# Analytical Calculation of Mutual Inductance of Finite-Length Coaxial Helical Filaments and Tape Coils

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Establishment of a Mathematical Model and the Magnetic Vector Potential of a Finite-Length Helical Filament

_{1}of coil-1. And then z-component of the magnetic field of ${I}_{1}$ is derived by H. Buchholz:

## 3. Mutual Inductance of Finite-Length Coaxial Helical Filaments

_{1}along the tangential direction of the conductor is

## 4. Mutual Inductance of Finite-Length Coaxial Helical Tape Coils

_{1}, and the angular coordinates of the intersections in which the two boundary lines intersect with the z = 0 plane are ${\alpha}_{1}$ and $-{\alpha}_{1}$, respectively.

_{1},

## 5. Mutual Inductance Calculation for Closely-Wound Tape Coils Using the Inverse Mellin Transform

## 6. Numerical Comparison and Simulation Verification

^{−2}, between two subsequent passes.

#### 6.1. Comparison with Respect to Finite-Length Coaxial Helical Filaments

#### 6.2. Comparison with Respect to Finite-Length Coaxial Helical Tape Coils and the Effect of the Inverse Mellin Transform

^{®}Core™ i5-5257U processor with 1866-MHz DDR3 memory (MacBook Pro, Apple Inc., Cupertino, California, U.S.). From Table 10, we can see that the series in Equation (48) only needs to take the 10 first terms to achieve the same as the first 9 or 10 digits of the numerical integration result, and when the $h$ is large, only 3 terms of the series are required to provide a number which is consistent with the first 10 significant digits of the numerical integration result. It is worth noting that, in most cases, only the first term of the series can lead to a result which fulfills the accuracy requirements of the practical engineering application. The reason why Equation (48) has such a characteristic is that this equation is an asymptotic series with extremely fast convergence. In aspects of time consumption, although the numerical integration has a very high calculation speed (about 200 ms), the series with the same accuracy is faster (about 5 ms) owing to the intrinsic property of itself. Compared to the first two analytical calculation methods, as shown in Table 11, FEM is extremely time consuming, especially when the coil length is long. The FEM was running on a workstation configured as an Intel

^{®}Xeon

^{®}Silver 4110 processor and 2666-MHz DDR4 memory (Precision 7820, Dell Inc., Round Rock, Texas, U.S.), however, the solution time was greater than 25 h and computing resources were considerably occupied.

## 7. Conclusion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Lateral surface of coil-1 in Figure 3.

**Figure 5.**Deformations of the integration path of $\frac{{\mu}_{0}}{16{\pi}^{3}i}{\displaystyle {\int}_{\epsilon -i\infty}^{\epsilon +i\infty}{I}_{1}\left(t\right)dt}$ and the poles on the real axis.

**Figure 6.**Deformations of the integration path of $\frac{{\mu}_{0}}{16{\pi}^{3}i}{\displaystyle {\int}_{\epsilon -i\infty}^{\epsilon +i\infty}{I}_{2}\left(t\right)dt}$ and the poles on the real axis.

**Figure 10.**Pitch length (h) dependence of the mutual inductance (${M}_{12}$) of coils with equal radius.

**Figure 12.**Pitch length (h) dependence of the mutual inductance (${M}_{12}$) of coils with different heights.

**Figure 15.**Pitch length (h) dependence of the mutual inductance (${M}_{12\mathrm{band}}$) of tape coils with equal radius.

**Figure 16.**Pitch length ($h$) dependence of the mutual inductance (${M}_{12\mathrm{band}\pi}$) of closely wound tape coils.

Functions/Symbols | Name |
---|---|

${I}_{\lambda}\left(x\right)$ | Modified Bessel function of the first kind |

${K}_{\lambda}\left(x\right)$ | Modified Bessel function of the second kind |

$\mathrm{\Gamma}\left(x\right)$ | Gamma function |

${}_{2}F_{1}\left(a,b;c;x\right)$ | Gauss hypergeometric function |

${}_{2}F_{1}^{(1,0,0,0)}\left(a,b;c;x\right)$ | First-order differentiation of the Gauss hypergeometric function with respect to $a$ |

${}_{2}F_{1}^{(0,1,0,0)}\left(a,b;c;x\right)$ | First-order differentiation of the Gauss hypergeometric function with respect to $b$ |

