Name: transfer functions from signal-flow graphs
Uploaded: 2024-11-21T14:54:07.000Z
Duration: 1 min 20 s
Channel: JoVE

梅森法则通过信号流图简化了传递函数的计算，用于评估各种元件。
通过从一个节点出发并返回该节点的路径来计算环路增益，将各支路增益相乘。
前向路径增益涉及从输入节点到输出节点的路径，计算各增益的乘积。
不接触回路是指没有公共节点的回路。其增益为这些独立回路增益的乘积。
利用梅森法则从这些元件计算传递函数。
Delta 由环路增益及两个或更多不接触环路增益的交替和序列推导而来。Delta_k 通过从 Delta 中排除与之相交的环路增益而形成 kth 正向通路
为了计算系统的传递函数，需要确定前向通路增益，并评估环路增益。
然后，识别不接触的回路及其相应的增益。
计算 Delta，并通过去除相交环路增益来评估 Delta_k。
最后，将这些值代入梅森增益公式，以得到系统的传递函数。

梅森公式

梅森公式是控制系统和信号处理中的一种强大工具。它简化了从信号流图计算传递函数的过程。该方法利用各种元素（包括环路增益、前向路径增益和非接触环路）来高效确定传递函数。

环路增益是通过识别并追踪从节点回到自身的路径来确定的。这涉及计算环路沿线分支增益的乘积。每个环路的增益对于进一步的计算至关重要，并有助于整个系统的行为。

前向路径增益是通过追踪从输入节点到输出节点的路径来计算的。与环路增益一样，它涉及沿此路径的增益的乘积。前向路径表示输入对输出的直接影响，对于确定传递函数至关重要。

非接触环路是信号流图中不共享任何公共节点的环路。非接触环路的增益是各个环路增益的乘积。这些非接触环路非常重要，因为它们会影响梅森公式中使用的行列式 Delta (Δ) 的计算。

Delta (Δ) 是通过对环路增益和两个或多个非接触环路增益进行交替求和而得出的。从数学上讲，它可以表示为:

<img alt="Equation1" src="/files/ftp_upload/16108/16108_Equation_PT_001.svg" style="height: 50px;">

Δ_k 是 Δ 的修正版本，排除了与第 k 条前向路径相交的环路增益。这种排除对于准确确定系统的传递函数至关重要。

要使用梅森公式计算系统的传递函数，请遵循以下步骤:
<ol style="padding-left: 35px; margin: 0;">
 <li>确定从输入到输出的所有前向路径增益（T_k）。</li>
 <li>评估所有环路增益（L_i）并识别非接触环路。</li>
 <li>按照所述，通过对环路增益和非接触环路增益的乘积进行求和和相减来计算 Δ。</li>
 <li>对于每条前向路径，通过排除相交的环路增益来计算 Δ_k。</li>
 <li>将这些值代入梅森公式，得出:</li>
</ol>

<img alt="Equation2" src="/files/ftp_upload/16108/16108_Equation_PT_002.svg" style="height: 34px;">

通过这些步骤，梅森公式提供了一种有组织的、系统的方法来推导复杂系统的传递函数，使其成为控制理论和信号处理中不可或缺的一部分。

梅森法则通过信号流图简化传递函数的计算，用于评估各个元件。  
环路增益通过追踪路径计算……

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Diagrams and Signal Flow Graphs

图表与信号流图

KEY TERMS AND CONCEPTS

关键术语与概念

Mason&#39;s Rule

transfer functions from signal-flow graphs

Mason's rule helps students find transfer functions from signal-flow graphs in control systems and signal processing. It gives a clear path for solving graphs that can look complicated at first. The method uses loop gains, forward-path gains, and non-touching loops to organize the calculation.

A loop gain comes from tracing a path from a node back to the same node. The gain is the product of the branch gains along that loop. Each loop gain matters because it feeds into the later transfer-function steps and affects the behavior of the system.

A forward-path gain is found by tracing a path from the input node to the output node. It is also the product of the gains along that route. Forward paths show the direct effect of the input on the output, so they are essential for the final result.

Non-touching loops are loops that do not share any common nodes. Their combined gain is the product of the individual loop gains. These loop groups matter because they are used when building the determinant, Delta (&amp;#916;), in Mason's rule.

Delta is formed from an alternating series of sums and subtractions. It uses loop gains and the gains of non-touching loops taken two or more at a time. For each forward path, a modified value, Delta k, is also found. Delta k leaves out any loop gains that intersect with the kth forward path, which is needed for an accurate transfer function.

To use Mason's rule, first identify all forward-path gains, Tk, from input to output. Then evaluate all loop gains, Li, and find any non-touching loops. Next, calculate Delta, find each Delta k by removing intersecting loop gains, and substitute the values into Mason's rule. This gives a systematic way to determine the transfer function of complex systems in control theory and signal processing.

Transfer Functions from Signal-Flow Graphs

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mason&#39;s rule

Mason's rule helps students find transfer functions from signal-flow graphs in control systems and signal processing. It gives a clear path for solving graphs that can look complicated at first. The method uses loop gains, forward-path gains, and non-touching loops to organize the calculation. A loop gain comes from tracing a path from a node back to the same node. The gain is the product of the branch gains along that loop. Each loop gain matters because it feeds into the later...

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key terms and concepts

Learn how Mason's rule uses forward paths, loop gains, and non-touching loops to calculate transfer functions from signal-flow graphs in control systems.

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electrical engineering

Transfer Functions from Signal-Flow Graphs

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