1 Introduction

Inductive power transfer (IPT) devices have attracted great interest with their inherent advantages such as electrical isolation, safety and reliability [1, 2]. Additionally, IPT is less time-consuming, causes less pollution and is safe to use in a wet environment than the plug-in charging system, which has disadvantages of safety and reliability issues such as electric sparks, contact losses and mechanical corrosion [3]. However, IPT devices provide less efficiency than plug-in charging systems due to the low coupling coefficient [4, 5]. Despite these situations mentioned above, IPT systems have been widely preferred to make efficient battery charging in fields such as mobile phones, wireless charging medical apparatus, and electric vehicles (EV) [3, 6, 7].

In the discussed applications above, the lithium-ion (Li-ion) battery has been generally preferred due to their reliability, high-performance, high-power density and longevity [6, 8]. However, constant current (CC) and constant voltage (CV) charging modes are required for efficient charging and prolonging the lifetime of Li-ion batteries [9]. Generally, the battery charging starts with the CC charging process and moves to the CV process when the battery voltage reaches a defined value [6]. At CC charging mode, the battery voltage rises depending on the battery characteristics, but the battery current remains the same. On the other hand, at CV charging mode, the battery current starts to decrease depending on the battery requirement, but the charging voltage remains the same. Eventually, the charging is terminated when the battery current decreases to a specific level.

Since the Li-ion battery is expensive, an efficient charging strategy is necessary to decrease charging time, optimal charging modes and energy efficiency for battery health [4]. To achieve desired CC and CV charging profiles in IPT systems, various conventional circuitries have been studied to perform CC/CV charging, such as variable frequency control, special tuning control, back-end and front-end DC/DC converter, and different compensation topologies [1, 10, 11]. An IPT-based charging system generally consists of three sub-circuitries: the high frequency (HF) inverter, the loosely magnetic coupled transformer and the compensation networks [12].

In IPT charging systems, the compensation networks are very important for the overall performance since they constitute resonant tanks with loosely connected inductors to decrease the system’s VA rating while increasing power transfer capabilities [6].

The system phase and frequency responses under CC and CV modes are different. By the compensation topology, the system’s input impedance can be made independent of the nonlinear load characteristic of batteries. That requires providing zero phase angle (ZPA) of the impedance. Therefore, the overall impedance acts like a resistor. As a result of that, the voltage-current rating of the power supply can be reduced during the charging process, which decreases power transfer losses and eliminates power stress on the HF inverter. [13, 14].

Various compensation circuitries, such as basic and high-order types under different load levels, have been investigated in the literature [15]. Based on the location and connection types of the resonant capacitors, the basic compensation topologies can be divided into four categories: series–series (SS), series–parallel (SP), parallel–series (PS) and parallel–parallel (PP) [16, 17]. However, these four compensated topologies have limited design parameters and cannot simultaneously assure the unity power factor with ZPA for both CC and CV modes [18]. To overcome these drawbacks of the four basic compensation topologies, high-order compensation circuitries are introduced, such as LCL, Double-sided LCC, LCC-P, LLC-P and LCCL-S.

In a double-sided LCL topology, the magnitude of the output voltage can be regulated using phase modulation techniques [19]. However, the design of an LCL topology requires additional capacitors and inductors on the transmitter and receiver sides of the transformer.

Different LCC topology can also perform CC and CV charging modes with ZPA conditions by its single circuitry. It also allows flexibility to adjust configuration parameters [2, 20]. LCC circuit structure is designed by adding a capacitor into LCL topology, located in series with a transmitter and/or receiver coil in LCL circuitry. The LCC topology not only provides the same advantages as the LCL, but it also results in the freedom of tuning resonance frequencies for the system. The double-sided LCC (or LCC-LCC) has received significant attention due to its constant current output characteristic, low voltage stress, and excellent efficiency throughout a wide variety of operating range. A double-sided LCC topology with variable coupling coefficient approach is introduced to implement zero-voltage switching (ZVS) in [5]. Furthermore, a double-sided LCC topology is analyzed considering the rectifier equivalent inductance of the rectifier in [21], which makes the circuit to obtain output power closer to the desired level. In [18], another proposed double-sided topology realizes CC and CV modes with reducing AC switches by changing the resonant frequency of the circuit. In [22], an efficient and practical control trajectory is proposed to operate the inverter side of the circuit in half-bridge and full-bridge mode to enable ZVS operation over a wide current range.

Using an LCC compensation tank on the transmitter side is widely preferred in some IPT applications, such as LCC-P and LCC-S types [2]. LCC-P and LCC-S compensation tanks have fewer components on the receiver side than double-sided LCC. An LCC-P compensation topology is proposed for underwater vehicles, including a single capacitor on the receiver side connected in parallel to the receiver coil [11]. An LCC-S topology is realized for CC and CV wireless charging mode with ZPA condition by having two different resonance frequencies in [6]. In [23], an LCC-S circuit is used to perform WPT without any information on the receiver side parameters by using PWM-synchronized samples from the front-end values of the transmitter side of the LCC-S circuit. In [24], another modified LCC-S topology is alternatively presented, which can perform CC and CV charging operations with ZPA through a single switch in two switching frequencies. This approach requires the control circuit complex and makes difficult to eliminate EMI problems. To have a single frequency, another modified LCC-S is also presented for CC and CV charging modes by connecting capacitance in series on the receiver side by two AC switches in [25]. The LCC-P and LCC-S have not only the same advantages of the double-sided LCC topology, but also they reduced size and lower the cost due to requiring fewer components on the receiver side than double-sided LCC topology. In [26], a modified hybrid topology based on double-sided LCC and LCC-S is presented, which performs CC and CV modes with ZPA by modifying circuit types through two AC switches. In [27], the tolerances of LCC-P and LCC-S are compared for the misalignment condition for the pads of the transformer. It was found that LCC-P demonstrates better resistance offsets, while LCC-S can achieve ZPA, and it reduces the reactive part of the input power.

Additionally, higher-order compensation topologies such as LCCL-S have also been studied to improve power transfer efficiency and lead system design freedom. In [28], the performance of the SS and LCCL-S topology is compared in terms of the output characteristics and power transfer efficiencies in both tuning and disrupting states. The results show that the LCCL-S have higher efficiency power transfer at lower power levels. In [29], the output characteristics of SS and LCCL-S topologies are also analyzed for different voltage gain levels.

