§6.1 Ideal op-amp constraints and negative feedback

Unit 6, Lesson 1: Ideal op-amp constraints and negative feedback

Model: V+≈V− when linear, stable negative feedback Outcome: Analyze the operating principle, calculate a reference case, identify the limit of the approximation, and plan an independent check.

Physical mechanism

An op amp magnifies the difference between its input terminals. High open-loop gain plus properly connected negative feedback can force that difference to a small value. Virtual equality does not mean current can flow directly between input terminals. The approximation fails when the amplifier clips, violates its common-mode range, slews too slowly or becomes unstable.

Draw the relevant circuit or band diagram and name the source of energy, the signal that controls the result, the return path and the reference directions. A model of an individual element must be used with conservation of current and voltage around the complete circuit. This distinction is critical because a device characteristic measured under one set of bias conditions may not hold with a different load. Describe which parameters are external controls and which are intrinsic or empirical device parameters. In operational amplifiers and active functions, an especially important issue is closed-loop constraints; make it explicit before calculating anything.

Derivation and calculation

Start from V+≈V− when linear, stable negative feedback. Explain the assumptions required to use the expression and distinguish DC current, incremental gain, instantaneous waveform, average power and RMS quantities wherever they arise. Write the associated Kirchhoff equations and translate all input data to compatible SI units. Verify that the final expression has the dimensions of the required output. If the expression is nonlinear, first identify the valid region; if it is an incremental result, identify the linearization point. An elegant equation is not proof that the circuit's bias or signal swing permits the result.

Worked example. With open-loop gain 10^5 and unsaturated output 2 V, the differential input is around 20 μV. This is not a guarantee of zero input offset.

Two independent checks. First examine the zero-drive or zero-load limit where physically applicable. Second change one stated parameter by 10% while holding the others fixed. Explain the sign of the resulting response from the device physics rather than exclusively from differentiation. If this relation predicts a very large current, voltage gain or output power, inspect the supply, load and dissipation limits. The provided numbers are illustrative and are not claims of verified measured devices.

Limitations and expected departures

Output rails, input bias, finite gain-bandwidth, noise and slew rate limit real behavior.

This is the boundary of the teaching model, not evidence that all equations are unreliable. To judge model quality, identify the dominant neglected effect and whether it becomes important at high frequency, high drive, high temperature, or a specific operating region. Component spread may require using lower and upper device ratings rather than a single nominal value. Record whether an observed deviation is likely to arise from a different bias condition, instrument impedance or a real phenomenon the simple expression omits.

Experimental reasoning

Trace the sign of feedback after a small positive disturbance at the non-inverting input.

Before energizing hardware, calculate expected terminal voltages and resistor power dissipation and confirm the fixture uses an isolated current-limited low-voltage supply. Measure from a defined ground and avoid exposing test equipment to uncontrolled potentials. Build a table of controlled inputs, model predictions, observed outputs and measurement uncertainty. A measured curve should have labelled axes and should reveal when the hypothesized relation stops being useful. A single matching value does not validate the whole operating range.

Design analysis and independent validation

Treat ideal op-amp constraints and negative feedback as a testable physical claim about closed-loop constraints. Before selecting components, define the source, load, reference arrows and nominal temperature. Then make two different predictions: one based on V+≈V− when linear, stable negative feedback, and another based on a conservation law, an alternative model or a limiting case. If the predictions disagree, do not average the answers. Resolve whether there is a sign error, a unit mistake, a different operating region, or an inaccurate assumption. The specified relation models one aspect of feedback function, not every physical detail.

Challenge the nominal circuit by varying rail voltage, bandwidth and offset. Explain which quantity should increase and which should decrease based on output feasibility. The fact that a classroom formula gives an answer does not establish that a component can deliver the required power or survive the heat. In this example, an important warning is: Output rails, input bias, finite gain-bandwidth, noise and slew rate limit real behavior. Specify what extra model parameters would be necessary to remove that uncertainty.

