Analysis of LC-LC² Compensated Inductive Power Transfer for High Efficiency and Load Independent Voltage Gain

Abstract

A novel LC-LC² compensated resonant converter topology with high efficiency and good controllable voltage gain is presented in this paper. An additional receiving side inductor working together with the receiving coil has the contribution to work with a large range of air gap distance. Due to this property, proposed compensation technique is effective for IPT based EV charging application. Voltage gain with independent of load and input impedance having ZPA of the proposed resonant converter are observed by the frequency domain analysis. On the other hand, time domain analysis gives the circuit operation. A 500 W LC-LC² compensated resonant converter prototype is built to testify the theoretical analysis. To observe the efficiency-comparison, an S-SP compensated resonant converter with a similar amount of output power under different air gap is also presented. In order to justify the effectiveness, the proposed compensation method is verified by the laboratory results. The highest efficiency of the proposed compensated resonant converter is 93% with output power of 500 W at 140-mm air gap between the two sides of the IPT (inductive power transfer) transformer.

1. Introduction

Technology of inductive power transfer (IPT) delivers power to the utility load via magnetic coupling, is more suitable and versatile energy transfer method in comparison with traditional methods. Due to the low maintenance cost and highly reliable operation, IPT system is more popular for a large area of applications including battery charging of electric vehicles [1,2,3,4], home appliances [5], biomedical treatments [6,7] and other areas associated with industries [8]. IPT technology normally handles a large variation of loading and magnetic coupling coefficient. In order to achieve maximum efficiency and enhanced controllable output, researcher’s works have been observed on the basis of wide variation of IPT parameters [3,9,10,11,12,13,14,15,16,17,18].

The ruling parameters that affect the efficiency and controllability of output power are: coupling factor and misalignment tolerance. In order to obtain high coupling factor and misalignment tolerance, most of the researchers have focused on the optimization of IPT-parameters [9,19,20]. Another key factor that affects the efficiency and output power controllability of IPT system is the compensation network. Various types of compensation networks by the aid of control technique have been presented and explained in details [16,17,18,21,22,23,24,25,26,27]. Four conventional compensation techniques that are widely used named as S-S (series-series), S-P (series-parallel), P-S (parallel-series), and P-P (parallel-parallel). Among four compensation techniques, P-S and P-P compensated IPT systems are free from danger for the source if the pick-up coil is not present. But these compensation techniques are not suitable to transfer the rated power in case of misalignment condition, as resonant capacitor value strongly depends on the magnetic coupling coefficient and the output load [12]. Consequently, S-S and S-P topologies are widely used in many applications like [10,11,25]. The efficiency of the S-S compensated converter is high when self-inductances of the transmitter and receiver are tuned [10]. Comparison between self-inductance compensation and leakage inductance compensation of S-S topology is presented in [7,25]. However, self-inductance tuned by the S-S compensation presents output voltage which is very sensitive to the output load changes [10]. In order to improve the sensitivity of the output voltage under wide load variation, S-S compensated converter is designed to operate around the frequency of load independent voltage gain [7,25]. On the other hand, input impedance having ZPA is not maintained at a frequency of voltage gain where load is independent. Therefore, efficiency may be lower because of circulating current. In [10], a tradeoff between output voltage controllability and efficiency of the S-S topology is presented. Moreover, the problem still exists as tuning of compensating capacitors is not easy to obtain. Since, S-P compensation provides input impedance with ZPA at a frequency of load independent voltage gain [16]; this technique is suitable for high power applications like [26]. Furthermore, S-P compensation is not effective for a wide range of coupling coefficient and misalignment as load independent voltage gain is inversely related to the coupling coefficient [18]. To maintain a unique output power under the changes of coupling, SP-S compensation network was presented in [17]. There is a large number of research works based on LCL compensation [13,14,15,28]. In [28], a series-parallel LCL tuned unity pick up was presented, that provides uninterrupted power though it reflects reactive power to the source [29]. Another compensation topology with high efficiency named LCC was used in the areas of EV (electric vehicle) charging [23,27]. Here, load independent voltage gain under coupling changes is very much difficult to obtain because of self-coupling between the compensation inductor and main coil. At a frequency of load independent voltage gain, input impedance with ZPA and high power transfer efficiency can be obtained in S-SP compensated resonant converter [18]. Higher-order harmonics are the main limitation of this compensation technique, injected into the rectifier network. These higher order harmonics create the problems for filter design.