Over existing studies, this research proposes a novel hybrid topology based on the SS and LCC-S circuitries, including a variable inductor. Unlike the studies discussed in [6] and [25], the proposed system performs CC and CV modes with load-independent ZPA condition at a single resonant frequency by using single AC switches without varying resonant frequency for CC and CV modes. Moreover, the proposed topology includes fewer capacitors to achieve ZPA condition in both charging modes at a single resonant frequency compared to [21] and [26]. Hence, this leads the topology to extend the circuit’s lifespan due to reduced capacitor usage. In the proposed topology, a tunable inductor is introduced, and it is connected in parallel with the series capacitor as a novelty of the study. Besides that, a single switch is connected in series with the input parallel capacitor on the primary side of the circuit to implement the CC and CV modes at a single resonant frequency. Hence, the operating frequency of the proposed IPT systems was fixed throughout all charging modes, reducing the problems caused by frequency hopping. Also, this provides reduced complexity and low-cost circuitry compared to the other existing topologies, which use two or more ACSs to realize CC or CV mode at the same frequency.

The rest of the manuscript is organized as follows: Section 2 introduces a detailed analysis of each operation mode for the SS and LCC-S topology based on the transformer π model. The design consideration for the implementation of the proposed topology is mentioned in Section 3. In Section 4, analytical results are validated via simulations using a 2.4 kW modified hybrid topology, and a comprehensive topology comparison is discussed in detail. Finally, the conclusions are summarized in Section 5.

2 Theoretical analysis of the proposed circuit

In an IPT system, reducing the VA rating of the dc source is the fundamental criterion of the resonant network structure. On the receiver side, the resonant network adjusts the secondary coil’s inductance to enhance transmission capabilities. In general, inductors and capacitors are the two main components found in resonant networks. In IPT systems, LC and CL resonant networks are frequently applied [24]. The type of AC power source, such as an AC voltage source (ACVS) or an AC current source (ACCS), that is supplied to the particular resonant network determines the significance of the CV or CC output characteristics to the load impedance Z in these passive resonant networks [25]. Hereby, Fig. 1 shows two types of the several passive resonant networks, which provide CC and CV output possible for IPT applications.

Fig. 1
figure 1

Resonant network configurations. a V-C type, b C-V type

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Figure 1a presents a V–C type resonant network, where the resonant frequency is the same as the angular frequency (\({\omega }_{v-c}\)) of the input AC voltage source and can be defined as in Eq. (1).

$${\omega }_{v-c}=\frac{1}{\sqrt{{L}_{v-c}{C}_{v-c}}}$$
(1)

Based on Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL), the passive resonant current output in Fig. 1a can be given as in Eq. (2).

$${I}_{v-c}=-j{V}_{v-c}\sqrt{\frac{{C}_{v-c}}{{L}_{v-c}}}=-j\frac{{V}_{v-c}}{{\omega }_{v-c}{L}_{v-c}}$$
(2)

In (2), \({V}_{v-c}\) represents the input voltage of the V–C resonant network and \({I}_{v-c}\) is the value of the output current of the network circuit. According to (2), if the RMS value of \({V}_{v-c}\) is constant, the circuit operates at resonance frequency as well, \({I}_{v-c}\) is independent of \({Z}_{v-c}\). Thus, the V–C resonant network shows CC output characteristic.

Additionally, the C–V passive resonant network presented in Fig. 1(b) has similar analytic procedures when the resonant frequency of the C–V network is the same as the angular frequency (\({\omega }_{c-v}\)) of the input AC current source and can be defined as in Eq. (3).

$${\omega }_{c-v}=\frac{1}{\sqrt{{L}_{c-v}{C}_{c-v}}}$$
(3)

On the basis of Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL), the passive resonant voltage output in Fig. 1b can be given as in Eq. (4).

$${V}_{c-v}=j{I}_{c-v}\sqrt{\frac{{L}_{c-v}}{{C}_{c-v}}}=j{\omega }_{c-v}{L}_{c-v}{I}_{c-v}$$
(4)

In (4), \({I}_{c-v}\) is the input current of the C–V resonant network and \({V}_{c-v}\) is defined as the output voltage of the network circuit. According to (4), if the RMS value of \({I}_{c-v}\) is constant, the circuit operates at resonance frequency as well, \({V}_{c-v}\) is independent of \({Z}_{c-v}\). Therefore, the C–V resonant network presents CV output characteristic. As a result, the resonant frequency of the V–C network is adjusted as the resonant frequency of the V–C network to provide the CC and CV output characteristic for the designed IPT topology:

$${\omega }_{v-c}={\omega }_{c-v}$$
(5)

2.1 Analysis of the proposed hybrid topology

As discussed earlier, the LCC-S topologies perform better efficiency by applying the ZPA condition in lower power condition and are used in reducing reactive power. However, SS topologies demonstrate better performance for high-power applications [29]. Hence, SS and LCC-S-based hybrid topologies have become widely preferred in the literature. They include the SS and LCC-S advantages, such as better efficiencies for high and low power conditions and CC/CV charging mode with ZPA ability. In [6], SS combined with LCC-S performs ZPA conditional CC and CV operations without using any switches at two different resonant frequencies. However, another SS and LCC-S-based topology is demonstrated in [25], with ZPA capable of performing the CC and CV charge modes at the same resonant frequency, but using a total of two AC switches on each side of the circuit.

In this study, a modified IPT topology, derived from [6] and [25], is demonstrated by adding a tunable inductor to the primary side of the circuit. The proposed topology can perform each charging operation at the same resonant frequency and also includes only one AC switch on the primary side to select CC or CV operation depending on the equivalent battery load. The structure of the proposed WPT system based on the modified hybrid topology is shown in Fig. 2. The mutual inductance is \(M\), and the inductances of the transmitter and receiver coils are \({L}_{p}\) and \({L}_{s}\), respectively. \({C}_{p}\) is the parallel compensation capacitor, \({L}_{in}\) is the series compensation inductor, \({C}_{f}\) is the series compensation capacitor, \({L}_{x}\) is the adjustable shunt inductor connected in parallel with the \({C}_{f}\), and \({C}_{s}\) is the series compensation capacitor on the receiver side. Additionally, the proposed circuit includes a full-bridge inverter at the input of the transmitter stage, a rectifier at the output of the receiver stage and an extra AC switch (ACS) (\({S}_{5}\)) used to switch between the CC and CV charging modes of the circuit.

