Block Diagrams and Signal Flow Graphs

Block-diagram reduction and Mason's signal-flow-graph rule find the overall transfer function of interconnected systems — the starting point of the ~9-mark Control Systems section. GATE EC tests both routinely (1–2 marks).

Key results

• Negative feedback closed-loop: T(s) = G/(1 + GH); positive feedback: G/(1 − GH). Unity feedback H = 1.
• Series blocks multiply; parallel blocks add; a summing junction adds/subtracts.
• Mason's gain formula: T = (Σ_k P_k Δ_k)/Δ, where Δ = 1 − Σ(loop gains) + Σ(non-touching pairs) − Σ(non-touching triples)…; P_k = k-th forward path gain, Δ_k = Δ with paths touching P_k removed.
• Moving a takeoff/summing point requires compensating gain.

Interconnection	Equivalent
Series G₁, G₂	G₁G₂
Parallel G₁, G₂	G₁ + G₂
Negative feedback	G/(1+GH)
Positive feedback	G/(1−GH)

Exam Tricks & Tips

• 🎯 Closed-loop with negative feedback = G/(1+GH); the "+" comes from negative feedback, "−" from positive.
• 🎯 Use Mason's rule to avoid tedious block-reduction — identify all forward paths and loops first.
• 🎯 Δ = 1 − (sum of all loop gains) + (sum of products of non-touching loop pairs) − …
• 🎯 Two loops are "non-touching" if they share no node — only these appear in the higher-order Δ terms.
• 🎯 Series blocks multiply, parallel add — collapse these before applying feedback formulas.
• ❌ Common mistake: forgetting to include non-touching loop-pair products in Δ (dropping the +Σ term).

Expected exam pattern

1–2 marks: reduce a block diagram or apply Mason's rule to find C/R; determine loop gains and the graph determinant Δ.

Quick recap

Neg. feedback T=G/(1+GH). Series ×, parallel +. Mason: T=ΣP_kΔ_k/Δ, Δ=1−Σloops+Σ(non-touching pairs)−…; Δ_k drops loops touching path k. Non-touching = no shared node.

Cover the answer, recall, then check. 10 cards on block diagrams and signal flow graphs for GATE EC.

Q1. Closed-loop transfer function with negative feedback?
A1. T(s) = G/(1 + GH).

Q2. Closed-loop transfer function with positive feedback?
A2. T(s) = G/(1 − GH).

Q3. How do series and parallel blocks combine?
A3. Series blocks multiply (G₁G₂); parallel blocks add (G₁ + G₂).

Q4. State Mason's gain formula.
A4. T = (Σ_k P_k Δ_k)/Δ, where P_k are forward-path gains, Δ is the graph determinant, and Δ_k excludes loops touching path k.

Q5. Write the graph determinant Δ.
A5. Δ = 1 − Σ(individual loop gains) + Σ(products of non-touching loop pairs) − Σ(non-touching triples) + …

Q6. When are two loops "non-touching"?
A6. When they share no common node.

Q7. What is Δ_k in Mason's formula?
A7. The determinant Δ evaluated after removing all loops that touch the k-th forward path.

Q8. For unity feedback, what is H?
A8. H = 1, so T = G/(1 + G).

Q9. What happens when you move a takeoff point past a block?
A9. You must add a compensating block (multiply/divide by the gain) to keep signals equivalent.

Q10. A single forward path with no loops gives what transfer function?
A10. Just the forward-path gain (Δ = 1, Δ_k = 1).

Before you can analyse a control system, you must reduce its tangle of blocks and feedback loops to a single transfer function. Block-diagram algebra and Mason's gain formula are the two tools for that — and Mason's formula, in particular, is a GATE favourite because it gets the answer without messy step-by-step reduction.

Core concept: A block diagram or signal flow graph represents signal flow through gains and feedback; reduction (algebra or Mason's rule) collapses it to the overall transfer function C(s)/R(s).

