In a negative-feedback block diagram, the forward path has the transfer function G(s) and the feedback path has the transfer function H(s). Which of the following is the closed-loop transfer function Y(s)/R(s) from the setpoint R(s) to the output Y(s)?

Answer: C. G(s)/(1 + G(s)H(s))

The error is E(s) = R(s) − H(s)Y(s) and the output is Y(s) = G(s)E(s). Eliminating E(s) from the two equations gives Y(s) = G(s)R(s) − G(s)H(s)Y(s), and so Y(s)/R(s) = G(s)/(1 + G(s)H(s)). The term G(s)H(s) in the denominator is the transfer function that a signal sees going once around the loop, called the loop transfer function (open-loop transfer function). With unity feedback, H(s) = 1, it becomes G(s)/(1 + G(s)).

• A. A denominator of 1 − GH applies to positive feedback, where the fed-back signal is added.
• B. The GH in the numerator is wrong. The numerator is only G, the forward path from input to output.
• C. Correct. "Forward transfer function ÷ (1 + loop transfer function)" is a handy way to remember it.