A Design Methodology for EV-WPT Systems to Resonate at Arbitrary Given Bands

Abstract

Due to the effects of splitting frequency and cross coupling, the resonant frequency of the WPT system usually deviates from the given frequency band, and the system operating at the given frequency band suffers a very low output power. Ensuring that electric vehicle wireless power transfer (EV-WPT) systems operate at a resonant state is the prerequisite for efficient energy transfer. For this purpose, a novel design method by manipulating the eigenstate parameters is proposed in this paper. The proposed system can make a EV-WPT system with arbitrary coil successfully to resonate at any given bands, not just a single band. Therefore, the method designed in this article cannot only eliminate the problem of low power caused by frequency deviation, but also realize the application requirements of multiple frequency bands. Firstly, this article establishes an accurate state space model of an n-coil fully coupled EV-WPT system, and after that, the analytical current response on each circuit is derived. Based on that, a detailed frequency spectrum analysis is presented, along with several essential spectrum parameters’ derivations, including center frequencies and bandwidths. Then, with the center frequency and bandwidth as the design indexes, a novel methodology of designing to make EV-WPT systems achieve resonant-state at arbitrary given bands is derived. Finally, simulation and experimental verification are carried out. Simulation and experimental results show that whether it is a single-band or multi-band system, the accuracy of the value under designed resonant frequency is less than 0.01, which can effectively eliminate the frequency deviation phenomenon and obtain the maximum power output at the given frequency band.

1. Introduction

Magnetic coupling wireless power transfer (MC-WPT) system transfer energy from power sources to electric loads by coupled magnetic field. Due to the electric isolation between the sources and loads, this technology has advantages such as enough flexibility, full reliability, and high security. New research hotspots have emerged since the inspiring work in which researchers transferred 60 W power to a lamp over a distance of 2 m in 2007 [1,2]. Considerable research of this technology developed over the past decades has paved the way towards practical applications of this technology, such as electric vehicles (EV) [3,4], consumer electronics [5,6], and so on. With the development of MC-WPT technology and the expansion of the application field, more complicated multi-coils MC-WPT systems, such as multi-transmitter systems [7,8], multi-receiver systems [8,9], and multi-repeater systems [10,11,12], has received great attention in recent years.

For an electric vehicle wireless power transfer (EV-WPT) system, multiple industry consortia are developing corresponding standards for the components of EV-WPT systems, including the standard for operating frequency [13,14,15]. It is well known that the overwhelming majority of EV-WPT system designs are on the basis of the “resonant coupling principle” proposed in paper [1], which aims to make the natural frequency of each oscillating circuit in the system and the operating frequency of the system equal to the given frequency for the maximum power transfer [16,17,18]. However, since the phenomenon termed as “frequency splitting”, there are multiple resonant points deviated from the designed circuits’ natural frequency. Hence, the system operating at the given frequency suffers a much lower energy efficiency [19,20,21]. This ensures that the system operating at resonant state is a key point to efficient power transfer [22].

At present, frequency tracking [23], adding impedance matching network [24,25] and adjusting the coupling strength [19,20] are the most common methods used to compensate the energy efficiency against the frequency splitting. However, frequency tracking makes the system capable of operating at a varying optimal frequency, but not a given frequency. The essence of adding an impedance matching network and adjusting the coupling strength is adjusting some inner parameters to eliminate frequency splitting phenomenon. Since the number and position of the resonant points are determined by several parameters such as self-inductances, capacitors and resistances, the specific impedance matching network or adjusting method of coupling strength is different from system to system, and then they are difficult to apply in multi-coil EV-WPT systems. So far, there are no systematic design methods for the EV-WPT system to achieve resonant-state at a given band, which is a bottleneck problem that needs to be solved.

In addition, the multi-band EV-WPT systems have been receiving greater attention in recent years. The multi-band EV-WPT systems can not only satisfy multiple frequency standards, but also can transfer power and data simultaneously. The traditional multi-band EV-WPT systems are usually achieved by subjoining extra LC circuits [26,27] or extra coils [28,29,30]. The LC circuits or coils must increase with the increase of bands, which will make the system become complicated. Since the traditional design process of multi-band EV-WPT systems is based on the actual characteristic equation of maximum power transfer points versus the operating frequency, it is only suitable for simple EV-WPT systems with a few bands, but not for complicated multi-coil or multi-band EV-WPT systems.

In this paper, a novel design method by manipulating the eigen parameters is proposed in this paper. The proposed system can design an arbitrary coil EV-WPT system to resonate at arbitrary given bands, thereby not only eliminating the problem of low power caused by frequency deviation, but also realizing the application requirements of multiple frequency bands. The proposed method in this paper does not have to add any extra LC circuits or coils, and it suits arbitrary coil EV-WPT systems with arbitrary given bands (no more than the coils). Moreover, unlike the traditional design method of EV-WPT systems, in which the design indexes usually only consider the center positions of bands, here, the indexes of the proposed method include center frequency and bandwidth. Since the purpose of this paper is to design lumped parameters, not geometric parameters, there is no in-depth discussion on coil geometric parameters, voltage and load characteristic.

The arrangement of this paper is as follows: In Section 2, the state-space model of a general n-coil fully-coupled EV-WPT system with any number of transmitters, repeaters and receivers is established, and the current response on each circuit is derived at the same time. Based on that, the analytical expression of frequency spectrum parameters, including center frequencies and corresponding bandwidths, is deduced. In Section 3, the detailed design process with the center frequencies and bandwidths as the design indexes is presented, and then a systematic design criterion which suits designing arbitrary coil EV-WPT systems to resonate at arbitrary given bands is derived. According to the derived design criteria, a practical two-coil EV-WPT system with dual-band and single band is designed in Section 4. Finally, in Section 5, this article validates the proposed design method by building prototypes and tested it.

