§8.1 SPICE-based circuit verification

Unit 8, Lesson 1: SPICE-based circuit verification

Primary relation: Residual=Ax−b Unit outcome: Combine ideal theory with actual instrument limitations, numerical methods, uncertainty, and responsible engineering judgement.

What the relation means

Modified nodal analysis solves algebraic circuit models; transient solvers add integration and nonlinear iteration. Matching one SPICE run to a hand calculation does not verify all device models or component tolerances. The expression Residual=Ax−b is meaningful only when the input, output, reference arrows, and component idealizations are stated. Draw one simple circuit or signal diagram, label every referenced terminal, and identify which terms can be measured and which are inferred from the model. Distinguish an algebraic sign from a physical polarity; a sign reversal after changing an arrow is usually a coordinate choice, not a violation of energy conservation.

Detailed engineering reasoning

A reliable simulation has a test plan. Start with an analytically solvable small circuit. Compare node voltages, branch currents, stored energy, and steady-state values, and refine integration step size. A simulation solver reporting convergence does not prove that the netlist represents the physical device. Publish input values, model equations and numerical residuals.

For spice-based circuit verification, begin with a simplest-case prediction before substituting values. If an excitation vanishes, a passive zero-initial-state model must not invent an output. If one parameter becomes very large or small, interpret the corresponding limit with the relevant source and load held fixed. For an AC circuit, a magnitude-only answer is incomplete whenever phase matters, and for a transient circuit the initial state changes the waveform even if the final DC steady state is the same.

Worked example and interpretation

Compare a 1 kΩ–1 μF RC pole predicted at 159.15 Hz against an AC sweep near 159 Hz. Start from the stated equation, convert any milli-, micro- or kilo- prefixes to SI units, and substitute only quantities that share a consistent peak/RMS convention. Report the numerical quantity with physical units. Then explain what the sign or phase of the result tells an observer. Use another representation—for example rectangular versus polar complex notation or time versus frequency response—to check the result without simply repeating the first arithmetic.

The model predicts a conditional answer. It does not predict tolerance drift, insulation integrity, wiring parasitics, thermal runaway, bandwidth of an instrument, or safe operation unless those effects have been explicitly included. A nominal value should therefore be reported alongside its assumed operating conditions.

Failure mode to detect

A simulation is not empirical evidence of component accuracy or circuit safety. To diagnose this error, ask which assumption was silently changed. Trace the computation back to the definition of the source, the selected terminals, and the units of the constitutive law. A useful check is to repeat the calculation with a parameter changed by a factor of two and decide whether the direction of the output change is consistent with physical intuition. Where the network has several branches, apply KCL, KVL, and a power balance rather than trusting a single number.

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Conceptual: Explain in your own words why Residual=Ax−b is appropriate for the stated system and list two situations where the result would not apply without modification.
Quantitative: Rework the numerical example after increasing one independent component value by 20%; explicitly state which variables remain fixed, calculate the altered result, and explain whether a first-order sensitivity estimate would work.
Verification: Construct an independent check based on dimensions, a limiting condition, a KCL/KVL residual, or a conservation law. If it disagrees with the direct calculation, identify the first step where the models diverge.
Extension: Propose a simulation with three parameter values and a graph that could reveal a modeling assumption. Mark the horizontal/vertical units and explain the limits of the conclusion.

Connection to the wider book

This topic is a building block for analyzing networks in which one part shapes phase, another changes loading, and a measurement stage perturbs the circuit. The central habit is to distinguish a device's constitutive relation from the system-level network constraints. In later units the same pattern appears in three-phase neutral currents, coupled coils, transfer functions, and power-quality measurement. Keeping a consistent reference convention across these representations prevents apparently contradictory results.

§8.2 Operational amplifiers as circuit blocks

Unit 8, Lesson 2: Operational amplifiers as circuit blocks

Primary relation: Vout≈Aol(V+−V−) Unit outcome: Combine ideal theory with actual instrument limitations, numerical methods, uncertainty, and responsible engineering judgement.