A novel LC-LC² compensation network for IPT system is proposed in this paper which provides input impedance with ZPA and high efficiency at a frequency of load independent voltage gain like S-SP compensated resonant converter. Here, an additional receiving side inductor working parallel to the magnetizing inductance has the great contribution to work under a large range of air gap distance. Because, the parallel effect of additional inductor and magnetizing inductance minimizes the air gap variation effect.

This paper is presented as like: Proposed LC-LC² compensation is introduced by Section 2. Section 3 provides the frequency domain analysis of the proposed LC-LC² compensation based resonant converter. Section 4 presents time domain analysis of the proposed LC-LC² compensation based resonant converter. Section 5 represents the analysis of the experimental results. Finally, Section 6 is summarized.

2. Proposed LC-LC² Compensation

Analysis of input impedance with ZPA and voltage gain at a frequency where load is independent is described in this section. In this case, T-model for IPT transformer having turn ratio 1: n is chosen. In the T-model, L_l1 and L_l2 depict the leakage inductances of the transmitter and receiver coil of IPT transformer respectively whereas L_M is the magnetizing inductance. In case of S-SP compensated resonant converter, Ref. [18] capacitor C_f2 for compensating L_M is introduced to the S-S compensated resonant converter. Although input impedance with ZPA and voltage gain with load independent are obtained, higher order harmonics are injected to the rectifier network. To minimize the air gap variation effect as well as the effect of higher order harmonics, an inductor L_f2 parallel with L_M is launched to the S-SP compensated resonant converter. In this case, the equivalent parallel inductance of n²L_M and L_f2 are compensated by C_f2. This equivalent inductance is expressed as,

$$L_{Mf}=n^2{L}_M\parallel L_{f2}=\frac{n^2{L}_M{L}_{f2}}{n^2{L}_M+L_{f2}}$$

(1)

For a resonant frequency, a variation of the air gap (i.e., mutual inductance changes) does not affect the load independent voltage gain and the efficiency so much, because of the parallel effect of L_f2 with L_M. This is observed in Figures 6 and 8.

The equivalent circuit of the LC-LC² compensation network by the help of T-model is depicted in Figure 1. If above equivalent inductance as well as all leakage inductances of the proposed IPT transformer are compensated at a time,

$${\omega }_r=\frac{1}{\sqrt{L_{l1}{C}_1}}=\frac{1}{\sqrt{L_{l2}{C}_2}}=\frac{1}{\sqrt{L_{Mf}{C}_{f2}}}$$

(2)

where the resonant frequency is ω_r. L_l1, L_l2 and L_Mf are compensated by C₁, C₂ and C_f2 respectively.

It can be observed that at the resonant frequency, the output voltage gain of this resonant converter is independent of load and coupling coefficient changes. Also, ZPA of the input impedance is maintained under the full compensation condition.

3. LC-LC² Compensated Resonant Converter Using Frequency Domain Analysis

In this section, voltage gain with load independence as well as input impedance with ZPA of the proposed LC-LC² compensated resonant converter have been explained in brief. The proposed LC-LC² compensated resonant converter is shown in Figure 2.

Here, high frequency inverter is operated by the input dc supply of V_i whereas, V₀ is the voltage across the battery having resistance R_L. L_f and C_f are used for filtering purpose in order to get pure dc after the high frequency rectifier.

By the help of Kirchhoff’s voltage law, fundamental components of V_LM, V_AB and V_ab can be expressed as,

$$V_{L_M}=V_{AB}\frac{j\omega L_M\parallel \frac{Z_2+\left(-j\frac{1}{\omega C_{f2}}\right)\parallel \left(j\omega L_{f2}\right)\parallel R_E}{n^2}}{j\omega L_M\parallel \frac{Z_2+\left(-j\frac{1}{\omega C_{f2}}\right)\parallel R_E\parallel \left(j\omega L_{f2}\right)}{n^2}+Z_1}$$

(3)

$$V_{ab}=n{V}_{L_M}\frac{\left(-j\frac{1}{\omega C_{f2}}\right)\parallel R_E\parallel \left(j\omega L_{f2}\right)}{Z_2+\left(-j\frac{1}{\omega C_{f2}}\right)\parallel R_E\parallel \left(j\omega L_{f2}\right)}$$

(4)

where $R_E=\left(\frac{8}{{\pi }^2}\right)R_L$ = equivalent ac load resistance across the high frequency rectifier side, $Z_1=j\left(\omega L_{l1}-\frac{1}{\omega C_1}\right)$ and $Z_2=j\left(\omega L_{l2}-\frac{1}{\omega C_2}\right)$.

