Operational Amplifiers and Feedback Systems

Introduction

Operational amplifiers (op-amps) are high-gain differential amplifiers designed to operate with external feedback. Their versatility arises less from their open-loop behavior than from the way negative or positive feedback configures them as amplifiers, arithmetic circuits, filters, comparators, waveform processors, and sensor interfaces.

An ideal model makes circuit design tractable, but practical designs must also account for finite gain, bandwidth, slew rate, input offset, bias currents, noise, common-mode range, output swing, loading, and stability. These limitations become particularly important in precision instrumentation, high-speed acquisition, control systems, medical electronics, and industrial sensing.

Learning Objectives

After studying this chapter, students should be able to:

• apply ideal and nonideal op-amp models;
• analyze negative-feedback systems using loop gain;
• design inverting, noninverting, summing, differential, and instrumentation amplifiers;
• design integrators, differentiators, active filters, comparators, and precision circuits;
• evaluate gain-bandwidth, slew-rate, offset, common-mode, and output-range constraints;
• assess feedback stability using loop gain and phase margin; and
• translate sensor-interface requirements into a practical op-amp design.

1. Op-Amp Fundamentals

An op-amp has an inverting input, a noninverting input, and an output. Its open-loop differential relation is

where v_o is output voltage, v_+ and v_- are input voltages, and A_OL is open-loop differential gain. Voltages are measured in volts; A_OL is dimensionless.

Because practical open-loop gains may exceed 10⁵ at low frequency, even a small differential input can drive the output into saturation. Linear applications therefore normally use negative feedback.

Figure: An op-amp with a fraction β of its output returned to the inverting input.

1.1 Ideal op-amp model

For an ideal op-amp:

1. A_OL → ∞;
2. input resistance is infinite, so i_+ = i_- = 0;
3. output resistance is zero;
4. bandwidth and slew rate are infinite;
5. input offset voltage and noise are zero; and
6. common-mode rejection and power-supply rejection are infinite.

For a stable circuit operating linearly under negative feedback, infinite gain implies the virtual-short condition

This condition does not mean the input terminals are physically shorted. It is invalid when the output is saturated, feedback is positive, or the circuit is unstable.

1.2 Practical model and data-sheet parameters

“Rail-to-rail” must be interpreted carefully. Rail-to-rail input and rail-to-rail output are separate properties, and neither normally means exact operation at both supply rails under every load.

2. Negative Feedback

Let a forward amplifier have gain A and let the feedback network return βv_o to a subtracting junction. With v_e = v_s - βv_o and v_o = Av_e,

Here A_f is closed-loop gain and T=Aβ is loop gain. This relation assumes linear operation and negative feedback with the indicated sign convention.

If |Aβ| ≫ 1, then

Thus accurate passive components can determine closed-loop gain even when open-loop gain varies substantially. The relative sensitivity to forward gain is

Negative feedback also reduces distortion and output impedance, and generally increases usable bandwidth. These benefits apply only while adequate loop gain and stability margin remain.

2.1 Feedback and bandwidth

A dominant-pole voltage-feedback op-amp approximately satisfies

where f_CL is the small-signal closed-loop bandwidth and G_N is noise gain—the closed-loop gain from an equivalent noninverting input voltage to the output. For a noninverting amplifier, noise gain equals signal gain. For an inverting amplifier,

The approximation assumes a dominant-pole, unity-gain-stable op-amp and a substantially resistive feedback network.

3. Fundamental Amplifier Configurations

3.1 Inverting amplifier

With the noninverting terminal grounded, the inverting terminal is at virtual ground. Applying Kirchhoff’s current law gives

and therefore

The input resistance seen by the source is approximately R_in.

Worked example. Design a gain of −10 with an input resistance of 20 kΩ. Choose R_in=20\ kΩ and

For v_in=0.20\ V, the ideal output is −2.0 V. The noise gain is 11, not 10. An op-amp with GBW = 2.2 MHz therefore has an approximate small-signal bandwidth of 200 kHz.

3.2 Noninverting amplifier and voltage follower

For resistors R_g from the inverting input to ground and R_f from output to the inverting input,

The input resistance is ideally infinite. Setting R_f=0 and removing R_g produces a unity-gain voltage follower. Only an op-amp specified as unity-gain stable should be used this way.

