Negative Feedback, Loop Gain & Amplifier Stability

Feedback samples output, transforms it, and compares it with the input. If the returned signal opposes the error over the relevant frequencies, it is negative feedback. The loop can stabilize gain, reduce distortion, widen bandwidth, and reshape impedance — but those benefits exist only while the loop remains stable.

Closed-loop algebra #

For forward gain $A(s)$ and feedback factor $\beta(s)$,

$$y=A(x-\beta y), \qquad \frac{y}{x}=\frac{A}{1+A\beta}.$$

The product $T=A\beta$ is loop gain. When $|T|\gg1$, closed-loop gain is approximately $1/\beta$ and is insensitive to $A$. The sensitivity to fractional open-loop-gain change is

$$S_A=\frac{1}{1+T}.$$

Thus a loop gain of 99 reduces a 10% open-loop variation to about 0.1% in closed-loop gain — at that frequency and operating point.

flowchart LR
  X["input x"] --> SUM(("Σ"))
  SUM --> A["forward path A(s)"]
  A --> Y["output y"]
  Y --> B["feedback β(s)"]
  B -->|"subtract"| SUM

What the connection topology changes #

Sampling output voltage or current and mixing feedback in series or shunt determines impedance transformation. Series mixing raises input impedance; shunt mixing lowers it. Voltage sampling lowers output impedance; current sampling raises it. This gives four classical feedback topologies, but loop stability still depends on the complete return ratio, including source and load impedances.

Stability is a frequency-domain condition #

Each pole adds phase lag. If total loop phase reaches $-180°$ while loop magnitude is at least one, the returned signal reinforces the error. Phase margin is the distance from $-180°$ at the unity-loop-gain frequency; gain margin is the attenuation remaining when phase reaches $-180°$.

A single dominant pole lets loop magnitude fall at roughly 20 dB/decade through crossover and is usually well behaved. Additional poles near crossover reduce phase margin, causing overshoot, ringing, or oscillation. Capacitive loads, sensor capacitance, transistor Miller capacitance, PCB parasitics, and the measurement probe itself can add those poles.

Feedback tradeoffs #

Negative feedback can reduce in-band nonlinear error approximately by $1+T$, but cannot fix output clipping, slew-rate limiting, input common-mode violation, current limiting, or recovery from saturation. It redistributes gain and bandwidth; for a dominant-pole voltage-feedback amplifier, the gain-bandwidth product is approximately constant.

Compensation deliberately shapes loop gain: a dominant pole, lead/lag network, feedback capacitor, isolation resistor, or snubber moves crossover and phase. The correct method is to identify the actual loop, include source/load/parasitic impedances, plot or measure loop gain, then verify step response and worst-case loading.

Diagnostic signatures #

Monotonic but slow settling suggests ample phase margin and limited bandwidth. Overshoot and decaying ringing indicate low positive margin. Sustained oscillation means the loop has crossed into instability. Bursts only at large amplitude may be slew recovery, output current limiting, or supply interaction rather than small-signal instability. Diagnose the regime before changing compensation.