Mason’s Gain Rule (Solved Example 1)

Summary

In this tutorial, the process of finding the transfer function of a system using Mason's Gain Rule is explained in detail. The speaker guides the audience through converting a block diagram into a signal flow graph, calculating forward path gain, loop gains, and the determinant of the signal flow graph (Δ). Mason’s Gain Formula is then applied to determine the transfer function, showcasing how to handle systems with multiple loops and takeoff points. The session concludes with a practical example and offers a solid understanding of using Mason’s Gain Rule for control system analysis.

Takeaways

• 😀 The lecture explains how to calculate the transfer function of a system using Mason's Gain Rule.
• 😀 The system’s block diagram is converted into an equivalent Signal Flow Graph (SFG) for the analysis.
• 😀 The process begins by identifying the forward path from the input node to the output node in the signal flow graph.
• 😀 Two summing points are identified in the SFG, with three forward path blocks (g1, g2, g3) connecting them.
• 😀 The transfer function can be calculated by applying Mason's Gain Rule to the equivalent signal flow graph.
• 😀 The forward path gain is the product of the branch gains in the forward path, specifically g1, g2, and g3.
• 😀 The system contains two individual loops, each with their respective loop gains: one involving g1, g2, h1 and the other involving g2, g3, h2.
• 😀 The number of non-touching loops in the signal flow graph is zero, as the loops share common nodes.
• 😀 The determinant of the signal flow graph (Δ) is calculated as 1 + g1g2h1 + g2g3h2.
• 😀 The transfer function formula is applied using Mason’s Gain Rule: C(s)/R(s) = P1 * Δ1 / Δ, where P1 is the forward path gain and Δ1 is the associated path factor.
• 😀 The final transfer function of the system is derived as C/R = g1g2g3 / (1 + g1g2h1 + g2g3h2), based on the given SFG.

Q & A

What is the main goal of the presentation?

The main goal of the presentation is to demonstrate how to calculate the transfer function of a system using Mason's Gain Rule, with a focus on converting a block diagram into a signal flow graph (SFG).

What are the key components in the block diagram mentioned in the script?

The block diagram includes two summing points, three blocks (G1, G2, G3) in the forward path, two takeoff points with gains (H1 and H2), and the reference input 'r' which leads to the output 'c'.

What is the first step in applying Mason's Gain Rule?

The first step is to convert the given block diagram into an equivalent signal flow graph (SFG), identifying nodes, paths, and gains associated with each branch in the system.

What are the nodes used in the signal flow graph (SFG)?

The nodes in the SFG are labeled as node A (first summing point), node B (second summing point), node C (takeoff point after block G2), and node D (takeoff point after block G3).

How is the forward path in the signal flow graph determined?

The forward path is determined by tracing the signal flow from the input node 'r' to the output node 'c', passing through the nodes A, B, C, and D. The gain for this forward path is the product of the gains associated with each branch in the path.

How do you calculate the loop gains in the signal flow graph?

Loop gains are calculated by multiplying the gains of the branches that form individual loops. For example, the loop gain for loop L1 (A-B-C-A) is -G1 * G2 * H1, and the loop gain for loop L2 (B-C-D-B) is -G2 * G3 * H2.

What is the significance of non-touching loops in the application of Mason's Gain Rule?

Non-touching loops are important when calculating the determinant in Mason's Gain Rule. They allow for combinations of loop gains to be included in the formula. However, in this specific example, there are no non-touching loops in the signal flow graph.

How is the determinant (delta) of the signal flow graph calculated?

The determinant delta is calculated using the formula: delta = 1 - (sum of individual loop gains) - (sum of the products of the gains of non-touching loops). In this case, since there are no non-touching loops, delta simplifies to: delta = 1 + G1 * G2 * H1 + G2 * G3 * H2.

What is the associated path factor in this example, and how is it calculated?

The associated path factor is calculated by erasing the forward path and checking for isolated loops. In this case, since erasing the forward path destroys all loops, the associated path factor (delta1) is equal to 1.

How is the overall transfer function determined using Mason's Gain Rule?

The overall transfer function is determined using the formula: (C(s) / R(s)) = P1 * Del1 / Delta, where P1 is the gain of the forward path, Del1 is the determinant associated with the forward path, and Delta is the determinant of the entire signal flow graph. Substituting the values results in the transfer function: C/R = G1 * G2 * G3 / (1 + G1 * G2 * H1 + G2 * G3 * H2).