Negative feedback
Treat negative feedback as a physical comparison process before invoking op-amp shortcuts. Identify the error signal, returned output fraction and net loop sign, then analyze canonical non-inverting and inverting op-amp circuits with real resistor networks. Derive their ideal closed-loop gains only after checking the conditions that justify the virtual-short approximation, and show how feedback trades raw gain for accuracy, linearity and controlled impedances without removing saturation or stability limits.
- Identify source, error signal, forward path, output and feedback path in a negative-feedback system without relying on a memorized block formula.
- Determine the net feedback sign by perturbing the output and observing whether the returned signal reduces or reinforces the error.
- Draw and analyze the canonical non-inverting op-amp amplifier with a real R1–R2 feedback divider.
- Derive the ideal non-inverting gain 1+R2/R1 from the divider relation only after the op-amp model gate is satisfied.
- Draw and analyze the canonical inverting amplifier and derive vo/vs=−R2/R1 from KCL at the summing node.
- Define noise gain for the inverting configuration and distinguish it from signed signal gain.
- Explain why negative feedback reduces sensitivity to uncertain open-loop gain and nonlinear transfer details without eliminating device limits.
- Explain qualitatively why voltage-series feedback raises input resistance and lowers output resistance in the non-inverting voltage-amplifier case.
- Recognize failure modes: wrong loop sign, open feedback path, output saturation, invalid common-mode range, current limiting and instability.
- Use a solve-check workflow: topology/sign → operating limits → approximation → result → consistency checks.
Negative feedback is not the rule v+=v−. It is a physical loop connection in which a returned portion of the output acts to reduce the error that produced the output. The virtual-short approximation is one consequence of that loop when gain, operating range and stability conditions are satisfied.
03.1Determine the feedback sign by perturbing the output
For an op-amp define the differential error e=v+−v−. Hold the independent source fixed and imagine a small positive perturbation of vo. Follow that perturbation through the return path. If it changes the inputs so that the forward amplifier commands a change opposing the original perturbation, the net feedback is negative around that operating point.
A wire connected to the terminal marked “−” is not, by itself, proof of negative feedback. Additional inversions can exist elsewhere in the loop. The physically meaningful test is the net return effect on the error.
03.2Non-inverting amplifier: the feedback signal is a real divider voltage
Figure 3.1 — Non-inverting amplifier
The one question is: how does R1–R2 return a real fraction of vo to the inverting input?
The signal enters the + input. The feedback node vf is the physical node shared by R1, R2 and the − input. If vo rises, vf rises, reducing vd=v+−v− and commanding an opposing correction.
Neglecting op-amp input current, R2 and R1 form an unloaded divider from vo to ground. The returned voltage is therefore
The source imposes v+=vs and the loop returns vf to v−. If vo rises, vf rises; therefore e=vs−vf falls and the amplifier commands vo downward. This is a negative loop.
Only after checking that the loop is negative, stable, unsaturated, inside input/output ranges and operating with sufficiently large loop gain may we use v−≈v+=vs. Then vf≈vs and
The resistor ratio does not create gain by itself. The op-amp supplies the output power and voltage; the divider tells the loop what output is required to make the returned voltage approximately equal to the source.
03.3Finite open-loop gain explains why the equality is approximate
With a finite low-frequency forward gain A0, the non-inverting circuit obeys
Thus the differential error is nonzero: a small error must remain to generate the output. As A0β becomes large, the result approaches 1/β. Chapter 04 generalizes this result with loop gain, sensitivity and exact closed-loop error.
03.4Inverting amplifier: KCL at the summing node
Figure 3.2 — Inverting amplifier
The one question is: how does KCL at the real summing node impose the R2/R1 ratio?
The + input is grounded. The node vΣ is the physical node shared by R1, R2 and the − input. Under the validated ideal approximation its voltage is near zero, but it is not physically connected to ground.