For an empirical check, Trace the sign of feedback after a small positive disturbance at the non-inverting input. Choose at least five observations over the safe input range, label instruments and units, and record what was controlled rather than inferred. A data point that differs from the model is useful: first test the instrument's loading and connections, then check thermal drift and component tolerances. Explain which observed trend would falsify the original assumption. This is more informative than reporting a nominal error percentage without an uncertainty budget.

Exercises (attempt before consulting references)

Describe the carrier, field or feedback mechanism underlying V+≈V− when linear, stable negative feedback and show why the predicted output has its stated sign.
Reproduce the numerical case above using SI units, and calculate the percent change in output when the dominant input parameter rises by 20%.
Identify the operating-region or frequency assumption and test one boundary where the formula would be misleading.
Construct a five-point measurement or numerical sweep based on trace the sign of feedback after a small positive disturbance at the non-inverting input.; give axes, expected trend and uncertainty.
Propose a modification that reduces sensitivity to rail voltage, bandwidth and offset. Which cost, efficiency, gain or bandwidth tradeoff does it create?
Compare a nominal condition and a defensible worst case, and state which additional device-specific data would be required before hardware release.

Review standard. A complete solution includes a referenced schematic, units, at least one independent plausibility check, and a clearly separated statement of prediction versus observation. Do not certify a real circuit or safety-critical product solely from these simplified cases.

§6.2 Inverting and non-inverting amplifier design

Unit 6, Lesson 2: Inverting and non-inverting amplifier design

Model: Av,inv≈−Rf/Rin; Av,non≈1+Rf/Rg Outcome: Analyze the operating principle, calculate a reference case, identify the limit of the approximation, and plan an independent check.

Physical mechanism

The inverting topology holds a summing node near a reference through feedback and develops an output proportional to input through a resistor ratio. The non-inverting topology uses feedback division and has high nominal input impedance. Resistor matching affects closed-loop gain accuracy and input bias current may produce DC output error.

Draw the relevant circuit or band diagram and name the source of energy, the signal that controls the result, the return path and the reference directions. A model of an individual element must be used with conservation of current and voltage around the complete circuit. This distinction is critical because a device characteristic measured under one set of bias conditions may not hold with a different load. Describe which parameters are external controls and which are intrinsic or empirical device parameters. In operational amplifiers and active functions, an especially important issue is closed-loop constraints; make it explicit before calculating anything.

Derivation and calculation

Start from Av,inv≈−Rf/Rin; Av,non≈1+Rf/Rg. Explain the assumptions required to use the expression and distinguish DC current, incremental gain, instantaneous waveform, average power and RMS quantities wherever they arise. Write the associated Kirchhoff equations and translate all input data to compatible SI units. Verify that the final expression has the dimensions of the required output. If the expression is nonlinear, first identify the valid region; if it is an incremental result, identify the linearization point. An elegant equation is not proof that the circuit's bias or signal swing permits the result.

Worked example. For Rin=10 kΩ and Rf=47 kΩ, inverting gain is −4.7. For Rg=10 kΩ and Rf=47 kΩ, non-inverting gain is +5.7, provided output remains within its allowed swing.

Two independent checks. First examine the zero-drive or zero-load limit where physically applicable. Second change one stated parameter by 10% while holding the others fixed. Explain the sign of the resulting response from the device physics rather than exclusively from differentiation. If this relation predicts a very large current, voltage gain or output power, inspect the supply, load and dissipation limits. The provided numbers are illustrative and are not claims of verified measured devices.

Limitations and expected departures

These equations assume sufficient open-loop gain and stable operation at the signal frequency.

This is the boundary of the teaching model, not evidence that all equations are unreliable. To judge model quality, identify the dominant neglected effect and whether it becomes important at high frequency, high drive, high temperature, or a specific operating region. Component spread may require using lower and upper device ratings rather than a single nominal value. Record whether an observed deviation is likely to arise from a different bias condition, instrument impedance or a real phenomenon the simple expression omits.

Experimental reasoning

Apply a 0.5 V sine and predict phase, output peak and clipping risk on a ±3 V output-limited model.