2. Theoretical Analysis

2.1. The Necessary Concepts

Before the theoretical analysis, several essential concepts related to the frequency spectrum of EV-WPT systems need to be clarified:

• Frequency spectrum curve: Frequency spectrum curve is the curve of output power versus the operating frequency, which usually is a multi-peak curve.
• Band: The frequency range when the current amplitude is reduced to 0.707 times the maximum current amplitude (the power is reduced to the half of the maximum output power), referred to as the band.
• Center frequency: The frequency of the peak is termed as the center frequency of the band.
• Multi-band system: Multi-band system means that the frequency spectrum of the system is a multi-peak curve. The number of peaks equal to the number of bands. Single-band system indicates that the frequency spectrum curve is a single-peak curve, dual-band system indicates that the frequency spectrum curve is a dual-peak curve, and tri-band system indicates that the frequency spectrum curve is a tri-peak curve.
• Cut-off frequencies: the frequencies at which the current amplitudes equal to 0.707 times the maximum current amplitude in a band term as the cut-off frequencies of the band. In a band, there are usually two half-power frequencies, one of which is greater than the center frequency, and the other is less than the center frequency.
• Bandwidth: the frequency range of the two cut-off frequencies is termed as the bandwidth of the band.

2.2. System Modeling

Figure 1 shows the equivalent circuit diagram of an n-coil fully coupled EV-WPT system with arbitrary number of transmitters, repeaters and receivers. The system consists of n coils and each of these is connected with a capacitor in series, resulting in forming an oscillating circuit.

In Figure 1, the subscript m represents the parameters in the m-th oscillating circuit (m = 1, 2, …, n). R_pm and L_m represent the equivalent series internal resistance (ESR) and self-inductance of the m-th coil respectively, M_mr is defined as the mutual inductance between the m-th coil and r-th coil (m ≠ r and M_mr = M_rm). C_m and u_m stand for the compensating capacitor and corresponding capacitor voltage of the m-th circuit. v_m is the supply voltage and R_Lm is the load’s resistance of the m-th oscillating circuit, respectively. It must be noticed that v_m and R_Lm cannot exist simultaneously. The role of the coil varies with the values of v_m and R_Lm. The role of the coil varies with the values v_m and R_Lm, as shown in

Table 1. The role of coils.

Condition	The m-th Circuit
vm ≠ 0 and RLm = 0	transmitter
vm = 0 and RLm ≠ 0	receiver
vm = 0 and RLm = 0	repeater

.

For mathematic analysis, assume that the power is a sinusoidal power supply voltage with an angular frequency of ω. Based on the Kirchhoff’s voltage law (KVL), the mathematic model of the system in Figure 1 can be established as

$$\{\begin{array}{c}{L}_1{C}_1{\ddot{u}}_1+M_{12}{C}_2{\ddot{u}}_2+\cdots M_{1n}{C}_n{\ddot{u}}_n+R_1{C}_1{\dot{u}}_1+u_1=v_1\\ M_{12}{C}_1{\ddot{u}}_1+L_2{C}_2{\ddot{u}}_2+\cdots M_{2n}{C}_n{\ddot{u}}_n+R_2{C}_2{\dot{u}}_2+u_2=v_2\\ ⋮\\ M_{1n}{C}_1{\ddot{u}}_1+M_{2n}{C}_2{\ddot{u}}_2+\cdots L_n{C}_n{\ddot{u}}_n+R_n{C}_n{\dot{u}}_n+u_n=v_n\end{array}$$

(1)

where R_m = R_pm + R_Lm (m ≠ n) and R_m = R_pm + R_Lm (m = n).

Let $v={\left[\begin{array}{cccc}{v}_1& v_2& \cdots & v_n\end{array}\right]}^T$, $i={\left[\begin{array}{cccc}{i}_1& i_2& \cdots & i_n\end{array}\right]}^T$ and $u={\left[\begin{array}{cccc}{u}_1& u_2& \cdots & u_n\end{array}\right]}^T$ be the supply voltage vector, current vector and voltage vector for capacitor in the system illustrated above, respectively. Moreover, $x={[\begin{array}{cc}{u}^T& {\dot{u}}^T\end{array}]}^T$ and $v_{in}={\left[\begin{array}{cc}{v}^T& 0_{1\times n}\end{array}\right]}^T$ are defined as the state vector and input vector of the system respectively; the model established in Equation (1) can be equivalent to the new formula as

$$\mathbf{A}\dot{\mathbf{x}}+\mathbf{B}\mathbf{x}=v_{in}$$

(2)

where

$$\begin{array}{cc}A=\left[\begin{array}{cc}RC& LC\\ LC& 0_{n\times n}\end{array}\right],& \mathbf{B}=\left[\begin{array}{cc}{I}_{n\times n}& 0_{n\times n}\\ 0_{n\times n}& -LC\end{array}\right]\end{array}$$

(3)

$0_{n\times m}$ stands for $n\times m$ order zero matrix and $I_{n\times m}$ is expressed as a $n\times m$ unit matrix. L, C and R are the inductance matrix, capacitor matrix and resistance matrix of the system, which are given by

$$\begin{array}{l}\begin{array}{lll}L=\left[\begin{array}{cccc}{L}_1& M_{12}& \cdots & M_{1n}\\ M_{21}& L_2& \cdots & M_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ M_{n1}& M_{n2}& \cdots & L_n\end{array}\right],\hfill & C=\left[\begin{array}{cccc}{C}_1& 0& \cdots & 0\\ 0& C_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & C_n\end{array}\right],\hfill & R=\hfill \end{array}\hfill \end{array}\left[\begin{array}{cccc}{R}_1& 0& \cdots & 0\\ 0& R_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & R_n\end{array}\right]$$

(4)