Fig. 2
figure 2

Proposed LCC-based compensation topology circuit

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Also, the First Harmonic Approximation (FHA) is used to analyze the circuit in the frequency domain. A 50% percent fixed duty ratio control signal is used to regulate the switches on the inverter side in Fig. 2 to achieve maximum fundamental output voltage. T-model can accomplish load-independent voltage output and load-independent current output, respectively, when the input source is a sinusoidal voltage source and the aforementioned particular conditions are met. The T-model AC equivalent circuit of the suggested topology is also shown in Fig. 3 to make circuit analysis easier [18].

Fig. 3
figure 3

T-model AC equivalent circuit of the proposed topology

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The primary and secondary sides of the coils are considered to be equal for the sake of simplicity. The leakage inductance of the transmitter and receiver side coils is defined as \({L}_{T}\) and \({L}_{R}\). Using the relationships between the self-inductances \({L}_{p}\) and \({L}_{s}\), the coupling coefficient factor \(k\) can be stated as Eq. (6), where \(M\) represents mutual inductance [18].

$$k=\frac{M}{\sqrt{{L}_{p}{L}_{s}}}$$
(6)

T-model parameters such as \({L}_{T}\) and \({L}_{R}\) leakage inductances can also be calculated as follows [1]:

$$\left\{\begin{array}{c}{L}_{T}={L}_{p}-M\\ {L}_{R}={L}_{s}-M\end{array}\right.$$
(7)

In most WPT systems, the compensation circuits have very high-quality factors. As a result, the input current and voltage harmonics are relatively low. Therefore, FHA can be employed in mathematical analysis without error [25]. The first-order harmonics based on the circuit model illustrated in Fig. 3 can be produced using FHA as in Eq. (8) [18].

$$\left\{\begin{array}{c}{v}_{AB}\left(t\right)=\frac{4}{\pi }{V}_{\mathrm{in}}\mathrm{sin}\left(2\pi ft\right)\\ {v}_{ab}\left(t\right)=\frac{4}{\pi }{V}_{o}\mathrm{sin}\left(2\pi ft+\theta \right)\\ {i}_{ab}\left(t\right)=\frac{\pi }{2}{I}_{o}\mathrm{sin}\left(2\pi ft+\theta \right)\end{array}\right.$$
(8)

The input DC voltage, output charging voltage, and current are represented by the variables \({V}_{\mathrm{in}}\), \({V}_{o}\) and \({I}_{o}\), respectively. \(\theta \) is the degree of the phase difference between \({v}_{AB}\) and \({v}_{ab}\). On the receiver side, the bridge rectifier using \({D}_{1}\)~\({D}_{4}\) diodes converts the AC output voltage (\({{\varvec{V}}}_{{\varvec{a}}{\varvec{b}}}\)) and current (\({{\varvec{I}}}_{{\varvec{s}}}\)) to the battery charging DC voltage (\({V}_{o}\)) and current (\({I}_{o}\)) [25]. However, since the output low-pass filter only has one capacitance (\({C}_{o}\)), the relationships between the AC output voltage and current and the battery voltage and current can be stated in Eqs. (9) and (10), respectively [30].

$${V}_{o}=\frac{\pi \sqrt{2}}{4}{V}_{ab}$$
(9)
$${I}_{o}=\frac{2\sqrt{2}}{\pi }{I}_{s}$$
(10)

\({V}_{ab}\) and \({I}_{s}\) are the RMS values of \({{\varvec{V}}}_{{\varvec{a}}{\varvec{b}}}\) and \({{\varvec{I}}}_{{\varvec{s}}}\), respectively. Depending on Eqs. (9) and (10), the output AC equivalent resistance (\({R}_{AC}\)) depending on battery resistance can be defined as in Eq. (11) according to [28].

$${R}_{AC}=\frac{8}{{\pi }^{2}}{R}_{o}$$
(11)

In the following sections, the proposed hybrid topology is theoretically investigated to examine how to perform CC and CV charging operations for the load-independent condition.

2.2 CC Mode Implementation of the Proposed Hybrid Topology

In the case of CC charging mode, the \({S}_{5}\) is switched off, as shown in Fig. 3. Also, each component’s parasitic resistance is ignored. Therefore, the proposed hybrid circuit converts to the SS compensation topology. Figure 4 depicts the equivalent T model circuit for the CC mode charging, where \({L}_{p}{\prime}\) represents the equivalent series inductance of \({L}_{p}\) and \({L}_{\mathrm{in}}\) defined as (12).

Fig. 4
figure 4

Simplified equivalent T-model in CC mode charging

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$${L}_{p}{\prime}={L}_{p}+{L}_{\mathrm{in}}={L}_{T}+M+{L}_{\mathrm{in}}$$
(12)

In CC charging mode, the parallel connection of \({L}_{x}\) and \({C}_{f}\) is regarded as an equivalent capacitance \({C}_{f}{\prime}\) when the value of \({C}_{f}\) is large enough, according to [1].

$${C}_{f}{\prime}=\frac{{{\omega }_{cc}}^{2}{L}_{x}{C}_{f}-1}{{{\omega }_{cc}}^{2}{L}_{x}}$$
(13)

To achieve ZPA condition for the CC mode in the transmitter side of the topology regardless of the load, the \({C}_{f}{\prime}\) must be selected to resonate with \({L}_{p}{\prime}\). Depending on this requirement, the following equations must be satisfied:

$$0=j\left({\omega }_{\mathrm{cc}}{L}_{p}{\prime}-\frac{1}{{\omega }_{\mathrm{cc}}{C}_{f}{\prime}}\right)$$
(14)
$${{\omega }_{\mathrm{cc}}}^{2}=\frac{1}{{L}_{p}{\prime}{C}_{f}{\prime}}$$
(15)

