How it works
A signal flow graph has nodes representing variables and directed branches carrying transmittances (gain values, transfer functions, or constants). A forward path is any path from input node to output node that does not cross any node twice. A loop gain is the product of all branch transmittances around a closed path. Mason's gain formula: T = (1/Δ) Σ Pₖ·Δₖ, where Pₖ is the gain of the k-th forward path, Δ = 1 − ΣLᵢ + ΣLᵢLⱼ − ... (with non-touching loop products), and Δₖ is Δ calculated excluding loops that touch forward path k.
Key points to remember
Non-touching loops are loops that share no common node — their products appear with a + sign in the graph determinant Δ. Two loops touching even a single node are considered touching and do not appear as a product term. For a simple system with one forward path P₁ = G₁G₂ and one loop L₁ = −G₁G₂H, Δ = 1 + G₁G₂H and Δ₁ = 1 (the loop touches the forward path), giving T = G₁G₂/(1+G₁G₂H) — matching the standard feedback formula. Self-loops (a branch from a node back to itself) must be included in the loop gain calculation. Convert block diagrams to SFG by assigning a node to each summing junction and take-off point.
Exam tip
The examiner always asks you to apply Mason's gain formula to a given signal flow graph — list every forward path and every loop gain explicitly in a table before substituting into the formula, because partial credit depends on showing those intermediate steps.