Name: three-phase line impedance matrix
Uploaded: 2024-11-21T14:54:23.000Z
Duration: 1 min 27 s
Channel: JoVE

抵抗や誘導性リアクタンスなどの直列インピーダンスを計算するには、三相架空送電線用に定期的に接地された三相導体とN中性導体のネットワークを考えてみましょう。
キルヒホッフの法則を使用して、ネットワークの積分微分方程式が導出されます。
不平衡な位相電流は、最小インピーダンスの経路に従って、中性線とアースを介してリターン電流を誘導する可能性があります。
架空導体の負電流をミラーリングするアースリターン導体は、アースを置き換えることができます。
これらの数値は、オーバーヘッドの対応するものと同じ幾何平均半径と抵抗を持っています。
架空導体は、相導体から始まり、中性導体と番号が付け直されます。
伝送線路では、導体間の合計電流はゼロであり、各架空導体をつなぐ磁束を定義します。
1メートルのセクションの回路表現を使用して、導体間の電圧降下のベクトルが計算されます。
方程式は分割され、解かれ、相導体と相電流、およびそれらの関係にわたる電圧降下ベクトルが得られます。
これにより、3 行 3 列の直列位相インピーダンス行列でラインの動作を理解できます。
完全に転置された線の場合、対角要素と非対角要素を平均化すると、一般化された線の動作ビューが提供されます。

直列インピーダンス: 3 相線

Series Impedances: Three-Phase Line

3 相架空線の直列インピーダンスを計算するには、一定の間隔で接地された 3 相および複数の中性導体を含むネットワーク内の抵抗と誘導リアクタンスを評価する必要があります。 
キルヒホッフの法則を使用して、ネットワークの積分微分方程式が導出されます。この方程式は、不平衡位相電流を考慮しており、不平衡位相電流は、最小インピーダンス経路を求めて、中性線とアースを通るリターン電流を誘導する可能性があります。アースリターン導体は、計算で実際のアースを置き換えることができ、架空導体の負の電流を反映し、同一の幾何平均半径 (GMR) と抵抗を持ちます。 
架空導体は、相導体から始まり、中性導体の順に番号が付けられます。送電線路のすべての導体を流れる電流の合計はゼロであり、これが各架空導体の磁束鎖交を定義する条件です。導体間の電圧降下のベクトルは、回路の 1 m のセクションを表すことによって計算できます。 
キルヒホッフの法則から導かれた方程式を分割して解くと、相導体の電圧降下ベクトルと対応する相電流が得られます。これらの関係により、3 x 3 の直列相インピーダンス行列を作成できます。この行列は、線路の電気的動作をカプセル化し、相導体とそのインピーダンス特性間の相互作用に関する重要な洞察を提供します。 
完全に転置された線路では、導体の位置がインダクタンスと静電容量のバランスをとるために定期的に回転するため、インピーダンス行列をさらに簡略化できます。対角要素と非対角要素を平均化することで、線路の動作の一般的なビューが得られます。このアプローチは、さまざまな動作条件下での伝送線路のパフォーマンスを予測するのに役立ち、効率的で信頼性の高い電力伝送を保証します。 
詳細なモデリングと複雑な方程式の解決を通じて 3 相線の直列インピーダンスを理解することは、効率的な電力伝送システムを設計および運用するために不可欠です。

抵抗や誘導性リアクタンスなどの直列インピーダンスを計算するには、接地された三相導体とN中性導体のネットワークを考えてみましょう...

Education

JoVE Core

Electrical Engineering

Unit 1

Transmission Line Parameters

KEY TERMS AND CONCEPTS

three-phase line impedance matrix

Three-phase line impedance matrix calculations describe how resistance and inductive reactance behave in an overhead power line. The model includes three phase conductors and multiple neutral conductors that are grounded at regular intervals. It also accounts for unbalanced phase currents, which can send return current through the neutral wires and the earth along the lowest-impedance path.

Kirchhoff’s laws are used to build an integro-differential equation for the conductor network. To make the calculation practical, the earth can be replaced by an earth return conductor in the model. That conductor carries the negative current of the overhead conductors and is given the same geometric mean radius, or GMR, and resistance.