Equation (2) refers to a state space model. From Equation (2), the system matrix (S) can be derived as

$$S=-A^{-1}B=\left[\begin{array}{cc}0& I_{n\times n}\\ -{\left[LC\right]}^{-1}& -{\left[LC\right]}^{-1}RC\end{array}\right]$$

(5)

2.3. Frequency Spectrum Analysis

According to [21], we can find out that there are at most n bands in an n-coil EV-WPT systems. If the number of bands is less than the number of coils in a EV-WPT system, it indicates that in the system exists a band overlap phenomenon. In general, for an actual physical system, even if the center frequencies are very close, the overlapping bands do not completely coincide because the bandwidths and peak heights are difficult to be equal completely at the same time. In other words, the system matrix of an actual physical system usually does not have the exact same eigenvalues and it can be normalized to a diagonal matrix. In this case, the current response i_m in the overlapping bands is superimposed by multiple vectors with different amplitudes and phases [21], and then the bandwidths of the overlapping bands are difficult to determine.

Aiming to quickly and accurately determine the bandwidths of bands and simplify the design process, in this paper, the overlapping bands are designed to be coincident completely and the non-overlapping bands are designed to be separated from each other. The existence of completely overlapping bands indicates that the system matrix has identical eigenvalues. In this case, the system matrix cannot be normalized to a diagonal matrix but to a Jordan matrix. Supposing that there are k $\left(k\le n\right)$ different eigenvalues labeled as ${\lambda }_1,{\lambda }_2,\cdots {\lambda }_k$, and the multiplicity of eigenvalue ${\lambda }_q$ is $r_q$ $\left(q=1,2,\cdots k\right)$, then we can get that ${\displaystyle \sum _{q=1}^k{r}_q}=n$. On the basis of the practical system’s physical properties, the real part of eigenvalue ${\lambda }_q$ is a negative number with a small absolute value [31]. Then ${\lambda }_q$ and ${\lambda }_q^{*}$ can be written as two conjugate complex numbers ${\lambda }_q=-{\alpha }_q+j{\omega }_q$ and ${\lambda }_q^{*}=-{\alpha }_q-j{\omega }_q$, where ${\alpha }_q$ and ${\omega }_q$ are positive number, moreover, ${\alpha }_q$ is ordinarily pretty small.

Under this circumstance, according to the matrix theory, we can get that there are two non-singular matrixes $\mathbf{\Lambda }=diag{\left[\begin{array}{llllllll}{\mathbf{\Lambda }}_1\hfill & {\mathbf{\Lambda }}_2\hfill & \cdots \hfill & {\mathbf{\Lambda }}_k\hfill & {\mathbf{\Lambda }}_1^{*}\hfill & {\mathbf{\Lambda }}_2^{*}\hfill & \cdots \hfill & {\mathbf{\Lambda }}_k^{*}\hfill \end{array}\right]}_{2n\times 2n}$ and $\Phi ={\left[{\phi }_{ij}\right]}_{2n\times 2n}$, which makes $S\Phi =\Phi \Lambda$. The operator $diag\left[\right]$ represents the generation of an diagonal matrix with the elements in the parentheses as the diagonal elements, ${\Lambda }_q$ and ${\Lambda }_q^{*}$ are conjugate complex matrix and ${\Lambda }_q$ is a Jordan matrix as shown in Equation (6).

$${\Lambda }_q={\left[\begin{array}{lllll}{\lambda }_q\hfill & 1\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ 0\hfill & {\lambda }_q\hfill & 1\hfill & \cdots \hfill & 0\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ 0\hfill & 0\hfill & 0\hfill & \cdots \hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill & \cdots \hfill & {\lambda }_q\hfill \end{array}\right]}_{r_q\times r_q}$$

(6)

According to $S\Phi =\Phi \Lambda$, the state space model in Equation (2) can be equivalent to the new formula as

$$\dot{y}=\Lambda y+v^{\prime }$$

(7)

where $y={\Phi }^{-1}x$ and $v^{\prime }={\Phi }^{-1}{A}^{-1}{v}_{in}$. According to $\dot{y}=\Lambda y+v^{\prime }$, $x={[\begin{array}{cc}{u}^T& {\dot{u}}^T\end{array}]}^T=\Phi y$ and the relationship between current vector $i$ and capacitor voltage current $\dot{u}$ ($i=C\dot{u}$), the current response on the m-th circuit (i_m) can be derived as

$$i_m=\mathrm{Im}\left(I_m{e}^{j\left(\omega t+\pi /2\right)}\right)$$

(8)

where

$$\{\begin{array}{c}{I}_m={\displaystyle \sum _{q=1}^k{}_q{}^{}{I}_m_{}^{}}\\ {}_q{}^{}{I}_m_{}^{}=\frac{R_{mq}}{{\left(\frac{{\mathsf{\Omega }}_q^2}{\omega }-\omega +j2{\alpha }_q\right)}^{r_q}}\end{array}$$

(9)

The derivation process of i_m in Equation (8) is given in Appendix A. The operator $I_m\left(\right)$ represent the imaginary part of the variable value, ${\mathsf{\Omega }}_q=\sqrt{{\alpha }_q^2+{\omega }_q^2}$ and $R_{mq}$ is a plural as shown in Equation (A11) in the Appendix A. Equation (8) revealed that the amplitude of current $i_m$ is equal to the modulus of plural $I_m$, that is $\left|i_m\right|=\left|I_m\right|$. Meanwhile, from Equation (9) we can know that plural $I_m$ is superimposed by the k vectors ${}_q{}^{}{I}_m_{}^{}\left(q=1,2,\cdots k\right)$ and the amplitude of ${}_q{}^{}{I}_m_{}^{}$ can be deduced as

$$\left|{}_q{}^{}{I}_m_{}^{}\right|=\frac{\left|R_{mq}\right|}{{\left({\left(\frac{{\mathsf{\Omega }}_q^2}{\omega }-\omega \right)}^2+{\left(2{\alpha }_q\right)}^2\right)}^{\frac{r_q}{2}}}$$

