A Constant Voltage WPT System for AM-Modulated Implant Neural Stimulation

1. Introduction

Neural stimulation has long been practiced as an effective treatment for chronic diseases and pain. In recent years, advancements in electronic engineering have brought significant attention to implantable stimulators with wireless power transfer (WPT) capabilities.

Among a series of WPT techniques, resonance inductive coupling technology is most commonly employed for neural stimulation[1]. The basic principle is to utilize LC resonance characteristic to generate higher voltage or current and compensate the gain deduction caused by the enlarged gap. However, there are drawbacks of the traditional structure. The resonant frequency splits with varying coupling coefficient [2], and the voltage gain is affected by load impedance.

It is a popular direction for improvement to added compensation networks to the original circuit and form hybrid topologies. Numerous structures have been proposed, such as LCC[3,4,5], LCL[6,7] and others. Most of these topologies are designed for kW-level WPT[8] and are modeled at frequency around 100kHz. For medical use, however, the WPT systems work at several hundred kilohertz to a few megahertz to ensure more compact size and biocompatibility[1,9,10]. The parasite parameters like the stray capacitance become remarkable in this range, but are often neglected since they are trivial in lower frequency and in high-power system [3,4,11,12]. Additionally, researches from the perspective of power electronics typically focus on efficiency, which is not the top priority in low-power medical use. Therefore, the MRC structure should be remodeled and better tuned for medical applications.

For AM or ASK modulated implant stimulators[13], insensitivity to misalignment is crucial because relative movement between the transmitter and receiver may lead to varying coupling coefficient, and consequently, changes in received voltage amplitude. The transmitter should be able to monitor the received voltage and implement feedback control. Besides complex wireless communication, one solution is to track the resonant frequency to maintain constant voltage gain[14], yet the narrow ISM band strictly limit the adjustment range. Another approach is to adaptively compensate the MRC circuit by a selective impedance matching network[15], but it is only capable for discrete control and involves bulky relays.

The purpose of this paper is to analyze and design the LCL-compensated WPT structure for constant voltage transmission, primarily for medical use. Section 2 derives the characteristics of the structure by modeling the circuit and theoretically proves the feasibility of CV-WPT. Section 3 presents the experimental results, and Section 4 discusses the application methods.

2. Materials and Methods

The proposed LCL-compensated WPT structure is shown in Figure 1a. The coils with self-inductance $L_1$ and $L_2$ are coupled with mutual inductance $M=k\sqrt{L_1{L}_2}$, where k denotes the coupling coefficient. $R_L$ depicts the equivalent AC load. $R_L=8R_{dc}/{\pi }^2$ when the output is rectified and filtered. The circuit is re-drawn as Figure 1b for analysis. $R_1$ and $R_2$ depict ESR of the coils. $C_{p1}$ is split into $C_{p11}$ and $C_{p12}$ using the conclusion of Li et-al[12] to provide extra degree of freedom and achieve load-independent constant voltage transmission. Parasite capacitance of both coils are absorbed by $C_{p1}$ and $C_{p2}$. The composited capacitance can be measured and tuned to ideal in practice, hence the parasite capacitance is neglected in analysis.

2.1. Load-and-Matching-Independent Resonant Frequency

Denote the mesh currents in Figure 1b as $I_1$ to $I_5$, the KVL matrix is written as:

$$\left(\begin{array}{c}{V}_{in}\\ 0\\ 0\\ 0\\ 0\end{array}\right)=\left(\begin{array}{ccccc}j{Z}_1& -\frac{1}{j\omega C_{p11}}& 0& 0& 0\\ -\frac{1}{j\omega C_{p11}}& j{Z}_2& -\frac{1}{j\omega C_{p12}}& 0& 0\\ 0& -\frac{1}{j\omega C_{p12}}& j{Z}_3& -j\omega M& 0\\ 0& 0& -j\omega M& j{Z}_4& -\frac{1}{j\omega C_{p2}}\\ 0& 0& 0& -\frac{1}{j\omega C_{p2}}& j{Z}_5+R_L\end{array}\right)\left(\begin{array}{c}{I}_1\\ I_2\\ I_3\\ I_4\\ I_5\end{array}\right)=\mathbf{ZI}$$

