Name: delta-wye network conversion
Uploaded: 2025-05-22T09:06:14.000Z
Duration: 1 min 16 s
Channel: JoVE

K12 Standards

Arizona Science Standards

High School

Physical Science

HS+Phy.P4

HS+Phy.P4U1.8

Equivalent Resistance

实际应用中，例如三相电力传输和电机，会使用可相互转换的三端Y形（或T形）网络和三角形（或π形）网络。
考虑一个由三个电阻组成的Δ形网络。将其转换为Y形网络时，需通过添加一个中心节点，将相应的Y形网络叠加到Δ形网络上。
由于Δ网络中每对节点之间的电阻与Y网络中相应节点对之间的电阻相等，因此可建立一组三个方程。求解这些方程即可得到Y网络的电阻值。
在Y型网络中，每个电阻的阻值等于相邻两个Δ型支路电阻的乘积，再除以三个Δ型电阻的总和。
类似地，Y形网络可以转换为等效的Δ形网络。在Δ形网络中，每个电阻的阻值等于Y形网络中所有两两组合电阻乘积之和，再除以与之相对的Y形电阻。
对于Y形和三角形结构中电阻相等的平衡网络，其转换公式可简化。

delta网络和y网络

在电路分析中，经常会出现电阻既不是串联也不是并联配置的情况。为了能够解决这种情况，三端等效网络中如wye (Y)（图1(a)）或tee (T) 和delta (Δ)（图1 (b)）或pi (π) 网络便会开始发挥作用。这些网络提供了通用的解决方案，并且会在各种应用中经常遇到，其中包括三相电气系统、电气滤波器和匹配网络。 <img alt="Figure1" src="/files/ftp_upload/15074/15074_PT_Figure_1.svg" style="text-align: center; height: 160px;"> 这些网络的本质在于它们的适应性。它们能够单独使用或集成到更复杂的电路中。例如，当处理Delta网络时但发现使用Y形网络却更方便时，则可以将Y形网络叠加到现有的Delta电路中。为了能够确定Y形网络内的等效电阻，确保Delta网络中每对节点之间的电阻等于Y（或T）网络中同一对节点之间的电阻是至关重要的。 这些网络之间的转换过程涉及到与电阻相关的数学关系。通过引入额外的节点“n”并遵守其中的转换规则来促进转换：在Y网络中，每个电阻器是由两个相邻的Δ (delta) 支路中的电阻器的乘积除以三个Δ之和电阻器所得出的。相反，如果要将Y形网络转换为等效的Delta网络，可以使用如下转换规则：网络中的每个电阻器都是通过同时取两个Y电阻器中的所有可能乘积来除以相反的Y电阻器中所有可能的乘积之和。 这些转换公式简化了平衡网络，其中Y和Δ配置中的电阻是相等的。这种转换不会涉及到在电路中添加或删除组件，而是来替换数学上等效的三端网络模式。它有效地转换了电路，并允许对电阻器进行分析，就像它们是串联或并联的一样，从而使其在必要时能够快速的计算其中的等效电阻。

Delta网络和Y网络

在电路分析中，常会出现电阻既非串联也非并联的情况。为解决此类问题，可采用三端...

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Basics of Electric Circuits

电路基础

KEY TERMS AND CONCEPTS

关键术语与概念

delta-wye network conversion

Delta-wye network conversion helps analyze resistor circuits that are not arranged in simple series or parallel paths. These three-terminal equivalent networks use the wye (Y), also called tee (T), and the delta (Δ), also called pi (π), as practical circuit models.

These networks are useful in many circuit settings. They appear in three-phase electrical systems, electrical filters, and matching networks. They can also be used on their own or combined with more complex circuits.

A delta network can be replaced with an equivalent wye network when that makes the circuit easier to solve. The key rule is that the resistance between each pair of nodes must stay the same in both networks. To find the wye resistors, each one is the product of the two adjacent delta resistors divided by the sum of all three delta resistors.

A wye network can also be converted into an equivalent delta network. In this case, each delta resistor is found from the sum of all possible pairs of wye-resistor products, divided by the opposite wye resistor. These formulas become simpler for balanced networks, where the resistances in the Y and Δ forms are equal. The transformation does not add or remove parts from the circuit. It replaces one mathematically equivalent three-terminal pattern with another, which can make later equivalent-resistance calculations easier.

Delta-Wye Network Conversion

charge and current

Electric Charge in Circuits

Electric charge in circuits explains how charged particles behave in matter and how current is produced. Everyday static effects, like a wool sweater sticking to the body or a shock from a carpet, show charge in action. Charge is an inherent property of atomic particles and is measured in coulombs (C). Matter is made of atoms with electrons, protons, and neutrons. Electrons carry a negative charge (-e), and protons carry a positive charge (+e). These two charges have the same magnitude. An...

voltage

Voltage in Electric Circuits

Voltage in electric circuits is the energy needed to move electric charge between two points. It is also called electromotive force, or EMF. In most circuits, an external force such as a battery provides this energy for electron movement in a conductor. The voltage between points a and b is written as v_ab. It is defined as the work, or energy, needed to move a unit charge from point a to point b. In the equation, w is the energy in joules and q is the charge in coulombs. Voltage is measured...

