📚 6 Electrical Circuits: The Cutset Method | 电路分析：割集方法
Graph theory provides a powerful systematic approach to analysing electrical circuits, and the cutset method is one of its most elegant applications. In this article we develop the cutset technique from first principles: we convert a circuit into a directed graph, identify trees, define fundamental cutsets, and use the cutset matrix to express Kirchhoff’s current law. A step-by-step worked example shows how the method yields a complete solution for branch currents and voltages. This topic bridges discrete mathematics and physics, and appears in the IB Mathematics: Analysis and Approaches and Applications and Interpretation syllabuses under network and graph theory.
图论为分析电路提供了一种强大的系统化方法，而割集方法正是其中最优雅的应用之一。本文将从基本原理出发，详细讲解割集技术：将电路转化为有向图、确定树、定义基本割集，并利用割集矩阵表达基尔霍夫电流定律。通过逐步求解的示例，展示如何用该方法求得支路电流和电压的完整解。这一主题连接了离散数学和物理学，是 IB 数学：分析与方法以及应用与解释课程中网络与图论部分的重要内容。
1. Graph Representation of a Circuit | 电路的图表示
To apply the cutset method we first replace an electrical circuit by a directed graph, or digraph. Every two-terminal element (resistor, voltage source, current source) becomes a branch, and every junction becomes a node. A direction is assigned arbitrarily to each branch; this direction is used as the reference for current and will determine the signs in the cutset matrix.
要应用割集方法，我们首先用一个有向图（或称有向图）来代替电路。每一个二端元件（电阻、电压源、电流源）变成一条支路，每一个连接点变成一个节点。我们为每条支路任意指定一个方向；这个方向将作为电流的参考方向，并决定割集矩阵中各项的符号。
For instance, a simple circuit with two resistors, one voltage source and one current source might be drawn on paper, and its graph extracts the topology by ignoring the nature of the components while retaining the way they are connected. The graph then contains b branches and n nodes, and we are ready to apply topological analysis.
例如，一个由两个电阻、一个电压源和一个电流源组成的简单电路，可以画出对应的图；该图忽略了元件的物理性质，只保留它们的连接方式。这样得到的图含有 b 条支路和 n 个节点，我们就可以进行拓扑分析了。
2. Trees, Cotrees, and Branch Classification | 树、补树及支路分类
In a connected graph, a tree is a connected subgraph that contains all n nodes but has exactly n − 1 branches and no closed loops. The branches belonging to the tree are called twigs or tree branches. The remaining b − (n − 1) branches form the cotree, and each of these is called a link or chord.
在一个连通图中，树是一个包含所有 n 个节点、恰好有 n−1 条支路且没有闭合回路的连通子图。属于树的支路称为树支（twigs）；剩下的 b−(n−1) 条支路构成补树，每一条称为连支（link 或 chord）。
The choice of a tree is not unique, but once it is selected the entire analysis rests on it. Every link, when added to the tree, creates a unique loop; and every tree branch, together with some links, defines a unique cutset – the key concept we will now explore.
树的选取并不唯一，但一旦选定，整个分析都将建立在该树之上。每一条连支加入到树中就会形成一个唯一的回路；而每一条树支，与某些连支一起，则定义了一个唯一的割集——这正是我们接下来要探讨的核心概念。
3. What is a Cutset? | 什么是割集？
A cutset of a connected graph is a minimal set of branches whose removal divides the graph into two separate connected subgraphs. “Minimal” means that deleting all branches in the set disconnects the graph, but leaving any one of them untouched would keep the graph connected.
连通图的一个割集是最小的支路集合，移去这些支路将把图分成两个分离的连通子图。“最小”意味着删除该集合中所有支路会导致图不连通，但只要留下其中任意一条，图仍然是连通的。
In circuit terms, a cutset isolates a part of the network, and the algebraic sum of currents crossing the cutset must be zero by Kirchhoff’s current law (KCL). This observation is the foundation of the cutset method.
从电路的角度看，割集将网络的一部分隔离开来；根据基尔霍夫电流定律（KCL），穿过割集的电流代数和必须为零。这一观察正是割集方法的基础。
4. Fundamental Cutsets Determined by a Tree | 由树确定的基本割集
For a given tree, each tree branch forms a fundamental cutset together with some of the links. The fundamental cutset associated with a tree branch consists of that tree branch and exactly those links which, if cut together with the tree branch, separate the graph into two parts. Crucially, each fundamental cutset contains precisely one tree branch.
对于给定的一棵树，每条树支与一些连支一起构成一个基本割集。与某条树支关联的基本割集由该树支以及那些连支组成，这些连支一旦与该树支一起被切割，就会把图分成两部分。关键在于，每个基本割集恰好包含一条树支。
Because there are n − 1 tree branches, there are exactly n − 1 fundamental cutsets, and they are linearly independent. This independence guarantees that the KCL equations written for the fundamental cutsets are algebraically independent and sufficient to solve the network.
