IB Mathematics: Drawing Signal Flow Graphs and Applications in System Modeling | IB数学：信号流图的绘制与系统建模应用

📚 IB Mathematics: Drawing Signal Flow Graphs and Applications in System Modeling | IB数学：信号流图的绘制与系统建模应用

Signal flow graphs (SFGs) are a powerful mathematical tool for representing the relationships between variables in a system. In IB Mathematics, especially in applications such as mathematical modelling, economics, and engineering, SFGs help us visualise how signals propagate through interconnected components and simplify the analysis of complex systems.

信号流图（SFG）是一种强大的数学工具，用于表示系统中变量之间的关系。在IB数学中，特别是在数学建模、经济学和工程应用等领域，信号流图帮助我们直观地看到信号如何在相互连接的组件中传播，并简化复杂系统的分析。

1. What Is a Signal Flow Graph? | 什么是信号流图？

A signal flow graph is a network of directed branches that connect nodes. Each node represents a system variable, and each branch has a gain that multiplies the signal passing along it. The direction of the arrow indicates the direction of signal flow.

信号流图是一种由有向支路连接节点组成的网络。每个节点代表一个系统变量，每条支路具有一个增益，对沿其传递的信号进行乘法运算。箭头的方向表示信号流动的方向。

Mathematically, if node \(X_j\) depends on \(X_i\), we draw an arrow from \(X_i\) to \(X_j\) with gain \(a_{ij}\), so that \(X_j = a_{ij} X_i\). This compact representation allows us to model linear systems as graphs and apply graph-theoretic techniques to solve them.

在数学上，如果节点 \(X_j\) 依赖于 \(X_i\)，我们就从 \(X_i\) 向 \(X_j\) 画一条增益为 \(a_{ij}\) 的箭头，使得 \(X_j = a_{ij} X_i\)。这种紧凑的表示方法让我们能够将线性系统建模为图，并运用图论技术进行求解。

2. Basic Elements: Nodes and Branches | 基本要素：节点和支路

There are two fundamental elements in an SFG: nodes and branches. A node represents a variable such as a voltage, current, position, or price. A branch is a directed connection between two nodes, carrying a gain coefficient.

信号流图中有两个基本元素：节点和支路。节点代表一个变量，例如电压、电流、位置或价格。支路是两个节点之间的有向连接，带有增益系数。

Source node: a node with only outgoing branches. It represents an independent input variable.

源节点：只有出射支路的节点，代表独立输入变量。
Sink node: a node with only incoming branches. It represents an output variable.

汇节点：只有入射支路的节点，代表输出变量。
Path: a continuous sequence of branches traversed in the arrow direction, without passing any node more than once.

路径：沿箭头方向连续经过的支路序列，且不重复经过任何节点。
Loop: a closed path that starts and ends at the same node, with no other node repeated.

回路：起点和终点为同一节点且不重复经过其他节点的闭合路径。

3. Rules for Drawing a Signal Flow Graph from Equations | 从方程组绘制信号流图的规则

When a system is described by linear equations, we can draw its SFG systematically. First, identify each variable as a node. Second, for each equation, express the dependent variable as a sum of other variables multiplied by coefficients. Third, draw a branch for each term.

当系统由线性方程描述时，我们可以系统地绘制其信号流图。首先，将每个变量识别为一个节点。其次，对每个方程，将因变量表示为其他变量乘以系数后的和。第三，为每一项绘制一条支路。

For example, consider the equation \(x_2 = a x_1 + b x_3\). We place nodes \(x_1\), \(x_2\), \(x_3\). Then we draw a branch from \(x_1\) to \(x_2\) with gain \(a\), and a branch from \(x_3\) to \(x_2\) with gain \(b\).

例如，考虑方程 \(x_2 = a x_1 + b x_3\)。我们放置节点 \(x_1\)、\(x_2\)、\(x_3\)，然后从 \(x_1\) 到 \(x_2\) 画一条增益为 \(a\) 的支路，从 \(x_3\) 到 \(x_2\) 画一条增益为 \(b\) 的支路。

x₂ = a x₁ + b x₃ → nodes: x₁, x₂, x₃; branches: x₁→x₂ (a), x₃→x₂ (b)

4. Constructing an SFG from a Block Diagram | 从框图构建信号流图

Block diagrams are common in control theory and systems modelling. Each block represents a transfer function or operation, and summing junctions combine signals. To convert a block diagram to an SFG, we replace each block by a branch with the same gain, and each summing junction by a node.

框图在控制理论和系统建模中很常见。每个方框代表一个传递函数或运算，求和点用于合并信号。要将框图转换为信号流图，我们只需用具有相同增益的支路替换每个方框，并用节点替换每个求和点。

Consider a simple feedback loop: the forward path has gain \(G\), the feedback path has gain \(H\). The summing junction subtracts the feedback from the input \(R\) to produce the error \(E\). In the SFG, we draw \(R \rightarrow E\) with gain 1, \(E \rightarrow C\) with gain \(G\), and \(C \rightarrow E\) with gain \(-H\).