(1)

In which $j{Z}_1$ to $j{Z}_5$ denote the complex impedance of each mesh. Parasite parameters are neglected temporarily. The voltage gain is then:

$$G_V=\frac{I_5{R}_L}{V_{in}}={\mathbf{Z}}_{(5,1)}^{-1}{R}_L$$

(2)

$$\begin{array}{c}\hfill G_V^{-1}=\frac{j{C}_{p2}{C}_{p11}{C}_{p12}}{M{R}_L}·\left(\begin{array}{c}{M}^2{Z}_1{Z}_2{Z}_5{\omega }^4-\frac{1}{C_{p11}^2}{M}^2{Z}_5{\omega }^2-Z_1{Z}_2{Z}_3{Z}_4{Z}_5{\omega }^2\hfill \\ -\frac{Z_1}{C_{p2}^2{C}_{p12}^2}-\frac{Z_3}{C_{p2}^2{C}_{p11}^2}+\frac{Z_1{Z}_4{Z}_5}{C_{p12}^2}+\frac{Z_3{Z}_4{Z}_5}{C_{p11}^2}+\frac{Z_1{Z}_2{Z}_3}{C_{p2}^2}\hfill \end{array}\right)\\ \hfill +\frac{C_{p2}{C}_{p11}{C}_{p12}}{M}\left(M^2{Z}_1{Z}_2{\omega }^4-Z_1{Z}_2{Z}_3{Z}_4{\omega }^2-\frac{M^2{\omega }^2}{C_{p11}^2}+\frac{Z_1{Z}_4}{C_{p11}^2}+\frac{Z_3{Z}_4}{C_{p11}^2}\right)\end{array}$$

(3)

It is obvious that when $Z_1=Z_3=Z_5=0$ there are $\mathrm{d}{\mathrm{G}}_{\mathrm{V}}/\mathrm{d}{\mathrm{R}}_{\mathrm{L}}=0$ and $\mathrm{d}{\mathrm{G}}_{\mathrm{V}}/\mathrm{d}\mathrm{k}=0$, making the voltage gain irrelevant to both the load and the matching condition. To set this fully independent resonant frequency to be ${\omega }_0$, the circuit should satisfies:

$$\begin{array}{c}{\left.Z_1\right|}_{{\omega }_0}={\omega }_0{L}_{s1}-\frac{1}{{\omega }_0{C}_{p11}}=0\hfill \\ {\left.Z_3\right|}_{{\omega }_0}={\omega }_0{L}_1-\frac{1}{{\omega }_0{C}_{p12}}=0\hfill \\ {\left.Z_5\right|}_{{\omega }_0}={\omega }_0{L}_{s2}-\frac{1}{{\omega }_0{C}_{p2}}=0\hfill \end{array}$$

(4)

Merge Equation (3) and (4), the voltage gain at ${\omega }_0$ only depends on k:

$$G_{V0}=-\frac{C_{p11}}{C_{p2}{C}_{p12}M{\omega }_0^2}=-\frac{1}{k}·\frac{L_{s2}}{L_{s1}}\sqrt{\frac{L_1}{L_2}}$$

(5)

Equation (5) demonstrates that the LCL-compensated voltage gain is inversely proportional to $\sqrt{L_2/L_1}$ i.e. the "turns ratio" of an ideally-coupled transformer, which makes smaller receiving coil possible. Notice that the electromagnetic attenuation is neglected here. However, the loss becomes dominant in loosely-coupled condition and (5) no longer holds. Non-ideal parameters also restrict $G_{V0}$ from increasing further with smaller k, which will be discussed later.

2.2. Online Estimation of Received Voltage

Although load-independent, voltage gain at ${\omega }_0$ still varies with k. A feedback loop with information of received voltage $V_{out}$ or the intermediate parameter k must be constructed. With LCL compensation, the needed signal can be extracted from the transmitting side. It avoids extra wireless communication from the receiver and enhances responding speed as well as robustness.