As discussed above, \({C}_{f}{\prime}\) resonates with \({L}_{p}{\prime}\) to eliminate the imaginary part of input impedance (\({Z}_{\mathrm{in}\_\mathrm{cc}}\)). Therefore, the current of the receiver coil (\({I}_{s}\)) is calculated as in (16) based on the Kirchhoff’s voltage and Kirchhoff’s current laws for \({Z}_{\mathrm{in}\_\mathrm{cc}}\), according to [1]:

$${I}_{s}=-j{V}_{AB}\sqrt{\frac{{C}_{f}{\prime}}{M}}=\frac{{V}_{AB}}{j{\omega }_{\mathrm{cc}}M}$$
(16)

Under the CC mode, \({I}_{o}\) can be calculated by substituting Eq. (16) in Eq. (10) according to [1]:

$${I}_{o}=\frac{2\sqrt{2}}{\pi }{I}_{s}=\frac{2\sqrt{2}{V}_{AB}}{{\omega }_{cc}\pi M}$$
(17)

Hence, the transconductance gain (\({G}_{T}\)) for the CC charging mode regardless of the load can be derived as (18):

$${G}_{T}=\frac{{I}_{o}}{{V}_{AB}}=\frac{2\sqrt{2}}{{\omega }_{\mathrm{cc}}\pi M}$$
(18)

The impedance parameters described in Fig. 4, according to the proposed LCC circuit, are given as follows:

$${Z}_{A}={R}_{AC}+\frac{1}{j\omega {C}_{s}}+j\omega {L}_{R}$$
(19)
$${Z}_{B}=\frac{{Z}_{A}j{\omega }_{\mathrm{cc}}M}{{Z}_{A}+j{\omega }_{\mathrm{cc}}M}$$
(20)

Moreover, the input impedance of the proposed topology is required to detect the ZPA condition in CC charging mode. Therefore, the input impedance (\({Z}_{in\_cc}\)) of the proposed topology can be calculated as (21) depending on \({Z}_{A}\) and \({Z}_{B}\) impedance parameters.

$${Z}_{\mathrm{in}\_\mathrm{cc}}={Z}_{B}+\frac{1}{j\omega {C}_{f}{\prime}}+j\omega {L}_{T}+j\omega {L}_{\mathrm{in}}$$
(21)

To conduct ZPA at the resonant frequency, the phase angle of the input impedance (\({\theta }_{\mathrm{in}\_\mathrm{cc}}\)) must be equal to zero. The input phase angle of the proposed topology is defined as (22). In this manner, zero-voltage switching can be adequately achieved for the CC mode.

$${\theta }_{\mathrm{in}\_\mathrm{cc}}=\frac{180^\circ }{\pi }{\mathrm{tan}}^{-1}\left[\frac{Im\left({Z}_{\mathrm{in}\_\mathrm{cc}}\right)}{\mathrm{Re}\left({Z}_{\mathrm{in}\_\mathrm{cc}}\right)}\right]=0$$
(22)

2.3 CV mode implementation of the proposed hybrid topology

In Fig. 3, the whole circuit model of the proposed topology is demonstrated. When the \({S}_{5}\) is switched on, the T–model corresponding circuit for the CV mode shown in Fig. 5 transforms to the LCC-S topology. The transmitter coil current \({I}_{p}\) is computed as follows to guarantee that \({L}_{\mathrm{in}}\) resonates with \({C}_{p}\).

Fig. 5
figure 5

Simplified equivalent T-model in CV mode charging

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Based on the Kirchhoff’s voltage and Kirchhoff’s current laws, the current of the transmitter coil is defined as in Eq. (23) according to [25]:

$${I}_{p}=-j{V}_{AB}\sqrt{\frac{{C}_{p}}{{L}_{\mathrm{in}}}}=-j\frac{{V}_{AB}}{{\omega }_{cv}{L}_{\mathrm{in}}}$$
(23)

Under constant \({V}_{AB}\) and \({L}_{\mathrm{in}}\), the primary current \({I}_{p}\) has a fixed value and is regardless of the parameters. To achieve ZPA condition for the CV mode in the receiver side of the topology regardless of the load, the \({C}_{s}\) must be selected to resonate with \({L}_{s}\). Depending on this requirement, the following equations must be satisfied:

$$0=j\left({\omega }_{\mathrm{cv}}{L}_{s}-\frac{1}{{\omega }_{\mathrm{cv}}{C}_{s}}\right)$$
(24)
$${{\omega }_{\mathrm{cv}}}^{2}=\frac{1}{{L}_{s}{C}_{s}}$$
(25)
$${V}_{ab}=j{\omega MI}_{p}=\frac{M{V}_{AB}}{{L}_{\mathrm{in}}}$$
(26)

Therefore, the voltage gain (\({G}_{V}\)) for the CV charging mode regardless of the load can be derived as (27):

$${G}_{V}=\frac{{V}_{ab}}{{V}_{AB}}=\frac{M}{{L}_{\mathrm{in}}}$$
(27)

Moreover, the receiver side impedance can be calculated as in (28).

$${Z}_{R}={j\omega }_{\mathrm{cv}}{L}_{R}-j\frac{1}{{\omega }_{\mathrm{cv}}{C}_{s}}+{R}_{AC}$$
(28)

Using Eq. (28), the input impedance \({Z}_{\mathrm{in}\_\mathrm{cv}}\) of the simplified equivalent T – model of the proposed LCC topology shown in Fig. 5 can be calculated as in (29):

$$\begin{aligned}& Z_{{{\text{in}}\_{\text{cv}}}} = \left( {\left[ {\left( {Z_{R} ||j\omega _{{{\text{cv}}}} M} \right) + \left( {j\omega _{{{\text{cv}}}} L_{T} + \frac{1}{{j\omega _{{{\text{cv}}}} C_{f} \prime }}} \right)} \right]} \right.\\&\quad||\left. {\left[ {\frac{1}{{j\omega _{{{\text{cv}}}} C_{p} }}} \right]} \right) + \left( {j\omega _{{{\text{cv}}}} L_{{{\text{in}}}} } \right)\end{aligned} $$
(29)