(10)

$\left|{}_q{}^{}{I}_m_{}^{}\right|$ will achieve the maximum value ${\left|{}_q{}^{}{I}_m_{}^{}\right|}_{\max }$ while the operating angular frequency ω equals to ${\mathsf{\Omega }}_q$ ($\omega ={\mathsf{\Omega }}_q$), i.e.,

$${\left|{}_q{}^{}{I}_m_{}^{}\right|}_{\max }={\left|{}_q{}^{}{I}_m_{}^{}\right|}_{\omega ={\mathsf{\Omega }}_q}=\frac{\left|R_{mq}\right|}{{\left(2{\alpha }_q\right)}^{r_q}}$$

(11)

Since ${\alpha }_q$ is generally pretty small, ${\left|{}_q{}^{}{I}_m_{}^{}\right|}_{\max }$ will be pretty large. In addition, since the overlapping bands are designed to be coincident completely and non-overlapping bands are designed to be separated from each other, that is, the k different eigenvalues ${\lambda }_1,{\lambda }_2,\cdots {\lambda }_k$ (${\mathsf{\Omega }}_1,{\mathsf{\Omega }}_2,\cdots {\mathsf{\Omega }}_k$) are far apart from each other, hence $\left|{}_q{}^{}{I}_m_{}^{}\right|$ will be much larger than other items $\left|{}_q{}^{}{I}_m_{}^{}\right|$ (j = 1, 2, …, k and j ≠ q) when the operating angular frequency ω is close to ${\mathsf{\Omega }}_q$. In this case, $I_m$ can be approximate as

$${I_m|}_{\omega \to {\mathsf{\Omega }}_q}\approx {}_q{I}_m$$

(12)

The analysis indicates that there are k bands in the system, whose center angular frequencies are ${\mathsf{\Omega }}_1$, ${\mathsf{\Omega }}_2$, …, ${\mathsf{\Omega }}_k$. The plural $I_m$ in the q-th band can be approximated as ${I_m|}_{\omega \to {\mathsf{\Omega }}_q}{\approx }_q{I}_m$. From Equation (10) we can obtain that the amplitude of ${}_q{I}_m$ versus the operating angular frequency ω is an unimodal curve, which can be roughly plotted as

In Figure 2, the points a and b are half-power points. On the basis of the definition of half-power point as elucidated in Section 2.1, we have

$${\left|{}_q{I}_m\right|}_{\omega ={\omega }_{q,-}}={\left|{}_q{}^{}{I}_m_{}^{}\right|}_{\omega ={\omega }_{q,+}}=\frac{1}{\sqrt{2}}{\left|{}_q{}^{}{I}_m_{}^{}\right|}_{\max }$$

(13)

According to Equations (10), (11) and (13), the frequencies of the two half-power points can be derived as shown in Equation (14). The derivation process is given in Appendix B.

$$\{\begin{array}{c}{\omega }_{q,+}=\sqrt{\left(2^{1/r_q}-1\right){\alpha }_q^2+{\mathsf{\Omega }}_q^2}+\left(\sqrt{2^{1/r_q}-1}\right){\alpha }_q\\ {\omega }_{q,-}=\sqrt{\left(2^{1/r_q}-1\right){\alpha }_q^2+{\mathsf{\Omega }}_q^2}-\left(\sqrt{2^{1/r_q}-1}\right){\alpha }_q\end{array}$$

(14)

Then the bandwidth B_q can be deduced as

$$B_q={\omega }_{q,+}-{\omega }_{q,-}=2\left(\sqrt{2^{1/r_q}-1}\right){\alpha }_q$$

(15)

Then we can get that, for a EV-WPT system with k different and separate eigenvalues ${\lambda }_q=-{\alpha }_q+j{\omega }_q$ $\left(q=1,2,\dots k\right)$, there are k-bands in the system, whose center angular frequency ${\omega }_{o,q}$ and corresponding bandwidth $B_q$ of the q-th band can be expressed as

$$\begin{array}{cc}\{\begin{array}{l}{\omega }_{o,q}={\mathsf{\Omega }}_q\hfill \\ B_q=2\left(\sqrt{2^{1/r_q}-1}\right){\alpha }_q\hfill \end{array}& \left(q=1,2,\dots k\right)\end{array}$$

(16)

3. Method to Design of Multi-Band EV-WPT Systems

It is well known that the center angular frequency and the bandwidth are two key indexes of frequency band. On the basis of the analysis in Section 2, we can know that the number of bands as well as the corresponding center frequencies and bandwidths of a EV-WPT system are determined by the values under eigenstate of the system matrix. Therefore, in turn, suitable eigenvalues can be determined according to the given center frequencies and bandwidths.

Supposing that k-band is required in an n-coil EV-WPT system and the required center frequencies and corresponding bandwidths are ${\omega }_{o,1},{\omega }_{o,2},\cdots {\omega }_{o,k}$ and $B_1,B_2,\cdots ,B_k$ respectively (k ≤ n). On the basis of the analysis in Section 2 we can be aware of that k bands indicate that there are k different eigenvalues of the system matrix (${\lambda }_1,{\lambda }_2,\cdots {\lambda }_k$). Supposing that the multiplicity of the eigenvalue ${\lambda }_q$ is $r_q$, which can be determined by the designer on the premise of ${\displaystyle \sum _{q=1}^k{r}_q}=n$. In this case, according to the given center angular frequency (ω_o,q) and it’s homologous bandwidth (B_q) of the q-th band, the suitable eigenvalues ${\lambda }_q={\alpha }_q+j{\omega }_q$ and ${\lambda }_q^{*}={\alpha }_q-j{\omega }_q$ can be quickly obtained as shown in Equation (17) by using Equation (16).

$$\{\begin{array}{cc}\begin{array}{l}{\mathsf{\Omega }}_q={\omega }_{o,q}\hfill \\ {\alpha }_q=\frac{B_q}{2\left(\sqrt{2^{1/r_q}-1}\right)}\hfill \\ {\omega }_q=\sqrt{{\mathsf{\Omega }}_q^2-{\alpha }_q^2}\hfill \end{array}& \left(q=1,2,\cdots k\right)\end{array}$$

(17)

By using Equation (17), all eigenvalues satisfying the performance requirements can be quickly and accurately determined. Then the next step is to design the system parameters so that the eigenvalues of the system matrix equals to the expected values as shown in Equation (17).

