Cut-Set Analysis in Electrical Circuits | 电路分析中的割集法

📚 Cut-Set Analysis in Electrical Circuits | 电路分析中的割集法

Cut-set analysis is a powerful graph-theoretic method used in electrical circuit analysis. It extends the fundamental principles of Kirchhoff’s Current Law (KCL) to a systematic matrix formulation, allowing engineers and mathematicians to solve complex circuits with efficiency and precision. This method is particularly valuable when dealing with circuits that have many nodes and branches, as it reduces the number of equations needed to determine unknown voltages and currents.

割集分析是电路分析中一种强大的图论方法。它将基尔霍夫电流定律（KCL）的基本原理推广为系统的矩阵公式，使工程师和数学家能够高效、精确地求解复杂电路。当处理具有许多节点和支路的电路时，该方法尤为有价值，因为它减少了确定未知电压和电流所需的方程数量。

1. Fundamental Concepts of Cut-Sets | 割集的基本概念

A cut-set in graph theory is defined as a minimal set of branches that, when removed from the graph, separates it into two connected components. ‘Minimal’ here means that if any branch in the set were to remain, the graph would still be connected. In circuit analysis, a cut-set represents a collection of circuit elements (resistors, voltage sources, current sources, etc.) that form a closed boundary through which net current must be conserved.

在图论中，割集被定义为这样一组最小支路集合：将这些支路从图中移除后，图被分离成两个连通分量。这里的”最小”意味着如果集合中的任何一条支路被保留，图仍然是连通的。在电路分析中，割集表示一组电路元件（电阻、电压源、电流源等）的集合，它们形成一个闭合边界，通过该边界的净电流必须守恒。

Mathematically, for a graph G = (V, E), a cut-set C is a subset of E such that:

G – C is disconnected, but G – (C – {e}) is connected for every e ∈ C

This definition ensures that the cut-set is the smallest possible set of branches that separates the graph, making it the graph-theoretic analogue of a physical cut through a circuit.

该定义确保割集是分离图的最小支路集合，使其成为物理切割电路在图论上的类比。

2. Kirchhoff’s Current Law and Cut-Sets | 基尔霍夫电流定律与割集

The fundamental principle underlying cut-set analysis is Kirchhoff’s Current Law, which states that the algebraic sum of currents entering a node equals the sum of currents leaving it. More generally, for any closed surface in a circuit, the net current crossing that surface must be zero. A cut-set precisely defines such a closed surface.

割集分析的基本原理是基尔霍夫电流定律，该定律指出：进入节点的电流的代数和等于离开节点的电流之和。更一般地，对于电路中的任何闭合面，穿过该面的净电流必须为零。割集精确定义了这样一个闭合面。

For a cut-set containing branches b₁, b₂, …, bₖ, KCL gives:

i₁ + i₂ + … + iₖ = 0

where each current iₖ is considered positive if it flows in a specified reference direction across the cut boundary, and negative otherwise.

其中每个电流 iₖ 若沿指定参考方向穿过割边界则为正，否则为负。

3. The Cut-Set Matrix Q | 割集矩阵 Q

The cut-set matrix Q is the mathematical backbone of this method. For a circuit with b branches and n nodes, the tree has (n-1) branches. The number of fundamental cut-sets equals the number of tree branches, i.e., (n-1). The matrix Q has dimensions (n-1) × b.

割集矩阵 Q 是该方法的核心数学工具。对于具有 b 条支路和 n 个节点的电路，树具有 (n-1) 条树支。基本割集的数量等于树支的数量，即 (n-1)。矩阵 Q 的维度为 (n-1) × b。

The entries of Q are defined as:

qᵢⱼ = +1 if branch j is in cut-set i and its direction agrees

qᵢⱼ = -1 if branch j is in cut-set i and its direction opposes

qᵢⱼ = 0 if branch j is not in cut-set i

By convention, each fundamental cut-set contains exactly one tree branch, called the “cut-set branch,” and the direction of the cut-set follows the direction of its associated tree branch. This convention ensures linear independence of the resulting equations.

按照惯例，每个基本割集恰好包含一条树支，称为”割集支路”，割集的方向与其关联树支的方向一致。这一惯例确保了所得方程的线性独立性。

4. Constructing a Tree | 构造树

The first step in cut-set analysis is selecting a spanning tree of the circuit graph. A spanning tree is a subgraph that connects all nodes without forming any loops. For a circuit with n nodes, the tree will have exactly (n-1) branches. The remaining (b – n + 1) branches are called “links” or “chords.”

