# Analysis of Dynamic Wireless Power Transfer Systems Based on Behavioral Modeling of Mutual Inductance

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MQS Model of Mutual Inductance

**L**, and a resistance matrix

**R**. However, this is strictly correct only if one of the following conditions hold: (i) absence of any passive structure (e.g., conducting shields); (ii) negligible ohmic losses. If not, the two-port can be introduced only in frequency domain as the impedance

**Z**(ω) =

**R**(ω) + j ω

**L**(ω).

_{D}almost equals the static mutual inductance M

_{S}.

**L**and

**R**, we can calculate the magnetic time constants as the eigenvalues of

**R**, and choose as the magnetic characteristic time, τ

^{−1}L_{M}, the largest of such eigenvalues, τ

_{M}= max [eig (

**R**)]. The characteristic time associated to the motion may be introduced by considering the simple case of an EV (and the associated RX coil) moving along a straight trajectory over a series of rectangular-shaped TX coils, with a constant speed,

^{−1}L**v**

_{RX}= v

_{0}. The characteristic time t

_{f}can be taken as the flight time of the RX coil on the TX coil, with longitudinal length (the longer side of the TX coil) equal to l

_{TX}, i.e., t

_{f}= l

_{TX}/v

_{0}.

_{D}≈ M

_{S}. First of all, we should impose that both the static and the dynamic WPTS can reach the sinusoidal steady-state condition at each position of the trajectory: this happens if t

_{f}is much larger than the duration of the MQS transient, usually set as 4 τ

_{M}. This happens if condition (1) is fulfilled:

_{TX}are about meters, hence any reasonable vehicle speed satisfies (2). Therefore, for the purposes of this paper only (1) applies. For typical automotive WPTSs, the inductance values range from tens to hundreds of µH, and the resistances from tens to hundreds of mΩ, hence τ

_{M}ranges from fractions to some tens of ms. Assuming δ = 0.1 in (1), we can compute the maximum vehicle speed values above which the dynamic effects must be taken into account (see Figure 1).

^{2}cross-section and 5-mm thickness. The inner size is 150 cm × 50 cm for the TX coil and 30 cm × 50 cm for the RX coil. The RX coil is installed on board the EV into an aluminum case. Two ferrite blocks with relative permeability μ

_{r}= 2000 are used to improve the magnetic coupling. 50 TX coils are embedded in the road pavement, making a 100 m long charging lane. The nominal vertical distance between the RX coil and the TX coils is 20 cm. The chassis conductivity is 33.4 MS/m. The system can transfer 11 kW maximum power at 85 kHz frequency [30].

_{TX}= 1.5 m and l

_{RX}= 0.3 m, at a frequency of 85 kHz. In the nominal position, where the centers of the TX and RX coils are aligned along the same vertical axis, the quasi-static parameters of coils are given by: R

_{TX}= 30 mΩ, R

_{RX}= 15 mΩ, L

_{TX}= 80 µH, L

_{RX}= 10 µH and M = 8.51 µH. From these values, it is τ

_{M}= 2.7 ms. According to (1), the maximum vehicle speed should be v

_{0}≈ 50 km/h. An electrodynamic model was built in ANSYS Maxwell, imposing the motion of the RX coil along the longitudinal axis of the TX coil, with a constant speed of 36 km/h (10 m/s). The values of M

_{D}computed at given time instants (corresponding to different longitudinal position of the RX coil) are reported in Table 1, along with the values of M

_{S}computed under the static limit in the positions corresponding to the same time instants. At the initial position (t = 0 ms), the RX coil is completely outside the TX one, whereas at the final position (t = 48 ms), the RX coil is completely inside the TX one and their axes are perfectly aligned (nominal position). The two solutions differ by a maximum relative error of about 5–6%, hence demonstrating the validity of the criterion (1). Of course, lower values of δ in (1) would provide a better accuracy.

## 3. Efficiency of Wireless Power Transfer Systems

_{1}and L

_{2}of the TX and RX coils are compensated by the capacitors C

_{1}and C

_{2}. The resistor R

_{1}includes the resistances of the L

_{1}-C

_{1}series and of the two inverter MOSFETs conducting simultaneously. Similarly, the resistor R

_{2}includes the resistances of the L

_{2}-C

_{2}series and of the two rectifier diodes. From Figure 2c and Figure 3, we have L

_{1}= L

_{TX}, L

_{2}= L

_{RX}, R

_{1}= R

_{TX}and R

_{2}= R

_{RX}. The diode-bridge rectifier at the receiver side is connected to the load (battery) through a boost converter, which regulates the equivalent load resistance R

_{L}at the boost input to a given optimal value ensuring the maximum power transfer at the nominal mutual inductance M

_{nom}. The inverter switching frequency f

_{s}is equal to the resonance frequency ${f}_{0}=1/(2\mathsf{\pi}\sqrt{{L}_{1}{C}_{1}})=1/(2\mathsf{\pi}\sqrt{{L}_{2}{C}_{2}})$ of the WPTS. The full-bridge inverter at the transmitter side adopts a phase-shift control, which modulates the phase-shift angle α between the complementary square-wave gate signal pairs driving the inverter MOSFETs. The goal of the phase-shift control is to achieve a regulation of the transmitter rms current I

_{1rms}at the desired value I

_{1rms,ref}. Table 2 lists the operating parameters and component values of the analyzed WPTS. Additional details on the power control in dynamic conditions and power demand regulation are available in [29,30].

