Name: supernode equations in nodal analysis
Uploaded: 2025-05-22T09:06:13.000Z
Duration: 1 min 11 s
Channel: JoVE

Узловой анализ может эффективно упростить анализ цепей с зависимыми или независимыми источниками напряжения.
Рассмотрим схему с четырьмя резисторами и двумя источниками напряжения, настроенную таким образом, что один источник напряжения подключен между узлом, не являющимся опорным, и опорным узлом.
Здесь напряжение на неопорном узле может быть записано равным напряжению источника.
Другой источник напряжения подключен между двумя узлами, не являющимися опорными. Они образуют подузел или обобщенный узел.
Чтобы проанализировать цепь с подузлом, необходимо применить как закон тока, так и закон напряжения Кирхгофа для определения напряжения узла.
Во-первых, закон тока Кирхгофа применяется к подузлу с учетом токов, проходящих через каждый элемент. Полученное уравнение можно записать в терминах узловых напряжений.
Затем схема перерисовывается, чтобы применить закон напряжения Кирхгофа к подузлу.
Обход цикла, содержащего подузел, по часовой стрелке дает уравнение ограничения.
Полученные три уравнения теперь могут быть решены для определения напряжений узлов.

Метод узловых напряжений с источниками напряжения

Метод узловых напряжений — чрезвычайно эффективный метод, используемый в электротехнике для упрощения анализа сложных цепей, в том числе с зависимыми или независимыми источниками напряжения. Своей эффективностью он обязан систематическому подходу к разбиению схем на меньшие компоненты, что упрощает понимание и решение инженерных задач.

Рассмотрим схему, содержащую четыре резистора и два источника напряжения, как показано на рисунке 1. Один из этих источников напряжения подключен между неопорным узлом и опорным узлом. В этой конфигурации напряжение в неопорном узле можно принять равным напряжению источника. Это упрощает задачу за счет уменьшения количества неизвестных.

<img alt="Figure1" src="/files/ftp_upload/15048/15048_PT_Figure_1.svg" style="text-align: center; height: 140px;" />

Другой источник напряжения подключен между двумя неопорными узлами. Эта конфигурация образует так называемый суперузел или обобщенный узел. Суперузлы — это уникальная концепция метода узловых напряжений, которая помогает обрабатывать ситуации, когда источники напряжения подключаются между неопорными узлами.

Анализ схемы с суперузлом требует применения как закона тока Кирхгофа (KCL), так и закона напряжения Кирхгофа (KVL). Сначала KCL применяется к суперузлу. Этот закон гласит, что сумма токов, входящих в узел, равна сумме токов, выходящих из него. Рассматривая токи через каждый элемент внутри суперузла, можно получить уравнение и записать его в терминах узловых напряжений.

Далее, чтобы применить KVL, схема перерисовывается, чтобы выделить контур, содержащий суперузел. KVL утверждает, что алгебраическая сумма падений напряжений по любому замкнутому контуру цепи равна алгебраической сумме ЭДС, действующих вдоль этого же контура.. Обход этого цикла по часовой стрелке дает уравнение ограничения.

С помощью этих шагов получаются три уравнения, которые описывают поведение схемы. Эти уравнения можно одновременно решить для определения узловых напряжений.

Узловой анализ может эффективно упростить анализ цепей с зависимыми или независимыми источниками напряжения.
Рассмотрим схему с четырьмя резисторами...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

DC Circuit Analysis

Анализ цепей постоянного тока

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

Nodal Analysis with Voltage Sources

supernode equations in nodal analysis

Nodal analysis uses node voltages and Kirchhoff’s laws to solve circuits with voltage sources. It is a systematic method for breaking a complex circuit into smaller parts that are easier to analyze. In this example, the circuit includes four resistors and two voltage sources.

One voltage source is connected between a non-reference node and the reference node. In that case, the non-reference node voltage can be written directly as the source voltage. This removes one unknown and makes the circuit easier to solve.

The other voltage source is connected between two non-reference nodes. That connection creates a supernode, also called a generalized node. A supernode is a special case in nodal analysis that helps when a voltage source sits between two nodes that are not the reference node.

To analyze the supernode, Kirchhoff’s Current Law (KCL) is used first. KCL says the total current entering a node equals the total current leaving it. The currents through the elements in the supernode are written in terms of the node voltages to form one equation.

Next, Kirchhoff’s Voltage Law (KVL) is applied. The circuit is redrawn to show the loop that includes the supernode, and the loop is traced in a clockwise direction. KVL states that the sum of the electromotive forces in a closed loop equals the sum of the voltage drops in that loop, which gives a second constraint equation.

These steps produce three equations for the circuit. The equations can then be solved together to find the node voltages. This approach shows how nodal analysis handles voltage sources connected to both the reference node and other nodes.

