27.3
Inductancia: Conductor cilíndrico sólido
Para calcular la inductancia de un conductor cilíndrico sólido, considere una sección de 1 metro de un conductor no magnético que transporta corriente…
Para calcular la inductancia de un conductor cilíndrico sólido, considere una sección de 1 metro de un conductor portador de corriente no magnético con radio r, sin tener en cuenta los efectos finales y asumiendo una densidad de corriente uniforme.
La ley de Ampere calcula un campo magnético concéntrico constante dentro del conductor.
Dada una distribución de corriente uniforme, se determinan el campo magnético y la densidad de flujo.
Luego, se calcula el flujo diferencial por unidad de longitud, lo que determina los enlaces de flujo totales dentro del conductor.
Esto permite el cálculo de la inductancia interna del conductor por unidad de longitud.
Para encontrar el campo magnético fuera del conductor, se aplica la ley de Ampere a un contorno que encierra toda la corriente. Esto ayuda a calcular la densidad de flujo magnético, el flujo diferencial y el enlace de flujo fuera del conductor.
La integración entre dos puntos externos proporciona el enlace de flujo externo y la consiguiente inductancia externa por unidad de longitud.
El enlace de flujo total en un punto externo es la suma de los enlaces de flujo internos y externos, lo que determina la inductancia total.
Finalmente, considerando una matriz de conductores cilíndricos, cada uno de los cuales lleva una corriente, con su corriente total igual a cero, se calcula el enlace de flujo.
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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.