$\Re \mathfrak{e}\left(x\right)$ | Real part of $x$ |

${\mathbb{N}}^{*}$ | Positive integers |

**Table 2.**Geometric parameters of the coils in Figure 2 (1st comparison).

Parameters | Coil-1 | Coil-2 |
---|---|---|

Number of turns | 10 | 10 |

Pitch length (m) | h | h |

Coil length (m) | $10\times h$ | $10\times h$ |

Angular coordinate of the midpoint of the conductor (rad) | 0 | 0 |

Radius (m) | 0.4 | 0.5 |

**Table 3.**Geometric parameters of the coils in Figure 2 (2nd comparison).

Parameters | Coil-1 | Coil-2 |
---|---|---|

Number of turns | 10 | 10 |

Pitch length (m) | 0.629 | 0.629 |

Coil length (m) | 6.29 | 6.29 |

Angular coordinate of the midpoint of the conductor (rad) | 0 | θ |

Radius (m) | 0.4 | 0.5 |

**Table 4.**Geometric parameters of the coils in Figure 2 (3rd comparison).

Parameters | Coil-1 | Coil-2 |
---|---|---|

Number of turns | 10 | 10 |

Pitch length (m) | h | h |

Coil length (m) | $10\times h$ | $10\times h$ |

Angular coordinate of the midpoint of the conductor (rad) | 0 | π |

Radius (m) | 0.5 | 0.5 |

**Table 5.**Geometric parameters of the coils in Figure 2 (4th comparison).

Parameters | Coil-1 | Coil-2 |
---|---|---|

Number of turns | 10 | 5 |

Pitch length (m) | $h$ | $2h$ |

Coil length (m) | $10\times h$ | $5\times 2h$ |

Angular coordinate of the midpoint of the conductor (rad) | $0$ | $\pi $ |

Radius (m) | 0.4 | 0.5 |

**Table 6.**Geometric parameters of the coils in Figure 2 (5th comparison).

Parameters | Coil-1 | Coil-2 |
---|---|---|

Number of turns | 10 | 10 |

Pitch length (m) | $0.629$ | $h$ |

Coil length (m) | $6.29$ | $10\times h$ |

Angular coordinate of the midpoint of the conductor (rad) | 0 | 0 |

Radius (m) | 0.4 | 0.5 |

**Table 7.**Geometric parameters of the coils in Figure 3 (6th comparison).

Parameters | Coil-1 | Coil-2 |
---|---|---|

Number of turns | 10 | 10 |

Pitch length ($\mathrm{m}$) | $0.629$ | $0.629$ |

Coil length ($\mathrm{m}$) | $6.29$ | $6.29$ |

Angular coordinate of the midpoint of the center lines (rad) | 0 | 0 |

Radius ($\mathrm{m}$) | 0.4 | 0.5 |

**Table 8.**Geometric parameters of the coils in Figure 3 (7th comparison).

Parameters | Coil-1 | Coil-2 |
---|---|---|

Number of turns | 10 | 10 |

Pitch length (m) | $h$ | $h$ |

Coil length (m) | $10\times h$ | $10\times h$ |

Angular coordinate of the midpoint of the center lines (rad) | 0 | π |

Radius (m) | 0.5 | 0.5 |

**Table 9.**Geometric parameters of the closely-wound coils in Figure 3 (8th comparison).

Parameters | Coil-1 | Coil-2 |
---|---|---|

Number of turns | 10 | 10 |

Pitch length (m) | $h$ | $h$ |

Coil length (m) | $10\times h$ | $10\times h$ |

Angular coordinate of the midpoint of the center lines (rad) | 0 | 0 |

Radius (m) | 0.4 | 0.5 |

**Table 10.**Pitch length (h) dependence of the time consumption and calculation result between methods of integral and series with different partial sum of the terms.

Pitch Length | h (m) | 0.223 | 0.315 | 0.445 | 0.629 | 0.888 |
---|---|---|---|---|---|---|

Equation (39) of this paper | Time (ms) | 212.6 | 222.2 | 217.0 | 224.6 | 220.7 |

Value (μH) | 23.87046728 | 18.47706415 | 14.61415102 | 12.25892005 | 11.43827601 | |

Equation (48) of this paper (n = 1) | Time (ms) | 4.104 | 4.359 | 5.733 | 5.172 | 4.369 |

Value (μH) | 23.88430910 | 18.47956070 | 14.61459156 | 12.25899553 | 11.43828841 | |

Equation (48) of this paper (n = 3) | Time (ms) | 5.271 | 4.710 | 5.854 | 5.639 | 4.213 |

Value (μH) | 23.87051615 | 18.47706641 | 14.61415112 | 12.25892005 | 11.43827601 | |

Equation (48) of this paper (n = 5) | Time (ms) | 4.704 | 4.654 | 5.397 | 4.603 | 5.450 |

Value (μH) | 23.87046768 | 18.47706416 | 14.61415102 | 12.25892005 | 11.43827601 | |

Equation (48) of this paper (n = 10) | Time (ms) | 4.977 | 4.875 | 6.529 | 5.025 | 4.745 |

Value (μH) | 23.87046729 | 18.47706416 | 14.61415102 | 12.25892005 | 11.43827601 |

h (m) | 0.223 | 0.315 | 0.445 | 0.629 | 0.888 |

FEM (h) | 1.286 | 2.588 | 3.965 | 7.256 | 25.56 |

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**MDPI and ACS Style**

Zhou, X.; Chen, B.; Luo, Y.; Zhu, R.
Analytical Calculation of Mutual Inductance of Finite-Length Coaxial Helical Filaments and Tape Coils. *Energies* **2019**, *12*, 566.
https://doi.org/10.3390/en12030566

**AMA Style**

Zhou X, Chen B, Luo Y, Zhu R.
Analytical Calculation of Mutual Inductance of Finite-Length Coaxial Helical Filaments and Tape Coils. *Energies*. 2019; 12(3):566.
https://doi.org/10.3390/en12030566

**Chicago/Turabian Style**

Zhou, Xinglong, Baichao Chen, Yao Luo, and Runhang Zhu.
2019. "Analytical Calculation of Mutual Inductance of Finite-Length Coaxial Helical Filaments and Tape Coils" *Energies* 12, no. 3: 566.
https://doi.org/10.3390/en12030566