1. Introduction
In recent years, inductive power transfer (IPT) technology has made rapid progress [
1,
2]. As a promising technology, IPT technology can transfer energy over an air gap of a certain size via a high frequency magnetic field. IPT technology has been successfully applied in several low power applications, such as mobile phones [
3], sensors [
4] and so on. For high power applications (rail locomotives, electric cars, buses and other hybrid electric vehicles) [
5,
6], it is a huge challenge for the single-receiver IPT system to get enough power from the transmitter other than from multiple-receiver systems, which could dramatically increase the capacity of receiving power.
Generally, output power and system efficiency are two of the most important concerns for IPT systems. However, the limitation of semiconductor devices’ capacity makes it hard for a traditional single-receiver IPT system to meet the heavy load demand. Therefore, the multiple-receiver approach may be a promising solution to meet the high power demands of railway charging applications. A long transmitter track is usually employed to generate high magnetic fields in IPT systems for railway applications [
7]. As the train body is normally long, parallel-connected multiple receivers could be mounted together side by side along the track to pick up the magnetic power to feed the load as shown in
Figure 1. These parallel-connected receivers can pick up more power in total than a single receiver does without increasing the capacity of semiconductors.
Many researchers have paid attention to multiple-receiver IPT systems. The IPT system with dual secondary windings is analyzed in [
8] and a general tuning method with any combination of secondary side compensation schemes is also proposed. The closed form for realizing maximum power transfer and maximum efficiency transfer of the IPT system, which adopted one transmitter and two receivers, is investigated in [
9] and an optimal range of load resistance was determined accordingly. An equivalent model for any multiple-receiver system is given in [
10] and the optimal load is analyzed.
The researches above are based on the assumption that the cross coupling between the multiple receivers are zero. However, receivers are placed very close to each other due to the limited available space of the train body. The cross coupling between receivers could have great influence on the adjacent one and should not be ignored. If cross coupling exists between the multiple receivers, the resonant frequency of the coupled receivers is changed [
11] and the system efficiency also decreases accordingly [
11,
12].
In order to suppress the adversity of the cross coupling between the receivers, some methods have been proposed to improve the performance of IPT systems. An impedance matching method considering the cross coupling between the source coil and receivers is proposed and a design procedure for a single source coil and the optimal load impedance are presented in [
13]. The operational frequency for maximum efficiency transfer under the cross coupling between receivers is illustrated by adjusting the operational frequency to tune the receivers in [
11]. A load impedance matching algorithm is proposed in [
14] to control power allocation between two receivers, so that system efficiency is improved. Additional inductive or capacitive reactance is employed to suppress the influence of cross coupling between receivers in [
12].
The efficiency improvement methods for multiple-receiver IPT system mentioned above assume a constant optimal load value. However, the equivalent resistance of the load, for example, in a battery, can go from 3.6 Ω to 560 Ω depending on its charging profile [
15]. As a result, it is hard for the system to work in the efficiency-optimized load range all the time. Some maximum efficiency point tracking (MEPT) methods for single-receiver IPT systems have been proposed [
16]. A boost converter is employed in the secondary side to control the input voltage of the rectifier to maintain high efficiency in [
17]. A DC/DC converter either on the primary side or the secondary side is applied to search for the maximum efficiency point by adjusting the duty cycle in [
18,
19]. In addition, an active rectifier can be applied to maintain constant the current ratio between the transmitter coil and receiver coils in order to optimize system efficiency without an additional DC/DC converter in [
20].
From the discussion above, some progress has been made in the analysis and efficiency improvement of multiple-receiver IPT system in a static state, but there are two main problems that need to be addressed for a practical high power multiple-receiver IPT system under dynamic conditions:
- (1)
When the receivers move along the transmitter track, the relative positions between the receivers and the transmitter track change from time to time. Therefore, the induced voltage on each receiver coil changes and this leads to diversity in the receiver currents. As a result, receiver currents will impose an induced voltage on each other. Then, the whole system will suffer from detuned conditions. As a result, both the system efficiency and output power capacity are affected.
- (2)
With the different power demands during operation, the equivalent load of the train changes randomly, and may deviate from the designed optimal load value. The random load will greatly influence the system efficiency.
