Amplifiers and feedback

In short

An amplifier is a two-port circuit that uses DC power to produce an output signal that is a scaled copy of its input; feedback returns a fraction of the output to the input to control that scaling. Negative feedback trades raw gain for gain stability, wider bandwidth, lower distortion and input/output impedances that can be set by resistor ratios — the reason every op-amp, audio stage, sensor front-end and regulator is a feedback loop. This topic is one of the heaviest scorers in GATE EC (Analog Circuits), in university papers on Electronic Devices and Circuits, and in written tests at TI India, Analog Devices, NXP, Infineon, Qualcomm, BEL, ISRO and DRDO. Interviewers almost always ask you to identify a feedback topology from a schematic and say what happens to Rin, Rout and bandwidth. The mathematics is small — one equation and four variants — but it must be applied to a real circuit, not recited.

Explained

Amplifier models, gain and specifications

• Amplifier: a device whose output is Av·vi (or Ai·ii) with the extra power drawn from the DC supply; it is a controlled source, not an energy source.
• Modelled as a unilateral two-port: input resistance Ri, a dependent source of gain A, and output resistance Ro in series (Thevenin) or parallel (Norton).
• Four amplifier types, each with its own gain units:

• Loading: a source with resistance Rs and a load RL reduce the gain to Av·[Ri/(Ri+Rs)]·[RL/(RL+Ro)]. Terminal gain is never the open-circuit gain.
• Decibels: voltage/current gain in dB = 20 log₁₀|A|; power gain in dB = 10 log₁₀(Po/Pi). 0 dBm = 1 mW; cascaded gains add in dB.
• Other specifications examined: bandwidth, THD (total harmonic distortion, %), input-referred noise, linearity/dynamic range, output swing and efficiency. IEC 60268-3 (adopted in the Indian IS/IEC series) is the standard method of measuring audio amplifier gain, THD and noise; mains-operated audio/video apparatus is covered by IS 616 (from IEC 60065), with safety being migrated to IS/IEC 62368-1, and consumer electronics need BIS registration under the MeitY Compulsory Registration Scheme.

Frequency response, Miller effect and gain–bandwidth

• A real amplifier's gain is flat only in a midband; it falls at low frequency because of coupling and bypass capacitors (which act as high-pass elements) and at high frequency because of device and wiring capacitances (low-pass).
• Lower cut-off fL and upper cut-off fH are the −3 dB (0.707×) points; bandwidth BW = fH − fL ≈ fH when fH ≫ fL.
• Each capacitor contributes a pole; the dominant pole (largest time constant at the high end, highest fL at the low end) decides the cut-off.
• Bode plot: each simple pole gives −20 dB/decade slope and −90° asymptotic phase; each zero gives +20 dB/decade.
• Miller effect: a capacitance Cf bridging input and output of an inverting stage of gain −Av appears at the input as Cf(1+Av) and at the output as Cf(1+1/Av). This is why Cμ (BJT) or Cgd (MOSFET) dominates the high-frequency roll-off of a common-emitter/common-source stage.
• Gain–bandwidth product: for a single-pole amplifier, |A₀|·fH is a constant of the device. Trading gain for bandwidth is therefore free; increasing both is not.
• Unity-gain frequency fT (device) and rise time relation tr ≈ 0.35/fH for a single-pole response.

Feedback: the block diagram and the basic equation

• A feedback amplifier has four blocks: source, open-loop amplifier A (forward gain), feedback network β (sampling + feedback factor) and mixer/comparator at the input.
• Sampling is across the load (voltage sampling → shunt connection at output) or in series with it (current sampling → series connection at output). Mixing is in series (subtracting voltages) or in shunt (subtracting currents) at the input.
• Signal flow: xi = xs − xf, xf = β·xo, xo = A·xi ⇒ Af = A/(1 + Aβ).
• Loop gain T = Aβ (dimensionless, as A and β have reciprocal units); D = 1 + Aβ is the desensitivity factor or amount of feedback (also quoted as 20 log D dB).
• If Aβ ≫ 1, Af ≈ 1/β — the gain is set by the passive feedback network alone and is independent of the transistor. This is the whole point.
• Negative feedback: Aβ positive and real at midband, |1+Aβ| > 1, gain falls. Positive (regenerative) feedback: Aβ negative, gain rises; at Aβ = 1∠0° (Barkhausen condition) sustained oscillation begins, and Aβ > 1 with a non-linearity gives comparators, Schmitt triggers and multivibrators.
• Negative feedback can be identified by inspection: inject a positive step at the input and trace the sign round the loop; if the returned signal opposes the original, it is negative.

What negative feedback buys — and what it costs

• Gain stability (desensitivity): a fractional change in A becomes D times smaller in Af. Formally dAf/Af = (1/D)(dA/A), so S = (dAf/Af)/(dA/A) = 1/D.
• Bandwidth extension (single-pole A): fHf = fH(1 + A₀β) and fLf = fL/(1 + A₀β); gain falls by D and bandwidth rises by D, so A₀fH = Af·fHf — the gain–bandwidth product is unchanged.
• Distortion reduction: non-linear distortion generated *inside* the loop is divided by D (THDf ≈ THD/D). Distortion already present in the source is not reduced.
• Noise: noise injected inside the loop is reduced by D along with the gain, so the signal-to-noise ratio at the input stage is not improved; feedback does not fix a noisy first stage.
• Impedance control: Rin and Rout are multiplied or divided by D depending on topology (table below) — a 100 Ω output becomes milliohms, a 100 kΩ input becomes megohms.
• Costs: (i) gain is lost and must be paid for with extra stages; (ii) the loop can oscillate if the loop phase reaches −180° while |T| > 1; (iii) extra components and loading on the output; (iv) possible reduction of output swing and increase of input offset paths.
• Reversal to remember: whatever quantity is sampled is stabilised. Voltage sampling stabilises the output voltage (low Rout → good voltage source); current sampling stabilises the output current (high Rout → good current source).

The four feedback topologies and how to analyse them

• Named as (input mixing)–(output sampling), e.g. series–shunt = series mixing, shunt sampling.

• Memory rule: series mixing raises Rin, shunt mixing lowers Rin; voltage sampling lowers Rout, current sampling raises Rout — each by the factor D.
• Note that Rif and Rof are the resistances *looking into the amplifier including the feedback network*; the input resistance seen by the source may be modified further by a series source resistance or a shunt bias network.
• Standard analysis procedure (the exam method):
  1. Identify the feedback element(s) and classify sampling (across RL or in series with RL) and mixing (series = feedback into emitter/source or the second op-amp input; shunt = feedback into the same node as the input current).
  2. Draw the open-loop (A-circuit): kill the feedback but keep the loading. For shunt mixing, short the feedback network's input port to ground; for series mixing, open it. For voltage sampling, short the output port of the feedback network; for current sampling, open it.
  3. Compute A, Ri, Ro of this loaded open-loop circuit using the small-signal model.
  4. Compute β from the feedback network alone (β = xf/xo with the amplifier input open or shorted as appropriate) — usually a resistor ratio, e.g. β = R1/(R1+R2) or β = 1/Rf or β = RE.
  5. Form D = 1 + Aβ, then Af, Rif, Rof; finally convert Af back to the gain the question asks for (often vo/vs, which needs the source resistance included).
• Practical identification tips: a resistor from emitter to ground (unbypassed) = series–series; a resistor from collector/output back to base/input node = shunt–shunt; a potential divider from output to the inverting input = series–shunt; a divider sampling an emitter resistor and feeding the first base node = shunt–series.
• Feedback in real blocks: op-amp circuits (inverting = shunt–shunt, non-inverting = series–shunt), emitter follower (series–shunt, D ≈ 1+gmRE, gain just under 1), linear voltage regulators (the error amplifier samples the output voltage → very low output impedance), audio power amplifiers, PLL and AGC loops.

Stability, margins and compensation

• The closed-loop gain A/(1+Aβ) blows up when Aβ = −1, i.e. |T| = 1 with loop phase −180°. Stability is judged on the loop gain T(jf), not on Af.
• Gain margin: the amount (in dB) by which |T| is below 0 dB at the frequency where the loop phase is −180°. Phase margin: 180° minus the loop phase lag at the frequency where |T| = 1 dB-crossover. Designs target PM ≥ 45°–60° (PM 45° ⇒ ~1.3× peaking; PM 60° ⇒ nearly flat response).
• A single-pole amplifier can never oscillate (max 90° lag). Two poles are conditionally safe; three poles with high loop gain will oscillate unless compensated.
• Rate of closure on the Bode plot: if the |A| curve meets the 1/β line (20 log(1/β)) at a difference of slopes of 40 dB/decade, the circuit is marginal; 20 dB/decade is stable.
• Low closed-loop gain is the hard case — a unity-gain follower has the largest loop gain, so a device stable at Af = 100 may ring at Af = 1.
• Compensation methods: dominant-pole (lag) compensation — add a capacitor to push the first pole down so |T| crosses unity before the second pole; Miller compensation — a small capacitor across the high-gain stage (splits the poles, as in the 741's 30 pF); lead compensation — a zero (series RC) to cancel a pole and add phase; pole–zero cancellation; and reducing loop gain.
• Frequency compensation costs bandwidth and slew rate; the trade is deliberate.

Key points

• Amplifier - a controlled source: output = A × input, with the extra power from the DC supply.
• Gain in dB - 20 log₁₀ for voltage/current, 10 log₁₀ for power; cascade gains add in dB.
• Feedback equation - Af = A/(1 + Aβ); loop gain T = Aβ; desensitivity D = 1 + Aβ.
• Af ≈ 1/β when Aβ ≫ 1 - gain fixed by passive components, not by the device.
• Desensitivity - fractional gain change is reduced by D: S = 1/D.
• Bandwidth - fHf = D·fH, fLf = fL/D; gain–bandwidth product stays constant for a single-pole amplifier.
• Distortion - only distortion generated inside the loop is divided by D; source distortion is not.
• Noise - divided by D along with the gain, so input-stage SNR is unchanged.
• Series mixing - raises input resistance by D; shunt mixing lowers it by D.
• Voltage sampling - lowers output resistance by D; current sampling raises it by D.
• Sampled quantity is the stabilised quantity - voltage sampling makes a voltage source, current sampling a current source.
• Four topologies - series–shunt (voltage amp), shunt–shunt (transresistance), series–series (transconductance), shunt–series (current amp).
• Emitter/source degeneration - series–series feedback; Av = −gmRC/(1+gmRE), Rin rises by (1+gmRE).
• Miller effect - bridging capacitance appears as Cf(1+|Av|) at the input; main cause of high-frequency roll-off in CE/CS stages.
• Barkhausen criterion - Aβ = 1 at 0° phase gives sustained oscillation; |Aβ| = 1 with −180° loop phase is the instability boundary for negative feedback.
• Phase margin / gain margin - stability measures on the loop gain; target PM 45°–60°.
• Compensation - dominant-pole and Miller compensation buy stability by sacrificing bandwidth.
• Open-loop circuit rules - shunt port shorted, series port opened, while keeping all loading.

Formulas and examples

Example 1 — desensitivity. An amplifier has A = 1,000 and feedback β = 0.01.

• T = Aβ = 10, D = 11 → Af = 1,000/11 = 90.91.
• Device spread drops A by 10% to 900: D = 1 + 9 = 10 → Af = 90.00, a fall of (90.91−90.00)/90.91 = 1.0%.
• Check by formula: dAf/Af = 10%/11 = 0.91% (the small-signal approximation; the exact answer is 1.0%). A 10% device change becomes a 1% circuit change.

Example 2 — bandwidth, distortion and GBW. An amplifier has A₀ = 1,000, fL = 20 Hz, fH = 20 kHz and 5% THD. Apply β = 0.024.

• D = 1 + 1,000 × 0.024 = 25 → Af = 1,000/25 = 40.
• fHf = 20 kHz × 25 = 500 kHz; fLf = 20/25 = 0.8 Hz.
• GBW check: 1,000 × 20 kHz = 20 MHz; 40 × 500 kHz = 20 MHz — unchanged.
• THD = 5%/25 = 0.2%. To keep the same output, the input must now be 25× larger.

Example 3 — series–shunt op-amp. A = 10⁴ V/V, Ri = 100 kΩ, Ro = 100 Ω, non-inverting connection with R1 = 1 kΩ, R2 = 9 kΩ.

• β = 1,000/(1,000+9,000) = 0.1 → T = 1,000, D = 1,001.
• Af = 10⁴/1,001 = 9.990 (ideal 1/β = 10, so the gain error is −0.1%).
• Rif = 100 kΩ × 1,001 = 100.1 MΩ; Rof = 100/1,001 = 0.0999 Ω ≈ 100 mΩ.

Example 4 — series–series (emitter degeneration). BJT at IC = 1 mA, β₀ = 100, RC = 1 kΩ, VT = 25 mV, ro neglected.

• gm = 1 mA/25 mV = 40 mA/V, re = 25 Ω, rπ = 100 × 25 = 2.5 kΩ.
• RE = 0: Av = −gmRC = −40; Rin = (1+β₀)re = 101 × 25 = 2.525 kΩ.
• RE = 100 Ω: loop gain gmRE = 4, D = 5 → Av = −1,000/(25+100) = −8 (= −40/5 ✓), Gm = 40/5 = 8 mA/V, Rin = 101 × 125 = 12.625 kΩ (= 2.525 kΩ × 5 ✓).
• Current sampling also raises the resistance seen at the collector, which is why degeneration is used in current sources.

Example 5 — Miller effect sets the bandwidth. A CS stage with Av = −20, Cgs = 20 pF, Cgd = 5 pF, driven from Rsig = 10 kΩ.

• Cin = 20 + 5(1+20) = 20 + 105 = 125 pF.
• fH = 1/(2π × 10 × 10³ × 125 × 10⁻¹²) = 1/(7.854 × 10⁻⁶) = 127.3 kHz.
• Rise time tr ≈ 0.35/127.3 kHz = 2.75 µs. Without the Miller term (Cin = 25 pF) fH would be 637 kHz — a 5× loss caused by one 5 pF capacitor.

Example 6 — phase margin. An amplifier has A₀ = 10⁵, poles at f1 = 10 Hz and f2 = 1 MHz.

• Unity-gain follower (β = 1): T₀ = 10⁵, |T| = 1 at 10⁵ × 10 Hz = 1 MHz, exactly f2. Loop phase = −90° − 45° = −135° → PM = 45° (stable, with slight peaking).
• Af = 100 (β = 0.01): T₀ = 1,000, crossover at 10 kHz, phase ≈ −90° − 0.6° → PM ≈ 89° (very stable).
• Conclusion: the low-gain (high-feedback) connection is the critical one for stability — hence "unity-gain stable" in op-amp datasheets.

Where it is used

• Analog IC design (TI India, Analog Devices, NXP, Infineon, Qualcomm, Microchip, STMicroelectronics — Bengaluru/Hyderabad/Noida): op-amp and LDO loop design, feedback factor and compensation, phase-margin sign-off in SPICE/Cadence Spectre.
• Instrumentation and sensors: transimpedance (shunt–shunt) amplifiers for photodiodes and pulse oximeters, strain-gauge and thermocouple front-ends, 4–20 mA transmitters using series–series feedback as a V-to-I converter.
• Power and control: linear regulators, switch-mode converter error amplifiers, motor-current loops, UPS and inverter feedback — stability margin is a design deliverable.
• Communication and defence electronics (BEL, ISRO, DRDO labs, HAL): LNAs and IF amplifiers with resistive/inductive degeneration for gain flatness and input matching, AGC loops in receivers.
• Audio and consumer products: amplifier THD and damping factor set by loop gain; measured per IEC 60268-3 and sold under BIS registration.
• Job roles: analog/mixed-signal design engineer, characterisation and validation engineer, applications engineer, hardware design engineer (test questions in these interviews are mostly identify-the-topology and bandwidth problems).

