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

A regra de Mason simplifica o cálculo da função de transferência a partir de gráficos de fluxo de sinal, avaliando vários elementos.
O ganho de loop é calculado traçando um caminho de um nó de volta para si mesmo, pegando o produto dos ganhos de ramificação.
O ganho de caminho direto envolve seguir um caminho do nó de entrada para o nó de saída, calculando o produto dos ganhos.
Loops sem toque são loops sem nós comuns. O ganho é o produto dos ganhos de loop desses loops separados.
A função de transferência é calculada a partir desses elementos usando a regra de Mason.
Delta é derivado de séries alternadas de somas de ganhos de loop e ganhos de loop sem toque tomados dois ou mais de cada vez. Delta_k é formado pela exclusão de ganhos de loop do Delta que se cruzam com o késimo caminho para frente.
Para calcular a função de transferência de um sistema, os ganhos do caminho de avanço são identificados e os ganhos do loop são avaliados.
Em seguida, os loops sem toque e seus ganhos correspondentes são identificados.
O delta é calculado e Delta_k é avaliado removendo os ganhos de loop de interseção.
Finalmente, esses valores são substituídos na regra de Mason para produzir a função de transferência do sistema.

regra de mason

A regra de Mason é uma ferramenta poderosa em sistemas de controle e processamento de sinais. Ela simplifica o cálculo de funções de transferência a partir de gráficos de fluxo de sinal. Este método alavanca vários elementos, incluindo ganhos de loop, ganhos de caminho direto e loops não-tocantes, para determinar a função de transferência eficientemente.


 O ganho de loop é determinado pela identificação e rastreamento de um caminho de um nó de volta para si. Isso envolve calcular o produto de ganhos de ramificação ao longo do loop. O ganho de cada loop é crucial para cálculos posteriores e contribui para o comportamento geral do sistema.


 O ganho do caminho direto é calculado rastreando um caminho do nó de entrada para o nó de saída. Como o ganho de loop, ele envolve o produto dos ganhos ao longo deste caminho. Os caminhos diretos representam a influência direta da entrada na saída e são essenciais para determinar a função de transferência.


 Os loops não-tocantes são loops no gráfico de fluxo de sinal que não compartilham nenhum nó comum. O ganho de loops não-tocantes é o produto dos ganhos de loop individuais. Esses loops não-tocantes são significativos por afetarem o cálculo do determinante, delta (Δ), usado na regra de Mason.


 Delta (Δ) é derivado de uma série alternada de somas envolvendo ganhos de loop e os ganhos de loops não-tocantes tomados dois ou mais de cada vez. Matematicamente, pode ser expresso como:


 <img alt="Equation1" src="/files/ftp_upload/16108/16108_Equation_PT_001.svg" style="height: 50px;">


 Δk é uma versão modificada de Δ, excluindo ganhos de loop que cruzam com o k-ésimo caminho direto. Essa exclusão é crucial para determinar com precisão a função de transferência do sistema.


 Para calcular a função de transferência de um sistema usando a regra de Mason, as seguintes etapas são usadas:

<ol style="padding-left: 35px; margin: 0;">
 <li>
 Identifique todos os ganhos do caminho direto (T_k) da entrada para a saída.
 </li>
 <li>
 Avalie todos os ganhos de loop (L_i) e identifique loops não-tocantes.
 </li>
 <li>
 Calcule Δ somando e subtraindo os produtos dos ganhos de loop e ganhos de loop não tocantes, conforme descrito.
 </li>
 <li>
 Para cada caminho direto, calcule Δ_k excluindo os ganhos de loop de intersecção.
 </li>
 <li>
 Substitua esses valores na regra de Mason, dada por:
 </li>
</ol>

 <img alt="Equation2" src="/files/ftp_upload/16108/16108_Equation_PT_002.svg" style="height: 34px;">


 Por meio dessas etapas, a regra de Mason fornece uma abordagem organizada e sistemática para derivar a função de transferência de sistemas complexos, tornando-a indispensável na teoria de controle e no processamento de sinais.

Regra de Mason

A regra de Mason simplifica o cálculo da função de transferência a partir de gráficos de fluxo de sinal, avaliando vários elementos.
O ganho do loop é calculado traçando um caminho ...

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

Unidade 1

Diagrams and Signal Flow Graphs

Diagramas e Grafos de Fluxo de Sinais

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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'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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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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