What the relation means

Under negative feedback with adequate bandwidth and output headroom, the virtual-short approximation may be useful. Real op-amps have finite open-loop gain, input bias currents, noise, slew rate and output limits. The expression Vout≈Aol(V+−V−) is meaningful only when the input, output, reference arrows, and component idealizations are stated. Draw one simple circuit or signal diagram, label every referenced terminal, and identify which terms can be measured and which are inferred from the model. Distinguish an algebraic sign from a physical polarity; a sign reversal after changing an arrow is usually a coordinate choice, not a violation of energy conservation.

Detailed engineering reasoning

Physical instruments interact with circuits. An oscilloscope probe adds resistance and capacitance; a current meter adds burden; a real signal source has finite output impedance. This introduces a tradeoff between observability and perturbation. If the measured result differs from the ideal model, quantify instrument loading before treating the discrepancy as material failure.

For operational amplifiers as circuit blocks, begin with a simplest-case prediction before substituting values. If an excitation vanishes, a passive zero-initial-state model must not invent an output. If one parameter becomes very large or small, interpret the corresponding limit with the relevant source and load held fixed. For an AC circuit, a magnitude-only answer is incomplete whenever phase matters, and for a transient circuit the initial state changes the waveform even if the final DC steady state is the same.

Worked example and interpretation

An ideal inverting amplifier with Rin=10 kΩ and Rf=100 kΩ has gain −10. Start from the stated equation, convert any milli-, micro- or kilo- prefixes to SI units, and substitute only quantities that share a consistent peak/RMS convention. Report the numerical quantity with physical units. Then explain what the sign or phase of the result tells an observer. Use another representation—for example rectangular versus polar complex notation or time versus frequency response—to check the result without simply repeating the first arithmetic.

The model predicts a conditional answer. It does not predict tolerance drift, insulation integrity, wiring parasitics, thermal runaway, bandwidth of an instrument, or safe operation unless those effects have been explicitly included. A nominal value should therefore be reported alongside its assumed operating conditions.

Failure mode to detect

The virtual short is an approximation; it never licenses unlimited output voltage or bandwidth. To diagnose this error, ask which assumption was silently changed. Trace the computation back to the definition of the source, the selected terminals, and the units of the constitutive law. A useful check is to repeat the calculation with a parameter changed by a factor of two and decide whether the direction of the output change is consistent with physical intuition. Where the network has several branches, apply KCL, KVL, and a power balance rather than trusting a single number.

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Conceptual: Explain in your own words why Vout≈Aol(V+−V−) is appropriate for the stated system and list two situations where the result would not apply without modification.
Quantitative: Rework the numerical example after increasing one independent component value by 20%; explicitly state which variables remain fixed, calculate the altered result, and explain whether a first-order sensitivity estimate would work.
Verification: Construct an independent check based on dimensions, a limiting condition, a KCL/KVL residual, or a conservation law. If it disagrees with the direct calculation, identify the first step where the models diverge.
Extension: Propose a simulation with three parameter values and a graph that could reveal a modeling assumption. Mark the horizontal/vertical units and explain the limits of the conclusion.

Connection to the wider book

This topic is a building block for analyzing networks in which one part shapes phase, another changes loading, and a measurement stage perturbs the circuit. The central habit is to distinguish a device's constitutive relation from the system-level network constraints. In later units the same pattern appears in three-phase neutral currents, coupled coils, transfer functions, and power-quality measurement. Keeping a consistent reference convention across these representations prevents apparently contradictory results.

§8.3 Tolerance and Monte Carlo reasoning

Unit 8, Lesson 3: Tolerance and Monte Carlo reasoning

Primary relation: σy²≈∑(∂y/∂xi)² σxi² Unit outcome: Combine ideal theory with actual instrument limitations, numerical methods, uncertainty, and responsible engineering judgement.

What the relation means

Linear sensitivity propagation connects small independent parameter uncertainties to predicted output variance. Correlation and nonlinearity require a more careful model or repeated random sampling. The expression σy²≈∑(∂y/∂xi)² σxi² is meaningful only when the input, output, reference arrows, and component idealizations are stated. Draw one simple circuit or signal diagram, label every referenced terminal, and identify which terms can be measured and which are inferred from the model. Distinguish an algebraic sign from a physical polarity; a sign reversal after changing an arrow is usually a coordinate choice, not a violation of energy conservation.

