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For the given block diagram, which of the following options is a correct replacement for moving a summing point ahead of a block?

Block diagram: input X into a block P, then a summing point adding Y (with plus or minus sign) to give output Z

  • A.P in each of the X and Y paths, then summed to give Z
  • B.X and Y summed, with Y first passing through 1/P, followed by block P to give Z
  • C.X and Y summed directly, followed by block P to give Z
  • D.X branched to block P giving output Y, and a second block P in the lower path producing Y

Answer: B

The original diagram has the block \(P\) before the summing point: \(Z = PX \pm Y\). To move the summing point ahead of (before) the block, the signal \(Y\) that is added after \(P\) must be divided by \(P\) so that it has the same effect once it goes through the block.


\[ Z = P\left(X \pm \frac{Y}{P}\right) = PX \pm Y \]


So the correct replacement is to put the block \(1/P\) in the path of \(Y\), then sum with \(X\), and then pass the sum through \(P\), which is figure (B). In figure (A) the second \(P\) multiplies \(Y\) by \(P\), giving \(PX \pm PY\), which is wrong. In (C) the output is \(P(X \pm Y) \ne PX \pm Y\). Figure (D) is the rule for moving a take-off point, not a summing point. The answer is option B.

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