Also, the input impedance in CV mode \({Z}_{\mathrm{in}\_\mathrm{cv}}\) at ZPA can be calculated using the condition indicated in Eq. (21) and Eq. (9) when combined with Eq. (9) as follows:

$${Z}_{\mathrm{in}\_\mathrm{cv}}=\frac{8}{{\pi }^{2}}\frac{{{L}_{\mathrm{in}}}^{2}{R}_{o}}{{M}^{2}}$$
(30)

Additionally, the phase angle (\({\theta }_{\mathrm{in}\_\mathrm{cv}}\)) of the input impedance must be equal to zero to perform ZPA at the resonant frequency. Therefore, the input phase angle of the proposed topology in CV mode is defined as in Eq. (31). In this manner, the zero-voltage switching can be achieved properly for the CV mode.

$${\theta }_{\mathrm{in}\_\mathrm{cv}}=\frac{180^\circ }{\pi }{\mathrm{tan}}^{-1}\left[\frac{Im\left({Z}_{\mathrm{in}\_\mathrm{cv}}\right)}{Re\left({Z}_{\mathrm{in}\_\mathrm{cv}}\right)}\right]=0$$
(31)

3 Design procedures of the topology parameters

In general, compensation circuit parameters are important factors that influence the IPT system’s efficiency, output characteristics, and voltage/current stress [28]. The converter operates in the ZPA state when the resonant circuit parameters are designed according to the previous requirements. Therefore, the parameter design technique for the proposed switching hybrid compensation topology’s CC/CV output implementation is detailed below. As presented in Fig. 3, the proposed topology is composed of the SS and LCC-S compensation topology. To achieve the CC and CV charging modes, a tunable \({L}_{x}\) inductor is connected in parallel to the \({C}_{f}\) and an additional \({S}_{5}\) ACS is attached in series to the parallel capacitor on the primary side of the circuit. When \({S}_{5}\) is turned on, as shown in Fig. 4, the proposed circuit forms to LCC-S topology, which performs CV charging mode at the same resonant frequency as CC charging. Also, the equivalent capacitance values of \({C}_{f}{\prime}\) satisfying Eqs. (21) and (29) must be calculated for the ZPA condition in CC and CV mode, respectively.

When the value of \({C}_{f}\) is big enough, the parallel connection of \({L}_{x}\) and \({C}_{f}\) is regarded as an equivalent capacitance \({C}_{f}{\prime}\). As a result, as illustrated in Fig. 6, \({L}_{x}\) and \({C}_{f}\) are connected in parallel to fulfill the needed value of \({C}_{f}{\prime}\) for the ZPA requirement.

Fig. 6
figure 6

Equivalent circuit scheme for \({C}_{f}{\prime}\) with the parallel connection of \({L}_{x}\) and \({C}_{f}\)

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In order to achieve ZPA condition for each charging modes, the tunable \({L}_{x}\) must be adjusted to meet the proper value of \({C}_{f}{\prime}\). Accordingly, the tunable \({L}_{x}\) inductor value should be calculated for each charging mode depending the required value of \({C}_{f}{\prime}\) as defined in Eq. (32).

$${L}_{x}=\frac{1}{{\omega }^{2}\left({C}_{f}-{C}_{f}{\prime}\right)}$$
(32)

In this paper, a 2.4 kW WPT system is designed to implement the proposed topology circuit. The proposed system operates with a fixed frequency of 85 kHz for the CC and CV charging modes. The input DC voltage (\({V}_{\mathrm{DC}}\)) is 350 V, and the required battery current is 6 A. When the battery’s voltage exceeds 400 V, the charging mode switches from CC to CV. At this point, the \({S}_{5}\) is turned on and \({L}_{x}\) inductor value is changed, which satisfies the ZPA condition for the CV mode. Table 1 lists the required output qualities of the studied IPT charging. In addition, The CC/CV charging curve and equivalent impedance of the Li-ion battery utilized in this work are shown in Fig. 7.

Table 1 Specifications of the IPT battery charger
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Fig. 7
figure 7

CC/CV charging profile and equivalent circuit of the used Li-ion battery

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Depending on the IPT coupler side and coil geometry of the existing transformer, which is planning to be used for this study, the self-inductance value of each coil is measured as 252 µH for the primary and 244.6 µH for the secondary side.

From Eq. (18), the transconductance gain (\({G}_{T}\)) in CC mode can be calculated as follows:

$${G}_{T}=\frac{{I}_{s}}{{V}_{\mathrm{in}}}=\frac{{I}_{o}\times 2\sqrt{2}}{{V}_{\mathrm{in}}\times \pi }\approx 0.02$$
(33)

Using the IPT charger specifications presented in Table 1, the mutual inductance from Eq. (18) is calculated as follows:

$$M=\frac{2\sqrt{2}}{{\omega }_{\mathrm{cc}}\pi {G}_{T}}\approx 89 \, {\upmu {\mathrm H}}$$
(34)

Then, \({L}_{\mathrm{in}}\) can be determined from the voltage gain (\({G}_{V}\)) in the CV charging mode shown in Eq. (26) as follows:

$${L}_{\mathrm{in}}=\frac{M}{{G}_{V}}=\frac{M\times {V}_{\mathrm{in}}}{{V}_{\mathrm{out}}}\approx 78.7 \, {\upmu {\mathrm H}}$$
(35)

Additionally, the serial capacitor value of the secondary part \(({C}_{s})\) can be obtained from Eq. (25) as follows:

$${C}_{s}=\frac{1}{{L}_{s}{\omega }_{\mathrm{cv}}^{2}}\approx 14.33 nF$$
(36)

In Fig. 7, four operation points (25 Ω, 67 Ω, 90 Ω and 200 Ω) are selected due to observing the all over the performance of the proposed design. To realize the required battery charging conditions, the prototype’s designed circuit resonant circuit parameters are listed in Table 2.

