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Systems8 min read
Negative Feedback
See how feeding part of the output back can trade raw gain for stability, bandwidth and accuracy.
Mental model
A thermostat is a feedback system: it compares the actual temperature with the target and corrects the error instead of heating blindly.
The feedback path measures the output and subtracts it from the input to reduce error.UNDERSTAND THE BEHAVIOR
01
Feedback forms a loop: the forward system creates an output, a fraction β is measured, and that value is compared with the input. Negative feedback uses the error to correct the output.
02
When loop gain Aβ is large, closed-loop gain approaches 1/β. This makes the result depend mostly on accurate passive components instead of uncertain transistor or op-amp gain.
03
At high frequency, delays add phase shift. If the returned signal arrives with enough phase rotation, intended negative feedback can behave like positive feedback and create ringing or oscillation.
Why it matters
Feedback is the central idea behind amplifiers, voltage regulators, motor control and automatic control systems.
Core ideas
Negative feedback subtracts a fraction of output from the input.
It makes closed-loop gain less sensitive to component variation.
It often increases bandwidth and reduces distortion, but poor phase margin can cause oscillation.
Key equationsClosed-loop gain: Af = A / (1 + Aβ)Loop gain: T = Aβ
EXAMPLE
Worked example — gain stabilization
1Open-loop gain A = 100,000 and β = 0.1.
2Af = 100,000 / (1 + 10,000).
3Af ≈ 9.999, almost exactly 1/β = 10.
4Even a large change in A has little effect on Af.
Common mistake
More feedback is not automatically better. Stability must be checked with loop gain and phase margin.
RememberNegative feedback subtracts a fraction of output from the input.
DEEPER EXPLANATION
Why does feedback make gain less dependent on the amplifier?
Imagine a system with open-loop gain A. Feed a fraction β of the output back to its input and subtract it from the command. The amplifier acts on the remaining error, so as the output rises the returned signal reduces the drive that made it rise. For a simple stable loop, the closed-loop gain is Af = A/(1 + Aβ).
If Aβ is much larger than one, the denominator is approximately Aβ, and Af approaches 1/β. For A = 1000 and β = 0.1, Af = 1000/101 ≈ 9.901. Doubling A to 2000 gives 2000/201 ≈ 9.950, a modest change despite doubling the raw gain. The accuracy of β now matters: resistor tolerance in a feedback network affects the final gain.
The equation is a low-frequency starting point. Real amplifiers have frequency-dependent gain and phase, which changes loop gain Aβ. At some frequencies, enough phase shift can turn an intended correction into reinforcement. Check stability, phase margin and the device's recommended feedback configuration before applying the result at high frequency.
Walk through another example
A system has A = 100 and β = 0.1. Its loop gain is 10 and Af = 100/(1 + 10) ≈ 9.09.
Increase A to 1000 while keeping β fixed: loop gain is 100 and Af ≈ 9.901.
Both results approach 1/β = 10, but neither equals it exactly; a larger loop gain improves the approximation.
Check your intuition
If β increases, the strong-feedback approximation 1/β decreases: returning a larger output fraction requires less output to balance the input.
Where this model can mislead you
Do not conclude that more feedback always guarantees a stable circuit. Delay and phase shift can cause oscillation.
TEST YOUR UNDERSTANDING
Three quick checks
Choose one answer per question, then check your work. Each answer includes an explanation.
Practice and check your understanding
Try each question before opening the worked answer.
1. For A = 1000 and β = 0.1, calculate the closed-loop gain.
Show answer and reasoning
Af = A/(1 + Aβ) = 1000/(1 + 100) = 9.901. It is close to 1/β = 10, but finite open-loop gain produces a small error.
2. Does adding negative feedback guarantee stability at every frequency?
Show answer and reasoning
No. Frequency-dependent phase shift can make the feedback reinforce a disturbance. Loop gain and phase margin must be checked; the low-frequency gain equation alone is insufficient.