By the aid of Fourier analysis, the fundamental component of V_AB can be depicted as

$$V_{AB}=\frac{4V_i}{\pi }$$

(5)

Due to L_f2, majority portion of higher-order harmonics those are tending to inject into the rectifier, can be eliminated. In later, experimental results of LC-LC² compensated resonant converter will compare the efficiency with the results of S-SP compensated resonant converter. Furthermore, V_ab can be regarded as a sine wave. Therefore, the simplified voltage across the battery can be written as,

$$V_0=\frac{1}{\pi }\underset{0}{\overset{\pi }{{\displaystyle \int }}}{V}_{ab}(\sin \omega t)d\omega t=\frac{2}{\pi }{V}_{ab}$$

(6)

From Equations (3)–(6), voltage gain of the LC-LC² compensated resonant converter can be represented as,

$$\begin{array}{l}{G}_V(\omega )=\frac{V_0}{V_i}=\left|\frac{\frac{2V_{ab}}{\pi }}{\frac{\pi V_{AB}}{4}}\right|=\frac{8}{{\pi }^2}\left|\frac{V_{ab}}{V_{AB}}\right|=\frac{8}{{\pi }^2}\left|\frac{V_{ab}}{V_{L_M}}·\frac{V_{L_M}}{V_{AB}}\right|\\ =\frac{8}{{\pi }^2}\left|\frac{n}{1+\frac{C_{f2}\left[\left(Z_2+n^2{Z}_1\right)j\omega L_M+Z_1{Z}_2\right]}{L_M}-\left(n^2{L}_M+L_{f2}\right)\left(Z_3+j\frac{1}{\omega C_{f2}}\right)\left(\frac{Z_1}{{\omega }^2{L}_M{L_{f2}}^2}-j\frac{Z_1{Z}_2}{{\omega }^3{L_M}^2{L_{f2}}^2{n}^2}+\frac{Z_2}{{\omega }^2{L}_M{L_{f2}}^2{n}^2}+\frac{Z_1}{{\omega }^2{L_M}^2{L}_{f2}{n}^2}\right)+\frac{\delta }{j{\omega }^3{L}_M{R}_E{C}_1{C}_2}}\right|\end{array}$$

(7)

where $Z_3=j\left(\omega L_{Mf}-\frac{1}{\omega C_{f2}}\right)$ and

$$L_T=L_{l1}+L_M$$

(8)

$$L_R=L_{l2}+n^2{L}_M$$

(9)

$$\delta ={\omega }^4{C}_1{C}_2\left(n^2{L_M}^2-L_T{L}_R\right)+{\omega }^2\left(L_T{C}_1+L_R{C}_2\right)-1$$

(10)

Equation (10) can be expressed as,

$$\delta ={\omega }^2{C}_1{C}_2\left[Z_1{Z}_2+j\omega L_M\left(Z_2+n^2{Z}_1\right)\right]$$

(11)

From (7), it is observed that voltage gain is not dependent of the load when δ = 0.

Using δ = 0, frequencies at load independent voltage gain are:

$${\omega }_L=\sqrt{\frac{{{\omega }_T}^2+{{\omega }_R}^2-\sqrt{{({{\omega }_T}^2+{{\omega }_R}^2)}^2-4\left(1-k^2\right){{\omega }_T}^2{{\omega }_R}^2}}{2\left(1-k^2\right)}}$$

(12)

$${\omega }_H=\sqrt{\frac{{{\omega }_T}^2+{{\omega }_R}^2+\sqrt{{({{\omega }_T}^2+{{\omega }_R}^2)}^2-4\left(1-k^2\right){{\omega }_T}^2{{\omega }_R}^2}}{2\left(1-k^2\right)}}$$

(13)

where ${\omega }_T=\frac{1}{\sqrt{L_T{C}_1}}$ and ${\omega }_R=\frac{1}{\sqrt{L_R{C}_2}}$.

