Energy Storage: Capacitors and Inductors
Comprehensive university textbook unit covering fundamental principles, mathematical formulations, and practical engineering applications of Energy Storage: Capacitors and Inductors.
§5.1 Capacitor Constitutive Relations
U05-L01 · Capacitor Constitutive Relations
Unit: 5 — Energy-Storage Elements Core relation: q=Cv Essential ideas: electric field storage, charge–voltage relation, and capacitance.
Learning objectives
After this lesson, explain capacitor constitutive relations and its circuit context, assign reference polarities and directions, derive or use q=Cv with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.
Physical and mathematical foundation
An ideal capacitor voltage cannot jump instantaneously under finite current. The selected current arrow and voltage polarity give meaning to the signs in an equation; they do not force a physical quantity to flow in that direction. Nodes define equal-potential connection points in the lumped model. Branches define component connections, and a chosen ground provides a mathematical reference rather than a physical energy sink. Interpreting the equation q=Cv requires knowing both the connected terminals and the applicable ideal-component assumptions.
Within energy-storage elements, distinguish a source-imposed quantity from a circuit response. A voltage source constrains one terminal difference, while a current source constrains a branch current. An element's constitutive relation describes its response but cannot replace Kirchhoff's laws. If a diagram changes, the electrical topology may change even when the drawn symbols and their numerical values remain the same.
Concept 1 — Electric field storage. Identify the branch or terminals to which this idea applies, state its units, and connect it with q=Cv. Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 2 — Charge–voltage relation. Identify the branch or terminals to which this idea applies, state its units, and connect it with q=Cv. Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 3 — And capacitance. Identify the branch or terminals to which this idea applies, state its units, and connect it with q=Cv. Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Worked case and result
Case. A 100 µF capacitor is charged to 20 V.
Result. It stores 2 mC of charge and 0.020 J of energy. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. An ideal capacitor voltage cannot jump instantaneously under finite current. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
A reliable solution method
- Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
- Combine the component relation
q=Cvwith KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers. - Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
- Identify the dominant idealization. Beware replacing microfarads by farads without the 10⁻⁶ factor. If any predicted value approaches a component rating, the analysis is only a starting point for a safety-qualified design review.
Exercises
1. Concept: Explain why electric field storage needs a clear reference direction, voltage polarity, or system boundary. Provide one counterexample.
2. Numerical: Reproduce the worked case, showing each unit conversion. Increase one supplied value by 20% and compute the resulting change.
3. Verification: Construct a KCL, KVL, energy, or state-continuity check appropriate to this circuit; explain which assumption the check tests.
4. Application: Specify a current-limited low-voltage demonstration and describe how meter input impedance would change the result.
5. Extension: Change one boundary condition and explain whether `q=Cv` remains valid and which previous shortcut fails.
Key limitation: Beware replacing microfarads by farads without the 10⁻⁶ factor.
§5.2 Capacitor Connections and Energy
U05-L02 · Capacitor Connections and Energy
Unit: 5 — Energy-Storage Elements Core relation: E_C=½Cv² Essential ideas: series capacitance, parallel capacitance, and voltage sharing.
Learning objectives
After this lesson, explain capacitor connections and energy and its circuit context, assign reference polarities and directions, derive or use E_C=½Cv² with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.
Physical and mathematical foundation
Equal series voltage sharing requires equal capacitance and no other stored charge or leakage effect. The selected current arrow and voltage polarity give meaning to the signs in an equation; they do not force a physical quantity to flow in that direction. Nodes define equal-potential connection points in the lumped model. Branches define component connections, and a chosen ground provides a mathematical reference rather than a physical energy sink. Interpreting the equation E_C=½Cv² requires knowing both the connected terminals and the applicable ideal-component assumptions.
Within energy-storage elements, distinguish a source-imposed quantity from a circuit response. A voltage source constrains one terminal difference, while a current source constrains a branch current. An element's constitutive relation describes its response but cannot replace Kirchhoff's laws. If a diagram changes, the electrical topology may change even when the drawn symbols and their numerical values remain the same.
Concept 1 — Series capacitance. Identify the branch or terminals to which this idea applies, state its units, and connect it with E_C=½Cv². Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 2 — Parallel capacitance. Identify the branch or terminals to which this idea applies, state its units, and connect it with E_C=½Cv². Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 3 — And voltage sharing. Identify the branch or terminals to which this idea applies, state its units, and connect it with E_C=½Cv². Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Worked case and result
Case. Two identical 10 µF capacitors in series have 5 µF equivalent capacitance.