因为有 n−1 条树支，所以恰好有 n−1 个基本割集，并且它们是线性无关的。这种无关性保证了为基本割集写出的 KCL 方程是代数独立的，并且足以求解整个网络。
5. Orientation and the Cutset Matrix | 割集的方向与割集矩阵
We assign each cutset the same orientation as its defining tree branch. When a branch belongs to the cutset and its direction agrees with the cutset orientation, the corresponding entry in the cutset matrix is +1; if it opposes, the entry is −1; if the branch is not part of the cutset, the entry is 0.
我们赋予每个割集与其定义树支相同的方向。当一条支路属于该割集且其方向与割集方向一致时，割集矩阵中相应的元素为 +1；若方向相反，元素为 −1；若该支路不属于该割集，元素为 0。
The information for all fundamental cutsets is collected in the fundamental cutset matrix Qf. Its rows correspond to the cutsets (tree branches) and its columns to all branches of the graph. By ordering the columns so that tree branches appear first, Qf takes the block form: Qf = [U | Qℓ], where U is an identity matrix of size (n − 1) × (n − 1).
所有基本割集的信息被收集在基本割集矩阵 Qf 中。矩阵的行对应各个割集（即各条树支），列对应图的所有支路。若将列按先树支后连支的顺序排列，Qf 具有分块形式：Qf = [U | Qℓ]，其中 U 是一个 (n−1)×(n−1) 的单位矩阵。
6. Kirchhoff’s Current Law in Cutset Form | KCL 的割集形式
Let the column vector Ib contain all branch currents arranged in the same order as the columns of Qf. Kirchhoff’s current law then becomes the single matrix equation:
Qf · Ib = 0
This compact equation represents n − 1 independent scalar equations – one for each fundamental cutset – expressing that the net current crossing the cutset is zero.
设列向量 Ib 包含所有支路电流，且排列顺序与 Qf 的列相同。基尔霍夫电流定律即可写成单一的矩阵方程：
Qf · Ib = 0
这个紧凑的方程代表了 n−1 个独立的标量方程——每个基本割集一个——表达了穿越该割集的净电流为零。
7. Combining with Branch Constitutive Relations | 结合支路特性关系
KCL alone gives only n − 1 equations for b unknowns (branch currents). To obtain a complete system we also need b relationships between branch voltages and currents, known as the branch constitutive equations. For resistors, Ohm’s law gives Vk = Rk Ik; for voltage sources, Vk is specified and the current is an unknown; for current sources, Ik is known and the voltage is an unknown.
仅靠 KCL 只能为 b 个未知数（支路电流）提供 n−1 个方程。为了得到完整的系统，我们还需要支路电压与电流之间的 b 个关系，这些关系称为支路特性方程。对于电阻，欧姆定律给出 Vk = Rk Ik；对于电压源，Vk 是给定的，电流是未知数；对于电流源，Ik 是已知的，电压是未知数。
Furthermore, because the cutset matrix also expresses KVL indirectly through the link between branch voltages and tree-branch voltages, the full set of equations can be assembled. In practice, the cutset method often works directly with KCL and the constitutive equations, solving simultaneously for all branch currents and voltages without explicitly writing loop equations.
此外，由于割集矩阵也通过支路电压与树支电压之间的联系间接表达了 KVL，完整的方程组就可以组合起来。在实际应用中，割集方法常常直接使用 KCL 和特性方程，同时求解所有的支路电流和电压，而无需显式地写出回路方程。
8. Worked Example – Part 1: The Circuit and Its Graph | 示例第一部分：电路及其图
Consider a DC circuit with four branches: a 10 V voltage source (branch 1, polarity + to −), and three resistors: R2 = 5 Ω, R3 = 10 Ω, R4 = 10 Ω, connected to form a simple network. The graph has n = 3 nodes and b = 4 branches. We direct branch 1 away from the positive terminal, and assign directions to branches 2, 3, 4 arbitrarily. A convenient tree consists of branches 1, 2, and 3; then branch 4 is the sole link.
考虑一个含有四条支路的直流电路：一个 10 V 的电压源（支路 1，极性 + 到 −），以及三个电阻：R2 = 5 Ω，R3 = 10 Ω，R4 = 10 Ω，它们构成一个简单网络。对应的图有 n = 3 个节点和 b = 4 条支路。我们令支路 1 的方向从正极指向外，并任意规定支路 2、3、4 的方向。一个方便的树由支路 1、2 和 3 组成；于是支路 4 是唯一的连支。
9. Worked Example – Part 2: Fundamental Cutsets and Qf | 示例第二部分：基本割集与 Qf
The tree has tree branches 1, 2, 3. Inspection shows:
- Cutset of tree branch 1: includes tree branch 1 and link 4 (since cutting 1 and 4 separates a part of the network).
- Cutset of tree branch 2: includes tree branch 2 and link 4.
- Cutset of tree branch 3: includes tree branch 3 and link 4.
