### I. Introduction

### II. Design Theory

*d*. The loops, having inductances, are loaded with capacitors for LC resonance. The power is transferred from Tx to Rx via magnetically coupled resonant loops.

*R*

*and*

_{i}*L*

*are the resistance and inductance of the Tx (*

_{i}*i*= 0) and Rx (

*i*= 1) loops, respectively.

*C*

*is the capacitance of the lumped capacitor for Tx (*

_{i}*i*= 0) and Rx (

*i*= 1), respectively.

*r*

*is the radius of the*

_{i}*i*

^{th}loop and

*r*

*is the inner radius of the*

_{in,i}*i*

^{th}loop wire.

*V*

_{0}is the voltage at the input. The single-pole single-through (SPST) switch is operated between off (for an optimum load) and on (for a shorted load). The usual function of the Rx matching circuit is to transform the usually small Rx optimum load to the large device load (say, 50 Ω). Conventional WPT has been made with the switch off to achieve maximum efficiency. The Rx status information (such as a device identity, energy charged levels, etc.) can be transferred from Rx to Tx by switching between the on and off states of the SPST. Based on this equivalent circuit, the system is usually formulated by a Z-matrix given by

*k*is the coupling coefficient between Tx and Rx loops [17]. It is well known that

*k*approaches 1 as the distance between Tx and Rx becomes 0 and it approaches 0 as the distance between Tx and Rx becomes very large.

*V*

_{0}is the source voltage,

*R*

*is the Rx load resistance, and*

_{L}*ω*is the angular frequency.

*I*

_{0}and

*I*

_{1}are the currents on loop 1 and 2, respectively. The ratio of currents on Tx and Rx loops is given by [18]:

*R*

*increases, (2) becomes small and approaches 0 as*

_{L}*R*

*goes to infinity (open). The phase of the current flowing on Rx (*

_{L}*I*

_{1}) is 90° ahead of that flowing on Tx (

*I*

_{0}) following Faraday’s law. The input impedance is defined as the ratio of

*V*

_{0}and

*I*

_{0}on a Tx terminal and expressed as

*F*) as

*R*

*can be re-written as*

_{L}*b*is the deviation factor against the Rx optimum load

*R*

*defined by*

_{L,opt}*F*) and

*b*, and a large

*F*is merely a necessary condition for a high efficiency. The maximum efficiency is guaranteed with the additional condition of

*R*

*is just like monitoring the current on Tx (*

_{L}*I*

_{0}) depending on

*R*

*since*

_{L}*I*

_{0}=

*V*

_{0}/

*Z*

*. When*

_{in}*R*

*= 0 (short), the system is at the over-coupled limit, no power is transferred to Rx, and the efficiency (7) is 0. However, the largest current is induced on Rx and the magnetic flux back to Tx affects the Tx input impedance most significantly, and the input impedance becomes the largest:*

_{L}*R*

_{0}(1 +

*F*

_{2}). With this, the Tx current

*I*

_{0}becomes the smallest. When

*R*

*= ∞ (open), no current is induced on the Rx loop, the system is at the under-coupled limit, no power is transferred to Rx, the efficiency (7) is also 0, the Tx is as if being isolated from Rx, and the Tx input impedance becomes that of its own:*

_{L}*R*

_{0}. When

*R*

*(open, short, optimum) can be used for the information transfer from Rx to Tx. These input impedances (6) and the power transfer efficiencies (7) are summarized in Table 1.*

_{L}*Z*

*at the Tx terminal occurring between the two load states of Rx, but mostly at a close proximity of Tx to Rx. Using the results in Table 1, we can assess the transfer capability of Rx information for the usual resonant WPT systems.*

_{in}*F*

*) as a ratio of the Tx*

_{I}*Z*

*when*

_{in}*R*

*= 0 and the same when*

_{L}*R*

*=*

_{L}*R*

*in Table 1, given by*

_{L,opt}*F*

*) as defined in (10) also corresponds to the current (*

_{I}*I*

_{0}) variation at the Tx terminal depending on Rx loads of “short” and “optimum”

*R*

*since*

_{L}*I*

_{0}=

*V*

_{0}/

*Z*

*. The metric (10) is plotted in Fig. 2 together with the maximum efficiencies as a function of the figure of merit for power transfer (*