Before energizing hardware, calculate expected terminal voltages and resistor power dissipation and confirm the fixture uses an isolated current-limited low-voltage supply. Measure from a defined ground and avoid exposing test equipment to uncontrolled potentials. Build a table of controlled inputs, model predictions, observed outputs and measurement uncertainty. A measured curve should have labelled axes and should reveal when the hypothesized relation stops being useful. A single matching value does not validate the whole operating range.

Design analysis and independent validation

Treat inverting and non-inverting amplifier design as a testable physical claim about closed-loop constraints. Before selecting components, define the source, load, reference arrows and nominal temperature. Then make two different predictions: one based on Av,inv≈−Rf/Rin; Av,non≈1+Rf/Rg, and another based on a conservation law, an alternative model or a limiting case. If the predictions disagree, do not average the answers. Resolve whether there is a sign error, a unit mistake, a different operating region, or an inaccurate assumption. The specified relation models one aspect of feedback function, not every physical detail.

Challenge the nominal circuit by varying rail voltage, bandwidth and offset. Explain which quantity should increase and which should decrease based on output feasibility. The fact that a classroom formula gives an answer does not establish that a component can deliver the required power or survive the heat. In this example, an important warning is: These equations assume sufficient open-loop gain and stable operation at the signal frequency. Specify what extra model parameters would be necessary to remove that uncertainty.

For an empirical check, Apply a 0.5 V sine and predict phase, output peak and clipping risk on a ±3 V output-limited model. Choose at least five observations over the safe input range, label instruments and units, and record what was controlled rather than inferred. A data point that differs from the model is useful: first test the instrument's loading and connections, then check thermal drift and component tolerances. Explain which observed trend would falsify the original assumption. This is more informative than reporting a nominal error percentage without an uncertainty budget.

Exercises (attempt before consulting references)

Describe the carrier, field or feedback mechanism underlying Av,inv≈−Rf/Rin; Av,non≈1+Rf/Rg and show why the predicted output has its stated sign.
Reproduce the numerical case above using SI units, and calculate the percent change in output when the dominant input parameter rises by 20%.
Identify the operating-region or frequency assumption and test one boundary where the formula would be misleading.
Construct a five-point measurement or numerical sweep based on apply a 0.5 v sine and predict phase, output peak and clipping risk on a ±3 v output-limited model.; give axes, expected trend and uncertainty.
Propose a modification that reduces sensitivity to rail voltage, bandwidth and offset. Which cost, efficiency, gain or bandwidth tradeoff does it create?
Compare a nominal condition and a defensible worst case, and state which additional device-specific data would be required before hardware release.

Review standard. A complete solution includes a referenced schematic, units, at least one independent plausibility check, and a clearly separated statement of prediction versus observation. Do not certify a real circuit or safety-critical product solely from these simplified cases.

§6.3 Summing, differential and instrumentation stages

Unit 6, Lesson 3: Summing, differential and instrumentation stages

Model: Vo≈−Rf Σ(Vi/Ri) Outcome: Analyze the operating principle, calculate a reference case, identify the limit of the approximation, and plan an independent check.

Physical mechanism

An op-amp summing node permits weighted analog addition, but every input drives through an impedance that must be included in the loading calculation. A differential stage subtracts two signals; its rejection of common-mode voltage depends on resistor-ratio matching. Instrumentation amplifiers increase input impedance and can improve common-mode rejection while retaining programmable gain.

Draw the relevant circuit or band diagram and name the source of energy, the signal that controls the result, the return path and the reference directions. A model of an individual element must be used with conservation of current and voltage around the complete circuit. This distinction is critical because a device characteristic measured under one set of bias conditions may not hold with a different load. Describe which parameters are external controls and which are intrinsic or empirical device parameters. In operational amplifiers and active functions, an especially important issue is closed-loop constraints; make it explicit before calculating anything.

