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

Die Knotenanalyse kann die Analyse von Schaltkreisen mit abhängigen oder unabhängigen Spannungsquellen effektiv vereinfachen.
Stellen Sie sich eine Schaltung mit vier Widerständen und zwei Spannungsquellen vor, die so konfiguriert sind, dass eine Spannungsquelle zwischen einem Nicht-Referenzknoten und dem Referenzknoten verbunden ist.
Dabei kann die Spannung am Nicht-Referenzknoten gleich der Spannung der Quelle geschrieben werden.
Die andere Spannungsquelle ist zwischen zwei Nicht-Referenzknoten geschaltet. Diese bilden einen Superknoten oder verallgemeinerten Knoten.
Um eine Schaltung mit einem Superknoten zu analysieren, muss sowohl das Kirchhoffsche Strom- als auch das Spannungsgesetz angewendet werden, um die Knotenspannungen zu bestimmen.
Zunächst wird das aktuelle Gesetz von Kirchhoff auf den Superknoten angewendet, wobei die Ströme durch jedes Element berücksichtigt werden. Die erhaltene Gleichung kann in Bezug auf die Knotenspannungen geschrieben werden.
Dann wird die Schaltung neu gezeichnet, um das Kirchhoffsche Spannungsgesetz auf den Superknoten anzuwenden.
Wenn man die Schleife, die den Superknoten enthält, im Uhrzeigersinn umgeht, erhält man eine Einschränkungsgleichung.
Die daraus resultierenden drei Gleichungen können nun gelöst werden, um die Knotenspannungen zu bestimmen.

knotenanalyse mit spannungsquellen

Die Knotenanalyse ist eine bemerkenswert effektive Methode, die in der Elektrotechnik verwendet wird, um die Analyse komplexer Schaltungen zu vereinfachen, einschließlich solcher mit abhängigen oder unabhängigen Spannungsquellen. Ihre Stärke liegt in ihrem systematischen Ansatz, Schaltungen in handhabbare Komponenten zu zerlegen, was es Ingenieuren erleichtert, diese zu verstehen und zu lösen. Betrachten wir eine Schaltung, die vier Widerstände und zwei Spannungsquellen enthält, wie in Figure 1 gezeigt. Eine dieser Spannungsquellen ist zwischen einem Nicht-Referenzknoten und dem Referenzknoten geschaltet. In dieser Konfiguration kann die Spannung am Nicht-Referenzknoten direkt als gleich der Spannung der Quelle geschrieben werden. Dies vereinfacht das Problem, indem die Anzahl der Unbekannten reduziert wird. <img alt="Figure1" src="/files/ftp_upload/15048/15048_PT_Figure_1.svg" style="text-align: center; height: 140px;"> Die andere Spannungsquelle ist zwischen zwei Nicht-Referenzknoten geschaltet. Diese Konfiguration bildet das, was als Supernode oder generalisierter Knoten bekannt ist. Supernodes sind ein einzigartiges Konzept in der Knotenanalyse, das hilft, Situationen zu behandeln, in denen Spannungsquellen zwischen Nicht-Referenzknoten verbunden sind. Die Analyse einer Schaltung mit einem Supernode erfordert die Anwendung sowohl des Kirchhoffschen Stromgesetzes (KCL) als auch des Kirchhoffschen Spannungsgesetzes (KVL). Zuerst wird KCL auf den Supernode angewendet. Dieses Gesetz besagt, dass die Summe der in einen Knoten einfließenden Ströme gleich der Summe der aus ihm herausfließenden Ströme ist. Durch Berücksichtigung der Ströme durch jedes Element innerhalb des Supernode kann eine Gleichung erhalten und in Bezug auf die Knotenspannungen geschrieben werden. Um KVL anzuwenden, wird der Schaltkreis neu gezeichnet, um die Schleife mit dem Supernode hervorzuheben. KVL besagt, dass die Summe der elektromotorischen Kräfte in jeder geschlossenen Schleife oder Masche in einem Netzwerk gleich der Summe der Potentialabfälle in dieser Schleife ist. Wenn man diese Schleife im Uhrzeigersinn durchläuft, ergibt sich eine Einschränkungsgleichung. Mit diesen Schritten werden drei Gleichungen erhalten, die das Verhalten der Schaltung darstellen. Diese Gleichungen können gleichzeitig gelöst werden, um die Knotenspannungen zu bestimmen.

Knotenanalyse mit Spannungsquellen

Die Knotenanalyse kann die Analyse von Schaltkreisen mit abhängigen oder unabhängigen Spannungsquellen effektiv vereinfachen.
Stellen Sie sich eine Schaltung mit vier Widerständen vor ...

Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

DC Circuit Analysis

DC-Schaltungsanalyse

KEY TERMS AND CONCEPTS

SCHLÜSSELBEGRIFFE UND KONZEPTE

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...

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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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