Table 2 Component parameters of the designed hybrid topology
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However, the circuit parameters must be determined to implement the ZPA in CC and CV mode charging at the same resonant frequency. The imaginary components of Eqs. (21) and (29) should be set to zero to achieve ZPA for both charging modes. Depending on the specifications in Table 2, also, \({C}_{f}{\prime}\) is determined as 10.6 nF for the CC mode and 20.23 nF for the CV mode. Hence, the tunable \({L}_{x}\) inductor value can be calculated by using Eq. (32) for both modes as follows:

$${L}_{x(\mathrm{cc})}=\frac{1}{{\left(2\pi \times 85\right)}^{2}\times \left(30-10.6\right)\times {10}^{-9}}\cong 180 \, {\upmu {\mathrm H}}$$
(37)
$${L}_{x(\mathrm{cv})}=\frac{1}{{\left(2\pi \times 85\right)}^{2}\times \left(30-20.23\right)\times {10}^{-9}}\cong 360 \, {\upmu {\mathrm H}}$$
(38)

Moreover, the value of each circuit component is necessary to be calculated depending on the desired output current–voltage levels and selected resonant frequency. Hence, the design procedures to determine component values require some calculating iterations, as presented in Fig. 8, to meet the required CC and CV modes for the better charging. To calculate the exact values of the component parameters, a feasible flowchart for the proposed hybrid topology is illustrated in Fig. 8 to perform the CC charge with the following CV charge profile.

Fig. 8
figure 8

The flowchart to specify the topology parameters of the proposed system

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However, the battery equivalent load changes the entire charging process, affecting the efficiency of the whole topology. Therefore, in the topology design, the input impedance should be adjusted to be independent of the equivalent load, and the reactive power component should be minimized. Hence, this provides to reduce the switching losses of the inverter side of the circuit and improves system performance for each charging process. To verify the proposed design, a mathematical analysis was performed referring to the circuit parameters listed in Table 2. In this context, Fig. 9 depicts the transconductance gain (\({G}_{T}\)) and the input phase angle (\({\theta }_{in\_cc}\)) under the CC charging mode for various load conditions of 20 Ω, 40 Ω, 60 Ω and 80 Ω at different frequencies. Figure 10 shows the voltage gain (\({G}_{V}\)) and the input phase angle (\({\theta }_{in\_cv}\)) at different frequencies under the CV charging mode. Figures 9b and 10a show that \({\mathrm{G}}_{\mathrm{T}}\) and \({\mathrm{G}}_{\mathrm{V}}\) are constant at the same at 85 kHz regardless of the load for the CC and CV modes, respectively.

Fig. 9
figure 9

a Voltage gain (\({G}_{V}\)), b transconductance gain (\({G}_{T}\)), c phase angle (\({\theta }_{\mathrm{in}\_\mathrm{cc}}\)) of the proposed topology for the CC mode

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Fig. 10
figure 10

a Voltage gain (\({G}_{V}\)), b transconductance gain (\({G}_{T}\)), c phase angle (\({\theta }_{\mathrm{in}\_\mathrm{cv}}\)) of the proposed topology for the CV mode

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As demonstrated in Fig. 9c and Fig. 10c, the phase of the input impedances equals to zero for both charging modes. And, it provides fixed current and voltage output levels regardless of the battery load at the 85 kHz resonant frequency. Thus, ZPA is achieved successfully for each charging mode at the same operating frequency, which improves system efficiency and VA rating of the designed topology.

4 Simulation results and system topology comparison

4.1 Simulation results

A model of 2.4 kW IPT was created to simulate and verify theoretical results for CC and CV operation modes to confirm the investigated hybrid topology circuit and designs. In the simulation study, a DC voltage source with 350 V input was used as the input of the proposed hybrid topology, and the assigned battery charge voltage (\({V}_{o}\)) and current (\({I}_{o}\)) values are 400 V and 6 A, respectively. Furthermore, the proposed circuit is built to provide the maximum power for the critical charging point between CC and CV modes.

Consequently, Fig. 11 depicts the proposed hybrid topology, which is built using the parameters stated in Table 2. Also, the required simulation analysis was realized in PLECS with MATLAB/Simulink environment between CC and CV modes, referring to Table 2. To be more accurate to the real application of the circuit, non-ideal switches, transformers and inductors are utilized in the simulation. The output power and system efficiency achieved for various battery resistance values are assessed in this manner. According to the topology’s mathematical analysis, the ZPA condition is satisfied for both CC and CV modes at a single resonant frequency of 85 kHz, as shown in Fig. 9 and Fig. 10.

Fig. 11
figure 11

Simulation setup for the proposed LCC compensation topology

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As illustrated in Fig. 11, the proposed topology contains a single ACS, which generally consists of two series-opposite connected MOSFETs in the WPT applications. In addition, the operating mode of the circuit changes according to whether S5 is logically high or low, and its logic state is selected as on or off by detecting the battery output voltage. When the output voltage is measured below as 400 V reference voltage, the logic state becomes logic low, and S5 is turned off. Otherwise, the S5 becomes logic high and turned on. The control signal waveform of S5 is presented in Fig. 12.

Fig. 12
figure 12

Logic control waveform for the S5 in CC and CV modes

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To obtain variable inductor, there are different ways such as changing the ferromagnetic core operating points [31] or relay-controlled inductor. For simplicity, the second control approach is implemented. The tunable \({L}_{x}\) inductor is also controlled in the same trajectory as the S5. In this regard, Fig. 13 shows the equivalent circuit model of the \({L}_{x}\) inductor. As presented in Fig. 13, the tunable \({L}_{x}\) inductor contains two \(180\mathrm{ \mu H}\) inductors and a controllable relay. Depending on the logical state of S5 illustrated in Fig. 12, the relay is enabled depending on whether CC or CV mode is selected. In CC charging operation, a single \(180\mathrm{ \mu H}\) inductor is normally connected to the circuit in parallel with \({C}_{f}\). In CV mode, two \(180\mathrm{ \mu H}\) are connected in series in parallel with \({C}_{f}\) by activating the controllable relay.

Fig. 13
figure 13

Equivalent circuit model for the tunable \({L}_{x}\) inductor

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As a result, a detailed control flowchart for controlling the single S5 ACS switch used in the proposed circuit and the relay used to set the inductor \({L}_{x}\) is shown in Fig. 14. As presented in Fig. 14, the suggested controller structure has a straightforward design, making it easy to implement and control. According to the control method, the battery reference voltage is set to 400 V.