According to the derived k different eigenvalues ${\lambda }_1,{\lambda }_2,\cdots {\lambda }_k$ that are as shown in Equation (17), the expected characteristic formula which satisfy the performance requirements can be obtained as

$$\begin{array}{cc}f{\left(\lambda \right)}_{expcted}& ={\left(\lambda -{\lambda }_1\right)}^{r_1}{\left(\lambda -{\lambda }_1^{*}\right)}^{r_1}\cdots {\left(\lambda -{\lambda }_k\right)}^{r_k}{\left(\lambda -{\lambda }_n^{*}\right)}^{r_k}\\ & ={\lambda }^{2n}+a_{2n-1}{\lambda }^{2n-1}+a_{2n-2}{\lambda }^{2n-2}\cdots a_1\lambda +a_0=0\end{array}$$

(18)

In addition, on the basis of the definition of eigenvalues, we have

$$\left|S-\lambda I_{n\times n}\right|=0$$

(19)

Substituting Equation (5) into Equation (19), we can get that

$$\left|\begin{array}{cc}-\lambda I_{n\times n}& I_{n\times n}\\ -{\left[LC\right]}^{-1}& -{\left[LC\right]}^{-1}RC-\lambda I_{n\times n}\end{array}\right|=0$$

(20)

Evaluating Equation (20) by block matrices, we can get

$$\left|{\lambda }^2\left[LC\right]+\lambda \left[RC\right]+I_{n\times n}\right|=0$$

(21)

Replacing the inductance matrix L, capacitor matrix C and resistance matrix R into Equation (21), the characteristic formula of the system can be deduced as

$$f{\left(\lambda \right)}_{actual}={\lambda }^{2n}+b_{2n-1}{\lambda }^{2n-1}+b_{2n-2}{\lambda }^{2n-2}\cdots b_1\lambda +b_0=0$$

(22)

By comparing Equation (18) with Equation (22), we have

$$\{\begin{array}{c}{a}_{2n-1}=b_{2n-1}\\ a_{2n-2}=b_{2n-2}\\ ⋮\\ a_1=b_1\\ a_0=b_0\end{array}$$

(23)

From Equation (23), the suitable system parameters that satisfy the performance requirements can be quickly and accurately determined.

According to the analysis referred above, the design criteria of multi-band EV-WPT systems can be summarized as

S1.

Determining the number of coils (n), inductance matrix L, capacitor matrix C and resistance matrix R according to the practical application, where the parameters need to be designed are represented by corresponding symbols as shown in Figure 1.

S2.

According to the application requirement, determining the number of bands (k) and corresponding center frequencies ${\omega }_{o,1},{\omega }_{o,2},\cdots {\omega }_{o,k}$ and bandwidths $B_1,B_2,\cdots ,B_k$.

S3.

Determining the multiplicity $r_q$ of the q-th band. The value of r_q is freely determined by the designer, but it need to satisfy ${\displaystyle \sum _{q=1}^k{r}_q}=n$.

S4.

Determining the expected eigenvalues (${\lambda }_q={\alpha }_q+j{\omega }_q$ and ${\lambda }_q^{*}={\alpha }_q-j{\omega }_q$) of the q-th band by using

$$\{\begin{array}{cc}\begin{array}{l}{\mathsf{\Omega }}_q={\omega }_{o,q}\hfill \\ {\alpha }_q=\frac{B_q}{2\left(\sqrt{2^{1/r_q}-1}\right)}\hfill \\ {\omega }_q=\sqrt{{\mathsf{\Omega }}_q^2-{\alpha }_q^2}\hfill \end{array}& \left(q=1,2,\cdots k\right)\end{array}$$

(24)

S5.

Calculating the expected characteristic equation by using

$$\begin{array}{cc}f{\left(\lambda \right)}_{expcted}& ={\left(\lambda -{\lambda }_1\right)}^{r_1}{\left(\lambda -{\lambda }_1^{*}\right)}^{r_1}\cdots {\left(\lambda -{\lambda }_k\right)}^{r_k}{\left(\lambda -{\lambda }_n^{*}\right)}^{r_k}\\ & ={\lambda }^{2n}+a_{2n-1}{\lambda }^{2n-1}+a_{2n-2}{\lambda }^{2n-2}\cdots a_1\lambda +a_0=0\end{array}$$

(25)

S6.

Substituting matrix L, C and R into $\left|{\lambda }^2\left[LC\right]+\lambda \left[RC\right]+I_{n\times n}\right|=0$, the actual characteristic formula of the system can be calculated as

$$f{\left(\lambda \right)}_{actual}={\lambda }^{2n}+b_{2n-1}{\lambda }^{2n-1}+b_{2n-2}{\lambda }^{2n-2}\cdots b_1\lambda +b_0=0$$

(26)

S7.