Voltage gain for the above frequencies can be expressed as,

$$\begin{array}{l}{G}_V\left({\omega }_L\right)=\\ \frac{8}{{\pi }^2}\frac{n}{\left|1+\frac{C_{f2}\left[\left(Z_2+n^2{Z}_1\right)j{\omega }_L{L}_M+Z_1{Z}_2\right]}{L_M}-\left(n^2{L}_M+L_{f2}\right)\left(Z_3+j\frac{1}{{\omega }_L{C}_{f2}}\right)\left(\frac{Z_1}{{{\omega }_L}^2{L}_M{L_{f2}}^2}-j\frac{Z_1{Z}_2}{{{\omega }_L}^3{L_M}^2{L_{f2}}^2{n}^2}+\frac{Z_2}{{{\omega }_L}^2{L}_M{L_{f2}}^2{n}^2}+\frac{Z_1}{{{\omega }_L}^2{L_M}^2{L}_{f2}{n}^2}\right)\right|}\end{array}$$

(14)

$$\begin{array}{l}{G}_V\left({\omega }_H\right)=\\ \frac{8}{{\pi }^2}\frac{n}{\left|1+\frac{C_{f2}\left[\left(Z_2+n^2{Z}_1\right)j{\omega }_H{L}_M+Z_1{Z}_2\right]}{L_M}-\left(n^2{L}_M+L_{f2}\right)\left(Z_3+j\frac{1}{{\omega }_H{C}_{f2}}\right)\left(\frac{Z_1}{{{\omega }_H}^2{L}_M{L_{f2}}^2}-j\frac{Z_1{Z}_2}{{{\omega }_H}^3{L_M}^2{L_{f2}}^2{n}^2}+\frac{Z_2}{{{\omega }_H}^2{L}_M{L_{f2}}^2{n}^2}+\frac{Z_1}{{{\omega }_H}^2{L_M}^2{L}_{f2}{n}^2}\right)\right|}\end{array}$$

(15)

However, frequency at load independent voltage gain is ω_r. To obtain a fixed voltage gain at ω_r, the subsequent equation should be fulfilled:

$$\begin{array}{l}\frac{C_{f2}\left[\left(Z_2+n^2{Z}_1\right)j{\omega }_r{L}_M+Z_1{Z}_2\right]}{L_M}-\left(n^2{L}_M+L_{f2}\right)\left(Z_3+j\frac{1}{{\omega }_r{C}_{f2}}\right)(\frac{Z_1}{{{\omega }_r}^2{L}_M{L_{f2}}^2}-j\frac{Z_1{Z}_2}{{{\omega }_r}^3{L_M}^2{L_{f2}}^2{n}^2}+\\ \frac{Z_2}{{{\omega }_r}^2{L}_M{L_{f2}}^2{n}^2}+\frac{Z_1}{{{\omega }_r}^2{L_M}^2{L}_{f2}{n}^2})=0\end{array}$$

(16)

By putting $Z_1=j\left(\omega L_{l1}-\frac{1}{\omega C_1}\right),Z_2=j\left(\omega L_{l2}-\frac{1}{\omega C_2}\right), Z_3=j\left(\omega L_{Mf}-\frac{1}{\omega C_{f2}}\right)$ and δ = 0 into (16), gives

$$\begin{array}{c}\frac{C_{f2}\left[\left(Z_2+n^2{Z}_1\right)j{\omega }_r{L}_M+Z_1{Z}_2\right]}{L_M}-\left(n^2{L}_M+L_{f2}\right)\left(Z_3+j\frac{1}{{\omega }_r{C}_{f2}}\right)(\frac{Z_1}{{{\omega }_r}^2{L}_M{L_{f2}}^2}-j\frac{Z_1{Z}_2}{{{\omega }_r}^3{L_M}^2{L_{f2}}^2{n}^2}+\\ \frac{Z_2}{{{\omega }_r}^2{L}_M{L_{f2}}^2{n}^2}+\frac{Z_1}{{{\omega }_r}^2{L_M}^2{L}_{f2}{n}^2})=\frac{\delta }{{{\omega }_r}^2{C}_1{C}_2}-\frac{L_M}{C_{f2}}\left(\frac{j{Z}_1{n}^2}{{\omega }_r{L}_{f2}}+\frac{Z_1{Z}_2}{{{\omega }_r}^2{L}_M{L}_{f2}}+j\frac{Z_2}{{\omega }_r{L}_{f2}}+j\frac{Z_1}{{\omega }_r{L}_M}\right)=0\end{array}$$

(17)

Therefore, $Z_1=Z_2=0$. Substituting $Z_1=Z_2=0$ into (11), we get

$$\delta ={{\omega }_r}^2{C}_1{C}_2\left[Z_1{Z}_2+j{\omega }_r{L}_M\left(Z_2+n^2{Z}_1\right)\right]=0$$

(18)

So that, load independent fixed voltage gain with zero effect of coupling factor is obtained when, $Z_1=Z_2=0$.