Supernode Equations in Nodal Analysis

nodal analysis

Using Node Voltages in Circuit Analysis

Nodal analysis uses node voltages to simplify circuit analysis. In this method, the voltage at each node becomes the main variable. Once those voltages are known, they can be used to find other values, such as the current through specific circuit parts. The process begins by choosing a reference node, also called a datum node. This node is selected for convenience and assigned a voltage of zero. The other nodes, called non-reference nodes, are then measured relative to that point. In the...

nodal analysis with voltage sources

Nodal analysis uses node voltages and Kirchhoff’s laws to solve circuits with voltage sources. It is a systematic method for breaking a complex circuit into smaller parts that are easier to analyze. In this example, the circuit includes four resistors and two voltage sources. One voltage source is connected between a non-reference node and the reference node. In that case, the non-reference node voltage can be written directly as the source voltage. This removes one unknown and makes the...

mesh analysis

Mesh Currents for Circuit Analysis

Mesh currents give students a simpler way to analyze planar circuits. In mesh analysis, the unknowns are currents, not voltages. The method uses Kirchhoff's voltage law, or KVL, to build the equations needed to solve the circuit. A mesh is a closed loop in a circuit that does not contain any smaller loops inside it. Each independent mesh is assigned a mesh current, and this current is usually drawn in the clockwise direction. These mesh currents are used as the main circuit variables during...

mesh analysis with current sources

Current Sources in Mesh Circuit Analysis

Current sources make mesh circuit analysis easier because they can reduce the number of equations needed. This applies to both independent and dependent current sources. The method depends on where the current source is placed in the circuit. When a current source appears in only one mesh, the setup is simple. Assign mesh currents as usual, but leave out the mesh that contains the current source from the equation process. Then apply Kirchhoff's voltage law, or KVL, to the remaining mesh to get...

source transformation

Circuit Source Equivalents

Circuit source equivalents make it easier to simplify electrical circuits in circuit analysis. A source transformation replaces a voltage source in series with a resistor by a current source in parallel with a resistor, or it replaces a current source in parallel with a resistor by a voltage source in series with a resistor. The key idea is that the equivalent resistance at the circuit terminals stays the same when the original sources are turned off. When making the change, the current source...

linear circuits

Circuit Linearity and Resistor Behavior

Circuit linearity describes how a circuit responds when its input changes. A linear circuit has an output that is directly proportional to its input. This behavior follows two rules: homogeneity and additivity. Homogeneity means that if the input, or excitation, is multiplied by a constant factor, the response is multiplied by the same factor. For example, if the current is increased by a constant k, the voltage also increases by k times. Additivity means that the response to a sum of inputs...

superposition theorem

Analyzing Circuits with Superposition

Superposition is a circuit analysis method for linear circuits with more than one independent source. It lets you find the voltage across or current through an element by adding the separate effects of each source acting alone. To use superposition, keep one independent source active and turn off the others. A voltage source is replaced by a short circuit, and a current source is replaced by an open circuit. Then calculate the output from the single active source. Repeat the process for each...

thevinin's theorem

Thevenin Circuit Equivalents

Thevenin circuit equivalents simplify linear two-terminal circuits by replacing the fixed part of a network with one equivalent source and one resistor. This method is useful when a circuit has a fixed section and a load that can change. A household electrical outlet is a familiar example, because different appliances can be plugged in or removed. Each change in load can require a new circuit analysis, so an equivalent circuit makes the work easier. Thevenin's theorem states that a linear...

norton's theorem

Norton Equivalent Circuit in DC Analysis

Norton equivalent circuit in DC analysis shows how a linear two-terminal circuit can be replaced by a simpler network. The replacement uses a current source, called I_N, in parallel with a resistor, called R_N. I_N is the short-circuit current through the terminals, and R_N is the input or equivalent resistance seen at the terminals when the independent sources are turned off. To find Norton's resistance, all independent sources in the circuit are set to zero. The resistance measured at the...

maximum power transfer

Load Matching in Thevenin Circuits

Load matching in Thevenin circuits helps a source deliver the most power to a connected load. This matters in engineering fields such as telecommunications, where power delivery must be optimized. Even then, internal losses can be significant and may equal or exceed the power sent to the load. The Thevenin equivalent circuit is a useful way to study this situation. It replaces the full linear circuit with a simpler source and resistance while keeping the load in place. In this setup, the load...

design example: strain gauge bridge or wheatstone bridge

design-example-strain-gauge-bridge-or-wheatstone-bridge

Wheatstone Bridge Output for Strain Gauge Design

A strain gauge and Wheatstone bridge work together to turn mechanical strain into an electrical signal. In engineering, this setup is often used to measure force or pressure. Each part of the bridge has a resistance that changes slightly when strain is applied. The main goal is to turn a very small voltage output into a larger, easier-to-read signal. A voltmeter can then measure that output more clearly. To do this, an engineer designs the strain gauge circuit and figures out how much...

key terms and concepts

Learn how nodal analysis uses supernodes, KCL, and KVL to solve circuits with voltage sources and find unknown node voltages.

Supernode Equations in Nodal Analysis | 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

Supernode Equations in Nodal Analysis

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