Sensitivity analysis of L-type impedance matching circuits for inductively coupled wireless power transfer

Abstract

L-type impedance matching circuits (IMCs) can maximize the power transfer efficiency (PTE) by converting the load to the optimum one in the wireless power transfer system. However, their effectiveness is inadequately studied. In this work, the applicability of the L-type IMCs is carefully examined under the restriction of capacitance realization. The sensitivities with respect to the deviation of the capacitance in mass production as well as the variations of the load and the mutual inductance are analyzed in detail and validated by experiments. By comparison with the basic-type IMCs, the L-type IMCs have higher, nearly equal and lower sensitivities with respect to the deviation of the capacitance, the variation of the load, and the variation of the mutual inductance, respectively.

Keywords

L-type impedance matching circuit sensitivity power transfer efficiency

1. Introduction

Wireless power transfer (WPT) technology has attracted significant attention in recent years owing to the potential of removing the cumbersome wires when energizing the electronic devices [1–3]. Although capacitively coupled WPT [4] and ultrasonically coupled WPT [5] technologies are also developed, inductively coupled WPT (IPT) [6] is arguably the most popular approach in reality. Typically, an IPT system transfers the power from a transmitter (Tx) coil to a receiver (Rx) coil with the help of mutual magnetic coupling between coils.

The most interesting property of IPT links is the power transfer efficiency (PTE) or the power transfer distance (PTD). In general, PTE decreases with PTD. For a fixed PTD, optimizations of the coil geometries and the impedance matching circuits (IMCs) are two main tasks in maximizing PTE. The coils are usually firstly designed because the design of IMCs depends on the electrical parameters of the coils (e.g., self-inductances).

Two IMCs are, respectively, inserted at Tx and Rx, as shown in Fig. 1, in which V _S and R _S are the power source and its output impedance, respectively, R _L is the equivalent load, Z ₁ and Z _L,IMC are the impedance seen into the Tx coil and the load, L ₁ and L ₂ are the self-inductances of the Tx and Rx coils, respectively, and k is the coupling coefficient. In order to maximize PTE, an optimum value of Z _L,IMC (i.e., Z _L,opt) and the corresponding optimum PTE (i.e., 𝜂_opt) are derived in [7], providing that IMCs are lossless, $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}Z_{\text{L},\text{opt}}=\text{Re}(Z_{\text{L},\text{opt}})+j\text{Im}(Z_{\text{L},\text{opt}})=R_{2}\sqrt{1+k^{2}Q_{1}Q_{2}}-j{\omega}L_{2}\\[6.0pt] \displaystyle {\eta}_{\text{opt}}=\frac{k^{2}Q_{1}Q_{2}}{(1+\sqrt{1+k^{2}Q_{1}Q_{2}})^{2}},\end{array} & & \displaystyle\end{eqnarray}$ (1) where R ₂ is the parasitic resistance of the Rx coil [8], ω is the operating frequency, Q ₁₍₂₎ is the quality factors of the Tx (Rx) coil, and Re(Z _L,opt) and Im(Z _L,opt) are the real and imaginary parts of Z _L,opt respectively. Note that IMC converts R _L to Z _L,opt. Although π-, T-, or L-type IMC, even a resonant coil [9], can be used, L-type IMC is a good choice [10] if efficiency is of the primary concern. Only two components (inductors or capacitors) are required for an L-type IMC [11–15]. Because capacitors introduce less loss than inductors, capacitors are apt to be used [12–15] although inductors may be adopted occasionally [11].

Fig. 1.

A diagram of two-coil inductively coupled WPT system with IMCs.

Although IMCs seem to be very attractive in WPT applications, their effectiveness is inadequately examined. The maximum PTE is actually reachable only in some special cases. From (1), one can observe that the IMC design depends on the electrical parameters of the coil, the load, and the mutual inductance that may vary in reality (according to specific applications). In addition to the impacts from other portions of the system, the variation of the IMC itself (e.g., its element variation) is usually not negligible either, especially in mass production. For instance, the deviation of the capacitance from the rated value (i.e., tolerance) in a general application that employs E12 or E6 series corresponds to ±10% and ±20% deviation, respectively, according to the standard IEC60063 [16].