The proposed resonant converter’s input impedance is obtained as,

$$\begin{array}{l}{Z}_i\left(\omega \right)\\ =Z_1+\frac{j\omega L_M\left[j\omega L_{f2}\left(j\omega C_{f2}{Z}_2{R}_E+Z_2+R_E\right)+Z_2{R}_E\right]}{j\omega n^2{L}_M\left[j\omega L_{f2}\left(j\omega C_{f2}{R}_E+1\right)+R_E\right]+j\omega L_{f2}\left(j\omega C_{f2}{Z}_2{R}_E+Z_2+R_E\right)+Z_2{R}_E}\end{array}$$

(19)

Substituting $Z_1=Z_2=0$ into (19), yields

$$Z_i\left({\omega }_r\right)=\frac{j{\omega }_r{L}_M{L}_{f2}{R}_E}{j{\omega }_r{n}^2{L}_M{L}_{f2}+\left(L_{f2}{R}_E+n^2{L}_M{R}_E-{{\omega }_r}^2{n}^2{L}_{f2}{C}_{f2}{L}_M{R}_E\right)}$$

(20)

In order to get ZPA of Z_i at ω_r, the imaginary part must be equal to zero. The input impedance’s imaginary part of (20) can be expressed as,

$$Im\left[Z_i\left({\omega }_r\right)\right]=\frac{-{{\omega }_r}^2{L}_M{L}_{f2}{C}_{f2}{R_E}^2\left(n^2{L}_M+L_{f2}\right)Z_3}{{({\omega }_r{n}^2{L}_M{L}_{f2})}^2+{\left(L_{f2}{R}_E+n^2{L}_M{R}_E-{{\omega }_r}^2{n}^2{L}_{f2}{C}_{f2}{L}_M{R}_E\right)}^2}$$

(21)

Under the full compensation, (12) and (13) yields,

$${\omega }_L=\frac{1}{\sqrt{L_T{C}_1+n^2{L}_M{C}_2}}$$

(22)

$${\omega }_H=\frac{1}{\sqrt{L_{l1}{C}_1}}$$

(23)

Equation (23) follows the Equation (2). Therefore, constant voltage gain and ZPA of input impedance both can be obtained, under the full compensation ($Z_1=Z_2=Z_3=0$) condition. This is a desired criterion to get high efficiency and output voltage having good controllability. From (7), the required voltage gain can be obtained as,

$$G_V\left({\omega }_r\right)=\frac{8n}{{\pi }^2}$$

(24)

That means with the help of the proposed compensation technique, load independent constant voltage gain is achievable which is desirable for IPT based EV application.

4. LC-LC² Compensated Resonant Converter Using Time Domain Analysis

To explain the time domain analysis, the following hypotheses are made:

(a)

All semiconductor switches and diodes are ideal.

(b)

Receiver current, i_S is almost sinusoidal since proposed converter operates near the resonant frequency.

Figure 3a–d depicts the equivalent circuits during different periods of operation of the proposed converter for the operating waveforms shown in Figure 4.

Considering the time instant t₀, diagonal switch pair (S₁ and S₄) of the proposed converter is in conduction and another diagonal switch pair (S₂ and S₃) is in off mode. At that time V_AB is positively equal to V_i. Firstly, transmitter current i_P passes through the body diodes of S₁ and S₄, after that S₁ and S₄ start to conduct with the changing of current direction. Consequently, ZVS is obtained.

Throughout this period, current flowing into L_M can be written as,

$$i_{L_M}\left(t\right)=\frac{V_i}{L_M}\left(t-\frac{T_S}{4}\right)$$

(25)

where T_S is the time period.