割集分析的第一步是选择电路图的一棵生成树。生成树是连接所有节点但不形成任何回路子图。对于具有 n 个节点的电路，树恰好有 (n-1) 条树支。其余 (b – n + 1) 条支路称为”连支”。

For example, consider a simple circuit with 4 nodes and 6 branches:

Select a tree: edges AB, BC, CD (3 = n-1 branches)
Links: edges AC, AD, BD (3 = b-n+1 branches)
Each tree branch defines one fundamental cut-set

例如，考虑一个具有 4 个节点和 6 条支路的简单电路：

选择树：边 AB、BC、CD（3 = n-1 条树支）
连支：边 AC、AD、BD（3 = b-n+1 条连支）
每条树支定义一个基本割集

5. Fundamental Cut-Sets | 基本割集

For each tree branch, the corresponding fundamental cut-set is constructed by taking that tree branch plus all links that connect the two sides of the cut. To find the cut-set for tree branch ‘t’: remove ‘t’ from the tree, which divides the nodes into two sets — those reachable from one end of ‘t’ and those reachable from the other end. The fundamental cut-set consists of ‘t’ plus all links that cross between these two sets.

对于每条树支，相应的基本割集通过取该树支加上连接割面两侧的所有连支来构造。为了找到树支 ‘t’ 的割集：从树中移除 ‘t’，这将节点分为两组——从 ‘t’ 一端可达的节点和从另一端可达的节点。基本割集由 ‘t’ 加上所有跨接在这两组之间的连支组成。

Example fundamental cut-sets for the 4-node circuit:

Cut-Set	Tree Branch	Included Links	Complete Cut-Set
C₁	AB	AC, AD	{AB, AC, AD}
C₂	BC	AC, BD	{BC, AC, BD}
C₃	CD	AD, BD	{CD, AD, BD}

Note that each fundamental cut-set contains exactly one tree branch, ensuring the (n-1) cut-sets are linearly independent.

注意每个基本割集恰好包含一条树支，确保 (n-1) 个割集是线性独立的。

6. Writing Cut-Set Equations | 建立割集方程

Once the fundamental cut-sets are identified, we apply KCL to each cut-set. For a circuit with (n-1) fundamental cut-sets, we obtain (n-1) independent equations. Each equation sums the currents through branches in that cut-set to zero.

一旦确定了基本割集，我们对每个割集应用 KCL。对于具有 (n-1) 个基本割集的电路，我们得到 (n-1) 个独立方程。每个方程将穿过该割集中支路的电流之和设为零。

In matrix form, this is expressed as:

Q · i = 0

where Q is the (n-1) × b cut-set matrix and i is the b × 1 branch current vector. Expanding this matrix equation yields the individual cut-set equations.

其中 Q 是 (n-1) × b 割集矩阵，i 是 b × 1 支路电流向量。展开该矩阵方程即可得到各个割集方程。

For the example circuit, the matrix equation becomes:

[1 0 0 1 1 0] [i_AB] [0]

[0 1 0 1 0 1] [i_BC] = [0]

[0 0 1 0 1 1] [i_CD] [0]

[i_AC]

[i_AD]

[i_BD]

7. Expressing Currents in Terms of Voltages | 用电压表示电流

To make the cut-set equations solvable, we substitute each branch current with its voltage-current relationship. For a resistor, this is Ohm’s Law: i = V/R = G·V, where G is conductance. For a voltage source with series resistance, the relationship includes the source voltage. For a current source, the current is directly known.

为了使割集方程可解，我们将每条支路的电流替换为其电压-电流关系。对于电阻，这就是欧姆定律：i = V/R = G·V，其中 G 是电导。对于带有串联电阻的电压源，关系式中包含源电压。对于电流源，电流是直接已知的。

This substitution transforms the KCL equations into a system in terms of tree branch voltages v(t₁), v(t₂), …, v(tₙ₋₁):

Q · G · Qᵗ · v_tree = Q · i_sources

where G is a diagonal matrix of branch conductances, Qᵗ is the transpose of Q, v_tree is the vector of tree branch voltages, and i_sources represents current sources and voltage source contributions.

其中 G 是支路电导对角矩阵，Qᵗ 是 Q 的转置，v_tree 是树支电压向量，i_sources 表示电流源和电压源的贡献。

8. Solving a Complete Example | 完整实例求解

Consider the following circuit: a 12 V voltage source connected to three resistors — R₁ = 2 Ω between nodes A and B, R₂ = 4 Ω between B and C, and R₃ = 6 Ω between C and A. Take the tree as branches AB and BC, with CA as the link.