The input number is denoted as $I_1$ in Figure 1b. Let Equation (4) into Equation (1), $I_1$ at ${\omega }_0$ is obtained:

$${\left.I_1\right|}_{{\omega }_0}\stackrel{\Delta }{=}{I}_{10}=V_{in}{\left.{\mathbf{Z}}_{(1,1)}^{-1}\right|}_{{\omega }_0}=V_{in}\left(\frac{L_1^2{L}_{s2}^2}{R_L{L}_{s1}^2{M}^2}+j\frac{L_1^2\left(L_2-L_{s2}\right)-M^2\left(L_{s1}+L_1\right)}{L_{s1}^2{M}^2{\omega }_0}\right)$$

(6)

Where the effect of $R_L$ cannot be eliminated, but can be reduced to negligible. Let $A=\Re \left(I_1\right)$, $B=\Im \left(I_1\right)$, it is obvious that B is irrelevant to $R_L$. Then the effect of $R_L$ on $I_{10}$ is:

$$\begin{array}{c}\hfill \frac{\mathrm{d}\left|I_{10}\right|}{\mathrm{d}{R}_L}=\frac{\mathrm{d}\sqrt{A^2+B^2}}{\mathrm{d}{R}_L}={\left(A^2+B^2\right)}^{-\frac{1}{2}}A\frac{\mathrm{d}A}{\mathrm{d}{R}_L}=\frac{A}{\left|I_{10}\right|}\frac{\mathrm{d}A}{\mathrm{d}{R}_L}\\ \hfill =-{\left(R_L+\frac{R_L^3}{V_{in}{\omega }_0}·\frac{L_2{L}_{s1}^2{k}^2}{L_1^2{L}_{s2}^4}\left(L_1\left(L_2-L_{s2}\right)-k^2{L}_2\left(L_1+L_{s1}\right)\right)\right)}^{-1}\end{array}$$

(7)

$\mathrm{d}\left|I_{10}\right|/\mathrm{d}{R}_L$ diminishes with larger $R_L$. For K$\Omega$-level load, the load adjustment rate can be further reduced by tuning the inductors ratio. Although a small $L_2$ is desired for more compact receiver, it is practical to have a larger $L_{s1}$ and smaller $L_{s2}$. Notice that $L_2\ne L_{s2}$, otherwise $I_1$ becomes irrelevant to k from Equation (6).

At light load, Equation (6) becomes a function of k. $\left|I_{10}\right|$ monotonically deceases when $k<k_b$ and increases otherwise. The boundary $k_b$ is obtained by:

$$\begin{array}{c}{\left.\frac{\mathrm{d}\left|I_{10}\right|}{\mathrm{d}k}\right|}_{k=k_b}={\left.\frac{\mathrm{d}\sqrt{A^2+B^2}}{\mathrm{d}k}\right|}_{k=k_b}=0\end{array}$$

(8)

$$\begin{array}{c}{k}_b=\frac{\sqrt{L_1{L}_2\left(L_2-L_{s2}\right)\left(L_1+L_{s1}\right)\left({\left(L_2-L_{s2}\right)}^2+\frac{L_{s2}^4{\omega }_0^2}{R_L^2}\right)}}{L_2(L_2-L_{s2})(L_1+L_{s1})}\approx \sqrt{\frac{L_1\left(L_2-L_{s2}\right)}{L_2\left(L_1+L_{s1}\right)}}\end{array}$$

(9)

When $k_b$ is not within the working range, $\left|I_{10}\right|\left(k\right)$ is a always a monotonic function. Therefore the inverse function $\widehat{k}\left(I_1\right)$ is single-valued and can be used to estimate k. Notice $k_b$ is also where the maximum $\mathrm{d}\left|I_{10}\right|/\mathrm{d}{R}_L$ lies. So $k_b$ should be put as far as possible from the possible range of k.

