Signal flow graphs and Mason's gain formula

Why it matters

Block diagram reduction becomes slow and error-prone once loops overlap, which is common in multi-sensor mechatronic systems (a robot joint with current, speed and position loops plus a disturbance path). A signal flow graph (SFG) shows the same linear equations as nodes and branches, and Mason's gain formula gives the overall transfer function in one step, without redrawing the diagram. It is also the quickest way to get the transfer function from a set of simultaneous equations.

Key ideas

What an SFG is. A signal flow graph is a picture of a set of linear algebraic equations in the s-domain, each written in cause-and-effect form xⱼ = Σ aᵢⱼ·xᵢ.

• Node: a system variable (signal). The value at a node is the sum of all signals entering it, so summing points become nodes.
• Branch: a directed line from one node to another, carrying a gain (transmittance) — the transfer function of a block. A signal travels only in the arrow direction.
• Input (source) node: only outgoing branches. Output (sink) node: only incoming branches; if the output node has outgoing branches, add a dummy branch of gain 1 to a new output node.
• Path: a continuous sequence of branches in the arrow direction, passing no node twice.
• Forward path: a path from the input node to the output node, touching no node more than once.
• Loop: a path that starts and ends on the same node, touching no other node more than once. A self-loop is a single branch from a node back to itself.
• Path gain / loop gain: the product of branch gains along the path or loop. Feedback signs are inside the loop gain: a negative feedback branch makes the loop gain negative.
• Touching: two loops (or a loop and a path) touch if they share at least one node. Non-touching loops share no node.

From a block diagram. Every summing point and every take-off point becomes a node; each block becomes a branch with the block's transfer function; a subtraction at a summing point becomes a negative sign on that branch. Input and output become source and sink nodes.

Mason's gain formula gives the gain from one source node to one sink node. The graph determinant Δ collects all loops: subtract every individual loop gain, add the products of every pair of non-touching loops, subtract the products of every triple of mutually non-touching loops, and so on. For each forward path k, the cofactor Δₖ is Δ evaluated with every loop that touches path k removed. If every loop touches a path, its Δₖ = 1.

Checks. Δ for a single loop L reduces to 1 − L, so a single negative feedback loop gives G/(1 + GH) — the same as block reduction. Δ = 0 is the characteristic equation of the system. Mason's formula gives input-to-output gain only; between two internal nodes, divide the gains from the input to each.

Formulas

T = C(s)/R(s) = (1/Δ) · Σₖ Pₖ·Δₖ

• T: overall transfer function from source to sink (units of output per unit input).
• Pₖ: gain of the k-th forward path.
• Δ: graph determinant; Δₖ: cofactor of path k.

Δ = 1 − ΣLᵢ + ΣLᵢLⱼ − ΣLᵢLⱼLₖ + …

• ΣLᵢ: sum of all individual loop gains.
• ΣLᵢLⱼ: sum of products of gains of all pairs of non-touching loops.
• ΣLᵢLⱼLₖ: sum over all triples of mutually non-touching loops.

Δₖ = Δ with all loops touching the k-th forward path removed

Δ = 0 — characteristic equation (same as 1 + G·H = 0 for a single loop).

Worked examples

Example 1 (standard — check against block reduction). A forward path has G₁(s) followed by G₂(s). A minor feedback H₁(s) (negative) runs from the output of G₂ back to the input of G₂, and the output C is fed back with unity negative feedback to the input summing point. Find C/R.

1. Nodes: R, E (after first summing point), X (after second summing point), C.
2. Forward path: R → E → X → C, so P₁ = G₁·G₂.
3. Loops: inner loop X → C → X, L₁ = −G₂·H₁; outer loop E → X → C → E, L₂ = −G₁·G₂.
4. L₁ and L₂ share nodes X and C, so they touch; there is no non-touching pair.
5. Δ = 1 − (L₁ + L₂) = 1 + G₂H₁ + G₁G₂.
6. Both loops touch P₁, so Δ₁ = 1.
7. C/R = G₁G₂ / (1 + G₂H₁ + G₁G₂). Reducing the inner loop first and then the unity loop gives the same result.

Example 2 (GATE level — non-touching loops and a non-unit cofactor). An SFG has nodes y₁ (input) to y₆ (output) with branches: y₁→y₂ = 2, y₂→y₃ = 3, y₃→y₄ = 4, y₄→y₅ = 5, y₅→y₆ = 1, a feedback branch y₃→y₂ = −0.5, a feedback branch y₅→y₄ = −0.2, and a feed-forward branch y₁→y₄ = 1. Find y₆/y₁.

1. Forward paths: P₁ = 2 × 3 × 4 × 5 × 1 = 120 (y₁y₂y₃y₄y₅y₆); P₂ = 1 × 5 × 1 = 5 (y₁y₄y₅y₆).
2. Loops: L₁ = 3 × (−0.5) = −1.5 (y₂y₃y₂); L₂ = 5 × (−0.2) = −1.0 (y₄y₅y₄).
3. L₁ uses nodes y₂, y₃; L₂ uses y₄, y₅. They are non-touching, so L₁L₂ = 1.5.
4. Δ = 1 − (−1.5 − 1.0) + 1.5 = 1 + 2.5 + 1.5 = 5.0.
5. P₁ touches both loops: Δ₁ = 1. P₂ passes through y₄ and y₅, so it touches L₂ but not L₁: Δ₂ = 1 − L₁ = 2.5.
6. T = (P₁Δ₁ + P₂Δ₂)/Δ = (120 × 1 + 5 × 2.5)/5 = 132.5/5.
7. y₆/y₁ = 26.5.

Common mistakes

• Treating any two loops as non-touching. Check node by node — two loops that share even one node touch.
• Setting every Δₖ = 1 out of habit. Δₖ = 1 only when every loop touches path k.
• Forgetting the sign: Δ = 1 − ΣL, so a negative-feedback loop of gain −0.5 adds +0.5 to Δ.
• Missing a loop formed through a feed-forward branch combined with a feedback branch, or missing self-loops.
• Counting a forward path that revisits a node, or one that goes against a branch arrow.
• Using a sink node that still has outgoing branches without adding a unity dummy branch.
• Applying Mason's formula between two internal nodes directly.

For GATE ME

Typical questions: list the number of forward paths and individual loops in a given SFG; count pairs of non-touching loops; find Δ or a specific Δₖ; compute C/R for a graph with numeric gains (often with one non-unit cofactor); convert a small block diagram to an SFG and find the characteristic equation. Practise marking node sets for every loop and path before computing — almost all wrong answers come from touching/non-touching errors, not from arithmetic.

Quick check

1. A graph has one forward path P = 10 and one loop L = −4 that touches it. Find T.
2. Two non-touching loops have gains −2 and −3, and there are no other loops. Find Δ.
3. What does a node represent in an SFG?
4. When is Δₖ equal to 1?
5. What equation do you get by setting Δ = 0?

Answers: 1. 10/(1 + 4) = 2. 2. 1 + 5 + 6 = 12. 3. A system variable (signal). 4. When every loop in the graph touches the k-th forward path. 5. The characteristic equation of the system.