考虑如下电路：一个 12 V 电压源连接到三个电阻——R₁ = 2 Ω 连接节点 A 和 B，R₂ = 4 Ω 连接 B 和 C，R₃ = 6 Ω 连接 C 和 A。取树为支路 AB 和 BC，CA 为连支。

Step 1: Fundamental cut-sets

步骤 1：基本割集

Cut-set C₁: {AB, CA} — separates {A} from {B, C}
Cut-set C₂: {BC, CA} — separates {A, B} from {C}

Step 2: Write Q matrix (branches in order: AB, BC, CA)

步骤 2：写出 Q 矩阵（支路顺序：AB、BC、CA）

Q = [1 0 1]

[0 1 1]

Step 3: Branch conductance matrix

步骤 3：支路电导矩阵

G = diag(1/2, 1/4, 1/6) = diag(0.5, 0.25, 0.1667)

Step 4: Compute Q · G · Qᵗ

步骤 4：计算 Q · G · Qᵗ

Q · G · Qᵗ = [0.5+0.1667 0.1667 ] = [0.6667 0.1667]

[0.1667 0.25+0.1667] [0.1667 0.4167]

Step 5: Right-hand side (current contributions)

步骤 5：右端项（电流贡献）

Assuming the 12 V source is in branch CA oriented from C to A, and the tree branch voltages are v₁ (across AB) and v₂ (across BC):

假设 12 V 电源在支路 CA 中，方向从 C 到 A，树支电压为 v₁（AB 两端）和 v₂（BC 两端）：

[0.6667 0.1667] [v₁] = [0.1667 × 12] = [2.000]

[0.1667 0.4167] [v₂] [0.1667 × 12] [2.000]

Step 6: Solve the system

步骤 6：求解方程组

v₁ = 2.143 V, v₂ = 4.286 V

Step 7: Find branch currents

步骤 7：求支路电流

i_AB = v₁/2 = 1.071 A
i_BC = v₂/4 = 1.071 A
i_CA = (v₁ + v₂ – 12)/6 = (2.143 + 4.286 – 12)/6 = -0.929 A (flows from A to C)

The negative sign for i_CA indicates the current direction is opposite to the assumed reference direction, which is consistent with the source polarity.

i_CA 的负号表示电流方向与假设的参考方向相反，这与电源极性一致。

9. Key Advantages of Cut-Set Method | 割集法的主要优势

The cut-set method offers several significant advantages over other circuit analysis techniques:

割集法相比其他电路分析技术具有若干显著优势：

Reduced system size: Only (n-1) equations are needed, versus b equations for branch-current analysis.
Reduced system size approach: 只需 (n-1) 个方程，而支路电流法需要 b 个方程。

Systematic formulation: The matrix approach eliminates the need for ad-hoc equation writing, reducing human error.
Systematic formulation 系统性建模： 矩阵方法避免了临时编写方程的随意性，减少人为错误。

Handles voltage sources naturally: Voltage sources are readily incorporated into the tree, eliminating the need to convert them to equivalent current sources.
自然处理电压源： 电压源可以直接纳入树中，无需将其转换为等效电流源。

Computational efficiency: The Q·G·Qᵗ matrix is symmetric and positive definite for passive networks, enabling efficient numerical solution algorithms.
计算效率： 对于无源网络，Q·G·Qᵗ 矩阵是对称正定的，可以采用高效的数值求解算法。

10. Relation to Other Methods | 与其他方法的关系

The cut-set method is closely related to the more familiar node-voltage method. In fact, the node-voltage method can be viewed as a special case of cut-set analysis where every node (except the reference node) is paired with the cut-set consisting of all branches incident to that node.

割集法与我们更熟悉的节点电压法密切相关。实际上，节点电压法可以视为割集分析的一种特例：每个节点（除参考节点外）对应一个由连接到该节点的所有支路组成的割集。

The dual of the cut-set method is the loop (mesh) analysis method: while cut-set analysis applies KCL to fundamental cut-sets, mesh analysis applies KVL to fundamental loops. The loop analysis uses the tie-set matrix B, where Q·Bᵗ = 0, showing the orthogonality between cut-set and loop spaces.