In this work, the sensitivities of PTE with respect to the deviation of capacitance and the variations of the load and the mutual inductance are analyzed and validated by experiments when the L-type IMCs are used in the inductively coupled WPT systems. Moreover, comparisons between the basic-type IMCs and the L-type IMCs are also presented [17]. The sensitivity analysis is not only useful for selection and design of the IMCs, but also useful for design of the PTE tracking system to realize adaptive control [18,19].

The rest of this work is organized as follows. In Section 2, the applicability of the L-type IMCs is carefully examined. In Section 3, the sensitivity of PTE with respect to the deviation of the capacitance is analyzed in detail. In Section 4, the sensitivities of PTE with respect to the variations of the load and the mutual inductance are studied. Finally, a conclusion is drawn in Section 5.

2. Applicability of L-type IMCs

Figure 2 shows four types of IMCs, which are either basic-type or L-type IMCs. The four types of IMCs are named as S-S, S-P, S-SP and S-PS IMCs. The S-S and S-P IMCs are basic-type IMCs, whereas the S-SP and S-PS IMCs are L-type IMCs. The letters “P” and “S” represent the connecting types of the capacitance (in parallel and in series, respectively), while the letters before and after hyphen correspond to the Tx and Rx sides, respectively. Without loss of generality, the structure of one capacitor connecting to the Tx coil in series is considered, and thus the IMC in the Tx side has no impact on the power input to the Tx coil since the IMC in the Tx side is lossless. Consequently, the IMC in the Tx side does not impact the sensitivity analysis of PTE.

Fig. 2.

Four different types of IMCs. (a) S-S, (b) S-P, (c) S-SP, and (d) S-PS.

For basic-type IMCs shown in Fig. 2a and 2b, the resonance is realized by the compensation capacitance and inductance. Compared with the condition of the optimum load given by (1), it is unable to convert R _L to Z _L,opt with only one capacitance [11]. More complicated circuits, such as the L-type IMCs including at least two components, have to be used. The S-SP and S-PS IMCs, shown in Fig. 2c and 2d, are two types of the L-type IMCs in which the components are restricted to be capacitors since the capacitors have lower losses than the inductors in practice. Although the L-type IMCs have been used in many applications [12–15], their applicability has not been well analyzed. In fact, the L-type IMCs may not be applicable to some of the IPT links.

According to Fig. 2, the PTE 𝜂 = P _L∕P _IN can be expressed as $\begin{eqnarray}\displaystyle {\eta}=\frac{P_{\text{L}}}{P_{\text{IN}}}=\frac{|I_{\text{L}}|^{2}R_{\text{L}}}{|I_{1}|^{2}\text{Re}(Z_{1})}, & & \displaystyle\end{eqnarray}$ (2) where P _L and P _IN are the power delivered into the load and the power input to the Tx coil, respectively, Re(Z ₁) is the real part of the impedance Z ₁ (see Fig. 2c and 2d), and I _L and I ₁ are the currents flowing through R _L and Z ₁, respectively. The parameters in (2) can be obtained as $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle Z_{1}=\frac{1}{j{\omega}C_{1}}+j{\omega}L_{1}+R_{1}+\frac{({\omega}M)^{2}}{Z_{\text{L,IMC}}+j{\omega}L_{2}+R_{2}}\\[11.0pt] \displaystyle \frac{I_{\text{L}}}{I_{1}}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{j{\omega}M}{Z_{\text{L},\text{IMC}}+j{\omega}L_{2}+R_{2}}\frac{1}{1+j{\omega}C_{\text{2P}}R_{\text{L}}} & \text{S-SP}\\[11.0pt] \displaystyle \frac{j{\omega}M}{Z_{\text{L},\text{IMC}}+j{\omega}L_{2}+R_{2}}\frac{1}{1+j{\omega}C_{\text{2P}}R_{\text{L}}+C_{\text{2P}}/C_{\text{2S}}} & \text{S-PS}\end{array}\right.\\[24.0pt] \displaystyle Z_{\text{L,IMC}}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{j{\omega}C_{\text{2S}}}+\frac{1}{j{\omega}C_{\text{2P}}}\Vert R_{\text{L}} & \text{S-SP}\\[11.0pt] \displaystyle \frac{1}{j{\omega}C_{\text{2P}}}\Vert \left(R_{\text{L}}+\frac{1}{j{\omega}C_{\text{2S}}}\right) & \text{S-PS},\end{array}\right.\end{array} & & \displaystyle\end{eqnarray}$ (3) where M is the mutual inductance between the Tx and Rx coils [20].

