# Numerically Stable and Computationally Efficient Expression for the Magnetic Field of a Current Loop

^{*}

## Abstract

**:**

## 1. Introduction and Motivation

## 2. Numerical Stability

#### 2.1. Fundamentals and Annotation

#### 2.2. The Classical Solution

#### 2.3. Numerical Stability of the Classical Solution

#### 2.4. Dipole Approximation

#### 2.5. Taylor Approximation

#### 2.6. Binomial Expansion

#### 2.7. An Exact and Stable Representation

#### 2.8. Loss of Precision at Sign Change

## 3. Performance

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Algorithms

#### Appendix A.1. Straightforward Implementation

#### Appendix A.2. Taylor Series Implementations

**Table A1.**Precision of various Taylor implementations in relation to the number of terms included in the respective sums.

Br_taylor_k ($\mathit{k}\le $ value) | Br_taylor_q ($\mathit{k}\ge $ value) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

sig.figs | k-order included in series | sig.figs | q-order included in series | ||||||||

21 | 31 | 41 | 51 | 61 | 20 | 30 | 40 | 50 | 60 | ||

8 | 0.46 | 0.62 | 0.71 | 0.77 | 0.81 | 8 | 0.77 | 0.67 | 0.60 | 0.55 | 0.50 |

10 | 0.35 | 0.53 | 0.63 | 0.70 | 0.75 | 10 | 0.85 | 0.76 | 0.69 | 0.63 | 0.58 |

12 | 0.27 | 0.45 | 0.56 | 0.64 | 0.69 | 12 | 0.90 | 0.82 | 0.75 | 0.70 | 0.65 |

Bz_taylor_k ($k\le $ value) | Bz_taylor_q ($k\ge $ value) | ||||||||||

sig.figs | k-order included in series | sig.figs | q-order included in series | ||||||||

21 | 31 | 41 | 51 | 61 | 20 | 30 | 40 | 50 | 60 | ||

8 | 0.43 | 0.59 | 0.68 | 0.74 | 0.78 | 8 | 0.86 | 0.76 | 0.70 | 0.64 | 0.59 |

10 | 0.33 | 0.50 | 0.61 | 0.68 | 0.73 | 10 | 0.91 | 0.83 | 0.76 | 0.71 | 0.66 |

12 | 0.26 | 0.43 | 0.54 | 0.62 | 0.68 | 12 | 0.94 | 0.87 | 0.81 | 0.76 | 0.72 |

#### Appendix A.3. Original Implementation of Bulirsch’s cel Algorithm

#### Appendix A.4. Implementation of cel *

#### Appendix A.5. Implementation of cel **

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**Figure 1.**(

**a**) Sketch of a current loop and the observed first quadrant. (

**b**) The number of correct significant figures of a straightforward implementation of the textbook expression for the radial component of the B-field of a current loop on a log–log scale.

**Figure 2.**(

**a**) Sketch of typical current loop positioning with the package Magpylib. (

**b**) Demonstrating the relevance of the numerical instability, which becomes visible when a current loop rotates about an observer.

**Figure 3.**Relative error of a straightforward implementation of the textbook expressions for the B-field of a current loop. (

**a**) Radial and (

**b**) axial components, as well as (

**c**) vectorwise analysis reveal a high level of numerical instability. (

**d**) The relation between the cylindrical coordinates and the important quantity k.

**Figure 4.**(

**a**) Two different implementations of the function ${\xi}_{0}$ and (

**b**) the relative difference when comparing them to each other.

**Figure 5.**Vectorwise relative error of the dipole approximation (

**a**) and a Taylor approximation (

**b**) of the B-field of a current loop. The colored contour lines show the numerical error from a straightforward implementation of the classical textbook expressions.

**Figure 6.**Vectorwise relative error of Taylor approximations of the B-field of a current loop. (

**a**) Expansion for small k. (

**b**) Expansion for small q.

**Figure 8.**(

**a**) Computation times of various methods with respect to the fastest one (dipole) as a function of ${k}^{2}$. (

**b**) The respective vectorwise relative errors.

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**MDPI and ACS Style**

Ortner, M.; Leitner, P.; Slanovc, F.
Numerically Stable and Computationally Efficient Expression for the Magnetic Field of a Current Loop. *Magnetism* **2023**, *3*, 11-31.
https://doi.org/10.3390/magnetism3010002

**AMA Style**

Ortner M, Leitner P, Slanovc F.
Numerically Stable and Computationally Efficient Expression for the Magnetic Field of a Current Loop. *Magnetism*. 2023; 3(1):11-31.
https://doi.org/10.3390/magnetism3010002

**Chicago/Turabian Style**

Ortner, Michael, Peter Leitner, and Florian Slanovc.
2023. "Numerically Stable and Computationally Efficient Expression for the Magnetic Field of a Current Loop" *Magnetism* 3, no. 1: 11-31.
https://doi.org/10.3390/magnetism3010002