考虑一个简单的反馈回路：前向路径增益为 \(G\)，反馈路径增益为 \(H\)。求和点从输入 \(R\) 中减去反馈以产生误差 \(E\)。在信号流图中，我们绘制 \(R \rightarrow E\) 增益为 1，\(E \rightarrow C\) 增益为 \(G\)，以及 \(C \rightarrow E\) 增益为 \(-H\)。

R → E (1) → C (G); C → E (−H)

5. Example: Electrical Circuit Modelling | 示例：电路建模

Operational amplifier circuits and RLC networks can be modelled using SFGs. For instance, an integrator circuit with input voltage \(V_{in}\) and output voltage \(V_{out}\) has the relationship \(V_{out} = – \frac{1}{RC} \int V_{in} \, dt\). In Laplace domain, this becomes \(V_{out}(s) = – \frac{1}{RCs} V_{in}(s)\).

运算放大器电路和RLC网络可以使用信号流图建模。例如，输入电压 \(V_{in}\)、输出电压 \(V_{out}\) 的积分器电路满足关系 \(V_{out} = – \frac{1}{RC} \int V_{in} \, dt\)。在拉普拉斯域中，这变为 \(V_{out}(s) = – \frac{1}{RCs} V_{in}(s)\)。

We draw a single branch from \(V_{in}\) to \(V_{out}\) with gain \(-1/(RCs)\). For a more complex circuit, multiple state variables such as capacitor voltages and inductor currents become nodes, and their interconnections form branches according to Kirchhoff’s laws.

我们绘制一条从 \(V_{in}\) 到 \(V_{out}\) 的支路，增益为 \(-1/(RCs)\)。对于更复杂的电路，电容器电压和电感电流等多个状态变量成为节点，并根据基尔霍夫定律形成支路连接。

6. Example: Mechanical System Modelling | 示例：机械系统建模

Mechanical systems with masses, springs, and dampers are often modelled by second-order differential equations. For a mass-spring-damper system, the equation is \(m \ddot{x} + c \dot{x} + kx = F(t)\). By taking the Laplace transform with zero initial conditions, we get \((ms^2 + cs + k) X(s) = F(s)\).

包含质量、弹簧和阻尼器的机械系统通常用二阶微分方程建模。对于质量-弹簧-阻尼系统，方程为 \(m \ddot{x} + c \dot{x} + kx = F(t)\)。在零初始条件下取拉普拉斯变换，得到 \((ms^2 + cs + k) X(s) = F(s)\)。

This can be represented as an SFG with nodes \(F(s)\), intermediate variable \(A(s)\) (acceleration), and \(X(s)\). The branches reflect the algebraic relationships: \(A = F/m – (c/m)sX – (k/m)X\), and \(X = A/s^2\).

这可以表示为具有节点 \(F(s)\)、中间变量 \(A(s)\)（加速度）和 \(X(s)\) 的信号流图。支路反映代数关系：\(A = F/m – (c/m)sX – (k/m)X\)，以及 \(X = A/s^2\)。

7. Mason’s Gain Formula | 梅森增益公式

Mason’s gain formula gives the overall transfer function from an input node to an output node directly from the SFG without simplification. It is especially valuable when the graph contains many loops.

梅森增益公式直接从信号流图给出从输入节点到输出节点的总传递函数，无需逐步化简。当图中包含许多回路时，它尤其有价值。

T = (Σₖ Pₖ Δₖ) / Δ

Here, \(P_k\) is the gain of the \(k\)-th forward path, \(\Delta\) is the determinant of the graph, and \(\Delta_k\) is the cofactor of path \(k\). The determinant is computed as:

这里，\(P_k\) 是第 \(k\) 条前向路径的增益，\(\Delta\) 是图的特征行列式，\(\Delta_k\) 是第 \(k\) 条路径的余因子。特征行列式计算如下：

Δ = 1 − ΣL₁ + ΣL₂ − ΣL₃ + …

ΣL₁ is the sum of all individual loop gains.

ΣL₁ 是所有独立回路增益之和。
ΣL₂ is the sum of products of gains of all pairs of non-touching loops.

ΣL₂ 是所有互不接触回路对的增益乘积之和。
ΣL₃ is the sum of products of gains of all triples of non-touching loops, and so on.

ΣL₃ 是所有互不接触回路三元组的增益乘积之和，依此类推。

8. Applying Mason’s Rule to a Feedback System | 应用梅森法则于反馈系统

Let us apply Mason’s rule to a classic feedback system with forward gain \(G\) and feedback gain \(H\). There is one forward path from \(R\) to \(C\): gain \(P_1 = G\). There is one loop: gain \(L_1 = -GH\).

