Mason Gain Formula

Mason's gain formula provides a direct method to compute the transfer function of a system from its signal flow graph without performing step-by-step block diagram reduction. It is especially powerful for systems with multiple loops and forward paths, where manual reduction becomes error-prone and time-consuming.

Core Concept Explanation

Mason's gain formula states that the overall transfer function T from a source node to a sink node is: T = (1/Δ) × sum over all forward paths k of (Pk × Δk). Here Pk is the gain of the k-th forward path, Δ is the graph determinant, and Δk is the path cofactor for the k-th forward path.

The graph determinant Δ (also called delta) is computed as: Δ = 1 - (sum of all individual loop gains) + (sum of gain products of all pairs of non-touching loops) - (sum of gain products of all sets of three mutually non-touching loops) + ... This alternating sum captures the contribution of all loops in the SFG to the overall system behavior.

The path cofactor Δk is the value of Δ computed for the part of the SFG that does not touch the k-th forward path. Two elements are said to touch if they share at least one common node. If a forward path touches all loops, then Δk = 1, simplifying the calculation significantly.

Mathematical Expression

The full statement of Mason's formula is: T = [sum_k (Pk × Δk)] / Δ, where Δ = 1 - L1 - L2 - L3 ... + L1L2 (non-touching) + L1L3 (non-touching) ... - L1L2L3 (mutually non-touching) + ... Each Li is the gain of the i-th loop in the SFG. The product terms include only loops that share no common node (non-touching loops).

To compute Δk for path Pk, remove all nodes and branches that belong to Pk from the graph determinant formula. The remaining graph's determinant gives Δk. If the forward path touches every loop in the graph, all loop terms disappear and Δk = 1. This scenario is very common in GATE problems where simple SFGs are given.

Practical Understanding

The advantage of Mason's formula over block diagram reduction is that it can be applied directly to the SFG without any intermediate simplification. The procedure is: first identify all forward paths and compute their gains, then identify all loops and compute loop gains, then identify non-touching loop pairs, then calculate Δ and each Δk, and finally substitute into the formula.

For a simple system with one forward path P1 and one loop L1, the formula reduces to T = P1 / (1 - L1). If L1 = -GH (as in negative feedback), then T = P1 / (1 + GH), which matches the standard closed-loop formula. This confirms that Mason's formula is consistent with block diagram methods.

Given:
SFG with nodes x1, x2, x3, x4
Branch gains: x1→x2: 1, x2→x3: G1=5, x3→x4: G2=2, x4→x2: -H=-0.4
Also a direct branch x2→x4: G3=3

Why this formula applies:
Multiple forward paths exist, so Mason's formula is used directly.

Forward Paths:
P1: x1→x2→x3→x4 = 1 × 5 × 2 = 10
P2: x1→x2→x4 = 1 × 3 = 3

Loops:
L1: x2→x3→x4→x2 = 5 × 2 × (-0.4) = -4

(No other loops, no non-touching loop pairs)

Graph Determinant:
Δ = 1 - L1 = 1 - (-4) = 5

Path Cofactors:
Δ1: path P1 touches loop L1 (shares x2, x3, x4) → Δ1 = 1
Δ2: path P2 touches loop L1 (shares x2, x4) → Δ2 = 1

Mason's Formula:
T = (P1×Δ1 + P2×Δ2) / Δ
= (10×1 + 3×1) / 5
= 13/5

Final Answer:
Transfer function T = 13/5 = 2.6

Exam Tip: In GATE, the most common mistake in Mason's formula is incorrectly identifying non-touching loops. Two loops are non-touching only if they share NO common node at all. Also, Δk = 1 whenever the forward path touches all loops in the graph, which is true in most GATE-level SFGs with one or two loops.

Quick Revision

• Mason's formula: T = [Σ Pk Δk] / Δ
• Δ = 1 - ΣLi + Σ(LiLj non-touching) - Σ(LiLjLk mutually non-touching) + ...
• Pk = gain of k-th forward path (product of all branch gains along path).
• Δk = graph determinant of the part of SFG not touching k-th forward path.
• Non-touching loops share NO common node. Only such pairs contribute product terms in Δ.
• For single forward path and single loop: T = P1 / (1 - L1). For negative feedback L1 = -GH → T = G/(1+GH).
• GATE trap: If a forward path touches all loops, Δk = 1. Students often re-include loop terms in Δk unnecessarily, which gives a wrong answer.