Then the parameters in expected system can be defined by using

$$\{\begin{array}{c}{a}_{2n-1}=b_{2n-1}\\ a_{2n-2}=b_{2n-2}\\ ⋮\\ a_1=b_1\\ a_0=b_0\end{array}$$

(27)

4. Practical System Design

In Section 3, the design method for EV-WPT systems to achieve the state of resonance at given frequencies is proposed, which is suitable for arbitrary numbers of coils EV-WPT systems with arbitrary number of resonant points (no more than the coils). Aiming to verify the method proposed above, designing examples of tri-coil EV-WPT systems are illustrated in this Section.

4.1. The Nonlinear Programming Model of Tri-Coil EV-WPT Systems

According to the above analysis, we can obtain that a tri-coil EV-WPT system can be designed as the system with a single given resonant point, two given resonant points, or three given resonant points. No matter the system with how many resonant points, the design steps are the same expected for seeking the expected eigenvalues.

Supposing that the expected eigenvalues of system matrix of the tri-coil EV-WPT systems which determined by practical performance requirements are ${\lambda }_1=-{\alpha }_1+j{\omega }_1$, ${\lambda }_2=-{\alpha }_2+j{\omega }_2$, ${\lambda }_3=-{\alpha }_3+j{\omega }_3$ and ${\lambda }_1^{*}=-{\alpha }_1-j{\omega }_1$, ${\lambda }_2^{*}=-{\alpha }_2-j{\omega }_2$, ${\lambda }_3^{*}=-{\alpha }_3-j{\omega }_3$. Substituting the expected eigenvalues into Equation (18), the characteristic equation of the system can be derived as

$$f{\left(\lambda \right)}_{expcted}={\lambda }^6+a_5{\lambda }^5+a_4{\lambda }^4+a_3{\lambda }^3+a_2{\lambda }^2+a_1\lambda +a_0=0$$

(28)

where

$$\{\begin{array}{l}{a}_5=2{\alpha }_1+2{\alpha }_2+2{\alpha }_3\hfill \\ a_4=4{\alpha }_1{\alpha }_2+4{\alpha }_1{\alpha }_3+4{\alpha }_2{\alpha }_3+{\mathsf{\Omega }}_1^2+{\mathsf{\Omega }}_2^2+{\mathsf{\Omega }}_3^2\hfill \\ a_3=8{\alpha }_1{\alpha }_2{\alpha }_3+2{\alpha }_1\left({\mathsf{\Omega }}_2^2+{\mathsf{\Omega }}_3^2\right)+2{\alpha }_2\left({\mathsf{\Omega }}_1^2+{\mathsf{\Omega }}_3^2\right)+2{\alpha }_3\left({\mathsf{\Omega }}_1^2+{\mathsf{\Omega }}_2^2\right)\hfill \\ a_2=4{\alpha }_1{\alpha }_2{\mathsf{\Omega }}_3^2+4{\alpha }_1{\alpha }_3{\mathsf{\Omega }}_2^2+4{\alpha }_2{\alpha }_3{\mathsf{\Omega }}_1^2+{\mathsf{\Omega }}_1^2{\mathsf{\Omega }}_2^2+{\mathsf{\Omega }}_1^2{\mathsf{\Omega }}_3^2+{\mathsf{\Omega }}_2^2{\mathsf{\Omega }}_3^2\hfill \\ a_1=2{\alpha }_1{\mathsf{\Omega }}_2^2{\mathsf{\Omega }}_3^2+2{\alpha }_2{\mathsf{\Omega }}_1^2{\mathsf{\Omega }}_3^2+2{\alpha }_3{\mathsf{\Omega }}_1^2{\mathsf{\Omega }}_2^2\hfill \\ a_0={\mathsf{\Omega }}_1^2{\mathsf{\Omega }}_2^2{\mathsf{\Omega }}_3^2\hfill \end{array}$$

(29)

For tri-coil EV-WPT systems, the inductance matrix L, capacitor matrix C and resistance matrix R can be expressed as

$$L=\left[\begin{array}{ccc}{L}_1& M_{12}& M_{13}\\ M_{12}& L_2& M_{23}\\ M_{13}& M_{23}& L_3\end{array}\right],C=\left[\begin{array}{ccc}{C}_1& 0& 0\\ 0& C_2& 0\\ 0& 0& C_3\end{array}\right],R=\left[\begin{array}{ccc}{R}_1& 0& 0\\ 0& R_2& 0\\ 0& 0& R_3\end{array}\right]$$

(30)

Substituting Equation (30) into $\left|{\lambda }^2\left[LC\right]+\lambda \left[RC\right]+I_{n\times n}\right|=0$, the practical characteristic equation of two-coil EV-WPT systems can be established as

$$f{\left(\lambda \right)}_{actual}={\lambda }^6+b_5{\lambda }^5+b_4{\lambda }^4+b_3{\lambda }^3+b_2{\lambda }^2+b_1\lambda +b_0=0$$

(31)

