Question

Using masons gain formula, find the non-touching loops in terms of loop gains:

• A. acbd, efgh
• B. acdhfe, bc
• C. acdhfe, fg
• D. bc, fg

Hint:

Signal Flow Graph Loops. A loop starts and ends at the same node, following branch directions. Two loops are non-touching if they share no common nodes. Mason's Gain Formula uses sums of individual loop gains, sums of products of pairs of non-touching loop gains, etc.

Answer 1

The Correct Option is D

First, identify the individual loops in the signal flow graph and their gains (assuming forward path gains are represented by the letters): - Loop 1: Starts at node after 'a', goes b \(\rightarrow\) c \(\rightarrow\) back to node after 'a'
The gain is \(L_1 = bc\)
- Loop 2: Starts at node after 'e', goes f \(\rightarrow\) g \(\rightarrow\) back to node after 'e'
The gain is \(L_2 = fg\)
Next, determine if these loops are non-touching
Two loops are non-touching if they do not share any common nodes
- Loop 1 (bc) involves the nodes between a/b, b/c, and c/d
- Loop 2 (fg) involves the nodes between e/f, f/g, and g/h
Observing the diagram, these two sets of nodes are distinct; the loops do not share any nodes
Therefore, loops \(L_1\) (with gain bc) and \(L_2\) (with gain fg) are non-touching loops
Mason's Gain Formula often requires identifying pairs (or higher orders) of non-touching loops
The pair of non-touching loops has gains bc and fg
Option 4 lists these two loop gains
The other options list paths or combinations that aren't pairs of non-touching loop gains