In order to overcome these problems, an analysis of a dual-receiver IPT system is carried out in this paper and a compensation approach to suppress cross coupling influence is proposed. To improve the overall system efficiency, the optimal load reactance is derived according to the relationship between the parasitic resistances and the cross coupling between receivers. Then, a MEPT control scheme is proposed to track the maximum efficiency point.
This paper is organized as follows:
Section 2 analyzes the effect of cross coupling between receivers under various operational conditions and an approach to improve overall system efficiency is presented. A MEPT control scheme, in which the inverter searches for the maximum efficiency point and the active rectifiers of the dual receivers regulate the output voltage, is proposed in
Section 3. A 2 kW dual-receiver IPT system is set up in
Section 4 to verify the performance of MEPT control scheme. Finally, the conclusions of this paper is drawn in
Section 5.
2. Analysis on Dual-Coil Receiver IPT System
As illustrated in
Figure 2a, the primary side consists of a DC voltage source
E, a full-bridge inverter, the transmitter coil
LP with the equivalent series resistance (ESR)
RLP and the resonant capacitor
CP.
LP and
CP are supposed to be tuned to be resonant at the switching frequency of the inverter. Each of the two secondary sides consists of the receiver coil
LS1 (
LS2) with the ESR
rLS1 (
rLS2) and an active rectifier. The components of both secondary sides are supposed to be the same. The two active rectifiers are parallel-connected to the DC load
Rdc. The primary side circuit generates a high frequency current in the transmitter coil, which is loosely coupled with the two receiver coils. The current in the transmitter coil excites an alternative magnetic field and a high frequency voltage is induced in the receiver coil of each receiver.
MP1 and
MP2 are the mutual inductances between the transmitter and two receivers. The cross coupling between two receivers is
M12.
Figure 2b shows the equivalent circuit of the dual-receiver IPT system in which each receiver is connected with one equivalent load resistance seen from the active rectifiers
R1 (
R2). The equivalent load resistance seen from the active rectifiers can be described as follows:
Rac is defined as the AC equivalent resistance.
2.1. Effect of the Cross Coupling between Two Coils
From the equivalent circuit in
Figure 2b, the induced receiver voltages can be expressed as:
As shown in Equation (2), when M12 = 0, the induced receiver voltages are independent. When M12 ≠ 0, the amplitude and phase of induced receiver voltages interact with cross coupling between two receivers, so the cross coupling between receivers could impact the performance of the system and it is necessary to find a way to eliminate the effects of this cross coupling between receivers.
2.2. Self-Inductance Compensated Scenario
Based on the assumption that the capacitors in the primary side and the secondary sides fully compensate the self-inductances of the transmitter coil and receiver coils, the ESRs of the coils are
rLP,
rLS1 and
rLS2. The receiver coils are identical, so it is assumed that
rLS1 =
rLS2 =
rLS. The relationships between the input voltage
, the currents
,
and
, can be described according to the Kirchhoff’s voltage law as:
where:
.
By solving Equation (3), the input voltage and receiver currents can be provided by:
Assuming the self-inductances of receivers are the same (
LS1 =
LS2 =
LS), and the mutual inductances between the transmitter and receivers are equal to each other (
MP1 =
MP2 =
MP), the equivalent load resistances are identical as follows:
By substituting Equation (5) into (4), the input voltage and receiver currents can be obtained by:
According to Equation (6), the primary side and secondary sides are detuned, so the performance of the system is impacted. From Equation (6) with the assumption that
, the output voltage can be described as:
When the mutual inductances between the transmitter and two receivers are the same, the cross coupling between receivers will reduce the system output voltage amplitude. The larger the mutual inductance M12 is, the more the output voltage will be affected.
The voltage-current gain (the receiver voltage versus transmitter current) is given by:
In a well-tuned IPT system
GVI is not related to the load.
Figure 3 shows the relationship between voltage-current gain ratio
and
. The voltage-current gain ratio decreases with the increase of ρ. In other words, the cross coupling between receivers has a serious effect on output voltages under heavy load conditions (small load resistance).
Besides, the output power can be expressed as:
where * is the conjugate symbol.
Figure 4 shows the relationship between output power ratio
and ρ. According to
Figure 4, the output power ratio decreases with the increase of ρ. When ρ = 1, the output power is equal to 80% of a well-tuned one.
From Equations (8) and (9), the cross coupling between receivers reduces both the output power and the voltage-current gain (the receiver voltage versus transmitter current). The cross coupling between receivers would seriously affect IPT system in heavy load situations. Therefore, it is necessary to find a way to suppress the adverse effects of cross coupling between receivers.