3.3 Summing amplifier

For several voltages applied through separate input resistors to an inverting summing node,

The result assumes ideal linear operation. Equal input resistors create a scaled negative sum. Applications include audio mixing, digital-to-analog conversion, and weighted control signals.

3.4 Difference amplifier

For a four-resistor difference amplifier with matched ratios

the output is

High common-mode rejection requires accurate ratio matching, not merely accurate individual resistor values. Integrated resistor networks often outperform four unrelated discrete resistors.

3.5 Instrumentation amplifier

A three-op-amp instrumentation amplifier provides high input impedance and adjustable differential gain. For the common symmetric arrangement,

This expression assumes the two first-stage resistors are equal to R, the gain resistor is R_G, and the second-stage difference-amplifier ratios are precisely matched.

Instrumentation amplifiers do not permit arbitrary common-mode voltage. Their input range and internal first-stage output swings must be checked at the selected gain and supply voltage.

4. Dynamic Signal-Processing Circuits

4.1 Integrator

An ideal inverting integrator uses input resistor R and feedback capacitor C. Its transfer function is

where s has units s⁻¹. With zero initial capacitor voltage,

Any DC offset can cause an ideal integrator to drift into saturation. A practical integrator places resistor R_f in parallel with C. Its transfer function is

At frequencies well above 1/(2πR_fC), it approximates an integrator; at low frequency its gain is limited to -R_f/R.

4.2 Differentiator

An ideal differentiator has an input capacitor C and feedback resistor R_f:

Thus v_o=-R_fC(dv_in/dt). Since gain rises with frequency, noise and instability can become severe. Adding a series input resistor R₁ limits high-frequency gain:

A feedback capacitor may be added across R_f for further high-frequency roll-off.

5. Active Filters

Active filters combine op-amps with resistors and capacitors, avoiding inductors at low and moderate frequencies. Their realization must account for op-amp GBW, component tolerance, output loading, and stability.

5.1 First-order low-pass filter

A buffered RC low-pass filter has

Its −3 dB cutoff is f_c=1/(2πRC).

5.2 Unity-gain Sallen–Key low-pass filter

Figure: Unity-gain Sallen–Key low-pass section; the op-amp is connected as a voltage follower.

For the topology shown, assuming an ideal voltage follower,

Comparison with

gives

For R₁=R₂=R and C₁=C₂=C,

The equal-component unity-gain case therefore has two coincident real poles. Its natural frequency f₀=1/(2πRC) is not its −3 dB frequency. For Q=1/2, f_−3dB=f₀\sqrt{\sqrt{2}-1}≈0.6436f₀.

Worked example. With R₁=R₂=15.9\ kΩ and C₁=C₂=100\ nF,

The corresponding −3 dB frequency is approximately 64.4 Hz.

5.3 Component tolerance

Since

the small-error differential approximation is

For worst-case bounded tolerances, absolute contributions add approximately linearly:

For independent, zero-mean random component uncertainties with relative standard deviations u_i, root-sum-square propagation is appropriate:

Thus four equal 5% bounded tolerances imply approximately 10% worst-case frequency error. Treating 5% as independent one-standard-deviation random uncertainty instead gives 5% RSS uncertainty. RSS must not be presented as a worst-case tolerance calculation.

6. Comparators and Positive Feedback

An op-amp used open-loop as a comparator drives toward a positive or negative saturation level according to the sign of v_+-v_-. Dedicated comparators are usually preferable because they recover faster from saturation and may provide logic-compatible outputs.

Noise near a single threshold can cause repeated switching. A Schmitt trigger introduces positive feedback and two thresholds. For an inverting Schmitt trigger with the noninverting input receiving a fraction β of the output and zero reference, the thresholds are

where V_OH and V_OL are actual high and low output levels. For symmetric saturation ±V_sat, the hysteresis width is 2βV_sat.

7. Precision Circuits

A diode’s forward drop makes an ordinary rectifier inaccurate for small signals. In a precision rectifier, the diode is placed inside an op-amp feedback path, so feedback compensates for much of the diode drop while the diode conducts. Applications include envelope detection, absolute-value circuits, and low-level AC measurement.