Result. At 12 V total, the pair stores 0.00036 J and divides voltage equally if initially uncharged. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. Equal series voltage sharing requires equal capacitance and no other stored charge or leakage effect. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
A reliable solution method
- Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
- Combine the component relation
E_C=½Cv²with KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers. - Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
- Identify the dominant idealization. Before asserting capacitor energy conservation, account for dissipation when connections change. If any predicted value approaches a component rating, the analysis is only a starting point for a safety-qualified design review.
Exercises
1. Concept: Explain why series capacitance needs a clear reference direction, voltage polarity, or system boundary. Provide one counterexample.
2. Numerical: Reproduce the worked case, showing each unit conversion. Increase one supplied value by 20% and compute the resulting change.
3. Verification: Construct a KCL, KVL, energy, or state-continuity check appropriate to this circuit; explain which assumption the check tests.
4. Application: Specify a current-limited low-voltage demonstration and describe how meter input impedance would change the result.
5. Extension: Change one boundary condition and explain whether `E_C=½Cv²` remains valid and which previous shortcut fails.
Key limitation: Before asserting capacitor energy conservation, account for dissipation when connections change.
§5.3 Inductor Constitutive Relations
U05-L03 · Inductor Constitutive Relations
Unit: 5 — Energy-Storage Elements Core relation: v=L di/dt Essential ideas: flux linkage, inductive voltage, and current continuity.
Learning objectives
After this lesson, explain inductor constitutive relations and its circuit context, assign reference polarities and directions, derive or use v=L di/dt with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.
Physical and mathematical foundation
An ideal inductor current cannot jump under finite applied voltage. The selected current arrow and voltage polarity give meaning to the signs in an equation; they do not force a physical quantity to flow in that direction. Nodes define equal-potential connection points in the lumped model. Branches define component connections, and a chosen ground provides a mathematical reference rather than a physical energy sink. Interpreting the equation v=L di/dt requires knowing both the connected terminals and the applicable ideal-component assumptions.
Within energy-storage elements, distinguish a source-imposed quantity from a circuit response. A voltage source constrains one terminal difference, while a current source constrains a branch current. An element's constitutive relation describes its response but cannot replace Kirchhoff's laws. If a diagram changes, the electrical topology may change even when the drawn symbols and their numerical values remain the same.
Concept 1 — Flux linkage. Identify the branch or terminals to which this idea applies, state its units, and connect it with v=L di/dt. Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 2 — Inductive voltage. Identify the branch or terminals to which this idea applies, state its units, and connect it with v=L di/dt. Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 3 — And current continuity. Identify the branch or terminals to which this idea applies, state its units, and connect it with v=L di/dt. Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Worked case and result
Case. A 50 mH inductor current rises from 0 A to 2 A over 10 ms.
Result. Assuming constant slope, the induced voltage is 10 V. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. An ideal inductor current cannot jump under finite applied voltage. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
A reliable solution method
- Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
- Combine the component relation
v=L di/dtwith KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers. - Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
- Identify the dominant idealization. A physical coil has winding resistance, parasitic capacitance, saturation, and heating. If any predicted value approaches a component rating, the analysis is only a starting point for a safety-qualified design review.
Exercises
1. Concept: Explain why flux linkage needs a clear reference direction, voltage polarity, or system boundary. Provide one counterexample.
2. Numerical: Reproduce the worked case, showing each unit conversion. Increase one supplied value by 20% and compute the resulting change.
3. Verification: Construct a KCL, KVL, energy, or state-continuity check appropriate to this circuit; explain which assumption the check tests.
4. Application: Specify a current-limited low-voltage demonstration and describe how meter input impedance would change the result.
5. Extension: Change one boundary condition and explain whether `v=L di/dt` remains valid and which previous shortcut fails.
Key limitation: A physical coil has winding resistance, parasitic capacitance, saturation, and heating.
§5.4 Inductor Networks and Stored Energy
U05-L04 · Inductor Networks and Stored Energy
Unit: 5 — Energy-Storage Elements Core relation: E_L=½Li² Essential ideas: series inductance, ideal energy storage, and transient hazards.
