# Simplified Mutual Inductance Calculation of Planar Spiral Coil for Wireless Power Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mutual Inductance Equation Derivation

#### 2.1. Two Perfect Aligned the Planar Spiral Coil

_{1}lies on the x–y plane with the center at the origin, and coil C

_{2}is placed above C

_{1}at a distance h apart, as shown in Figure 1b. The direction of the y-axis is into the page, not shown here. The primary coil’s C

_{1}radius is R

_{A}, the secondary coil C

_{2}radius is R

_{B}, the distance between them is h, P and Q are the tangential elements on the C

_{1}and C

_{2}, respectively, the gap between P and Q is represented by R, θ

_{1}is the angle of R

_{A}relative to x-axis of C

_{1}, and the angle of R

_{B}is θ

_{2}.

_{0}is the vacuum permeability, dl

_{1}and dl

_{2}are line elements, and R is the separation between it. Eventually, the mutual inductance of the planar circular spiral coil can be determined by finding dl

_{1}, dl

_{2}, and R in Equation (1).

_{i}is an initial radius, N is the number of turns, s is the gap between turns, a is pitch factor, and θ

_{i}and θ

_{o}are initial and final angle of Archimedean spiral coil. The pitch factor a affects the gap between turns.

_{i}, outer radius R

_{o}, number of turns N, and the gap between them s, are calculated from the spiral coil Equation (5).

_{1}and C

_{2}can be expressed as Equations (6) and (7).

_{1}and dl

_{2}can be described as

_{A}and R

_{B}represent the distance from the origin to the tangential elements C

_{1}and C

_{2}, respectively.

_{1}and dl

_{2}can be denoted by using cosine law.

#### 2.2. Lateral Misalignment of the Planar Spiral Coil

## 3. Simulation Verification

_{o}and the number of turns N increased, the error increased slightly, as indicated in Table 2 and Table 3, respectively. However, for all cases the errors are below 4% which is in acceptable range. Thus, it proves the accuracy of the simplified equation.

## 4. Experimental Verification

_{i}is the inner radius of the planar spiral coil, and R

_{o}is the outer radius.

#### 4.1. Selection of Wire and Its Measurement

#### 4.2. Mutual Inductance Measurement Method

#### 4.3. Mutual Inductances of Different Spiral Coil Configurations for Aligned Distances and Their Error Comparison

_{p}and N

_{s}denote the number of turns of primary coil and secondary coil, respectively.

#### 4.4. Behavior of Mutual Inductance between Two Large Primary and Large Secondary Spiral Coils under Lateral Misalignment and Their Error Comparison

#### 4.4.1. Behavior of Mutual Inductance between Two Small Primary and Small Secondary Spiral Coils under Lateral Misalignment and Their Error Comparison

#### 4.4.2. Behavior of Mutual Inductance between Large Primary and Small Secondary Spiral Coils under Lateral Misalignment and Their Error Comparison

#### 4.5. Coupling Coefficient of Spiral Coil

_{p}is the self-inductance of primary coil, and L

_{s}is the self-inductance of secondary coil. Self-inductance of the coil can be found by replacing the axial distance parameter h in the denominator of Equation (12) with the wire diameter w. Using a modified Equation (12), the self-inductance of the sampled 10 turns and 16 turns spiral coil were 4.584 µH and 18.734 µH, respectively. It can also be measured by an LCR meter. The measured self-inductance values are 4.712 µH and 19.310 µH, alternately. k relative of axial distance is depicted in Figure 9.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Experimental setup for measuring mutual inductance. (

**a**) Bobbin. (

**b**) Spiral coil with bobbin. (

**c**) Measuring components.

**Figure 4.**Two different connection mode for measuring mutual inductance. (

**a**) Forward mode. (

**b**) Backward mode.

**Figure 5.**Mutual inductance. (

**a**) Comparison between different coils configuration at various axial distances. (

**b**) Error comparison of coils relative to calculation and FEM.

**Figure 6.**(

**a**) Mutual inductance comparison between large primary and large secondary coils at various lateral misalignment distances. (

**b**) Calculation variation relative to simulation (FEM) for N = 16. (

**c**) Measurement variation relative to calculation.

**Figure 7.**(

**a**) Mutual inductance comparison between small primary and small secondary coils at various lateral misalignment distances. (

**b**) Calculation variation relative to simulation (FEM) for N = 10. (

**c**) Measurement variation relative to calculation.

**Figure 8.**Mutual inductance comparison between large primary and small secondary coils at various lateral misalignment distances. (

**a**) Primary large and secondary small coil. (

**b**) Calculation variation relative to simulation (FEM) for N

_{p}= 16, N

_{s}= 10. (

**c**) Measurement variation relative to calculation.

Parameters | Results | |||||
---|---|---|---|---|---|---|

h (mm) | s (mm) | R_{i}(mm) | R_{o}(mm) | FEM (µH) | (11) (µH) | Error (%) |

10 | 7.5 | 10 | 85 | 5.85 | 5.67 | 3.07 |

10 | 6.25 | 10 | 85 | 8.25 | 8.03 | 2.66 |

10 | 5.35 | 10 | 85 | 11.24 | 11.00 | 2.13 |

10 | 4.68 | 10 | 85 | 14.56 | 14.29 | 1.85 |

Parameters | Results | |||||
---|---|---|---|---|---|---|

h (mm) | N (mm) | R_{i}(mm) | R_{o}(mm) | FEM (µH) | (11) (µH) | Error (%) |

10 | 10 | 10 | 85 | 5.85 | 5.67 | 3.07 |

10 | 10 | 10 | 95 | 6.61 | 6.40 | 3.17 |

10 | 10 | 10 | 105 | 7.39 | 7.15 | 3.24 |

10 | 10 | 10 | 115 | 8.19 | 7.91 | 3.41 |

Parameters | Results | |||||
---|---|---|---|---|---|---|

h (mm) | N (mm) | R_{i}(mm) | s (mm) | FEM (µH) | (11) (µH) | Error (%) |

10 | 10 | 10 | 7.5 | 5.85 | 5.67 | 3.07 |

10 | 12 | 10 | 7.5 | 9.87 | 9.56 | 3.14 |

10 | 14 | 10 | 7.5 | 15.86 | 15.29 | 3.59 |

10 | 16 | 10 | 7.5 | 23.48 | 22.67 | 3.44 |

Coil Size | N (mm) | R_{i}(mm) | s (mm) | R_{o}(mm) |
---|---|---|---|---|

Small Coil | 10 | 10 | 4 | 50 |

Large Coil | 16 | 10 | 4.68 | 85 |

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**MDPI and ACS Style**

Hussain, I.; Woo, D.-K.
Simplified Mutual Inductance Calculation of Planar Spiral Coil for Wireless Power Applications. *Sensors* **2022**, *22*, 1537.
https://doi.org/10.3390/s22041537

**AMA Style**

Hussain I, Woo D-K.
Simplified Mutual Inductance Calculation of Planar Spiral Coil for Wireless Power Applications. *Sensors*. 2022; 22(4):1537.
https://doi.org/10.3390/s22041537

**Chicago/Turabian Style**

Hussain, Iftikhar, and Dong-Kyun Woo.
2022. "Simplified Mutual Inductance Calculation of Planar Spiral Coil for Wireless Power Applications" *Sensors* 22, no. 4: 1537.
https://doi.org/10.3390/s22041537