# Precise Analysis on Mutual Inductance Variation in Dynamic Wireless Charging of Electric Vehicle

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emission reduction, and also extensive deployment of green energy sources [1,2,3,4]. Moreover, general public interest has motivated car manufacturers towards the development of various electric vehicles [2,5], such as battery electric vehicles (BEV), hybrid electric vehicles (HEV), fuel cell electric vehicles (FCEV), and fuel cell hybrid vehicles (FCHV) [6]. The development and improvement of battery storage systems as well as implementation of various techniques for the monitoring and control of battery states facilitates the attractiveness of EVs [7,8]. EV charging can be accomplished either using cables (wired) or wirelessly. The wireless charging technique compared to the traditional wired approach has recently become more engaging in terms of safety, suitability and simplicity [9,10]. Since charging cables are not required for Wireless Power Transfer (WPT); the hazards of tripping, arcing or getting electrocuted while plugging or unplugging batteries in a wet environment are eliminated. The safety and efficiency of WPT systems can be attained by the installation of new protective devices, interconnection protection systems and earthing arrangements [11,12] that monitor various faults and respond to them as quickly as possible. The applications of this technology can be found in wireless chargers for both small-scale appliances such as toothbrushes, cellular phones [13,14,15] and medical implants [16,17] or even for high-power charging of EV batteries [18,19].

## 2. DWPT System Model

_{DC}. The full-bridge inverter with four switches S

_{1}, S

_{2}, S

_{3}and S

_{4}was used to convert input DC voltage into AC voltage since the nature of magnetic resonance coupling method is based on AC current. Moreover, in order to achieve proper resonance state and increase the system efficiency, each transmitting coil is connected to the LCC compensation circuit. The compensation elements such as inductor and two capacitors are denoted as L

_{t1}, C

_{1}, C

_{1t}for the upper transmitter and L

_{t}

_{2}, C

_{2}, C

_{2t}for the lower transmitter, respectively. The corresponding stray resistances are represented as R

_{t}

_{1}, R

_{1}, R

_{t}

_{2}, and R

_{2}. In addition, capacitors C

_{t}

_{1}and C

_{t}

_{2}are included into the circuit in order to adjust inverter’s reactance. The two transmitter coils’ self-inductances L

_{1t}and L

_{2t}successively couple with the receiver coil’s self-inductance L

_{1r}as it moves over them and create mutual inductances M

_{1}and M

_{2}, respectively. The overall stray resistances of the first and the second transmitters are designated as R

_{1t}and R

_{2t}. Moreover, switches S

_{11}and S

_{22}are used to switch on and off the transmitter 1 or transmitter 2 when it is required. Currents I

_{t}

_{1}and I

_{t}

_{2}flow through the compensation circuits, while I

_{1}and I

_{2}are the transmitting currents and I

_{L}is the induced current, which flows through the secondary-side compensation capacitor C

_{1r}connected in series with its stray resistance R

_{1r}and equivalent load resistance R

_{Load}of system.

_{1}is applied to the primary-side coil with inductance L

_{1}. It transfers power to the transmitter coil and then voltage V

_{2}is induced in the secondary-side coil with inductance L

_{2}. As a result, the mutual inductance M is created between them. The primary and secondary side capacitors of the compensation circuits, which create perfect resonance, are denoted as C

_{1}and C

_{2}, respectively. In addition, the terms I

_{1}and I

_{2}denote the currents flowing through primary and secondary inductors.

