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

ノード解析は、依存または独立した電圧源を持つ回路の解析を効果的に簡素化できます。
4つの抵抗と2つの電圧源が構成され、1つの電圧源が非リファレンスノードとリファレンスノードの間に接続される回路を考えてみましょう。
ここで、非リファレンスノードの電圧は、ソースの電圧に等しいものとして書き込むことができます。
もう一方の電圧源は、2つの非リファレンスノード間に接続されています。これらは、スーパーノードまたは一般化ノードを形成します。
スーパーノードを持つ回路を解析するには、キルヒホッフの電流法則と電圧法則の両方を適用して、ノード電圧を決定する必要があります。
まず、キルヒホッフの電流法則がスーパーノードに適用され、各要素を通過する電流が考慮されます。得られた方程式は、ノード電圧で書くことができます。
次に、回路を再描画して、キルヒホッフの電圧則をスーパーノードに適用します。
スーパーノードを含むループを時計回りに一周すると、制約方程式が得られます。
結果として得られる3つの方程式を解いて、ノード電圧を決定できます。

電圧源を使用したノード解析

ノード解析は、独立電圧源および従属電圧源を含む複雑な回路の解析を簡素化するために電気工学で使用される非常に効果的な方法です。その強みは、回路を管理可能なコンポーネントに分割する体系的なアプローチにあり、エンジニアが理解しやすく、解決しやすくなります。

Figure 1 に示すように、4 つの抵抗と 2 つの電圧源を含む回路を考えてみましょう。これらの電圧源の 1 つは、ノードと基準ノードの間に接続されています。この構成では、ノードの電圧はソースの電圧と等しいものとして直接書き込むことができます。これにより、不明な点の数が減り、問題が単純化されます。

<img alt="Figure1" src="/files/ftp_upload/15048/15048_PT_Figure_1.svg" style="text-align: center; height: 140px;">

もう 1 つの電圧源は 2 つのノード間に接続されます。この構成は、スーパーノードとして知られるものを形成します。スーパーノードはノード解析における独自の概念であり、ノード間に電圧源が接続されている状況を処理するのに役立ちます。

スーパーノードを使用した回路を解析するには、「キルヒホッフの第一法則（電流則）」および「キルヒホッフの第二法則（電圧則）」 の両方を適用する必要があります。まず、KCL がスーパーノードに適用されます。この法則は、ノードに入る電流の合計はノードから出る電流の合計に等しい、と述べています。スーパーノード内の各要素を流れる電流を考慮することで、Figureを取得し、ノード電圧に関して記述することができます。

次に、KVL を適用するために、回路が再描画されて、スーパーノードを含むループが強調表示されます。 KVL は、ネットワーク内の閉ループまたはメッシュ内の電源電圧の合計は、そのループ内の電圧降下の合計に等しいと述べています。このループを時計回りに周回すると、制約式が得られます。

これらの手順により、回路の動作を表す 3 つのFigureが得られます。これらのFigureを同時に解くことでノード電圧を決定できます。

ノード解析は、依存または独立した電圧源を持つ回路の解析を効果的に簡素化できます。
抵抗が4つある回路を考えてみましょう...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

DC Circuit Analysis

直流回路解析

KEY TERMS AND CONCEPTS

主要用語および概念

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

Supernode Equations in Nodal Analysis

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