2.3. Cross Coupling Compensated Scenario
In order to compensate the cross coupling between the two receivers, additional reactance is applied in the secondary sides to overcome the drawbacks caused by the cross coupling between receivers as shown in following equation:
where:
XS1 and
XS2 are the additional reactances supposed to be added in the secondary sides. By solving Equation (10), the input voltage and receiver currents can be obtained by:
According to Equation (11), the apparent power transferred from the transmitter to the secondary sides can be described as follows:
Let the imaginary part of Equation (12) be zero:
Then,
XS can be expressed as:
By substituting Equation (14) into
Z1 and
Z2, the compensated capacitor is derived as:
Then Equation (11) can be simplified as:
From Equation (16), the output voltage can be described as:
From Equation (17), when the mutual inductances between the transmitter and two receivers are the same, additional capacitors could be added to tune the IPT system.
The voltage-current gain (the receiver voltage versus transmitter current):
The output power is attained by:
The adverse influence of the mutual inductance between receivers can be eliminated by additional capacitors added to the secondary side. Not only the voltage-current gain (the receiver voltage versus transmitter current) can be improved, but also more power can be transferred from the transmitter to the receivers.
2.4. Different Mutual Inductance Scenario
The mutual inductances between the transmitter and receivers would change as the receivers move along the transmitter track. Letting
MP1 = λ
MP2 and substituting Equation (15) into (10), Equation (10) can be simplified as follows:
By solving Equation (20), the input voltage and receiver currents can be provided by:
where:
According to Equation (21), when the mutual inductances between the transmitter and two receivers are different, the amplitude and phase of receiver currents are different from each other. Reactive power is transferred amongst the transmitter and receivers. The additional capacitor compensation only works under the condition that two receiver currents are identical.
With these relationships, the apparent output power is given by:
Let the imaginary component of Equation (22) be zero:
R1 >>
rLS and
R2 >>
rLS, then
R1 can be solved as:
From Equations (1) and (24),
R1 and
R2 can be expressed as:
By substituting Equation (25) into (21), we have:
Hence, in order to tune the system with different mutual inductances
MP1 and
MP2, the receiver currents should be controlled to be the same as shown in Equation (26). When Equation (26) is met, the output voltage and the output power of two receivers can be described as:
It is clear that the difference of the two mutual inductances between the transmitter and the receivers changes the resonance characteristics of the system. Only when receiver currents are identical, the cross coupling between receivers can be compensated by a capacitor. Then the whole system is under resonant conditions. Not only under light load conditions, but also under heavy load conditions, the whole system can be operated under resonant conditions.
2.5. Adjust the Receiver Current
The RMS value of the first order harmonic of one active rectifier’s input voltage can be expressed as [
21]:
where α is the pulse width of the active rectifier.
As shown in
Figure 5, the output DC current can be described as:
According to Equations (28) and (29), the equivalent input resistance of each active rectifier can be described as:
From Equation (30), the resistance seen from the active rectifiers changes according to the pulse widths of active rectifiers. When the mutual inductances between the transmitter and receivers are different, active receiver currents are different from each other. According to Equations (26) and (30), the reduction of the pulse widths of active rectifiers could decrease its equivalent resistance and increases its current. Therefore, by adjusting the pulse width of active rectifier, active receiver currents could be regulated to be identical and a tuned IPT system is obtained.
2.6. Maximum Efficiency Point
In order to analyze the maximum efficiency point of the system, ESRs of the coils (
rLP,
rLS1 and
rLS2) are taken into account. When receiver currents are the same at all the same. Letting
,
rLS1 =
rLS2 =
rLS and substituting Equation (24) into (20), Equation (20) can be simplified as follows:
The input and output currents can be solved as:
The output and input power are obtained by:
Then the efficiency of the system can be derived by:
By substituting Equation (25) into Equation (35), the system efficiency can be expressed as:
According to Equation (36), the system efficiency is related to the load resistance value. The relationship between the system efficiency and the load resistance value is plotted in
Figure 6. It is clear that a maximum efficiency point exists with the variation of the load. The resistance value corresponding to the maximum efficiency point can be solved by setting the derivative of the system efficiency to be zero as follows:
Then the optimum load can be solved as: