Control Engineering Basics
A negative-feedback system has a forward gain G = 100 and a feedback gain H = 0.1. When G halves to 50 because a component degrades, which of the following is closest to the fractional decrease in the closed-loop gain?
- AAbout 4.5%
- BAbout 8.3%
- C50%
Show answer and explanation
Answer: B. About 8.3%
The closed-loop gain is 100/(1 + 100 × 0.1) = 100/11 ≈ 9.09 before the degradation and 50/(1 + 50 × 0.1) = 50/6 ≈ 8.33 after it. The fractional decrease is 1 − 8.33/9.09 = 1/12 ≈ 8.3%, much smaller than the 50% change in G. Negative feedback reduces the effect of changes in the forward gain by a factor of about 1 + GH. However, dividing by this factor is an approximation for small changes; for a large change such as halving, the closed-loop gain must be calculated directly.
- A. This is 50% divided by 1 + GH = 11. This sensitivity formula holds only for small changes, and for halving the error is large.
- B. Correct. The gain falls from 9.09 to 8.33, exactly one twelfth.
- C. This is the change without feedback (open loop). Negative feedback makes the change smaller.