Optimum Receiver-Side Tuning Capacitance for Capacitive Wireless Power Transfer

Abstract

This paper reveals the optimum capacitance value of a receiver-side inductor-capacitor (LC) network to achieve the highest efficiency in a capacitive power-transfer system. These findings break the usual convention of a capacitance value having to be chosen such that complete LC resonance happens at the operating frequency. Rather, our findings in this paper indicate that the capacitance value should be smaller than the value that forms the exact LC resonance. These analytical derivations showed that as the ratio of inductor impedance divided by plate impedance increased, the optimum Rx capacitance decreased. This optimum capacitance maximized the TX-to-RX transfer efficiency of a given set of system conditions, such as matching inductors and coupling plates.

1. Introduction

Capacitive wireless power-transfer systems wirelessly transmit electrical energy without the use of actual wire coils. Instead, thin metallic plates form a capacitor through which current can flow. Such a system has previously been investigated for biomedical applications [1], electric vehicles [2,3,4,5], mobile devices [6], and constant-current applications [7]. Although a variety of circuit topologies are available for capacitive power system [1,2,3,4,5,6,7,8], an inductor-capacitor (LC) section in a receiver (RX) is the simplest topology for systems with small coupling capacitances [2,3]. Additional matching inductors, such as those in [5], require a large inductance value (~240 μH), which is too bulky. Parasitic capacitances due to nearby metals can also be merged with a parallel tuning capacitor [2] to form another type of LC section.

Although much work has been done on the LC matching network design, only a few works have focused on efficiency maximization, which is an important key requirement in an effective wireless power-transfer system. Reference [3] proposes operating near the resonance frequency of an inductor and capacitor, either for constant current or constant voltage operation. Their operating frequency would slightly deviate from the self-resonant frequency of an LC matching network for constant voltage or current operation, where the amount of deviation is determined by the strength of the capacitive coupling. At weak coupling, the operating frequency would approach the LC resonant frequency.

While the LC matching design of [3] successfully achieves either constant voltage or current operation, this design does not focus on efficiency maximization. Reference [9] analyzes the effect of matching detuning and proposes a design method to operate over a wide frequency bandwidth. Although this design successfully operated over this wider frequency range via inverter soft-switching, the optimum matching capacitance for maximum efficiency has not yet been discussed.

Reference [8] proposed a resonance-matching network to improve the power factor. This is equivalent to enhancing the real part of Z_RX, as seen in Figure 1. The large resistive impedance of the receiver increased the power factor here because the CP impedances were highly imaginary. A matching network design [10] also aimed toward power-factor maximization. Unfortunately, as will be discussed later in this paper, power-factor maximization does not necessarily maximize efficiency. Hence, any design method that focused on achieving maximum efficiency would be different from the methods in [8,10].

Reference [11] proposed that the matching capacitor should be small in order to reduce sensitivity to parameter variations and voltage stress. Through this method, the drift of system performance against load or component variation would be minimized.

While the various design methods mentioned above aimed to achieve different goals, such as constant output, wide bandwidth, high power factor, or reduced sensitivity, none of them explicitly defined the optimum RX capacitance for maximum efficiency. This paper investigates exactly what optimum capacitance value could maximize the transfer efficiency for a given set of system parameters, such as coupling plates, load, and matching inductors. The results showed that the optimum RX capacitor value should be smaller than that of a value that achieves complete LC resonance. A quantitative closed-form equation predicted the optimum capacitor value as a function of coupling plates, load, and inductors.

2. Optimum Resonant Matching Capacitor

2.1. Circuit Topology of Capacitive Power System

Figure 1 shows the equivalent circuit of capacitive power transfer. This work used a differential Class-E amplifier that produced a sinusoid output voltage and current. Its schematic and measured waveform is presented in Figure 8. However, this mathematical derivation was also applicable to a square-wave voltage source as well (e.g., voltage-mode Class-D inverter) because the first-harmonic approximation was valid due to the high-Q of the matching network. In other words, due to the high selectivity of L_TX—C_TX resonance, higher frequency components of I_TX were suppressed and I_TX became sinusoidal. Since the impedance of capacitive interface, X_P = (ωC_P/2)⁻¹, is extremely high in noncontact applications, it is common to boost the receiver load R_L using an L_RX–C_RX network. The parallel resonance of L_RX–C_RX–R_L increases the real part of Z_RX while minimizing the reactance of Z_RX.