_{ac}is the equivalent resistance seen at the diode rectifier input, given by

_{1}and P

_{2}, and the resulting efficiency η = P

_{2}/P

_{1}, are given by (8) and (9):

_{1}is fixed by the inverter phase-shift control, i.e., I

_{1}= $\sqrt{2\text{}}$I

_{1rms,ref}. Equation (9) highlights the impact of mutual inductance M on the WPTS efficiency. In particular, (9) shows that the efficiency increases with higher mutual inductance.

## 4. Mutual Inductance Behavioral Modeling for WPTS Dynamic Charging

_{0}= 0, z

_{0}= 0), corresponding to the center of the left-side TX coil.

_{TX}= 359 mΩ and R

_{RX}= 128 mΩ. Accordingly, the largest time constant of the MQS problem is estimated in 0.94 ms. On the other side, the RX coil requires about 133 ms to cover the distance of 3.7 m over the two TX coils at a constant speed of 100 km/h. In practical cases, this time is much longer than the MQS time constant. Consequently, any dynamic effect can be neglected in the mutual inductance evaluation, allowing us to consider the case studies in Table 3 under static limit.

_{tx}

_{1,bhv}(Δy, Δz), between the RX coil and the left side TX coil of Figure 4, and use such a model to evaluate the total mutual inductance M

_{tot}, given in (10):

_{tx}

_{2}, i.e., the mutual inductance between the RX coil and the right side TX coil, while Δy

_{mid}= 1.052 m is the middle point between the two TX coils. In addition, as M

_{tx}

_{1}is symmetric with respect to the left side TX coil center, the TDS of case # 1 can be increased up to 138 points, by mirroring the M

_{tx}

_{1}values obtained for positive Δy values to negative symmetric Δy values.

_{tx}

_{1}= M

_{tx}

_{1,bhv}(Δy, Δz) as a function of Δy and Δz. The details regarding the setup and execution of the GPA developed for this study are discussed in [32]. Here, we put the focus on the criteria adopted to drive the GPA in the generation of candidate functions M

_{tx}

_{1,bhv}. In particular, a first fundamental choice concerns the discrimination between Δy and Δz, as these two geometric variables play different roles. Indeed, while Δy is associated to the vehicle movement direction, and then it is the main variable influencing the time variation of the mutual inductance, Δz is rather a bias factor, as it is associated to the EV lateral drift, which is not expected to change too much during the vehicle transit along the charging lane. For this reason, the GPA has been set up to generate functions M

_{tx}

_{1,bhv}(Δy,

**p**(Δz)), where the analytical structure M

_{tx}

_{1,bhv}is determined by the way M

_{tx}

_{1}changes along Δy, and the coefficients

**p**are functions of Δz and are determined by the way M

_{tx}

_{1}changes along Δz. This separation of variables greatly helps in keeping the behavioral model simple and suitable for the application purpose. Among the best candidate models generated by the GPA discussed in [32], the following model provides a good trade-off among complexity, accuracy and repeatability:

_{tx}

_{1,bhv}are expressed in meters and μH, respectively. The model given in (11) is characterized by high repeatability over GPA runs and small error on the TDS and VDS. The coefficients p

_{k}(k = 0, 1, …,6) have a regular and monotonous trend, and can be analytical represented by arctangent functions of Δz (in meters):

_{k,}

_{0}, a

_{k,}

_{1}, a

_{k,}

_{2}, a

_{k,}

_{3}} are listed in Table 5.

_{tot}obtained by combining the formulas given in Equations (10)–(12), for the TDS trajectories of case #1.

_{tx}

_{1,bhv}and M

_{tot}

_{,bhv}(dotted lines) to the corresponding FEM-based data (square markers), for the case #2 VDS trajectory. The plots of Figure 5 and Figure 6 confirm the good accuracy and the generalization capability of the behavioral model given in Equations (10)–(12).

## 5. Behavioral Model Experimental Validation

_{TX}= 28.5 A. The RX coil is mounted within a movable mechanical framework that provides the possibility of fine-tuning the receiver structure position along the three axes by means of tuning screws. The trajectory shown in Figure 8 has been considered for the test (case #3).