_{1}and output V

_{2}voltages are obtained as:

## 3. Coil Design and Simulation

#### 3.1. Description of Coil Design

#### 3.2. Simulation Results and Analysis

#### 3.2.1. Case 1: Comparison of Self and Mutual Inductances

^{2}cross section was selected for the coils design. The vertical distance Z between the transmitter coils and the moving pick-up coil is set constant, 65 mm, for all three cases. The total length L of the conductor used for the design of the circular coil and square coils are 14.15 m and 14.6 m, respectively, which are approximately the same. The number of turns for the transmitter N

_{1}and receiver N

_{2}coils is equal to 17 and 18, respectively. The circular coils’ inner r

_{1}and outer r

_{2}radii are selected to be 106 mm and 156 mm. On the other hand, the distance between the origin and the first turn of the square coils is equal to 106 mm and the length of each square coil is chosen to be 126 mm. The 3-D models created in ANSYS Maxwell software for the two described shapes are illustrated in Figure 5a,b.

_{1}and L

_{2}of the circular shaped coils are 115.714 μH and 115.715 μH respectively, while the corresponding values for the square shaped coils are 106.001 μH and 106.027 μH. In addition, the mutual inductance between the transmitter and the receiver coils is equal to 40.068 μH for the circular shape and 32.392 μH for the square shape, respectively. Consequently, practical results have shown a theoretical prediction that the circular shaped coils have higher mutual inductance value than square shaped coils of the same size.

#### 3.2.2. Case 2: Identification of Proper Distance between Transmitter Coils

#### 3.2.3. Case 3: Observation of Mutual Inductance Change with Respect to Lateral Misalignment

## 4. Practical Study on Variable Mutual Inductance

#### 4.1. Method for Measuring Mutual Inductance

_{1}and L

_{2}coupled with magnetic fields in one direction (Figure 10).

_{1}and V

_{2}are voltages induced by L

_{1}and L

_{2}, respectively. V

_{M}

_{,1}and V

_{M}

_{,2}are voltages induced by M in these two coils. They have positive sign due to aiding magnetic fields. Since the current flowing through the coils is the same, V

_{M}

_{,1}and V

_{M}

_{,2}are equal. If the current flowing through coils is I and the induced electromotive force (EMF) by Lenz’s law is—LdI/dt, then the expression (4) becomes:

_{1}and L

_{2}are known, then the mutual inductance M can be obtained as:

#### 4.2. Simulation and Experiment Results

_{13}), and another between the coil 2 and the receiver (M

_{23}). The behavior of each of these mutual inductances with respect to the relative receiver coil position is similar to that of Figure 13 and is shown in Figure 14. In this case, transmitter coils 1 and 2 have center coordinates at (0, 0, 0) and (0, 50 cm, 0), respectively. As noticed, M

_{23}behavior is similar to M

_{13}with the peak being at the time when receiver coil center is at (0, 50 cm, 6 cm), i.e., when transmitter coil 2 and receiver coil are in perfect alignment.

_{13}+ M

_{23}. Then, the behavior of M is as in Figure 16.

## 5. Dynamic WPT Simulation

#### 5.1. DWPT Simulation

_{11}and S

_{22}, as shown in Table 5. The time interval between the case transitions is selected arbitrarily and assumed as 30 ms. It should be noted that, when the transmitter coil is active, its mutual inductance with the vehicle-side coil is 40 μH in all three cases. For instance, in case 2 both transmitter coils, each having 40 μH mutual inductance value, will be coupled with the receiver coil for 30 ms, whereas in cases 1 and 3 only one transmitter coil with 40 μH mutual inductance is energized.

_{t}

_{1}, I

_{t}

_{2}flowing from the inverter output are shown in Figure 17a–d. It can be observed from Figure 17a, I

_{t}

_{1}in case 1 has a stable value which is equal to 40 A, but in case 2 this current has decreased to 37 A. It is decreased due to switch on of the second switch and the input current from the inverter is distributed between the two transmitter coils. The reverse situation can be seen in Figure 17c, the current I

_{t}

_{2}is initially equal to zero and starts to increase to approximately 75 A in case 2. In case 3, I

_{t}

_{2}becomes higher since all the input current is flowing through the second transmitter coil. Moreover, the nature of these curves is the sinusoidal curve as shown in Figure 17b,d for the currents I

_{t}

_{1}, I

_{t}

_{2}, respectively. These graphs are the large scale views of the corresponding currents at the instant when transmitter coil 1 becomes de-energized (at 0.06 s). At 0.06 s, I

_{t}

_{1}immediately reaches zero because an open circuit is created, while a transient is created in the I

_{t}

_{2}waveform which stabilizes after some amount of time.