割集法的对偶方法是回路（网孔）分析法： 割集分析对基本割集应用 KCL，而回路分析对基本回路应用 KVL。回路分析使用关联矩阵 B，其中 Q·Bᵗ = 0，表明割集空间与回路空间的正交性。

For the example circuit, the dual relationship yields:

Q · Bᵗ = [1 0 1] · [1 1 1]ᵗ = 1×1 + 0×1 + 1×(-1) = 0

This orthogonality condition demonstrates the fundamental duality in circuit theory between voltage and current laws, and it underpins the equivalence of different analysis methods.

这种正交性条件展示了电路理论中电压定律与电流定律之间的基本对偶性，并为不同分析方法之间的等效性提供了数学基础。

11. Practical Applications | 实际应用

Cut-set analysis finds extensive application in various fields of electrical engineering and beyond:

割集分析在电气工程的各个领域及其他学科中都有广泛的应用：

Power systems: Fault analysis in large transmission networks uses cut-set concepts to identify critical lines whose removal would cause system separation.
电力系统： 大型输电网中的故障分析使用割集概念来识别关键线路，这些线路的断开会导致系统解列。

Circuit simulation: SPICE and other circuit simulators internally use modified nodal analysis, which builds upon cut-set equation formulations.
电路仿真： SPICE 和其他电路仿真器内部使用修正节点分析，该分析建立在割集方程公式的基础之上。

Network reliability: The reliability of communication networks is assessed by identifying minimum cut-sets — the smallest sets of links whose failure disconnects the network.
网络可靠性： 通信网络的可靠性通过识别最小割集来评估——即故障会导致网络断开的最小的链路集合。

Mixed-signal electronics: Modern integrated circuit design uses graph-theoretic methods such as cut-set analysis to optimize circuit topologies and verify design correctness.
混合信号电子： 现代集成电路设计使用图论方法（如割集分析）来优化电路拓扑并验证设计正确性。

12. Error Analysis and Numerical Considerations | 误差分析与数值考虑

When implementing the cut-set method numerically, several considerations ensure accuracy and computational stability:

在数值实现割集法时，有几个考虑因素可确保精度和计算稳定性：

Scaling: Using conductance values rather than resistance values significantly improves the conditioning of the system matrix, especially when branch values vary widely. For example, a circuit containing both 1 mΩ and 1 MΩ resistors would have conductances spanning 10⁹ in magnitude, and scaling helps prevent numerical overflow.

缩放： 使用电导值而非电阻值可以显著改善系统矩阵的条件数，尤其是当支路值变化范围较大时。例如，同时包含 1 mΩ 和 1 MΩ 电阻的电路，其电导值跨越 10⁹ 量级，缩放有助于防止数值溢出。

Ill-conditioning: When the node-to-reference voltages are very small compared to branch voltages, the matrix Q·G·Qᵗ can become ill-conditioned. In such cases, using a different tree selection or employing double-precision arithmetic is recommended.

病态条件： 当节点到参考电压相对于支路电压非常小时，矩阵 Q·G·Qᵗ 可能变得病态。在这种情况下，建议选择不同的树或采用双精度运算。

Sparse matrix techniques: For circuits with many nodes, the cut-set matrix Q is typically sparse. Advanced algorithms exploit this sparsity to reduce computation time from O(n³) to approximately O(n¹·⁵).

稀疏矩阵技术： 对于具有大量节点的电路，割集矩阵 Q 通常是稀疏的。高级算法利用这种稀疏性将计算时间从 O(n³) 减少到大约 O(n¹·⁵)。

Sensitivity analysis: The cut-set formulation allows straightforward computation of circuit sensitivities. If we denote the branch conductance vector as g, the derivative of any output voltage v_out with respect to a particular conductance gₖ can be computed directly from the inverse of the Q·G·Qᵗ matrix.

灵敏度分析： 割集公式允许直接计算电路灵敏度。如果我们把支路电导向量记为 g，则任何输出电压 v_out 对特定电导 gₖ 的导数可以直接从 Q·G·Qᵗ 矩阵的逆计算得出。

∂v_out/∂gₖ = -(vₖ)² · [row_out · (Q·G·Qᵗ)⁻¹ · column_k]

These numerical techniques extend the practical applicability of cut-set analysis to large, realistic circuits where manual solution methods would be impractical.

这些数值技术将割集法的实际应用扩展到大型真实电路，在这些电路中手工求解方法是不切实际的。

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Cut-Set Analysis in Electrical Circuits | 电路分析中的割集法