Block Diagrams

Block diagrams show how signals flow through a system. Learn series, parallel and feedback reduction rules, moving summing points and take-off points, and Mason's gain formula for complex signal-flow graphs.

A block diagram draws each subsystem as a block containing its transfer function, connected by arrows carrying signals, with summing junctions and take-off points. It is the standard way to describe control architectures — from a thermostat to a flight-control system.

Reducing a diagram to a single transfer function uses a handful of algebraic rules, or Mason's formula for tangled graphs.

An analogy

A block diagram is like a factory flowchart: each machine (block) transforms what flows in, conveyors (arrows) connect them, and merge points (summers) combine streams.

Elements

Blocks multiply the signal by a transfer function; summing junctions add or subtract signals (signs marked on each input); take-off points copy a signal to several places without changing it.

Reduction rules

Use these moves to simplify a diagram step by step.

• Combine series blocks by multiplying and parallel blocks by adding.
• Replace each feedback loop with G/(1 ± GH).
• Moving a summing point ahead of a block: divide the moved branch by that block.
• Moving a take-off point past a block: divide the branch by the block (or multiply when moving it backwards).
• Reduce inner loops first, then work outward.

Signal-flow graphs and Mason's formula

Represent nodes as signals and branches as gains. Find all forward paths P_k and loops. Δ = 1 − (sum of loop gains) + (sum of products of pairs of non-touching loops) − …; Δ_k is Δ with loops touching path k removed. It gives the transfer function in one step, even for interlinked loops.

Practical architectures

Real controllers combine feedback, feed-forward, cascade loops, disturbance observers and pre-filters. Block diagrams make the signal flow and each path's role explicit before any algebra, and map directly onto Simulink or Xcos models.

Formulas

• Series: G = G₁ · G₂
• Parallel: G = G₁ ± G₂
• Negative feedback: G_cl = G / (1 + G·H)
• Mason's gain formula: T = (Σ P_k · Δ_k) / Δ, Δ = 1 − ΣL₁ + ΣL₂ − …

Worked example: Single-loop reduction

Given: G₁ = 2 in series with G₂ = 5/(s + 1); Feedback H = 0.5

Find: Closed-loop transfer function and DC gain.

• Forward G = 2 × 5/(s + 1) = 10/(s + 1)
• T = G/(1 + GH) = [10/(s+1)] / [1 + 5/(s+1)] = 10/(s + 6)
• DC gain = 10/6 = 1.67

Answer: T(s) = 10/(s + 6)

Worked example: Mason's formula, two parallel paths

Given: Paths P₁ = G₁G₂, P₂ = G₃; One loop L₁ = −G₂H touching both paths

Find: T.

• Δ = 1 − L₁ = 1 + G₂H
• Both paths touch the loop → Δ₁ = Δ₂ = 1
• T = (G₁G₂ + G₃)/(1 + G₂H)

Answer: T = (G₁G₂ + G₃)/(1 + G₂H)

Common mistakes

• Wrong: Ignoring the sign at a summing junction. — Right: Negative feedback gives 1 + GH; positive feedback gives 1 − GH.
• Wrong: Moving a take-off point without compensating. — Right: Multiply or divide the moved branch by the block it passes.

Applications

• Drive control: Motor drives are drawn as cascaded current, speed and position loops with feed-forward paths.
• Process plants: P&IDs and control narratives translate into block diagrams for tuning and simulation.
• Simulation tools: Simulink and similar tools are block-diagram editors that simulate the equations directly.

Key takeaways

• Series multiply, parallel add, feedback → G/(1 ± GH).
• Reduce inner loops first; compensate when moving summers or take-offs.
• Mason's formula handles complex signal-flow graphs in one step.
• Block diagrams describe architecture before algebra.

Going further

• MIMO block diagrams: With multiple inputs and outputs, blocks become transfer-function matrices and order matters (matrices don't commute). Feedback becomes (I + GH)⁻¹G.
• Algebraic loops: In simulation, a loop with no dynamics (direct feedthrough all the way round) forms an algebraic loop the solver must iterate; adding realistic dynamics or a unit delay breaks it.
• Linear fractional transformations: Robust-control theory writes any interconnection as an LFT, separating the nominal plant from uncertainty blocks, which is the starting point for µ-analysis.