Derivation and calculation

Start from Vo≈−Rf Σ(Vi/Ri). Explain the assumptions required to use the expression and distinguish DC current, incremental gain, instantaneous waveform, average power and RMS quantities wherever they arise. Write the associated Kirchhoff equations and translate all input data to compatible SI units. Verify that the final expression has the dimensions of the required output. If the expression is nonlinear, first identify the valid region; if it is an incremental result, identify the linearization point. An elegant equation is not proof that the circuit's bias or signal swing permits the result.

Worked example. With Rf=20 kΩ, R1=10 kΩ and R2=20 kΩ, inputs V1=0.1 V and V2=0.4 V produce Vo≈−(0.2+0.4)=−0.6 V.

Two independent checks. First examine the zero-drive or zero-load limit where physically applicable. Second change one stated parameter by 10% while holding the others fixed. Explain the sign of the resulting response from the device physics rather than exclusively from differentiation. If this relation predicts a very large current, voltage gain or output power, inspect the supply, load and dissipation limits. The provided numbers are illustrative and are not claims of verified measured devices.

Limitations and expected departures

Bias currents, input range and matching error can cause significant offsets for very small sensor signals.

This is the boundary of the teaching model, not evidence that all equations are unreliable. To judge model quality, identify the dominant neglected effect and whether it becomes important at high frequency, high drive, high temperature, or a specific operating region. Component spread may require using lower and upper device ratings rather than a single nominal value. Record whether an observed deviation is likely to arise from a different bias condition, instrument impedance or a real phenomenon the simple expression omits.

Experimental reasoning

Design a weighted summer with a 2:1 contribution ratio, then calculate its output when input signs reverse.

Before energizing hardware, calculate expected terminal voltages and resistor power dissipation and confirm the fixture uses an isolated current-limited low-voltage supply. Measure from a defined ground and avoid exposing test equipment to uncontrolled potentials. Build a table of controlled inputs, model predictions, observed outputs and measurement uncertainty. A measured curve should have labelled axes and should reveal when the hypothesized relation stops being useful. A single matching value does not validate the whole operating range.

Design analysis and independent validation

Treat summing, differential and instrumentation stages as a testable physical claim about closed-loop constraints. Before selecting components, define the source, load, reference arrows and nominal temperature. Then make two different predictions: one based on Vo≈−Rf Σ(Vi/Ri), and another based on a conservation law, an alternative model or a limiting case. If the predictions disagree, do not average the answers. Resolve whether there is a sign error, a unit mistake, a different operating region, or an inaccurate assumption. The specified relation models one aspect of feedback function, not every physical detail.

Challenge the nominal circuit by varying rail voltage, bandwidth and offset. Explain which quantity should increase and which should decrease based on output feasibility. The fact that a classroom formula gives an answer does not establish that a component can deliver the required power or survive the heat. In this example, an important warning is: Bias currents, input range and matching error can cause significant offsets for very small sensor signals. Specify what extra model parameters would be necessary to remove that uncertainty.

For an empirical check, Design a weighted summer with a 2:1 contribution ratio, then calculate its output when input signs reverse. Choose at least five observations over the safe input range, label instruments and units, and record what was controlled rather than inferred. A data point that differs from the model is useful: first test the instrument's loading and connections, then check thermal drift and component tolerances. Explain which observed trend would falsify the original assumption. This is more informative than reporting a nominal error percentage without an uncertainty budget.

Exercises (attempt before consulting references)

Describe the carrier, field or feedback mechanism underlying Vo≈−Rf Σ(Vi/Ri) and show why the predicted output has its stated sign.
Reproduce the numerical case above using SI units, and calculate the percent change in output when the dominant input parameter rises by 20%.
Identify the operating-region or frequency assumption and test one boundary where the formula would be misleading.
Construct a five-point measurement or numerical sweep based on design a weighted summer with a 2:1 contribution ratio, then calculate its output when input signs reverse.; give axes, expected trend and uncertainty.
Propose a modification that reduces sensitivity to rail voltage, bandwidth and offset. Which cost, efficiency, gain or bandwidth tradeoff does it create?
Compare a nominal condition and a defensible worst case, and state which additional device-specific data would be required before hardware release.