An explicit expression of PTE can be expressed by substituting (3) into (2). By letting the derivatives of 𝜂 with respect to C _2S and C _2P be zero, i.e., d𝜂∕dC _2S = 0 and d𝜂∕dC _2P = 0, one can find out the solutions of C _2S and C _2P corresponding to the extremum of PTE.

Under the constraints of C _2S ≥ 0 and C _2P ≥ 0, three solutions and their corresponding PTEs can be obtained for the S-SP IMC as follows: $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\left\{\begin{array}{@{}l@{}}C_{\text{2P}\_1}=0\\[8.0pt] \displaystyle C_{\text{2S}\_1}=\frac{1}{{\omega}^{2}L_{2}}\end{array}\right.\\[24.0pt] \left\{\begin{array}{@{}l@{}}\displaystyle C_{\text{2P}\_2}=\frac{1}{{\omega}R_{\text{L}}}\sqrt{\frac{R_{\text{L}}-R_{\text{L},\text{opt}}}{R_{\text{L},\text{opt}}}}\\[15.0pt] \displaystyle C_{\text{2S}\_2}=\frac{1}{{\omega}}\frac{{\omega}L_{2}+\sqrt{R_{\text{L},\text{opt}}(R_{\text{L}}-R_{\text{L},\text{opt}})}}{|Z_{\text{L},\text{opt}}|^{2}-R_{\text{L},\text{opt}}R_{\text{L}}}\end{array}\right.\\[34.0pt] \left\{\begin{array}{@{}l@{}}\displaystyle C_{\text{2P}\_3}=\frac{1}{{\omega}^{2}L_{2}}\frac{Q_{2}^{2}}{1+k^{2}Q_{1}Q_{2}+Q_{2}^{2}}\\[15.0pt] C_{\text{2S}\_3}=\infty ,\end{array}\right.\end{array} & & \displaystyle\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle {\eta}_{\text{opt}\_1}=\frac{k^{2}Q_{1}Q_{\text{2L}}}{1+k^{2}Q_{1}Q_{\text{2L}}}\cdot \frac{R_{\text{L}}}{R_{2}+R_{\text{L}}}\\[15.0pt] \displaystyle {\eta}_{\text{opt}\_2}={\eta}_{\text{opt}}=\frac{k^{2}Q_{1}Q_{2}}{(1+\sqrt{1+k^{2}Q_{1}Q_{2}})^{2}}\\[15.0pt] \displaystyle {\eta}_{\text{opt}\_3}=\frac{k^{2}Q_{1}Q_{2\text{L}}(1+k^{2}Q_{1}Q_{2}+Q_{2}^{2})}{(1+k^{2}Q_{1}Q_{2}+Q_{2}Q_{\text{2L}})(1+k^{2}Q_{1}Q_{\text{2L}}+Q_{2}Q_{\text{2L}})}\cdot \frac{R_{\text{L}}}{R_{2}+R_{\text{L}}},\end{array} & & \displaystyle\end{eqnarray}$ (5) where Q ₁₍₂₎ = ωL ₁₍₂₎∕R ₁₍₂₎ is the unloaded quality factor of the Tx (Rx) coil, Q _2L = ωL ₂∕(R ₂ + R _L) is the loaded quality factor of the Rx coil, R ₁₍₂₎ is the parasitic resistance of the Tx (Rx) coil, and R _L,opt is the real part of the optimal impedance Z _L,opt.

From (4) and (5), in the first solution, C _{2P_1} = 0 implies that the capacitance C _2P should be removed and then the S-SP IMC degenerates to the S-S IMC (see Fig. 2a). Similarly, in the third solution, C _{2S_3} = ∞ implies that the capacitance C _2S should be shorten and the S-SP IMC degenerates to the S-P IMC (see Fig. 2b). Only the second solution is the meaningful solution of the S-SP IMC. The PTE corresponding to the second solution, i.e., 𝜂_{opt_2}, is the same as the result of the optimum PTE in (1). However, the second solution is unattainable if the condition R _L,opt ≤ R _L ≤ |Z _L,opt|²∕R _L,opt is not satisfied, because C _2P and C _2S must be a positive real number. That is, not all the load value can be realizable by using the L-type IMCs. In this load range, 𝜂_{opt_2} has the highest value by comparing to 𝜂_{opt_1} and 𝜂_{opt_3}.

