▌01 无线传输系统最大能量效率跟踪

Paper Link: Maximum Energy Efficiency Tracking for Wireless Power Transfer Systems

In the section II:Theoretical Analysis:

1. 最大传输效率负载条件

Theoratical Analysis of condition for maximum energy efficiency:

^{▲ Circuit Model of Two Coils of WPT}

Energy Efficiency(η\etaη):

The ratio of I1/I2I_1 /I_2I1/I2can be obtained from (2):

I1I2=−RP2+RL+jX2jωM12{{I_1 } \over {I_2 }} = - {{R_{P2} + R_L + jX_2 } \over {j\omega M_{12} }}I2I1=−jωM12RP2+RL+jX2

(I1I2)2=−(RP2+RL)2+X22(ωM12)2\left( {{{I_1 } \over {I_2 }}} \right)^2 = - {{\left( {R_{P2} + R_L } \right)^2 + X_2^2 } \over {\left( {\omega M_{12} } \right)^2 }}(I2I1)2=−(ωM12)2(RP2+RL)2+X22

η=ω2M122RL[(RP2+RL)2+X22]RP1+ω2M122(RP2+RL)\eta = {{\omega ^2 M_{12}^2 R_L } \over {\left[ {\left( {R_{P2} + R_L } \right)^2 + X_2^2 } \right]R_{P1} + \omega ^2 M_{12}^2 \left( {R_{P2} + R_L } \right)}}η=[(RP2+RL)2+X22]RP1+ω2M122(RP2+RL)ω2M122RL

by solving the differentiaing η\etaηto X2 and RL can obtain the MEE load condition:X2_OPT_η=0X_{2\_OPT\_\eta } = 0X2_OPT_η=0

RL_OPT_η=RP21+ω2M122RP1RP2R_{L\_OPT\_\eta } = R_{P2} \sqrt {1 + {{\omega ^2 M_{12}^2 } \over {R_{P1} R_{P2} }}}RL_OPT_η=RP21+RP1RP2ω2M122

如果，RP1,RP2R_{P1} ,R_{P2}RP1,RP2相比ωM,\omega M,ωM,很小，则根据 - [ ] Applying LCC Compensation Network to Dynamic Wireless EV Charging System 附件推导，其中公式（A6）将RL_OPT_ηR_{L\_OPT\_\eta }RL_OPT_η定义为：

RL_OPT_η≈ωM⋅RP2RP1≈ωMR_{L\_OPT\_\eta } \approx \omega M \cdot \sqrt {{{R_{P2} } \over {R_{P1} }}} \approx \omega MRL_OPT_η≈ωM⋅RP1RP2≈ωM

上面公式中，假设发送和接收线圈具有相同的RP1,RP2R_{P1} ,R_{P2}RP1,RP2。

2.最大传输功率负载条件

Optimal Load Condition for Maixmum power Operations.

PO=I22RL=ω2M122Vin2RL(ω2M122+RP1RP2+RP1RL−X1X2)2+(RP1X2+RP2X2+RLX1)2P_O = I_2^2 R_L = {{\omega ^2 M_{12}^2 V_{in}^2 R_L } \over {\left( {\omega ^2 M_{12}^2 + R_{P1} R_{P2} + R_{P1} R_L - X_1 X_2 } \right)^2 + \left( {R_{P1} X_2 + R_{P2} X_2 + R_L X_1 } \right)^2 }}PO=I22RL=(ω2M122+RP1RP2+RP1RL−X1X2)2+(RP1X2+RP2X2+RLX1)2ω2M122Vin2RL

As ：X1_OPT_Po=0;X2_OPT_Po=0X_{1\_OPT\_Po} = 0;\,\,\,\,X_{2\_OPT\_Po} = 0X1_OPT_Po=0;X2_OPT_Po=0

By solving： ∂Po∂RL=0{{\partial P_o } \over {\partial R_L }} = 0∂RL∂Po=0， obtain the optimum load for MPO:

RL_OPT_Po=RP2(1+ω2M122RP1RP2)R_{L\_OPT\_Po} = R_{P2} \left( {1 + {{\omega ^2 M_{12}^2 } \over {R_{P1} R_{P2} }}} \right)RL_OPT_Po=RP2(1+RP1RP2ω2M122)

对比最大效率和最大功率条件，可以看到：
RL_OPT_η<RL_OPT_PoR_{L\_OPT\_\eta } < R_{L\_OPT\_Po}RL_OPT_η<RL_OPT_Po

3.最大效率跟踪算法

下图给出了最大功率跟踪控制框图：

^{▲ Schematic of a two-winding WPT system with ac-dc and dc-dc converters}

（1）阻抗变换

全桥整流负载阻抗变化：
RL_OPT_η=8π2RL_eqR_{L\_OPT\_\eta } = {8 \over {\pi ^2 }}R_{L\_eq}RL_OPT_η=π28RL_eq

全桥整流之后的阻抗变化系数8/π28/\pi ^28/π2在下面的博文通过理论和实测进行验证。（不过最终还是没有能够推导出8/π28/\pi ^28/π2系数）：

全桥整流等效负载阻抗是多少？
整流电路对应的阻抗是多少？

DC-DC阻抗变换：如果输入电压与输出电压的比值：γ=Vin/Vout\gamma = V_{in} /V_{out}γ=Vin/Vout，那么DC-DC变换之后的电阻：
RL_eq=γ2RLR_{L\_eq} = \gamma ^2 R_LRL_eq=γ2RL

（2）最优阻抗搜索算法

Details refer to the origin Paper.

^{▲ 最优负载搜索算法}

4.实验结果

（1）Paameters of WPT

^{▲ Parameters of WPT system with weak coupling}

^{▲ Couple Coils and misalignment woth couple coefficient:0.1}

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