Review standard. A complete solution includes a referenced schematic, units, at least one independent plausibility check, and a clearly separated statement of prediction versus observation. Do not certify a real circuit or safety-critical product solely from these simplified cases.

§6.4 Integrators and differentiators

Unit 6, Lesson 4: Integrators and differentiators

Model: Vo(t)=−(1/RC)∫Vi(t)dt Outcome: Analyze the operating principle, calculate a reference case, identify the limit of the approximation, and plan an independent check.

Physical mechanism

An ideal inverting integrator accumulates input over time using feedback capacitance. A differentiator emphasizes rapid input changes but can amplify high-frequency noise. Practical integrators require DC feedback leakage paths or reset controls to prevent tiny offsets from driving output saturation. Practical differentiators limit both low and high frequency gains.

Draw the relevant circuit or band diagram and name the source of energy, the signal that controls the result, the return path and the reference directions. A model of an individual element must be used with conservation of current and voltage around the complete circuit. This distinction is critical because a device characteristic measured under one set of bias conditions may not hold with a different load. Describe which parameters are external controls and which are intrinsic or empirical device parameters. In operational amplifiers and active functions, an especially important issue is closed-loop constraints; make it explicit before calculating anything.

Derivation and calculation

Start from Vo(t)=−(1/RC)∫Vi(t)dt. Explain the assumptions required to use the expression and distinguish DC current, incremental gain, instantaneous waveform, average power and RMS quantities wherever they arise. Write the associated Kirchhoff equations and translate all input data to compatible SI units. Verify that the final expression has the dimensions of the required output. If the expression is nonlinear, first identify the valid region; if it is an incremental result, identify the linearization point. An elegant equation is not proof that the circuit's bias or signal swing permits the result.

Worked example. For a constant 0.1 V input, R=100 kΩ and C=1 μF, ideal integrator slope is −1 V/s. The output reaches −2 V after 2 s from a zero initial state.

Two independent checks. First examine the zero-drive or zero-load limit where physically applicable. Second change one stated parameter by 10% while holding the others fixed. Explain the sign of the resulting response from the device physics rather than exclusively from differentiation. If this relation predicts a very large current, voltage gain or output power, inspect the supply, load and dissipation limits. The provided numbers are illustrative and are not claims of verified measured devices.

Limitations and expected departures

Finite rails, capacitor leakage, offset and frequency limits prevent indefinite ideal integration.

This is the boundary of the teaching model, not evidence that all equations are unreliable. To judge model quality, identify the dominant neglected effect and whether it becomes important at high frequency, high drive, high temperature, or a specific operating region. Component spread may require using lower and upper device ratings rather than a single nominal value. Record whether an observed deviation is likely to arise from a different bias condition, instrument impedance or a real phenomenon the simple expression omits.

Experimental reasoning

Explain how a small 1 mV input offset accumulates over ten seconds in an unreset ideal integrator.

Before energizing hardware, calculate expected terminal voltages and resistor power dissipation and confirm the fixture uses an isolated current-limited low-voltage supply. Measure from a defined ground and avoid exposing test equipment to uncontrolled potentials. Build a table of controlled inputs, model predictions, observed outputs and measurement uncertainty. A measured curve should have labelled axes and should reveal when the hypothesized relation stops being useful. A single matching value does not validate the whole operating range.

Design analysis and independent validation

Treat integrators and differentiators as a testable physical claim about closed-loop constraints. Before selecting components, define the source, load, reference arrows and nominal temperature. Then make two different predictions: one based on Vo(t)=−(1/RC)∫Vi(t)dt, and another based on a conservation law, an alternative model or a limiting case. If the predictions disagree, do not average the answers. Resolve whether there is a sign error, a unit mistake, a different operating region, or an inaccurate assumption. The specified relation models one aspect of feedback function, not every physical detail.

Challenge the nominal circuit by varying rail voltage, bandwidth and offset. Explain which quantity should increase and which should decrease based on output feasibility. The fact that a classroom formula gives an answer does not establish that a component can deliver the required power or survive the heat. In this example, an important warning is: Finite rails, capacitor leakage, offset and frequency limits prevent indefinite ideal integration. Specify what extra model parameters would be necessary to remove that uncertainty.