Interview questions

• Q: What does negative feedback actually trade away? A: Gain. Everything it improves — gain accuracy, bandwidth, distortion, impedance control — is bought with a factor D = 1+Aβ of gain reduction.
• Q: Why does a feedback amplifier's gain become almost independent of the transistor? A: When Aβ ≫ 1, Af ≈ 1/β, and β is set by a passive network (usually a resistor ratio) that is far more stable than device gm or β₀.
• Q: Does negative feedback improve the signal-to-noise ratio? A: No. Noise generated inside the loop is reduced by the same factor D as the signal, so SNR is unchanged; a quieter input stage is the only fix.
• Q: A resistor connects collector to base. What topology is it and what happens to Rin? A: Shunt–shunt (voltage-sampled, shunt-mixed) feedback; both Rin and Rout fall by D — and the input looks like Rf/(1+A), the Miller-reduced value.
• Q: How do you get a good current source from an amplifier? A: Use current sampling (series at the output), e.g. an unbypassed emitter resistor: Rout is multiplied by D, stabilising the output current.
• Q: Why is the gain–bandwidth product constant? A: For a single-pole open-loop response, feedback divides the midband gain by D and multiplies the pole frequency by D, so their product is unaffected.
• Q: Which closed-loop configuration is hardest to stabilise and why? A: The lowest gain (unity-gain follower), because β = 1 gives the maximum loop gain, so |T| crosses unity at a higher frequency where extra poles have already added phase lag.
• Q: What is phase margin and what value do you design for? A: The extra phase lag the loop could tolerate at the frequency where |Aβ| = 1; typically 45°–60°, with 60° giving a nearly flat, non-peaking response.
• Q: What is the Miller effect and how do you defeat it? A: Feedback capacitance Cf appears at the input as Cf(1+|Av|); it is reduced by a cascode or common-base/common-gate stage, lower source resistance, or by reducing the stage gain.
• Q: Does feedback reduce distortion in the input signal? A: No — only distortion generated within the loop. Anything already present at the input is amplified faithfully.
• Q: How do you tell negative from positive feedback by inspection? A: Apply a positive-going test signal at the input and follow the sign through the amplifier and feedback path; if the returned signal subtracts from the input, it is negative feedback.
• Q: Emitter degeneration reduces gain — why use it? A: It stabilises gain against gm/temperature variation, linearises the stage (lower distortion), raises input resistance and output resistance, and widens bandwidth.

Common mistakes

• Writing Af = A/(1−Aβ) for negative feedback, or forgetting the "1 +" and quoting Af = 1/β when Aβ is only a few.
• Mixing up the units of A and β in shunt–shunt or series–series problems — if Aβ is not dimensionless, the topology has been misidentified.
• Applying fHf = D·fH to an amplifier with two or more comparable poles; the simple result holds only for a single-pole response.
• Omitting loading when building the open-loop A-circuit (the feedback resistors load both input and output), then wondering why SPICE disagrees.
• Reversing the impedance rules: series mixing raises Rin, voltage sampling lowers Rout — not the other way round.
• Claiming feedback reduces noise and improves SNR, or that it reduces distortion already present in the source.
• Ignoring the Miller multiplication of Cμ/Cgd and over-estimating the bandwidth of a common-emitter/common-source stage; and judging stability from the closed-loop gain plot instead of the loop gain.

Related topics

BJT and MOSFET · FET amplifiers and biasing · Multistage and differential amplifiers · Operational amplifiers · Oscillators and power supplies · Network analysis: transients, resonance and two-port networks