_{1}, I

_{2}as well as through the receiver coil I

_{L}are presented in Figure 18a–f.

_{1}and I

_{2}, which flow through L

_{1t}and L

_{2t}. coils, respectively. In case 1, only the switch S

_{11}is activated; thus, the total input current of 6.75 A is supplied to the first ground-side coil from 0 until 30 ms. In case 2, both currents I

_{1}and I

_{2}become 8.2 A. This is caused by activating both switches S

_{11}and S

_{22}, which has resulted to the equal distribution of input current between two ground-sided coils. Similar behavior to case 1 is observed for current I

_{2}in case 3. Precisely, the entire input current is solely delivered to the second transmitter coil. This is explained by the fact that only S

_{22}switch is in operation for the case 3. Also, current I

_{1}must equal to zero in case 3 and current I

_{2}must equal to 0 in case 1. Moreover, current transients are observed between the case transitions. They can be caused by the switching operation between transmitter coils. Large scale views of some of these transients are shown as an example in Figure 18b,d,f. These transients have different time of stabilization for different cases. For instance, it took about 1 ms for both currents I

_{1}, I

_{2}to reach a steady state from case 1 to case 2, and about 1 μs from case 2 to case 3.

_{L}, which is delivered to the receiver coil L

_{1}

_{r}. The stable current of 9 A is flowing through the receiver coil in cases 1 and 3. However, it is observed that in case 2, I

_{L}is higher than in other cases and equals to approximately 15 A. This is because both transmitters coils are energized in case 2 and transmit power to the receiver.

_{L}with respect to the EV’s travelling time is shown in Figure 19.

_{L}are achieved at 11.429 ms and 29.524 ms, when the receiver is in perfect alignment with the first and the second transmitters, respectively. As the EV begins the motion, it drives over the first transmitter coil. At the same time, the current starts to grow, peaks at zero lateral misalignment and, after that, it decreases again (see Figure 19). The same behavior is observed in case when the receiver moves over the second transmitter coil.

#### 5.2. Short Discussion

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 6.**Effect of displacement between transmitters on the mutual inductance for circular shaped coils.

**Figure 7.**Effect of displacement between transmitters on the mutual inductance for square shaped coils.

**Figure 8.**(

**a**) 3-D view of circular shaped coils; (

**b**) Plot of mutual inductance change with respect to lateral misalignment.

**Figure 9.**(

**a**) 3-D view of square shaped coils; (

**b**) Plot of mutual inductance change with respect to lateral misalignment.

**Figure 12.**Experimental setup, (

**a**) sliding setup for transmitter and receiver; (

**b**) measurement equipment.

**Figure 15.**Mutual inductance between two transmitters and one receiver vs. X coordinate of the receiver coil center.

**Figure 16.**Mutual inductance between two transmitters and one receiver vs. X coordinate of the receiver coil center.

**Figure 17.**Simulation results for: (

**a**) Input current of transmitter 1, I

_{t}

_{1}; (

**b**) large scale view of I

_{t}

_{1}when coil 1 turns off; (

**c**) Input current of transmitter 2, I

_{t}

_{2}; (

**d**) large scale view of I

_{t}

_{2}when coil 1 turns off.

**Figure 18.**Simulation results for: (

**a**) Current through first transmitter coil, I

_{1}; (

**b**) large scale view of I

_{1}when both coils are off; (

**c**) Current through second transmitter coil, I

_{2}; (

**d**) large scale view of I

_{2}when coil1 turns on; (

**e**) Current through receiver, I

_{L}; (

**f**) large scale view of I

_{L}when coil 2 turns on.