_{OC}at the receiver coil terminals by means of a differential voltage probe, according to (13):

_{tot,bhv}obtained by using the model given in Equations (10)–(12), compared to the experimental measured values M

_{tot,exp}(green square markers). These results further validate the proposed behavioral model and confirm its good accuracy and generalization capability.

## 6. WPTS Performance Sensitivity Analysis

_{max}ensuring an efficiency de-rating no greater than a given drop Δη, compared to the nominal efficiency η

_{nom}achieved when the vehicle trajectory coincides with the TX coils symmetry axis. As the efficiency is function of M according to (9), the proposed behavioral model allows calculating the derivative of the efficiency with respect to Δz. By using the first-order linear approximation (14), it is possible to obtain the function Δz

_{max}(Δy) given in (15):

_{max}(Δy) for Δη = −1% (blue dashed line) and Δη = −2% (red dashed line) efficiency drops. The green dashed line represents the plot of Δz

_{max}(Δy) for the −(1% + 1%) nested efficiency drop, calculated by determining the derivative of the efficiency with respect to Δz along the Δz

_{max}(Δy) corresponding to −1% efficiency drop, and then determining the new Δz

_{max}(Δy) curve corresponding to a further −1% efficiency drop.

_{av}of Δz

_{max}(Δy) over the Δy range (solid lines in Figure 10), resulting in average lateral drifts of about 7 cm for −1%, 10 cm for −(1% + 1%) and 14 cm for −2% efficiency drop. The efficiency plots in Figure 11 and Figure 12 are calculated using the formula (9) and the model given in Equations (10)–(12) over the Δz

_{max}(Δy) trajectories (dashed lines) and over the Δz

_{av}trajectories (solid lines).

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Dynamic wireless power transfer system (WPTS): maximum vehicle speed vs. the magnetic characteristic time. For a given transmitting (TX) coil length, the corresponding curve is the boundary between the regions where the dynamic effects are/are not negligible.

**Figure 2.**(

**a**) Magnetic structure of the dynamic WPTS considered for this investigation. The left-side picture shows the receiver structure mounted under the vehicle and the transmitter coils embedded under the road pavement. The right-side picture shows the laboratory prototype. (

**b**) Geometry of the coil pair system, made of a TX coil (red), a receiving (RX) coil (green), a metallic shield (blue) and ferrite blocks (magenta and cyan). (

**c**) Circuit schematic of the coil pair two-port model.

**Figure 4.**Trajectories of the RX coil (green) along the two TX coils (red): (

**a**) nominal trajectory; (

**b**) case #1: trajectory parallel to the nominal one, with a lateral displacement Δz; (

**c**) case #2: trajectory crossing diagonally the nominal one.

**Figure 5.**Total mutual inductance M

_{tot}values obtained by combining the model given in Equations (10)–(12) (solid lines) vs. Finite Element Method (FEM)-based M

_{tot}data (square markers), for the case #1 TDS trajectories.

**Figure 6.**Mutual inductances M

_{tx1}and M

_{tot}values obtained by means of the model given in Equations (10)–(12) (dotted lines) vs. FEM-based data (square markers), for the case #2 VDS trajectory.

**Figure 8.**Analyzed trajectory of the RX coil along the two TX coils used for experimental validation.

**Figure 9.**Total mutual inductance: predicted values (red square markers) vs. experimental measured values (green square markers), for the trajectory sample positions given in Table 6.

**Figure 10.**Maximum lateral misalignment Δz

_{max}vs. longitudinal displacement Δy for maximum allowable WPTS efficiency derating Δη: blue dashed line = Δz

_{max}(Δy) for Δη = −1% η

_{nom}; red dashed line = Δz

_{max}(Δy) for Δη = −2% η

_{nom}; green dashed line = Δz

_{max}(Δy) for Δη = − (1% + 1%) η

_{nom}; blue solid line = Δz

_{av}for Δη = −1% η

_{nom}; red solid line = Δz

_{av}for Δη = −2% η

_{nom}; green solid line = Δz

_{av}for Δη = − (1% + 1%) η

_{nom}.

**Figure 11.**WPTS efficiency η vs. longitudinal displacement Δy for different lateral misalignments Δz

_{max}corresponding to efficiency derating Δη: blue dashed line = η(Δy,Δz

_{max}

_{,−1%}); red dashed line = η(Δy,Δz

_{max}

_{,−2%}); green dashed line = η(Δy,Δz

_{max}

_{,}

_{−}

_{(}

_{1% + 1%)}); blue solid line = η(Δy,Δz

_{av}

_{,−1%}); red solid line = η(Δy,Δz

_{av}

_{,−2%}); green solid line = η(Δy,Δz

_{av}

_{,}

_{−}

_{(1% + 1%)}).