Detailed engineering reasoning

Students should separate three kinds of evidence: mathematics, numerical simulation, and experimental measurement. Mathematics verifies an ideal model; a unit test verifies code against expected numbers; a bench measurement tests a real assembled circuit within equipment limits. In any design report, mark unmeasured parameters as assumptions and never claim electrical safety certification from software alone.

For tolerance and monte carlo reasoning, begin with a simplest-case prediction before substituting values. If an excitation vanishes, a passive zero-initial-state model must not invent an output. If one parameter becomes very large or small, interpret the corresponding limit with the relevant source and load held fixed. For an AC circuit, a magnitude-only answer is incomplete whenever phase matters, and for a transient circuit the initial state changes the waveform even if the final DC steady state is the same.

Worked example and interpretation

For fc=1/(2πRC), a 5% R tolerance and 10% C tolerance combine to about 11.2% RSS relative cutoff uncertainty under independence. Start from the stated equation, convert any milli-, micro- or kilo- prefixes to SI units, and substitute only quantities that share a consistent peak/RMS convention. Report the numerical quantity with physical units. Then explain what the sign or phase of the result tells an observer. Use another representation—for example rectangular versus polar complex notation or time versus frequency response—to check the result without simply repeating the first arithmetic.

The model predicts a conditional answer. It does not predict tolerance drift, insulation integrity, wiring parasitics, thermal runaway, bandwidth of an instrument, or safe operation unless those effects have been explicitly included. A nominal value should therefore be reported alongside its assumed operating conditions.

Failure mode to detect

Adding all tolerance percentages is a worst-case bound, not automatically a 1σ statistical prediction. To diagnose this error, ask which assumption was silently changed. Trace the computation back to the definition of the source, the selected terminals, and the units of the constitutive law. A useful check is to repeat the calculation with a parameter changed by a factor of two and decide whether the direction of the output change is consistent with physical intuition. Where the network has several branches, apply KCL, KVL, and a power balance rather than trusting a single number.

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Conceptual: Explain in your own words why σy²≈∑(∂y/∂xi)² σxi² is appropriate for the stated system and list two situations where the result would not apply without modification.
Quantitative: Rework the numerical example after increasing one independent component value by 20%; explicitly state which variables remain fixed, calculate the altered result, and explain whether a first-order sensitivity estimate would work.
Verification: Construct an independent check based on dimensions, a limiting condition, a KCL/KVL residual, or a conservation law. If it disagrees with the direct calculation, identify the first step where the models diverge.
Extension: Propose a simulation with three parameter values and a graph that could reveal a modeling assumption. Mark the horizontal/vertical units and explain the limits of the conclusion.

Connection to the wider book

This topic is a building block for analyzing networks in which one part shapes phase, another changes loading, and a measurement stage perturbs the circuit. The central habit is to distinguish a device's constitutive relation from the system-level network constraints. In later units the same pattern appears in three-phase neutral currents, coupled coils, transfer functions, and power-quality measurement. Keeping a consistent reference convention across these representations prevents apparently contradictory results.

§8.4 AC measurement and instrument loading

Unit 8, Lesson 4: AC measurement and instrument loading

Primary relation: Measured signal=signal × instrument transfer Unit outcome: Combine ideal theory with actual instrument limitations, numerical methods, uncertainty, and responsible engineering judgement.

What the relation means

Oscilloscopes and multimeters report quantities according to bandwidth, coupling, probe impedance and RMS measurement method. Calibration and stated instrument uncertainty bound conclusions. The expression Measured signal=signal × instrument transfer is meaningful only when the input, output, reference arrows, and component idealizations are stated. Draw one simple circuit or signal diagram, label every referenced terminal, and identify which terms can be measured and which are inferred from the model. Distinguish an algebraic sign from a physical polarity; a sign reversal after changing an arrow is usually a coordinate choice, not a violation of energy conservation.