power and energy

Circuit Power, Absorption, and Supply

Circuit power, absorption, and supply explain how much energy an element uses or delivers in an electric circuit. These ideas matter in electrical engineering because power and energy calculations help describe circuit behavior. A 100-watt light bulb, for example, emits more light than a 60-watt bulb, so wattage is a useful measure in real devices. Power is the time rate of expending or absorbing energy. It is measured in watts, written as W. The relation between power and energy uses the...

electric circuit elements

Circuit Elements: Passive, Active, and Linear

Circuit elements are the building blocks of an electric circuit. An electric circuit is the connection of these elements. In this context, circuit elements are often grouped by how they behave and whether they supply energy. One major group is passive elements. These elements do not generate energy. Common passive elements include resistors, capacitors, and inductors. The other group is active elements, which can generate energy. Generators, batteries, and operational amplifiers are typical...

circuit terminology

Circuit Parts: Branches, Nodes, and Loops

Circuit parts like branches, nodes, loops, and meshes help describe how electrical networks work. An electrical network is a system of connected elements, such as resistors, capacitors, inductors, and voltage or current sources. Unlike a circuit, a network does not have to form a closed path. A circuit is also made of connected electrical elements, but it must include one or more closed paths. These loops let current flow continuously, which is what makes the circuit operate. Because of this,...

independent and dependent sources

Circuit Source Symbols and Controls

Circuit sources in electrical circuits can be independent or dependent. They provide the power a circuit needs to operate. Independent voltage sources supply a fixed voltage, and independent current sources supply a fixed current. Their output does not change because of other parts in the circuit. These sources use standard symbols in circuit diagrams. An independent voltage source is shown with + and - signs to mark the potential difference. An independent current source is shown with an...

ohm's law

Resistance and Conductance in Circuits

Resistance and conductance help explain how electric current moves through a circuit. Resistors are basic circuit parts made from metallic alloys or carbon compounds. They are used to model how a material resists current and to control and regulate electrical power in a circuit. Ohm's law describes the link between voltage and current across a resistor. The voltage across a resistor is directly proportional to the current through it. The constant that sets this relationship is resistance.

kirchhoff's current law

Kirchhoff's Current Law at Circuit Nodes

Kirchhoff's Current Law explains how current behaves at a circuit node. A node is the point where two or more circuit elements meet. The law says the current entering a node equals the current leaving it. That balance follows the rule of charge conservation, which means electric charge is not lost at the junction. When applying KCL, currents entering the node are treated as positive and currents leaving the node are treated as negative. The currents can be added algebraically at the node, and...

kirchhoff's voltage law

Voltage Balance in Closed Circuit Loops

Kirchhoff's Voltage Law explains how voltage is balanced in a closed circuit loop. It is a basic rule in electrical engineering and comes from the idea of energy conservation. Energy can change form, but it cannot be created or destroyed. KVL states that the algebraic sum of all voltages around a closed path, or loop, is zero. In other words, the total voltage supplied in a loop equals the total voltage drop across the components in that loop. A loop can be traced clockwise or anticlockwise,...

voltage dividers

Series Resistors and Voltage Split

Series resistors split voltage across a circuit. In a series setup, resistors are linked one after another. The same current flows through each resistor in the chain. To analyze this behavior, Ohm's law is used to relate voltage, current, and resistance. Kirchhoff's voltage law says the voltages around a series loop add up to the source voltage. That means the current in the circuit equals the source voltage divided by the total resistance in series. The current value can then be used in Ohm's...

current dividers

Parallel Resistors and Current Division

Parallel resistors share the same two nodes, so each resistor has the same voltage across it. In this circuit setup, Kirchhoff's current law applies. The source current equals the sum of the currents through each branch. Ohm's law helps find the branch currents from the source voltage and the resistor values. The source current is equal to the source voltage times the sum of the reciprocals of the resistances. This makes it easier to calculate how current divides in the circuit. The current...

equivalent resistance

Delta-wye network conversion helps analyze resistor circuits that are not arranged in simple series or parallel paths. These three-terminal equivalent networks use the wye (Y), also called tee (T), and the delta (Δ), also called pi (π), as practical circuit models. These networks are useful in many circuit settings. They appear in three-phase electrical systems, electrical filters, and matching networks. They can also be used on their own or combined with more complex circuits. A delta...

design example: resistive touchscreen

Resistive Touchscreen Voltage Mapping

A resistive touchscreen uses voltage mapping to locate a finger touch on a mobile device. It is built from two metallic layers. The top layer is flexible, and the bottom layer is rigid. A narrow gap separates them, and the high resistance between the layers is an important part of the design. When a user touches the screen, the two layers meet at one point. That contact point is called the touchpoint. The touch lowers the resistance between the layers at that spot and changes the electrical...

key terms and concepts

Learn how delta and wye resistor networks are converted into equivalent forms to simplify circuit analysis when resistors are not in series or parallel.

Delta-Wye Network Conversion | 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.

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Delta-Wye Network Conversion

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