The theory

Beginner — block diagram algebra

Basic reductions:

• Series (cascade): gains multiply, G1·G2.
• Parallel: gains add, G1 + G2.
• Negative feedback loop: C/R = G/(1 + G·H); positive feedback: G/(1 − G·H). Here G is the forward gain and H the feedback gain.
  Moving summing/pickoff points across blocks follows rules that keep the input-output relation unchanged.

Intermediate — signal flow graphs (SFG)

An SFG is a directed graph: nodes are signals, branches are gains. Terms:

• Forward path: a path from input to output touching no node twice.
• Loop: a closed path; loop gain = product of its branch gains.
• Non-touching loops: loops sharing no node.

Advanced — Mason's gain formula

The overall gain:
T = (1/Δ)·Σ Pk·Δk,
where:

• Pk = gain of the k-th forward path.
• Δ = 1 − (Σ all loop gains) + (Σ products of two non-touching loops) − (Σ products of three non-touching loops) + …
• Δk = the value of Δ with all loops touching path k removed (i.e., keep only loops that don't touch Pk).
  This single formula replaces iterative block reduction and is far less error-prone for multi-loop systems.

Worked example

A single-loop system: forward gain G, negative feedback H. Apply Mason's formula.
One forward path P1 = G. One loop with gain L1 = −G·H (negative feedback).
Δ = 1 − (−GH) = 1 + GH. Δ1 = 1 (the loop touches the path, so removed → 1).
T = P1·Δ1/Δ = G·1/(1 + GH) = G/(1 + GH) — matches the standard feedback formula, confirming Mason's rule.

GATE relevance

1–2 mark NAT/MCQ: reduce a block diagram or apply Mason's formula to get C/R, identify forward paths / loops / non-touching loops, and simplify multi-loop systems. Mason's-formula problems appear regularly.

Exam tricks and shortcuts

• Standard loop: C/R = G/(1 ± GH) (− for positive feedback, + for negative).
• Mason's Δ = 1 − Σloops + Σ(two non-touching) − …; Δk keeps only loops not touching path k.
• Series multiply, parallel add.
• Mnemonic: "Paths over Delta" (T = ΣPkΔk/Δ).

Key formulas

• Closed-loop transfer function: T(s) = G/(1 + GH) (negative feedback).
• Series blocks multiply; parallel add.
• Mason's gain formula: T = (1/Δ)Σ Pₖ Δₖ, where Δ = 1 − Σloops + Σ(non-touching loop products) − …
• Pₖ = kth forward-path gain; Δₖ = Δ with loops touching path k removed.

Mason's gain formula gives the transfer function directly from a signal-flow graph without block reduction.

Worked Example

Problem: A control system has forward-path transfer function G(s) = 10/[s(s + 2)] with unity negative feedback H(s) = 1. (a) Reduce the block diagram to find the closed-loop transfer function T(s). (b) Identify the undamped natural frequency ωₙ and damping ratio ζ.

Solution:
(a) For a single loop with negative feedback, the closed-loop transfer function is:
T(s) = G(s) / [1 + G(s)H(s)].
Substitute G = 10/[s(s+2)], H = 1:
T(s) = {10/[s(s+2)]} / {1 + 10/[s(s+2)]}.
Multiply numerator and denominator by s(s+2):
T(s) = 10 / [s(s+2) + 10] = 10 / (s² + 2s + 10).

(b) Compare the denominator with the standard second-order form s² + 2ζωₙs + ωₙ²:
ωₙ² = 10 ⇒ ωₙ = √10 ≈ 3.16 rad/s.
2ζωₙ = 2 ⇒ ζ = 1/ωₙ = 1/√10 ≈ 0.316.

Since 0 < ζ < 1 the system is underdamped (oscillatory step response).

Answer: T(s) = 10/(s² + 2s + 10); ωₙ = √10 ≈ 3.16 rad/s and ζ ≈ 0.316 (underdamped).