleq \left\{\begin{array}{l}(\eta ,C):\eta \le (1\rho ){P}_{\text{S}}{\eta}_{{i}_{\mathrm{max}}},\text{\hspace{0.17em}\hspace{0.17em}}\\ C\le \rho {\displaystyle \sum _{i=1}^{N}{\mathrm{log}}_{2}\left(1+\frac{{p}_{i}^{\text{WF}}{\eta}_{i}}{{N}_{0}}\right)},\text{\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{i=1}^{N}{p}_{i}\le {P}_{\text{S}}},\bm{p}\text{\hspace{0.17em}}\succcurlyeq 0\end{array}\right\}.$$
V. Simulation Results and Discussion
In our simulations, we consider identical Tx and Rx coils with a radius of 0.3 m and an internal resistance of 0.5 Ω. The selfinductance of the coils is chosen depending on the value of Q, such as L = Q_{r}/ω_{o}. In addition, the selfcapacitance of the coils is chosen depending on the value of selfinductance, such as C = 1/ω_{o}^{2}L, to make the two coils resonate at the same frequency of 10 MHz. Also, we use the optimal load quality factor Q_{L}^{*} at all distances. We set P_{S} to 1 W, so the transferred power can simply be considered to be the PTE. The bandwidth is 250 kHz, and there are nine subchannels. We assume that the total noise power is equal to one; thus, the noise power of each subchannel is N_{0} = 1/9. For comparative purposes, we use the EPA scheme, which allocates power equally to all subchannels, as a conventional scheme.Figure 3 shows the PTE ƞ versus frequency. To verify the magneticinductive channel, we perform both circuit and EM simulations using the Agilent ADS and the Ansoft HFSS, respectively. As Q increases, the strong resonance between the Tx and Rx coils occurs. As a result, subchannels near ω_{o} have higher PTE and the variation of PTE among subchannels becomes large. On the other hand, as k increases, the coupling strength between the Tx and Rx coils becomes large. Consequently, the PTE of all subchannels increases generally, which relieves the variation of PTE among subchannels. This means that the magneticinductive channel has different characteristics depending on the values of Q and k. There are some differences between the analytical and simulation results, as the subchannels are apart from ω_{o} because we approximated jωL_{t} + 1/jωC_{t} ≈ j2(Δω)(Q_{t})r_{lt} and jωL_{r} + 1/jωC_{r} ≈ j2(Δω)(Q_{r})r_{lr} in (4) using the firstorder Taylor series expansion. However, the analytical results are relatively wellmatched to both simulation results.
Figure 4 shows the PTE ƞ versus the load quality factor Q_{L} for different k. At a given k, there is an optimal load quality factor where the maximum ƞ is achieved. For example, the maximum ƞ at Q_{L} = 100 is 0.90 when k = 0.01, which is indicated as ƞ_{Peak}(Q_{L}^{*} = 100) : 0.90 in Fig. 4. Since the optimal load quality factor is inversely proportional to k in (9), the value of Q_{L}^{*} is small for large k.
Figure 5 shows the capability in power and information transfer. In ƞC regions, the boundary point (C^{TPA}, ƞ^{TPA}) can be obtained using TPA. Another boundary point, (C^{WF}, ƞ^{WF}), can be obtained using WFPA. Here, ƞ^{TPA} is the maximum achievable PTE, which can be obtained when C ≤ C^{TPA}. Similarly, C^{WF} is the maximum achievable capacity, which can be obtained when ƞ ≤ ƞ^{WF}. The ƞ achieved by OPA lies on the boundary line between two boundary points; for example, ƞ^{WF} ≤ ƞ ≤ ƞ^{TPA} and C^{TPA} ≤ C ≤ C^{WF}; however, its exact position depends on the required information capacity constraint. In the ƞC regions, there are four circular points that indicate the ƞ obtained using PRA. These points are close to the boundary of the ƞC regions, which implies that the PRA achieves nearoptimal ƞ while reducing the computational complexity considerably. In addition, when C_{m} is small, the PRA uses a narrow bandwidth. So, power is allocated intensively on the subchannels with higher PTE and that are close to ω_{o}. On the other hand, the PRA uses a broad bandwidth at larger values of C_{m}. So, power is allocated to even the subchannels with lower PTE, which are far from ω_{o}, to guarantee C_{m}. As a result, the ƞ of the PRA decreases as C_{m} increases. PATD can use all subchannels for transferring both information and power, and can control ρ adaptively depending on C_{m}. On the other hand, PAFD cannot use the subchannel i_{max} for transferring information although large C_{m} is required. This means that the PAFD is inefficient when C_{m} is large; as a result, the ƞ of the PAFD is lower than that of the PATD as C_{m} increases. EPA cannot allocate power to subchannels while considering C_{m}, so the ƞ of the EPA is constant regardless of the achieved capacity. On the other hand, the PAFD and PATD can use power efficiently for transferring power rather than information when C_{m} is small. As a result, the PAFD and PATD achieve higher ƞ than the EPA when the achieved capacity is small.