The non-inverting input is grounded. After the same model gate has been satisfied, vΣ=v−≈0 and the op-amp input current is negligible. KCL at the physical summing node gives
Using vΣ≈0,
The minus sign is a real polarity inversion. The node is called a virtual ground only because its voltage is approximately ground potential; current still flows through R1 and R2 and must satisfy KCL there.
03.5Signal gain and noise gain are different quantities
The inverting signal path has the signed gain −R2/R1. An input-referred error that acts non-invertingly, such as a simple offset-voltage model, sees the closed-loop factor
This noise gain is positive and is not the same as the signal gain. It later becomes important for bandwidth and stability because the loop “sees” the feedback network through this closed-loop noise-gain requirement.
03.6What negative feedback buys — and what it does not
Negative feedback trades the very large but uncertain open-loop gain for a smaller gain set mainly by the external network. While sufficient loop gain remains, changes in device gain and moderate nonlinear errors are corrected by the loop, improving gain accuracy and linearity.
Feedback cannot create unavailable output swing, output current or input common-mode range. Once an internal or output stage saturates, the forward path cannot supply the correction demanded by the loop and the differential error grows.
03.7Input and output impedances depend on feedback topology
In the non-inverting voltage-amplifier connection, the returned voltage is compared in series at the input and the output voltage is sampled across the output port. This voltage-series topology tends to increase the input resistance seen by the source and decrease the output resistance seen by the load.
That statement is not universal. Other feedback topologies mix in shunt or sample current and therefore transform input/output impedances differently. Chapter 04 will attach quantitative factors to these changes.
03.8Negative at low frequency does not guarantee stable at all frequencies
Real amplifiers contain poles and delay. A return signal that opposes the error at low frequency can accumulate enough phase lag to reinforce it at a higher frequency. Therefore the low-frequency schematic sign is necessary but not sufficient for stability.
Chapter 05 develops phase-margin intuition, while AE-04 later connects those poles to transistor capacitances, Miller multiplication and high-frequency circuit structure.
03.9Gate before using ideal feedback formulas
- Trace the actual return path and verify the net sign around the intended operating point.
- Check that the feedback path is closed and connected to the intended physical node.
- Check input common-mode range, output swing and output-current capability.
- Check that the loop is stable over the frequencies of interest.
- Check that available loop gain is large enough for the required accuracy.
- Only then apply the virtual-short and zero-input-current idealizations.
03.10Mastery exercises
- For a non-inverting amplifier with R1=2 kΩ and R2=18 kΩ, find β and the ideal closed-loop gain.
- With A0=10 000 for the same circuit, compute the finite-gain closed-loop gain from A0/(1+A0β) and compare it with the ideal value.
- For an inverting amplifier with R1=10 kΩ and R2=47 kΩ, find signal gain and noise gain.
- In Figure 3.1, explain the sign of the correction caused by a small positive perturbation of vo.
- Why is the inverting summing node a virtual ground but not a physical ground?
- An ideal gain formula asks for vo=14 V from an op-amp whose loaded output can reach only 11 V. Which assumption fails first?
- Give one reason a loop that is negative at low frequency can become unstable at high frequency.
Solutions
1. β=2/(2+18)=0.1; ideal gain=1/β=10 V/V.
2. Gain=10000/(1+1000)=10000/1001≈9.990 V/V, slightly below 10 V/V.
3. Signal gain=−47/10=−4.7 V/V; noise gain=1+47/10=5.7.
4. Higher vo raises vf and therefore v−, reducing e=v+−v−; the amplifier responds by driving vo downward.
5. Its voltage is held near the ground-reference voltage by feedback, but it is not wired to ground and resistor currents flow through the node.
6. The output-swing constraint is violated; the amplifier saturates and the virtual-short approximation no longer predicts the circuit.
7. Poles and delay add phase lag, so the returned signal can become reinforcing near the frequency where loop gain is still significant.
Next: Chapter 04 turns these qualitative correction ideas into loop gain, exact closed-loop gain, sensitivity and impedance-change formulas.