$I_1$ is extracted by measuring the amplitude of voltage across $L_{s1}$.

$$V_m=\left|I_{10}\right|{\omega }_0{L}_{s1}$$

(10)

$$\widehat{k}=\sqrt{-\frac{L_1{V}_{in}\left(V_m{L}_{s1}\left|L_2-L_{s2}\right|-V_{in}\left(L_1+L_{s1}\right)\left(L_2-L_{s2}\right)\right)}{L_2\left(L_1{V}_{in}+L_{s1}{V}_{in}+L_{s1}{V}_m\right)\left(L_1{V}_{in}+L_{s1}{V}_{in}-L_{s1}{V}_m\right)}}$$

(11)

Where $V_{in}$ and $V_m$ denotes the amplitude of the input and measured voltage respectively.

From Equation (5), the overall voltage gain at ${\omega }_0$ is a single-valued function of k. Therefore for any measured $\left|V_m\right|$ there is only one corresponding $G_{V0}$. Hence the received voltage $V_{out}$ can be estimated by Equation (12). The relationship between $V_m$, $G_{V0}$ and k is shown in Figure 2.

$${\widehat{V}}_{out}=V_{in}{G}_{v0}\left(\widehat{k}\left(V_m\right)\right)$$

(12)

Notice that Equation (11) and (12) only serve as a proof of the feasibility of the k estimation. It is recommended to plot ${\widehat{V}}_{out}\left(V_m\right)$ in experiment to cover the disturbance of parasite parameters, which will be discussed later.

2.3. Minimization of Effect of Frequency Error

By selecting compensation component value as given in Equation (4), it is straightforward to demonstrate that ${\omega }_0$ is the independent resonant frequency where $\Im \left(G_V\right)=0$. However, ${\omega }_0$ is not necessarily where the maximum voltage gain is achieved. A high Q-value in the LCL-compensated WPT circuit may results in significant overvoltage with minor deviations in frequency, posing risks especially in medical applications. Therefore, it is advisable to smoothen the gain curve around ${\omega }_0$.

To obtain $G_V$ as a function of $\omega$, expand $Z_i$ in Equation (3):

$$G_V^{-1}=\left(\begin{array}{cccccc}j{\omega }^5\hfill & {\omega }^4& j{\omega }^3& {\omega }^2& j{\omega }^1& 1\end{array}\right)\mathbf{W}$$

(13)

Where $\mathbf{W}$ is the weight matrix of the polynomial function $G_V^{-1}$. It is omitted here for simplicity. The analytical solution of peaks of $G_V^{-1}$ is hard to obtained, but it is still easy to investigate whether ${\omega }_0$ is one of the solution. With the resonance characteristic that $\Im \left(G_V\right)=0$, $\mathrm{d}\left|G_V\right|/\mathrm{d}\omega =0$ as long as $\Re \left(\mathrm{d}{G}_V^{-1}/\mathrm{d}\omega \right)=0$.

Obtain the derivative of Equation (13) to $\omega$, then substitute $\omega$ with ${\omega }_0$:

$$\begin{array}{cc}\hfill {\left.\frac{\mathrm{d}{G}_V^{-1}}{\mathrm{d}\omega }\right|}_{{\omega }_0}=& -\frac{2L_1^2{L}_{s2}-2L_2{L}_1^2+2L_{s1}{L}_1{L}_{s2}-2L_2{L}_{s1}{L}_1+2L_1{M}^2+4L_{s1}{M}^2}{L_1{L}_{s2}M{\omega }_0}\hfill \\ \hfill & +\frac{j{L}_1^2{L}_{s2}+j{L}_{s1}{L}_1{L}_{s2}+j{L}_{s1}{M}^2}{L_1{R}_{L}M}\hfill \end{array}$$

(14)

The real part of Equation (14) is variable with k. Thus ${\omega }_0$ cannot be a static peak. However, when an average coupling coefficient $k_0$ is determined, the inductors can be tuned to minimize the gain slope around ${\omega }_0$ as Equation (15). The effect is demonstrated in Figure 3.