Also, receiver current, i_S can be obtained as,

$$i_S\left(t\right)=\frac{4n}{\pi }{V}_i\times \left\{j\left(\omega C_{f2}-\frac{1}{\omega L_{f2}}\right)+\frac{1}{R_E}\right\}\sin \left(\omega t+{tan}^{-1}\left(\omega C_{f2}-\frac{1}{\omega L_{f2}}\right)R_E\right)$$

(26)

Therefore, transmitter current, i_P can be calculated as,

$$\begin{array}{cc}\hfill i_P\left(t\right)=i_{L_M}\left(t\right)+& n{i}_S\left(t\right)\hfill \\ & =\frac{V_i}{L_M}\left(t-\frac{T_S}{4}\right)\hfill \\ & +n\frac{4n}{\pi }{V}_i\hfill \\ & \times \left\{j\left(\omega C_{f2}-\frac{1}{\omega L_{f2}}\right)+\frac{1}{R_E}\right\}\sin \left(\omega t+{tan}^{-1}\left(\omega C_{f2}-\frac{1}{\omega L_{f2}}\right)R_E\right)\hfill \end{array}$$

(27)

Diagonal switch pair (S₁ and S₄) is in off mode at t₁. So that i_P charges the C_S1 and C_S4 and discharges C_S2 and C_S3. Next, S₁ and S₄ are turned off at zero voltage as C_S1 and C_S4 limit the rising voltage rate across the switches. In this case, the direction of i_P can be considered as unchanged because of the short duration of this mode. At time instant t₂, the voltage across the switches S₂ and S₃ is fully discharged whereas complete charging of the capacitors of S₁ and S₄ occurs from 0 to V_i. Then, to maintain the continuity of i_P, D₂ and D₃ start to conduct. So V_AB becomes to −V_i.

After the conduction of D₂ and D₃, diagonal switch pair (S₂ and S₃) of the proposed converter is in conduction with ZVS. Here, V_AB equals to −V_i. During this interval, as like to mode 1 current changes its direction. At first, body diodes of S₂ and S₃ conduct, after that S₂ and S₃ start to conduct with the changing of current direction. Throughout this period, current flowing into L_M can be written as,

$$i_{L_M}\left(t\right)=\frac{V_i}{L_M}\left(-t+\frac{3T_S}{4}\right)$$

(28)

Using (26), transmitter current, i_P can be calculated as,

$$\begin{array}{cc}\hfill i_P\left(t\right)=\frac{V_i}{L_M}(-t+& \frac{3T_S}{4})\hfill \\ & +n\frac{4n}{\pi }{V}_i\hfill \\ & \times \left\{j\left(\omega C_{f2}-\frac{1}{\omega L_{f2}}\right)+\frac{1}{R_E}\right\}\sin \left(\omega t+{tan}^{-1}\left(\omega C_{f2}-\frac{1}{\omega L_{f2}}\right)R_E\right)\hfill \end{array}$$

(29)

Diagonal switch pair (S₂ and S₃) is in off mode with ZVS at t₃. At time instant t₄, the voltage across the switches S₁ and S₄ is fully discharged whereas complete charging of the capacitors of S₂ and S₃ occurs from 0 to V_i. Then, to maintain the continuity of i_P, D₁ and D₄ start to conduct. Therefore, ZVS turn on condition occurs for S₁ and S₄. So, V_AB becomes to +V_i.

5. Analysis and Experimental Results

The specifications for 500 W LC-LC² compensated resonant converter are as follows: V_in = 211 V, V₀ = 171 V, P₀ = 500 W, an air gap of 140 mm. The loosely coupled transformer with a rectangular shape having I shape ferrite core is used in this resonant converter. Although voltage gain is independent of the mutual inductance, capacitors are designed in case of maximum coupling air gap to make resonance with the corresponding inductances. Therefore, at resonant frequency ZVS can be obtained with the specific frequency control. Detailed specifications of the proposed resonant converter with a constant resonant frequency are shown in

Table 1. Design specifications of the LC-LC2 converter.

Design Specifications	Value
Resonant frequency	f = 80 kHz
Parameters of IPT transformer	n = 1, Ll1 = 82.95 µH, Ll2 = 82.05 µH, LM = 25.25 µH–13.1 µH, k = 0.234–0.140
Compensating inductor	Lf2 = 20 µH
Resonant capacitors	C1 = 47.71 nF, C2 = 48.24 nF, Cf2 = 354.65 nF
Load resistance	RL = 33–200 Ω
Filter parameters	Lf = 1000 µH and Cf = 500 nF
Load power	PL = 500 W
Air gap	H = 140–180 mm

.

