Based on the LCC-S resonant compensation topology, the structure of the electric vehicle DWPT system is shown in
Figure 1.
represents the series compensating inductance at the transmitter end, while
represents the parallel compensating capacitance.
indicates the inductance of the receiving coil, and
represents the series compensating capacitance at the receiver end. M represents the mutual inductance between the power supply rail and the receiving coil.
,
, …
,
denote the series compensating capacitance at the transmitter end. In the centralized power−supply rail mode, the series compensating capacitance at the transmitter end adopts a distributed compensation structure, effectively preventing a steep increase in voltage at both ends of the power supply rail due to long-distance laying.
The resonant compensation topology at the transmitter end, as shown in
Figure 2, can be equivalently represented by the circuit model illustrated in
Figure 1.
,
, …
,
is each segment of the power supply rail, while
,
, …
,
and
,
, …
,
can be combined and represented as a total inductance
Lp for ease of analysis and calculation. Accordingly, the transmitter end of the LCC-S topology is depicted in
Figure 3, in which
represents the inverter output voltage,
represents the inverter output current, and
represents the power supply rail current.
By disregarding the parasitic resistances of compensating inductance and transmitter coil inductance and considering the existence of higher-order impedances in the system, the high-order impedance at the transmitter end is exclusively odd-order owing to the frequency response characteristics and non-linear properties of inductance, capacitance, and resistance elements. The high-order impedance of the system is denoted as
, where “2n − 1” represents the odd-order impedance, with “n” being a positive integer. Then, the expressions
and I
in are as follows:
2.1. Unilateral Resonant Compensation Topology
It is widely acknowledged that when wires or cables are buried underground or in close proximity to the ground, the charge distribution of these conductors near the ground induces an electric field, leading to the generation of capacitive currents. The power supply rail at the transmitter end of the DWPT system, being essentially one or more long, straight wires, inevitably exhibits a capacitive coupling with the earth, resulting in a capacitance to ground [
20]. Moreover, for the power supply rail to effectively couple with the magnetic field at the receiver end and achieve wireless power transfer, it must carry high-frequency AC current exclusively. Therefore, a high-frequency voltage difference, referred to as the voltage to earth, constantly exists between the power supply rail and the ground during its operation. This voltage to earth serves as the primary cause of leakage current generation.
In the practical application of the DWPT system, when the length of the power rail at the transmitting end remains constant and the vertical distance between the rail and the ground remains constant, the capacitance between the power rail and the ground can be considered constant. According to the formula “
i = Cdu/dt”, the voltage of the power rail with respect to the ground is another key factor affecting the magnitude of leakage current. As shown in
Figure 4, by neglecting the parasitic capacitance between the transmitter’s switch power supply and the heat sink, the ground capacitance between the power supply rail and the earth, denoted by
and
, emerges as the primary factor contributing to the generation of capacitive currents at the transmitter end. Here,
represents the DC bus voltage, and A and B represent the two endpoints of the power supply rail, while point C denotes the negative terminal of the inverter’s DC bus. Although a certain amount of noise current is generated by the negative pole of the inverter circuit with respect to ground, it can be reasonably assumed that point C is at the same potential as PE concerning high-frequency voltage coupling, given that point C is adequately grounded through the Y capacitor [
21]. Consequently, the voltage to earth of the power supply rail can be regarded as the voltage between points A and B relative to point C.
In accordance with the operational principle of the full-bridge inverter circuit, the Fourier series expansion of can be expressed as follows:
Based on
Figure 4, the duration of one working period for the inverter circuit is designated as T. Switches S
1 and S
4 are opened in the interval of 0 to T/2 and closed in the interval of T/2 to T, while switches S
2 and S
3 operate in the opposite manner. The voltage between points B and C, denoted as
, represents the voltage across switch S
4. When switch S
4 is closed in the inverter circuit, the voltage between points B and C is zero. When switch S
4 is open, the voltage between points B and C becomes
.