If R _L < R _L,opt, the second solution has to be removed. The fist solution should be chosen, because 𝜂_{opt_1} > 𝜂_{opt_3} can be observed. In contrast, if R _L > |Z _L,opt|²∕R _L,opt, the third solution should be chosen because of 𝜂_{opt_1} < 𝜂_{opt_3}. That is, if the load resistance is too small or too large, the S-S IMC or the S-P IMC, instead of the S-SP IMC, is the optimum choice.

Similar to the case of the S-SP IMC, three effective solutions can be derived for the S-PS IMC (see Fig. 2d) as shown in (6) where only the second solution is given because the first and the third solutions are the same with the corresponding ones in (4). $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle C_{\text{2P}\_2}=\frac{1}{{\omega}}\frac{R_{\text{L}}-R_{\text{L},\text{opt}}}{{\omega}L_{2}R_{\text{L}}+\sqrt{R_{\text{L}}R_{\text{L},\text{opt}}}\sqrt{|Z_{\text{L},\text{opt}}|^{2}-R_{\text{L}}R_{\text{L},\text{opt}}}}\\[24.0pt] \displaystyle C_{\text{2S}\_2}=\frac{1}{{\omega}R_{\text{L}}}\sqrt{\frac{R_{\text{L}}R_{\text{L},\text{opt}}}{|Z_{\text{L},\text{opt}}|^{2}-R_{\text{L}}R_{\text{L},\text{opt}}}}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (6) The S-PS IMC degenerates to the S-S IMC or the S-P IMC if the second solution is unattainable in practice when the load resistance is out of the range R _L,opt ≤ R _L ≤ |Z _L,opt|²∕R _L,opt. The PTEs corresponding to the three solutions in (6) are the same as the PTEs of the S-SP IMC in (5), i.e., 𝜂_{opt_1}, 𝜂_{opt_2}, and 𝜂_{opt_3}, respectively.

From these analyses, one can conclude that the S-SP and S-PS IMCs have the same applicable load range, i.e., R _L,opt ≤ R _L ≤ |Z _L,opt|²∕R _L,opt. This applicable range depends on the load, the electrical parameters of the coils, and the mutual inductance between the coils.

Table 1

Parameters of the IPT link in the example

Parameters	Values
Self-Inductance	7.30 μH
Parasitic resistance @10 MHz	3.36 Ω
Mutual inductance	0.25 μH
R _L,opt	16.06 Ω
\|Z _L,opt\|²∕R _L,opt	13.12 kΩ

Herein, an IPT link between two coils with the same geometrical parameters is taken as an example. The coils are 11-turn rectangular spiral coils that are printed on FR4-substrate with 1-oz copper, 1.7-mm trace width, and 0.8-mm trace space. The operating frequency is set to 10 MHz. The electrical parameters of the coils are shown in Table 1, which are obtained from measurements by using a Vector Network Analyzer (VNA, Agilent E5071C) as shown in Fig. 3a. The characteristic impedance of the VNA is 50 Ω. In order to measure the case with a load larger or smaller than 50 Ω, an additional resistance (R _L1 or R _L2) is connected in series or in parallel, respectively, as shown in Fig. 3b. For instance, if an additional 3300-Ω resistance is connected in series with VNA (i.e., R _L1 = 3300 Ω), the total load for the Rx side is 3350 Ω. Consequently, the power on the 3350-Ω load is obtained by multiplying the measured power of 50-Ω load from VNA by 67.

Fig. 3.

Experimental setup and measurement circuit. (a) Experimental setup for a WPT system with two coils, (b) Circuit for measuring the case with the load not equal to 50 Ω.

