27.3
电感:实心圆柱导体
要计算实心圆柱形导体的电感，请考虑一段半径为 r 的非磁性载流导体，长度为 1 米。忽略端部效应并假设电流密度均匀，安培定律有助于确定导体内部的磁场。该定律指出，磁场强度 H 在导体内是同心且恒定的。
给定均匀电流分布，计算导体内部的磁场 H_x 和磁通密度 B_x。然后计算导体内部每单位长度的差分…
为计算实心圆柱导体的电感，需考虑一段长度为1米、半径为r的非磁性载流导体，忽略端部效应，并假设电流密度均匀。
安培环路定律计算导体内部恒定的同心磁场。
在电流分布均匀的情况下，可确定磁场和磁通密度。
然后计算单位长度的差动磁通量，以确定导体内部的总磁通链。
这允许计算导体单位长度的内电感。
为了求解导体外部的磁场，将安培环路定律应用于包围全部电流的闭合回路。这有助于计算导体外部的磁通密度、微分磁通以及磁通链。
在两个外部点之间进行积分可得到单位长度的外部磁通链以及相应的外部电感。
外部一点的总磁链是内磁链和外磁链之和，该值决定了总电感。
最后，考虑一组圆柱形导体，每个导体均载有电流，且它们的总电流等于零，此时可计算其磁通链。
View the full transcript and gain access to JoVE Core videos
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.