Hence, the Fourier series expansion of
can be expressed as follows:
The Fourier series expansion of U
S2 can be represented as follows:
The Fourier series expansion of U
Lf can be represented as follows:
Simplify Equation (7) to:
Let
, then the Fourier series expansion of
can be represented as follows:
In DWPT systems, because
Lf =
Lp and
LfCf = 1/ω2, the expression “
a” can be reduced to:
“
a” gradually approaches 1 as “n” increases. Combining
Figure 5 and Equation (8), it can be found that when a unilateral resonant compensation structure is adopted, the voltage to ground at the A end of the power supply rail contains more high-order harmonics.
2.2. Bilateral Resonant Compensation Topology
Referring to
Figure 6, the series compensating inductor
Lf in the LCC-S resonant compensation topology is decomposed into a bilateral compensating structure
and
, in which
. Consequently, the impedance observed at points A and B, with respect to point C, remains consistent. Therefore, the voltage across points A and B, relative to point C, can be denoted as
and
, respectively.
and
are the leakage currents. At this point, the LCC-type topology is symmetric on its upper and lower edges and is therefore called a symmetric compensation structure.
When the LCC-type symmetrical compensation topology is used, the flow direction of
in
Figure 4 is also taken as the positive direction. The voltages
and
to ground at points A and B in this case are expressed as:
According to Kirchhoff’s voltage law, it is known that when the symmetric compensation topology is used, it does not affect the state of the internal system currents
and
, and in order to better equalize the track-to-ground voltage, let:
Then the formula for
is:
Taking Equation (12) as a condition for full resonance at the transmitter end of the system, Equation (12) can be simplified to:
Then, according to Equations (4), (6) and (13), the
,
Fourier series expansion is:
First, according to Equations (14) and (15), according to Equation (4), with the bilateral structure, the structure is symmetrical, but according to the circuit analysis, the voltage to ground
and
are different, so the currents
and
are different. Afterward, according to Equations (9), (14) and (15), the Fourier series expansion of the voltage to ground at the two ends of A and B consists of two main parts: the dc component term
and the high-order harmonic Fourier series term. However, the coefficients in front of the Fourier series term determine the magnitude of each order harmonic, and in order to visualize and analyze the difference between the higher harmonics in the voltage to ground at points A and B at the two ends of the coil of the type unilateral/symmetric resonance topology, such that
in Equation (9), let the coefficient before the Fourier series term in Equation (4) be 1. Let
,
in Equation (14), and
in Equation (15).
,
,
, and
represent the constant term coefficients of the higher harmonics of the voltage to ground, respectively.
Figure 6 and
Figure 7 below show the trends of
,
,
, and
with n, respectively.
In this article, (2n − 1) is used to denote the order of the higher harmonics, and n is an integer greater than 1. According to Equations (5) and (10), the voltage to ground at both ends of the supply rail under the LCC-type symmetrical compensation topology is smaller than that under its unilateral compensation topology if only the voltage-to-ground fundamental is considered. According to
Figure 6 and
Figure 7, without considering the ground voltage fundamental,
and
are greater than or equal to 1 regardless of the value of n, indicating that a large number of high harmonics exists in the supply-rail-to-ground voltage under the LCC-type unilateral compensation topology.
and
tend to be close to 0 with the increase of n, and the absolute values of
and
are only about 0.14 when n = 1, which indicates that the supply-rail-to-ground voltage under the LCC-type bilateral compensation topology substantially reduces the high harmonic content of the supply-rail-to-ground voltage by more than 90. However, according to
Figure 7, it can be found that
is always less than 0, which indicates that the high harmonics of the voltage to ground at point B are negative under the LCC-type bilateral compensation topology, which means that the high harmonics of the voltage to ground at this time are all negative sequence harmonics. The positive and negative high harmonics only indicate different phases, and the harmonic high harmonics generate negative sequence harmonics because the nonlinear impedance (series inductance
) is added between point B and ground under the bilateral compensation topology, which leads to the phase shift of the high harmonics of the voltage to ground.
According to the harmonic superposition theorem, when the higher harmonic content of the ground voltage is less, the waveform tends to be closer to a sinusoidal waveform, which shows that the ground voltage waveforms at both ends of the supply rails under the LCC-type bilateral compensation topology are similar to a sinusoidal waveform, which is subsequently verified by simulation and experiment.