Figure 6 shows the PTE ƞ versus the quality factor Q when k = 0.01 and C_{m} = 80 kbps. The PTE of subchannels becomes higher as Q increases, so the performance of all schemes increases. The PTE among subchannels is relatively constant at small Q, so the proposed algorithms allocate power to all subchannels evenly, which is similar to EPA. However, the PTE of subchannels varies significantly as Q increases, so power is allocated intensively to the subchannels near ω_{o}. Consequently, the PAFD, PATD, and OPA all achieve higher ƞ than the EPA for large Q. In addition, the proposed algorithms can perform power allocation adaptively when the PTE of subchannels is changed with the variation of Q. Therefore, the ƞ of the proposed algorithms is improved more rapidly than that of the EPA schemes as Q increases.Figure 7 shows the PTE ƞ versus the coupling coefficient k when Q = 2,000 and C_{m} = 80 kbps. The PTE of all subchannels becomes better as k increases, which causes a little variation of PTE among subchannels. This indicates that the performance gain that can be achieved by the proposed algorithms becomes small; consequently, the ƞ of EPA becomes higher than that of PAFD and PATD for large k. The EPAFD cannot adapt to the variation of channel quality or C_{m}, while EPATD can adjust the ratio of transmission time, ρ, adaptively depending on the situation. Consequently, there is little variation in ƞ in the EPAFD; however, ƞ increases as k increases in the EPATD. Also, we can show that the proposed algorithms can achieve higher ƞ than the EPA schemes in both the coreceiver and the separatedreceiver cases, in Figs. 6 and 7.
In this paper, we considered the maximization of transferred power while ensuring the required information capacity within the total available power constraint in a wireless information and power transfer system. We constructed an equivalent circuit model for our magnetic resonance–based system and provided a magneticinductive channel model. Based on a formulated convex optimization problem, we derived an optimal power allocation (OPA) strategy in the case of the coreceiver. In addition, we found the conditions for the existence and boundedness of the OPA. To reduce the computational complexity of the OPA, we also proposed a lowcomplexity power reallocation algorithm. For practical consideration, we also proposed two heuristic algorithms in the case of the separated receiver. In our simulation results, we verified the characteristic of the magneticinductive channel. Also, we provided the ƞC regions to compare the performance of the proposed algorithms with that of the OPA. In short, we provided not only a theoretical performance bound of achievable transferred power and capacity pairs but also the algorithms that can be implemented in real time. For further works, it is necessary to address some practical issues, such as the effect of noise, the experimental verification of the MI channel, and electromagnetic compatibility with other existing systems.
This research was funded by the MSIP (Ministry of Science, ICT & Future Planning), Rep. of Korea in the ICT R&D Program 2014
BIO
Corresponding Authorkslee851105@gmail.comKisong Lee received his BS degree in electrical engineering from the Information and Communications University, Daejeon, Rep. of Korea, in 2007 and his MS and PhD degrees in electrical engineering from the Korea Advanced Institute of Science and Technology, Daejeon, Rep. of Korea, in 2009 and 2013, respectively. He is currently with the Electronics and Telecommunications Research Institute as a researcher. His research interests include femtocell networks, selforganizing networks, radio resource management, magnetic induction communication, energy harvesting, and wireless power transfer.
dhcho@ee.kaist.ac.krDongHo Cho received his PhD degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Rep. of Korea, in 1985. From 1987 to 1997, he was a professor at the Department. of Computer Engineering, Kyunghee University, Seoul, Rep. of Korea. Since 1998, he has been a professor at the Department of Electrical Engineering, KAIST, and he was a director of the KAIST Institute for Information Technology Convergence from 2007 to 2011. He has been a director of the KAIST Online Electric Vehicle Project since 2009, and he has been serving as a head of The Cho Chun Shik Graduate School for Green Transportation since 2010. He was also an IT Convergence Campus vice president of KAIST from 2011 to 2013. His research interests include mobile communication, Online Electric Vehicle systems based on wireless power transfer, and bio informatics.
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@article{ HJTODO_2014_v36n5_808}
,title={Simultaneous Information and Power Transfer Using Magnetic Resonance}
,volume={5}
, url={http://dx.doi.org/10.4218/etrij.14.0114.0161}, DOI={10.4218/etrij.14.0114.0161}
, number= {5}
, journal={ETRI Journal}
, publisher={Electronics and Telecommunications Research Institute}
, author={Lee, Kisong
and
Cho, DongHo}
, year={2014}
, month={Aug}
TY  JOUR
T2  ETRI Journal
AU  Lee, Kisong
AU  Cho, DongHo
SN  12256463
TI  Simultaneous Information and Power Transfer Using Magnetic Resonance
VL  36
PB  Electronics and Telecommunications Research Institute
DO  10.4218/etrij.14.0114.0161
PY  2014
UR  http://dx.doi.org/10.4218/etrij.14.0114.0161
ER 
Lee, K.
,
&
Cho, D. H.
( 2014).
Simultaneous Information and Power Transfer Using Magnetic Resonance.
ETRI Journal,
36
(5)
Electronics and Telecommunications Research Institute.
doi:10.4218/etrij.14.0114.0161
Lee, K
,
&
Cho, DH
2014,
Simultaneous Information and Power Transfer Using Magnetic Resonance,
ETRI Journal,
vol. 5,
no. 5,
Retrieved from http://dx.doi.org/10.4218/etrij.14.0114.0161
[1]
K Lee
,
and
DH Cho
,
“Simultaneous Information and Power Transfer Using Magnetic Resonance”,
ETRI Journal,
vol. 5,
no. 5,
Aug
2014.
Lee, Kisong
and
,
Cho, DongHo
and
,
“Simultaneous Information and Power Transfer Using Magnetic Resonance”
ETRI Journal,
5.
5
2014:
Lee, K
,
Cho, DH
Simultaneous Information and Power Transfer Using Magnetic Resonance.
ETRI Journal
[Internet].
2014.
Aug ;
5
(5)
Available from http://dx.doi.org/10.4218/etrij.14.0114.0161
Lee, Kisong
,
and
Cho, DongHo
,
“Simultaneous Information and Power Transfer Using Magnetic Resonance.”
ETRI Journal
5
no.5
()
Aug,
2014):
http://dx.doi.org/10.4218/etrij.14.0114.0161