Feedback Systems
Learning Objectives
- Define feedback and distinguish between positive and negative feedback by their effect on a system
- Derive and apply the closed-loop transfer function H(s) = G(s) / [1 + βG(s)]
- Explain the benefits negative feedback provides: gain stability, reduced distortion, bandwidth extension, impedance modification
- State the stability criteria (phase margin, gain margin) and explain what happens when they're violated
- Compare series and shunt feedback topologies and their effect on input/output impedance
- Apply feedback concepts to real op-amp and voltage regulator circuits
Quick Answer
A feedback system takes a portion of a circuit's output and routes it back to combine with the input, shaping how the overall system behaves. Negative feedback (output fed back out of phase with the input) is the workhorse of analog design: it sacrifices some raw gain in exchange for stability, linearity, reduced distortion, and predictable behavior set mostly by external components rather than the amplifying device itself. Positive feedback (output fed back in phase) does the opposite — it reinforces changes, which is unstable for amplifiers but exactly what's needed to sustain oscillation. Nearly every op-amp circuit, voltage regulator, and stable amplifier relies on negative feedback as its core operating principle.
What a Feedback System Actually Is
Every feedback system has two paths: the forward path (gain G(s)) that carries the input to the output, and the feedback path (feedback factor β) that samples the output and returns a fraction of it to be combined with the input. Whether that combination adds to or subtracts from the input determines whether the feedback is positive or negative.
Negative feedback: the fed-back signal is subtracted from the input. If the output rises too much, more signal gets fed back and subtracted, pulling the output back down — this is a self-correcting, stabilizing mechanism. Op-amps, voltage regulators, and most linear amplifier stages use negative feedback.
Positive feedback: the fed-back signal adds to the input. If the output rises, more gets fed back and added, pushing the output up further still — this is a self-reinforcing, potentially runaway mechanism. Used deliberately in oscillators (to sustain a stable waveform) and comparators with hysteresis (Schmitt triggers), but destabilizing if it appears unintentionally in an amplifier design.
The Closed-Loop Gain Formula
For a system with forward gain G(s) and feedback factor β, the overall (closed-loop) transfer function is:
H(s) = G(s) / [1 + β G(s)]
This single formula explains almost everything useful about negative feedback. If the "loop gain" βG(s) is much larger than 1 (a common design goal), the formula simplifies to:
H(s) ≈ 1 / β
This is a remarkable result: the overall gain becomes almost entirely determined by the feedback network (β) — typically a pair of precision resistors — rather than by G(s), the amplifier's own internal, temperature-sensitive, unit-to-unit varying open-loop gain. This is exactly what happens in the op-amp inverting and non-inverting amplifier configurations from earlier chapters: massive open-loop gain (G) is traded away so the resistor ratio (which sets β) determines the precise closed-loop gain.
Worked example: An op-amp has open-loop gain G = 200,000. A feedback network sets β = 0.01 (1%). Loop gain βG = 2,000, comfortably ≫ 1. Closed-loop gain H ≈ 1/β = 100. Even if the op-amp's open-loop gain varies wildly from unit to unit (say, from 100,000 to 500,000), the closed-loop gain stays almost exactly 100, because it's set by the resistor-defined β, not by G.
What Negative Feedback Buys You
- Gain stability: As shown above, closed-loop gain depends mainly on the feedback network, not the (unreliable) open-loop gain of the active device.
- Reduced distortion: Any nonlinearity or distortion introduced by the amplifier itself gets divided down by the same (1 + βG) factor as the gain, so feedback significantly cleans up the output waveform.
- Bandwidth extension: While DC gain is reduced by the factor (1 + βG), the usable bandwidth is extended by roughly the same factor — this is the origin of the "gain-bandwidth product" being roughly constant for an op-amp.
- Impedance modification: Negative feedback that samples the output voltage and feeds back in series with the input increases input impedance and decreases output impedance — ideal for a voltage amplifier. Different combinations of series/shunt sampling and mixing produce different impedance-transformation effects, useful for current amplifiers, transconductance amplifiers, and transresistance amplifiers.
Feedback Topologies
| Topology | Sampled Quantity | Mixed With Input | Effect on Input Impedance | Effect on Output Impedance | Typical Use |
|---|---|---|---|---|---|
| Series-voltage (series-shunt) | Output voltage | In series | Increases | Decreases | Non-inverting op-amp amplifier |
| Shunt-voltage (shunt-shunt) | Output voltage | In parallel (shunt) | Decreases | Decreases | Transresistance amplifier |
| Series-current (series-series) | Output current | In series | Increases | Increases | Transconductance amplifier |
| Shunt-current (shunt-series) | Output current | In parallel (shunt) | Decreases | Increases | Current amplifier |
Real-world example: A voltage regulator IC (like the LM7805) uses negative feedback: it continuously samples its own output voltage, compares it against an internal precision reference, and adjusts an internal pass transistor to correct any deviation — holding the output at a stable 5 V despite changes in input voltage or load current. This is series-voltage negative feedback in action, prioritizing a stable, low-output-impedance voltage source.
Stability: When Feedback Goes Wrong
Negative feedback is only stabilizing if the loop phase shift stays well below 180° at the frequency where loop gain drops to 1. If internal delays or additional filtering stages (parasitic capacitances, multiple amplifier stages) add up to 180° of phase shift while loop gain is still ≥ 1, negative feedback effectively becomes positive feedback at that frequency — the system can begin to oscillate uncontrollably, exactly the failure mode engineers must design around.
Two standard stability margins quantify how much safety cushion a design has:
- Phase margin: how far the loop phase is from 180° at the frequency where loop gain = 1 (0 dB). A common target is phase margin > 45°.
- Gain margin: how far the loop gain is below 1 (0 dB) at the frequency where loop phase = 180°. A common target is gain margin > 6 dB.
Violating these margins doesn't necessarily cause outright oscillation immediately, but it produces excessive ringing, overshoot, and poor transient response — an amplifier that "rings" when driven with a fast step input is a classic symptom of insufficient phase margin.
Why It Matters
Feedback is the single most important concept separating "a circuit that happens to amplify" from "a circuit engineers can actually rely on." Without negative feedback, every amplifier's gain, linearity, and bandwidth would be at the mercy of uncontrolled, temperature- and batch-dependent transistor parameters. With well-designed negative feedback, engineers trade away some raw gain for a circuit whose behavior is set almost entirely by a couple of precision resistors — reliable, repeatable, and predictable across temperature, supply voltage, and manufacturing variation.
Common Mistakes
Misconception 1: "More negative feedback (higher β) is always better because it makes the circuit more stable." Why it's wrong: Increasing β (feedback factor) does increase loop gain, which does reduce distortion and stabilize DC gain — but it also increases the risk of instability if it pushes the loop gain-crossover frequency into a region where phase shift approaches 180°, and it always reduces the closed-loop gain (since H ≈ 1/β). Correct understanding: Feedback amount is a design tradeoff between gain, bandwidth, distortion reduction, and stability margin — "more" isn't automatically better; the feedback network must be designed alongside the amplifier's frequency response to guarantee adequate phase margin.
Misconception 2: "Positive feedback is always bad and should be avoided in circuit design." Why it's wrong: Positive feedback is only undesirable when it's unintentional in a circuit meant to be a stable, linear amplifier. It is deliberately and usefully applied in oscillators (to sustain oscillation) and in comparators with hysteresis (Schmitt triggers, to create clean, noise-immune switching). Correct understanding: Whether feedback is "good" or "bad" depends entirely on the circuit's intended function — negative feedback for stable linear amplification, positive feedback for oscillators and hysteresis-based switching circuits.
Misconception 3: "The closed-loop gain formula H = G/(1+βG) means the amplifier's open-loop gain G doesn't matter at all once feedback is added." Why it's wrong: G still matters — it determines the loop gain βG, and a large loop gain is exactly what's needed for the approximation H ≈ 1/β to hold well. If G is too small (or drops off at high frequency, as all real amplifiers eventually do), the approximation breaks down and the closed-loop gain deviates noticeably from 1/β. Correct understanding: G(s) still needs to be large enough (across the frequencies of interest) for the feedback-dominated behavior to hold; the formula shows G's effect is minimized when loop gain is large, not that G becomes irrelevant.
Comparison and Connections
| Feature | Negative Feedback | Positive Feedback |
|---|---|---|
| Fed-back signal phase | Out of phase with input (subtracts) | In phase with input (adds) |
| Effect on gain | Reduces and stabilizes gain | Increases gain, can lead to runaway |
| Effect on distortion | Reduces distortion | Can increase distortion if uncontrolled |
| Typical application | Amplifiers, voltage regulators, control systems | Oscillators, Schmitt trigger comparators |
| Stability | Generally stabilizing (if phase margin adequate) | Generally destabilizing unless deliberately controlled |
| Stability Margin | What It Measures | Typical Healthy Target |
|---|---|---|
| Phase margin | Distance from 180° phase shift at 0 dB loop gain | > 45° |
| Gain margin | Distance below 0 dB loop gain at 180° phase shift | > 6 dB |
Practice Questions
Recall 1: Write the closed-loop transfer function formula for a negative feedback system in terms of G(s) and β. Answer guidance: H(s) = G(s) / [1 + βG(s)].
Recall 2: State the difference between positive and negative feedback in terms of signal phase. Answer guidance: Negative feedback: the fed-back signal is out of phase with (subtracted from) the input. Positive feedback: the fed-back signal is in phase with (added to) the input.
Understanding 1: Explain why closed-loop gain H ≈ 1/β when loop gain βG is much greater than 1, and why this is useful in practice. Answer guidance: When βG ≫ 1, the "+1" term in H = G/(1+βG) becomes negligible compared to βG, so H ≈ G/(βG) = 1/β. This is useful because β is set by stable, precise external components (resistors), so the overall gain becomes largely independent of the amplifier's own variable, temperature-sensitive open-loop gain G.
Understanding 2: Why does negative feedback reduce distortion introduced by the amplifier itself? Answer guidance: Any distortion appears as an unwanted deviation in the output, which the feedback loop "sees" as an error relative to the desired output and corrects for by adjusting the effective drive to the amplifier — the same (1+βG) factor that suppresses gain variation also divides down distortion appearing at the output.
Application 1: A negative feedback amplifier has open-loop gain G = 50,000 and a feedback network with β = 0.02. Calculate the closed-loop gain using the exact formula and compare it to the 1/β approximation. Answer guidance: Loop gain βG = 1,000. H = G/(1+βG) = 50,000/1,001 ≈ 49.95. The approximation 1/β = 1/0.02 = 50. The two values are very close (within 0.1%), confirming the approximation is valid when loop gain is large.
Application 2: A voltage regulator's output droops when a heavy load is connected. Explain, using feedback concepts, how adding negative feedback (sensing output voltage and adjusting a pass transistor) fixes this. Answer guidance: The feedback loop continuously compares the sampled output voltage to a reference; any droop caused by increased load current is detected as an error signal, which the loop uses to increase the pass transistor's conduction and restore the output voltage — effectively lowering the regulator's output impedance and holding the voltage steady despite load changes.
Analysis 1: An amplifier with substantial internal phase shift at high frequencies is wrapped in negative feedback with high loop gain, and the output begins to ring and eventually oscillate when driven with a fast step input. Diagnose this using the concepts of phase margin and the Barkhausen criterion. Answer guidance: At some high frequency, the amplifier's internal phase shift approaches 180°, converting the intended negative feedback into effective positive feedback at that frequency. If loop gain is still ≥ 1 (0 dB) at that same frequency (i.e., phase margin is too low or negative), the Barkhausen criterion for oscillation is satisfied and the circuit rings or oscillates instead of behaving as a stable amplifier — the fix is to reduce loop gain at high frequency (compensation) to ensure gain drops below 1 before phase shift reaches 180°.
Analysis 2: Compare series-voltage feedback and shunt-current feedback in terms of their effect on input and output impedance, and explain which would be preferred for building an ideal voltage amplifier versus an ideal current amplifier. Answer guidance: Series-voltage feedback increases input impedance and decreases output impedance — exactly the impedance profile wanted for an ideal voltage amplifier (high input impedance so it doesn't load the source, low output impedance so it can drive a load without voltage droop). Shunt-current feedback decreases input impedance and increases output impedance, matching the profile wanted for an ideal current amplifier (low input impedance to accept a current-source-like input, high output impedance to behave like a current source at the output).
FAQ
Q1: Why would an engineer deliberately reduce an amplifier's gain by adding feedback, when more gain sounds better? A: Raw open-loop gain in real devices is unreliable — it varies with temperature, supply voltage, and manufacturing tolerance, and is usually nonlinear. Trading away some of that raw gain via negative feedback buys precise, stable, linear, and predictable behavior instead, which is almost always more valuable than maximum raw gain.
Q2: Is it possible for feedback to be neither purely positive nor purely negative? A: At a single frequency, the sign is fixed by the loop's total phase shift, but that phase shift changes with frequency in every real circuit. This is precisely why a feedback loop designed as negative feedback at low/mid frequencies can turn into positive feedback at some higher frequency where internal delays add up to 180° — the root cause of feedback-induced oscillation if not managed with compensation.
Q3: What does "loop gain" mean, and why is it different from closed-loop gain? A: Loop gain (βG) is the gain around the entire feedback loop, and it determines how "strong" the feedback correction is. Closed-loop gain (H) is the overall gain from input to output of the assembled system — closed-loop gain is typically much smaller than open-loop gain G, while loop gain βG is what determines how close closed-loop gain gets to the ideal 1/β value.
Q4: How is feedback related to oscillators, which seem to be the "opposite" of a stable amplifier? A: Oscillators are feedback systems too — but designed with positive feedback and a loop gain/phase combination that satisfies the Barkhausen criterion (loop gain = 1, phase shift = 0°/360°) at one specific frequency, deliberately sustaining oscillation rather than suppressing deviations.
Q5: What real component sets β in a typical op-amp negative feedback amplifier? A: A resistor divider — specifically the feedback resistor(s) connecting the output back to the inverting input, as seen in the inverting and non-inverting amplifier configurations. Because resistors are cheap, stable, and precise, β (and therefore the closed-loop gain) can be set very accurately.
Quick Revision
- Feedback systems combine a sampled fraction of the output (via feedback factor β) with the input.
- Negative feedback: fed-back signal subtracts from input — stabilizing, reduces distortion, sets gain via 1/β.
- Positive feedback: fed-back signal adds to input — reinforcing, used deliberately in oscillators and Schmitt triggers.
- Closed-loop transfer function: H(s) = G(s)/(1+βG(s)); when βG ≫ 1, H ≈ 1/β.
- Negative feedback benefits: gain stability, reduced distortion, extended bandwidth, tailored input/output impedance.
- Series-voltage feedback → high input impedance, low output impedance (ideal voltage amplifier profile).
- Shunt-current feedback → low input impedance, high output impedance (ideal current amplifier profile).
- Stability requires adequate phase margin (>45°) and gain margin (>6 dB) to avoid unwanted oscillation/ringing.
- If loop phase shift reaches 180° while loop gain is still ≥1, negative feedback effectively becomes positive feedback (Barkhausen condition met unintentionally).
- Voltage regulators use series-voltage negative feedback to hold output voltage constant despite load changes.
Related Topics
Prerequisites: Operational Amplifiers, Filters and Amplifiers
Related: Oscillators, Noise and Interference
Next: Oscillators, Analog Measurement Techniques