Questions: Signal Flow Graphs and Mason's Gain Formula

Question 1

A signal flow graph has three loops: L₁, L₂, and L₃. Loops L₁ and L₂ share a node; L₂ and L₃ share a node; but L₁ and L₃ share no nodes. How do L₁ and L₃ appear in the graph determinant Δ?

• A. They appear as a sum: −(L₁ + L₃), added with a negative sign like all individual loops
• B. They appear as a product: +L₁·L₃, added with a positive sign as a non-touching pair
• C. They are ignored because they are not directly connected to each other
• D. They appear only in the cofactor Δₖ for paths that touch L₂

Answer: (B)

In Δ = 1 − ΣLᵢ + ΣLᵢLⱼ − ···, pairs of non-touching loops enter as *products* with a positive sign. Since L₁ and L₃ share no nodes, they contribute the term +L₁·L₃. Individual loops contribute −ΣLᵢ; non-touching pairs contribute +ΣLᵢLⱼ; non-touching triples contribute −ΣLᵢLⱼLₖ; and so on in alternation. Option A is the most common misconception — students add rather than multiply non-touching loops. L₁ and L₂ touching (and L₂ and L₃ touching) means those pairs do NOT contribute product terms.

Question 2

A simple negative feedback system has forward path gain G and feedback loop gain L₁ = −GH (negative sign from the summing junction). Applying Mason's formula, what is the transfer function?

• A. T = G / (1 − GH)
• B. T = G / (1 + GH)
• C. T = GH / (1 + G)
• D. T = G·GH

Answer: (B)

Applying Mason's formula: Δ = 1 − L₁ = 1 − (−GH) = 1 + GH. The single forward path P₁ = G touches the loop (they share nodes), so the cofactor Δ₁ = 1 (no untouched loops remain). Therefore T = P₁Δ₁/Δ = G·1/(1+GH) = G/(1+GH). This recovers the familiar negative feedback formula directly. If the feedback were positive (L₁ = +GH), Δ = 1−GH and T = G/(1−GH). The sign of the loop gain is critical and must be computed carefully.

Question 3

If all loops in a signal flow graph touch a particular forward path k, then the cofactor Δₖ equals 1.

• T. True
• F. False

Answer: True

The cofactor Δₖ is computed by removing from Δ all loops (and their non-touching combinations) that touch forward path k. If every loop touches path k, nothing remains after deletion: Δₖ = 1. This is why the simple feedback system T = G/(1+GH) has Δ₁ = 1 — the single loop touches the single forward path, so no independent loop dynamics remain. When some loops are untouched by path k, Δₖ captures those independent dynamics as a sub-determinant.

Question 4

Different valid signal flow graph representations of the same linear system can yield different transfer functions when Mason's gain formula is correctly applied.

• T. True
• F. False

Answer: False

While the same system can be drawn as multiple different SFGs (different node placement, different intermediate signals), they all encode the same set of linear equations. Mason's formula, when correctly applied to any valid representation, extracts the same input-output transfer function. If two correct SFGs of the same system give different transfer functions, there is a bookkeeping error — a missed loop, wrong loop gain, or incorrect non-touching pair identification. This consistency is why comparing against block diagram reduction serves as a useful check.

Question 5

What is the graph determinant Δ in Mason's formula, and why do non-touching loops contribute as products rather than sums?

Model answer: Δ = 1 − ΣLᵢ + ΣLᵢLⱼ − ΣLᵢLⱼLₖ + ···, where sums alternate in sign over all loops, all pairs of non-touching loops, all non-touching triples, etc. Non-touching loops contribute as products because their gains multiply independently — two loops sharing no nodes create compounded feedback that is neither captured by summing them nor by treating them as a single loop. The product term ΣLᵢLⱼ represents second-order interactions between independent feedback paths, and the alternating signs follow an inclusion-exclusion structure to prevent double-counting.

Non-touching loops are independent subsystems, and their combined gain contribution to the denominator is multiplicative, analogous to how independent probabilities multiply. Summing them would undercount the feedback from independent loops operating simultaneously; the correct accounting requires their product.