For the given L_RX inductance, the C_RX capacitance would normally be chosen such that L_RX–C_RX was resonant at the operating frequency. This was to maximize the real part of Z_RX (i.e., Re{Z_RX}) while minimizing the reactive part of Z_RX (Im{Z_RX}). In this paper, however, we revealed that an exact LC resonance was not the optimum design for efficiency maximization. Rather, the capacitance should be slightly smaller, such that there exists significant inductive impedance in Z_RX. This Im{Z_RX} partially cancels out the larger capacitive X_P. Although the Re{Z_RX} obtained via the proposed shifted resonance was lower than that obtained with exact LC resonance, it is also analyzed in this paper that a higher Re{Z_RX} is not always beneficial: there was an optimum Re{Z_RX}.

2.2. Analytical Derivation for Optimum C_RX

The Tx-to-Rx efficiency of Figure 1 was defined as

$$\begin{array}{c}{\eta }_{TX\text{-}to\text{-}RX}=\frac{\text{Load power}}{\text{Supplied power by source}}\\ =\frac{I_L^2{R}_L}{I_{TX}^2\left(\mathrm{Re}\left\{Z_{TX}\right\}+R_{P,T}\right)}\end{array}$$

(1)

The Z_TX is the equivalent impedance with regard to the C_TX capacitance, and R_P,T and R_P,R are the parasitic resistance of the L_TX and L_RX inductors, respectively. Equation (1) can be separated into a two-stage equation. The first stage, which was the transmitter efficiency, consisted of the power entering the capacitive interface $I_{TX}^2\mathrm{Re}\left\{Z_{TX}\right\}$, divided by the total power supplied from our power source $I_{TX}^2(\mathrm{Re}\{Z_{TX}\}+R_{P,T})$. The second stage, which was the receiver efficiency, consisted of the power dissipation at final load divided by the power dissipation across the whole receiver. Hence, Equation (1) could be written as

$${\eta }_{TX\text{-}to\text{-}RX}=\frac{I_{TX}^2\mathrm{Re}\left\{Z_{TX}\right\}}{I_{TX}^2\left(\mathrm{Re}\left\{Z_{TX}\right\}+R_{P,T}\right)}\times \frac{I_L^2{R}_L}{I_L^2\left(R_L+R_{P,R}\right)}$$

(2)

thereby arriving at the impedance ratio equation of

$${\eta }_{TX\text{-}to\text{-}RX}=\frac{\mathrm{Re}\left\{Z_{TX}\right\}}{\mathrm{Re}\left\{Z_{TX}\right\}+R_{P,T}}\times \frac{R_L}{R_L+R_{P,R}}$$

(3)

Since the power delivered to the receiver was equal to the power dissipated at the Re{Z_TX}, it was important that we obtained a large value of Re{Z_TX} to maximize transmission efficiency. In other words, the power delivered to RX is $P={\left|I_{TX}\right|}^2\mathrm{Re}\{Z_{TX}\}$, whereas the power dissipated at TX parasitic is $P={\left|I_{TX}\right|}^2{R}_{P,T}$. Hence, the Re{Z_TX} should have been higher than R_P,T. The Re{Z_TX} is defined found as follows:

$$\mathrm{Re}\{Z_{TX}\}=\frac{X_{CTX}^2\mathrm{Re}\left\{Z_{RX}\right\}}{\mathrm{Re}{\left\{Z_{RX}\right\}}^2+{(X_{CTX}+X_P-\mathrm{Im}\{Z_{RX}\})}^2}$$

(4)

where Re{Z_RX} and Im{Z_RX} is

$$\mathrm{Re}\{Z_{RX}\}=\frac{X_{CRX}^2{R}_L}{R_L^2+{(X_{LRX}-X_{CRX})}^2}$$

(5)

$$\mathrm{Im}\{Z_{RX}\}=-\frac{X_{CRX}\left\{X_{LRX}\right(X_{LRX}-X_{CRX})+R_L^2\}}{R_L^2+{(X_{LRX}-X_{CRX})}^2}$$

(6)

and $X_{CTX}={(\omega C_{TX})}^{-1}$, $X_{CRX}={(\omega C_{RX})}^{-1}$, $X_P={(\omega C_P/2)}^{-1}$, and $X_{LRX}=\omega L_{RX}$.

