How it works
Three basic reduction rules: series blocks G₁(s)·G₂(s) combine by multiplication; parallel blocks G₁(s) ± G₂(s) combine by addition or subtraction; a negative feedback loop G(s) in forward path and H(s) in feedback reduces to G(s)/(1+G(s)H(s)). To move a summing junction ahead of a block G(s), insert a 1/G(s) block on the branch being moved. To move a take-off point behind a block, insert a G(s) block on the new branch. These moves are always needed when summing junctions and take-off points are interleaved.
Key points to remember
The standard closed-loop transfer function is T(s) = G(s) / (1 + G(s)H(s)) for negative feedback. For unity feedback H(s) = 1, this simplifies to G/(1+G). Order of reduction matters: always eliminate inner loops before outer loops. Moving a summing junction before a block requires adding 1/G to the moved signal branch; moving a take-off point after a block requires adding G to the branch. Mason's gain formula is faster for complex diagrams but requires identifying all forward paths and loops. The characteristic equation is 1 + G(s)H(s) = 0, and its roots are the closed-loop poles.
Exam tip
The examiner always asks you to reduce a multi-loop block diagram to a single transfer function — tackle inner feedback loops first, then work outward, labelling intermediate results at each step to avoid sign errors.