Detailed engineering reasoning

A reliable simulation has a test plan. Start with an analytically solvable small circuit. Compare node voltages, branch currents, stored energy, and steady-state values, and refine integration step size. A simulation solver reporting convergence does not prove that the netlist represents the physical device. Publish input values, model equations and numerical residuals.

For ac measurement and instrument loading, begin with a simplest-case prediction before substituting values. If an excitation vanishes, a passive zero-initial-state model must not invent an output. If one parameter becomes very large or small, interpret the corresponding limit with the relevant source and load held fixed. For an AC circuit, a magnitude-only answer is incomplete whenever phase matters, and for a transient circuit the initial state changes the waveform even if the final DC steady state is the same.

Worked example and interpretation

A 10 MΩ probe loading a 1 kΩ output has negligible DC divider effect; a 10 kΩ probe does not. Start from the stated equation, convert any milli-, micro- or kilo- prefixes to SI units, and substitute only quantities that share a consistent peak/RMS convention. Report the numerical quantity with physical units. Then explain what the sign or phase of the result tells an observer. Use another representation—for example rectangular versus polar complex notation or time versus frequency response—to check the result without simply repeating the first arithmetic.

The model predicts a conditional answer. It does not predict tolerance drift, insulation integrity, wiring parasitics, thermal runaway, bandwidth of an instrument, or safe operation unless those effects have been explicitly included. A nominal value should therefore be reported alongside its assumed operating conditions.

Failure mode to detect

A non-true-RMS meter may misreport non-sinusoidal current even with the correct nominal range. To diagnose this error, ask which assumption was silently changed. Trace the computation back to the definition of the source, the selected terminals, and the units of the constitutive law. A useful check is to repeat the calculation with a parameter changed by a factor of two and decide whether the direction of the output change is consistent with physical intuition. Where the network has several branches, apply KCL, KVL, and a power balance rather than trusting a single number.

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Conceptual: Explain in your own words why Measured signal=signal × instrument transfer is appropriate for the stated system and list two situations where the result would not apply without modification.
Quantitative: Rework the numerical example after increasing one independent component value by 20%; explicitly state which variables remain fixed, calculate the altered result, and explain whether a first-order sensitivity estimate would work.
Verification: Construct an independent check based on dimensions, a limiting condition, a KCL/KVL residual, or a conservation law. If it disagrees with the direct calculation, identify the first step where the models diverge.
Extension: Propose a simulation with three parameter values and a graph that could reveal a modeling assumption. Mark the horizontal/vertical units and explain the limits of the conclusion.

Connection to the wider book

This topic is a building block for analyzing networks in which one part shapes phase, another changes loading, and a measurement stage perturbs the circuit. The central habit is to distinguish a device's constitutive relation from the system-level network constraints. In later units the same pattern appears in three-phase neutral currents, coupled coils, transfer functions, and power-quality measurement. Keeping a consistent reference convention across these representations prevents apparently contradictory results.

§8.5 Integrated filter and power-quality design

Unit 8, Lesson 5: Integrated filter and power-quality design

Primary relation: Design=specification→model→test→revision Unit outcome: Combine ideal theory with actual instrument limitations, numerical methods, uncertainty, and responsible engineering judgement.

What the relation means

A defensible capstone defines frequency, amplitude, load, safety and tolerance requirements; tests multiple alternatives; and reports both performance and uncertainty. Never confuse a synthetic demonstration with deployed certified equipment. The expression Design=specification→model→test→revision is meaningful only when the input, output, reference arrows, and component idealizations are stated. Draw one simple circuit or signal diagram, label every referenced terminal, and identify which terms can be measured and which are inferred from the model. Distinguish an algebraic sign from a physical polarity; a sign reversal after changing an arrow is usually a coordinate choice, not a violation of energy conservation.

Detailed engineering reasoning

Physical instruments interact with circuits. An oscilloscope probe adds resistance and capacitance; a current meter adds burden; a real signal source has finite output impedance. This introduces a tradeoff between observability and perturbation. If the measured result differs from the ideal model, quantify instrument loading before treating the discrepancy as material failure.