**Figure 19.**Simulation results for: (

**a**) of I

_{L}current; (

**b**) large scale view between time intervals of 0.005–0.007 s.

Parameter | Transmitter & Receiver Coils |
---|---|

Number of turns | 18 |

Inner diameter | 140 mm |

Conductor diameter | 4 mm |

Turn spacing | 3 mm |

Outer diameter | 400 mm |

Coil Type | Simulation Results | Experiment Results |
---|---|---|

Transmitter coil | 86.8 μH | 93.4 μH |

Receiver coil | 86.8 μH | 90.7 μH |

Position Number | Receiver Coil Center Coordinates, cm | Mutual Inductance (Experiment), μH | Mutual Inductance (Simulation), μH |
---|---|---|---|

1 | (0, −50, 6) | 1.30 | 1.10 |

2 | (0, −40, 6) | 2.70 | 2.22 |

3 | (0, −30, 6) | 3.05 | 2.65 |

4 | (0, −20, 6) | 4.70 | 6.63 |

5 | (0, −10, 6) | 28.2 | 27.8 |

6 | (0, 0, 6) | 40 | 40.96 |

7 | (0, 10, 6) | 29.4 | 27.8 |

8 | (0, 20, 6) | 6.35 | 6.62 |

9 | (0, 30, 6) | 3.75 | 2.65 |

10 | (0, 40, 6) | 3.15 | 2.22 |

11 | (0, 50, 6) | 2.20 | 1.10 |

Parameter | Value |
---|---|

${L}_{1t},{L}_{2t},{L}_{1r}$ | 115.4 μH, 115.5 μH, 115.4 μH |

${R}_{1t},{R}_{2t},{R}_{1r}$ | 60 mΩ, 60 mΩ, 60 mΩ |

${C}_{1t},{C}_{2t},{C}_{1r}$ | 190 pF, 190 pF, 43 pF |

${L}_{t1},{L}_{t2}$ | 40 μH, 40 μH |

${R}_{t1},{R}_{t2}$ | 12 mΩ, 12 mΩ |

${C}_{t1},{C}_{t2}$ | 190 pF, 190 pF |

${R}_{1},{R}_{2}$ | 12 mΩ, 12 mΩ |

${C}_{1},{C}_{2}$ | 260 pF, 260 pF |

${M}_{1},{M}_{2},{M}_{12}$ | 40 μH, 40 μH, 2.3 μH |

${R}_{L}$ | 15 Ω |

Case | Switches ON | Time Duration |
---|---|---|

1 | S_{11} | 0–30 ms |

2 | S_{11} and S_{22} | 30 ms–60 ms |

3 | S_{22} | 60 ms–90 ms |

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## Share and Cite

**MDPI and ACS Style**

Rakhymbay, A.; Khamitov, A.; Bagheri, M.; Alimkhanuly, B.; Lu, M.; Phung, T.
Precise Analysis on Mutual Inductance Variation in Dynamic Wireless Charging of Electric Vehicle. *Energies* **2018**, *11*, 624.
https://doi.org/10.3390/en11030624

**AMA Style**

Rakhymbay A, Khamitov A, Bagheri M, Alimkhanuly B, Lu M, Phung T.
Precise Analysis on Mutual Inductance Variation in Dynamic Wireless Charging of Electric Vehicle. *Energies*. 2018; 11(3):624.
https://doi.org/10.3390/en11030624

**Chicago/Turabian Style**

Rakhymbay, Ainur, Anvar Khamitov, Mehdi Bagheri, Batyrbek Alimkhanuly, Maxim Lu, and Toan Phung.
2018. "Precise Analysis on Mutual Inductance Variation in Dynamic Wireless Charging of Electric Vehicle" *Energies* 11, no. 3: 624.
https://doi.org/10.3390/en11030624