让我们将梅森法则应用于经典反馈系统，其前向增益为 \(G\)，反馈增益为 \(H\)。从 \(R\) 到 \(C\) 有一条前向路径：增益 \(P_1 = G\)。有一个回路：增益 \(L_1 = -GH\)。

The graph determinant is \(\Delta = 1 – L_1 = 1 + GH\). The forward path touches the loop, so its cofactor is \(\Delta_1 = 1\). Therefore, the transfer function is:

图的特征行列式为 \(\Delta = 1 – L_1 = 1 + GH\)。前向路径与该回路接触，因此其余因子为 \(\Delta_1 = 1\)。因此，传递函数为：

C/R = P₁ Δ₁ / Δ = G / (1 + GH)

This matches the familiar closed-loop transfer function of a negative feedback system.

这与负反馈系统的经典闭环传递函数一致。

9. SFG Simplification Techniques | 信号流图化简技巧

Although Mason’s formula is powerful, sometimes we need to simplify an SFG step by step, especially for interpreting intermediate variables. Several elementary operations preserve the overall input-output relationship.

尽管梅森公式非常强大，但有时我们需要逐步化简信号流图，特别是为了解释中间变量。有几种基本操作可以保持整体输入输出关系不变。

Series branches: two branches with gains \(a\) and \(b\) in sequence can be replaced by one branch with gain \(ab\).

串联支路：增益为 \(a\) 和 \(b\) 的两条支路依次连接，可以用一条增益为 \(ab\) 的支路替代。
Parallel branches: two branches from the same node to the same node with gains \(a\) and \(b\) combine into one branch with gain \(a+b\).

并联支路：从同一节点到同一节点的两条增益分别为 \(a\) 和 \(b\) 的支路，可合并为一条增益为 \(a+b\) 的支路。
Node elimination: an intermediate node with one incoming and one outgoing branch can be removed, multiplying the two gains.

节点消除：具有一条入支路和一条出支路的中间节点可以移除，并将两条增益相乘。

These techniques are analogous to reducing block diagrams, but they must be applied carefully to avoid creating fictitious loops.

这些技巧类似于简化框图，但必须小心应用，以避免产生虚假回路。

10. SFG in Economic and Statistical Modelling | 信号流图在经济与统计建模中的应用

Beyond engineering, SFGs appear in recursive economic models and path analysis in statistics. In a simple Keynesian income model, consumption \(C\) depends on income \(Y\), and investment \(I\) is exogenous: \(Y = C + I\), \(C = bY\) where \(b\) is the marginal propensity to consume.

除工程外，信号流图还出现在递归经济模型和统计学中的路径分析中。在一个简单的凯恩斯收入模型中，消费 \(C\) 依赖于收入 \(Y\)，投资 \(I\) 是外生的：\(Y = C + I\)，\(C = bY\)，其中 \(b\) 是边际消费倾向。

The SFG has nodes \(I\), \(Y\), \(C\); branches \(I \to Y\) with gain 1, \(C \to Y\) with gain 1, and \(Y \to C\) with gain \(b\). Applying Mason’s rule gives the income multiplier \(1/(1-b)\).

该信号流图有节点 \(I\)、\(Y\)、\(C\)；支路 \(I \to Y\) 增益为 1，\(C \to Y\) 增益为 1，\(Y \to C\) 增益为 \(b\)。应用梅森法则得到收入乘数 \(1/(1-b)\)。

11. Advantages and Limitations | 优点与局限性

The main advantages of SFGs are clarity, compactness, and the direct application of Mason’s formula. They also highlight causal relationships between variables in a way that algebraic equations do not always reveal.

信号流图的主要优点是清晰、紧凑以及可直接应用梅森公式。它还以代数方程不一定能揭示的方式突出变量之间的因果关系。

However, SFGs are limited to linear time-invariant systems. Non-linearities and time-varying coefficients cannot be represented directly without approximation or linearisation. In addition, large graphs can become visually complex, although the underlying computation remains systematic.

然而，信号流图仅适用于线性时不变系统。非线性和时变系数无法直接表示，除非进行近似或线性化。此外，大型图可能变得视觉上复杂，尽管其底层计算仍然是系统化的。

12. Conclusion and Exam Tips | 结论与考试技巧

Signal flow graphs are a versatile modelling tool in IB Mathematics. Mastering the rules for drawing, simplifying, and applying Mason’s gain formula enables you to analyse complex feedback systems efficiently.

信号流图是IB数学中一种通用的建模工具。掌握绘制、化简和应用梅森增益公式的规则，使您能够高效地分析复杂的反馈系统。

For exams, always label nodes clearly, check branch directions, and identify all loops before applying Mason’s formula. Remember that non-touching loops are those that share no nodes. Practise with block diagrams and algebraic equations to build fluency.

在考试中，务必清晰标记节点，检查支路方向，并在应用梅森公式之前找出所有回路。记住互不接触回路是指没有任何共同节点的回路。通过框图和代数方程进行练习，以培养熟练度。

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Mathematics: Drawing Signal Flow Graphs and Applications in System Modeling | IB数学：信号流图的绘制与系统建模应用