**Figure 12.**Zoom of Figure 11. WPTS efficiency η vs. longitudinal displacement Δy for different lateral misalignments Δz

_{max}corresponding to efficiency derating Δη: blue dashed line = η(Δy,Δz

_{max}

_{,−1%}); red dashed line = η(Δy,Δz

_{max}

_{,−2%}); green dashed line = η(Δy,Δz

_{max}

_{, − ( 1% + 1%)}); blue solid line = η(Δy,Δz

_{av}

_{,−1%}); red solid line = η(Δy,Δz

_{av}

_{,−2%}); green solid line = η(Δy,Δz

_{av}

_{, − ( 1% + 1%)}).

Time Instant (ms) | M_{D} (µH) | M_{S} (µH) | Error (%) |
---|---|---|---|

0 | −1.45 | −1.41 | 2.76 |

10 | 0.54 | 0.55 | 1.85 |

20 | 3.39 | 3.21 | 5.31 |

30 | 5.99 | 6.30 | 5.17 |

48 | 8.60 | 8.51 | 1.05 |

f_{s}[kHz] | V_{in}[V] | I_{1}_{rms,ref}[A] | R_{L}[Ω] | L_{1}[μH] | L_{2}[μH] | C_{1}[nF] | C_{2}[nF] | R_{1}[Ω] | R_{2}[Ω] |
---|---|---|---|---|---|---|---|---|---|

85 | 500 | 28.5 | 8 | 281.4 | 119.8 | 12.5 | 29.2 | 1.157 | 0.555 |

[cm] | Case #1 | |||||||||

Δy | {0, 23.4, 35.0, 46.8, 55.0, 70.1, 93.5, 116.9, 140.3, 163.6, 187.0, 210.4} | |||||||||

Δz | {0, 6, 12, 18, 24, 30} | |||||||||

[cm] | Case #2 | |||||||||

Δy | −80.2 | −39 | 2.2 | 43.4 | 84.6 | 125.8 | 167 | 208.2 | 249.4 | 290.6 |

Δz | −30.2 | −26.8 | −23.5 | −20.1 | −16.8 | −13.4 | −10.1 | −6.7 | −3.4 | 0 |

Parameter | FEM Simulation | Experimental Values |
---|---|---|

L_{TX} (µH) | 278.6 | 281.4 |

L_{RX} (µH) | 115.4 | 119.8 |

M (µH) | 18.1 | 18.3 |

Coefficients | a_{k,}_{0} | a_{k,}_{1} | a_{k,}_{2} | a_{k,}_{3} |
---|---|---|---|---|

p_{0} | 13.3 | 7.35 | 0.190 | −17.7 |

p_{1} | 0.136 | 20.2 | 0.257 | 2.93 |

p_{2} | −0.05 | 8.40 | 0.234 | −0.484 |

p_{3} | 9.92 | 7.32 | 0.187 | −14.0 |

p_{4} | 0.12 | 8.46 | 0.263 | −1.5 |

p_{5} | 1.08 | 7.28 | 0.323 | −2.73 |

p_{6} | −13.2 | 7.40 | 0.189 | 17.9 |

[cm] | Case #3 | ||||||
---|---|---|---|---|---|---|---|

Δy | −80.2 | 0 | 60 | 110 | 160 | 230 | 291 |

Δz | −30.2 | −23.7 | −18.8 | −14.7 | −14.7 | −14.7 | −14.7 |

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## Share and Cite

**MDPI and ACS Style**

Di Capua, G.; Maffucci, A.; Stoyka, K.; Di Mambro, G.; Ventre, S.; Cirimele, V.; Freschi, F.; Villone, F.; Femia, N.
Analysis of Dynamic Wireless Power Transfer Systems Based on Behavioral Modeling of Mutual Inductance. *Sustainability* **2021**, *13*, 2556.
https://doi.org/10.3390/su13052556

**AMA Style**

Di Capua G, Maffucci A, Stoyka K, Di Mambro G, Ventre S, Cirimele V, Freschi F, Villone F, Femia N.
Analysis of Dynamic Wireless Power Transfer Systems Based on Behavioral Modeling of Mutual Inductance. *Sustainability*. 2021; 13(5):2556.
https://doi.org/10.3390/su13052556

**Chicago/Turabian Style**

Di Capua, Giulia, Antonio Maffucci, Kateryna Stoyka, Gennaro Di Mambro, Salvatore Ventre, Vincenzo Cirimele, Fabio Freschi, Fabio Villone, and Nicola Femia.
2021. "Analysis of Dynamic Wireless Power Transfer Systems Based on Behavioral Modeling of Mutual Inductance" *Sustainability* 13, no. 5: 2556.
https://doi.org/10.3390/su13052556