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

Mason&#39;s rule simplifies transfer function calculation from signal-flow graphs, assessing various elements.
Loop gain is calculated by tracing a path from a node back to itself, taking the product of branch gains.
Forward-path gain involves following a path from the input to the output node, computing the product of gains.
Non-touching loops are loops with no common nodes. The gain is the product of loop gains from these separate loops.
The transfer function is computed from these elements using Mason&#39;s rule.
Delta is derived from alternating series of sums of loop gains and nontouching-loop gains taken two or more at a time. Delta_k is formed by excluding loop gains from Delta that intersect with the kth forward path.
To calculate the transfer function of a system, the forward-path gains are identified, and the loop gains are evaluated.
Then, the non-touching loops and their corresponding gains are identified.
Delta is calculated, and Delta_k is evaluated by removing intersecting loop gains.
Finally, these values are substituted into Mason&#39;s rule to yield the system&#39;s transfer function.

de regel van mason

De regel van Mason is een krachtig hulpmiddel in controlesystemen en signaalverwerking. Het vereenvoudigt de berekening van overdrachtsfuncties uit signaalstroomgrafieken. Deze methode maakt gebruik van verschillende elementen, waaronder lusversterkingen, voorwaartse padversterkingen en niet-rakende lussen, om de overdrachtsfunctie efficiënt te bepalen.

Lusversterking wordt bepaald door het identificeren en traceren van een pad van een knooppunt terug naar zichzelf. Dit omvat het berekenen van het product van takversterkingen langs de lus. De versterking van elke lus is cruciaal voor verdere berekeningen en draagt ​​bij aan het algehele systeemgedrag.

De voorwaartse padversterking wordt berekend door een pad te traceren van de invoerknooppunt naar de uitvoerknooppunt. Net als lusversterking, omvat het het product van de versterkingen langs dit pad. Voorwaartse paden representeren de directe invloed van de invoer op de uitvoer en zijn essentieel voor het bepalen van de overdrachtsfunctie.

Niet-rakende lussen zijn lussen in de signaalstroomgrafiek die geen gemeenschappelijke knooppunten delen. De versterking van niet-rakende lussen is het product van de individuele lusversterkingen. Deze niet-rakende lussen zijn significant omdat ze de berekening van de determinant, Delta (Δ), beïnvloeden die wordt gebruikt in de regel van Mason.

Delta (Δ) wordt afgeleid van een afwisselende reeks sommen met betrekking tot lusversterkingen en de versterkingen van niet-rakende lussen die twee of meer tegelijk worden genomen. Wiskundig kan het worden uitgedrukt als:

<img alt="Equation1" src="/files/ftp_upload/16108/16108_Equation_PT_001.svg" style="height: 50px;">

Δk is een aangepaste versie van Δ, waarbij lusversterkingen die het k-de voorwaartse pad kruisen, worden uitgesloten. Deze uitsluiting is cruciaal voor het nauwkeurig bepalen van de overdrachtsfunctie van het systeem.

Om de overdrachtsfunctie van een systeem te berekenen met behulp van de regel van Mason, worden de volgende stappen gebruikt:

<ol style="padding-left: 35px; margin: 0;">
 <li>Identificeer alle voorwaartse padversterkingen (T_k) van de invoer naar de uitvoer.</li>
 <li>Evalueer alle lusversterkingen (L_i) en identificeer de lussen die elkaar niet raken.</li>
 <li>Bereken Δ door de producten van de lusversterkingen en de niet-rakende lusversterkingen op te tellen en van elkaar af te trekken, zoals beschreven.</li>
 <li>Bereken voor elk voorwaarts pad Δ_k door de kruisende lusversterkingen uit te sluiten.</li>
 <li>Vervang deze waarden door de regel van Mason, gegeven door:</li>
</ol>

<img alt="Equation2" src="/files/ftp_upload/16108/16108_Equation_PT_002.svg" style="height: 34px;">

Met behulp van deze stappen biedt de regel van Mason een georganiseerde en systematische aanpak voor het afleiden van de overdrachtsfunctie van complexe systemen, waardoor deze onmisbaar is in de regeltheorie en signaalverwerking.

De regel van Mason

Mason&#39;s rule simplifies transfer function calculation from signal-flow graphs, assessing various elements.
Loop gain is calculated by tracing a path ...

Education

JoVE Core

Electrical Engineering

Elektrotechniek

Unit 1

Diagrams and Signal Flow Graphs

Diagrammen en signaalstroomgrafieken

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

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

transfer-functions-from-signal-flow-graphs

elements of block diagrams

Block Diagrams in Control Systems

Block diagrams show the input-output relationships in a system. They help students see how one signal affects another. A heating system is a simple example. The set temperature turns the furnace on, and the room warms to the desired level. Block diagrams can model both linear and nonlinear systems. Linear systems often use Laplace transform variables. Nonlinear systems are usually described with time domain variables. This makes block diagrams useful for many kinds of control problems. A...

relation between mathematical equations and block diagrams

relation-between-mathematical-equations-and-block-diagrams

Spring-Mass-Damper Laplace Block Diagram

A spring-mass-damper system can be written as a second-order differential equation. This equation describes how the system changes over time. When the equation is transformed into the Laplace domain with zero initial conditions, it becomes easier to analyze and rearrange. The Laplace transform changes a differential equation into an algebraic equation. That makes it simpler to isolate the output. For the standard spring-mass-damper model, the transformed equation gives the output in a form...

block diagram reduction

Simplifying Control System Block Diagrams

Simplifying control system block diagrams helps make a system easier to analyze. The goal is to reduce several connected blocks to one block so the transfer function can be found more easily. This process keeps the system behavior the same while making the diagram simpler to work with. A branch point is the place where one signal splits into multiple paths. To simplify the diagram, the branch point can be moved to a new location. The new position must preserve the mathematical relationship...

multi-input and multi-variable systems

Cruise Control and Aircraft System Models

Cruise control systems show how a control system can handle more than one input. In a car, the driver sets a desired speed, and the system also responds to outside disturbances such as changes in road slope. The throttle is adjusted so the vehicle speed stays as close as possible to the target value. When the disturbance is ignored, the cruise control block diagram can be reduced to a transfer function. This transfer function describes how the desired speed input, R(s), affects the actual...

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...

signal flow graphs

Mason’s Rule in Signal Flow Graphs

Signal-flow graphs are a clear way to represent control systems. They offer an alternative to traditional block diagrams. In these graphs, branches stand for systems and nodes stand for signals. Arrows show the direction of signal flow, and the transfer function is written next to each arrow. Signal-flow graphs also handle subtraction in a different way from block diagrams. In a block diagram, summing junctions often use negative signs. In a signal-flow graph, those negative signs are built...

sfg algebra

Signal Flow Graph Branch Rules

Signal flow graph (SFG) branch rules show how signals move through a control system. Each node in an SFG stands for a variable. The value at a node is the sum of all signals entering it. That summed value then travels through every branch leaving the node. Branches connect the nodes and show the relationship between variables. Each branch has a gain, which tells how strongly the signal is passed along. When several branches enter the same node, their signals are added together. When one node...

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.

Transfer Functions from Signal-Flow Graphs | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Transfer Functions from Signal-Flow Graphs

Description

Transcript

Tags