2.1. Factors Affecting System Transmission Efficiency
The S-S topology is used in magnetic coupling resonant WPT frequently, because its load receiving power and system transmission efficiency are better when the system is resonant [
20,
21]. A series–series(S-S) MCR-WPT system was selected as the object of study, and the theory of mutual inductance model was used to analyze the transmission characteristics of the system and determine the factors affecting the transmission efficiency of the system.
Figure 1 demonstrates a typical S–S-type MCR-WPT system mutual sensing model topology [
2].
Where, L1 refers to the transformer inductances on the primary side, while L2 indicates the one on the secondary side. Similarly, C1 and C2 refer to the primary and secondary compensation capacitors, respectively, which are used for enhancing energy transferred from an AC source to an output loading resistance RL. u0 indicates the high frequency AC voltage source and R0 is the internal resistance of the power source. R1 refers to the equivalent resistance of the primary coil, while R2 indicates the equivalent resistance of the secondary coil.
To simplify the analysis, this study makes the resonant compensation network structure symmetrical, so that the primary and secondary coils are identical (the mutual inductance of the two coils is equal, i.e., M12 = M21 = M). Since the system is powered by an adjustable power supply, the internal resistance R0 of the power supply is ignored.
The input power and output power of the system [
22] can be respectively expressed as
where
indicates the phase angle between the input voltage and the primary current. When the circuit is in resonance,
.
The transmission efficiency of the system [
22] is given by
Since the coil equivalent resistance is not easily changed after the coil winding is completed, its value is considered as a constant and is analyzed further. Therefore, the main factors affecting the transmission efficiency of the MCR-WPT system are the resonant frequency of the system ω, the mutual inductance of the coupling coil M and the load equivalent resistance RL.
2.2. Parameter Calculation of FSP Analysis
When the distance between the two coils is larger than a critical value or if the mutual inductance is too large, the characteristics of the transferred power will change from a single-peaked curve to a double-peaked curve [
23,
24]. The WPT system reaches its power peak when operating at non-resonant frequencies, and the power decreases at resonant frequencies instead. This is called the frequency splitting phenomenon (FSP), which seriously affects the system safety [
25]. The mechanism of FSP generation will be analyzed from the perspective of input impedance and a basis for system parameter optimization is provided.
(a) Normally, the input impedance for WPT topology [
26] can be given by
where
indicates the equivalent impedance of the primary circuit,
denotes the impedance of the secondary circuit equivalent to the primary and M is mutual inductance.
The real and imaginary parts of Equation (4) are separated by
(b) Because the system is symmetric, this study lets
L1 =
L2 =
L and
C1 =
C2 =
C, which leads to the derivation that
=
. Therefore, the mode of the imaginary part of the input impedance can be simplified as
where
denotes the detuning factor and
.
(c) When the circuit resonates, the imaginary part of the input impedance is zero and the input impedance of the circuit is equal to the resistance value of the resistor [
27]. When the system undergoes FSP, the circuit resonance condition is met (the imaginary part of the input impedance is zero and the system output power reaches its maximum). However, the operating frequency does not coincide with the resonant frequency, and the output power drops at the resonant frequency. For the case that both FSP and circuit resonance may occur (
), this study makes the imaginary part of the input impedance (Equation (6)) zero. The results of calculation can be expressed as
After transformation, Equation (8) can be expressed as
where
, k denotes the coupling coefficient,
.
According to the definition of M, the coefficient of
on the left side of Equation (8) is less than zero, so the right side of the equation must also be less than zero. Similarly the left side of Equation (9) should be greater than 0. The three inequalities can be expressed as
where both
Q1 and
Q2 denote the quality factor,
.
In summary, conditions that make Equations (8) and (9) have solutions that are given by
(d) Critical mutual inductance, load resistance and coupled coefficient are given by
Based on the analysis above, two conclusions can be drawn as follows: (i) When the load resistance is less than the critical load resistance and the mutual inductance M is greater than the critical mutual inductance , the system is in an overcoupled state and FSP occurs (both conditions of Equation (11) must be satisfied at the same time). (ii) On the contrary, when the load resistance is larger than the critical load resistance or the mutual inductance is smaller than the critical mutual inductance, the system will not experience FSP (only one of the two conditions must be satisfied).
To verify the correctness of theoretical analysis (i) and (ii), the system simulation parameters are assumed to be as shown in
Table 1, where resonant capacitance is obtained by performing the calculation
and
.
is calculated according to Equation (12).
Figure 2 shows the different cases of FSP in the system with different load resistance values. When
= 26 Ω, the critical coupling coefficient
calculated according to Equation (12) is 0.15. Since the load resistance is less than the critical load resistance (
= 195 Ω) and the system in the
Figure 2a undergoes FSP when k is greater than 0.15, it is consistent with the analysis of conclusion (i). When
= 195 Ω and 300 Ω, since the load resistance is already greater than the critical load resistance, the two conditions of Equation (12) cannot be satisfied simultaneously no matter what value of K is taken. In both cases, the systems in the
Figure 2b,c do not undergo FSP, which is consistent with the analysis in conclusion (ii).