For integrated filter and power-quality design, begin with a simplest-case prediction before substituting values. If an excitation vanishes, a passive zero-initial-state model must not invent an output. If one parameter becomes very large or small, interpret the corresponding limit with the relevant source and load held fixed. For an AC circuit, a magnitude-only answer is incomplete whenever phase matters, and for a transient circuit the initial state changes the waveform even if the final DC steady state is the same.

Worked example and interpretation

Choose a 1 kHz low-pass target, 1 kΩ load, ±10% components, and quantify both ideal and loaded cutoff before claiming compliance. Start from the stated equation, convert any milli-, micro- or kilo- prefixes to SI units, and substitute only quantities that share a consistent peak/RMS convention. Report the numerical quantity with physical units. Then explain what the sign or phase of the result tells an observer. Use another representation—for example rectangular versus polar complex notation or time versus frequency response—to check the result without simply repeating the first arithmetic.

The model predicts a conditional answer. It does not predict tolerance drift, insulation integrity, wiring parasitics, thermal runaway, bandwidth of an instrument, or safe operation unless those effects have been explicitly included. A nominal value should therefore be reported alongside its assumed operating conditions.

Failure mode to detect

An apparently perfect Bode curve can hide an unstable driver, unsafe mains connection or noncompliant component rating. To diagnose this error, ask which assumption was silently changed. Trace the computation back to the definition of the source, the selected terminals, and the units of the constitutive law. A useful check is to repeat the calculation with a parameter changed by a factor of two and decide whether the direction of the output change is consistent with physical intuition. Where the network has several branches, apply KCL, KVL, and a power balance rather than trusting a single number.

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Conceptual: Explain in your own words why Design=specification→model→test→revision is appropriate for the stated system and list two situations where the result would not apply without modification.
Quantitative: Rework the numerical example after increasing one independent component value by 20%; explicitly state which variables remain fixed, calculate the altered result, and explain whether a first-order sensitivity estimate would work.
Verification: Construct an independent check based on dimensions, a limiting condition, a KCL/KVL residual, or a conservation law. If it disagrees with the direct calculation, identify the first step where the models diverge.
Extension: Propose a simulation with three parameter values and a graph that could reveal a modeling assumption. Mark the horizontal/vertical units and explain the limits of the conclusion.

Connection to the wider book

This topic is a building block for analyzing networks in which one part shapes phase, another changes loading, and a measurement stage perturbs the circuit. The central habit is to distinguish a device's constitutive relation from the system-level network constraints. In later units the same pattern appears in three-phase neutral currents, coupled coils, transfer functions, and power-quality measurement. Keeping a consistent reference convention across these representations prevents apparently contradictory results.

Unit 8, Lesson 1: SPICE-based circuit verification

What the relation means

Detailed engineering reasoning

Worked example and interpretation

Failure mode to detect

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Connection to the wider book

Unit 8, Lesson 2: Operational amplifiers as circuit blocks

What the relation means

Detailed engineering reasoning

Worked example and interpretation

Failure mode to detect

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Connection to the wider book

Unit 8, Lesson 3: Tolerance and Monte Carlo reasoning

What the relation means

Detailed engineering reasoning

Worked example and interpretation

Failure mode to detect

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Connection to the wider book

Unit 8, Lesson 4: AC measurement and instrument loading

What the relation means

Detailed engineering reasoning

Worked example and interpretation

Failure mode to detect

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Connection to the wider book

Unit 8, Lesson 5: Integrated filter and power-quality design

What the relation means

Detailed engineering reasoning

Worked example and interpretation

Failure mode to detect

Design exercise: a three-stage method

1. Prediction. Before any calculator is used, describe the expected response for low, nominal, and high excitation. Write down the reference direction and the output to measure.

2. Calculation. Derive the governing expression from the ideal element relations. Recalculate a second operating point and show intermediate values, SI conversions, and the choice of complex argument branch if needed.

3. Assessment. Identify a parasitic or nonideality absent from the calculation. Explain how an engineer would test it with a safe low-voltage setup, without assuming that software itself validates a built device.

Problems

Connection to the wider book

Step-by-Step Circuit Analysis II Solutions