After substituting Equations (5) and (6) into Equation (4), Equation (4) became a function of the receiver parameters, such as R_L, X_LRX, and X_CRX. The typical complete resonance, $\omega =1/\sqrt{L_{RX}{C}_{RX}}$, almost cancelled out the Im{Z_RX}, whereas the opposite was true for the Re{Z_RX}, which was maximized.

In this paper, we tested the theory that there may be an optimum C_RX,opt to maximize the efficiency of Equation (3) for any given set of system parameters. Our efficiency maximization was realized by a maximum Re{Z_TX} resistance and a corresponding minimum I_TX current, thereby suppressing power losses at the inverter and TX passive components thanks to a minimum of I_TX current.

Differentiating Equation (4) with respect to X_CRX and setting this differentiation to zero, i.e., $\partial \mathrm{Re}\left\{Z_{TX}\right\}/\partial X_{CRT}=0$, the optimum X_CRX was derived:

$$\begin{array}{|c|}\hline X_{CRT,opt}=\frac{({(R_L+R_{P,R})}^2+X_{LRX}^2)(X_P+X_{CTX})}{X_{LRX}(X_P+X_{CTX})-{(R_L+R_{P,R})}^2-X_{LRX}^2}\\ \hline\end{array}$$

(7)

Note that the derivation of Equation (7) does not involve any approximations and therefore was generally applicable for any given set of system parameters e.g., load, L_RX, C_P, C_TX etc. Equation (7) can be simplified because the coupling plate impedance, X_P, is usually a much higher value than the X_CTX. [2,3]. Moreover, the Rx inductor reactance, X_LRX, is also usually designed as a much higher value than R_L in order to boost a small R_L into a large Re{Z_RX}, generally because $\mathrm{Re}\left\{Z_{RX}\right\}\approx X_{LRX}^2/R_L$. Under these conditions, Equation (7) was simplified as follows:

$$\begin{array}{|c|}\hline X_{CRX,opt}\cong X_{LRX}{\left(1-\frac{X_{LRX}}{X_P}\right)}^{-1}\\ \hline\end{array}$$

(8)

Equation (8) indicated that the optimum X_CRX impedance, which maximized the Re{Z_TX} and our efficiency, should be higher than the inductor impedance X_LRX. The ratio between inductor impedance and coupling plate impedance, i.e. X_LRX/X_P, determined the level of deviation from the complete LC canceling condition of X_CRX = X_LRX. Equation (8) indicated that a higher ratio of X_LRX/X_P required a larger deviation of X_CRX from the X_LRX.

2.3. Discussion

Figure 2b is the Z_RX representation of Figure 2a at a conventional resonance. Conventional RX cancelled the Im{Z_RX} while maximizing the R_L into a high Re{Z_RX} so that the power factor of Z_CAP was maximized. However, higher Re{Z_RX} was not always beneficial for TX-to-RX efficiency. As seen in Figure 1, the I_TX supplied from the inverter was directed toward two separate paths: one was through C_TX (which did not contribute to power delivery), and the other was through I_P flowing into the receiver. If Re{Z_RX} was too high, then most of the I_TX was circulated to C_TX and only limited current could flow through I_P, which resulted in a reduced power efficiency. The bottom graph of Figure 2d shows that at conventional resonance the I_TX required to deliver a specified I_L should have been increased.

However, the proposed C_RX detuning in Figure 2c did not maximize the Re{Z_RX} and, at the same time, intentionally generated +Im{Z_RX}. This partially cancelled X_P by detuning L_RX–C_RX. Its impedance, as seen in Figure 2d, was a frequency of 7.1 MHz. The overall impedance |Z_CAP| = Re{Z_RX} + j(Im{Z_RX} − X_P) was significantly reduced compared to conventional L_RX–C_RX. As a result, the bottom graph of Figure 2d shows that the I_TX current required to deliver a given load current I_L could be minimized, which in turn could reduce the losses in the transmitter.

The exact amount of detuning of L_RX–C_RX was quantitatively obtained from Equations (7) and (8). Figure 3 illustrates the design trade-off. In Figure 3a, while Re{Z_RX} should have been high to maximize the load power per unit I_P of current, the Re{Z_RX} should not have been so excessively high that the I_P current per unit I_TX could not be maintained. At the same time, in Figure 3b, $j\mathrm{Im}\left\{Z_{RX}\right\}-j{X}_P$ was minimized by maximizing the +jIm{Z_RX} so that the I_P was increased per given I_TX. However, as seen in Figure 3c, excessively high Im{Z_RX} may have compromised the achievable Re{Z_RX}. The proposed Equations (7) and (8) optimized the trade-offs of Figure 3 and produced an optimum Re{Z_RX} and Im{Z_RX} that maximized power efficiency.