Learning objectives
After this lesson, explain inductor networks and stored energy and its circuit context, assign reference polarities and directions, derive or use E_L=½Li² with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.
Physical and mathematical foundation
Connecting a switching device across an inductor requires a path for decaying current. The selected current arrow and voltage polarity give meaning to the signs in an equation; they do not force a physical quantity to flow in that direction. Nodes define equal-potential connection points in the lumped model. Branches define component connections, and a chosen ground provides a mathematical reference rather than a physical energy sink. Interpreting the equation E_L=½Li² requires knowing both the connected terminals and the applicable ideal-component assumptions.
Within energy-storage elements, distinguish a source-imposed quantity from a circuit response. A voltage source constrains one terminal difference, while a current source constrains a branch current. An element's constitutive relation describes its response but cannot replace Kirchhoff's laws. If a diagram changes, the electrical topology may change even when the drawn symbols and their numerical values remain the same.
Concept 1 — Series inductance. Identify the branch or terminals to which this idea applies, state its units, and connect it with E_L=½Li². Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 2 — Ideal energy storage. Identify the branch or terminals to which this idea applies, state its units, and connect it with E_L=½Li². Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 3 — And transient hazards. Identify the branch or terminals to which this idea applies, state its units, and connect it with E_L=½Li². Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Worked case and result
Case. A 200 mH inductor carries 3 A.
Result. Its magnetic energy is 0.9 J. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. Connecting a switching device across an inductor requires a path for decaying current. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
A reliable solution method
- Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
- Combine the component relation
E_L=½Li²with KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers. - Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
- Identify the dominant idealization. High opening-switch voltages are possible; the simulation is not a hardware safety design. If any predicted value approaches a component rating, the analysis is only a starting point for a safety-qualified design review.
Exercises
1. Concept: Explain why series inductance needs a clear reference direction, voltage polarity, or system boundary. Provide one counterexample.
2. Numerical: Reproduce the worked case, showing each unit conversion. Increase one supplied value by 20% and compute the resulting change.
3. Verification: Construct a KCL, KVL, energy, or state-continuity check appropriate to this circuit; explain which assumption the check tests.
4. Application: Specify a current-limited low-voltage demonstration and describe how meter input impedance would change the result.
5. Extension: Change one boundary condition and explain whether `E_L=½Li²` remains valid and which previous shortcut fails.
Key limitation: High opening-switch voltages are possible; the simulation is not a hardware safety design.
§5.5 Initial and Final Conditions
U05-L05 · Initial and Final Conditions
Unit: 5 — Energy-Storage Elements Core relation: v_C(0+)=v_C(0−); i_L(0+)=i_L(0−) Essential ideas: state variables, continuity, and DC steady state.
Learning objectives
After this lesson, explain initial and final conditions and its circuit context, assign reference polarities and directions, derive or use v_C(0+)=v_C(0−); i_L(0+)=i_L(0−) with SI units, solve a numerical example, and evaluate its validity using conservation laws and a practical limiting case.
Physical and mathematical foundation
The state of an energy-storage component provides the initial condition for differential equations. The selected current arrow and voltage polarity give meaning to the signs in an equation; they do not force a physical quantity to flow in that direction. Nodes define equal-potential connection points in the lumped model. Branches define component connections, and a chosen ground provides a mathematical reference rather than a physical energy sink. Interpreting the equation v_C(0+)=v_C(0−); i_L(0+)=i_L(0−) requires knowing both the connected terminals and the applicable ideal-component assumptions.
Within energy-storage elements, distinguish a source-imposed quantity from a circuit response. A voltage source constrains one terminal difference, while a current source constrains a branch current. An element's constitutive relation describes its response but cannot replace Kirchhoff's laws. If a diagram changes, the electrical topology may change even when the drawn symbols and their numerical values remain the same.
Concept 1 — State variables. Identify the branch or terminals to which this idea applies, state its units, and connect it with v_C(0+)=v_C(0−); i_L(0+)=i_L(0−). Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 2 — Continuity. Identify the branch or terminals to which this idea applies, state its units, and connect it with v_C(0+)=v_C(0−); i_L(0+)=i_L(0−). Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Concept 3 — And dc steady state. Identify the branch or terminals to which this idea applies, state its units, and connect it with v_C(0+)=v_C(0−); i_L(0+)=i_L(0−). Explain what changes when the source or component parameter increases; predict the direction of the change before doing arithmetic. Consider whether reference directions, instrument loading, or tolerance could distort the interpretation.
