22.5
Mason's Rule
Mason'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'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's rule to yield the system's transfer function.
Mason's rule is a powerful tool in control systems and signal processing. It simplifies the calculation of transfer functions from signal-flow graphs. This method leverages various elements, including loop gains, forward-path gains, and non-touching loops, to determine the transfer function efficiently.
Loop gain is determined by identifying and tracing a path from a node back to itself. This involves computing the product of branch gains along the loop. Each loop's gain is crucial for further calculations and contributes to the overall system behavior.
The forward-path gain is calculated by tracing a path from the input node to the output node. Like loop gain, it involves the product of the gains along this path. Forward paths represent the direct influence of the input on the output and are essential for determining the transfer function.
Non-touching loops are loops in the signal-flow graph that do not share any common nodes. The gain of non-touching loops is the product of the individual loop gains. These non-touching loops are significant as they affect the computation of the determinant, Delta (Δ), used in Mason's rule.
Delta (Δ) is derived from an alternating series of sums involving loop gains and the gains of non-touching loops taken two or more at a time. Mathematically, it can be expressed as:
Δk is a modified version of Δ, excluding loop gains that intersect with the kth forward path. This exclusion is crucial for accurately determining the system's transfer function.
To calculate the transfer function of a system using Mason's rule, the following steps are used:
- Identify all forward-path gains (Tk) from the input to the output.
- Evaluate all loop gains (Li) and identify non-touching loops.
- Calculate Δ by summing and subtracting the products of loop gains and non-touching loop gains as described.
- For each forward path, compute Δk by excluding intersecting loop gains.
- Substitute these values into Mason's rule, given by:
Through these steps, Mason's rule provides an organized and systematic approach to deriving the transfer function of complex systems, making it indispensable in control theory and signal processing.
Mason'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'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's rule to yield the system's transfer function.
From Chapter 22:
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