Fig. 14
figure 14

Flowchart of the CC and CV mode control of the proposed circuit

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In CC charging mode, the S5 is turned off, and the topology transforms into an SS compensation circuit. Hence, the primary parallel capacitor \({C}_{p}\) is disconnected and neglected for the CC mode operation. Additionally, the tunable \({L}_{x}\) inductor value is set to \(180\mathrm{ \mu H}\). Figure 15 demonstrates the main waveforms of the hybrid LCC circuit for the CC mode charging with 350 V DC input. As observed in Fig. 15a, the \({I}_{o}\) charging current remains at 6 A, when the equivalent resistance of the battery (\({R}_{o}\)) varies from 25 Ω to 67 Ω, and the switching frequency remains as same 85 kHz resonant frequency. As a result, the CC mode charging is met for various battery equivalent resistances, as illustrated in Fig. 7. In addition, the input current and voltage waveforms of the hybrid compensation circuit are also shown in Fig. 15b. As investigated in Fig. 15b, the ZPA condition is succeed for the CC charging mode since the zero-voltage-switching (ZVS) condition of the S1 ~ S4 switches is realized.

Fig. 15
figure 15

The CC mode operation waveforms for the LCC circuit, a output voltage and current waveforms, b input current and voltage waveforms

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In CV charging mode, the S5 is turned on, and the topology transforms into an LCC-S compensation circuit. Therefore, the primary parallel capacitor \({C}_{p}\) is connected in parallel and considered for the CV mode operation. Also, the tunable \({L}_{x}\) inductor value is set to \(360\mathrm{ \mu H}\). The proposed LCC circuit’s waveforms during CV mode charging with a 400 V DC input are shown in Fig. 16. In CV mode operation, the equivalent resistance varies from 90 to 200 Ω. When the equivalent resistance of the battery (\({R}_{o}\)) increases from 90 to 200 Ω, and the output battery charging voltage remains at 400 V, as shown in Fig. 16a. The switching frequency remained constant at 85 kHz, the same as the CC mode resonant frequency. As a result, the CV mode charging is achieved for various output resistances. The input voltage and current waveforms of the hybrid topology are also demonstrated in Fig. 16b. As investigated in Fig. 16b, the ZPA condition is succeed for the CV charging mode since the zero-voltage-switching (ZVS) condition of the S1 ~ S4 switches is realized.

Fig. 16
figure 16

The CV mode operation waveforms for the LCC circuit, a output voltage and current waveforms, b input current and voltage waveforms

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When \({R}_{o}\) changes from 25 to 67 Ω in CC mode, the output charge current \({I}_{o}\) remains virtually constant while the output charge voltage \({V}_{o}\) rises steadily. In the CV charging mode, the output charge voltage \({V}_{o}\) remains almost constant while the output charge current \({I}_{o}\) continues to drop, when \({R}_{o}\) is varied from 90 to 200 Ω. Furthermore, the ZPA condition is achieved with a nearly zero phase angle. Figures 15 and 16 show the transient waveforms for both CC and CV charging modes as the \({R}_{o}\) varies. As a result, the possible significant damages caused by high current/voltage spikes can be avoided due to its fast response time according to the load changes. Hence, the fundamental advantage of the designed hybrid topology is that the CC and CV charging modes can be achieved at a single resonance frequency regardless of the battery load.

Figure 17 presents the overall efficiency and output power in the simulation analysis of the proposed topology for four different loads investigated before. As examined from the efficiency results, it can be observed that the average efficiency of the CV mode is better than the CC mode due to the switching of the circuit from SS to LCC-S topology. At the maximum power of the circuit, the average efficiency is measured as 98.66%, and the output power equals to 2.4 kW approximately. Also, the average efficiency increases as the output load increases in CV mode.

Fig. 17
figure 17

Overall system efficiency and output power for the CC and CV charging modes

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4.2 Effect of coupling factor change

The considerable degree of coil freedom presents a challenge for IPT applications. Therefore, misalignment and distance variations are inevitable between the primary and secondary pads of the transformer. It can also cause undesirable variations in the output voltage and current levels. Consequently, the operation of the designed topology should be examined under varying coupling factors. Therefore, the proposed topology was also examined for various coupling factors, including k = 0.3 and k = 0.4. For each charging mode, the ZPA condition was also investigated.

Figure 18 depicts the output current and voltage waveforms of the proposed LCC circuit for CC and CV modes with an input voltage of 350 V and a coupling factor of k = 0.3. As a result, the output current reaches 7.1 A, as depicted in Fig. 18a, which is slightly greater than the desired 6 A and remains constant regardless of load variations. In addition, the output voltage remains constant regardless of load shown in Fig. 18c. Consequently, the CC and CV modes are attained, albeit at various current and voltage levels. In Fig. 18b and d, current and voltage simultaneously cross the zero point. Therefore, the ZPA condition is specified for both charging phases.

Fig. 18
figure 18

The output current and voltage waveforms of the LCC circuit for k = 0.3 coupling factor, a output voltage and current waveforms in CC mode, b input current and voltage waveforms in CC mode, c output voltage and current waveforms in CV mode, d input current and voltage waveforms in CV mode

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Figure 19 illustrates the waveforms of the proposed LCC circuit for the CC and CV modes with a coupling factor of k = 0.4. Figure 19a reveals that the output current approaches 5.37 A, which is lower than the desired 6 A, and remains constant regardless of load variations. In CV mode, as depicted in Fig. 19c, the output voltage remains constant at 441.7 V, which is substantially higher than the 400 V desired value. In Fig. 19b and d current and voltage simultaneously cross the zero point. Therefore, the ZPA condition is specified for both charging phases.

Fig. 19
figure 19

The output current and voltage waveforms of the LCC circuit for k = 0.4 coupling factor, a output voltage and current waveforms in CC mode, b input current and voltage waveforms in CC mode, c output voltage and current waveforms in CV mode, d input current and voltage waveforms in CV mode

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In the simulation analysis of the proposed topology, Fig. 20 depicts the overall efficiency for various coupling factor values, such as 0.3 and 0.4. Due to the misalignment and distance variations of transformer pads, the coupling factor parameter becomes a crucial issue. Examining the efficiency results, it can be seen that the average efficiency for the CV and CC modes has nearly remained constant in comparison with its original target value. At the circuit’s maximum power, the average efficacy is measured to be nearly 98.6%. Additionally, as the output load increases in CV mode, the average efficiency increases. Consequently, the average efficacy of the CV mode is greater than that of the CC mode as a result of the circuit’s transition from SS to LCC-S topology.