where

$$\{\begin{array}{l}{b}_5=\frac{L_1{L}_2{R}_3+L_1{L}_3{R}_2+L_2{L}_3{R}_1-M_{12}^2{R}_3-M_{13}^2{R}_2-M_{23}^2{R}_1}{L_1{L}_2{L}_3-L_1{M}_{23}^2-L_2{M}_{13}^2-L_3{M}_{12}^2+2M_{12}{M}_{13}{M}_{23}}\hfill \\ b_4=\frac{C_1{C}_2{C}_3\left(L_1{R}_2{R}_3+L_2{R}_1{R}_3+L_3{R}_1{R}_2\right)+C_1{C}_2\left(L_1{L}_2-M_{12}^2\right)+C_1{C}_3\left(L_1{L}_3-M_{13}^2\right)+C_2{C}_3\left(L_2{L}_3-M_{23}^2\right)}{C_1{C}_2{C}_3\left(L_1{L}_2{L}_3-L_1{M}_{23}^2-L_2{M}_{13}^2-L_3{M}_{12}^2+2M_{12}{M}_{13}{M}_{23}\right)}\hfill \\ b_3=\frac{C_1{C}_2{C}_3{R}_1{R}_2{R}_3+C_1{C}_2\left(L_1{R}_2+L_2{R}_1\right)+C_1{C}_3\left(L_1{R}_3+L_3{R}_1\right)+C_2{C}_3\left(L_2{R}_3+L_3{R}_2\right)}{C_1{C}_2{C}_3\left(L_1{L}_2{L}_3-L_1{M}_{23}^2-L_2{M}_{13}^2-L_3{M}_{12}^2+2M_{12}{M}_{13}{M}_{23}\right)}\hfill \\ b_2=\frac{C_1{C}_2{R}_1{R}_2+C_1{C}_3{R}_1{R}_3+C_2{C}_3{R}_2{R}_3+C_1{L}_1+C_2{L}_2+C_3{L}_3}{C_1{C}_2{C}_3\left(L_1{L}_2{L}_3-L_1{M}_{23}^2-L_2{M}_{13}^2-L_3{M}_{12}^2+2M_{12}{M}_{13}{M}_{23}\right)}\hfill \\ b_1=\frac{C_1{R}_1+C_2{R}_2+C_3{R}_3}{C_1{C}_2{C}_3\left(L_1{L}_2{L}_3-L_1{M}_{23}^2-L_2{M}_{13}^2-L_3{M}_{12}^2+2M_{12}{M}_{13}{M}_{23}\right)}\hfill \\ b_0=\frac{1}{C_1{C}_2{C}_3\left(L_1{L}_2{L}_3-L_1{M}_{23}^2-L_2{M}_{13}^2-L_3{M}_{12}^2+2M_{12}{M}_{13}{M}_{23}\right)}\hfill \end{array}$$

(32)

According to Equations (29) and (32), the nonlinear equation of tri-coil EV-WPT systems can be established by

$$\{\begin{array}{l}{b}_5=a_5\hfill \\ b_4=a_4\hfill \\ b_3=a_3\hfill \\ b_2=a_2\hfill \\ b_1=a_1\hfill \\ b_0=a_0\hfill \end{array}$$

(33)

The nonlinear equation of tri-coil EV-WPT systems in Equation (33) is too complex to be solved precisely. Then, the nonlinear equation of the tri-coil EV-WPT systems can be converted into

$$\begin{array}{cc}\underset{x}{\min }& f\left(x\right)={\displaystyle \sum _{i=0}^5{\left(a_i-b_i\right)}^2}\end{array}$$

(34)

The self-inductances L₁, L₂, L₃ and corresponding inner resistance R_p1, R_p2, R_p3 are coil dependent and can be predetermined. In the set of examples, the self-inductance L₁, L₂ and L₃ are fixed at 57 μH and the inner resistance R_p1, R_p2 and R_p3 are fixed at 0.13 Ω. In addition, for tri-coil EV-WPT system with given coils, if the mutual inductance M₁₂ and M₂₃ are determined, then the mutual inductance M₁₃ is fixed. All compensation capacitors are usually set to be equal in the actual systems, that is C₁ = C₂ = C₃ = C. Then, the nonlinear programming model of the EV-WPT system with tri-coil can be established as

$$\begin{array}{ll}\min \hfill & f\left(x\right)={\displaystyle \sum _{i=0}^5{\left(a_i-b_i\right)}^2}\hfill \\ s.t.\hfill & \{\begin{array}{l}{L}_1=L_2=L_3=52 \mathsf{\mu }\mathrm{H}\hfill \\ R_{p1}=R_{p2}=R_{p3}=0.1 \mathsf{\Omega }\hfill \\ C_1=C_2=C_3\hfill \\ M_{13}=f\left(M_{12},M_{23}\right)\hfill \\ M_{ij}\le 52 \mathsf{\mu }\mathrm{H}\hfill \end{array}\hfill \end{array}$$

(35)

There are also four parameters waiting for the setting of the system: compensating capacitor C, mutual inductance M₁₂, M₂₃ and load resistance R_L, which are able to be determined accurately by using Equations (32) and (35).

4.2. Parameter Design

In the set of examples, the self-inductance L₁, L₂ and L₃ are fixed at 57 μH, and the inner resistance R_p1, R_p2 and R_p3 are fixed at 0.13 Ω. A tri-coil EV-WPT system can be designed as a single-band system, a dual-band system, or a tri-band system. This is providing that, on the basis of the actual requirement, the given center frequency and its corresponding bandwidth of the three systems are given, as shown in

Table 2. The given spectrum parameters.

	Center Frequency (kHz)	Bandwidth (kHz)
The single-band system (k = 1, r1 = 3)	fo,1 = 150	b1 = 10
The dual-band system (k = 2, r1 = r2 =2)	fo,1 = 100	b1 = 10
	fo,2 = 150	b2 = 20
The tri-band system (k = 3, r1 = r2 = r3=1)	fo,1 = 100	b1 = 10
	fo,2 = 150	b2 = 30
	fo,3 = 200	b3 = 10

.