Assigning each cutset the orientation of its tree branch, and ordering columns as (tree 1,2,3 ; link 4), the fundamental cutset matrix is:
该树有树支 1、2、3。观察可知：
- 树支 1 的割集：包含树支 1 和连支 4（因为切割 1 和 4 会把网络的一部分分离出来）。
- 树支 2 的割集：包含树支 2 和连支 4。
- 树支 3 的割集：包含树支 3 和连支 4。
赋予每个割集与其树支相同的方向，并将列按（树支 1,2,3 ; 连支 4）的顺序排列，得到基本割集矩阵为：
| 1 | 0 | 0 | −1 |
| 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 |
The sign of the link 4 entry depends on whether the branch direction aligns with the cutset direction. In this example we have set branch 4 direction opposite to cutset 1, giving −1, and aligned with cutsets 2 and 3, giving +1.
连支 4 对应元素的符号取决于支路方向是否与割集方向一致。在此例中我们设定支路 4 的方向与割集 1 相反，故为 −1；与割集 2 和 3 相同，故为 +1。
10. Worked Example – Part 3: KCL Equations and Solution | 示例第三部分：KCL 方程与求解
Write the branch current vector as Ib = [I1, I2, I3, I4]ᵀ. KCL in cutset form gives:
Qf · Ib = 0 ⇒
I1 − I4 = 0
I2 + I4 = 0
I3 + I4 = 0
将支路电流向量写为 Ib = [I1, I2, I3, I4]ᵀ。割集形式的 KCL 给出：
I1 − I4 = 0
I2 + I4 = 0
I3 + I4 = 0
Now add the branch equations. Branch 1 is a voltage source: V1 = 10 V, I1 unknown. Ohm’s law for the resistors: V2 = 5 I2, V3 = 10 I3, V4 = 10 I4. We also need KVL around a loop containing the source and resistors, but with the cutset method we can use the fact that the cutset matrix also relates branch voltages to tree-branch voltages: the relation Qf · Vb = 0 gives a set of loops indirectly. In practice, one can write a loop equation using the known source: going through the source and resistors, say V1 − V2 − V3 = 0 (depending on actual circuit). For brevity, assuming the loop is formed by branches 1,2,3, we obtain 10 − 5 I2 − 10 I3 = 0. Combined with the KCL expressions I2 = −I4, I3 = −I4, and I1 = I4, we solve: 10 − 5(−I4) − 10(−I4) = 10 + 15 I4 = 0 ⇒ I4 = −2/3 A. Then I1 = −2/3 A, I2 = 2/3 A, I3 = 2/3 A. Voltages follow from Ohm’s law.
现在加入支路方程。支路 1 为一个电压源：V1 = 10 V，I1 未知。电阻的欧姆定律：V2 = 5 I2，V3 = 10 I3，V4 = 10 I4。我们还需要围绕含源和电阻的回路使用 KVL；但在割集方法中，可利用割集矩阵将支路电压与树支电压关联起来：关系 Qf · Vb = 0 间接提供了一组回路。实践中，可以写出一个包含已知源的回路方程：例如经过源和电阻的回路，有 V1 − V2 − V3 = 0（取决于实际电路）。为简洁起见，假设回路由支路 1、2、3 构成，得到 10 − 5 I2 − 10 I3 = 0。结合 KCL 表达式 I2 = −I4，I3 = −I4 以及 I1 = I4，解得：10 − 5(−I4) − 10(−I4) = 10 + 15 I4 = 0 ⇒ I4 = −2/3 A。于是 I1 = −2/3 A，I2 = 2/3 A，I3 = 2/3 A。电压可由欧姆定律求得。
11. Comparison with Loop/Mesh Analysis and Summary | 与回路/网孔分析法的比较与总结
The cutset method is the dual of loop (mesh) analysis. While loop analysis uses KVL around independent loops to write equations for loop currents, the cutset method employs KCL on independent cutsets to write equations for tree-branch voltages or branch currents. In planar networks, mesh analysis is often more intuitive, but for non-planar networks or when given current sources abound, the cutset approach can be more efficient and systematic.
割集方法与回路（网孔）分析法互为对偶。回路分析利用独立回路的 KVL 写出关于回路电流的方程，而割集方法则在独立的割集上运用 KCL，写出关于树支电压或支路电流的方程。在平面网络中，网孔分析通常更直观，但对于非平面网络或当存在大量电流源时，割集方法可能更为高效和系统化。
In summary, the cutset method translates a circuit’s topology into an algebraic form that clearly separates KCL from the branch equations. Mastering it deepens understanding of linear algebra applications and the structure of networks. For IB Mathematics students, it serves as a perfect illustration of how abstract graph theory connects to real-world engineering problems.
总而言之，割集方法将电路的拓扑结构转化为代数形式，清晰地将 KCL 与支路方程分离开来。掌握该方法能加深对线性代数应用以及网络结构的理解。对于 IB 数学的学生而言，它完美地展示了抽象图论如何与现实中的工程问题相联系。
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