27.3
Inductance : conducteur cylindrique solide
Pour calculer l'inductance d'un conducteur cylindrique solide, considérons une section d'un mètre d'un conducteur non magnétique et porteur de courant…
Pour calculer l'inductance d'un conducteur cylindrique solide, considérons une section de 1 mètre d'un conducteur porteur de courant non magnétique de rayon r, sans tenir compte des effets finaux et en supposant une densité de courant uniforme.
La loi d'Ampère calcule un champ magnétique concentrique constant à l'intérieur du conducteur.
Étant donné une distribution uniforme du courant, le champ magnétique et la densité de flux sont déterminés.
Ensuite, le flux différentiel par unité de longueur est calculé, ce qui détermine les liaisons de flux totales à l’intérieur du conducteur.
Cela permet de calculer l'inductance interne du conducteur par unité de longueur.
Pour trouver le champ magnétique à l'extérieur du conducteur, la loi d'Ampère est appliquée à un contour englobant tout le courant. Cela permet de calculer la densité de flux magnétique, le flux différentiel et la liaison de flux à l’extérieur du conducteur.
L’intégration entre deux points externes donne la liaison de flux externe et l’inductance externe conséquente par unité de longueur.
La liaison de flux total en un point externe est la somme des liaisons de flux internes et externes, qui détermine l’inductance totale.
Enfin, en considérant un ensemble de conducteurs cylindriques, chacun transportant un courant, avec leur courant total égal à zéro, la liaison de flux est calculée.
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Q1: How does Ampere's law determine the magnetic field inside a solid cylindrical conductor?
Ampere's law calculates a constant concentric magnetic field inside a non-magnetic conductor carrying uniform current. The magnetic field intensity H is concentric and constant throughout the conductor's cross-section. Given uniform current distribution, this determines the magnetic field Hx and flux density Bx inside the conductor, which are essential for calculating internal inductance per unit length.
Q2: What is the difference between internal and external inductance in a cylindrical conductor?
Internal inductance results from magnetic flux linkages within the conductor itself, calculated using Ampere's law on an internal contour. External inductance arises from flux linkages outside the conductor, determined by integrating between two external points. The total inductance combines both components, with external inductance typically dominating in practical transmission line applications.
Q3: How is the differential flux per unit length calculated inside a conductor?
The differential flux per unit length is computed from the magnetic flux density Bx determined by Ampere's law inside the conductor. This differential flux, integrated over the conductor's cross-section, yields total internal flux linkages. These flux linkages directly determine the conductor's internal inductance per unit length.
Q4: Why is uniform current density assumed when calculating cylindrical conductor inductance?
Uniform current density simplifies the mathematical analysis by ensuring consistent magnetic field distribution throughout the conductor's cross-section. This assumption, combined with disregarding end effects, allows Ampere's law to calculate a constant concentric magnetic field. These simplifications make the inductance calculation tractable while maintaining accuracy for practical conductor analysis.
Q5: How does the total inductance relate to internal and external flux linkages?
Total inductance at an external point equals the sum of internal and external flux linkages. Internal flux linkages originate from the magnetic field within the conductor, while external flux linkages result from the field outside. This comprehensive approach ensures accurate inductance determination for solid cylindrical conductors used in transmission systems.
Q6: How is inductance calculated for an array of cylindrical conductors with zero total current?
For multiple cylindrical conductors where total current equals zero, flux linkage is calculated by summing contributions from each conductor individually. This method accounts for the magnetic interactions between conductors in complex configurations. Accurate inductance calculation for conductor arrays is essential for optimizing transmission line performance and efficiency.
Q7: What role does the conductor radius play in determining inductance?
Conductor radius r directly affects both internal and external inductance calculations. Internal inductance depends on the conductor's cross-sectional area and current distribution within radius r. External inductance is determined by integrating magnetic flux density from the conductor surface outward, making radius a critical parameter in the overall inductance formula.