For an empirical check, Explain how a small 1 mV input offset accumulates over ten seconds in an unreset ideal integrator. Choose at least five observations over the safe input range, label instruments and units, and record what was controlled rather than inferred. A data point that differs from the model is useful: first test the instrument's loading and connections, then check thermal drift and component tolerances. Explain which observed trend would falsify the original assumption. This is more informative than reporting a nominal error percentage without an uncertainty budget.

Exercises (attempt before consulting references)

Describe the carrier, field or feedback mechanism underlying Vo(t)=−(1/RC)∫Vi(t)dt and show why the predicted output has its stated sign.
Reproduce the numerical case above using SI units, and calculate the percent change in output when the dominant input parameter rises by 20%.
Identify the operating-region or frequency assumption and test one boundary where the formula would be misleading.
Construct a five-point measurement or numerical sweep based on explain how a small 1 mv input offset accumulates over ten seconds in an unreset ideal integrator.; give axes, expected trend and uncertainty.
Propose a modification that reduces sensitivity to rail voltage, bandwidth and offset. Which cost, efficiency, gain or bandwidth tradeoff does it create?
Compare a nominal condition and a defensible worst case, and state which additional device-specific data would be required before hardware release.

Review standard. A complete solution includes a referenced schematic, units, at least one independent plausibility check, and a clearly separated statement of prediction versus observation. Do not certify a real circuit or safety-critical product solely from these simplified cases.

§6.5 Active filters and frequency response

Unit 6, Lesson 5: Active filters and frequency response

Model: |H(f)|=A0/sqrt(1+(f/fc)²) Outcome: Analyze the operating principle, calculate a reference case, identify the limit of the approximation, and plan an independent check.

Physical mechanism

A first-order low-pass stage passes lower frequencies and increasingly attenuates higher frequencies, with a phase lag approaching 90 degrees in the ideal limit. Active realizations use an amplifier to buffer or provide gain but may be restricted by amplifier bandwidth. Cascading sections changes order and the shape near cutoff.

Draw the relevant circuit or band diagram and name the source of energy, the signal that controls the result, the return path and the reference directions. A model of an individual element must be used with conservation of current and voltage around the complete circuit. This distinction is critical because a device characteristic measured under one set of bias conditions may not hold with a different load. Describe which parameters are external controls and which are intrinsic or empirical device parameters. In operational amplifiers and active functions, an especially important issue is closed-loop constraints; make it explicit before calculating anything.

Derivation and calculation

Start from |H(f)|=A0/sqrt(1+(f/fc)²). Explain the assumptions required to use the expression and distinguish DC current, incremental gain, instantaneous waveform, average power and RMS quantities wherever they arise. Write the associated Kirchhoff equations and translate all input data to compatible SI units. Verify that the final expression has the dimensions of the required output. If the expression is nonlinear, first identify the valid region; if it is an incremental result, identify the linearization point. An elegant equation is not proof that the circuit's bias or signal swing permits the result.

Worked example. For A0=2, fc=1 kHz and f=1 kHz, the magnitude is 2/√2≈1.414. At 10 kHz it drops to approximately 0.199.

Two independent checks. First examine the zero-drive or zero-load limit where physically applicable. Second change one stated parameter by 10% while holding the others fixed. Explain the sign of the resulting response from the device physics rather than exclusively from differentiation. If this relation predicts a very large current, voltage gain or output power, inspect the supply, load and dissipation limits. The provided numbers are illustrative and are not claims of verified measured devices.

Limitations and expected departures

Component tolerance, loading, op-amp bandwidth and multiple-pole response can alter the ideal single-pole curve.

This is the boundary of the teaching model, not evidence that all equations are unreliable. To judge model quality, identify the dominant neglected effect and whether it becomes important at high frequency, high drive, high temperature, or a specific operating region. Component spread may require using lower and upper device ratings rather than a single nominal value. Record whether an observed deviation is likely to arise from a different bias condition, instrument impedance or a real phenomenon the simple expression omits.