$$\left(L_1+L_{s1}\right)\left(L_{s2}-L_2\right)+k_0^2{L}_2\left(L_1+2L_{s1}\right)=0$$

(15)

2.4. Analyzing the Effect of Parasite Resistance

Denote ESR of $L_1$ as $r_1$, and likewise $r_2$, $r_{s1}$ and $r_{s2}$. The output impedance of the power source is absorbed by $r_{s1}$. $r_{s2}$ and $R_L$ forms a frequency-independent voltage divider, hence temporarily assume $r_{s2}=0$ for simplicity. Keep the values of the LC components so that the Equation (4) still stands. The $\mathbf{Z}$ matrix in Equation (1) is re-written as Equation (16).

$$\mathbf{Z}=\left(\begin{array}{ccccc}j{Z}_1+r_{s1}& -\frac{1}{j\omega C_{p11}}& 0& 0& 0\\ -\frac{1}{j\omega C_{p11}}& j{Z}_2& -\frac{1}{j\omega C_{p12}}& 0& 0\\ 0& -\frac{1}{j\omega C_{p12}}& j{Z}_3+r_1& -j\omega M& 0\\ 0& 0& -j\omega M& j{Z}_4+r_2& -\frac{1}{j\omega C_{p2}}\\ 0& 0& 0& -\frac{1}{j\omega C_{p2}}& j{Z}_5+r_{s2}+R_L\end{array}\right)$$

(16)

Merge Equation (2), (4) and (16). $G_{v0}^{-1}$ with parasite resistance considered is obtained.

$$\begin{array}{cc}\hfill G_{V0}^{-1}=& -\frac{\frac{L_1^2{L}_{s2}^2\left(r_1+r_{s1}\right){\omega }_0^4}{R_L}+L_{s1}^2{M}^2{\omega }_0^4+L_1^2{r}_2{r}_{s1}{\omega }_0^2+L_{s1}^2{r}_1{r}_2{\omega }_0^2-Z_2{Z}_4{r}_1{r}_{s1}}{L_1{L}_{s1}{L}_{s2}M{\omega }_0^4}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & -j\frac{\frac{L_{s2}^2{Z}_2{r}_1{r}_{s1}{\omega }_0^2}{R_L}+L_1^2{Z}_4{r}_{s1}{\omega }_0^2+L_{s1}^2{Z}_4{r}_1{\omega }_0^2+M^2{Z}_2{r}_{s1}{\omega }_0^2+Z_2{r}_1{r}_2{r}_{s1}}{L_1{L}_{s1}{L}_{s2}M{\omega }_0^4}\hfill \end{array}$$

(18)

$$\frac{\mathrm{d}{G}_{V0}^{-1}}{\mathrm{d}{R}_L}=-\frac{1}{G_{V0}^2}\frac{\mathrm{d}{G}_{V0}}{\mathrm{d}{R}_L}=\frac{L_{s2}}{L_2{L}_{s1}M}\left(\frac{r_{s1}}{R_L^2}{L}_2^2+\frac{r_1}{R_L^2}{L}_{s1}^2+j\frac{r_1{r}_{s1}}{{\omega }_0^2{R}_L^2}{Z}_2\right)$$

(19)

Observed from Equation (19), the derivative of $G_{V0}^{-1}$ to $R_L$ is controlled by $r_x/R_L$. With K$\Omega$-level load and $r_x$ less than 10$\Omega$, the load adjustment rate $\mathrm{d}\left|{\mathrm{G}}_{\mathrm{V}0}\right|/\mathrm{d}{\mathrm{R}}_{\mathrm{L}}$ is normally less than 0.1%.

When take $r_{s2}$ into consideration, $G_{V0}$ remains the same except of an extra voltage-division factor $R_L/(R_L+r_{s2})$. $L_{s2}$ usually has a lower Q-factor compared to $L_{s1}$ for the strict size limitation of the implant receiver. However, for a reasonable $Q=25$ and $L_{s2}=1\mu$H at 10MHz, $r_{s2}=2.5\Omega$ is still much smaller than $R_L$, leading to a division factor of greater than $99.7\%$. Therefore the voltage gain at ${\omega }_0$ can still be considered as load-independent.