Worked case and result
Case. Immediately before a switching event, a capacitor holds 8 V and an inductor carries 0.5 A.
Result. Immediately after switching, ideal state continuity requires the same 8 V and 0.5 A. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. The state of an energy-storage component provides the initial condition for differential equations. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
A reliable solution method
- Draw the topology, identify nodes, and choose polarities and branch-current arrows; write down which source and element quantities are known.
- Combine the component relation
v_C(0+)=v_C(0−); i_L(0+)=i_L(0−)with KCL, KVL, and state continuity where relevant; solve symbolically before substituting numbers. - Convert units, calculate, and cross-check using a conservation equation or limiting case. Explain why the result is physically plausible.
- Identify the dominant idealization. At a settled DC state, an ideal capacitor is open and an ideal inductor is a short, under valid assumptions. If any predicted value approaches a component rating, the analysis is only a starting point for a safety-qualified design review.
Exercises
1. Concept: Explain why state variables needs a clear reference direction, voltage polarity, or system boundary. Provide one counterexample.
2. Numerical: Reproduce the worked case, showing each unit conversion. Increase one supplied value by 20% and compute the resulting change.
3. Verification: Construct a KCL, KVL, energy, or state-continuity check appropriate to this circuit; explain which assumption the check tests.
4. Application: Specify a current-limited low-voltage demonstration and describe how meter input impedance would change the result.
5. Extension: Change one boundary condition and explain whether `v_C(0+)=v_C(0−); i_L(0+)=i_L(0−)` remains valid and which previous shortcut fails.
Key limitation: At a settled DC state, an ideal capacitor is open and an ideal inductor is a short, under valid assumptions.
Step-by-Step Circuit Analysis I Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Case. A 100 µF capacitor is charged to 20 V.
Result. It stores 2 mC of charge and 0.020 J of energy. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. An ideal capacitor voltage cannot jump instantaneously under finite current. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. A 100 µF capacitor is charged to 20 V.
Result. It stores 2 mC of charge and 0.020 J of energy. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. An ideal capacitor voltage cannot jump instantaneously under finite current. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. Two identical 10 µF capacitors in series have 5 µF equivalent capacitance.
Result. At 12 V total, the pair stores 0.00036 J and divides voltage equally if initially uncharged. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. Equal series voltage sharing requires equal capacitance and no other stored charge or leakage effect. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. Two identical 10 µF capacitors in series have 5 µF equivalent capacitance.
Result. At 12 V total, the pair stores 0.00036 J and divides voltage equally if initially uncharged. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. Equal series voltage sharing requires equal capacitance and no other stored charge or leakage effect. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. A 50 mH inductor current rises from 0 A to 2 A over 10 ms.
Result. Assuming constant slope, the induced voltage is 10 V. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. An ideal inductor current cannot jump under finite applied voltage. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. A 50 mH inductor current rises from 0 A to 2 A over 10 ms.
Result. Assuming constant slope, the induced voltage is 10 V. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. An ideal inductor current cannot jump under finite applied voltage. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. A 200 mH inductor carries 3 A.
Result. Its magnetic energy is 0.9 J. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. Connecting a switching device across an inductor requires a path for decaying current. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. A 200 mH inductor carries 3 A.
Result. Its magnetic energy is 0.9 J. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. Connecting a switching device across an inductor requires a path for decaying current. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. Immediately before a switching event, a capacitor holds 8 V and an inductor carries 0.5 A.
Result. Immediately after switching, ideal state continuity requires the same 8 V and 0.5 A. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. The state of an energy-storage component provides the initial condition for differential equations. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.
Case. Immediately before a switching event, a capacitor holds 8 V and an inductor carries 0.5 A.
Result. Immediately after switching, ideal state continuity requires the same 8 V and 0.5 A. First translate all prefixes into SI units and state the formula before substituting. Verify that the dimensions of the answer agree with the quantity being requested. Reverse the calculation, or test a meaningful limiting case, rather than trusting the displayed calculator result alone.
Interpretation. The state of an energy-storage component provides the initial condition for differential equations. A negative signed answer may only mean that a current or voltage points opposite to its reference. It does not indicate an impossible circuit. When multiple elements appear, check their node-current balance, the signed sum of drops around a loop, and the sum of absorbed and delivered powers.