The maximum achievable PTEs inside the applicable load range by using the L-type IMCs are shown in Fig. 4a. The PTEs of the S-S IMC and the S-P IMC (i.e., 𝜂_{opt_1} and 𝜂_{opt_3}, respectively) are shown together in Fig. 4a. It is clearly shown that R _L,opt and |Z _L,opt|²∕R _L,opt are the low and high critical load resistances, respectively, to use the L-type IMCs. The PTEs of the S-S IMC and the S-P IMC reach the maximum values at R _L,opt and |Z _L,opt|²∕R _L,opt, respectively. The capacitances have to change with the load if the optimum load condition in (1) must be satisfied. One can find from Fig. 4b that when the load is out of the L-type applicable load range, the capacitances become zero or infinite, and accordingly the L-type IMCs degenerate to the basic-type IMCs.

Fig. 4.

(a) Maximum achievable PTEs for IPT links by using the L-type and the basic-type IMCs. (b) Capacitance changes with the load for the L-type IMCs when keeping the satisfaction of the optimum load condition in (1).

3. Sensitivity with respect to capacitance

From (1), the optimum PTE can be achieved only when the optimum load condition is satisfied. If the coils, the operating frequency, and the PTD are fixed, the capacitances must exactly satisfy (4) or (6). Although the L-type IMCs may give better PTE than the basic-type IMCs in this situation, if the deviation of the capacitance in mass production is considered, the basic-type IMCs can exhibit lower sensitivity than the L-type IMCs according to the following analysis.

In general applications, ±10% and ±20% deviations for capacitance are commonly used. In mass production, these deviations may result in large PTE variations from the expected value. Because the PTE expression in (2) is not concise enough when (4) is not satisfied, the PTE variations at different load conditions (50 Ω and 3350 Ω) with respect to the deviation of capacitance are plotted in Fig. 5, where the parameters of the IPT link listed in Table 1 are used.

Fig. 5.

PTE Variations at different-load conditions (50 Ω and 3350 Ω) with respect to the deviation of capacitance for the S-SP and S-PS IMCs.

For the relatively low-resistance load case (50 Ω), the PTE for the S-SP IMC is very sensitive to C _2S but not C _2P. Similar results can be observed for the relatively high-resistance load case (3350 Ω) for the S-PS IMC, in which the PTE is very sensitive to C _2P but not C _2S. For the relatively low-resistance load case of the S-PS IMC and the relatively high-resistance load case of the S-SP IMC, both the PTEs are sensitive to C _2P and C _2S.

The PTEs under the deviation of capacitance are further compared between the L-type and basic-type IMCs in Fig. 6, where C _2P and C _2S vary together with the same deviation. From Fig. 6, the S-P IMC for the relatively low-resistance load case and the S-S IMC for the relatively high-resistance load case result in very low PTE, which is consistent with the conclusion in [11] and [21]. Compared with the L-type IMCs, although the S-S IMC for the relatively low-resistance load case and the S-P IMC for the relatively high-resistance load case have lower maximum PTEs, the basic-type IMCs exhibit lower variations.

Fig. 6.

Comparison of sensitivity with respect to the deviation of the capacitance between the L-type and the basic-type IMCs for (a) R _L = 50 Ω, and (b) R _L = 3350 Ω.

For instance, the S-P IMC for the relatively high-resistance load has about 24% PTE variation, which is much smaller than about 54% PTE variation for the S-PS IMC. The measurement results shown in Fig. 6 reveal the same trend with the analysis although there exist some discrepancies between the results from the analyses and the measurements. In experiments, the capacitors with 1% accuracy (manufactured by Murata Electronics, Japan) are used and are approximated as exact values without deviation which will not change the trends of the curves in Fig. 6.

4. Sensitivity with respect to load and IPT link

Besides the deviations of capacitances, the IMC design also depends on the load, the parameters of the coils, and the mutual inductance between the coils. In some applications, the load is not fixed. For instance, the power supply of a sensor is rectified and regulated from an Rx coil of an IPT link. The equivalent load of the IPT link depends on the current consumption according to the working state of the sensor.

Under the variation of the load, the variation of PTEs for the L-type and basic-type IMCs are analyzed and compared in Fig. 7. Although the basic-type IMC designs are not dependent on the load (only dependent on the self-inductance and the operating frequency), there exists an optimum load corresponding to the maximum PTE, which actually is the critical load of the applicable range for each of the L-type IMCs, i.e., the optimum load is R _L,opt or |Z _L,opt|²∕R _L,opt for the S-S IMC or the S-P IMC, respectively. The L-type IMC design is optimized for a specific set of parameters. If the load varies, the PTE may decrease from the maximum PTE 𝜂_opt. In theory, the S-SP IMC and the S-PS IMC have the same PTE results when the load varies. Consequently, the calculated ones for the S-SP IMC and the S-PS IMC are denoted by “L-type” in Fig. 7.