Fig. 20
figure 20

Overall system efficiency for k = 0.3 and k = 0.4 coupling factors under the CC and CV charging modes

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4.3 System topology comparison

Some proposed LCC circuits, such as [6] and [25], perform CC and CV operations for different load levels requiring a frequency control module. Contrary to this, the proposed topology can realize CC and CV charging with ZPA condition at a single frequency level. In practice, varying frequency according to the charging mode is not necessary. Therefore, a switching module is not required to achieve the CC and CV charging modes. In addition, according to the mathematical analysis and simulation results of the proposed topology, it has been verified that it can operate in CC and CV mode with ZPA at 85 kHz resonance frequency for different load conditions.

Switching between CC and CV charging modes can be implemented by simply controlling only one ACS at the same resonant frequency. As a result, the control circuit is easier and more feasible to implement as it requires single ACS control with simple control logic between the charging modes. Also, the proposed topology is less complex compared to other fixed frequency controlled and coupling coefficient tuned resonant circuits due to its reduced number of components. In the proposed topology, using an extra capacitor on the transmitter side is not required to perform the CC and CV modes compared to the other studies. Moreover, the proposed topology includes fewer capacitors to achieve ZPA condition in both charging modes at a single resonant frequency. Hence, this provides the topology extending the lifespan of the circuit and making it less bulky due to the use of a reduced number of capacitors. Additionally, a comparison table among the proposed compensation topologies and discussed circuitries in the literature is demonstrated in Table 3.

Table 3 Comparison review between the proposed study and other related topologies
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Firstly, compared to the four main topologies such as SS, SP, PS and PP, the proposed topology is capable of performing CC and CV charge modes simultaneously with ZPA condition, but contains more components and increased circuit control complexity. Secondly, compared to the double-sided LCC topology demonstrated in [5] and [18], the proposed topology includes fewer components on the secondary side. Besides, the proposed system performs each charging mode without needing to change any mutual inductance value or resonant switching frequencies contrary to [5] and [18], simplifying the system’s controlling process. The maximum efficiencies for the proposed topologies in [5] and [18] are 97.7 and 96.1%, respectively. The simulated efficiency of the proposed circuit is slightly better than the topologies studied in [5] and [18]. Third, the LCC-S topology presented in [6] requires dual resonant frequency control to realize CC and CV output under ZPA condition, and the measured efficiency is 97.3% in [6]. This means different resonant frequency control for each charging mode, further complicating the control of the switches on the HV inverter side of the circuit. According to the study in [6], the proposed circuit requires the use of an additional tunable inductor and an ACS on the primary side of the circuit as a disadvantage. However, the proposed topology is lightly higher compared to [6]. Fourthly, the proposed design includes fewer capacitors according to the modified LCC-S circuitries presented in [24, 25] and [26]. In [24], the proposed topology uses a single switch, but it requires switching frequency tunning to perform CC/CV charging modes. Thus, it causes increased control complexity by adjusting the resonant frequency and encounters EMI problems. Additionally, the efficiency of the proposed system in [24] is 87%. In [25], the designed topology includes an additional capacitor and an ACS for each side of the circuit compared to the simple LCC-S circuit and the efficiency is measured as 89.28%. Moreover, two-sided-ACS control is required to provide stable CC and CV output, which causes sophisticated controlling and harms the WPT system’s robustness. Also, the topology shown in [26] contains two additional ACSs and a single capacitor and inductor components on the secondary side, converting LCC-S to double-sided LCC. Moreover, the measured efficiency is 91% in [26]. Although these modified LCC-S circuitries are capable of performing CC and CV charging modes regardless of the battery load at a single switching frequency, they comprise more capacitors according to our proposed topology, which makes the circuits bulkier and complicated and shortens their lifetime. Compared to the studies in [24, 25] and [26], the simulated system performance of this study is enhanced, and the efficiency level is improved.

Although the use of a variable inductor in the proposed hybrid topology may introduce some challenges in terms of increased size and complexity compared to alternative approaches, it is important to note that these considerations must be weighed against the unique benefits it offers. The proposed topology achieves the load-independent ZPA condition, which enables efficient and seamless transitions between CC and CV charging modes at a single resonant frequency using a single AC switch. This simplifies control and operation, enhancing the overall charging efficiency and prolonging the battery lifespan. In addition, the reduced number of capacitors required by the proposed topology results in cost savings and enhanced reliability. Furthermore, the tunable inductor offers flexibility in system design and optimization, allowing for the adjustment of performance parameters. These benefits, along with the fixed operating frequency and the simplicity of a single switch, contribute to the development of IPT systems for efficient and dependable battery charging. Therefore, despite the potential size and complexity considerations, the proposed hybrid topology offers notable advantages that justify its simple operation, increased reliability and further efficiency.

5 Conclusion

For wireless battery charging applications, load-independent CC and CV outputs at a single resonant frequency are possible by concentrating on the LCC-S compensation topologies. Hence, this study proposes a novel SS and LCC-S hybrid topology based on a tunable inductor and single ACS for inductive charging system to be used for inductive power transfer applications. By combining the SS and LCC-S topologies, the charging modes can be switched through a single ACS and a tunable inductor located on the primary side of the circuit. Therefore, the control complexity of the circuit can be reduced, and it makes the proposed circuit simpler. An additional benefit of the study is that CC and CV charging with ZPA condition can be provided at a fixed resonant frequency. Hence, this feature increases the reliability of the system.

In this paper, a 2.4 kW simulation prototype was built and analyzed extensively to validate the proposed circuit. The design techniques, and the mathematical analysis have been studied in detail. Furthermore, the simulation results of the output current–voltage levels have been verified for various battery equivalent resistances. As a result, the desired charging current value of 6 A and the desired charging voltage value of 400 V were effectively achieved regardless of the battery charge level in CC and CV modes, respectively. The simulated efficiency of the maximum system power is measured as 98.66%. Hence, the simulation experiments approved the high stability of the proposed topology and its feasibility.