The overhead conductors are then renumbered. The phase conductors come first, and the neutral conductors follow. The total current across all conductors in the transmission line is zero, and that condition is used to define the flux linkage for each overhead conductor.

A one-meter section of the circuit is used to form the vector of voltage drops across the conductors. The equations from Kirchhoff’s laws are then partitioned and solved for the phase conductors and their currents. These results lead to a three-by-three series-phase impedance matrix that describes how the phases interact electrically.

For fully transposed lines, the conductor positions are rotated in a regular pattern to balance inductance and capacitance. In that case, the impedance matrix can be simplified by averaging the diagonal and off-diagonal elements. This gives a general picture of line behavior under different operating conditions and helps support efficient, reliable power transmission.

Three-Phase Line Impedance Matrix

transmission line design considerations

Overhead Power Line Materials and Safety

Overhead transmission lines rely on the right materials, insulation, and support hardware to move electricity safely and efficiently. Aluminum is now the preferred conductor material because it is abundant and cost-effective. Copper was used more often in the past, but aluminum has become the common choice for modern lines. The most widely used conductor is aluminum conductor, steel-reinforced, or ACSR. It uses aluminum strands around a steel core for added strength. Other conductor types...

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How Transmission Lines Lose Energy

Transmission lines lose energy through resistance and conductance. DC resistance depends on a conductor’s resistivity, length, and cross-sectional area. Resistivity is a material property, and annealed copper is the international standard used for measurement. Material choice and conductor shape both matter. Hard-drawn aluminum at 20 degrees Celsius has a resistivity equal to 61% of the standard conductivity of annealed copper. Stranded conductors also gain a little extra resistance because...

inductance: solid cylindrical conductor

Magnetic Flux Linkage in Solid Conductors

Magnetic flux linkage in a solid cylindrical conductor is found by combining the field inside the wire with the field around it. The method uses a 1-meter section of a non-magnetic conductor with radius r. It also assumes the current is spread uniformly and ignores end effects. Inside the conductor, Ampere's law is used to find the magnetic field intensity H. The field is concentric and constant within the conductor. From H, the magnetic flux density B inside the wire is calculated. The...

inductance: single-phase and three-phase line

inductance-single-phase-and-three-phase-line

Transmission Line Inductance in Power Cables

Transmission line inductance helps explain how current-carrying conductors behave in power systems. It is especially important when studying single-phase two-wire lines and three-phase three-wire lines with equal phase spacing. These line types are common models for efficient transmission design and operation. A single-phase line uses two solid cylindrical conductors, labeled x and y. Each conductor carries a phasor current, ix and iy, and the sum of these currents is zero. The total flux...

series impedances: three-phase line

Three-phase line impedance matrix calculations describe how resistance and inductive reactance behave in an overhead power line. The model includes three phase conductors and multiple neutral conductors that are grounded at regular intervals. It also accounts for unbalanced phase currents, which can send return current through the neutral wires and the earth along the lowest-impedance path. Kirchhoff’s laws are used to build an integro-differential equation for the conductor network. To make...

capacitance: single-phase and three-phase line

capacitance-single-phase-and-three-phase-line

Transmission Line Capacitance in Power Systems

Transmission line capacitance affects how power lines store charge and behave in electrical power systems. It is an important part of line modeling because it helps describe efficient operation. For a single-phase line, the key idea is the charge on the conductors and the voltage between them. A single-phase transmission line uses two wires with equal spacing. One conductor carries a uniform positive charge, and the other carries an equal negative charge. From the voltage between the...

shunt admittances

Image Method for Line Capacitance

Image conductors help model line capacitance in transmission lines above Earth. A uniformly charged conductor induces an equal and opposite charge on the ground. This creates electric field lines between the conductor and the Earth. The method of images replaces the Earth with a matching conductor placed symmetrically below the line. The image conductor has the same radius and the same charge magnitude as the original conductor. That setup keeps the electric field and voltage above the Earth...

key terms and concepts

Learn how Kirchhoff’s laws, neutral conductors, and earth return paths are used to build a three-by-three impedance matrix for a three-phase overhead line.

Three-Phase Line Impedance Matrix | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Three-Phase Line Impedance Matrix

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