Figure 5 shows that measured phase angle of the input impedance for an air gap of 140 mm with the specifications listed in Table 1. In accordance with the Figure 5, it is observed that load independent input impedance with ZPA is achieved when the proposed converter is totally compensated.

The calculated voltage gain corresponding to the frequency is shown in Figure 6. From Figure 6, load independent voltage gain is found at 80 kHz which follows the Equation (23) as well as Equation (2). Comparing Figure 5 and Figure 6; voltage gain having load independence and input impedance with ZPA, both can be obtained at the frequency of ω_H. The load-regulation curve of the proposed LC-LC² converter from laboratory results is shown in Figure 7.

The experimental efficiency from dc source to the battery is calculated for proposed mutual inductance as well as mutual inductance (13.1 µH) for different air gap of 180 mm.

To compare the coupling tolerance of the proposed converter with S-SP compensated resonant converter, the specifications for 500 W S-SP compensated converter are: V_in = 211 V, V₀ = 171 V, P₀ = 500 W, C₁ = 47.71 nF, C₂ = 48.24 nF, C_f2 = 156.75 nF, an air gap of 140 mm, R_L = 33–200 Ω. In case of S-SP compensated resonant converter, C_f2 compensates the mutual inductance, L_M. On the other hand, equivalent inductance $L_{Mf}=n^2{L}_M\parallel L_{f2}$ is compensated by C_f2 in case of the proposed converter.

Comparing Figure 8 and Figure 9, when the mutual inductance is about 13.1 µH, efficiency is much affected in case of S-SP resonant converter. Because efficiency of the S-SP compensated resonant converter directly related to the mutual inductance whereas efficiency of the LC-LC² compensated resonant converter related to the equivalent inductance, $L_{Mf}=n^2{L}_M\parallel L_{f2}$. Therefore, LC-LC² compensated resonant converter provides better efficiency than S-SP compensated resonant converter in case of variant air gap condition by which mutual inductance changes. Based on this, it is clear that due to the parallel effect of magnetizing inductance and the additional inductance of the secondary side, phase angle of the input impedance is closely maintained to zero. That’s why efficiency is not so much affected by the air gap variation. Also higher order harmonics of i₂ from i_s are limited. Although the proposed compensation technique is built for 140–180 mm air gap variation, the laboratory setup for 140 mm air gap variation is shown in Figure 10.

At the defined air gap, the current i₁ is in phase with the high frequency inverter output voltage V_AB. ZVS condition is also observed in case of experimental result which is also shown in Figure 11. In this case, V_DS and I_DS are the switch voltage and current of the high frequency inverter; V_GS is the gate signal.

In case of higher-order harmonics injection to the rectifier side, LC-LC² is better than the S-SP compensation technique because of the effect of equivalent inductance $L_{Mf}$. Comparison between S-SP compensation and LC-LC² compensation is shown in

Table 2. Comparison of LC-LC2 compensation with S–SP² compensation.

Compensation Networks	Evaluation Criteria
	Maximum Efficiency	Power Rating	Air Gap	Efficiency Variation	Higher-Order Harmonics
S–SP [18]	93%	1500 W	200 mm	Affected by the mutual inductance changes	Injected to the rectifier side
LC-LC2	93%	500 W	140 mm	Less affected by the mutual inductance changes	Not injected to the rectifier side

.

Though above comparison table is for different power rating, efficiency variation by mutual inductance changes and injection of higher order harmonics are the main parameters to evaluate the proposed network. Efficiency variation is not so much affected by the mutual inductance changes in case of proposed compensation based resonant converter in comparison to the S-SP compensation based resonant converter, shown in Figure 8 and Figure 9. Because of an additional receiving side inductor, proposed compensation technique cannot inject higher order harmonics to the rectifier side.

6. Conclusions

The proposed LC-LC² compensated resonant converter provides the input impedance with ZPA and voltage gain having load independent characteristics at a common resonant frequency. Using a common resonant frequency, experimental results of the proposed LC-LC² compensated resonant converter are obtained. Laboratory results represent the efficacy of the proposed compensation method with a maximum efficiency of 93%. In IPT based EV charging application, efficiency is directly related to the air gap variation. Efficiency-comparison between LC-LC² and S-SP compensated resonant converter depicts that efficiency of the LC-LC² compensated resonant converter is not much sensitive to the air gap variation than the S-SP compensated resonant converter. Due to the less sensitivity of the air gap variation, proposed compensation technique is very much effective in the IPT based EV charging application.