Precision does not imply unlimited speed. During diode switching, the op-amp may enter saturation or encounter an open feedback loop. Recovery time, slew rate, diode capacitance, input common-mode range, and output swing can then dominate error. Modern designs may use improved two-op-amp absolute-value circuits or integrated precision detectors.

8. Nonideal Performance Analysis

8.1 Finite open-loop gain

For a noninverting feedback factor β,

If A_OLβ≫1, the fractional gain error relative to 1/β is approximately -1/(A_OLβ). Since A_OL falls with frequency, gain accuracy also worsens with frequency.

8.2 Slew rate and full-power bandwidth

For v_o=V_p\sin(2πft), the maximum slope is 2πfV_p. Avoiding slew-rate distortion requires

Hence the full-power bandwidth is

Example. With SR = 0.5 V/μs and peak output 5 V,

This large-signal limit is distinct from small-signal bandwidth.

8.3 Offset and bias-current errors

Input offset voltage is approximately amplified by noise gain. In an inverting amplifier,

Bias-current errors can be reduced by placing a compensation resistor at the noninverting input equal to

This cancels error due to equal bias currents but not input offset current, resistor mismatch, or temperature drift.

8.4 Common-mode rejection

Common-mode voltage is

The common-mode rejection ratio is

An approximate input-referred common-mode error is v_CM/CMRR, provided CMRR is expressed as a linear ratio. This estimate does not replace checking common-mode input limits.

8.5 Output loading and supply constraints

For a resistive load R_L, output current is approximately i_o=v_o/R_L. The required current must remain within source and sink ratings. Capacitive loads add phase lag and may destabilize the feedback loop; an isolation resistor or compensated driver may be necessary.

9. Stability and Phase Margin

Because an op-amp contains energy-storage elements, its open-loop gain has frequency-dependent phase lag. Oscillation becomes possible when the loop signal returns with sufficient magnitude and a net reinforcing phase.

The phase margin is evaluated at the frequency ω_c where |A(jω_c)β(jω_c)|=1:

This expression uses the negative-feedback loop-gain convention. A positive phase margin generally indicates small-signal stability; greater margin usually reduces ringing. Approximately 45°–60° is common, although required margin depends on transient-performance objectives and model accuracy.

Stability practices include:

• selecting an op-amp stable at the intended noise gain;
• minimizing parasitic capacitance at the inverting node;
• decoupling each supply locally;
• using short feedback paths and a sound ground layout;
• isolating substantial capacitive loads; and
• evaluating loop gain across component, load, supply, and temperature extremes.

A decompensated op-amp may offer greater speed but require a specified minimum closed-loop noise gain.

10. Illustrative Case: Bridge-Sensor Interface

Background

An industrial pressure bridge produces a differential signal from −10 mV to +10 mV around a 2.5 V common-mode level. A 0–5 V ADC should receive approximately 0.5–4.5 V. This is an illustrative design case.

Problem

The circuit requires differential gain, common-mode rejection, level shifting, noise filtering, and protection against ADC overdrive. Ordinary single-ended amplification would also amplify the 2.5 V common-mode component.

Analysis and approach

Mapping ±10 mV to ±2.0 V requires gain

A three-op-amp instrumentation amplifier is selected. With unity gain in its final difference stage and R=10.0\ kΩ,

so

The output reference terminal is connected to a low-impedance 2.5 V reference, producing a nominal output range of 0.5–4.5 V.

A unity-gain, equal-component Sallen–Key low-pass section is added after amplification. Choosing 15.9 kΩ resistors and 100 nF capacitors gives f₀≈100.1\ Hz, Q=0.5, and a −3 dB frequency near 64.4 Hz. If the system requires a −3 dB point of exactly 100 Hz, the natural frequency must instead be about 100/0.6436=155.4\ Hz, followed by recalculation of component values.

Nonideal checks

A 25 μV input offset can produce approximately 5 mV output error at gain 200. A CMRR of 100 dB corresponds to a linear ratio of 10⁵; 2.5 V common mode then represents an approximate 25 μV input-referred error, or another 5 mV after gain. These estimates motivate calibration or a lower-offset, higher-CMRR amplifier.

The instrumentation amplifier must support 2.5 V input common mode and the required internal node swings. Its output must reach 0.5 V and 4.5 V under the ADC load. The ADC sampling capacitor may require a buffer or series isolation resistor. Input protection must not introduce excessive leakage.

Outcome

The design converts a 20 mV full-span differential signal into a 4 V ADC span while suppressing common-mode voltage and high-frequency noise. Calibration can remove much of the fixed offset and gain error, but it cannot compensate for clipping, instability, inadequate settling, or uncontrolled temperature drift.

Learning points

• Gain calculations must be accompanied by range and bandwidth checks.
• CMRR matters when a small differential signal rides on a large common-mode voltage.
• Filter natural frequency and −3 dB frequency are not always identical.
• Error budgeting should include offset, drift, resistor tolerance, noise, and reference accuracy.

11. Professional Practice, Safety, and Contemporary Developments

A robust design should include supply decoupling, tolerance analysis, thermal assessment, input and output protection, test points, and documented worst-case assumptions. Simulation is useful but must employ appropriate macromodels and be confirmed by measurement.

In safety-related medical, automotive, or industrial systems, saturation and open-circuit sensor behavior must not create unsafe actuator commands. Applicable requirements may include IEC, ISO, automotive qualification, electromagnetic-compatibility, isolation, and functional-safety standards. Calibration records and traceability are important where circuit outputs affect diagnosis, billing, process control, or public safety.

Contemporary precision amplifiers use chopper stabilization or auto-zeroing to reduce offset and drift, although switching ripple and input-current artifacts must be managed. Fully differential amplifiers are increasingly used to drive high-resolution ADCs. Current-feedback amplifiers provide high speed but follow different gain and feedback-resistance rules from conventional voltage-feedback op-amps. Automated optimization and AI-assisted design can explore component choices, but engineering approval still requires verified models, stability analysis, tolerance evaluation, and laboratory testing.

Chapter Summary

Operational amplifiers become useful linear building blocks through feedback. Under ideal, stable negative feedback, input currents vanish and the input voltages are approximately equal. These principles lead to inverting, noninverting, summing, differential, and instrumentation-amplifier equations.

Frequency-dependent feedback produces integrators, differentiators, and active filters. The unity-gain Sallen–Key low-pass denominator is s²R₁R₂C₁C₂+sC₂(R₁+R₂)+1; equal components give Q=1/2. Filter tolerance must distinguish linearly combined worst-case bounds from RSS propagation of independent random uncertainties.

Practical design must address finite loop gain, GBW, slew rate, offset, bias currents, noise, common-mode range, output swing, loading, and stability. Phase margin, transient response, supply decoupling, component matching, range checks, and error budgets are therefore central to professional op-amp design.

Key Terms

Operational amplifier: A high-gain differential amplifier intended primarily for feedback applications.

Open-loop gain: Gain of an amplifier without external feedback.

Virtual short: Approximate equality of op-amp input voltages during stable linear negative-feedback operation.

Negative feedback: Return of an output fraction in opposition to the input error.

Loop gain: Product Aβ around a feedback loop.

Noise gain: Closed-loop gain applied to input-referred op-amp error or noise.

Gain-bandwidth product: Approximate product of closed-loop gain and bandwidth for a dominant-pole op-amp.

Slew rate: Maximum output-voltage rate of change.

Input offset voltage: Differential DC input required to force the ideal output condition.

Input bias current: DC current required by an op-amp input terminal.

Common-mode rejection ratio: Ratio of differential gain to common-mode gain.

Instrumentation amplifier: Precision differential amplifier with high input impedance and high CMRR.

Integrator: Circuit whose output is proportional to the time integral of its input.

Differentiator: Circuit whose output is proportional to the time derivative of its input.

Sallen–Key filter: Voltage-controlled-voltage-source active-filter topology.

Quality factor: Dimensionless parameter describing second-order pole damping or selectivity.

Comparator: Circuit that selects an output state according to the sign of an input difference.

Hysteresis: Separation between rising and falling switching thresholds.

Phase margin: Additional loop phase lag required to reach the oscillation phase condition at unity loop-gain magnitude.

Precision rectifier: Feedback circuit that reduces diode-threshold error in rectification.