Experimental reasoning

Plot magnitude in decibels versus logarithmic frequency and identify the −3 dB point relative to passband.

Before energizing hardware, calculate expected terminal voltages and resistor power dissipation and confirm the fixture uses an isolated current-limited low-voltage supply. Measure from a defined ground and avoid exposing test equipment to uncontrolled potentials. Build a table of controlled inputs, model predictions, observed outputs and measurement uncertainty. A measured curve should have labelled axes and should reveal when the hypothesized relation stops being useful. A single matching value does not validate the whole operating range.

Design analysis and independent validation

Treat active filters and frequency response as a testable physical claim about closed-loop constraints. Before selecting components, define the source, load, reference arrows and nominal temperature. Then make two different predictions: one based on |H(f)|=A0/sqrt(1+(f/fc)²), and another based on a conservation law, an alternative model or a limiting case. If the predictions disagree, do not average the answers. Resolve whether there is a sign error, a unit mistake, a different operating region, or an inaccurate assumption. The specified relation models one aspect of feedback function, not every physical detail.

Challenge the nominal circuit by varying rail voltage, bandwidth and offset. Explain which quantity should increase and which should decrease based on output feasibility. The fact that a classroom formula gives an answer does not establish that a component can deliver the required power or survive the heat. In this example, an important warning is: Component tolerance, loading, op-amp bandwidth and multiple-pole response can alter the ideal single-pole curve. Specify what extra model parameters would be necessary to remove that uncertainty.

For an empirical check, Plot magnitude in decibels versus logarithmic frequency and identify the −3 dB point relative to passband. Choose at least five observations over the safe input range, label instruments and units, and record what was controlled rather than inferred. A data point that differs from the model is useful: first test the instrument's loading and connections, then check thermal drift and component tolerances. Explain which observed trend would falsify the original assumption. This is more informative than reporting a nominal error percentage without an uncertainty budget.

Exercises (attempt before consulting references)

Describe the carrier, field or feedback mechanism underlying |H(f)|=A0/sqrt(1+(f/fc)²) and show why the predicted output has its stated sign.
Reproduce the numerical case above using SI units, and calculate the percent change in output when the dominant input parameter rises by 20%.
Identify the operating-region or frequency assumption and test one boundary where the formula would be misleading.
Construct a five-point measurement or numerical sweep based on plot magnitude in decibels versus logarithmic frequency and identify the −3 db point relative to passband.; give axes, expected trend and uncertainty.
Propose a modification that reduces sensitivity to rail voltage, bandwidth and offset. Which cost, efficiency, gain or bandwidth tradeoff does it create?
Compare a nominal condition and a defensible worst case, and state which additional device-specific data would be required before hardware release.

Review standard. A complete solution includes a referenced schematic, units, at least one independent plausibility check, and a clearly separated statement of prediction versus observation. Do not certify a real circuit or safety-critical product solely from these simplified cases.

Unit 6, Lesson 1: Ideal op-amp constraints and negative feedback

Physical mechanism

Derivation and calculation

Limitations and expected departures

Experimental reasoning

Design analysis and independent validation

Exercises (attempt before consulting references)

Unit 6, Lesson 2: Inverting and non-inverting amplifier design

Physical mechanism

Derivation and calculation

Limitations and expected departures

Experimental reasoning

Design analysis and independent validation

Exercises (attempt before consulting references)

Unit 6, Lesson 3: Summing, differential and instrumentation stages

Physical mechanism

Derivation and calculation

Limitations and expected departures

Experimental reasoning

Design analysis and independent validation

Exercises (attempt before consulting references)

Unit 6, Lesson 4: Integrators and differentiators

Physical mechanism

Derivation and calculation

Limitations and expected departures

Experimental reasoning

Design analysis and independent validation

Exercises (attempt before consulting references)

Unit 6, Lesson 5: Active filters and frequency response

Physical mechanism

Derivation and calculation

Limitations and expected departures

Experimental reasoning

Design analysis and independent validation

Exercises (attempt before consulting references)

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