It is obvious from Equation (17) that $G_{V0}$ is still a single-valued function to k, but is no longer monotonic with parasite resistance. $\left|G_{V0}\right|$ first increases with k and then declines after a critical point $k_{crit}$, which is written as:

$$k_{crit}={\left(\frac{\begin{array}{c}{L}_2^4{Z}_4^2{r}_{s1}^2{\omega }_0^4+L_2^4{r}_2^2{r}_{s1}^2{\omega }_0^4+2L_2^2{L}_{s1}^2{Z}_4^2{r}_1{r}_{s1}{\omega }_0^4+2L_2^2{L}_{s1}^2{r}_1{r}_2^2{r}_{s1}{\omega }_0^4\hfill \\ +L_{s1}^4{Z}_4^2{r}_1^2{\omega }_0^4+L_{s1}^4{r}_1^2{r}_2^2{\omega }_0^4+Z_2^2{Z}_4^2{r}_1^2{r}_{s1}^2+Z_2^2{r}_1^2{r}_2^2{r}_{s1}^2\hfill \end{array}}{L_1^2{L}_2^2\left(L_{s1}^4{\omega }_0^8+Z_2^2{r}_{s1}^2{\omega }_0^4\right)}\right)}^{1/4}$$

(20)

The intermediate derivation is omitted for simplicity. From Equation (20), The CV control is made easier as the gain around $k_{crit}$ becomes smoother.

Additionally, for $I_{10}$ which is used to extract k, $r_{s2}$ is indivisible from $R_L$. Equation (7) is re-written as:

$$\frac{\mathrm{d}{I}_{10}}{\mathrm{d}{R}_L}=-\frac{L_1^2{L}_{s1}^2{L}_{s2}^2{M}^2{\omega }_0^8}{{\left(\begin{array}{c}{L}_{s1}^2{M}^2{R}_L{\omega }_0^4+L_1^2{L}_{s2}^2{r}_{s1}{\omega }_0^4+L_{s1}^2{L}_{s2}^2{r}_1{\omega }_0^4+L_1^2{R}_L{r}_2{r}_{s1}{\omega }_0^2\hfill \\ +L_{s1}^2{R}_L{r}_1{r}_2{\omega }_0^2-R_L{Z}_2{Z}_4{r}_1{r}_{s1}+j{L}_{s2}^2{Z}_2{r}_1{r}_{s1}{\omega }_0^2+j{L}_1^2{R}_L{Z}_4{r}_{s1}{\omega }_0^2\hfill \\ +j{L}_{s1}^2{R}_L{Z}_4{r}_1{\omega }_0^2+j{M}^2{R}_L{Z}_2{r}_{s1}{\omega }_0^2+j{R}_L{Z}_2{r}_1{r}_2{r}_{s1}\hfill \end{array}\right)}^2}$$

(21)

With $r_1$, $r_2$ and $r_{s1}$ join the denominator, $I_{10}$ becomes less sensitive to $R_L$.

In conclusion, the impact of parasite resistance on resonant frequency and k estimation is negligible in practice. While the exact number of $G_V$ does change, it can be easily tracked with experiment measurement and numerical simulation. Moreover, $r_2$ is irrelevant to the resonant frequency, allowing a relatively low Q-value for the receiving coil.

2.5. Design Procedure

The circuit in Figure 1b has seven parameters to be determined. Given that the effect of the parasite resistance is limited, the circuit can be assumed as ideal when select the LC values. Based on experiment result, iteration may be needed to account for parasite parameters, including the stray capacitance of the coils.

