Block Diagrams and Signal Flow
Block diagrams represent interconnected transfer functions; algebra on them reduces complex loops to a single input-output relation.
Reading a Block Diagram
A block diagram shows blocks (each a transfer function), arrows (signals), summing junctions (adding or subtracting signals), and pickoff points (branching a signal). It is a graphical form of the algebra that relates system inputs to outputs in the Laplace domain.
Series and parallel
- Blocks in series multiply: G1 then G2 gives G1 * G2.
- Blocks in parallel add: G1 and G2 summed give G1 + G2.
- A feedback loop with forward G and feedback H reduces to G / (1 + G*H) for negative feedback.
The canonical loop
The most-used reduction is the single-loop formula. With forward transfer function G(s) and feedback path H(s), the closed-loop transfer function is G / (1 + GH). When H = 1 this is unity feedback, the case most textbooks develop first. The denominator 1 + GH set to zero gives the characteristic equation, whose roots are the closed-loop poles.
Block-diagram algebra
Complex diagrams are simplified by moving summing junctions and pickoff points past blocks using equivalence rules, then collapsing series, parallel, and feedback forms. Each move preserves the input-output relation while shrinking the diagram.
Mason's gain formula
For tangled multi-loop diagrams, Mason's gain formula computes the overall transfer function directly from the signal-flow graph without step-by-step reduction. It sums forward-path gains weighted by cofactors that account for every loop and their interactions.
Block diagrams are more than bookkeeping. They expose structure: where disturbances enter, where sensors sit, and which loops are nested. A plasma-control block diagram, for instance, separates the fast radial-position loop from the slower shape and current loops, making clear which dynamics each controller sees. Getting the diagram right is usually the first real step in any control design.