_{in}*F*). We can see that as the figure of merit for power (

*F*) increases, the figure of merit for information transfer (

*F*

*) also increases and the power transfer efficiency also increases. Notice that when*

_{I}*F*is roughly larger than 3, the efficiency is greater than 50% and

*F*

*almost ap proaches*

_{I}*F*. Taking an example, the transfer efficiency of about 80% is achieved with

*F*= 9, where

*F*

*is also about 9.*

_{I}### III. Design Examples Validated with Electromagnetic Simulations

*F*given by (5) are numerous. Several realizations for a system consisting of a Tx loop with

*r*

_{0}= 10 cm and a Rx loop with

*r*

_{0}= 5 cm are shown in Table 2, where the quality factor

*Q*is the geometrical mean of

*Q*

_{0}and

*Q*

_{1}(5). The coupling coefficients

*k*were obtained using [16].

*r*

_{0}=

*r*

_{1}= 20 cm, enabling power and information transfer in longer ranges. It is seen in Table 3 that an efficiency of 56% is achieved for the case of

*d*= 80 cm with

*F*

*≈*

_{I}*F*= 7.3. The figure of merit for power (

*F*), one important parameter in the resonant WPT efficiency, can now be understood as the Tx current variations used for Rx information transfer.

*d*= 15 cm and 25 cm in Table 2 and plot the real part (a) and imaginary part (b) of the input impedance (3) for different loads of

*R*

*= 0 and*

_{L}*Q*

_{0}= 388.8) and 5 cm (

*Q*

_{1}= 334.2), respectively. The Tx and Rx loop wire radii are 0.1 cm. The imaginary parts (b) of the input impedances for both cases of

*d*= 15 cm and

*d*= 25 cm are observed to be zero at the resonant frequency since the system is at resonance. The real parts (a) of the input impedances at the resonant frequency, given by (6) and in Table 1, have the maximums when

*R*

*= 0. The circuit-simulated figures of merit for information transfer (*

_{L}*F*

*) are 3.7/0.5 (= 7.38) for the case of*

_{I}*d*= 15 cm and 0.42/0.17 (= 2.48) for the case of

*d*= 25 cm. The EM-simulated figures of merit for information transfer (

*F*

*) are 3.26/0.45 (= 7.2) for the case of*

_{I}*d*= 15 cm and 0.33/0.16 (= 2.1) for the case of

*d*= 25 cm. The results are in agreement with the theoretical ones of 7.4 and 2.5 in Table 2. These

*F*

*s should be understood as the Tx current variations due to the Rx load changes. The overall EM-simulated input impedances are shown to be in good agreement with the theoretical ones.*

_{I}*R*

*= 0 and*

_{L}*d*= 15 cm and

*F*

*) with each Rx load has been found to be almost identical to the theoretical value of 7.4. Based on the transient analysis with increasing switching rates, Fig. 4(a) and 4(b) show that it takes about 0.5 ms for the Tx current to reach a steady state. The source voltage*

_{I}*V*

_{0}in Fig. 1 is assumed to be 1 V. For the shorted Rx load with

*R*

*= 0 Ω and optimum load with*

_{L}*R*

*= 0.25 Ω, the average Tx current envelopes are 0.33 A and 1.97 A, respectively, for the case of the 10 kbps switching rate (Fig. 4(a)). Thus, the ratio of the Tx currents is 6.0 (= 1.97/0.33). For the case of 20 kbps (Fig. 4(b)), the ratio of the Tx current envelopes is 5.0 (= 1.96/0.39). Even though the ratio of Tx currents somewhat deviates from the expected*

_{L}*F*

*due to the effects of the transients from switching operations.*

_{I}*d*= 80 cm and

*F*

*of 3.6). The source voltage*

_{I}*V*

_{0}in Fig. 1 is assumed to be 1 V. The short and optimum loads (

*R*

*s) are 0 Ω and 0.24 Ω. The ratio of Tx current envelopes is 3.3 for the case of 10 kbps and 3.1 for the case of 20 kbps, again close to the theoretical value (*

_{L}*F*

*) of 3.6.*

_{I}*F*

*) as defined in (10), expressed as a function of the figure of merit for power transfer (*

_{I}*F*), has been shown to be very useful in characterizing and estimating the resonant WPIT systems, especially in their capability of transferring Rx information to Tx.