§2.1 Phasor Kirchhoff laws

Unit 2, Lesson 1: Phasor Kirchhoff laws

Primary relation: ∑ Ĩnode=0; ∑ Ṽloop=0 Unit outcome: Use impedance networks to solve currents, power, and amplitude response with consistent complex arithmetic.

What the relation means

KCL and KVL remain valid in linear lumped networks at a common sinusoidal frequency because summation commutes with differentiation. Current reference arrows and passive signs still matter. The expression ∑ Ĩnode=0; ∑ Ṽloop=0 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

Frequency-domain network solving resembles DC solving, but every branch quantity is complex. An AC nodal analysis must use a single frequency at a time; if sources differ in frequency, solve each tone separately. A complex answer represents two real facts—amplitude and phase—not a mysterious imaginary current. Check KCL by adding rectangular phasors, not their magnitudes.

For phasor kirchhoff laws, 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 node with 2∠0° A entering and 1∠90° A leaving requires a third leaving current 2−j1 A. 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

Do not add phasor magnitudes as if they were signed scalar currents. 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 ∑ Ĩnode=0; ∑ Ṽloop=0 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.

§2.2 AC nodal and mesh solutions

Unit 2, Lesson 2: AC nodal and mesh solutions

Primary relation: Y Ṽ=Ĩ Unit outcome: Use impedance networks to solve currents, power, and amplitude response with consistent complex arithmetic.

What the relation means

Complex admittance transforms connected impedance into linear equations. Formulate node equations with conductances and susceptances, or mesh equations with impedances. The expression Y Ṽ=Ĩ 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 current-phasor power calculation should document the positive terminal and current reference of the load. For a series RL network, the current lags voltage, reactive absorption is positive, and real power is I²R. For a capacitor-dominated network, source-current phase leads voltage and Q is negative under passive convention. Apparent power magnitude is |S|=Vrms Irms, whereas P can be far smaller if reactance dominates.

For ac nodal and mesh solutions, 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 V RMS source across a series 30+j40 Ω branch yields 0.2∠−53.13° A. 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 matrix solver can return precise nonsense if source polarities were encoded incorrectly. 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 Y Ṽ=Ĩ 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.

§2.3 Real, reactive, and apparent power

Unit 2, Lesson 3: Real, reactive, and apparent power

Primary relation: S=Ṽ Ĩ*=P+jQ Unit outcome: Use impedance networks to solve currents, power, and amplitude response with consistent complex arithmetic.

What the relation means

The conjugate is essential for RMS complex power under passive-sign convention. Positive Q indicates net inductive reactive absorption; negative Q indicates net capacitive behavior. The expression S=Ṽ Ĩ*=P+jQ 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

Resonance creates a useful but dangerous tradeoff: cancellation of external reactances permits high current, while individual reactive voltages may exceed the source voltage by Q. Ideal lossless formulas miss coil resistance, capacitor ESR, and source impedance. Always test a resonance formula above, below, and exactly at its nominal frequency and state how a shifted component value alters the peak.

For real, reactive, and apparent power, 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 120 V RMS load drawing 5 A at 0.8 lagging power factor takes P=480 W and Q=360 var. 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 real power of an ideal inductor over a period is zero, despite nonzero instantaneous power. 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 S=Ṽ Ĩ*=P+jQ 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.

§2.4 Power-factor correction

Unit 2, Lesson 4: Power-factor correction

Primary relation: Qc=P(tanφ1−tanφ2) Unit outcome: Use impedance networks to solve currents, power, and amplitude response with consistent complex arithmetic.

What the relation means

A shunt capacitor can reduce source reactive power demanded by an inductive load without eliminating the real load power. Practical compensation must respect voltage, frequency, resonance, and switching constraints. The expression Qc=P(tanφ1−tanφ2) 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

Frequency-domain network solving resembles DC solving, but every branch quantity is complex. An AC nodal analysis must use a single frequency at a time; if sources differ in frequency, solve each tone separately. A complex answer represents two real facts—amplitude and phase—not a mysterious imaginary current. Check KCL by adding rectangular phasors, not their magnitudes.

For power-factor correction, 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 1 kW load at 0.8 lagging PF needs 750 var; improving to unity nominally requires 750 var of capacitive compensation. 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

Do not infer that correction reduces the mechanical power needed by a motor at unchanged shaft output. 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 Qc=P(tanφ1−tanφ2) 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.

§2.5 RLC resonance and selectivity

Unit 2, Lesson 5: RLC resonance and selectivity

Primary relation: ω0=1/√(LC); Q=ω0L/R Unit outcome: Use impedance networks to solve currents, power, and amplitude response with consistent complex arithmetic.

What the relation means

Series RLC resonance cancels inductive and capacitive reactances at one frequency, maximizing series current for a fixed source. Quality factor describes the sharpness of the response near resonance in the ideal lumped model. The expression ω0=1/√(LC); Q=ω0L/R 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 current-phasor power calculation should document the positive terminal and current reference of the load. For a series RL network, the current lags voltage, reactive absorption is positive, and real power is I²R. For a capacitor-dominated network, source-current phase leads voltage and Q is negative under passive convention. Apparent power magnitude is |S|=Vrms Irms, whereas P can be far smaller if reactance dominates.

For rlc resonance and selectivity, 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

L=10 mH and C=100 μF give ω0=1000 rad/s (159.15 Hz); with R=10 Ω, Q=1. 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 high-Q design can develop large internal reactive voltages even at modest source voltage. 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 ω0=1/√(LC); Q=ω0L/R 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 2, Lesson 1: Phasor Kirchhoff laws

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 2, Lesson 2: AC nodal and mesh solutions

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 2, Lesson 3: Real, reactive, and apparent power

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 2, Lesson 4: Power-factor correction

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 2, Lesson 5: RLC resonance and selectivity

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