22.5
Règle de Mason
La règle de Mason est un outil puissant dans les systèmes de contrôle et le traitement du signal. Elle simplifie le calcul des fonctions de transfert…
La règle de Mason simplifie le calcul de la fonction de transfert à partir de graphiques de flux de signaux, en évaluant divers éléments.
Le gain de boucle est calculé en traçant un chemin d’un nœud à lui-même, en prenant le produit des gains de branche.
Le gain vers l’avant consiste à suivre un chemin entre le nœud d’entrée et le nœud de sortie, en calculant le produit des gains.
Les boucles non tactiles sont des boucles sans nœuds communs. Le gain est le produit des gains de boucle de ces boucles séparées.
La fonction de transfert est calculée à partir de ces éléments à l'aide de la règle de Mason.
Delta est dérivé de séries alternées de sommes de gains de boucle et de gains de boucle sans contact pris deux ou plusieurs à la fois. Delta_k est formé en excluant les gains de boucle de Delta qui coupent le kème chemin aller.
Pour calculer la fonction de transfert d’un système, les gains du chemin aller et les gains de boucle sont évalués.
Ensuite, les boucles non tactiles et leurs gains correspondants sont identifiés.
Le delta est calculé et Delta_k est évalué en supprimant les gains de boucle qui se croisent.
Enfin, ces valeurs sont substituées dans la règle de Mason pour produire la fonction de transfert du système.
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Q1: What is loop gain in Mason's rule?
Loop gain is calculated by tracing a path from a node back to itself and computing the product of branch gains along that loop. Each loop's gain is crucial for determining system behavior and contributes to the overall transfer function calculation in Mason's rule.
Q2: How do you calculate forward-path gain in a signal-flow graph?
Forward-path gain is determined by tracing a path from the input node to the output node and computing the product of gains along this path. Forward paths represent the direct influence of the input on the output and are essential for determining the transfer function using Mason's rule.
Q3: What are non-touching loops and why do they matter?
Non-touching loops are loops in the signal-flow graph that do not share any common nodes. Their gain is the product of individual loop gains, and they significantly affect the computation of Delta, the determinant used in Mason's rule for transfer function calculation.
Q4: What is Delta and Delta_k in Mason's rule?
Delta is derived from an alternating series of sums involving loop gains and non-touching loop gains taken two or more at a time. Delta_k is a modified version of Delta that excludes loop gains intersecting with the kth forward path, crucial for accurately determining the system's transfer function.
Q5: What are the main steps for applying Mason's rule to find a transfer function?
First, identify all forward-path gains from input to output. Next, evaluate all loop gains and identify non-touching loops. Then calculate Delta using the alternating series formula. For each forward path, compute Delta_k by excluding intersecting loop gains. Finally, substitute these values into Mason's rule to yield the transfer function.
Q6: How does Mason's rule simplify transfer function calculation compared to other methods?
Mason's rule provides a systematic, organized approach to deriving transfer functions from signal-flow graphs by leveraging loop gains, forward-path gains, and non-touching loops. This method is indispensable in control theory and signal processing, making complex system analysis more efficient than manual algebraic manipulation.
Q7: Why is understanding the relation between mathematical equations and block diagrams important for Mason's rule?
Mason's rule operates on signal-flow graphs derived from system equations and block diagrams. Understanding the relation between mathematical equations and block diagrams helps students translate system models into graphical representations, enabling them to apply Mason's rule effectively to calculate transfer functions.