Block diagram reduction and signal flow graphs

Why it matters

A real instrument loop is drawn as many blocks — controller, actuator, process, transmitter, sometimes an inner cascade loop. To analyse stability or error you need one overall transfer function. Block-diagram reduction and Mason's gain formula are the two standard ways to get it quickly and without algebra mistakes, and GATE tests both every year in some form.

Key ideas

Block diagram elements

• Block: multiplies its input signal by its transfer function.
• Summing point: adds or subtracts signals; the sign on each arrow matters.
• Take-off (pick-off) point: copies a signal to another path without loading it.

Reduction rules

1. Blocks in series (cascade): multiply, G1·G2 — valid only if the second block does not load the first.
2. Blocks in parallel (feeding the same summing point): add, G1 ± G2.
3. A feedback loop: G/(1 ± GH) (plus sign in the denominator for negative feedback).
4. Moving a summing point ahead of a block G: put 1/G in the moved branch. Moving it after the block: put G in the moved branch.
5. Moving a take-off point ahead of a block G: put G in the moved branch. Moving it after the block: put 1/G in the moved branch.
6. Adjacent summing points can be interchanged; adjacent take-off points can be interchanged. A summing point and a take-off point next to each other cannot be swapped without adding a compensating branch. Reduce the innermost loop first, and move points only to un-nest interlocked loops.

Signal flow graph (SFG)

• Node: a system variable. Branch: a directed line with a gain (transmittance) from one node to another. A node's value is the sum of all incoming branch signals.
• Source node has only outgoing branches; sink node has only incoming branches.
• Forward path: from source to sink, visiting no node more than once.
• Loop: a closed path that starts and ends on the same node without passing any other node twice.
• Non-touching loops: loops that share no node.
• A block diagram converts to an SFG by making every summing point and take-off point a node; the gains of SFG branches carry the signs that the summing points had.

Mason's gain formula gives the transfer function directly from the graph. It is exact for linear graphs and usually faster than reduction when there are several forward paths or interlocked loops.

Formulas

• T = C(s)/R(s) = (Σ Pk·Δk) / Δ — Mason's gain formula.
• Δ = 1 − ΣLi + ΣLiLj − ΣLiLjLk + … — graph determinant: 1 − (sum of all individual loop gains) + (sum of gain products of every pair of non-touching loops) − (sum for every triple of mutually non-touching loops) + …
• Δk = Δ evaluated after removing every loop that touches the k-th forward path.
• Pk = gain of the k-th forward path.
• T = G/(1 + GH) — single negative-feedback loop (loop gain in SFG form L = −GH, so Δ = 1 + GH).
• Series: G1·G2; Parallel: G1 + G2.

All gains are transfer functions in s (or constants); T, Pk, Li carry the units of output/input of the path they describe, and Δ, Δk are dimensionless. The loop gain Li includes the sign of the feedback branch: for a negative-feedback loop, Li is negative.

Worked examples

Example 1 — minor loop, by reduction and by Mason Given: G1 = 10 in series with G2 = 5; a negative-feedback branch H = 2 from the output of G2 back to a summing point just before G2. No outer loop.

1. Reduction: inner loop G2/(1 + G2H) = 5/(1 + 10) = 5/11.
2. Series with G1: T = 10 × 5/11 = 50/11.
3. Mason: one forward path P1 = G1G2 = 50; one loop L1 = −G2H = −10 (the minus sign comes from the negative summing point).
4. Δ = 1 − L1 = 1 − (−10) = 11; the loop touches the path, so Δ1 = 1.
5. T = P1Δ1/Δ = 50/11.

Result: T = 50/11 = 4.55. A common wrong answer is 50/(1 − 10) = −5.56, which comes from forgetting the negative sign of the loop gain.

Example 2 — two forward paths and non-touching loops (GATE level) Given an SFG with nodes R, x1, x2, x3, C and branches: R→x1 gain 1, x1→x2 gain G1 = 2, x2→x3 gain G2 = 3, x3→C gain G3 = 4, x2→x1 gain −H1 = −0.5, C→x3 gain −H2 = −0.25, and a feed-forward branch R→x3 gain G4 = 1. Find C/R.

1. Loops: L1 = G1·(−H1) = −1 (nodes x1, x2); L2 = G3·(−H2) = −1 (nodes x3, C). They share no node, so they are non-touching.
2. Δ = 1 − (L1 + L2) + L1L2 = 1 − (−2) + 1 = 4.
3. Forward path 1: R→x1→x2→x3→C, P1 = G1G2G3 = 24. It touches both loops, so Δ1 = 1.
4. Forward path 2: R→x3→C, P2 = G4G3 = 4. It touches L2 but not L1, so Δ2 = 1 − L1 = 2.
5. T = (P1Δ1 + P2Δ2)/Δ = (24 × 1 + 4 × 2)/4 = 32/4.

Result: C/R = 8. (Solving the four node equations simultaneously gives the same value, which is a good way to check Mason answers.)

Common mistakes

• Dropping the minus sign of a negative-feedback branch when computing loop gains in Mason's formula.
• Counting two loops as non-touching when they share even one node.
• Using Δk = 1 automatically; check which loops the k-th path leaves untouched.
• Treating a branch that only enters the source node as a loop, or forgetting that the source node must have only outgoing branches (add a unity branch if needed).
• Multiplying blocks in series when the second loads the first (e.g. two RC sections connected directly).
• Moving a take-off point the wrong way and putting G where 1/G belongs.

For GATE IN

Expect questions that give a block diagram or SFG and ask for the overall transfer function, the number of forward paths or non-touching loop pairs, or the value of a gain that makes the output independent of a disturbance. Practise converting block diagrams into SFGs, listing loops systematically, and checking Mason results by writing node equations.

Quick check

1. Two blocks 4/s and s/(s + 2) are in cascade. What is the equivalent?
2. An SFG has a single loop of gain −0.2 touching the only forward path of gain 6. What is T?
3. Two loops share exactly one node. Are they non-touching?
4. When a summing point is moved from the input of block G to its output, what gain goes in the moved branch?

Answers: 1. 4/(s + 2); 2. 6/1.2 = 5; 3. No; 4. G.