Figure 4 compares the conventional and the proposed methods. The proposed method surpassed the upper limit imposed by conventional RX tuning.

The high Re{Z_TX} might also be obtained by using a small C_TX, as in Equation (4). However, a small C_TX demands a large L_TX, which increases inductor volume and parasitic R_P,T. As an example, in Figure 2d the bottom graph typical resonance still gives the same Re{Z_TX} = 9 Ω if the C_TX was reduced from 168.5 to 87.5 pF. However, then the required L_TX should have increased from 3.8–6.8 μH. Due to the increased parasitic R_P,T, the spice-simulated efficiency degraded from 77.4% to 69.9%. Hence, an optimum C_RX,opt becomes important in order to produce the highest Re{Z_TX} under the constraint of L_TX volume and parasitic resistance.

3. Results

Figure 5 shows the measurement setup using wireless charging of an unmanned aerial vehicle (Drone) prototype can be seen in Figure 5. The load condition was 36 V–1.8 A and resulted in a value of 64.8 W. A differential Class-E inverter and full-bridge rectifier were used. Efficiency in this paper was defined as from DC source to DC load.

A 0.2 mm thick copper plate was used for each plate. A transmit plate of 30 × 30 cm² was placed underneath the landing pad and a receiver plate of 13 × 1.5 cm² was attached under the landing foot of the UAV. The C_P was 23 and 14 pF for a 2 and 4 mm distance, respectively. These distances were due to electrical isolation by way of an acrylic sheet to prevent electrical shorts and mechanical damage of the TX plates that may have resulted from a collision with foreign objects. L_TX, L_RX, and C_TX were 3.8 μH, 7.13 μH, and 165 pF, respectively. A GS66508T FET and PMEG6045 diode were used as our inverter and rectifier, respectively. Please note that Equations (7) and (8) are generally applicable to different systems with different component parameters.

Table 1. Circuit parameters.

Parameter	Value	Parameter	Value
Cp	14~23 pF	Load	36 V, 64.8 W
LTX	3.8 μH	TX plate	30 × 30 cm2
LRX	7.13 μH	RX pad	13 × 1.5 cm2
CTX	165 pF	Switching freq.	6.78 MHz

provides circuit parameters.

Figure 6a presents the DC-to-DC efficiency for each RX capacitor value. A L_RX value of 7.13 μH was chosen because, as can be seen from Figure 4, efficiency could be maximized near ~7 μH at a Cp of 10 pF (worst coupling) using typical LC resonance. The C_RX of 77.3 pF corresponded to typical LC resonance, whose resonance frequency coincided with an operating frequency of 6.78 MHz. The optimum C_RX,opt was predicted by Equation (8) for different C_P coupling plates. As expected, our proposed C_RX,opt values achieved the highest efficiency for the given set of system constraints. Figure 6b presents the I_TX current required to deliver the given load power, which was minimized at the proposed C_RX,opt capacitor tuning. This result was expected because Equations (7) and (8) maximized the Re{Z_TX}, and therefore the power delivered to the receiver, which was $P={\left|I_{TX}\right|}^2\mathrm{Re}\left\{Z_{TX}\right\}$.

Figure 7 presents the loss analysis for the same load power. The proposed C_RX,opt greatly reduced the power loss of the transmitter. This was because the C_RX affected the Re{Z_TX}, which in turn determined the magnitude of current through the transmitter. The waveform in Figure 8 shows that the inverter achieved zero-voltage switching.

4. Conclusions

This paper thoroughly reveals the optimum parallel capacitance value of a receiver for a given set of system parameters. Our finding showed that a complete LC resonance at operating frequency did not result in the highest efficiency. Rather, the RX capacitor should have been be of a smaller capacitance value than the nominal resonance-tuning value. The optimal deviation from nominal resonance should have been proportional to the ratio between the RX inductor impedance and the coupling plate impedance, as formulated in Equation (8). This minimized our I_TX value and its associated losses in the transmitter, thereby increasing overall efficiency while not affecting receiver loss characteristics.