Fig. 7.

Comparison of sensitivity with respect to the variation of the load between the L-type and the basic-type IMCs for (a) R _L = 50 Ω, and (b) R _L = 3350 Ω.

Figure 7a plots the variations of the PTEs when the L-type IMCs are optimized for 50-Ω load. In Fig. 7a, the L-type IMCs are compared to the S-S IMC, whereas the S-P IMC is excluded because of its very low PTE for 50-Ω load. Both the L-type and S-S IMCs have large variations with the load. The difference is that the loads corresponding to the maximum PTEs for the S-S IMC and the L-type IMCs are different, i.e., 16 Ω and 50 Ω, respectively. Similarly, the variations of the PTEs when the L-type IMCs are optimized for 3350 Ω load are compared to those of the S-P IMC as shown in Fig. 7b. Both the L-type and S-P IMCs have large variations with the load but are with different loads corresponding to the maximum PTEs (i.e., 3350 Ω and 13.12 kΩ, respectively). All the analyses are verified by the experimental results. One can conclude that the sensitivities of the L-type and basic-type IMCs with respect to the load are similar.

Besides the load, the IMC design also depends on the electrical parameters of the coils and the mutual coupling between the Tx and Rx coils. The electrical parameters like the self-inductance and the parasitic resistance of the coils usually have good uniformity since the coils are generally fabricated on printed-circuit board (PCB). The mutual inductance, however, may have a big variation because the relative position or angle between the Tx and Rx coils in reality can be much different according to the applications.

Figure 8a and 8b depict the PTE variations with respect to the mutual inductance for cases that the L-type IMCs are optimized for 50-Ω load and 3350-Ω load, respectively (where a 75-mm PTD is assumed). As an example, the mutual inductance is varied by changing the PTD from 15 mm to 105 mm while keeping the other parameters fixed as shown in Table 1. In general, compared to the long PTD case, the short PTD case has higher mutual inductance and higher coupling coefficient, thereby resulting in higher PTE. Because the S-SP IMC and the S-PS IMC have the same variation with the mutual inductance, both their theoretical results are denoted as “L-type” in Fig. 8, which are also validated by experiments.

Fig. 8.

Comparison of sensitivity with respect to the mutual inductance between the L-type and the basic-type IMCs for (a) R _L = 50 Ω, and (b) R _L = 3350 Ω.

In Fig. 8a, one can observe that when the mutual inductance is higher than 0.25 μH (which corresponds to 75-mm PTD), although the capacitances deviate the optimum values, the more tight coupling between the Tx and Rx coil leads to higher PTE. However, the PTEs for the L-type IMCs increase more slowly than the PTE for the S-S IMC. When the mutual inductance decreases, the PTEs for the L-type IMCs decrease more slowly than the PTE for the S-S IMC. Similar variation trends can be observed in Fig. 8b for the L-type IMC and the S-P IMC. One can conclude that the sensitivity of the L-type IMCs with respect to the mutual inductance is lower than the basic-type IMCs.

5. Conclusion

Although the L-type IMCs have the potential to maximize the PTE of an IPT link, the L-type IMCs are not robust enough in reality. Firstly, the L-type IMCs can be used only when the applicable range, R _L,opt ≤ R _L ≤ |Z _L,opt|²∕R _L,opt, is satisfied. Secondly, an analyses on the PTE variations under the deviation of the capacitance, the variation of the load and the variation of the mutual inductance have been performed. According to the analyses and experiments, the L-type IMCs are more sensitive to the deviation of the capacitance, nearly sensitive to the variation of the load, and less sensitive to the variation of the mutual inductance than the basic-type IMCs.

Footnotes

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under grants No. 61771175, No. 61411136003, and No. 61331007, and in part by the Zhejiang Provincial Natural Science Foundation of China No. LY17F010018, and in part by the Talent Project of Zhejiang Association of Science and Technology under grant 2018YCGC017.

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