Substituting the given parameters and given spectrum parameters shown in Table 2 into Equation (17) to Equation (35), the nonlinear model of the single-band system, dual-band system and tri-band system can be obtained. Solving these nonlinear models with the help of the genetic algorithm in Matlab software, the expected electric parameters of the systems can be determined as shown in Equation (36).

$$\{\begin{array}{ll}\{\begin{array}{l}{C}_1=C_2=C_3=22.01 \mathrm{nF}\hfill \\ M_{12}=3.37 \mathsf{\mu }\mathrm{H}\hfill \\ M_{23}=9.51 \mathsf{\mu }\mathrm{H}\hfill \\ M_{13}=1.23 \mathsf{\mu }\mathrm{H}\hfill \\ R_L=18.46 \mathsf{\Omega }\hfill \end{array}\hfill & \left(k=1\right)\hfill \\ \{\begin{array}{l}{C}_1=C_2=C_3=32.03 \mathrm{nF}\hfill \\ M_{12}=17.59 \mathsf{\mu }\mathrm{H}\hfill \\ M_{23}=18.84 \mathsf{\mu }\mathrm{H}\hfill \\ M_{13}=7.56 \mathsf{\mu }\mathrm{H}\hfill \\ R_L=19.516 \mathsf{\Omega }\hfill \end{array}\hfill & \left(k=2\right)\hfill \\ \{\begin{array}{l}{C}_1=C_2=C_3=27.49 \mathrm{nF}\hfill \\ M_{12}=23.87 \mathsf{\mu }\mathrm{H}\hfill \\ M_{23}=25.19 \mathsf{\mu }\mathrm{H}\hfill \\ M_{13}=10.92 \mathsf{\mu }\mathrm{H}\hfill \\ R_L=9.68 \mathsf{\Omega }\hfill \end{array}\hfill & \left(k=3\right)\hfill \end{array}$$

(36)

5. Experimental Verification

5.1. Experimental Setup

A practical EV-WPT system with three coils is established as shown in Figure 3 to verify the design method proposed in this article. The schematic diagram of the circuit of the system is shown in Figure 4, where an H-bridge consisting of four MOSFETs is used as the voltage source. Meanwhile, it acts as the transmitter which is coupled to the intermediate coil. The intermediate coil serves as a repeater, and the right coil is the receiver with a load. The geometric parameters of the three coils are set to be equal to the value listed in

Table 3. The Geometric Parameters of the Practical Coils.

Parameters	Value
Number of turns	18
Inner radius of coil (m)	0.03
Width between per turn (m)	0.01
Radius of copper wire (m)	0.00125

. In addition, the coils are aligned to the coil centers r = 0 along the z-axis. The measured self-inductances of the three coils are 57 μH and the inner resistances of the three coils are 0.13 Ω depending on a LCR digital electric bridge.

5.2. Practical Measurements

In the first set of experiments, the compensating capacitor C₁, C₂ and C₃ are fixed at 22 nF and the load resistance R_L = 20 Ω. The transfer distance d₁₂ = 0.29 m and d₂₃ = 0.16 m. For the given transmission distance, the mutual inductance M₁₂, M₂₃ and M₁₃ are 3.34, 9.94 and 1.28 µH, which are pretty close to the optimal values of the single-band system as calculated in Equation (36). Meanwhile, simulation with the measured electric parameters are conducted. The supply voltage is adjusted at 8 V. And the operating frequency changes from 100 kHz to 200 kHz with a interval of 1 kHz. The power simulated and measured versus the operating frequency is plotted as shown in Figure 5a.

In the second set of experiments, the compensating capacitor C₁, C₂ and C₃ are fixed at 33 nF and the load resistance R_L = 20 Ω. The transfer distance d₁₂ = 0.1 m and d₂₃ = 0.09 m. For the given transmission distance, the mutual inductance M₁₂, M₂₃ and M₁₃ are 18.39, 20.42 and 7.56 µH, which are pretty close to the optimal values of the dual-band system as calculated in Equation (36). Similarly, simulations with the measured electric parameters are conducted. The supply voltage is fixed at 8 V. And the operating frequency changes from 100 kHz to 200 kHz with a interval of 1 kHz. The power simulated and measured versus the operating frequency is plotted as shown in Figure 5b.

In the third set of experiments, the compensating capacitors C₁, C₂ and C₃ are fixed at 27 nF, and the load resistance R_L = 10 Ω. The transfer distance d₁₂ = 0.08 m and d₂₃ = 0.07 m. For the given distance of energy transmission, the mutual inductance M₁₂, M₂₃ and M₁₃ are 22.7, 25.3 and 10.9 µH, which closes the optimal values of the three-band system as calculated in Equation (36). Meanwhile, simulations with the measured electric parameters are conducted. The supply voltage is fixed at 8 V. And the operating frequency changes from 100 kHz to 200 kHz with a interval of 1 kHz. The power simulated and measured versus the operating frequency is plotted as shown in Figure 5c.

As indicated in Figure 5a, only one band emerges in the system, whose measured center frequency is 150 kHz and its corresponding bandwidth is about 10 kHz. The measured results fully satisfy the required performance indexes, illustrating that the design method proposed in this article is effective and accurate. In Figure 5b, there are two bands, whose measured center frequencies are about 101 kHz and 151 kHz, and the corresponding measured bandwidth are about 10 kHz and 24 kHz, respectively. In Figure 5c, there are three bands, whose measured center frequencies are about 100 kHz, 150 kHz and 200 kHz, and the respective corresponding measured bandwidth are about 5 kHz, 20 kHz and 15 kHz. The performance indexes are pretty close to the given values, which indicates that the proposed method is effective and accurate. In Figure 5, there are some deviations between the simulated value and measured value, which mainly caused by the deviation between the practical value and optimal value of the electrical parameters.

6. Conclusions

Multi-band EV-WPT technology is one of the important research trends of EV-WPT technology. In this paper, a novel generalized design method of multi-band EV-WPT systems is proposed by employing and utilizing frequency splitting. The design theory and design criteria of multi-band multi-coil EV-WPT systems are presented and explained, and have been verified by practical prototypes. Compared to the traditional design methods, the method proposed in this paper has the following advantages:

• The design indexes of the method proposed in this article include not only the center frequencies of the bands, but also the bandwidths.
• The proposed method does not have to add any extra LC branch circuits or extra coils.
• The proposed method suits for arbitrary coil EV-WPT systems with an arbitrary number of bands (no more than the coils), including a multi-coil EV-WPT system with a single given band (without frequency splitting phenomenon).