Review Questions

Short Answer Questions

1. State the two principal ideal-op-amp rules used in negative-feedback analysis.
2. Distinguish signal gain from noise gain in an inverting amplifier.
3. Why is resistor-ratio matching important in a difference amplifier?
4. Define slew rate and explain how it differs from gain-bandwidth product.
5. Why is a practical integrator provided with a resistor in parallel with its feedback capacitor?

Long Answer Questions

1. Derive the closed-loop gain of an amplifier with forward gain A and negative feedback factor β. Discuss sensitivity reduction.
2. Derive the inverting and noninverting amplifier gain equations and compare their input resistances.
3. Explain the operation and limitations of a three-op-amp instrumentation amplifier.
4. Derive the unity-gain Sallen–Key low-pass transfer function and obtain expressions for ω₀ and Q.
5. Discuss finite gain, offset, bias current, CMRR, output swing, slew rate, and stability as practical op-amp design constraints.

Analytical / Application Questions

1. Design an inverting amplifier of gain −25 and input resistance 10 kΩ. Determine its noise gain and approximate bandwidth for GBW = 5.2 MHz.
2. A 2 V peak sine wave must be reproduced at 100 kHz. Calculate the minimum slew rate and discuss an appropriate engineering margin.
3. Design an equal-component unity-gain Sallen–Key section with natural frequency 2 kHz. Select a standard capacitor value, calculate the resistor value, and determine its −3 dB frequency.

Multiple Choice Questions with Answers

Question 1

For an ideal op-amp operating linearly with stable negative feedback, which statement is correct?

A. The input currents are zero and the two input voltages are equal.

B. The output voltage is always zero.

C. The open-loop gain is unity.

D. The two input terminals carry equal nonzero currents.

Correct Answer: A. The input currents are zero and the two input voltages are equal.

Question 2

An inverting amplifier has R_in=10\ kΩ and R_f=90\ kΩ. What are its signal gain and noise gain?

A. Signal gain −9; noise gain 10

B. Signal gain 9; noise gain 9

C. Signal gain −10; noise gain 9

D. Signal gain 10; noise gain 11

Correct Answer: A. Signal gain −9; noise gain 10

Question 3

A noninverting amplifier uses R_f=40\ kΩ and R_g=10\ kΩ. Its ideal voltage gain is:

A. 4

B. 5

C. −4

D. −5

Correct Answer: B. 5

Question 4

For the unity-gain Sallen–Key low-pass topology described in the chapter, equal resistors and equal capacitors produce what quality factor?

A. 1/3

B. 1/2

C. 1/√2

D. 1

Correct Answer: B. 1/2

Question 5

Which expression is the correct denominator of the unity-gain Sallen–Key low-pass transfer function used in the chapter?

A. s²R₁R₂C₁C₂+sC₂(R₁+R₂)+1

B. s²R₁R₂C₁C₂+sC₁(R₁+R₂+R₁R₂)+1

C. sR₁R₂C₁C₂+C₂(R₁+R₂)+1

D. s²(R₁+R₂)(C₁+C₂)+1

Correct Answer: A. s²R₁R₂C₁C₂+sC₂(R₁+R₂)+1

Question 6

Four components determining a filter natural frequency each have independent relative standard uncertainty of 2%. Under the RSS approximation in the chapter, the relative standard uncertainty of frequency is:

A. 1%

B. 2%

C. 4%

D. 8%

Correct Answer: B. 2%

Question 7

An op-amp has a slew rate of 1 V/μs. What is the approximate maximum frequency for an undistorted sine-wave output of 4 V peak, based only on slew rate?

A. 19.9 kHz

B. 39.8 kHz

C. 79.6 kHz

D. 250 kHz

Correct Answer: B. 39.8 kHz

Question 8

Why is positive feedback used in a Schmitt trigger?

A. To create two switching thresholds and improve noise immunity

B. To force both input currents to zero

C. To make closed-loop gain exactly unity

D. To eliminate output saturation

Correct Answer: A. To create two switching thresholds and improve noise immunity

Question 9

At the frequency where loop-gain magnitude is unity, the loop phase is −130°. Using the chapter convention, the phase margin is:

A. 30°

B. 50°

C. 130°

D. 180°

Correct Answer: B. 50°

Question 10

A bridge produces ±5 mV and must produce an output variation of ±1.5 V. What differential gain is required?

A. 30

B. 150

C. 300

D. 600

Correct Answer: C. 300