The coil $L_1$ and $L_2$ should be determined before designing the LCC compensation circuit. The parasite parameters of coils and their coupling coefficient range is then measured. The last 5 components can be solved by Equation (4),(5) and (15) to make the circuit resonant at ${\omega }_0$, smoothen the gain slope and acquire desired gain at $k_0$. The actual capacitors are selected as $C_{p1}^{\prime }=C_{p11}+C_{p12}+C_{coil1}$ and $C_{p2}^{\prime }=C_{p2}+C_{coil2}$. A numerical simulation based on Equation (3) to check weather the desired gain is achieved. when not, $L_{s2}$/$L_{s1}$ and the corresponding resonant capacitor must be adjusted. If the adjusted value is beyond acceptance, then roll-back to the coil designing.

Figure 4. Design flowchart

Figure 4. Design flowchart

3. Results

3.1. Validation of LCL-Compensated WPT and Voltage Gain Estimation

An experiment platform of LCL-Compensated WPT circuit that works on 6.78MHz ISM band is constructed, shown in Figure 5. The coils are implemented on FR4 PCBs and of circular shape. The detailed parameters are listed in Table 1 and Table 2. To adjust the coupling coefficient, i.e. the distance between the coils, the PCBs are fixed on a 3D-printed PLA material base, with the length of the nylon columns between them adjustable.

The transmitter is driven by signal generator via SMA cable. As 0805 chip inductors are used in the compensation circuit, the driving voltage is set to 300mVpp. $V_m$ and $V_{out}$ are monitored by oscilloscope. To cover the internal resistance of the signal generator, $V_{in}$ is also monitored, and $V_m$ data is normalized as $V_m/V_{in}$.

Two sets of data are recorded in experiment, with $1K\Omega$ pure resistive load and open-circuit respectively. The recorded result and theoretical $V_m-\left|G_V\right|$ curves are plotted in Figure 6.

4. Discussion

4.1. Discussion of experiment result

The recorded data in Figure 6 shows that the $V_m-G_V$ characteristic of experiment circuit follows the identical trend as that of the mathematical model proposed in section 2. The effect of load changes on the $G_V$ estimation is limited, but still observable. Data points of $1K\Omega$ load deviate from the theoretical curve farther, as the parasite parameters vary at the frequency higher than those at which they were measured.

Noises occurred in the measurement has affected the recorded data, creating inconsistent points. Such noises are expected to appear in applications as well. Averaging may be required to ensure the accuracy of the $G_V$ estimation. Additionally, as the independency of resonant voltage gain is slightly weaker than expected due to complex parasite parameters, $V_m$ should be measured during intervals of transmission with lower $V_{in}$, based on Equation (7). The new estimation strategy is shown in Figure 7.

4.2. Application

The proposed CV-WPT strategy is planned to be implemented in an implant neural stimulating system shown in Figure 8. To achieve bipolar current stimulation, the system uses amplitude modulation (AM) to transfer power. A threshold voltage is established in the implant stimulator. When the demodulated AM signal surpasses the threshold, the stimulator outputs positive current and vice versa. This procedure is referred as Polarity Recovery. The aforementioned CV-WPT techniques provides robust and precise control with varying k. We use a linear transmission circuit built by Direct Digital Synthesis (DDS) and High-Power Output Current Feedback Amplifier (CFA) for better accuracy, with a trade-off of efficiency.

The MCU uses two LUTs to estimate the received voltage. First of them is derived from Equation (11), it estimates $\widehat{k}$ basing on measured $V_m$. The other is derived from Equation (18) and outputs the corresponding ${\widehat{G}}_V$ of given $\widehat{k}$. The desired $V_{in}$ is calculated by $V_{in}=V_{set}/{\widehat{G}}_V$, and sent to the DDS. The intermediate $\widehat{k}$ is also used to detect misalignment or fault.

5. Conclusion

The LCL-Compensated WPT circuit is re-analysis for constant voltage transmission. The proposed CV-WPT system uses voltage measures on the transmitting side to estimate the coupling coefficient and voltage gain. Its feasibility is proved and the load-independent characteristic is validated through experiment. Compared to existing methods, our approach does not require frequency sweep or bulky components such as RF relays. The tuning strategy helps further shrink the size of the implant receiver.